Magnetic Rotation Study of the A ^ i ^ - X ^ j j " system of 7 9 B r 2 by Christopher D. Boone B.Sc, The University of New Brunswick, 1990 M.Sc, The University of New Brunswick, 1992 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Department of Physics) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA July 1999 © Christopher D. Boone, 1999 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of British Columbia Vancouver, Canada Abstract The A 3 I I i u - X 1 £ + system of 7 9 B r 2 has been studied using magnetic rotation spectroscopy. A new procedure has been introduced to simplify the calculation of magnetic rotation spectra. The A state was investigated over a range of vibrations 13 < v < 37, which extended to within 2 c m - 1 of the dissociation limit for the A state. A global analysis was performed on the A state, making use of near-dissociation expansions. From the global analysis, the following parameters were obtained: D (dissociation energy, relative to v = 0, J = 0 in the ground state) = 15,894.619 ± 0.007 c m - 1 , YD (effective vibration at dissociation) = 41.544 ± 0.013 and C5 (a parameter describing the long-range form of the potential) = 61,700 ± 900 c m - 1 A 5 . From the potential curve generated in the global analysis, the following spectroscopic parameters were determined for the A state: r e (equilibrium distance) = 2.7026 A, B e (equilibrium rotational constant) = 0.05849 c m - 1 , D e (binding energy) = 2147.82 c m - 1 , and T e (energy relative to the minimum of the X state potential) = 13,909.14 c m - 1 . From an analysis of the long-range potential of the A state, it was determined that the dissociation energy relative to v = 0, J = 0 in the ground state was 15,894.58 ± 0.01 c m - 1 ; this differed significantly from the value obtained in the global analysis, a consequence of the fact that the global analysis gave effective values for the parameters determined. The f2-type doubling constant in the A state was observed to level off as a function of v above v = 19. Structure in low-J lines was attributed to hyperfine structure in the A state. Extra lines were observed in the spectrum, including an entire series of extra lines associated with v = 27 in the A state that extended over a range of J from 2 to 32. Many of these extra lines, including the extra series, arise from perturbation-allowed transitions to levels in the A ' 3 ! ] ^ electronic state. Using the extra lines, an estimate was made for the energy of the A ' state (relative to the bottom of the potential in the ground state): 13,187 ± 20 c m - 1 . ii Contents Abstract " Table of Contents iii List of Tables vi List of Figures vii Acknowledgments xi 1 Introduction 1 1.1 Background 1 1.2 Thesis Organization 5 2 General Theoretical Considerations 6 2.1 Angular Momenta 6 2.2 Energy 11 2.2.1 The Born-Oppenheimer Approximation 11 2.2.2 Electronic States 12 2.2.3 Potential Energy Curves 12 2.2.4 Vibrational Energy 14 2.2.5 Rotational Energy 14 2.2.6 Total Energy 16 2.2.7 Rotational Distortion of the Potential 16 2.3 Symmetry 17 2.3.1 Molecular Symmetry and crv 18 2.3.2 Homonuclear Molecules 19 2.4 Perturbations 20 2.4.1 The Rotational Hamiltonian 21 2.4.2 Uncoupling Phenomena 23 2.4.3 A-doubling 23 2.5 Interaction with Magnetic Fields 25 2.5.1 Splitting of Levels in a Magnetic Field 27 2.6 Interaction with Electromagnetic Radiation 29 iii 2.7 Transitions 31 2.7.1 Franck-Condon Factor 34 2.7.2 Electronic Transitions 34 2.8 Index of Refraction 35 2.9 Widths of Spectral Lines 37 2.9.1 Natural Linewidth 37 2.9.2 Pressure Broadening 38 2.9.3 Doppler Broadening 39 2.10 Hyperfine effects 39 2.10.1 Ortho-Para Symmetry 40 2.10.2 Hyperfine Energies 41 3 Magnetic Rotation 47 3.1 Magnetic Circular Dichroism 48 3.2 Faraday Effect 52 3.3 Combination 53 3.4 Branches 56 3.5 Contributions From Intensity Perturbations 64 3.6 Doppler Effect 68 3.7 Examples of Calculated Signals 69 3.8 General Notes on Magnetic Rotation 74 3.9 Incorporating Quadratic Shifts 76 3.10 Static Magnetic Fields 76 3.11 Overlapping Lines 77 4 Experiment 79 4.1 Frequency Measurements 83 4.2 Doppler-Free Setup 84 5 Results and Analysis 87 5.1 Cause of High-J Signals 90 5.2 Assigning Uncertainties 96 5.3 Frequency Analysis 96 5.4 fi-type Doubling 97 iv 5.5 Global Analysis 105 5.6 Summary of Fitting Procedure 119 5.7 Results of Global Analysis 121 5.8 Long-Range Behaviour of the Potential 126 5.9 Lines Above the Dissociation Limit 131 5.10 Extra Lines . . 133 5.11 Building the A ' State 151 5.12 Hyperfine Effects 158 5.13 Doppler-free Magnetic Rotation 161 6 Future Work and Conclusions 165 References 166 Appendix 171 v List of Tables 2-1 Selection rules for perturbation (excluding hyperfine effects) 20 2- 2 Selection rules for transitions (excluding hyperfine effects) 32 3- 1 The ^-factors, Q(J",J',Q,'), in matrix elements 57 5-1 Second Differences in 23'-l"R-branch 88 5-2 Vibration-by-vibration results for 7 9 B r 2 A 3 I I i u 98 5-3 fi-doubling constants (All values in cm - 1 ) 103 5-4 Current state of knowledge for A state 105 5-5 The parameters resulting from the global analysis of the A state 121 5-6 Centrifugal distortion constants calculated from R K R potential. (All values in cm - 1 ) 123 5-7 Table 5-7: Turning points for the R K R potential, along with the values of G(v) and B„ resulting from the global analysis. 124 5-8 Results of fitting turning points in the long-range portion of the potential 128 5-9 Deviation of quasi-bound levels from expectation for v' = 24 (All values in cm - 1 ) . . 131 5-10 A selected set of transitions in the perturbed state v = 27 (All values in cm - 1 ) . . . 139 5-11 Results of fitting the extra series (All values in cm - 1 ) 142 5-12 Extra lines observed in the spectrum 144 5-13 Preliminary test of extrapolation procedure in the A state. (All values in cm - 1 ) . . 155 5-14 Extrapolation from dissociation in the A ' state. (All values in cm - 1 ) 157 A - l Data used in the global analysis (all units in cm - 1 ) 171 A-2 Q-lines used in the analysis of il-type doubling (all units in cm - 1 ) 214 A-3 Unblended lines from quasi-bound levels above the dissociation limit. These were not used in analysis. All units in c m - 1 220 A-4 Transitions in the extra series associated with v = 27 of the A state (all units in cm - 1).226 vi List of Figures 2-1 Vector model for the total electronic orbital angular momentum, L in the Hund's case (a) coupling scheme. The projection along the internuclear axis is Ah, and the vector precesses about the internuclear axis 7 2-2 Hund's case (a) coupling scheme. The electronic angular momenta, L and S, couple strongly to the internuclear axis, and the component of angular momentum along the axis, Q, adds to the angular momentum due to rotation of the nuclei, R, to give the total angular momentum (exclusive of nuclear spin) for the molecule, J 10 2-3 Hund's case (c) coupling scheme. The electronic angular momenta couple more strongly to each other than to the internuclear axis 10 2-4 Bound and dissociative potential energy curves for a diatomic molecule 13 2-5 Rotational distortion of the potential energy curve, the result of treating the rota-tional energy as an effective potential energy. 17 2-6 A-type doubling in n states, from an interaction with a E - state at higher energy. Only the f parity levels are perturbed, and as a result are lower in energy than their e parity counterparts 24 2-7 Splitting of magnetic sublevels in an external magnetic field. To first order, the energy spacing between levels is constant. The levels are degenerate in the absence of an external field 29 2-8 Orientation of linearly polarized light, for a phase difference of A between LHCP and RHCP light 31 2-9 Labelling of electric dipole transitions 33 2- 10 Lineshapes associated with the real and imaginary parts of the complex index of refraction 37 3- 1 An R(0) transition in the presence of an external magnetic field. The transition frequency in the absence of an external magnetic field is uQ 49 3-2 Frequency separation of absorption profiles for LHCP and RHCP light in the presence of an external magnetic field, for the field in the +Z direction. vQ is the transition frequency in the absence of external fields 50 3-3 Frequency separation of dispersion profiles for LHCP and RHCP light in the presence of an external magnetic field. vQ is the transition frequency in the absence of external fields 52 vii 3-4 The possible transitions with k along H for an R(2) line. The A m j = +1 transitions are driven by LHCP light, and the Amj = -1 transitions are driven by RHCP light. 58 3-5 Relative transition probabilities for the different possible transitions within an R(2) line. The vertical scale is in arbitrary units; only the relative heights are important. 58 3-6 Relative transition probabilities for transitions within a P(4) and within a Q(3) line. 60 3-7 Calculated frequency shift and intensity perturbation contributions to the first har-monic signal in Aa 71 3-8 Calculated first harmonic signal from A a 71 3-9 Calculated frequency shift and intensity perturbation contributions to the first har-monic signal in A n 72' 3-10 Calculated first harmonic signal from An 73 3-11 Calculated frequency shift and intensity perturbation contributions to the second harmonic signal 73 3-12 Calculated second harmonic signal, including both frequency shift and intensity per-turbation contributions 74 3-13 Calculated first harmonic signal for P(5). The phase is opposite to that of the R line. 75 3-14 Calculated first harmonic signal for a Q(4) line. Two contributions to the intensity perturbation signal nearly cancel, and the signal looks almost symmetric. The phase is the same as that of an R line 75 3- 15 Interference effect between two lines well-separated in frequency, the result of elec-tronic state mixing affecting the lineshapes. The signal shown was taken in second harmonic 78 4- 1 Potential energy curves for selected valence states of 7 9 B r 2 79 4-2 Experimental setup for Doppler-limited magnetic rotation spectroscopy 81 4- 3 Experimental setup for Doppler-free magnetic rotation spectroscopy. 85 5- 1 The frequency difference between R(J) and P(J) is a direct measure of the frequency separation between levels in the upper state. The frequency difference between R(J-1) and P(J+1) is a measure of the spacing in the lower state 89 5-2 Experimental traces for first harmonic signals from transitions in the 18'—1" band of the A - X system. The vertical scale is in arbitrary units, but the units are the same in all three plots 93 5-3 Calculated first harmonic signals in the 18'-1" band of the A - X system. The vertical scales are in arbitrary units, but the units are the same in all three plots 94 viii 5-4 Calculated first harmonic signal for P(54) with a different g-factor. The units on the vertical axis are the same as in Figure 5-3 94 5-5 Calculated second harmonic signals in the 18'-1" band of the A - X system 95 5-6 Comparison of first and second harmonic spectrum in the same frequency region. . . 100 5-7 The manually assigned center of gravity for a skewed first harmonic lineshape. . . . 101 5-8 Difference between the energies of f parity levels (E-^(J), determined from Q lines) and e parity levels (E e(J), determined from R and P lines) as a function of J(J+1). The f parity levels are lower in energy than the e parity levels 102 5-9 Electronic states involved in the fi-type doubling in the A state. There are two states that interact with f parity levels, and one state that interacts with e parity levels. . . 104 5-10 First differences for B-values. The B-values were determined from least-squares fits that included distortion constants up to H„ for all vibrations I l l 5-11 First differences for B-values. The B-values were determined from least-squares fits that included distortion constants up to H„ for all vibrations except v = 35, 36 and 37, for which the fits included distortion constants only up to D„ 112 5-12 Uppermost portion of the inner, repulsive wall of the calculated R K R potential. . . . 113 5-13 Plot to find the form of the smoothing function and the vibration above which smoothing should be performed 114 5-14 The difference between the vibrational energies used to calculate the R K R potential and the energies determined from the potential, in both the corrected and uncor-rected cases 119 5-15 Difference between the B-values used to calculate the R K R potential and the B-values determined from the potential, in both the corrected and uncorrected cases. . 119 5-16 First difference of B-values. The B-values come from least-squares fits with the distortion constants fixed to the values determined from the potential in the global analysis 125 5-17 Energy separation between the lowest two dissociation limits for 7 9 B r 2 129 5-18 Discrepancy of data above the dissociation limit from the results of the global analysis. 133 5-19 A selection of extra lines associated with transitions in v = 27 of the A state. For R(l) , it is not clear which line is "main" and which is "extra." 135 5-20 Separation between the lines assigned as main and extra lines in v = 27 of the A state. 136 ix 5-21 The relative intensity of the line assigned as the extra line as compared to the line assigned as the main line in v = 27 of the A state. The data point for J = 31 is speculative; see text 137 5-22 Structure in Q lines in v = 27 of the A state 138 5-23 First differences in B-values for various states in I2 and Br2 143 5-24 Example of an extra line observed with the v = 16 R(43) line 145 5-25 Examples of the extra lines observed in v = 20. Shown are 20'-l" R(71) and P(71). 145 5-26 Examples of extra lines observed in v = 23. Shown are 23'-l" R(15) and 23-2" P(17).146 5-27 Examples of extra lines observed in v = 24. Shown are 24'-2" R(17) and P(19). . . . 146 5-28 Extra lines associated with high-J lines in v = 24. Shown are 24'-l" R(57) and P(59).147 5-29 Examples of extra lines observed in v = 30. Shown are 30-1" R(42) and P(44). . . . 148 5-30 Extra lines in v = 31. Shown are 31'-2" R(23) and P(25) 148 5-31 Examples of extra lines observed with high-J transitions in v = 31. Shown are 31'-1" R(41) and P(43) 149 5-32 LeRoy-Bernstein plot for the A state of Br2 152 5-33 Expanded view of the LeRoy-Bernstein plot for the A state of Br2 for data near the dissociation limit 152 5-34 The slope of the LeRoy-Bernstein plot for the A state of 7 9 B r 2 as a function of v. . . 154 5-35 LeRoy-Bernstein plot for the A ' state of Br2 156 5-36 R(0) + R(l) spectra for several vibrations in the A state 159 5-37 Experimental and calculated spectra for selected low-J lines showing structure in the 21'-1" band of A - X 160 5-38 Doppler free magnetic rotation for 20'-l" R(70) in the B - X system of I2 163 5-39 Doppler free magnetic rotation for 20'-l" R(6) in the B - X system of I2 164 x Acknowledgments I would like to thank Dr. Irving Ozier for his patient, encouraging support. His ability to systematically mull over a problem from different angles serves as an excellent role model. I would also like to thank Dr. Bill Dalby for his encouraging and tireless (at least up to several hours into a group meeting) support. His intuition serves him as well as other people's knowledge, and his love for generating ideas is another excellent role model. I hope I came away with the best traits of both supervisors, since they complement each other very well. I would also like to thank Drs. Jim Booth and Wing Ho for the initial instructions with the laser system and with experimental procedures. I would also like to thank Dr. Alak Chanda, Shixin Wang and Jyrki Schroderus for making the workplace enjoyable. I would like to thank my parents (both biological and by marriage) for their support and belief. It is very much appreciated. I would especially like to thank my wife, Adrienne, for her above the call of duty contributions to the thesis preparation, for putting up with the extended stretch of thesis activity, and most of all for making my life a better place. xi 1 INTRODUCTION 1 1 Introduction The advent of tunable frequency lasers has resulted in increased interest and activity in the study of electronic states of molecules. The diatomic halogens and inter-halogens in particular have been fertile ground for study, revealing many previously unknown or unobserved molecular interactions. Investigation of the A 3 I I i u electronic state of Br2 using standard spectroscopic techniques such as absorption or fluorescence is hampered by the presence of the B 3 i l 0 + U electronic state. Transi-tions from the ground X X E^" electronic state to the B state are generally much stronger with these techniques than signals due to transitions from the X state to the A state. With magnetic rotation spectroscopy, however, the opposite is true: in the frequency region studied in the current work, the signals due to transitions from the X state to the A state are much stronger than the signals due to transitions from the X state to the B state. This makes magnetic rotation spectroscopy an ideal tool for studying the A state of Br2-The current foray into the exciting world of magnetic rotation spectroscopy began when this author was handed a magnetic rotation spectrum of 7 9 B r 2 , originally taken by Dr. Alak Chanda. In this 40 c m - 1 span, there were hundreds of unassigned lines (roughly half the total number), some of which were among the strongest signals in the spectrum. The lines were eventually assigned as coming from high-J transitions in the A 3 I I i u - X 1 E + system. These high-J signals, together with the suppression of signals from the overlapping B 3 n 0 + U -X 1 E^ system, opened up the possibility of investigating the A state of 7 9 B r 2 in more detail than has ever been achieved before. 1.1 Background Previous knowledge of the A 3 I I i u state of Br2 comes mainly from an absorption study by Coxon [1], with measurements that extended over the range of vibrations 7 < v < 24. Previous data for vibrations above v = 24 is minimal, coming from 10 overlapped transitions measured during the course of a multi-photon experiment [2]. The only other high-precision data available for the A state is in v = 0 [3]. The current work gives a significant improvement in precision in the range of vibrations that overlaps data from previous works, and the current work extends the measurements in the A state to higher v, very close to the dissociation limit. Magnetic rotation spectroscopy came into use for diatomic molecules early in the 1900s. The unexpected observation of magnetic rotations signals in the B 3 I I 0 + U -X 1 E+ system of I2 was inves-tigated in detail for six rotational lines by Wood and Ribaud [4]. The mechanism that gave rise to the signals was explained by Serber [5] as resulting from an interaction of the B state with the 1 INTRODUCTION 2 A 3 I I i u state of I2. The treatment of magnetic rotation theory of the time was only valid for frequen-cies well separated from the resonance frequency for a transition, and no quantitative calculations could be performed. Asymmetries in the lines were explained by Carroll [6] as a perturbation of transition intensities resulting from the interaction between the A and the B states. Again, the fact that the theory was not valid at resonance prevented a quantitative test of this hypothesis. The general arguments behind both of these explanations are correct, but it seems more likely that the signals in the B - X system of I 2 arise from the known [7] interaction of the B state with a 1Uiu state rather than through an interaction with the A state. The formalism developed in the current work accounts for the presence of magnetic rotation signals in the B - X system of I2 and for the asymmetry in magnetic rotation lines, and the expressions in the current work (unlike those in [5] and [6]) are valid at resonance. The expressions for magnetic rotation signals given by Serber in [5] were later extended to be valid at resonance. Notable is a paper by Stalder and Eberhardt [8] in which the magnetic rotation signals for transitions in the A - X system of IC1 were calculated. The expressions presented in [8] include the effect that caused the asymmetery discussed by Carroll, but did not include the effect that gave rise to magnetic rotation signals in the B - X system of I 2 . In [8], it was not clear how the lineshapes were determined. In the current work, the lineshapes are derived from first principles and explicitly account for the effects of phase-sensitive detection. Probably the most common procedure for calculating magnetic rotation signals currently in use is described in the paper by Liftin et al [9]. The expressions presented there do not account for the asymmetry of magnetic rotation lines or the magnetic rotation lines in the B - X system of I 2 . The expressions in [9] (and all of the previous works, for that matter) also omit important contributions to the magnetic rotation signal from dichroism induced by stress on cell windows. The formalism in the Liftin paper does, however, account for Doppler-broadening and also provides a systematic procedure for working out the effects of phase-sensitive detection. Another formulation for calculating magnetic rotation signals was derived by Nienhuis et al [10]. The formulation in this paper takes into account the effects that give rise to magnetic rotation signals in the B - X system of I2. In theory, the expressions could also explain the asymmetry of magnetic rotation lines explained by Carroll, except only the real parts of the equations are taken, whereas the asymmetry would come from the imaginary part. The Doppler effect was not accounted for. The effects of stress-induced birefringence were not accounted for. There was no provision made for the effects of phase-sensitive detection. A recent paper by Brecha et al [11] does include the effects of stress-induced birefringence 1 INTRODUCTION 3 (although it is attributed to imperfect polarizers, which would also give a contribution, but the effects are likely much, much smaller than the effects caused by the cell windows), as well as the Doppler effect. The formalism in [11] does not account for the presence of magnetic rotation signals in the B - X signals of I2, nor does it account for the asymmetry of magnetic rotation signals described by Carroll. The expressions presented, as well as the method used to account for the effects of phase-sensitive detection, are only valid for a P(l) transition or an R(0) transition, which limits the usefulness of the formalism. Note also that because it is the lower state that is magnetically active in O2, the molecule studied in this paper, population effects in the lower state (see e.g. [5]) should be taken into account, and they were not. In the current work, the formalism includes the aspects of magnetic rotation often neglected in other works: the presence of magnetic rotation in the B - X system of I2, the asymmetry of magnetic rotation lines, the effects of stress-induced dichroism and the Doppler effect. In addition, the effects of phase-sensitive detection are accounted for explicitly (rather than following the more complicated procedure described in [9]), which greatly simplifies the calculation procedure in comparison to the standard method. Other than the early studies on the B - X system of I2, there were several low-resolution magnetic rotation studies performed on diatomic molecules in the 1930s. See the references in [12] on this subject area. A higher resolution (±0.1 cm - 1 ) magnetic rotation study of diatomic inter-halogens were performed in the late 1950s, on the A - X systems of IC1 and IBr [13]. The signals from the A - X system of IC1 were later used to test the calculation procedure in [8]. There apparently were no signals observed from high-J transition in these systems (at least no mention was made of such signals), unlike the A - X system in Br2- The B - X system of IBr was measured with magnetic rotation spectroscopy [14] in the 1970s, although transitions to only a very narrow range of vibrations in the B state gave signals strong enough to observe (with the setup used). It was mentioned in this paper that magnetic rotation signals were looked for in the B - X system of IC1 as well, but were not observed. Also in the 1970s, there was a magnetic rotation study of transitions to dissociative states in I2, Br2 and CI2 [15]. No magnetic rotation studies have been performed on the diatomic halogens or inter-halogens using the improved sensitivity available from tunable lasers (although a variation of magnetic rotation spectroscopy has been demonstrated using transitions in the A - X system of IC1 [16]). A larger range of vibrations could likely have been studied in the B - X system of IBr using lasers than was possible for the study in [14], and at a significantly improved resolution. The superior sensitivity offered by lasers would also probably make possible measurement of the B - X system of 1 INTRODUCTION 4 IC1 with magnetic rotation spectroscopy. It was also mentioned that the signals in the A - X system of IBr were very weak; lasers would improve the sensitivity of those measurements, as well as the precision by more than an order of magnitude. Since the advent of tunable lasers, there have been few magnetic rotation studies of diatomic molecules. The molecule NO was used to demonstrate the sensitivity of magnetic rotation spec-troscopy in Liftin et al [9], and the same molecule was also studied in [17] with this method. The main role that magnetic rotation spectroscopy has played (using laser techniques) in diatomic molecules seems to be for simplifying and sorting complex spectra [18], [19]. Magnetic rotation was used to enhance the sensitivity in the study of C2 [20]. There has been an observational study done in molecular oxygen [11]. Magnetic rotation spectroscopy has been coupled with other spectroscopic techniques to enhance sensitivity. Molecular oxygen was used to demonstrate the combination of cavity ring-down spectroscopy and magnetic rotation spectroscopy [21]. PdH and NiH were used to demonstrate the combination of frequency-modulation spectroscopy and magnetic rotation spec-troscopy [22]. IBr was used to demonstrate a combination of magnetic rotation spectroscopy and polarization spectroscopy [23], a technique used for Doppler-free spectroscopy. The use of magnetic rotation signals in O2 has been proposed as a way to measure magnetic fields [24]. Magnetic rota-tion spectroscopy has also been proposed as a technique for measuring the concentrations of NO and N 0 2 in the atmosphere [25] and [26]. It is clear that magnetic rotation spectroscopy has not been used to its fullest advantage for investigating diatomic halogens and inter-halogens, or for investigating diatomic molecules in gen-eral. Magnetic rotation spectroscopy often seems to be viewed (and employed) almost as a curiosity. One of the main purposes of the current work is to demonstrate a strength of magnetic rotation spectroscopy that has not been exploited to its full potential. The selectivity of magnetic rotation spectroscopy can be used to suppress the signals from some molecular systems (such as the B - X system in B r 2 ) , and this allows the study of systems (such as the A - X system of Br2) for which absorption spectroscopy could not yield comparable results. One drawback to the approach in the current work was the fact that the data could not be taken all at once, as it could with Fourier transform spectroscopy; the data had to be taken in 1 c m - 1 segments. The gathering of the experimental data therefore required a major investment of time, but the amount of information the study provided on the A 3 I I i u state of Br2 made it worth the extra effort. 1 INTRODUCTION 5 1.2 Thesis Organization Section 2 gives background theory, following mainly a semi-classical approach in order to remain consistent with the derivations performed for magnetic rotation. The background includes the basic theory necessary for the understanding of current spectroscopic techniques. Section 3 derives the theory for magnetic rotation spectroscopy. A method of calculating mag-netic rotation signals is developed that is simpler and more convenient (for experiments that use an oscillating magnetic field) than the standard method used in the literature. The signal is explicitly divided into contributions from different harmonics of the magnetic field's oscillation frequency, and closed algebraic expressions are derived from which the signal can be directly calculated. Some example calculations are also provided. Section 4 describes the Doppler-limited experiment performed on 7 9 B r 2 , and the frequency measurements of the signals. A Doppler-free setup for magnetic rotation is also briefly described. Section 5 gives the results of the experiments. The data on the A 3 I i i u state is used to perform a global analysis on this state. Observations on the fi-type doubling in the A state are presented. Extra lines in the spectrum were used to deduce some information on the A ' 3 I l 2 u state of Br2-Hyperfine effects for low-J transitions were considered, and results from preliminary tests with the Doppler-free magnetic rotation technique are shown. Section 6 outlines some potential future work in diatomic halogens and some possible further investigations using magnetic rotation spectroscopy. 2 GENERAL THEORETICAL CONSIDERATIONS 6 2 General Theoretical Considerations In the field of molecular spectroscopy, it can get no simpler than a diatomic molecule, but even this relatively simple system has many factors to take into consideration. A brief description of some important physical quantities for diatomic molecules will be presented here, including angular momentum of the electrons and nuclei that compose the molecule, energies (both kinetic and potential) associated with these constituent particles, and symmetries for the quantum mechanical wavefunction describing the molecule. For more complete treatments of these, and related, topics, see [12], [27] and [28]. The theory given here follows a semi-classical approach in order to lead more directly into the derivations performed for magnetic rotation. The approach is not intended to be a rigorous treatment of the topics covered, but it gives a much better insight into magnetic rotation than would a pure quantum mechanical treatment. 2.1 Angular Momenta The various angular momenta in the molecule couple together to give a total angular momentum for the molecule, J (or F, if hyperfine effects are considered). The best description of how the angular momenta couple together depends on which interactions are the strongest. There are five different idealized coupling schemes that describe the possible ways in which the angular momenta can combine to give J. These coupling schemes are known as "Hund's cases." The two coupling schemes important for this thesis are Hund's case (a) and Hund's case (c). See [12] or [27] for discussions of the other coupling schemes. The majority of diatomic molecules are best approximated by Hund's case (a) coupling, and the theory developed in the current work will therefore use the quantum numbers and concepts appropriate to that coupling scheme. The molecule being studied, however, 7 9 B r 2 , tends more towards Hund's case (c) coupling. The theory developed will be applicable for this coupling scheme by simply treating the Hund's case (a) quantum numbers as effective parameters that must be determined from the experimental data. It will be pointed out where this treatment is necessary during the development of the theory and during the analysis. The description of the angular momenta begins with Hund's case (a). Consider first the angular momentum associated with the orbits of the electrons. Molecular electrons can exist only in certain, discrete orbits, so-called stationary states, and each state will have an associated angular momentum. Imagine the simple example of an electron executing a circular orbit around the centre 2 GENERAL THEORETICAL CONSIDERATIONS 7 of mass of the nuclei in the molecule: The angular momentum vector would be perpendicular to the plane of the orbit (its sense determined, by convention, using the right-hand rule), with a magnitude that was constant and quantized. Of course, the actual orbits are more complex than this, but the concept remains the same. The orbit of a molecular electron can only have certain shapes (see [12] for examples), and the orbital angular momentum associated with the electron must be some multiple of a unit quantity for the angular momentum. The unit quantity for angular momentum is h (= where h is the Planck constant. When there is more than one electron in the molecule, the angular momenta of these electrons couple together to give a total electronic orbital angular momentum, denoted here by L. The magnitude of L (|L|) is \JL(L + l)h, where L is the quantum number used to label this angular momentum and takes on the possible integer values L = 0,1,2,.... Figure 2-1: Vector model for the total electronic orbital angular momentum, L, in the Hund's case (a) coupling scheme. The projection along the internuclear axis is Ah, and the vector precesses about the internuclear axis. A molecule-fixed frame of reference is defined by taking the z-axis along the internuclear axis, as shown in Figure 2-1. An alternate frame of reference, the space-fixed frame, defines a Z-axis, which lies along an external electric or magnetic field. Note the use of capital letters for the space-fixed co-ordinate system and lower-case for the molecule-fixed. This convention will be followed throughout the current work. In a diatomic molecule, the electric field produced by protons in the two nuclei is cylindrically symmetric. L couples to this electric field, and because of the cylindrical symmetry of the field. 2 GENERAL THEORETICAL CONSIDERATIONS 8 the component of L along the internuclear axis is (to first order) conserved [29]. The component of L along the axis is m ^ , as shown in Figure 2-1, where the quantum number takes on the possible integer values: rriL = -L, -L + 1, ...,0, . . . ,L - 1,L. The energy of the molecule depends (to first order) only on the magnitude of m^, not the sign [12]. It is therefore convenient to introduce a new quantum number, A, such that A EE \mL\. The vector L does not remain fixed in direction, but rather precesses (i.e. rotates at a constant angle) about the internuclear axis [12], as indicated-by the dotted trajectory in Figure 2-1. The need for this precession is intimately related to the uncertainty principle. The z-component of the electrons' total angular momentum is known exactly (if A is a good quantum number), and so there has to be a large uncertainty in the x- and y-components of the angular momentum. Note that in reality, even for molecules whose angular momentum coupling is most closely approximated by Hund's case (a), L is typically not considered to be a good quantum number [12]. Besides the angular momentum associated with their orbital motion, electrons also possess an intrinsic spin angular momentum. Each electron carries the same spin angular momentum, of magnitude x ^h. Contributions from all molecular electrons sum together to give S, the total electronic spin angular momentum for the molecule. The magnitude of S is y/S(S + l)h. The quantum number S is either integer (if the number of electrons in the molecule is even) or half-integer (if the number of electrons is odd), unlike L, which can only have integer values. The electric field of the nuclei does not interact with magnetic moments (only electric ones), and so it does not directly constrain 5, as it does L. There can also be a magnetic field directed along the internuclear axis, however, due to the orbiting electrons. An electron moving along a circular path, very much analogous to a current loop, gives rise to a magnetic field. Since the orbits of the electrons produce a constant component of angular momentum along the internuclear axis, the magnetic field created by their motions must also have a constant component along that direction. S couples to this magnetic field created by the orbiting electrons, and the component of S along the field direction (i.e. along the internuclear axis) is Eft, where the quantum number E (not to be confused with the label for A = 0 electronic states) can take on the values E = - S , - S + 1, . . . ,S-1,S. Just as with L, S precesses about the internuclear axis. It would be expected that the energy of the molecule would depend only on the magnitude of the spin, not the component along the 2 GENERAL THEORETICAL CONSIDERATIONS 9 axis, which would lead to a 2S+1 degeneracy of the energy [29]. However, relativistic effects make the energy become a function of E, and so each component represents a different energy for the molecule. Unlike A, the quantum number £ can take on both positive and negative values. Note that if A = 0 (i.e. if there were no component of orbital angular momentum along the internuclear axis), then there would be no magnetic field along the internuclear axis, and S would not couple to the axis. There is angular momentum associated with the nuclei as well, both from end-over-end rotation of the nuclei, denoted here by R, and from the intrinsic nuclear spin angular momentum, / . Because the coupling of I to the other angular momenta is relatively small, the effects of nuclear spin (termed hyperfine effects) are neglected for now, but will be discussed in more detail in Section 2.10. The angular momentum associated with rotation of the nuclear frame, R, is perpendicular to the internuclear axis, as shown in Figure 2-2. A new quantum number, ft, is introduced to describe the total component of electronic angular momentum along the internuclear axis, ft ft, where ft is defined in Hund's case (a) by: ft = A + S. The component of angular momentum along the internuclear axis adds to R to give the total angular momentum (exclusive of nuclear spin), J, as shown in Figure 2-2. In Figure 2-2, the angular momenta, R and ft, where ft = VlTik and A; is a unit vector along z in the molecule-fixed frame (i.e. along the internuclear axis), both precess about the resultant angular momentum, J. The second coupling scheme considered here, Hund's case (c) coupling, occurs when the inter-action of a particular electron's intrinsic magnetic moment with its own orbital magnetic moment (an effect known as spin-orbit coupling) becomes larger than its interactions with other electrons or with the electric field generated by the nuclei. The quantities L and S are not well-defined in this coupling scheme. Instead, a total electronic angular momentum Ja (= L + S) is defined. The component of Ja along the internuclear axis is ftft, and the sum of this component with R to give J proceeds in the same manner as for case (a). The coupling scheme is illustrated in Figure 2-3. 2 GENERAL THEORETICAL CONSIDERATIONS 10 Figure 2-2: Hund's case (a) coupling scheme. The electronic angular momenta, L and S, couple strongly to the internuclear axis, and the component of angular momentum along the axis, ft, adds to the angular momentum due to rotation of the nuclei, R, to give the total angular momentum (exclusive of nuclear spin), J . » Z Figure 2-3: Hund's case (c) coupling scheme. The electronic angular momenta couple more strongly to each other than to the internuclear axis. Q = Q t i k R 2 GENERAL THEORETICAL CONSIDERATIONS 11 2.2 Energy 2.2.1 The Born-Oppenheimer Approximation The total energy of a system can be determined by solving the time-independent Schrodinger equation: H |tt„) = En \mn) (2-1) where H is the Hamiltonian operator, E n is a particular eigenvalue and |vl/n) represents the wave-function. Unfortunately, even for the relatively simple system of a diatomic molecule, coupling among the motions of the various constituents makes exact solutions of Eq 2-1 very difficult to determine. A standard technique in such a situation is to establish a basis set of wavefunctions that (hopefully) closely approximates the true wavefunctions of the system. The basis set for diatomic molecules derives from what is known as the Born-Oppenheimer (BO) approximation (see e.g. [30]). The electrons in a molecule typically move at a much faster rate than the nuclei. Consequently, many cycles of electron motion occur during a small portion of a cycle of nuclear motion, which means the electron can always be considered to have reached its equilibrium state corresponding to the nuclear configuration (the snapshot of the nucleus at that instant). In other words, the electron adjusts almost instantaneously, on time scales of nuclear rotation or vibration, to changes in the nuclear configuration. This allows the simplifying approximation that the electronic energy can be treated separately from vibration and rotation of the nuclei, and terms in the Hamiltonian operator in Eq 2-1 that represent coupling between nuclear and electronic angular momenta are excluded (to first order)'. As a result of this separation, a basis set of wavefunctions that shall be referred to as the "Born-Oppenheimer wavefunctions," ty30^, can be written as the product of two wavefunctions: * B O ) = |tte) \yvib,rot), (2-2) where |$ e) is an electronic wavefunction, and \tyVib,rot) represents the part of the wavefunction that describes the motion of the two nuclei. The nuclear part (i.e. for the nuclear framework, not to be confused with the nuclear part of the wavefunction that enters into the consideration of hyperfine effects) involves two different types of motion. The first type is oscillation of the bond length between the two nuclei, called vibration, and the second type is end-over-end rotation of the molecule. A third type of motion that could be attributed to the nuclei, translation of its center of mass in a space-fixed frame, need not be considered here. 2 GENERAL THEORETICAL CONSIDERATIONS 12 2.2.2 Electronic States Treating the electronic part separately allows the definition of an electronic state. For Hund's case (a) coupling, the quantum numbers A, Q, and S are used to label a particular electronic configuration. A symbol is assigned to the electronic state according to the value of A: A 0 1 2 3 .... symbol S n A $ .... The designation for an electronic state has the form: 2S+1symbol(A)n {e.g. 3HU3 H2,1 So)-The term 2S+1 is called the multiplicity. It is the number of possible values for the projection of S along the internuclear axis, and therefore represents the maximum number of values of for a particular value of A. For Hund's case (c), the only quantum numbers available to label the electronic state are J a and Q. However, only Q is considered to be a good quantum number [30], and the electronic states in case (c) are therefore labelled according to their value of Q (e.g. an electronic state labelled lUi in Hund's case (a) would simply be labelled "1" in Hund's case (c), a state would be labelled "2", etc.). 2.2.3 Potential Energy Curves In the BO approximation, a zeroth-order Hamiltonian operator can be defined as the sum of an electronic part and a nuclear part: HQ = He + HVibiTOt. (2-3) Consider first the effect of H 0 acting on the electronic part of the wavefunction in Eq 2-2. Keep in mind that H.Vib,rot does not operate on this part of the wavefunction. H0 |*e) - He |*e> = (Te + Ue(r)) |* e) = Ee |*e) . Te is the electron's kinetic energy, r denotes internuclear separation, and U e(r) is the potential energy for a given nuclear configuration (i.e. at a particular value of r). The basic premise of the BO approximation is that the nuclei can be taken as stationary when considering the instantaneous energy of the electron, and the electronic potential energy is therefore written as a function of r. The nuclear portion of the Hamiltonian operator can be written as a sum of the kinetic energy operator of the nuclei (T/v) and the potential energy operator associated with the nuclei (VN), Hvib,rot = T)\r + Vjv(r). • 2 GENERAL THEORETICAL CONSWERATIONS 13 The approximate solutions to Eq 2-1 using the BO basis set and zeroth-order Hamiltonian are determined by solving [TN + Te + [VN(r) + Ue{r)}] \^vib,rot) = E\^vib,rot) • (2-4) The term Vjv(r) + U e(r) describes a background potential in which the molecule can vibrate and rotate. Figure 2-4 shows the two possible varieties for the potential energy curves of electronic states in diatomic molecules. The top curve represents a dissociative (or repulsive) electronic state. The total energy of the molecule would be greater than the total energy of two separated atoms, and so the molecule is expected to dissociate into its constituent atoms when it enters such an electronic state. The curve at lower energy in Figure 2-4 represents a bound electronic state. Figure 2-4: Bound and dissociative potential energy curves for a diatomic molecule. J L_i I i I i I i L e Internuclear separation r (A) T e in Figure 2-4 is the electronic kinetic energy. At low r (small internuclear separation), the potential increases rapidly due to repulsion between the two nuclei. At large r, the two nuclei get so far apart that they are no longer bound together. The energy at which this separation occurs is called the dissociation limit. The minimum of the attractive potential, the potential for the bound state, occurs at the separation defined as r e, the equilibrium internuclear separation. The binding energy, D e , is the difference in energy between the dissociation limit and the minimum of the potential. 2 GENERAL THEORETICAL CONSIDERATIONS 14 2.2.4 Vibrational Energy To a first approximation, taking the simple model for a diatomic molecule of two point masses connected by a massless spring, vibration of the internuclear separation would be described by simple harmonic motion. Solving Schrodinger's equation for this "harmonic oscillator" gives evenly spaced energies for the system, defined by G(v) = hiuosc(v + ^), (2-5) where uiosc is 2-7T times the frequency of oscillation (i.e. the angular frequency of oscillation), and the vibrational quantum number, v, takes on the integer values: 0, 1, 2,.... The different solutions correspond to different allowed modes of the vibration. The potential for a harmonic oscillator, U(r), is a simple quadratic curve: U{r) = ^k{r-re)\ where k is the spring constant and r e is the equilibrium length of the spring. This potential is symmetric about r e. In the diatomic molecule, there is a strong repulsion at small r, while at large r, the force holding the two nuclei decreases as r increases (analogous to a decrease in spring constant k). These differences from the simple model of an oscillating spring, among other things, warp the symmetric harmonic potential into the lower curve shown in Figure 2-4. The portion of the lower potential in Figure 2-4 near the bottom is approximately harmonic, and so the vibrational levels in this region should approximately follow Eq 2-5. As energy increases, however, the energy spacing between vibrational levels get smaller and smaller as the potential deviates more and more from harmonic. Deviation of the vibrational level spacing from harmonic behaviour with increasing v is typically accounted for by adding anharmonic terms to Eq 2-5: G(y) = ue(v + )^ - wexe(v + ^)2+ueye(v + ^)3 + ueze(v + i ) 4 + (2-6) where ue —• hcoosc in the harmonic limit. toexe is a positive quantity and is much smaller than u>e. The coefficients of higher order terms (coeye, ujeze,...) could be either positive or negative and are expected to get successively smaller in magnitude. This treatment is not well-suited for vibrational levels near the dissociation limit, as will be seen later. 2.2.5 Rotational Energy Classically, the kinetic energy of a rotating object is expressed as: Erot = \Mot, (2-7) 2 GENERAL THEORETICAL CONSIDERATIONS 15 where I is the moment of inertia about the axis of rotation, and urot is the angular frequency of rotation. The molecule can be represented to first order by a rigid rotator, two point masses connected by a massless rod (rather than by the spring used in the previous section). The moment of inertia for end-over-end rotation of this rigid rotator is I = fir2, where fi is the reduced mass of the two point masses and r is the length of the rod (the separation between the two masses). Writing the rotational energy in Eq 2-7 in operator form, i.e. in terms of the angular momentum R defined previously, (where R = Iu>rot = hy/R(R + 1)), leads to the rotational Hamiltonian: HTot — 0 j , 2 ' (2-8) The operator in Eq 2-8 has off-diagonal matrix elements in the BO representation, which leads to a breakdown of the BO approximation, as will be discussed in Section 2.4. The rotational energy of the rigid rotator, analogous the classical rotational energy defined by Eq 2-7, will be derived explicitly later; it comes out to be Erot = F(J) = B [j(J + 1) - ft2] , (2-9) where B is known as the rotational constant. The present work will use the standard spectroscopic units of c m - 1 (wavenumber units). To change from energy units to wavenumber units, energies must be divided by he (h = the Planck constant, c = speed of light in a vacuum). The B-value in wavenumber units is therefore defined as: ^ 8n2ficr2' ^ ^ Note that the factor of h implicit in R has been pulled into this constant. To apply the rigid rotator model to the molecule, the reduced mass, fi, becomes the reduced mass for the two atoms. Now, consider what happens for the model with a spring connecting the two masses instead of a rod. As the molecule rotates faster (J increases), centrifugal effects cause the bond length to stretch, which increases the moment of inertia (fir2). This leads to a decrease in the rotational energy. This is accounted for by the introduction of "distortion terms" into Eq 2-9: FV{J) = Bv [ j ( J + 1) - ft2] -Dv [ j ( J + 1) - ft2]2+#„ [ j (J + 1) - ft2]3+L„ [ j ( J + 1) ft: 4 (2-11) where D^, H^, Lv, etc. are called centrifugal distortion constants. Dv is a positive quantity, but higher order terms could be either positive or negative. It has been explicitly assumed that the parameters in Eq 2-11 are functions of v. As v increases, i.e. moving toward higher energy in the 2 GENERAL THEORETICAL CONSIDERATIONS 16 potential depicted in Figure 2-4, the B-value is expected to decrease because the mean value of r increases. The distortion constants, on the other hand, increase with v, since the force between the two nuclei diminishes as they move farther apart. An equation analogous to Eq 2-6 can be written for the B-values, to parameterize their vibra-tional dependence: Bv — Be- ae(v + ^)+ (3e(v + ^)2 + le{v + + (2-12) where ae is a positive quantity, and higher order terms can be either positive or negative. 2.2.6 Total Energy The total energy of the molecular system (excluding hyperfine effects, which will be discussed later), calculated using the BO basis set, can be written as E = Te + G(v)+Fv(J). Typically, the electronic energy, T e , is the largest contribution and the rotational energy, F„(J), is the smallest. 2.2.7 Rotational Distortion of the Potential The rotational Hamiltonian (Eq 2-8) is part of the kinetic energy operator (T/v) in Eq 2-4, but the rotational energy in Eq 2-9 could be treated as an effective contribution to the potential energy. The radial part of the wavefunction can be determined by solving the following equation: [TN(r) +Te + ^ 2 [J (J + 1) - + VN(r) + Ue(r)} | ^ 6 , r o t ( r ) ) = ET |^ 6 , r o t ( r )> . See e.g. [30] for a derivation of this equation. The angular part of the wavefunction can be accounted for with the use of what are known as symmetric rotor functions [30], [31], but that will not be discussed here. The quantity k ;[J(J + l)-n2} + VN(r) + Ue(r) 8n2ficr21 represents an effective potential energy for the rotating system, the "rotationless" potential in Figure 2-4 plus an extra rotational term. So, within a given electronic state, there is a series of potential energy curves, one for each value of J (see Figure 2-5). Note that there can be quasi-bound levels above the dissociation limit. 2 GENERAL THEORETICAL CONSRJERATIONS 17 Figure 2-5: Rotational distortion of the potential energy curve, the result of treat-ing the rotational energy as an effective potential energy. 2.3 Symmetry A symmetry operation that plays an important role in molecules is inversion of space: the reflection of spatial coordinates through the origin (-X «-»• X , - Y <-> Y, and -Z Z). Since physical observables cannot change in magnitude as a result of this operation, the only possible effect on the wavefunction describing the molecule is a multiplication of the wavefunction by a phase factor. Since reflecting twice represents an identity transformation, the phase factor can only be ± 1 . A wavefunction that does not change sign as a result of this reflection is said to have even parity, while a wavefunction that does change sign is said to have odd parity. Al l symmetries correspond to some quantity being conserved [32]. The symmetry with respect to spatial inversion means that parity must be conserved. Parity is a somewhat esoteric concept compared to classical observables for which conservation laws exist, such as energy (whose con-servation law arises from a symmetry with respect to translation in time) or angular momentum (whose conservation law derives from an invariance under rotations in space). We are all familiar with rotations in space and the flow of time, but how often do we experience spatial inversion? However, parity is an important quantity in a quantum mechanical system, since the question of whether the wavefunction is symmetric or antisymmetric plays an important role in the selection 2 GENERAL THEORETICAL CONSIDERATIONS 18 rules for perturbations and transitions. The process of spatial inversion can be represented by an operator, L,,, such that J* |tt) = ± | t t> , (2-13) i.e. eigenvalues of +1 for even-parity states and —1 for odd-parity states. To put the law of conservation of parity into somewhat more formal words, the operator ISi commutes with the full Hamiltonian for the molecule, and so its eigenvalues (+1 or —1) must be rigorously good quantum numbers. 2.3.1 Molecular Symmetry and av Rather than working out the eigenvalues of ISi from Eq 2-13 in the space-fixed frame, it turns out that parity can be determined by considering symmetries in the molecule-fixed frame, a much simpler proposition. In the molecule-fixed frame, any plane passing through the two nuclei is a plane of symmetry: a reflection through this plane leaves the molecule unaltered. The operator av is used to denote such a reflection. This operation is equivalent to Is,[33], i.e. they have the same eigenvalues: +1 for + parity states and —1 for — parity states. To consider reflection of only electronic coordinates (somewhat confusingly, av is always used to denote this operation as well [30]), there is a special symmetry for a A = 0 (i.e. E) state that needs to be treated separately: av |A = 0) - ± |A = 0) = ( - l ) A s |A = 0), where As is either zero or one. There is a distinction between two types of £ states, and so a new label is added in order to keep track. For As = 0 ((-1)A e = +1), the state is labelled a E + state, and As = 1 corresponds to a £~ state. Following the phase convention of Condon and Shortley [34], the effect of av (i.e. the full reflection, not just the reflection of electronic coordinates used in the special case of a E state) on a case (a) basis set wavefunction is given by <7„|A,S,E;J,n) = ( - l ) A + A s + 5 - E + J - n | - A , 5 , - E ; J,-SI), (2-14) where As = 1 for a E _ state and 0 for all other states. Using Q — A -f E and the fact that 2(S-E) is an even integer gives • u; g <-> g E+ <-» £+; £ " «-» E " Rotational A J = 0 e <-> e; f <-> f There are standard techniques for the treatment of perturbations (see e.g. [29]). The Hamil-tonian operator is separated into the sum of a zeroth-order Hamiltonian (H„) and an interaction Hamiltonian: H = HQ + H', where H 0 has the following properties: BO Ho = En and (**° H0 ^ ° ) = 0 ( m ^ n ) . (2-18) E n is, of course, the (zeroth-order) energy of the level labelled by "n." For diatomic molecules, operators that provide a coupling between nuclear rotation and elec-tronic motion, along with any other operators excluded from the zeroth-order Hamiltonian, are collected into the interaction Hamiltonian, H' . An important property of the interaction Hamiltonian is that it can have non-zero off-diagonal matrix elements ((v%o H' *^°^) / 0 for m / n). The effect of these non-zero off-diagonal elements is to mix the states m and n, and the resultant wavefunctions for both states become hybrids of the two basis set wavefunctions, ,BO. *'Tn = -sin([3)^0 + cos((3)*BO w ^BO_^BO. 2 GENERAL THEORETICAL CONSIDERATIONS 21 where the mixing coefficient, (3, * m H' Ego-Ego has been assumed to be small (B l° the energy shift in a state n (6En) is given by second-order perturbation theory to be ;BOV2 H' 6En = BO H' n / Ego - E£0 (2-19) The perturbation provides a mutual repulsion: the state with higher energy is shifted upwards in energy by an amount 6En, and the lower state gets shifted down by the same amount. 2.4.1 The Rotational Hamiltonian The rotational Hamiltonian in Eq 2-8 can be written more explicitly as 1 Hrot = ;(RX +Ry), where R z = 0 because R is perpendicular to the internuclear axis, the z-axis in the molecule-fixed frame. In Hund's case (a), since J = R + L + S, this becomes 1 H rot 2fj,r2 (Jx Lx Sx) ~\~ (Jy Ly Sy) Using ladder operators, symmetric and antisymmetric combinations of the x and y components of angular momenta, J-i- = Jx ± ijy, L± = Lx ± iLy, and S± = Sx ± iSy, 2 GENERAL THEORETICAL CONSIDERATIONS 22 the rotational Hamiltonian can be written in a more convenient form: 1 Hrot — • 2yur2 {J2-J2) + {L2-L2z) + {S2-S2) + (L+S-+L_S+)-(J+L^+J-L+)-(J+S-+J-S+)\ . (2-20) The operators J 2 , L 2 , and S2 represent J • J, L • L, and S • S, respectively. The matrix elements associated with these three operators are diagonal, and give the square of the magnitude of the vectors: J 2 | J, ft; S, £, A) = J( J + l)h2 | J, ft; S, £, A) , L 2 | J , f t ;S ,£ ,A) = L(L + l ) f t 2 | J , f t ;S ,£ ,A) , and S 2 | J, ft; S, £, A) = S(S + l)h2 | J, ft; S, £, A ) . Note that the matrix elements shown here for L 2 and S2 (and later on for L 2 , S 2, L± and S±) are defined in pure Hund's case (a), and not in Hund's case (c). The operators J 2 , L z and Sz also have only diagonal matrix elements: Jz | J, Q; S, E, A) = Qh \ J, ft; 5, E, A ) , L 2 | J , Q ; 5 , E , A ) = Aft |J,f2;5,E,A), and 5 2 |J ,f t ;5,E,A) = Eft | J , f t ;5 ,E,A). The ladder operators have no diagonal elements. The rotational energy is given by the diagonal matrix elements of the operator in Eq 2-20: Erot = B [j(J + 1) - 0? + S(S + 1) - E 2 + L(L + 1) - A 2 ] , (2-21) where B is the rotational constant defined in Eq 2-10. The common practise is to write the rotational energy in the form already presented in Eq 2-9 Erot = B [J(J + l ) - f t 2 ] , and the remaining terms are absorbed into the electronic energy and not considered further. Only when comparing results for different isotopes of the same molecule does this simplification cause problems [12]. Just one isotopomer of Br2 was studied in the current work, and so isotope effects will not be considered. 2 GENERAL THEORETICAL CONSIDERATIONS 23 2.4.2 Uncoupling Phenomena L+ is a raising operator. Its effect on the wavefunction is to increase A (the eigenvalue of the —* projection of L along the internuclear axis) by one unit. L _ is a lowering operator, which means it will decrease this eigenvalue by unity. Similarly, S+ and S- are the raising and lowering operators corresponding to S. Explicitly, L± | J , ft; S, £, A) = \JL{L + 1) - A(A ± l)h | J, ft; S, S, A ± 1) and S± | J, ft; 5, £, A) = y/s(S + 1) - £ ( £ ± | J , f t ;S ,£ ± 1,A). The roles of the ladder operators for J are the opposite of those for L and 5, i.e. J_ is a raising operator (increases ft by one unit) and J+ is a lowering operator (decreases ft by one unit): J± | J, ft; 5, E, A) = + - f t ( f tTl ) f t | J, ft qF 1; S, E, A) . The 3±Szp operator in Eq 2-20 is called the S-uncoupling (or spin-uncoupling) operator, since it is responsible for the progressive uncoupling of the total electronic spin angular momentum from the internuclear axis as the energy spacing between rotational levels approaches the energy of the spin-axis interaction [29] (i.e. the interaction of S with the magnetic field along the internuclear axis). Similarly, J±Lzp is known as the L-uncoupling operator. The operator L±S^: plays no role in the current work, and will therefore not be discussed here. 2.4.3 A-doubling L-uncoupling gives rise to a phenomenon known as A-doubling (or as ft-doubling, in the case (c) coupling scheme). Since the L-uncoupling operator mixes (couples) states with A A = ±1 , it can couple a II state (A=l) to a E state (A=0). Recall that a II state is doubly degenerate, having two states of opposite parity that are degenerate in energy to first order. E states, on the other hand, are not doubly-degenerate, because there is only one possible value for A, A = 0. The situation is shown schematically in Figure 2-6. The labels on the left hand side show the +/- parity and the e/f parity of the levels. Because perturbations connect only states with the same J and the same parity, only half of the levels (the f parity levels) in the IT state are affected (i.e. pushed down) by the interaction with the E _ state, thereby lifting the degeneracy of the two states. The arrows in Figure 2-6 indicate the perturbations. Which of the two parity states affected by the perturbation depends on the symmetry of the E state. A E + state would perturb only the e parity levels in Figure 2-6. For a particular vibrational level (v) in the IT state, the shift from the 2 GENERAL THEORETICAL CONSIDERATIONS 24 Figure 2-6: A-type doubling in II states, from an interaction with a £~ state at higher energy. Only the f parity levels are perturbed, and as a result are lower in energy than their e parity counterparts. perturbation with the E state can be calculated from Eq 2-19, summing over contributions from all vibrational levels (v') in the E state: AE(v,J}) = ^ f En — E<£± v,J , f i + l , 5 , S , A = l f e j _ I + \v',J,Q,S,Y,,A = 0} v, J -h En,v,j - -^s±y,j v', j)I2 [J(J + 1) - ft(ft + 1)] [L(L + 1)] (2-22) Ej[,v,J ~ E-z±tV',j Note that the above equation assumes the energies are expressed in wavenumber units. The upper labels on the left side of the equation go with the upper labels on the right side of the equation. The value of (v, J\ 4j \v', J) varies slowly as a function of J. In addition, if the two states are far apart in energy, the energy difference, [Eu,v,J ~ E^±yj] is roughly constant as a function of J. With these conditions, the energy shift is AE(v, J})= qv[J(J + 1) - + 1)] « qv[J(J + 1) - ft2], (2-23) where qv is called the A-doubling (or ft-doubling) constant. The purpose of the approximation in Eq 2-23 was to write the energy shift with the same J-dependence as the rotational energy in Eq 2-9. Note again that the set of perturbed parity levels (e parity levels or f parity levels) depends 2 GENERAL THEORETICAL CONSIDERATIONS 25 (2-24) This constant can be positive or negative, depending on the direction of the push, i.e. on the sign of the energy denominator in Eq 2-22, which indicates whether the £ state is above or below the II state. Referring back to Eq 2-9, the energy shift in Eq 2-23 resembles a change in the rotational constant, B„. Thus, one set of rotational energy levels (the set not shifted) are characterised by the true B-value, and the set of perturbed rotational levels has an apparent B-value: Often, A-doubled states are incorrectly represented as being symmetrically split about the mean'. Bv-%jf. This is an expedient approximation when it is not known which set of levels is perturbed, but it must be realized that this does not give the true B-value. It is often the case that both levels are perturbed, but one is perturbed more than its partner. Under those circumstances, only an approximation to the true B-value can be determined. The L-uncoupling operator can also couple a II state to a A state (i.e. A = 2 state), but since both states are doubly degenerate, this would not (to first order) lead to a splitting of the two parity levels. The fact that a £ state is not doubly degenerate is what leads to the A-doubling in a II state. 2.5 Interaction with Magnetic Fields For magnetic rotation spectroscopy, it is important to consider what happens when the molecule is placed in an external magnetic field. As mentioned previously, the orbits of the electrons are analogous to a current loop, which produces a magnetic moment where I is the current and A is the area it encloses. Take the classical picture of an electron in a circular orbit of radius r. The electron making v revolutions per second is equivalent to a current of -eu, where e is the magnitude of the charge on the electron. The magnetic moment is therefore BP£rturbed = Bv + qv. (2-25) position, with the apparent B-values for the two different sets of parity levels written as Bv+^ and fi = —euivr, (2-26) 2 GENERAL THEORETICAL CONSIDERATIONS 26 where the magnetic moment is directed opposite the angular momentum vector, since it is a nega-tively charged particle. The classical angular momentum of this electron is L = mvunearr = 2nmvr2, (2-27) where m is the mass of the electron. Comparing Eq 2-26 and Eq 2-27 yields fl = »L = - ( ^ ) L . (2-28) Although the above expression was derived using classical methods, quantum mechanical treatments produce the same result [32]. There are higher order correction terms from relativistic effects and from the fact that a reduced mass (rather than electron mass) should be used [36], and these are accounted for by multiplying the right hand side of Eq 2-28 by a g-factor, g£. This is typically assumed to be equal to 1, since the correction terms are very small. The magnitude of L is y/L(L + l)h. The factor h multiplied by ^ from Eq 2-28 gives a quantity known as the Bohr magneton (/xg): W = (2-29) The magnetic moment associated with electron spin, S, is * s = ~ 9 S ^ ( 2 " 3 0 ) where the g-factor for electron spin, gs, is approximately 2.0023. For the magnetic moment produced by rotation of the nuclei, consider two bare nuclei (i.e. with no electrons), each possessing Z protons, and therefore having charge Ze. Similar to the orbiting electron, the magnetic moment produced is " " " O b s ; ) * ( 2 - 3 1 ) where mp is the mass of the proton, and ^ is the ratio of charge to mass number (ratio of number of protons to the number of protons plus neutrons). Note that it was assumed that the masses of neutrons and protons were equal. This magnetic moment is opposite in sign to the electronic magnetic moments, since it comes from the motion of positively charged protons as opposed to negatively charged electrons. There are also higher order contributions to this magnetic moment that arise from pertur-bations between electronic states caused by rotation of the molecule (i.e. the S-uncoupling and 2 GENERAL THEORETICAL CONSIDERATIONS 27 L-uncoupling). These effects actually play a major role in the magnetic rotation spectrum, as will be seen later. The quantity = 7,— n is called a nuclear magneton, a factor of roughly 1800 smaller than its electronic counterpart, the Bohr magneton. The magnetic moment associated with the nuclear spin, / , is the order of nuclear magnetons (as opposed to Bohr magnetons): ni = 9 N ^ (2-32) The sign of the nuclear magnetic moment (and therefore the sign of gjv) can be either positive or negative. It depends on the nucleus. 2.5.1 Splitting of Levels in a Magnetic Field The potential energy ( V m ) of a magnetic dipole (p) in a magnetic field (H) is defined as Vm = -jl-H. Neglecting the magnetic moments associated with the nuclei, since their interactions with the magnetic field produce much smaller effects than do the magnetic moments associated with the electrons, the Hamiltonian operator for this interaction can be written as: HZee = -p.H=^-(gLL + gsS) • H, (2-33) where H ^ e e is known as the Zeeman Hamiltonian. The operator in Eq 2-33 looks deceptively simple. Because L and S are coupled to other angular momenta, the coupling scheme needs to be taken into account to work out the effect of the Zeeman Hamiltonian. It is assumed that the precession of L and S around the internuclear axis is fast enough to consider only the components of their magnetic moments along the axis. However, the internuclear axis (i.e. the angular momentum U) also precesses around J , and J precesses around the external magnetic field (in the low-field limit). If the precession of J around the field is slow in comparison to the precession rates of the other angular momenta, then only the magnetic moment along J need be considered to first order. If the field strength is increased, the precession rate of J about the field direction also increases. As this precession rate approaches the precession rate of L and S about the internuclear axis, the 2 GENERAL THEORETICAL CONSIDERATIONS 28 magnetic field begins to interact directly with the magnetic moments along L and S themselves, rather than their components along J . In what is known as the high-field limit, L and S uncouple from the internuclear axis and precess around the field direction. The experiments performed for this thesis were in the low-field limit, but the fact that the magnetic moments are not actually aligned along J does play an important role because it leads to perturbations, as will be described later. The component of angular momentum along the direction of the magnetic field must be con-served. There are 2J + 1 possible orientations of J with respect to the magnetic field, corresponding to the different allowed projections, mjh, where the magnetic quantum number, mj, takes on the possible values —J, — J + 1,..., J — 1, J. The matrix elements of H ^ e e given in Eq 2-33, assuming that only the magnetic moment along J contributes, is [12] B*. = M M A + ^ ) ^ + 1 ) % / J ( 7 + 1 ) | g | = WS1mAH\. (2-34) For convenience, an effective rotational g-factor, gj, has been introduced. (9LA + gsE)n 9 J = J ( J + 1) " ( 2 " 3 5 ) For case (c) coupling, the quantity g^A + g^E is not well-defined. Although its value remains constant, it cannot be calculated a priori; it can be left as a parameter to be determined from experimental data. Because of the dot product in Eq 2-33, the Zeeman energy is different for different orientations of J with respect to the field, i.e. for states with different values of the magnetic quantum number mj. These 2J + 1 levels, which are degenerate in the absence of a field, are therefore split in energy when the molecule is brought into an external magnetic field, as depicted in Figure 2-7 for a level with J = 2. Note that the rotational g-factor is taken to be positive in Figure 2-7. Note from Eq 2-34 that the energy spacing between magnetic sublevels is a constant, a level with m ; = 0 does not shift as a result of the magnetic field, and a level with a positive value of m ; shifts in the opposite direction from a level with a negative value of m;. Also note that the splitting decreases as J increases. The magnetic field also leads to perturbations. In the presence of a magnetic field, the total angular momentum of the molecule need not be conserved; only the component of the total angular 2 GENERAL THEORETICAL CONSIDERATIONS 29 Figure 2-7: Splitting of magnetic sublevels in an external magnetic field. To first order, the energy spacing between levels is constant. The levels are degenerate in the absence of an external field. + 2 J = 2 -*€=l 0 -1 ^ - 2 m J field off field on momentum along the field direction need be considered. Treating this situation quantum mechani-cally, the magnetic field can cause perturbations off-diagonal in J by ±1 (and diagonal in mj). Al l other selection rules in Table 2-1 would remain the same except for the e/f parity selection rule. For A J = ±1 interactions, the selection rule is e <-> f. Interaction of the magnetic field with the magnetic moments along L and S directly, rather than with the components along J , can couple different electronic states. Importantly, for the current work, the Zeeman Hamiltonian can couple the same electronic states as the S-uncoupling and L-uncoupling Hamiltonians. 2.6 Interaction with Electromagnetic Radiation An electromagnetic (EM) wave can be described in terms of semi-classical travelling waves: E(r, t) = E0e^-?-^, and H(r, t) = H0e^-^. (2-36) The electric field, E, and the magnetic field, H, are functions of both position (r) and time (t). The two fields are perpendicular to each other and to the direction of propagation. The amplitudes of the fields oscillate with frequency v (= ^ ) and wavelength A. The wave vector k points in the direction of propagation and has a magnitude 2TT r l For simplicity, the fields are taken to be plane waves travelling the +Z direction in the space-fixed frame: E = E0ei(-kZ-^ and H = H0ei(-kZ~^. (2-37) The direction and magnitude of E M radiation's fields are important considerations for interac-tion with matter. The polarization of an E M wave is defined according to the electric field in Eq 2 GENERAL THEORETICAL CONSIDERATIONS 30 2-37. The definition could have been based on either field, but the electric field of the E M wave typically produces more significant effects than the magnetic field component when interacting with matter. About a given point in space (Z), the tip of the electric field vector traces out an ellipse in the X - Y plane. This is known as the polarization ellipse. In one limit, this ellipse collapses to a straight line, in which case the light is said to be linearly polarized. The opposite limit is circular polarization, where the electric field vector traces out a circle. Circularly polarized light whose electric field rotates clockwise as we view it traveling towards us is called right-handed circular polarization (RHCP), while light rotating counterclockwise is left-handed circular polarization (LHCP). Note that there is some disagreement as to the definitions for RHCP and LHCP light. The definitions given here follow the traditional optics definitions, but in some fields (such as quantum electrodynamics), the opposite definitions are used. For completely polarized light, any arbitrary polarization could be decomposed into a sum of RHCP and LHCP light. The electric field corresponding to the definition of RHCP light used here (including an arbitrary phase (fift) can be written as ER = EoR[x-iy}ei(kZ-«t+t>*\ and the equation for LHCP light with arbitrary phase (f>L is EL=EoL[x + iy]ei(kZ-"t+^\ where x is the unit vector along the X-axis and y is the unit vector along the Y-axis. The component of the electric field along the Z-axis (the direction of propagation) is zero because this is a transverse wave. If (f) = and Acp = (f)R — L, the electric field for a wave with an arbitrary elliptical polarization can be written E= (ER + EL) = [x (EoLe-^ + EoRei^) + iy ( f ^ e " ^ - J5 o f l e^ ) ] e^-^+V. (2-38) Setting one of the amplitudes (E0R or E0L) to zero obviously gives circular polarization. When the two amplitudes are equal, Eq 2-38 becomes E = 2EQ /A\ x c o s \ 2 ) + y s m \~Y) ei(kZ-ut+4>)^ ^2-39) This is linearly polarized light, with its plane of polarization oriented along the axis shown in Figure 2-8, at an angle ^ from the -l-X-axis. Note that the +Z-axis in Figure 2-8 is directly out of the page. 2 GENERAL THEORETICAL CONSIDERATIONS 31 Figure 2-8: Orientation of linearly polarized light, for a phase difference of between LHCP and RHCP light. Y A Polarization ^ axis A<]) 2.7 Transitions The Hamiltonian that describes the interaction of an electric field with the molecule, analogous to the Zeeman Hamiltonian for magnetic fields given in Eq 2-33, is Helectric = ~PE • E(f, t) (2-40) The electric dipole moment, /Ug, is defined by = YLwi, (2-41) i where qi is the charge of the ith particle, and fi is its position (usually taken relative the center of the molecule). The sum extends over both electrons and nuclei. Using the expression in Eq 2-36 for the electric field of E M radiation, the spatial part can be expanded: E(r, t) = Eoe'fc*-**) = Eae-iujt [l + ik • r- (k • r)2 + ...] . (2-42) The interaction region (i.e. the range over which f extends), governed by the size of the molecule, is the order of 1 0 - 1 0 m. Since this is typically very small compared to the wavelength of the radiation, a first approximation involves taking only the leading term (elk"r w 1) [34]. This is known as the dipole approximation. Higher order terms are electric quadrupole, electric octopole, etc. In the dipole approximation, the matrix elements of the operator in Eq 2-40 have the form V > m ( * ) • Eo\M*)) e~i2™\ (2-43) 2 GENERAL THEORETICAL CONSIDERATIONS 32-where m ^ n. The wavefunctions (VVn(t) and VVi(t)) represent a particular electronic, vibrational and rotational state. Working in the time-dependent picture, the wavefunctions can be written as i/>m(t) = ^ m e ' ^ and ipn(t) = ipne1^111, where Em and En are the respective energies of the two states. The matrix element for an electric dipole transition, in Eq 2-43, has no time-dependence for a frequency of E M radiation v = Em-En _ When this condition is satisfied, the matrix element in Eq 2-43 reduces to 4>m ~PE • E0 1p. (2-44) As a result of the mixing of two states m and n by the electric field, the molecule can actually change (i.e. make a transition) from one state to the other. One constraint that must be satified for a transition to occur is that his = E m - E n , which is equivalent to saying that the molecule can absorb or emit a photon of energy hu, and the energy of that photon must be equal to the difference in energy between the two states. The square of the matrix element in Eq 2-44 is proportional to the probability of a molecule making a transition from m to n (or vise-versa), but this is typically written in terms of what is called the electric dipole transition moment: (2-45) where a can be X , Y or Z. fia represents the component of JIB along a particular axis. See e.g. [28]. The selection rules associated with transitions are listed in Table 2-2. These are specific to the Hund's case (a) coupling scheme. Table 2-2: Selection rules for transitions (excluding hyperfine effects) Electronic A A = 0, ±1 Aft = 0, ±1 A S = 0; A E = 0 £ + *-+ £ + ; X T S -u <-> g Rotational A J = 0, ±1 A J = ±1 (ft = o -> n = o) + <-» -A J = 0 =^ e <-> f A J = ±1 => e <-> e; f «-» f 2 GENERAL THEORETICAL CONSIDERATIONS 33 For Hund's case (c), most of the quantum numbers in the electronic portion of Table 2-2 are not defined. The selection rules for A A, AS and A S do not apply in Hund's case (c). If this were not the case, then the 3 H i u «— x £ + transitions being studied in the current work would not/be allowed. Since there is no such thing as a E+ or £~ state in Hund's case (c) (only 0+ or 0~), the selection rule analogous to that in the fourth line of Table 2-2 would be 0 + 0 + ; C T <-> (T. Note that, unlike the selection rules for perturbations given in Table 2-1, J can change by ±1 in an electric dipole transition. This occurs because light can be considered (in the dipole approximation) to carry one unit of angular momentum, an important point for the magnetic rotation theory presented later on. Also notice in Table 2-2 that the +/- parity changes during an electric dipole transition. Previously, it has been tacitly assumed that only wavefunctions, not operators, change sign under the symmetry operation av. However, the electric dipole moment in Eq 2-45 does change sign under this operation, which leads to a change in parity in an electric dipole transition. Figure 2-9: Labelling of electric dipole transitions. Special names are given to the transitions according to the change in quantum number J. A transition for which J in the higher-energy state is larger by 1 than J in the lower-energy state is called an R-transition. When J in the higher-energy state involved in the transition is smaller by 1 than J in the lower-energy state, it is called a P-transition. When J remains unchanged in the 2 GENERAL THEORETICAL CONSIDERATIONS 34 transition, it is termed a Q-transition. These transitions are shown in Figure 2-9. Note that the value of J in Figure 2-9 has been assumed to be even. It is traditional to label the quantum numbers for the lower-energy state involved in a transition with a double-prime (e.g. J", v", etc.) and the quantum numbers for the higher-energy state with-a single prime (J', v',.--)-2.7.1 Franck-Condon Factor To paraphrase what is known as the Franck-Condon principle, the transition of an electron from one state to another occurs so quickly that the nuclear configuration does not change appreciably during the transition. This has significant implications for the probabilities of transitions. For example, an electronic transition that required a large change in internuclear separation would have only a very small probability of occurring. In Eq 2-2, the wavefunction was separated into an electronic and a nuclear part, where only the nuclear part depended on the nuclear coordinates. The nuclear part can be further separated ^vib,rot ^ ^pvib^rot The two wavefunctions are not actually separable, but within a given vibrational state, this separation is approximately correct. For an electronic transition (described below), the effect of the vibrational part of the wavefunction in Eq 2-44 can be factored out: (vm \-jtE • E0\ipn) = (cf i rnib) (vc'c\-fiE • E0\cv*) -The factor involving the vibrational portion of the wavefunction is known as an overlap integral, and the square of this overlap integral is called the Franck-Condon factor. 2.7.2 Electronic Transitions For the homonuclear molecules considered in the current work, there exists no permanent electric dipole moment to interact with the electric field. The electric field does, however, interact directly with the electrons in the molecule, and may therefore induce a change in the electronic state. This is known as an electronic transition. For consistency with the derivations to be performed for the magnetic rotation theory, the interaction of the electric field in E M radiation with the molecular electron will be described in classical terms. 2 GENERAL THEORETICAL CONSIDERATIONS 35 Consider, for simplicity, a single electron. Classically, the electron can be treated as a harmonic oscillator, with Coulomb attraction as the restoring force that acts to bring it back to equilibrium: d2x va—pr = —Fix) w —Dx, where D is the "spring constant" of this oscillating system, x is the displacement of the electron from equilibrium, and m is the electron's mass. A more realistic model would also include a damping term, proportional to When this oscillator is placed in an electric field, there is also an external driving force -eE, where -e is the electron charge. Using the dipole approximation, the equation of motion, including the damping and driving terms, becomes m^I + b^ + Dx = -eE0e-^t, (2-46) atz at where u — 2iru. The solution to this equation is —eE e~iult S= . 2 ° 2 . v (2-47) where 7 = £ and UJ2 = The solution in Eq 2-47 is the standard solution for a driven harmonic oscillator, but consider what it means. When the driving frequency (^) is close to some resonance frequency of the system (^) , there is a large amplitude response (assuming 7 is small). This means there is a large amplitude oscillation of the displacement of the electron from equilibrium, i.e. an oscillation of the electron's position. This is indicative of the electron making a transition from one electronic state to another. As an interesting analogy, this has the appearance of oscillating dipole moment being induced in the molecule. Classically, an oscillating dipole acts as an antenna for absorption or emission of E M radiation at the frequency of oscillation. 2.8 Index of Refraction Consider a gas composed of N molecules per unit volume. The induced oscillation of electrons' positions, described in Eq 2-47, leads to a macroscopic polarization, P, which is the sum of all dipole moments per unit volume: Ne2 E„e~iujt P= ( 2 ° 2 • y (2-48) The polarization, as derived from Maxwell's equations, should have the form [37], P = eoXE, (2-49) 2 GENERAL THEORETICAL CONSIDERATIONS 36 where e0 is the permittivity constant, and % l s the susceptibility. The complex index of refraction, n, is defined by the relation X = n2 - 1. A comparison of Eq 2-48 and Eq 2-49 leads to 2 1 Ne2 n1 = 1 + e0m(u)2 — LO2 — iju)' In a gaseous medium at sufficiently low pressures, the index of refraction is close to unity: n 2 - 1 = (n + l)(n - 1) « 2(n - 1), which gives Ne2 For an E M wave travelling through a medium with complex refractive index n, the frequency of the wave is the same as it would be in vacuum, but the wavelength changes to A n = where A 0 is the wavelength in vacuum. The wavenumber k = ^ = | ^n = k 0n, where kD = If the complex index of refraction is separated into real and imaginary parts, n = n0 + tK, the electric field component of E M radiation propagating in the Z-direction from Eq 2-37 becomes E = E0ei{k°nZ-^ = E0e:z2Zrei(k°n°z-UJt\ (2-51) The real part of the index of refraction, n 0 , describes the dispersion of the medium (the variation of the speed of light in the medium as a function of frequency), since the phase velocity of the wave in Eq 2-51 is given by v\0 c na no1 where c is the speed of light in a vacuum. The imaginary part of the index of refraction, K, is related to absorption of the wave. The intensity of the wave, I, is proportional to E* -E. The absorption coefficient, a, is defined according to Lambert's law: I(Z) = I0e-aZ, where Ia is the intensity at Z = 0. From Eq 2-51, a is a = -Xo-2 GENERAL THEORETICAL CONSIDERATIONS 37 Expressions for na and K come from evaluating the real and imaginary parts of Eq 2-50: na = 1 + Ne2 u20-u2 AC = 2e0m {UJ2 — u>2)2 + 72o>2' Ne2 70; 2e0m (to2 — UJ2)2 + ^2UJ2 ' For CO u>0, these expressions reduce to _ Ne2 u)0 — UJ U o = l + 4e0nuv0(u,0-u)* + qy] (2"52) _ Ne2 7 K ~ 8e0muj0 (u>0 - u)2 + (^)2 ' ^2"53^ The lineshapes in Eq 2-52 and Eq 2-53 are plotted in Figure 2-10. Figure 2-10: Lineshapes associated with the real and imaginary parts of the com-plex index of refraction. 2.9 Widths of Spectral Lines 2.9.1 Natural Linewidth A molecule in an excited state can transfer to a lower energy state through spontaneous emission. If the mean radiative lifetime of a state n (the average amount of time the molecule remains in the state before spontaneous emission occurs) is denoted by r n , the uncertainty principle imposes a limit on the ability to measure the state's energy, En [37]: 8En^-. (2-54) 2 GENERAL THEORETICAL CONSIDERATIONS 38 This affects the spectral distribution measured for a transition, for either emission or absorption; instead of a sharp line (i.e. a delta- function in frequency), a line profile centered about the transition frequency is measured. Note that this was already accounted for in previous sections, through the inclusion of the parameter 7. When one treats the electron as a classical oscillator, in order to consider spontaneous emission, the equation of motion is which is just Eq 2-46 without the driving term. The solution, x(t), will include a term e _ i * [37]. Classically, an oscillating electron emits radiation, and this term represents a damping of the electron's oscillation through radiative energy losses. The parameter 7 determines the frequency width of the spectral distributions shown in Figure 2-10. The absorption or emission of E M radiation as a function of frequency will resemble Figure 2-10(b), which is referred to as a Lorentzian lineshape. The parameter 7 is called the full width at half-maximum (FWHM)—or more succinctly, the linewidth—of this Lorentzian lineshape. It is a combination of uncertainties from the two states involved in the transition. From the constraint hu = En - Em, the uncertainty in the frequency for the transition is 2.9.2 Pressure Broadening There are other contributions to the widths of spectral lines. Collisions between molecules in a gas can also affect measured linewidths. A collision can change the energy state of a molecule. This change is referred to as a "radiationless transition" because it does not involve interaction with E M radiation. Collisions can also change the phase of an excited molecule, a process that alters the apparent lifetime of a state, even if the molecule does not physically change state. This is a complex subject area (see [28] for an excellent discussion on the topic), but the simple view shall be taken that the collisions are brief and strong, and their only effect is to change the apparent lifetime of a state—e.g. the collision might bump a molecule into a new state before it has a chance to decay by spontaneous emission. The "width" of a transition between m and n then becomes 6v « SEn + SE, h '771 Using Eq 2-54, the resulting expression for 7 becomes: 2n8v = 6u = 7 = ( — + — ) . 1 (2-55) 7 = -rad Tcoll -j-rad 771 'm n + •coll ' T; n 2 GENERAL THEORETICAL CONSIDERATIONS 39 where TRAD is the mean radiative lifetime, and TCO11 is the mean nonradiative lifetime, governed by the time between collisions. 2.9.3 Doppler Broadening Another significant contribution to the width of spectral lines comes from the thermal motion of the molecules in a gas. The fact that molecules with different velocities relative to a monochromatic light source perceive the light as having different frequencies (due to the Doppler shift) leads to an additional broadening of the line. Assuming a Maxwellian distribution for the velocities of molecules in a gas, and ignoring other contributions to the width, the intensity profile of a Doppler-broadened spectral line (for either emission or absorption) is [37] I e(»-»o)\2 I{u) = Iae K ">°vp ' , (2-56) where the most probable velocity, v p is defined by !2kT F V M Here, k is Boltzmann's constant (not to be confused with the wave vector k), T is the temperature in Kelvin, and M is the mass of the molecule. The frequency profile of the expression in Eq 2-56 is called a Gaussian lineshape. The F W H M of Eq 2-56 is u0 l8kT ln2 This linewidth depends on the frequency of the transition. This is often called the inhomogeneous linewidth, since it arises from the same transitions occurring at different frequencies in different molecules. The linewidth in Eq 2-55 is known as the homogeneous linewidth, since all molecules experience the same broadening for the given transition. For the experimental studies in the current work, the Doppler width was orders of magnitude greater (roughly 106) than the natural (radiative) linewidth and a factor of 10 larger than the pressure broadening linewidth. The lineshape in Eq 2-56 was derived under the assumption that 7 (the homogeneous linewidth) was zero. To be rigorous, the Doppler-broadened spectral line must take the homogeneous linewidth into account; this is accomplished by performing a convolution of the two contributions [37]. 2.10 Hyperfine effects Just as an electron has an intrinsic angular momentum (electronic spin), the nuclei has a nuclear spin, / . The interaction of this angular momentum with the other angular momenta in the molecule 2 GENERAL THEORETICAL C0NSR1ERATI0NS 40 is relatively weak. The effects arising from these interactions, called hyperfine effects, are therefore typically quite small. Splitting caused by L, S or R interacting with I are often too small to resolve with standard spectroscopic techniques. In the current work, there are some relatively large hyperfine effects at low-J, but for the most part, the underlying hyperfine structure in transitions is obscured by Doppler-broadening. Special experimental techniques must be used to diminish the contribution of the Doppler effect to the experimental signal in order to resolve these small splitting. One such technique will be presented in the current work. For a homonuclear diatomic molecule, the standard coupling scheme (Hund's case (a^) [28]) —* defines the total nuclear spin of a given state, I, as the vector sum of the nuclear spins of the two nuclei: T=ti+i2, (2-58) where i\ is the spin of nucleus 1 and i2 is the spin of nucleus 2. The total angular momentum for the molecule, F, is then F = J + I, (2-59) and the quantum number F takes on the possible values | J - / | , ... , J + I - l , J + I. In an external magnetic field, it is F (rather than J) that is constrained to have a constant component along the field direction (see Section 2.5.1). Analagous to the magnetic quantum number mj, the quantum number that labels the component of F along the magnetic field direction is nip. 2.10.1 Ortho-Para Symmetry For a homonuclear molecule, exchange of the two identical nuclei represents a symmetry operation. Recall that such a symmetry operation either leaves the wavefunction unaffected or changes only the sign of the wavefunction. However, the molecules considered in the current work, Br2 and I2, the nuclei have half-integral spins: i i = i2 = § for B r 2 , and i i = 12 = § for I2. Particles with half-integral spins are called fermions. It is a well-known property of fermions that the wavefunction describing them must be antisymmetric with respect to exchange of any two of them. For the symmetry operation of exchanging the two nuclei, therefore, the wavefunction must change sign. The wavefunction cannot remain the same under this operation. 2 GENERAL THEORETICAL CONSIDERATIONS 41 This constraint excludes some possible combinations of i" and J in Eq 2-59. The condition that must be fulfilled for the wavefunction to change sign is [38]: where Xug comes from the electronic symmetry operation that led to u/g labelling for the electronic states of homonuclear molecules. This parameter is equal to 0 for g electronic states and 1 for u electronic states. The term (-1) J ~ s + ^ is just the effect of crv described in Eq 2-15. Since h = 12 = § for 7 9 B r 2 , the possible values for the quantum number I (as calculated from Eq 2-58) are 0, 1, 2 and 3. In a 1 £ + state, for example, for even J states, only the even I levels (0 and 2) exist, while for odd J states, only the odd I levels (1 and 3) exist. J states with the greater spin-statistical weight (i.e. for which there exists more hyperfine levels) are called ortho states, and the J states with the smaller spin-statistical weight are called para states. In the example of a XE+ state in 7 9 B r 2 , odd-J states are ortho and even-J states are para. For electric dipole transitions, the ortho/para symmetry must be conserved. For perturbations, states with the same u/g electronic symmetry must have the same ortho/para symmetry to interact. States with the opposite u/g electronic symmetry must have opposite ortho/para symmetry in order to perturb each other. 2.10.2 Hyperfine Energies The hyperfine Hamiltonian for a diatomic molecule, H/j/, can be expressed as the sum of interactions between nucleus 1 and the molecular electrons, H^y(l); between nucleus 2 and the electrons, H^(2); and between nucleus 1 and nucleus 2, H/^(l,2): Hhf = Hhf(l) + Hhf(2) + Hhf(l,2). The electron-nuclear terms are considerably larger than the nuclear-nuclear terms [39], and con-tributions from H/j/(l,2) shall therefore be neglected. The matrix elements for the remaining two terms are related by symmetry considerations [39]: (u,I' \Hhf{l)\u,l) = (-l)I+r (u,I1 \Hhf(2)\u,I), (g,I'\Hhf(l)\g,I) = (-l)I+I'(g,I'\Hhf(2)\g,I) and (u,I' \Hhf(l)\g,I) = ( - l / + ' ' + 1 (u, I' \Hhf(2)\g, I). 2 GENERAL THEORETICAL CONSIDERATIONS 42 One contribution to the hyperfine energy comes from the interaction of the magnetic moment of I with the magnetic moments associated with the electrons (i.e. with L and S). These are known as magnetic dipole effects. In general, the effects are calculated by evaluating the interactions from each nucleus separately [27]. In practise, the matrix elements for magnetic dipole and electric quadrupole hyperfine interactions were calculated from Eq (3) in [39]; this expression explictly accounts for the coupling of i\ and i2 to give I. To illustrate the various magnetic dipole hyperfine interactions, however, the expressions given below give the contribution from a single nucleus. The interaction of nuclear spin with electron spin looks very much like a classical interaction between two dipoles [28]. The magnetic moments associated with S and i\ (the nuclear spin for nucleus number 1) are given in Eq 2-30 and Eq 2-32, respectively. The Hamiltonian describing the interaction between the two has the form: His(i) = gsgNVB^N (2-60) where r (not to be confused with internuclear separation) is the spatial separation between the two "dipoles." Note that for convenience, the expression above uses dimensionless angular momenta (e.g. |S| = y/S(S + 1) rather than y/S(S + l)h), as will the expressions that follow. When the electron has a non-zero probability of being at the nucleus, (|*(r = 0)|2 ^ 0), there is an additional contribution to the interaction of electron and nuclear spin. The Hamiltonian for this additional contribution, called the Fermi contact interaction, is usually written as [40]: 87T HpciX) = -j9s9NHBriN |*(0)| 2?i • S. (2-61) The interaction between the nuclear spin and the magnetic field associated with the orbit of the electron (defined in Eq 2-28) is 1 . £ HiL^) = '2gLgNfXBm-L^-- (2-62) The same first-order approximation is made that was made for interactions with an external magnetic field: the precession of L and S is rapid enough that only their components along the internuclear axis need be considered. With this assumption, the magnetic dipole contribution to the hyperfine energy becomes [27] EMD(1) = [aA + (b + cWj^j + iy (2-63) where A, £ and Q are the usual projections of angular momenta along the internuclear axis in Hund's case (a) coupling. An important thing to note is that the hyperfine energy from magnetic dipole 2 GENERAL THEORETICAL CONSIDERATIONS 43 effects decreases with increasing J, roughly as j . For Hund's case (c) coupling, the coefficient in Eq 2-63 obviously needs to be left as an effective parameter to be determined from the experimental data. The constants a, b and c, known as Frosch-Foley parameters [41], are defined as: where 9 is the angle between the internuclear axis and the radius r from nucleus number 1 to the electron. The averaging implied by the brackets ( ) is done over the electronic space coordinates for the state under consideration. ( ) s denotes an average over the electron spins contributing to the interaction with i\, and ( ) e denotes an average over the orbital angular momenta of electrons contributing to the interaction with i\. An expression for i\ • J , required for the calculation of E M D Eq 2-63, can be determined by the procedure outlined in [27]. The contribution from nucleus number 2 to the magnetic dipole diagonal matrix elements will have the same form as the expression for nucleus number 1 in Eq 2-63. The only difference for nucleus number 2 is the fact that the expression for i2 • J is slightly more complicated. Also recall that only certain combinations of i\ and i2 are allowed for a homonuclear molecule, because of ortho/para considerations. Again, actual calculations were performed using the formulation in Eq (3) of [39], which gives the matrix elements in terms more convenient for homonuclear molecules. For a nucleus with nuclear spin i > 5, the charge distribution within a nucleus is not spherically symmetric [40], and the nucleus therefore has an electric quadrupole moment associated with its charge distribution. An important contribution to the hyperfine energy occurs for nuclei with i > ^ when the electric quadrupole moment of the nucleus interacts with the electric field gradient at the nucleus caused by an electronic charge distribution which is anisotropic with respect to the centre of the nucleus. The Hamiltonian describing this interaction is most conveniently expressed in terms of irre-ducible tensor operators [42], as are the magnetic dipole terms discussed previously when it comes to evaluating off-diagonal matrix elements. The form of the electric quadrupole contribution to the hyperfine energy will not be derived explicitly here (see [28] for such a derivation), but a few important points will be discussed. b = bFC - -, 87T °FC = "g-9s9NLLBrLN |*(0)| 2 , and 3 /3cos2e-l\ (2-64) 2 GENERAL THEORETICAL CONSIDERATIONS 44 Electric quadrupole effects are expressed in terms of the parameters eQqj, where j can be 0, 1 or 2. eQ (where e is the charge of a proton) is called the quadrupole moment of the nucleus; it is defined in such a way as to correspond approximately to the classical quadrupole moment. The parameter q^ describes a particular component of the electric field gradient of the electrons. For example, qo corresponds to the component along the internuclear axis, and is defined as being twice the matrix element diagonal in A of the second rank tensor (TQ(VE)) built out of V E : f. As mentioned previously, the u/g symmetry is preserved unless the ortho/para symmetry breaks down at the same time. In the current work, the matrix elements for K^f are computed from the Eq (3) in [39]. Hyperfine effects also give additional selection rules for transitions. Besides the selection rules already listed in Table 2-2, electric dipole transitions must have A F = 0, ±1 and AI = 0. The transition moment (as defined in Eq 2-45 from a particular lower state (F",m^",J",I) to a particular upper state (F',m^,J',I) is proportional to v / (2F ' + l)(2F" + l)(2J'+l)(2J" + l ) ( - l ) / + J ' + - 7 " + F ' + F " - m ^ - n " + 1 (v"\fia\v1) ( J" 1 J'\ ( F" 1 F' \ [I J' F'\ ( 9 , a ] { -ft" ft"-ft' ft'J ^ -m"F m"F-m'F m'F) { 1 F" J"} " (2"66) 2 GENERAL THEORETICAL CONSIDERATIONS 46 The algebraic forms for the 3-j symbols (the quantities in ( )) and the 6-j symbols (in { }) can be found in [31]. In an electric dipole transition, the change in magnetic quantum number, A m f (mF - m'F), can be 0, +1 or —1. In the current work, it is the Amp = ± 1 transitions that are important, as will be discussed later. In general, the mixing of the wavefunction due to perturbations must be taken into account when calculating the transition probability. As a simple example, consider the transition between the two superposition states: ip" = aipi + bip2 and ip' = atpx + Pipy, (2-67) where the ' indicates the upper state and the " indicates the lower energy state involved in the transition, as usual. Assuming the superposition constants a, b, a and @ are real, then the transition probability is proportional to: (ip" j2-E0\i>,S) 2 = \aa(^i f2-E0 ipx^+ap(i)± p,-E0 ipy}+ba(tp2 fl-Ea ipx^+bfifa fi-Eo ipy) (2-68) It is important to notice is that the transition probability goes as the square of the sum, rather than the sum of the squares. For interactions with the magnetic field, the shift of a particular magnetic sublevel, mj?, can be written as Ezee = /-<-B9FMFH. This equation has the same form as Eq 2-34. Note that terms smaller by a factor such as the contribution from the magnetic moment along J, are once again being neglected. This equation represents the energy resulting from the interaction of H with the component of the electronic magnetic moment (from L and S) along F. From the vector model, making the standard first order approximation that the electronic magnetic moment is aligned along J (because of the precessing electronic angular momenta), the expression for the g-factor, gp, becomes [28]: rF{F + l) + J{J + 1) -1(1+1)" 9F = 9J- (2-69) 2F(F + l) Note that this g-factor is for the matrix elements of the Zeeman Hamiltonian diagonal in F, J and I. 3 MAGNETIC ROTATION 47 3 Magnetic Rotation When a molecule is placed in an external magnetic field, the finer details of how it interacts with E M radiation can alter. The magnetic field can shift the transition frequency of a molecule (as a result of the Zeeman effect), or perturbations caused by the magnetic field can change the probability of a transition. These (and other) effects can lead to the modification of an E M wave propagating through a gaseous medium in a magnetic field. Such modifications of the E M radiation shall be referred to as magnetic rotation. The theory behind magnetic rotation is well-established. Early papers on the subject (e.g. [5] and [6]) treated the observed magnetic optical activity in molecules as a special case of Faraday rotation. The theory was originally limited to frequencies well outside the half-width of the transi-tion, but was later extended to be valid near transition frequencies [8]. A more rigorous quantum mechanical treatment was developed [44], treating the effect as the simultaneous absorption and re-emission of a photon, but the relations between the parameters and molecular properties was not evident for this formalism. Currently, the methods used for calculating magnetic rotation spectra typically resemble the procedure described in [9]. This involves calculating a lineshape for every Amj = +1 and every A m j = -1 transition associated with a particular spectral line. Since the experiment employed an oscillating magnetic field (as do the experiments in the current work), the calculation had to be performed for several values of the magnetic field and numerical integration used to find the Fourier component of the calculated signal at the modulation frequency of the magnetic field. This gave the signal that would be measured using phase sensitive detection tuned to the modulation frequency of the field. An alternative procedure for calculating magnetic rotation signals is presented here, specific to the use of an oscillating magnetic field (although other possibilities are discussed). There are at least two distinct advantages to the current approach. Firstly, it is not necessary to calculate a different lineshape for every single A m j = ±1 transition involved, or any at all, for that matter. The resultant lineshape of the magnetic rotation signal is calculated explicitly from an analytical expression, and finding the contributions from all of the different Amj = ±1 transitions can be done by carrying out a fairly simple summation. It is not necessary to add and subtract lineshapes, as is required with the other method. Secondly, numerical integration to find Fourier components is not necessary with the approach presented here. Expressions are worked out for different possible measurement schemes, and the signal is calculated directly from these expressions. It is not necessary to do 3 MAGNETIC ROTATION 48 calculations at several different values of the magnetic field, since in the current approach, the amplitude of the field becomes merely a parameter in determining the strength (or the width, if the field is large enough) of the signal. With the current approach, deriving expressions from which the signal can be directly calculated, more insight is gained into the various contributions to the magnetic rotation signal. The lineshapes determined from this approach agree very well with the experimentally observed lineshapes. The approach is flexible enough to account for such things as modulation broadening and frequency shifts from perturbations caused by the oscillating field, while the other procedure would likely have to use ad hoc methods to account for these. In the discussions that follow, it will be assumed that only the upper (i.e. higher energy) state involved in the transition has a non-zero g-factor, to be consistent with the experiments performed for the current work. In the A - X system of B r 2 , levels in the X state have g-factors the order of nuclear magnetons, while levels in the A state have g-factors the order of Bohr magnetons, and this assumption is justified. For transitions where both states were paramagnetic, the approach would require minor modifications. 3.1 Magnetic Circular Dichroism Recall that any polarization can be decomposed into a sum of two orthogonal circular polarizations, LHCP and RHCP light. Placing a medium in a magnetic field can create a difference between the absorption coefficient for LHCP light (ax) and the absorption coefficient for RHCP light {an), (aL - OR) ^ 0. This effect is known as magnetic circular dichroism (MCD). When polarized light travels through a medium for which ax ^ a#, one circular component is absorbed more than the other. After traversing the medium, the ratio of the amplitudes EQL and E0JI in Eq 2-38 will have changed, thereby changing the degree of ellipticity of the polarization. For example, MCD changes linearly polarized light (E0L = E0R) into elliptically polarized light (EDL ^ EDR). There can be more than one cause for (ax — OCR) being non-zero, but the largest effect, and probably the easiest to picture, results from transition frequency differences induced by the Zeeman effect discussed in Section 2.5. Changes in transition probabilities due to state-mixing caused by the magnetic field can give a contribution to the magnetic rotation signal, as can population effects in certain circumstances. These two contributions will be discussed in more detail later, but the frequency-shift contributions to the magnetic rotation signal will be considered first and used to 3 MAGNETIC ROTATION 49 illustrate the situation. Consider an R-transition from J = 0 to J = 1 in the presence of a magnetic field, H, as pictured in Figure 3-1. Figure 3-1: An R(0) transition in the presence of an external magnetic field. The transition frequency in the absence of an external magnetic field is va A m , =+1 To first order, the J = 0, m ; = 0 state is unaffected by the magnetic field, while the J = 1 state splits into three components, with mj = 0 unshifted, m ; = +1 shifted upwards by an energy fj^BgjH., and mj = -1 shifted downwards by the same amount. Note that the upper state has been taken to have a positive g-factor (for the definition of the Zeeman energy given in Eq 2-34). For E M radiation propagating along the magnetic field direction, only Amj = ±1 transitions are possible; the Amj = 0 transition, indicated by the dotted line in Figure 3-1, cannot occur when k (the vector that defines the direction of propagation for the light) lies along H. The reason for this has to do with conservation of angular momentum. In the dipole approxi-mation, circularly polarized light carries one unit of angular momentum, directed either parallel or anti-parallel to k. Recall that mjh is the component of J along the space-fixed Z-axis. If the E M wave propagates in the +Z direction, then mj increases by 1 if the angular momentum of the light is along k, and it decreases by 1 if the angular momentum is against k. There can be no Amj = 0 transitions, however, since this would incur no change of angular momentum along Z. When working with magnetic fields, the field direction is typically used to define the +Z direc-tion, but the experiments described in the current work make use of an oscillating magnetic field, which defines a preferred line in space, but not the sense along this line. Therefore, the positive space-fixed Z axis will be defined to be along k, i.e. along to the direction of propagation of the light—the only sensible choice. Using the right-hand rule to determine the sense of the angular 3 MAGNETIC ROTATION 50 momentum associated with the E M wave, for the definitions of LHCP and RHCP light given in Section 2.6, LHCP light has angular momentum along k and RHCP light has angular momentum against k. There seems to be some confusion in the literature on this issue, but it probably comes back to the different possible definitions for the two circular polarizations. It is perhaps counter-intuitive to have RHCP light with angular momentum anti-parallel to the direction of propagation, as is the case for the traditional optics definition of RHCP light used here. With the +Z axis defined to be along k, LHCP light drives A m j = +1 transitions and RHCP light drives Amj = -1 transitions, regardless of the direction of the magnetic field. If the axis were chosen in the opposite sense, LHCP and RHCP light would switch roles. For half of the oscillating magnetic field's cycle, H points in the +Z direction, and for the other half of the cycle, the field is directed along -Z. When the magnetic field points in the positive Z direction, the absorption profile for LHCP light (the Amj = +1 transition in Figure 3-1) is shifted to a higher frequency, while the absorption of RHCP light (Amj = -1) is shifted to lower frequency, as shown in Figure 3-2. Note that when the field reverses direction, the frequency shifts are in opposite directions: LHCP absorption to lower frequency and RHCP to higher. Figure 3-2: Frequency separation of absorption profiles for LHCP and RHCP light in the presence of an external magnetic field, for the field in the +Z direction. vQ is the transition frequency in the absence of external fields. When the molecular electron interacts with a monochromatic light source, the absorption profiles in Figure 3-2 are sampled only at a particular frequency, indicated by the dotted line in Figure 3-2. At this frequency, there is a different amount of absorption for LHCP and RHCP light, i.e. 3 MAGNETIC ROTATION 51 AL 7^ OCR. By the definition of the absorption coefficient a, the amplitude of the electric field decreases as E0e~2z, where E 0 is the amplitude of the electric field at Z = 0. The amplitudes of the electric fields for two polarizations are therefore given by: EL(Z) = EoLe~^z and ER{Z) = EoRe~^z. (3-1) An average field is defined as E a(Z) = EL{Z)+ER{Z) ^ a n ( j difference between the two fields is defined as AE(Z) = EJr,(Z) - E#(Z). Starting from Eq 2-38, E0L and EQR in this equation are replaced by E^(Z) and ER(Z) defined in Eq 3-1. The expression for the electric field in the E M wave after travelling a distance Z through a medium is then AE(Z)\ E(Z) = {[(*(*>+*m) +(E.(Z) - * m ) *+ Ea{Z) + AE{Z)\ e~*2 - i Ea(Z) which reduces to E{Z) = \2Ea{Z) (A(j>\ _ . (A(j)\ ; cos \ — ) x+sin\^—J y^ V 2 2 ) +iAE{Z) AE{Z)\ ai(kZ-ut+) A(j) -sin{ x+cos(^^ )']}• ei(kZ-ujt+(p) (3-2) Note that changes in the relative phase of LHCP and RHCP light have ignored; they will be considered later on. The electric field in Eq 3-2 has a component along the axis defined by the unit vector cos (2 • fA\ x + sin I — I y, and a component along the axis defined by the unit vector . /A(j>\ . [A^ -sin I — jx + cos I — ] y, (3-3) (3-4) which is perpendicular to the unit vector in Eq 3-3. It was shown in Section 2.6 (see Eq 2-39) that the original direction of polarization for the linearly polarized light was, by construction, aligned along the axis defined by the unit vector in Eq 3-3. This axis shall be referred to hereafter as the initial axis (IA). The axis perpendicular to this, defined by the unit vector in Eq 3-4, shall be referred to as the perpendicular axis (PA). The electric field in Eq 3-2 is associated with an E M wave that is elliptically polarized. If A E is not too large, the major axis of the ellipse is aligned along IA. Taking a = and A a = <*L — AR, the component of electric field along PA, EPA, is EPA = i(AE(Z))ei{-kZ-"t+® = ie-%z \EoLe EoRe 4 0i{kZ-wt+) (3-5) 3 MAGNETIC ROTATION 52 To allow for some ellipticity in the original polarization, a parameter d is introduced such that e2d = g * . That is, eDEOL = e-DEOR = E c . Eq 3-5 then simplifies to EPA = -2ie-%zE0sinh [^Z + d) e ' ( ^ - ^ + < « ( 3 . 6 ) In the case where the initial polarization is linear (E0L = EOR), the parameter d is zero. Note that Eq 3-6 describes the contributions from all sources of differential absorption, i.e. including transition intensity and population effects, not just the frequency shift effect described in the example. 3.2 Faraday Effect In addition to causing differential absorption, the magnetic field can lead to differences in the real part of the index of refraction for LHCP and RHCP light: (noL - noR) ^ 0. The situation is very much analogous to that for differential absorption. Just as there is a frequency separation for the absorption profiles in Figure 3-2, there is a frequency separation for the dispersion curves, as shown in Figure 3-3. Figure 3-3: Frequency separation of dispersion profiles for LHCP and RHCP light in the presence of an external magnetic field. vD is the transition frequency in the absence of external fields. In Figure 3-3, the higher frequency curve is associated with the Amj = +1 transition in Figure 3-1, driven by LHCP light according to the chosen definition for the +Z axis. The curve shifted to 3 MAGNETIC ROTATION 53 lower frequency is associated with the A m j = -1 transition, driven by RHCP light. The magnetic field was assumed to be pointing in the +Z direction. Note the difference in nQ between LHCP and RHCP light at the frequency indicated by the dotted line. A difference in nQ means the two polarizations travel at different velocities through the medium, which leads to a change in their relative phase. Recall from Section 2.6 that the relative phase of LHCP and RHCP light, A(f>, determines the polarization direction for linearly polarized light (or, more generally, the orientation of the polarization ellipse). The average value of nG for the two polarizations is n a = n<>L+naR a n d the difference is An = noL — n-oR- Starting again from Eq 2-38 and making the substitution k = 2™a.^ the electric field in the E M wave after travelling a distance Z through the medium is given by -* f r • Ad> „• 7r An v • A<£ 7r A E(Z) = { [EoLe-l2el—Z + EoRel 2 e — 5 ; x + E o L e - ^ e i E ^ z - E o R e ^ e - ^ z ] y} e ^ 2 ^ ) . (3-7) Here differential absorption has been neglected; it was considered previously, and a combination of differential absorption and changes in relative phase will be considered later on. Rather than introducing the parameter d, as was done with magnetic dichroism, it is more instructive to consider the case where the E M wave is initially linearly polarized (EQL = E O R = E0). Eq 3-7 then reduces to E(Z) = 2E0 cos (3-8) Comparing Eq 3-8 to Eq 2-39, it is evident that the birefringence causes a rotation of the axis defining the direction of polarization. The angle of this rotation is — E^kZ, radians. The polar-ization of the light remains linear. Again, note that perturbation and population contributions to the birefringence, not just frequency shift effects, are implicitly included in Eq 3-8. 3.3 Combination A general expression may now be written that includes both differential absorption and birefringence effects induced by the magnetic field. The initial polarization of the light is taken to be elliptical, with the parameter d introduced earlier describing the degree of ellipticity. It is useful to consider an arbitrary axis, oriented at an angle 60 from PA ( | - 0o radians from IA). Starting again from the expression for the electric field in Eq 2-38, the component of the electric field along this arbitrary axis after the E M wave travels a distance Z through the medium is 3 MAGNETIC ROTATION 54 E{Z) = 2 £ 0 e " , , Aa \ f n 7r A n „ . cosh [ —Z + d)sin[90 — Z ] -4 / V A 0 i sinh (3-9) This component of electric field from the E M wave along the arbitrary axis can be selected for measurement by aligning the transmission axis of a polarizer in the appropriate direction (i.e. at an angle 9a from PA). Most measurement techniques record the intensity of the light rather than its electric field. The intensity is proportional to the square modulus of the field, i.e. to E* • E. The intensity, I(Z), transmitted through the polarizer aligned along the axis 9Q radians from PA, using the expression for the electric field in Eq 3-9, is I(Z) = pE2e -aZ cosh' • / A a , Z + d) sin2 9, IT An \ . , o / A a „, \ 9 / TT A n —Z^+sinh^—Z + d ) ^ - — : where p is a constant of proportionality, into which all constant factors not explicitly included have been collected. This equation reduces to 7r A n I(Z) = p^-e-*z cosh 2 —Z + d 4 cos 2 A 0 (3-10) The series expansions (cos# « 1 — ^ + ... and cosh# RS 1 + |T + •••) can be used to simplify 01 2! this. The first order terms from Eq 3-10 are I(Z) = pE2e-aZ IT An \2 / A a „ ,N 2 I(Z) = pEle -ccZ 7r2(An)2 A 2 ' 29, IT A n z + e20 + ^ z 2 + ^zd + d2 (3-11) Eq 3-11 is the intensity of light transmitted through the polarizer, where the initial polarization of the light at Z = 0 was taken to be elliptical. The quantity Z then represents the distance the E M wave travelled through the medium under the influence of an external magnetic field, e.g. the length of the cell for the experiments performed in the current work. Note that in order that higher order terms can be neglected, it has been assumed that A n , A a , 60 and d are small (<< 1); this assumption appears to be very well justified for the experimental conditions used in the current work. The terms d2 and 92 are "background" terms—they do not depend on properties of the molecules, but all of the other terms do provide information about the molecules. The question becomes how to extract this information. 3 MAGNETIC ROTATION 55 The classical derivations for n and K given in Eq 2-52 and Eq 2-53 cannot be applied directly to a quantum mechanical system. These expressions must be modified to take into account the fact that there are many allowed transitions (at various frequencies) from a particular state. This is done by introducing a factor known as the oscillator strength, In a sample of molecules all starting in a particular state (labelled i) and absorbing light at all possible frequencies, the oscillator strength is the fraction of molecules that would make a transition to a particular state (labelled j), excluding saturation effects. Using the electric dipole transition moment defined in Eq 2-45, the oscillator strength for a transition from state i to state j (of frequency w^) is [37] / . i = ^ l * W « ! 2 (3-12) Using the expressions in Eq 2-52 and Eq 2-53, the contributions to the absorption and dispersion at frequency u from a state i are Nje2fij uijj - UJ E Ni& fij 7ij where Nj is the number of molecules in state i , and 7^ is the linewidth for the transition. Substituting the oscillator strength defined in Eq 3-12 into the above equations, and including contributions from all transitions, yields \-nN(J",m"j) u0-u nQ = 1 + transitions and Air ° transitions \{v ,J ,mj,il \fj,a\v , J ,mj,ll )| (3-13) (3-14) 6e0h (aj0-u)* + qy where the sum extends over all transitions possible for the polarization in question, e.g. The sum for II0L would be over all transitions driven by LHCP light, all A m j — +1 transitions. The standard notation has been used of labelling the lower state's quantum numbers with double-primes and the upper state's quantum numbers with single-primes. Note that only the quantum numbers pertinent to the discussion have been shown in the wavefunctions, although there will be other quantum numbers associated with the states, such as v, S, A, etc. It has also been assumed that hyperfine effects are neglible. 3 MAGNETIC ROTATION 56 The values of 7 (the linewidth associated with the transition) and u0 (2n times the transition frequency) are implicitly assumed in the above equations to take the appropriate values for the different transitions in the sum. N(J",m'j) represents the number of molecules per unit volume in the lower state involved in the transition. This can be expressed as [45] N(r,mHJ) = ^e-WJ">'*fi, (3-15) Zp where is the total number of molecules per unit volume in the sample, ZP is the partition function, E(J",m'j) is the energy of the given lower state, (5 — k is Boltzmann's constant, and T is the temperature in Kelvin. For the sake of brevity, the notation t4i>y'iJ') = (v",r,m'S = m'jTl,tf'\l**\v', J',m'j,n') (3-16) is introduced, where the upper label on the left-hand side of the equation goes with the upper sign on the right-hand side, and the lower goes with the lower. The reason for the atypical approach of labelling the transition according to an upper state quantum number (m'j) comes from the fact that the upper state is magnetically active and the lower state is not. The value of the magnetic quantum number in the upper state (m'j) is therefore a more convenient label for the transition when considering magnetic rotation. 3.4 Branches Differences in the mj-dependence of electric dipole matrix elements for R-, P- and Q-transitions lead to significant differences in the measured magnetic rotation signals for the three branches. The matrix elements in Eq 3-16 for the three branches are presented below (these can be taken from [27], for example). In these expressions, the quantum number J refers to the value of J in the lower state, the traditional approach, while the value of mj refers to the magnetic quantum number for the upper state, for the reasons given above. To leave the expressions general, the portion of the matrix element involving ft was put into the function Q, the expressions for which are shown in Table 3-1. Note that only the quantum number dependence is included in the following expressions for / ^ j , (J", J'). The dipole moment (/j,) and the Franck-Condon vibrational overlap are not explicitly shown in the following expressions, but it should be remembered that they are present. 3 MAGNETIC ROTATION 57 R-transitions (J —> J + 1) Amj = +1 : Amj = -1 : ^ ( J , J + 1) = y/(J + mj)(J + mj + l) 4(J + 1)V(2J + 3)(2J + 1) y/{J-mj)(J-mj + l) 4(J + 1)^(2J + 3)(2J + 1) P-transitions (J —> J - 1) (3-17) S(J,Q", J+l,ft ') (3-18) A m j = +1 A m j = -1 n+ (J J — I) = _y/(J-mj)(J-mj + l) HnM-l V 4 J V ( 2 J + 1)(2J-1) y { J ' U ' J ^ ( J , J - 1 ) = + V ( J + / ! ^ ± ! ! ^ g ( J , f t " , J - i , f t ' ) M J K ' 4 J V ( 2 J + 1 ) (2J- 1) v ' ' ' ; Q-transitions (J —> J) A m j = +1: ^ (J, J) = + ^ - + 7 ; ^ ^ + 1 } Q(J, n", J, ft') (3-19) A 1 — / t x\ , V{J -mj){J + mj + Tj „ , Amj = -1: fj,mj(J,J) = + 4 j ( j + i) G(J,Sl ,J,Sl) (3-20) Of particular importance for the current work, note that the sign is the same for A m j = +1 and Amj = -1 in Q-transitions, but the signs are opposite for the other two branches. Table 3-1: The ft-factors, g(J",ft",J',ft'), in matrix elements. j ' = j " + i = j + i j ' = J " = J j ' = j " - i = j - i ft' = ft"+l = 12+1 ft' = ft" = ft ft' = ft"-i = n-i -y/(J + Q + l)(J + Sl + 2) + n + 2ft -y/(J-Q)(J-Q-l) 2y/(J + Q + l){J-n + l) -2y/(J-n)(J + Q.) V ( J - n + i ) ( j - n + 2) V ( J + ft)(J-ft + i) y j + n)(j + n-i) To examine some properties of the above expressions, consider the specific example of an R(2) transition, shown in Figure 3-4. Note that the magnetic sublevels in the lower state are all at the same energy, continuing with the assumption that the lower state is not magnetically active. 2 For this R(2) transition, Figure 3-5(a) shows a plot of //+ (J, J + l ) for A m j = +1 as a function of mj, the magnetic quantum number of the upper level in the transition. The heights of the lines are proportional to the probabilities of A m j = +1 transitions to the different magnetic sublevels. Note that it was assumed that the amplitudes of the electric fields for LHCP light (i.e. 3 MAGNETIC ROTATION 58 Figure 3-4: The possible transitions with k along H for an R(2) line. The Amj = +1 transitions are driven by LHCP light, and the A m j = -1 transitions are driven by RHCP light. R(2) +3 +2 +1 0 -1 -2 -3 / \ /1 I \ 1 \ 1 \ A -1 A m , = +1 -2,-l,0,+l,+2 mi Figure 3-5: Relative transition probabilities for the different possible transitions within an R(2) line. The vertical scale is in arbitrary units; only the relative heights are important. (a) Mmj(J,J+l)| as a function of m r for Amj= +1 transitions (b) lMmj(J,J+l)| as a function of mj, for Amj= -1 transitions + ^ 10 + ^ 10 3 - 2 - 1 0 1 2 3 3 - 2 - 1 0 1 2 3 m m 3 MAGNETIC ROTATION 59 l-Eoil) and RHCP light (|£^0R|) were equal. The plot in Figure 3-5(a) should be compared to.the corresponding results for A m j = -1, plotted in Figure 3-5 (b). 2 _ 2 Notice that fimj(J, J + 1) for Amj = +1 transitions is equal to l^^_mj^(J, J + 1) for Amj = -1. In other words, the transition probability for mj - 1 to mj (driven by LHCP light) is the same as the transition probability for -mj+1 to -mj (which is driven by RHCP light), a symmetry true for all three branches. These pairs of equal-probability transitions shall be referred to as complementary transitions. Figure 3-6 shows the relative A m j = ±1 transition probabilities in a P(4) line and in a Q(3) line, the transitions from the other two branches with the same value of J in the upper state (J = 3) as the R(2) transition. The vertical scale for the P(4) plots are not necessarily the same as the vertical scale of the Q(3) plots, nor do the scale of the P(4) plots or the scale of the Q(3) plots correspond to the scale of the R(2) plots in Figure 3-5. Note that the intensity pattern for the P line is opposite to that for the R line. Also note that the largest transition probability for a Q line occurs at low mj, while for R and P lines, the strongest transitions occur at the extremes, i.e. mj = ±J . These factors have significant consequences for the magnetic rotation signal, which will be explored in more detail later, once the procedure for calculating the signals has been established. Using the expressions in Eq 3-13 and Eq 3-14, the magnetic rotation signal given by Eq 3-11 can be calculated. However, each term on the right-hand side of Eq 3-11 has a different lineshape associated with it, and so the lineshape actually measured depends on experimental conditions. Some choices need to be made concerning experimental procedures before the expected lineshape can be determined. The experiments in the current work were performed using an oscillating magnetic field: H = H0cos(2nfHt) = H0cos(u>Ht), where f# is the oscillation frequency of the magnetic field. The purpose of using an oscillating field (as opposed to a DC field), was to make use of phase sensitive detection. Using detection equipment tuned to f# (or some multiple thereof), the signal-to-noise ratio can be improved by selecting only the output from the measuring devices that is modulated at that frequency. A signal that varies as cos(a;#t) (or as the sine) is called the first-harmonic signal, and is measured by tuning the detection equipment to frequency f#. A signal that varies as cos(2w//t) is called the second-harmonic signal and is measured by tuning the detection equipment to 2f#. In a system with a non-linear response, there can be contributions to more than 3 MAGNETIC ROTATION 60 Figure 3-6: Relative transition probabilities for transitions within a P(4) and within a Q(3) line. +3 - 3 - 2 - 1 0 1 2 3 - 3 - 2 - 1 0 1 2 3 m m one harmonic. See e.g. [46] for a more detailed discussion on phase-sensitive detection. Consider first the frequency-shift contributions to the signals in Eq 3-11. In the absence of a magnetic field, the transitions between two states always occur at the same frequency, regardless of which magnetic sublevels are involved. In the presence of a magnetic field, the magnetic sublevels shift in energy, and the frequency of a particular transition depends on the value of mj>, the magnetic quantum number in the upper state (assuming, as always, that only the upper state is magnetically active). The change to the parameter co0 in Eq 3-13 and Eq 3-14 resulting from the shifts of the magnetic sublevels can be expressed as . , , , AE(m'j) 1 Au0(m'j) = ^ = -h VBgjmj cos{u>Ht) = T]jm'jCos(ujHt), (3-21) 3 MAGNETIC ROTATION 61 where the parameter 77 j (= \^B9J H0 ) has been introduced for convenience. Using Eq 3-14 and keeping in mind that LHCP light drives Amj = +1 transitions and RHCP light drives Amj = -1, an expression can be written for magnetic circular dichroism: 47r 7riVj» A a = ax - a R = X0 Qe0h Amj=+1 7 (w0 - to + rijM+cos(u>Ht))2 + (2)2 - E I ^ V ' ) I % , , , , , ^ a (3-22) A m J = - l ( W q ~ W + TljM-COsiiVHt))' + (^)2 where M + represents the value of m' ; (the value of mj in the upper state) in a Amj = +1 transition, M _ represents m'j in a A m j = -1 transition, and (J", J') is defined according to Eq 3-16. The parameter Nj» represents N(J",mj) from Eq 3-14, where it has been assumed that the populations of all the magnetic sublevels are the same (since they are at the same energy), and the term has been pulled out of the summations. It is difficult to extract the contributions to different harmonics with the cosine in the denomi-nator of Eq 3-22. To remedy this problem, the first term in the denominator is rewritten: (u0 - to + r]jM±cos(ojHt))2 = (u0 - u)2 + 2{OJ0 - u)njM±cos{ujHt) + ^ j ( M ± ) 2 (1 + cos{2uHt)), where use was made of the trigonometric relation cos2(a;#t) = 1(1 + cos(2cjfjt)). If the maximum splitting of the magnetic sublevels is small compared to the HWHM (^), the series expansion « l - x + x 2 - . . . , | x | < l can be used to bring terms involving cosines into the numerator: 7 ^_ 1 (Uo - OJ + rjjMcosiiJHt))2 + (§)2 ~ ( W q _ w ) 2 + lv2jM2 + (2)2 2{uQ - ui)r]jMcos(uJHt) + \rjjM2cos(2u)Ht) ( a ; 0 - W ) 2 + l7? 2 M2 + (2)2 4(w0 - u)2(r]jMcos(ujHt))2 + 2(u0 - u)nzJM'icos{ujHt)cos{2uHt) + \rjJM4cos2(2ojHt) + [ K _ w ) 2 + 1 ^ 2 M 2 + ( i ) 2 ] 2 (3-23) for M equal to either M + or M _ . The \r]2M2 term in the denominator of Eq 3-23 looks like an increase in the linewidth. It represents a modulation broadening of the signal. For the Doppler-limited studies performed in the current work, there was no experimental evidence of the widths of the signals being sensitive to the amplitude of the magnetic field. In the derivations that follow, this term is therefore dropped, or more accurately, absorbed into the linewidth 7. With its contribution to the apparent linewidth, however, it could play a significant role in the Doppler-free technique presented here and should not be forgotten completely. 3 MAGNETIC ROTATION 62 Substituting Eq 3-23 into Eq 3-22, and keeping only the leading terms in cos(u;#t) and cos(2oij/t), yields 4"7T nNjn Aafs « -———-X0 oeah [(u>0-u;)2 + (2)2] A m j = + 1 + A m J = - l 1 2 , n J ( 3 K - W ) 2 - ( | ) 2 ) 2 [ ( W o _ w ) 2 + (2)2]<» x ( £ l ^ + ( J " , J ' ) | 2 ( M + ) 2 - J2 \»M_(J",J')\2(M-)2) +...| , (3-24) \Amj=+l Amj=-1 / where /s stands for frequency shift, since this term is the frequency shift contribution to A a . To obtain the magnetic rotation signal, this expression for A a / S must be substituted into Eq 3-11. From the symmetries of A m j = +1 and A m j = -1 transitions discussed earlier, it should be evident that for the subtractions of complementary transitions in Eq 3-24 (i.e. considering the sub-tractions on a pair-by-pair basis), all even functions of M± (f(M±) = f(-M±)) will cancel, while odd functions (f(M±) = -f(-M±)) will add to give double the A m j = +1 term. This means that only terms with odd powers of M± survive. In examining these surviving terms, the trigonometric iden-tities cos3(u>Ht) — 3 [cos(3o;#t) + 3cos(u;tft)], and cos(cj^i)cos(2a;^i) = ^[cos(u;#£) + cos(3wf/£)], and so on, are useful. In Eq 3-11, once A a / S is substituted from Eq 3-24, there will be contribu-tions only to odd harmonics (first, third,...) from the A a term. By the same token, the A a 2 term, calculated from the square of the expression for A a , only gives even harmonics. The frequency shift contribution of A a to the first harmonic signal (i.e. proportional to cos(wffi)) in Eq 3-11, keeping only the leading term from Eq 3-24, is SF„(Aa,.) = pEy-z^Zd - p E l e - ^ Z i ^ " 7 i W Z i ^ E\^(J",f)fM+, (3-25) A 0 ieah [ ( W o _ a , ) 2 + (2)2] A m j = + 1 where SFH denotes the first harmonic signal. The significance of contributions to the first harmonic signal from neglected higher order terms in Eq 3-24 depends on the relative magnitudes of \ and VJ-From Eq 3-25, the signal, S F H ( A a / S ) , depends on the intensity of the incident radiation (|.E0|2), the distance travelled through the medium (according to Ze~aZ, roughly proportional to Z for weak absorption), the pressure and the temperature (both of which enter through the number density Nj»), the magnetic field strength through rjj, and the degree of initial ellipticity d. The parameter d depends to some extent on the quality of polarizers used in the experiment. In addition, and 3 MAGNETIC ROTATION .63 likely more important, strain on the cell window can introduce ellipticity into light as it enters a cell. Now consider the contributions from the real part of the index of refraction, n 0 . Using n c from Eq 3-13, the general expression for An is A n = noL - noR ITNJII 3eQh u>„ — U E \VM+(J ,J)\ { U o _ u + njM+cos(u>Ht))2 + mi Amj=+1 - E \»MAJ",J')f UJ0 — L) (3-26) A m , = - i K - w + ruM-cos(uHt)y + (2)2 Following the same procedure that was used to determine A a / S l the frequency-shift contribution to Eq 3-26 is A n •fa' 3e0h [(3r)2-(u;0-u;)2] r)jcos(cvHt) [(u; 0-a,)2 + ( 2 ) 2 ] 2 [ K - a ; ) 2 - 3(2)2] { U q _ u ) \^M+(J",J')\2M+- £ | ^ J / , / ) | 2 K Amj=+1 Amj=-1 +^n2cos(2u;/ji) ^ 2 , ^ 2 1 3 2 [(w 0 -a;) 2 + (2)2] E | ^ + ( J " , J ' ) | 2 ( M + ) 2 - E \^M_{J",J')\\MA +.... (3-27) ^Amj=+l Amj=-1 / Again breaking the summations over Amj = +1 and Amj = -1 into a consideration of com-plementary transitions on a pair-by-pair basis, the terms that would give rise to even harmonics (i.e. oc cos(2a>i/t), cos(4o;/jt), etc.) cancel out in the subtraction, and only the odd harmonic terms survive. For the magnetic rotation signal proportional to An in Eq 3-11, the leading contribution to the first harmonic signal from Eq 3-27 is SFH(Anfs) = -pEle-«zW*-^Z = pE2oe-aZVj90Z 7T 47rATj» (CJ0-UJ)2 - (2)2 E \^+(J",J')\2M+. (3-28) A 0 3e0h [ K - a ; ) 2 + ( 2 ) 2 ] 2 A m ^ + i ' None of the other terms in Eq 3-11 give a contribution to the first harmonic signal. The terms in (Aa) 2 and (An) 2 only have even harmonics, just as the terms in Aa and An have only odd harmonics. Taking the first order term for A a / S from Eq 3-24 and the first order term for Anfs from Eq 3-27, the leading frequency shift contribution to the second harmonic magnetic rotation signal, SSH, is SsH((&afs)2) + SSH((Anfs)2) = 1 p E ^ W j Z 2 ^ * 7 ^ A 2 96^2 [(u}_UJo)2 + q)2y £ \^M+(J",J')\2M+ , (3-29) i Amj=+1 3 MAGNETIC ROTATION 64 where the contributions from the two terms have been combined into a simple, convenient form. The lineshape for the expression in Eq 3-29 resembles a Lorentzian. It is positive definite, unlike the first harmonic signal, which can be either positive or negative. The second harmonic signal goes as Z2e~aZ, is proportional to the square of the amplitude of the magnetic field, and does not depend on the "background" parameters 9a and d. 3.5 Contributions From Intensity Perturbations The calculation of the magnetic rotation signal from Eq 3-11 is not yet complete. There are also population and perturbation contributions to consider. Population effects could arise if the lower state involved in the transition were magnetically active (or if the upper state had significant population). Under appropriate conditions, a shift in the energy of a level would be accompanied by a change in population according to the Boltzmann factor, as defined in Eq 3-15. However, for an experiment with an oscillating field, where (at sufficiently low pressures) collisional processes would have little time to redistribute the populations of the magnetic sublevels, this effect would be small. For the experiments performed in the current work, there is very little magnetic activity in the lower state and essentially no population in the upper state, and so this contribution will be ignored. Perturbation effects, on the other hand, are expected to give significant contributions. When a molecule is placed in a magnetic field, J need not be conserved, which leads to perturbations between different J levels for electronic states with A ^ 0. Including all possible perturbations induced by the magnetic field, the perturbed wavefunction, \J,mj,Cl)', can be written as: where again only the pertinent quantum numbers are explicitly listed in the wavefunctions. The label "p" denotes the perturbing state. The perturbation could be between different electronic states, or internal to the electronic state (i.e. between J and J+ l or between J and J - l within a given state). For perturbations internal to the electronic state, the middle term on the right-hand side of Eq 3-30 would not appear. Note that the mixing is between states with the same value of mj, since m ; remains a good quantum number in an external magnetic field (neglecting hyperfine effects). It is changes in intensities as a result of this mixing that leads to a contribution to the magnetic rotation signal. (J, mj, Clp \HZee\ J, mj, Cl) EJ-EJ J,mj,Cl)' « \J,mj,Cl) + J,mj,Clp) + J + l,mj,Clp) + \J-l,mj,Clp), (3-30) 3 MAGNETIC ROTATION 65 The mj-dependence of the mixing coefficients is an important factor to consider. The magnetic rotation signal always derives from a difference between Amj = -1 and A m j = +1 transitions. With the approach taken in the current work, this translates to a difference between complementary transitions. In this case, it is the relative sign of matrix elements that comes into play. A simple rule to follow is that if changing the sign of mj in the term changes the sign of the term itself, that term then contributes to the magnetic rotation signal. Typically, each term is either even or odd under this operation. It is necessary, therefore, to consider the mj dependence of the terms in Eq 3-30. The different possible matrix elements for the Zeeman Hamiltonian are shown below. Perturbation of J with J + 1: (J + l ,mj , f t p \Hzee\ J,mj,Q) = x / ( J + mj + l ) ( J - m j + l ) > 2(J + 1 V ( 2 J + 3)(2J + 1) G(J + 1, ftp, J, ft) fiBcos(cjHt) (ftp \gsS + gLL\ ft) (3-31) Perturbation of J with J: (J,mj,Q,p\HZee\ J,mj,ft) = mj G(J, ftp, J, ft) nBcos(ujHt) | i? 0 | (ftp \gsS + gLL\ft) (3-32) 2J(J + 1)-Perturbation of J with J - 1: (J - l,mj,ftp \HZee\ J, raj, CI) = y/(J + mj)(J - mj) Q(J - 1, ftp, J, ft) jjLBcos{LoHt) H0\ (ftp gsS + gLL 2J N / (2J + 1 ) (2J -1 )" V " - > - T o - > - v r ^ ~ - v ~ n ~ j |««| yv\**~ - (3-33) The functions involving ft, Q, are given in Table 3-1. S and L are the appropriate operators for connecting the two states in question, and (Qp gsS + gjjj ft) must be left as a parameter to be determined from the experiment for a molecule described by Hund's case (c) coupling. Note that the vectors in the above equations are taken as dimensionless (e.g. \s\ = y/S(S + 1), not \/S(S + l)h). An important thing to notice is that the matrix elements for perturbations with A J = ± 1 , the only possible perturbations internal to the electronic state, are even functions of mj. A J = 0 perturbations come into play for perturbations between electronic states and will be discussed in more detail later. For convenience, the following shorthand is adopted for the mixing coefficient: ± , ,\ (J ±l,mj,n\HZee\ J,mj,Q) HZeeKmJ)) = p S . 3 MAGNETIC ROTATION 66 where the upper sign on the left hand side of the equation goes with the upper sign on the right hand side, and the lower signs go together. Consider first the effects of perturbations within the electronic state. Remember that there is no A J = 0 term for internal perturbations, and keep in mind also that magnetic effects in the lower state are assumed to be negligible. The mixing of the wavefunction described in Eq 3 - 3 0 leads to a perturbed transition probability: M M ± ( J V ) | 2 * \VM±(J",J') + (HtE(M±))^±(J",J' + 1) + ( ^ E E ( M ± ) ) ^ ± ( J " , J' - 1) Expanding this gives: M M ± ( J " > J')\2 = | ^ M ± ( ^ " > ^ ) | 2 + | ( ^ e e ( M ± ) ) M ± ± ( j " , J' + l ) | 2 + | ( i ^ e e ( M ± ) ) / 4 ± ( J " , J' - 1) + 2 (H+ee(M±)) /4 ±(J", J ' ) / 4 ± ( J" , J' + l) + 2 {H~ZEE{M±)) M ± ± ( J " , J ' ) / 4 ± ( A J' ~ 1) + 2 (H+EE(M±)) ( fT j e e (M ± ) ) a4 ±(J", J' + l) /4 ± (J", J' ~ 1) (3-34) Take an R-transition for example, for which J" = J and J' = J + 1. Since A J = 2 is forbidden for electric dipole transitions, Eq 3 -34 becomes: + VL'±±(J,J + 1)\ = | / 4 ± ( J , J + 1) ( i ^ e e ( M ± ) ) M ± ± ( J , J ) | 2 + 2 ( i ^ e e ( M ± ) ) ^ ( J , J + l ) / 4 ± ( J , J). (3-35) The first term in Eq 3-35 is just the normal transition moment from the unperturbed wavefunc-tion. The second term is an even function of mj, and therefore the only effect it has is to change the overall intensities of the magnetic rotation signals in Eq 3-25, Eq 3 -28 and Eq 3 -29, but it will not affect the lineshape. The third and final term in Eq 3-35 is an odd function of mj. This is because the matrix element / L ^ ± ( J , J + l ) has a different sign for A m j = + 1 than it does for A m j = -1 , while fJ^±(J, J) has the same sign for the two types of transition. This can be seen in Eq 3-17, Eq 3-18, Eq 3-19 and Eq 3-20. The result is that the third term in Eq 3-35 is opposite in sign for complementary transitions and gives rise to a new set of contributions to the magnetic rotation signal. This effect arises because the probability of one type of transition (Amj = + 1 or -1) is enhanced by the perturbation, while the probability of the opposite type of transition is diminished. The contribution of the intensity perturbations to the signal from A a are considered first. + 2 Starting from the equation for A a in Eq 3 -22, the transition probabilities, fi%± (J,", J1) , are replaced by the expression in Eq 3-34. Keeping in mind the fact that (H^ee{Mj)^ is proportional to cos(wj^t), the signals for the various harmonics are determined as they were before: the first 3 MAGNETIC ROTATION 67 harmonic signal is taken as the coefficient of cos(w#t), the second harmonic signal is the coefficient of cos(2u;#t), etc. As with A a / S , there are only odd harmonics for the intensity perturbation contribution to A a , A a ^ p (where ip stands for intensity perturbation); all of the terms that would give rise to even harmonic signals cancel out in the subtraction performed in Eq 3-22. The leading contribution to A a , p is: ^ ^ T ^ h (u JLf iv £ [F(M+,J' + l,J',J")+HM+,J'-l,J',J")), X0 5e0n {u>0 - coy + A t 7 ^ 1 + 1 (3-36) where the fact that the contribution from the Amj = -1 transitions merely doubles the A m j = +1 term has been used. The function T is defined as J-{M+,J ,J ,j ) = E > _ gW HM+\J ,J )HM+{J ,J ), where the matrix elements for Hzee are given in Eq 3-31, Eq 3-32 and Eq 3-33. The contribution to the first harmonic signal of Eq 3-11 is thus: SFH(^ip) = pE^Zd^-^f-. -7^2 A c 3eah (u0 -co)2 + (f)2 x [ ^ " ( A f + ^ ' + l . J ' . J ' O + ^ M + . J ' - l . r . J " ) ] , (3-37) Amj=+1 Now consider the intensity perturbation contribution to the magnetic rotation signal from An. Introducing the perturbed transition moment in Eq 3-34 into the expression for An in Eq 3-26, the resulting first order term is AnNj" (Up - LP) a n i p * 3eQh ( W o -u / )2 + (2)2 x J2 [F{M+,J' + 1,J',J")+F(M+,J' -1,J',J")} , (3-38) The intensity perturbation contribution to An has only even harmonics, the same as for the frequency-shift contribution to An. The contribution of Eq 3-38 to the magnetic rotation signal in Eq 3-11 is X0 3e0h (u)0 - cv)z + {^y Y [J7(M+,J'+l,J',J")+F(M+,J'-l,J',J")], (3-39) Amj=+1 The intensity perturbation contributions to the terms in A a 2 and A n 2 in Eq 3-11 have only even harmonics, as was the case for the frequency-shift contributions. With the inclusion of intensity 3 MAGNETIC ROTATION 68 perturbation effects, the second harmonic signal can no longer be written in so simple a form as Eq 3-29. The full expression will not be explicitly shown, but it is calculated according to: S 5 „( (Aa) 2 ) + SsM(An)2) = PE2e-aZ Z2 7T2 — ( A a / S + Aaipf + -rj ( A n / s + An i p ) (3-40) where Actfs is taken from the leading term in Eq 3-24, Actip is defined in Eq 3-36, A n / S is the leading term in Eq 3-27 and An^ p is defined in Eq 3-38. In addition to the perturbations off-diagonal in J from within the same electronic state, there can also be perturbations off-diagonal in the electronic state. To calculate the contributions from exter-nal perturbations to A a and An, three terms must be added to each of the sums in 3-36 and 3-38 (for each electronic state providing an external perturbation): ^ r e x (M + , J ' , J ' , J") , j r e x (M + , J '+ l , J ' , J " ) and .Fea^M+jJ'-ljJ'jJ"), where the subscript "ex" stands for external. Note that the perturbing electronic state must have a non-zero transition moment to produce an additional magnetic rotation signal from the intensity perturbation effect. It should be noted that perturbations off-diagonal in the electronic state also occur by means other than the Zeeman Hamiltonian, e.g. through the rotational Hamiltonian. Interference effects between the Zeeman Hamiltonian and the rotational Hamiltonian coupling the same electronic states can have a significant effect on magnetic rotation signals, as will be seen later. 3.6 Doppler Effect As was discussed in Section 2.9, thermal motion of molecules in the medium affects the measured signal. The Doppler-broadened signal can be calculated by taking a convolution of the natural linewidth with a Maxwellian distribution for the velocities of the molecules [37]. For every calculated signal, the lineshape factor must be determined from an integral of the following form f Jo i Q O 4 l n ( 2 ) _ ( * o — / ) 2 F(uj,J)e **£duj', (3-41) where 8U>D is the Doppler width defined in Eq 2-57. This integral can be evaluated numerically. The expression F(UJ,UJ') denotes the lineshape of the signal in question. For Sjr#(Aa/ s) from Eq 3-25, this lineshape factor is F(u,a/) = 1 { W - J ) 2 . (3-42) This just comes from the frequency-dependent factor in Eq 3-25, where the transition frequency (u>0) gets replaced by the variable of integration (UJ'). The general procedure for including the Doppler effect is to replace the frequency-dependent factor in a particular expression by an integral 3 MAGNETIC ROTATION 69 of that frequency-dependent factor of the form given in 3-41. For Sjr.ff(An/s) from Eq 3-28, the lineshape factor is F M = K - o ' - ( i ) ' [ K - u 0 * + ( i ) 2 ] 2 For SFH(Actip) from Eq 3-36, it would be For SFK(Ariip) from Eq 3-38, it would be F M = W^TWY The effect of Doppler broadening on second harmonic signals can be calculated from the above expressions. Ss#((Aa/ s) 2) involves the square of the lineshape in Eq 3-42, for example. The procedure for calculating Ss//((Aa/ s) 2) is to find the integral in Eq 3-41 using the lineshape in Eq 3-42, and then square the result. Similar procedures are used to calculate the other contributions to the second harmonic signal. 3.7 Examples of Calculated Signals To illustrate how the equations are applied, some example calculations are given here. To begin with, the first harmonic portion of the magnetic rotation signal in Eq 3-11 is given by: SFH = SpHi&ctfs) + SFH(&aip) + SFH(Anfs) + S F H ( A n i p ) . One thing to notice is that the magnetic dichroism (Aa) contributions to the first harmonic signal involve d, the parameter that describes the original degree of polarization of the light, while the magnetic birefringence (An) contributions involve 9a, the offset of the polarizer from PA. In practise, it is difficult to measure either d or 60, which makes calculating the lineshape a challenge. However, the experimental conditions can be chosen in such a way as to make the signal depend (approximately) on only one or the other of the parameters, rather than both. If 9Q is much larger than d, the first harmonic magnetic rotation signal will be predominantly from An. Conversely, if 9a is set to zero, the signal comes entirely from A a . To simplify the lineshape, it is best to choose one of these two regimes (9a — 0 or 0o 3> d). Consider first the case for 9D — 0. The contributions from An vanish, and the first harmonic signal is given by SFH = % ( A a / s ) + S>w (Aa i p ) . 3 MAGNETIC ROTATION YU The expression for the frequency shift contribution is given in Eq 3-25, and the intensity pertur-bation contribution is given in Eq 3-37. For a particular length of the cell (Z) and a particular lower-state population (Nj»), there is a constant common to all contributions to the first harmonic signal ^ g « — z i ^ = a (3-43) The constant C is just set to arbitrary values (e.g. 1) for the calculation of an isolated line. For the case where 0O is equal to zero, the parameter d is also set to 1. To give an example typical of the experiment observations for the current work, the transition is an = 0 —»Q = 1 R-transition to an upper state J = 4 (i.e. R(3)). The magnetic field is taken to be 60 gauss, the B-value is 900 MHz (« 0.03 cm - 1 ) . To change this into the appropriate units for the calculations that follow, this is multiplied by 2ir. Matrix elements are evaluated in the Hund's case (a) basis set. The choice of the homogeneous linewidth is an important one, since that determines the relative contributions of A a / S and Actip. The larger the linewidth is, the more significant the contribution from ActiP becomes relative to the contribution from Aotfs. For the experiments in the current work, the natural linewidths for transitions in the A — X system are the order of 1 kHz. The pressure in the Br2 cell used in the current work was 6.5 Torr ([47]). According to [48], the collision induced self-broadening of the homogeneous linewidth varies as 0.168 cm - 1 /atm, which would give a homogeneous linewidth of about 45 MHz. The parameter 7 was rounded up to 300 Mrads/sec (ss 2ir times 45) for the calculations shown. The Doppler width was taken to be 450 MHz (again multiplied by 2n to get into the appropriate units). The calculated frequency shift and intensity perturbation contributions to the first harmonic signal with 0o = 0 are shown in Figure 3-7. The signal from Aonp looks like a Lorentzian, as the lineshape in Eq 3-37 suggests. The lineshape for the signal from Actfs looks like a dispersion, which is just the derivative of a Lorentzian. This is not surprising, since the signal comes from subtracting two Lorentzians offset in energy. Subtracting point-by-point the two lineshapes in Figure 3-2 would lead to a dispersion lineshape. The relative amplitude of the two signals depends most sensitively on the proximity of the perturbing state, on the homogeneous linewidth for the transition, and on the transition moment of the perturbing state. The calculated AaiP signal in Figure 3-7 is from perturbations internal to the electronic state, where the B-value (and the quantum number J) determines the energy separation between perturbing levels. No external perturbations were included in the calculation. Figure 3-8 shows the sum of the frequency shift and intensity perturbation contributions, the magnetic rotation signal that would be measured for 0O = 0. MAGNETIC ROTATION 71 Figure 3-7: Calculated frequency shift and intensity perturbation contributions to the first harmonic signal in A a . Figure 3-8: Calculated first harmonic signal from A a . The signal still looks vaguely like a dispersion line, except one lobe is larger than the other. In the regime where 90 is much larger than d, the first harmonic magnetic rotation signal is SFH ~ SFH(&nfs) + SFH(Anip). (3-44) 3 MAGNETIC ROTATION 72 The expression for the frequency shift contribution is given in Eq 3-28, and the intensity pertur-bation contribution is in Eq 3-39. The constant C in Eq 3-43 is again set arbitrarily to 1. The parameter 60 is also set to 1, and d is set to zero. Using the same molecular constants as was used for calculating A a , the first harmonic signal from A n was calculated. The results for the R(3) transition are shown in Figure 3-9. Figure 3-9: Calculated frequency shift and intensity perturbation contributions to the first harmonic signal in An. S F H ( e 0 » d ) _ 4 I i . i i i 1 i . 1 . -10000 -5000 0 5000 10000 co (Mrads/sec) The lineshape for the signal from Anj P is dispersion, while the A n / S signal has a second deriva-tive lineshape. This is the lineshape that would be obtained by subtracting point-by-point the two dispersion lines offset in frequency in Figure 3-3. The signal that would be measured for 9a 3> d is shown in Figure 3-10. It is the sum of the frequency shift and intensity perturbation contributions. The lineshape for the second harmonic signal can be obtained simply by taking the sum of the squares of the signals in Figure 3-8 and Figure 3-10. The signal will be proportional to C 2 , where C is the constant defined in Eq 3-43. For the same example of an R(3) transition with all of the same molecular constants, the frequency shift contribution to the second harmonic signal, calculated from Eq 3-29, is shown in Figure 3-11, along with the new contributions to the second harmonic signal from intensity perturbation effects. This includes both the squared terms (i.e. oc A a - p and oc An 2 p ) and the cross terms (oc A a / s A a i P and oc A n / s A n ; p ) . The resultant second harmonic lineshape from taking the fs and ip contributions into account is shown in Figure 3-12. It is no longer a symmetric Lorentzian, as it was when only the frequency 3 MAGNETIC ROTATION Figure 3-10: Calculated first harmonic signal from An. S F H ( A n f s) + S F H ( A n i p) _ 4 I i i i i 1 1 1 1 1 • -10000 -5000 0 5000 10000 co (Mrads/sec) Figure 3-11: Calculated frequency shift and intensity perturbation contributions to the second harmonic signal. Second harmonic signal SSH((Aafs)2) + SSH((Anfe)2) i . ' i i 1 — -5000 0 5000 10000 co (Mrads/sec) shift contributions were considered. There is a skew to the lineshape, and the centre of gravity that would be measured from this line would not equal the true centre of gravity. The error induced in the frequency determined by fitting the lineshape in Figure 3-12 to a Lorentzian is significant, but is difficult to account for without performing a lineshape analysis. 3 MAGNETIC ROTATION Figure 3-12: Calculated second harmonic signal, including both frequency shift and intensity perturbation contributions. 74 c rs •e TO 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 -0.5 SS H((Aa f s+Aa i p)2) + S s H((An f s+An i p) ) centre of^ gravity -10000 -5000 0 5000 co (Mrads/sec) 10000 As a simplifying approximation, the frequencies determined in the current work neglected the intensity perturbation contributions, since no attempt was made to account for the apparent shift these contributions induce in the measured frequency. To account for this problem, slightly larger uncertainties were assigned to the frequency measurements, as will be discussed in more detail later. 3.8 General Notes on Magnetic Rotation A well known property of magnetic rotation is that R-branch transitions and P-branch transitions give (first-harmonic) signals that are opposite in phase [6]. Using the same molecular constants used earlier to calculate the R(3) magnetic rotation signal, the first harmonic magnetic rotation signal for 60 2> d for a P(5) line is shown in Figure 3-13. This should be compared to Figure 3-10. The phase difference for R and P lines is a good tool for identifying the type of transition in the spectrum. The reason the signals for R and P transitions are opposite in phase can be seen from the intensity patterns in Figure 3-5 and Figure 3-6. The strongest A m j = +1 transitions for an R branch occur for mj = +J, while the strongest A m j = +1 transitions for a P branch occur for mj = -J. There are two terms for the intensity perturbation contribution to Q-lines, since the upper state mixes with two states (J+l and J-l) for which there is a non-zero electric dipole transition moment. 3 MAGNETIC ROTATION 75 Figure 3-13: Calculated first harmonic signal for P(5). The phase is opposite to that of the R line. However, the two terms are opposite in sign and almost cancel each other out. The calculated Q(4) first harmonic signal with 6a » d is shown in Figure 3-14. Figure 3-14: Calculated first harmonic signal for a Q(4) line. Two contributions to the intensity perturbation signal nearly cancel, and the signal looks almost sym-metric. The phase is the same as that of an R line. Calculated Q(4) (e 0 » d) . 1 5 I i . i i i . i . 1 1 - 1 0 0 0 0 - 6 0 0 0 0 5 0 0 0 1 0 0 0 0 co (Mrads/sec) The J-dependence of the Q branch is much different than that of the R and P branches. As seen Figure 3-6, the strongest transitions for a Q line occur for small |mj|, unlike the P line in Figure 3 MAGNETIC ROTATION 76 3-6 or the R line in Figure 3-5, for which the strongest transitions occur for large |mj|. Since the frequency separation between the strongest transitions is smaller for Q lines than for the R and/P lines, the subtraction of A m j = -1 transitions from Amj = +1 transitions has a more significant effect in the Q branch as the spacing between magnetic sublevels decreases with increasing J . The result is that the first harmonic magnetic rotation signals for Q lines vary as J J , while the first harmonic signals for R and P lines vary roughly as j. 3.9 Incorporating Quadratic Shifts With the magnetic fields used in this thesis, the shifts induced by effects quadratic in the magnetic field are expected to be minimal, i.e. less than the uncertainties attributed to frequency measure-ments. When determining transition frequencies from magnetic rotation spectra, however, it is an important thing to keep in mind. The expressions derived for magnetic rotation signals assumed a frequency shift that was linear in the magnetic field, as explicitly defined in Eq 3-21. There can also be shifts that are quadratic in the magnetic field (and therefore quadratic in mj). For example, the energy shift resulting from mixing states off-diagonal in J is quadratic in the field. Since it goes as m 2 , the direction of the shift is the same for mj = +x as that for mj = -x (where x is some arbitrary value). The strength of the signal is not affected, but there is an apparent shift in the line position. To account for this in the calculated signal, another term is added to the shift defined in Eq 3-21: Aw 0(mj) = r)jmjcos(LOHt) + ^0™j + ^(jmjCos(2uHt), where (j is some function of the quantum numbers and is proportional to the square of the magnetic field amplitude. In the calculation of the signal, the denominator ([(u0 — to)2 +• (^) 2] n) in expressions such as Eq 3-25 is replaced by [LJ0 — to + \C,jm2j)2 + ( 2 ) 2 . The lineshape can no longer be pulled out of the sum over the transitions, and there is an extra contribution to the broadening of the apparent linewidth (i.e. on top of \ri2jrn2J), but everything else is the same. 3.10 Static Magnetic Fields During the experiments performed for this thesis, no measures were taken to circumvent or minimize the effect of the Earth's magnetic field. However, the maximum splitting resulting from the Earth's field was calculated to be very much less than the maximum splitting due to the oscillating magnetic 3 MAGNETIC ROTATION 77 field, and the effects of the Earth's field were therefore neglected. When there is a static magnetic field in addition to the oscillating magnetic field, the approach used here becomes less convenient. Complementary transitions are no longer degenerate when the oscillating field goes through zero, so the sum over A m j = -1 terms can no longer be combined with A m j = +1 terms. The general formulation still works, but as with the quadratic shifts, the lineshape expressions cannot be pulled out of the sums over the transitions, which makes the calculation more complicated. Also, the even functions of mj no longer cancel, so the Act and A n terms contribute to second harmonic (not just first harmonic), and the A a 2 and A n 2 terms have first harmonic components. With such additional complexity, the usefulness of employing a combination of static and oscillating magnetic fields is questionable. 3.11 Overlapping Lines Extra care must be taken in the measurement of frequencies in a magnetic rotation spectrum when there are overlapping lines. Because the signals can have different phases, the signals from overlapping lines do not necessarily add, as they would in absorption spectroscopy. An R and a P line at the same frequency would destructively interfere, for example, since they are opposite in phase. This is true in both first and second harmonics. The biggest problems can occur when the lines are separated by a frequency the order of the linewidth or less. There can be destructive or constructive interference in the overlap region, which will either "eat away" or enhance the signal in the region between the lines. Both situations would lead to erroneous frequency measurements. This must be kept in mind when analyzing magnetic rotation spectra. In the current work, there was an evolution of the lineshapes as a function of J that resulted from interactions between electronic states. This will be discussed in more detail later. As a consequence of this, for a particular range of J values, the long-range tails of the dispersion and Lorentzian curves did not cancel out in the signal as it usually does, and there were interesting effects between lines well-separated in frequency. Figure 3-15 shows an example of this for two lines from the A - X system, 33'-2" P(15) and 32'-2" R(12). Note that there is a plateau between the two lines where the signal does not go to zero. This is similar to the observations in the first harmonic studies of C2 in [20]. MAGNETIC ROTATION Figure 3-15: Interference effect between two lines well-separated in frequency, the result of electronic state mixing affecting the lineshapes. The signal shown was taken in second harmonic. i i i 1 1 — 15218.00 15218.05 15218.10 wavenumber units 4 EXPERIMENT 79 4 Experiment The potential energy curves for selected valence states in Br2 are shown in Figure 4-1. Note that the electronic states in I2 look qualitatively very similar (see [53]). This is not surprising, since both molecules are diatomic halogens and both are described fairly well by Hund's case (c) coupling. Theoretical calculations are available in I2 for electronic states that have not been measured experimentally in either Br2 or I2 (these states are not depicted in Figure 4-1). It will be assumed that the general features of the theory for I2 are also applicable to Br2. Figure 4-1: Potential energy curves for selected valence states of 7 9 B r 2 -7 9 Br 2 Electronic states 0 Internuclear separation r (A) As mentioned previously, when using absorption spectroscopy to study B r 2 , signals for transi-tions from the X state to the B state typically dominate over signals from X —• A transitions. Both the X state and the B state have £2 = 0, which means they are not (to first order) magnetically 4 EXPERIMENT 80 active. The A state, on the other hand, is magnetically active, which makes magnetic rotation spectroscopy an ideal tool for measuring the A - X spectrum, especially since the technique will suppress the signals from transitions in the B - X system, the signals that are the main impediment to studying the A state with absorption spectroscopy. The experimental setup for the Doppler-limited magnetic rotation study is shown in Figure 4-2. The acronyms in Figure 4-2 are defined as follows: Ar+-Coherent Innova 400-15 Argon ion laser; Dye-Coherent 699-21 Dye laser; LIA-Lock-in Amplifier; M-Mirror; BS-Beam Splitter; CHOP-Mechanical chopper; PE-Photo-emitter; PD-Photo-detector; 12-Iodine reference cell; PMT-Photo-multiplier tube; PO-Polarizer; L-Lens; MR-Magnetic rotation setup, consisting of a cell filled with 7 9 B r 2 placed inside a solenoid; C-capacitor; AC-Alternating current power supply; A/D-analog to digital converter; mVax-yuVax computer. A CR699-21 dye laser running with D C M dye was pumped with the 6.0 Watt, 514.5 nm output from the Innova 400-15 Argon ion laser. During the course of the Doppler-limited studies, the typical single-frequency power output from the dye laser was 300 to 350 mWatt to a maximum with fresh dye of about 550 mWatt. Approximately 10 percent of the output from the Dye laser was shunted off by a beam-splitter and sent through a mechanical chopper that chopped the beam at a frequency of 1.8 kHz. A photo-emitter, photo-detector setup was used to provide a square-wave reference signal modulated at the chopping frequency. The chopped beam was passed through an iodine reference cell. The I2 fluorescence was measured with an EMI 9558B photo-multiplier tube and the output was sent through an A / D converter to be recorded by a yuVax computer. It was found that the signals from the I2 fluorescence were so strong that no phase-sensitive detection was required; the signal was fed directly to the computer. It was tested whether the chopping of the light, or the time constants used on the lock-in amplifiers for the other two signals that were measured simultaneously, introduced any errors, but there were no discernible effects. The chopped beam that passed through the iodine reference cell then entered a 30 cm long etalon that gave frequency markers approximately every 0.01 c m - 1 . The frequency spacing between markers was re-calibrated for every scan using the iodine reference peaks, as will be described shortly. The output from the etalon was measured by a photo-detector, and the signal was sent to a lock-in amplifier tuned to the first harmonic of the chopping frequency. The time-constant on the lock-in was set to 10 ms. The signal measured by the lock-in was then sent through the A / D converter to the /xVax. 4 EXPERIMENT Figure 4-2: Experimental setup for Doppler-limited magnetic rotation spec-troscopy. 81 E t a l o n CHOP • • • « P E PMT PO MR PO L BS El 0 f M m V a x The beam that was transmitted through the beam splitter went through a lens of focal length 4 m. The purpose of this lens was to reduce the divergence of the beam. The output of the dye laser is polarized, and so the alignment of the first polarizer needed to be chosen to allow maximum transmission of the incident light. The second polarizer, the analyzer, was then aligned with its transmission axis at right angles to that of the first polarizer. The extinction coefficient for both polarizers was about 10~5. Strain on the cell windows caused significant transmission even when 4 EXPERIMENT 82 the two polarizers were crossed; clamps were used to squeeze the cell near the windows in order to minimize this effect. The signal transmitted through the analyzer was measured with a PMT, and. the output of the P M T was sent to a lock-in amplifier that was tuned to either the first harmonic or the second harmonic of the modulation frequency for the alternating current used in the solenoid. For the second harmonic measurements, the voltage used on the P M T was 900 or 1000 V, the lower end of (or just below) its range of linear operation. For first harmonic measurements with the polarizers slightly decrossed, the limit (to avoid saturation) on the voltage that could be used on the PMT was 800 V, below the range of linear operation. A neutral density filter could have been used to decrease the signal reaching the P M T and allow a higher voltage, but no quantitative analysis of signal strengths was carried out, and such measures were therefore not deemed necessary. The time-constant used on the lock-in was 100 ms for first harmonic measurements and 300 ms for second harmonic measurements. In the magnetic rotation setup, the cell was 2.0 m long, filled to 6.5 Torr with 99.6% isotopically pure 7 9 B r 2 [47]. The solenoid consisted of a single layer of 18-gauge wire wrapped around a 6 cm diameter form. A capacitance of 0.186 mF was connected in series with the solenoid. The AC power supply was tuned to the resonance of this LCR circuit, which was at a frequency of approximately 5 kHz. The amplitude of the current used in the solenoid was 5.5 A, which gave a magnetic field amplitude of about 60 gauss. To minimize pick-up problems, isolation transformers were used for all of the lock-in amplifiers and the voltage supplies for the PMTs. The detection equipment was also moved as far away as possible from the solenoid. Data was taken in 1 c m - 1 segments, the maximum scanning range for the dye laser. The scanning was controlled by an external ramp supplied by the /iVax. The typical duration of each scan was five minutes. Data was recorded on the y^Vax from three separate channels: the I2 reference spectrum, the interferometer peaks and the magnetic rotation spectrum in B r 2 . The data in each channel was taken with a grid of 4096 points over the 1 c m - 1 span. The magnetic rotation spectrum of 7 9 B r 2 was measured from 15,027 to 15,488.5 c m - 1 . The lower end of the range overlapped with the original spectrum taken by Dr. Alak Chanda, which covered from 14,988 to 15,027 c m - 1 . Measurements for this portion of the thesis were taken in second harmonic, since it is easier to extract systematically frequency information from the second harmonic lineshapes, as will be described later. Each scan taken overlapped with a lower frequency scan and a higher frequency scan (giving about g c m - 1 of unique data on each scan), to provide a consistency check for adjoining scans. 4 EXPERIMENT 83 4.1 Frequency Measurements A computer program was written in a combination of C and ASSEMBLY for the purpose of deter-mining the frequencies of measured lines. The program displays the magnetic rotation spectrum along the bottom of the screen and an inverted graph of the reference I2 B - X fluorescence lines along the top. The baselines of both spectra were stable and flat, and so the program found the baselines and fixed them in the least-squares fits that followed. Reference I2 peaks are selected by the user, and the program performs least squares fits to find the centers of gravity of the peaks. Both Lorentzian and Gaussian lineshapes were tested in this fitting procedure and no significant difference was observed. Lorentzian lineshapes were used for the final determinations, since they were found to be more stable than Gaussians when fitting blended lines in the magnetic rotation spectrum. In the reference spectrum, blended lines were not used. It was not suitable to use the intensity maximum (in either the reference spectrum or the magnetic rotation spectrum) to determine the position of a line, because hyperfine effects, as well as intensity perturbation contributions to magnetic rotation, can lead to asymmetries in the lines. After all of the reference I2 lines are chosen, the interferometer fringes are calibrated, based on the known frequency separations between the various reference lines in the scan. This calibration was performed separately for each scan. The reference lines and the interferometer peaks were then used to measure the frequencies of the Br2 lines. Initially, the experimentally measured frequencies [49] of the iodine lines were used in this process. The relative accuracy of these reference lines is 0.002 c m - 1 . It was found that there were often serious discrepancies for lines in the overlapping regions between scans. Measurements of the same Br2 line performed in two different (overlapping) scans often differed by 0.002 c m - 1 or more. The reason for the discrepancy was that different I2 lines were used as reference lines for the different scans. Most scans contained only one or two lines for which measured frequencies were tabulated in the iodine atlas. Consequently, calculated frequencies for the I2 reference lines, rather than experimental values, were used in the frequency measurements. The calculated frequencies of the I2 lines were based on the results of a global analysis performed on the B - X system by [50]. The average number of suitable reference lines available per scan increased to three or four. The agreement between scans became much better, with typical discrepancies less than 0.0010 c m - 1 . The average calibrated spacing between interferometer fringes was approximately 0.009975 c m - 1 . For the small percentage of scans that contained only one I2 reference line, the fringe 4 EXPERIMENT 84 spacing was fixed to this value. For the two or three rare occurrences where there were no reference lines in a scan, magnetic rotation lines in overlap regions were used as references, their frequen-cies having been determined previously from the adjoining scans. There was no indication of any systematic problems with the data from any of these scans in the subsequent analysis. The calibrated value for the interferometer fringe spacing varied from scan to scan, ranging from 0.00995 to 0.01000 c m - 1 . The pressure and temperature of the etalon were not controlled, and so variations of these conditions in the lab would cause differences in the fringe spacing. The main cause of the fluctuations in the calibrated value of the fringe spacing, however, was due to the varying reliability of the iodine reference lines used in the calibrations in the different scans. After the frequencies of lines in the magnetic rotation spectrum were measured, the spectrum was "smoothed" by ensuring the fringe calibration stayed roughly constant over a range of scans (e.g. all scans taken on a particular day). The smoothed spectrum was built into a database, and a program was written using C and ASSEMBLY to view the spectrum on screen (now available in a-complete, continuous, calibrated form rather than in 1 c m - 1 , uncalibrated chunks). Using this new program, the frequency of each peak in the spectrum was remeasured (via least-squares fitting). Differences from the original measurements were small as a rule, but did range as high as 0.0020 c m - 1 for a few cases. 4.2 Doppler-Free Setup The experimental setup used for the Doppler-free studies in the current work is shown in Figure 4-3. The acronyms common to Figure 4-2 are the same as before. The new acronyms are: B S l -Beam splitter number 1, a 70/30 splitter; BS2-Beam splitter number 2; QW-Quarter wave plate; SUM-A device that takes a sinusoidal input at 6.5 kHz and a square-wave input at 1.8 kHz and gives a sinusoidal output that is the sum of the two input frequencies. The logic is very similar to the previous setup. The major difference here is that the output of the dye laser is split into two beams that are sent in opposite directions through the cell. Different beam splitters were used for BS2. With a 50/50 splitter, the two counterpropagating beams in the cell had roughly the same intensity. With a 90/10 splitter, the intensity of the chopped beam was about twice as large as the intensity of the unchopped beam. Not shown in the diagram (but obviously present) is the A / D converter through which the signals are sent to the /iVax. On occasion, the diameters of the beams were controlled with a telescope arrangement (two converging lenses, the distance between them controlling the beam diameter), also not shown in Figure 4-3. 4 EXPERIMENT Figure 4-3: Experimental setup for Doppler-free magnetic rotation spectroscopy. 85 D y e ^ M CHOP o|a«PI E t a l o n B S 2 / E ] P O P M T MR p o m QW L M The two crossed polarizers were aligned such that transmission of the chopped beam through the first polarizer was a maximum. With this configuration, almost no light from the unchopped beam made it into the cell (because the two beams had the same polarization). Therefore, a quarter wave plate was used to change the polarization of the unchopped beam, a step which maximized the amount of unchopped light transmitted into the cell. 4 EXPERIMENT 86 A different solenoid was used in the Doppler-free studies. It was water-cooled, and consisted of five layers of 18-gauge wire wrapped around a form of diameter 2 cm. Using the same capacitors as previously, the resonance frequency for this solenoid was 6.5 kHz. Running a current of amplitude 6 A in the solenoid gave a magnetic field amplitude of about 350 gauss. The length of the cell used was 30 cm. In the studies in I2, the pressure in the cell was dictated by the room temperature, roughly 280 mTorr. The pressure in the Br2 studies was approximately 300 mTorr. Scans were performed over the span of a single line, the order of 0.033 c m - 1 for I2 and 0.03 c m - 1 for Br2- Scans in I2 lasted 3 to 5 minutes. The scans taken during the search for signals in Br2 were typically 10 to 20 minutes. 5 RESULTS AND ANALYSIS 87 5 Results and Analysis The magnetic rotation spectrum of 7 9 B r 2 was much more congested than originally expected, and many of the strong signals in the spectrum were initially unassigned, as mentioned in the Intro-duction. Strong signals were expected only for low-J transitions, a well-known characteristic of magnetic rotation spectroscopy often used to simplify complex spectra [18]. Notwithstanding this low-J selectivity, one of the most likely sources for the unassigned lines in the spectrum was high-J transitions. To investigate this, a FORTRAN computer program was written to search through a list of frequencies for series with constant second differences. The spacing between rotational states within a given branch follows a systematic pattern. Taking the simple expression F(J) = BJ(J+1) for the rotational energy, the frequencies of the various branches are given by "H(J) = «V'y + B'(J + 1)(J + 2) - B"J(J + 1), vP(j) = vvnv, + B'J{J - 1) - B"J(J + 1), and »Q{J) = "v»,v> + B'J(J + 1) - B"J(J + 1), where zv>' is the band origin, \ times the energy separation between the two vibrational states v" and v'. B ' is the rotational constant of the upper state involved in the transition, and B" is the rotational constant of the lower state. The quantum number J is the value from the lower state, J". The spacing between successive rotational levels in an R-branch is thus Afl(J) = - u m = 2{B' - B")(J + 1) + 2B', and the second differences (the differences of these differences) are AR(J + 1) - A f l ( J ) = 2(B' - B"), (5-1) which is a constant. Similar expressions can be written for P and Q, and the second differences turn out to be the same in all cases, 2(B' - B"). An example of second difference values is given in Table 5-1, from a series of R-lines in the 23'-l" band. Note that there is some fluctuation in the value of the second difference; the effect of experi-mental scatter is compounded by taking differences. A worse threat to finding a series, however, is if one frequency is off (e.g. if you have a blended line), it will affect three of the second differences. With a congested spectrum, where there is a large probability of having overlapping lines, one must 5 RESULTS AND ANALYSIS 88 make fairly liberal allowances for fluctuations in the second difference. For a spectral window as narrow as this one, the relaxed constraints on variations in second differences led to the program finding a large number of false series. Table 5-1: Second Differences in 23'-1" R-branch transition frequency (cm 1) AR (cm"1) second difference (cm x) R(33) 15,323.5577 R(34) 15,320.1743 -3.3834 R(35) 15,316.6895 -3.4848 -0.1014 R(36) 15,313.1033 -3.5862 -0.1014 R(37) 15,309.4092 -3.6941 -0.1079 R(38) 15,305.6148 -3.7944 -0.1003 R(39) 15,301.7126 -3.9022 -0.1078 R(40) 15,297.7095 -4.0031 -0.1009 R(41) 15,293.5919 -4.1176 -0.1145 R(42) 15,289.3742 -4.2177 -0.1001 R(43) 15,285.0535 -4.3207 -0.1030 A refinement was added to the program to filter out some of these false series. It was based on what are known as combination differences. For transitions with a common lower state, such as R(J) and P(J) shown in Figure 5-1, the frequency difference between the two transitions is a direct measure of the spacing between the J - l and J+l rotational levels in the upper state. In other words, the difference vR(j)-"p(j) = 4B'(J + ±) (5-2) involves the rotational constant of the upper state only. Similarly, there are transitions that have a common upper state [R(J-l) and P(J+1) in Figure 5-1], and their difference is a measure of rotational spacing in the lower state: vR{j-i) ~ vp(j+i) = 4f?"(J + \ ) (5-3) When the program finds a series of lines with approximately constant second differences, it calculates from the first and second differences a range of possible values for J. Selecting a vibration to ascribe to the ground (X1S^") state, B " is set to the value determined by a Fourier transform 5 RESULTS AND ANALYSIS 89 study of the B 3 n 0 + U -X 1 E+ system [51] The vibrational states 0 < v" < 2 were the only possibilities for the frequency region. Figure 5-1: The frequency difference between R(J) and P(J) is a direct measure of the frequency separation between levels in the upper state. The frequency difference between R(J-l) and P(J+1) is a measure of the spacing in the lower state. P(J) J+l A A J R(J-1) R(J) J-l P(J+1) J+l J J-l The combination relation in Eq 5-3 is then used to find "associated" lines (P-lines if you assume the series belongs to an R-branch, R-lines if you assume P-branch—both possibilities need to be checked), stepping through the possible values of J. This new set of associated lines must itself form a series with constant second differences and must be consistent with the combination relation in Eq 5-2. If no associated series can be found, the original series is considered to be a false series and is discarded. This procedure is fairly simple, but it is actually quite a powerful technique for finding series with a minimum of false positives. It gives two series at once, the lower state vibration, the J-numbering of the transitions, and an estimate of the rotational constant in the upper state, B ' . Note that this could be adapted to include combination differences with Q-lines, but Q transitions end on different parity states than R and P transitions (as described in Section 2.7), and the A 3 n~i u state is expected to have significant fi-type doubling. The program was used to search the list of frequencies taken from the magnetic rotation spec-trum, and it came up with a dozen real series, confirming that the strong, unassigned lines in the spectrum were indeed high-J transitions. Subsequent experiments have proven this beyond a doubt. 5 RESULTS AND ANALYSIS 90 5.1 Cause of High-J Signals Both the S-uncoupling operator {j^J±S^) and the Zeeman Hamiltonian (iiZee) couple the A 3 n i u state to the A / 3 n . 2 U state as well as the A state to the B 3 n 0 + U state. Second-order perturbation theory (Eq 2-19) can be used to estimate the shift of a particular magnetic sublevel. As an example, consider the effect of the A ' state on the energy of the A state. The energy shift in the A state, AE(A), is given by: = \(A\HZee + Hsu\A')\2 = \(A\HZee\A')\2 (A\HZee\A') (A'\HSU\A) , \(A\HSU\A')\2 1 > EA-EA, EA-EA, EA-EA, + EA-EA, ' (5-4) where H s u represents the S-uncoupling operator. The first term on the right hand side of Eq 5-4 is quadratic in the magnetic field. This leads to a shift in the measured frequency of the magnetic rotation signal, but it does not affect the magnitude of the signal because quadratic shifts do not cause a frequency separation between complementary transitions. The final term in Eq 5-4 just describes the normal S-uncoupling. The middle term is proportional to the magnetic field, and it does affect the magnitude of the magnetic rotation signal because it leads to a frequency separation between complementary transitions. It is instructive to examine the middle term in more detail. Using the Zeeman Hamiltonian from Eq 2-33 and using dimensionless operators, this term (expressed in cm - 1 ) becomes (in the Hund's case (a) coupling scheme): AA\HZee\A>) (A'\HSU\A) _ 2 (A\gsS-\A') (A' A) ' ( j + 2)(J - 1) EA-EA, E^E^, w w r j T j + 1 ) , ^^{A,v\S-\A'J)(A',r/\$S+\A,v) (J + 2 ) ( J -1 ) = ~ 2 9 S ^ E A , , - E A , v l , J ( J + 1 ) , (5-5) where the sum over contributions from different A ' vibrational states has been explicitly included. Note that for the Zeeman Hamiltonian, it has been assumed that only the S± operators couple the A and the A' states (and the L± operators do not couple the two states), as would be the situation for Hund's case (a) coupling. For Hund's case (c) coupling, the matrix element (A\gsS-\A') should be replaced by (A \gsS- + gLL-\A'), which must be left as a parameter to be determined from the experiment. Similarly, the L-uncoupling operator (not just the S-uncoupling operator) from the rotational Hamiltonian could have a non-zero matrix element between the two states for Hund's case (c) coupling. The matrix element (^A! &1T^cri S+ Aj would therefore be replaced by (A' 8n2^cr'i (S+ + L+) yl) for Hund's case (c) coupling, and once again the matrix element is left as a parameter to be determined from the experimental data. 5 RESULTS AND ANALYSIS 91 Note that the numerator in Eq 5-5 can be either positive or negative, unlike the first and third terms from the right hand side of Eq 5-4, which are positive definite. The total energy shift described in Eq 5-4 must behave like a repulsion from the perturbing state, but the middle term can either add to or subtract from this shift. To picture what is happening, the electronic spin and orbital angular momenta precess about the internuclear axis and provide a magnetic moment the order of one Bohr magneton along that axis for the A state. However, it is the component of magnetic moment along J that is important, to first order, for interaction of the molecule with an external magnetic field. As the end-over-end rotation of the molecule increases, the angle between J and the internuclear axis increases, and the component of magnetic moment along J therefore decreases. This is implicit in the expression for the rotational g-factor given in Eq 2-35. It leads to the expected drop in the magnetic rotation signal. However, rotation of the molecule also serves to uncouple the electronic angular momenta from the internuclear axis, through L- and S-uncoupling, and this gives an additional component of magnetic moment along J . Comparing Eq 5-5 to the form given in Eq 2-34, this can be accounted for adding an extra term to the effective rotational g-factor: [(gsE+gLA)n}eff ^ a f e (A,v \S-1 A', v>) (A', v> \^S+ | A, t,) (j + 2)(j _ 1 } 9 J = J(J + 1) ~ 2 9 8 \ EA,v,j - EA,v,j J ( J + 1) ' ( 5 " 6 ) where (gs£+g.£,A)r2 = 1 for a Hund's case (a) 3 n i electronic state, but must be left as an effective parameter for this case (c) molecule. Note that there will be another term almost identical to the second one in Eq 5-6 from coupling of the B3HQ+U state to the A state. The energy denominators are much larger, though (see Figure 4-1), and so the effect of coupling to the B state is expected to be less important. The second term in Eq 5-6 is analogous to the well-known contribution from valence electrons to the magnetic moment of a state [28]. The second term in Eq 5-6 can be either positive or negative, and the magnitude is expected to be at most a few nuclear magnetons. The first term in Eq 5-6 decreases as J J , while the second term is roughly independent of J, assuming no large variations in the energy difference in the denominator. Looking at the form of Eq 5-6, the energy separation between consecutive magnetic sublevels initially decreases at low-J, but then tends towards a constant at high-J. Interestingly, in every vibration measured for this thesis in the A 3 H i u state, the two terms in Eq 5-6 are opposite in sign, and the g-factor actually goes through zero and changes sign before levelling off. This will be discussed in more detail later. The levelling-off of gj has a profound effect on the magnetic rotation signal, particularly for R and P branches. Even though the energy separation between magnetic sublevels goes to a constant, 5 RESULTS AND ANALYSIS 92 there are more and more magnetic sublevels as J increases. The signals for P and R transitions, dominated by contributions from large \mj\, go back up at high-J, so much so that the signals from high-J transitions were often stronger than the signals from low-J transitions. The signals for Q branch transitions remain roughly constant (and weak) at high-J, since transitions to low \mj\ have the largest weight in the signal for a Q line. Note that the signal does not increase indefinitely for R and P branches, due to population effects in the lower state. As J increases, the population of molecules in the lower state decreases exponentially, and so the signal will level off and eventually decrease again. High-J magnetic rotation signals in the B 3 n 0 + u -X 1 S^" system of molecular iodine [4] arise from a similar mechanism. One would expect (to first order) no magnetic rotation signals for B<—X transitions, because there is no intrinsic magnetic activity for an electronic state with Cl = 0, as can be seen from the expression in Eq 2-34. However, the Zeeman Hamiltonian and the S-uncoupling Hamiltonian both couple the B state to the A 3 H i u state, as well as to a dissociative lH\u state. See Figure 4-1 (which is for B r 2 , but works qualitatively for I2 as well). Interference effects similar to those contained in Eq 5-5 (involving B and A or involving B and the 1 n i u state) give rise to magnetic rotation signals for B<—X transitions. The signal is very weak at low-J and increases with increasing J, for the same reasons high-J signals in the A - X system of Br2 show an increase. The original qualitative explanation for the presence of the signals was based on a coupling between the A and B states. As depicted in Figure 4-1, the dissociative 1 H i u state is in the same energy region as the B state. It has been shown [7] that there is an interaction between the B state and the x n i u state, leading to predissociation effects in B<—X transitions. Because of the larger energy difference between the A and B states, it would seem likely that the dominant contribution to magnetic rotation signals in the B - X system of I2 comes from coupling to the lH\u state. Coupling to the A state still gives a contribution, but the effects from this should be relatively small because of a large energy denominator. In the A - X system of B r 2 , an evolution of the lineshape as a function of J was observed for the first harmonic signal. Figure 5-2 shows a series of lines for v = 18 of the A state. Each plot spans roughly 0.03 c m - 1 . For P(8), the lineshape of the magnetic rotation signal resembles the one shown in Figure 3-10. It is opposite in phase to the calculated P line in Figure 3-13, presumably because the offset angle, 0o, was negative in the experiment. The calculated lineshapes that follow therefore assumed a negative value for this parameter. The strongest part of the signal points downwards, and the lineshape is skewed, with the high-frequency side enhanced relative to the low-frequency side. As 5 RESULTS AND ANALYSIS 93 Figure 5-2: Experimental traces for first harmonic signals from transitions in the 18'-1" band of the A - X system. The vertical scale is in arbitrary units, but the units are the same in all three plots. discussed previously, the lineshape would be symmetric if there were only frequency shift contribu-tions to the signal. The skew of the lineshape results from intensity perturbation effects. For P(21), the lineshape of the signal looks much different; it looks almost dispersion. This occurs because the two terms in Eq 5-6 have opposite signs, and the splitting of the magnetic sublevels in J = 20 of the A state is very small where the two terms are comparable in magnitude. When the splitting approaches zero, the intensity perturbation contributions to the signal in Eq 3-44 become larger than the frequency shift contributions. Recall that the intensity perturbation contribution has a dispersion lineshape (see Figure 3-9). For P(54), the signal comes almost entirely from the second term in Eq 5-6, and is once again dominated by the frequency shift contribution. Note that the signal is opposite in phase to that of P(8), a consequence of the fact that the two terms in Eq 5-6 are opposite in sign. The expected signal can be calculated following the same procedure used in Section 3.9, with the simple modification that the rotational g-factor has an extra term (taken to be a constant): ^ j f J T T ) - 0 ' 0 0 1 5 - <5-7> Using this g-factor, along with the same parameters described in Section 3.9, the calculated signals are shown in Figure 5-3. The agreement between the observed and calculated lineshapes for P(8) and P(21) is quite good. The relative intensities and the shapes reproduce very well. The enhanced lobe in the lineshape goes from the high-frequency side at low-J to the low-frequency side at high-J, in agreement with 5 RESULTS AND ANALYSIS 94 Figure 5-3: Calculated first harmonic signals in the 18'—1" band of the A - X system. The vertical scales are in arbitrary units, but the units are the same in all three plots. Calculated P(8) 1.0[ 0.5 0.0 - 0 . * -14 Calculated P(21) -10000 -5000 0 5000 10000 co (Mrads/sec) -10000 -5000 0 5000 10000 to (Mrads/sec) Calculated P(54) -10000 -5000 0 5000 10000 co (Mrads/sec) theory. This occurs because the intensity perturbation contribution does not change phase as the frequency shift contribution does. The calculated signal for P(54) is stronger (relative to P(8)) than the experimental observations suggest. However, taking a g-factor of the form 1 9J 0.001, J ( J + 1) the calculated signal for P(54) is shown in Figure 5-4. Figure 5-4: Calculated first harmonic signal for P(54) with a different g-factor. The units on the vertical axis are the same as in Figure 5-3. 4 k Re-calculated P(54) -10000 -5000 0 5000 co (Mrads/sec) 10000 5 RESULTS AND ANALYSIS 95 The need for different g-factors to reproduce P(21) and P(54) is an indication that the simple model of adding a constant to the g-factor is not sufficient (for this vibration, at any rate). In the magnetic rotation spectrum of the B - X system of I2, asymmetries in the measured lines were attributed to mixing with the A state [6], although no quantitative tests were performed for this hypothesis. For the A-system of B r 2 , mixing to different J levels within the A state itself are sufficient to explain the observations. Note that the calculated for P(8) is not sensitive to the value of the constant added to the g-factor, as long as it is not too large. The difference in g-factors is likely a result of the energy separation to the perturbing state(s) changing as a function of J. Because the lineshapes change so dramatically as a function of J, it is clear that a great deal of effort would be required to measure the frequencies in a rigorous manner. Even to estimate the frequency would be difficult to do systematically, since each transition will have a different lineshape. To measure frequencies, it was decided to use the second harmonic signal. The calculated second harmonic signals for P(8), P(21) and P(54) are shown in Figure 5-5. Figure 5-5: Calculated second harmonic signals in the 18'-1" band of the A - X system. Note the apparent shift in the centre of gravity is largest for P(21) in Figure 5-5. Simply fitting P(21) to a symmetric lineshape, such as a Lorentzian, would give a significant error in the determined frequency. There would also be errors in determining the centres of gravity of the other two lines by fitting them to Lorentzians, but the errors would be smaller than that for P(21). Unfortunately, it is not possible to perform an accurate lineshape calculation because of the uncertainty in the energy of the A ' state, one of the perturbing electronic states that gives rise to the extra term in Eq 5-6. The simplest systematic method to determine frequencies is to fit all of the lines to a symmetric lineshape, although it should be kept in mind that there will be errors in the measured frequencies, and these will be largest where the intensity perturbation 5 RESULTS AND ANALYSIS 96 contributions to the magnetic rotation signal is comparable to the frequency shift contributions. The discrepancies from the results of least-squares fits observed in the current work for signals dominated by intensity perturbation contributions were always less than 0.01 c m - 1 , with a typical discrepancy of about 0.0075 c m - 1 . As is evident in Figure 5-5, the apparent shift is smallest for high-J lines. Therefore, the apparent frequency shifts could be a major problem at high-v, where there is not a lot of high-J data to "anchor" the least-squares fits. 5.2 Assigning Uncertainties The Doppler width for 7 9 B r 2 in this frequency region, calculated from Eq 2-57, is about 450 MHz (~0.015 cm - 1 ) , but the observed spectral widths are about twice that, approximately 0.03 c m - 1 , because hyperfine structure broadens the lines. A typical signal-to-noise ratio for the second har-monic magnetic rotation lines was about 200 to 1. If the lines were symmetric, the accuracy to which the center of gravity could be found would be approximately c m - 1 . The skew in the sig-nal induced by the intensity perturbation contributions (see e.g. Figure 5-5), as well as unresolved hyperfine structure, means that this limit cannot be achieved. The error in the frequencies of the I2 reference lines is about 0.002 c m - 1 , the standard error of the global analysis performed in [50]. With the additional uncertainty arising from the asymmetry in the Br2 lineshapes, the uncertainty assigned to the frequency measurement of a sharp, strong, unblended line was set at 0.0035 c m - 1 . The analysis seemed to confirm this as a reasonable (or perhaps slightly overestimated) error, ex-cept for lines where the signal was evolving from low-J to high-J (e.g. P(21) in Figure 5-5). For these lines, the measured frequencies occasionally deviated from the frequencies calculated from the results of analysis by 0.008 c m - 1 or more, as discussed previously. 5.3 Frequency Analysis The measured lines were assigned through a combination of automation (using the FORTRAN program described earlier) and manually searching for lines, using calculations based on preliminary analysis of lines already assigned. A band is the set of transitions (R, P and Q) between particular vibrations in the ground X 1 £ ^ state and the A 3 H i u state (v" and v', respectively). The set of transition frequencies for a band can be fitted to the form (see Eq 2-11) v(v", J", v', J') = v0(i/, v") + BV,K' - DV,K'2 + HV,K'3 - BV„K" + DV>,K!12 - HV»K"3, (5-8) 5 RESULTS AND ANALYSIS 97 where, for the A - X system, K' and K" are defined as K ' = [J'(J' + 1) - 1] and K" = J"(J" + 1) The ground state rotational constants (B„», D„» and H„») were fixed in this analysis to the very accurate values determined in [51]. For each vibration in the A state, a least squares fit to the expression in Eq 5-8 was performed using data from the R- and P-transitions (transitions involving the e parity levels in the A state). Data from different lower states (v" = 0, 1, and 2) were combined into a single fit for the given upper state vibration (v7). The results of these fits are shown in Table 5-2. The Q-transitions (transitions involving the f parity levels in the A state) were fit separately, as will be discussed in more detail later. The least-squares routine used was based on the Levenberg-Marquardt method [52]. Following the standard procedure in the literature, the quoted errors are statistical errors determined from the diagonal elements of the covariance matrix, the la errors, where a stands for the standard deviation. The data used in the least-squares fits are compiled in Table A - l of Appendix A. Note that the calculated frequencies are from a global analysis to be described later, and not from the vibration-by-vibration analysis. 5.4 fi-type Doubling The rotational constants in Table 5-2 are for the e parity levels of the A state, the levels involved in R- and P-transitions in the A - X system. The f parity levels, the rotational levels involved in Q-transitions, were observed to be lower in energy than their e parity counterparts. From the discussion in Section 2.4, this means that L-uncoupling effects either push the f parity levels down or push the e parity levels up. The only E-type (i.e. Q, = 0) electronic state lower in energy than the A state is the ground X X E+ state. The L-uncoupling operator does not couple these two states because they have different u/g symmetry. Therefore, the push must come from a higher energy E state, which means the f parity levels must be pushed down in order to be lower in energy. In fact, both parity levels are likely pushed down due to interactions with different E-type states, but since the f parity levels are lower in energy, these levels must be pushed down farther than the e parity levels. The e levels should therefore give a better approximation of the true B-value, as discussed in Section 2.4. Following Eq 2-25, the B-value for the f levels (B^) can be related to the B-value for the e levels (B£) by introducing the ^-doubling parameter q„: Bf = B ev+qv. (5-9) 5 RESULTS AND ANALYSIS Table 5-2: Vibration-by-vibration results for the e parity levels of 7 9 B r 2 A3Uiu a E(v') b 107T>V, c -10UILy c #d T e 3 D f 13 15,291.3759(25) 0.045311(7) 1.266(35) 0.272(43) 41 73 117 14 15,358.7395(12) 0.043758(5) 1.522(28) 0.244(38) 70 72 113 15 15,419.2088(10) 0.042129(3) 1.714(17) 0.258(22) 101 74 108 16 15,473.2780(11) 0.040473(2) 1.821(7) 0.357(8) 132 85 104 17 15,521.6133(9) 0.038855(1) 1.959(5) 0.379(5) 142 90 100 18 15,564.9797(10) 0.037286(2) 2.051(6) 0.425(6) 136 85 96 19 15,604.0748(9) 0.035779(1) 2.103(5) 0.524(4) 150 92 92 20 15,639.4925(8) 0.034347(1) 2.220(6) 0.569(7) 154 82 88 21 15,671.7031(7) 0.032953(1) 2.297(6) 0.704(7) 172 80 84 22 15,701.0470(6) 0.031597(1) 2.418(7) 0.846(8) 172 76 80 23 15,727.7847(7) 0.030254(2) 2.538(8) 1.073(10) 159 76 76 24 15,752.0982(7) 0.028921(2) 2.717(9) 1.331(13) 171 71 71 25 15,774.1346(14) 0.027574(3) 2.917(13) 1.686(20) 136 67 67 26 15,793.9937(8) 0.026212(2) 3.182(14) 2.151(24) 176 63 63 27 15,811.7789(7) 0.024795(2) 3.402(17) 2.919(35) 175 59 59 28 15,827.5110(9) 0.023390(3) 3.915(25) 3.640(57) 156 54 54 29 15,841.3149(11) 0.021899(4) 4.300(36) 5.144(97) 137 50 50 30 15,853.2392(11) 0.020349(5) 4.787(59) 7.24(18) 110 46 46 31 15,863.3573(13) 0.018750(9) 5.57(12) 9.76(45) 85 41 41 32 15,871.7522(13) 0.017092(10) 6.53(18) 13.89(86) 70 37 37 33 15,878.5453(13) 0.015356(13) 7.58(28) 21.2(17) 59 33 33 34 15,883.8584(17) 0.013603(24) 9.95(59) 26.9(59) 38 28 28 35 15,887.8527(19) 0.011800(17) 15.3(3) — 27 24 24 36 15,890.7002(14) 0.009927(21) 19.5(5) — 26 20 20 37 15,892.5995(29) 0.007899(65) 22.3(24) — 18 16 16 "Values in parentheses indicate la uncertainty in units of the final digit(s) 6 Energy (in cm - 1 ) of V," J ' = 0" in the A state relative to v" = 0, J" = 0 in the X state c Constants (in cm - 1 ) for the e-levels. The fits include only R- and P-transitions. (See text.) d Number of unblended R- and P-transitions used in the fit e Maximum upper state J for R- and P-transitions included in the fit f Highest J below dissociation hmit. Only levels below the dissociation Hmit were used in the 5 RESULTS AND ANALYSIS 99 Recall that the magnetic rotation signal for Q-transitions does not increase at high-J, as the signal for R- and P-transitions does; the signal for Q-transitions levels off at high-J instead of increasing. In the second harmonic spectrum, the signal for Q-transitions tends to a level that is comparable to the noise, such that it was typically difficult to follow the Q-lines above J of 25. No measurements of Q lines were made from the second harmonic spectrum above J = 40, because they were too weak. Due to the low signal-to-noise for Q lines above J = 25, the uncertainties in the frequencies were very large ( | to \ the linewidth). The scarcity of high-J Q-branch data imposes a severe limitation on the determination of the fi-doubling constant. The omega-doubling constant (q„) is listed in Table 5-3 for several vibrations, and is the order of one part in one thousand of the B-value itself. The frequency separation between the two parity levels therefore does not become significantly larger than the uncertainty in frequency measurements until J > 25 or so. As an added complication, hyperfine effects cause a separation of the e and f parity levels at low-J. This "hyperfine ^-doubling" is roughly constant (the order of 0.01 cm - 1 ) , and the normal fi-type doubling must be larger than this separation in order for q„ to be measured accurately. Thus, a special effort was made to measure the signals from high-J Q-transitions for two bands. To measure these very weak transitions, selected portions of the spectrum were re-investigated using first harmonic instead of second harmonic. Figure 5-6 shows the same region taken in both first and second harmonic. Note that the signal to noise ratio is much better in first harmonic, a factor of 50 to 100 times better. In particular, there are several peaks in the first harmonic version that do not rise above the noise in second harmonic. Because second harmonic signals are proportional to the squares of various quantities (see Eq 3-29), the relative heights of the strongest signals to the weakest is amplified in comparison to first harmonic. Every unblended line up to J = 65 in the 19'—1" Q-branch was measured in first harmonic. For the 20'-l" Q-branch, there were many blended Q lines for J > 52. In fact, the only Q lines beyond Q(52) that could be measured were Q(59) and Q(64). The latter two lines were left out of the final fit for q20' because the gaps of missing data were considered to be too large. High-J Q lines from different bands (21'—1" through 26'-l") that happened to fall in the first harmonic scan windows (i.e. close to a 19'-1" or a 20'-l" Q line) were also measured, but no other bands were investigated. There is significant variation in lineshape for first harmonic signals, as discussed before, so that extracting frequency information in a rigorous manner from scans like the first harmonic spectrum shown in Figure 5-6 could be a daunting task. It was decided to estimate manually centres of gravity. Starting with stronger lines that also appeared in second harmonic, criteria for measuring 5 RESULTS AND ANALYSIS 100 Figure 5-6: Comparison of first and second harmonic spectrum in the same fre-quency region. First harmonic spectrum 1 8 , _ r 24'-1" R(77) 24'-2" Q(8) Second harmonic spectrum 24'-1"R(77) 18'-1"R(54) i i i i i i i i i i i 15104.0 15104.1 15104.2 15104.3 15104.4 15104.5 frequencies from the first harmonic signals were chosen in such a way as to get the best agreement with measurements from second harmonic. For example, the center of gravity for a typical lineshape is shown by the marker in Figure 5-7. For lineshapes more skewed than the one in Figure 5-7, the deviation between first and second harmonic measurements ranged as high as 0.0025 c m - 1 . The minimum uncertainty was set to 0.0050 c m - 1 (rather than the 0.0035 c m - 1 precision assigned to second harmonic measurements) to account for the imprecision in finding the center of gravity. Since the data extended to such high-J, the accuracy of the frequency measurements was more than sufficient to determine a good value for q^. A good consistency check was provided by the presence of 7 9 B r 2 B - X transitions in 5 RESULTS AND ANALYSIS 101 Figure 5-7: The manually assigned center of gravity for a skewed first harmonic lineshape. centre of gravity the first harmonic scans. Frequencies calculated from the results of the global analysis of the B - X system in [51] always agreed with the measured frequencies for these lines to within 0.0020 c m - 1 , the stated uncertainty for the B - X transitions. The separation between e and f levels as a function of J(J+1) for v = 19 is shown in Figure 5-9. The relationship is linear, as expected, except for the portion near the beginning where the hyperfine effects cause it to level off. To determine the fi-doubling constant, a least-squares fit was performed to data from all three branches (R, P and Q). The fit was again to the expression in Eq 5-8, except the B-value for the f parity levels was given by Eq 5-9. Allowing for different band origins (i.e. v0(v',v")) for the e and f parity levels did not always yield reasonable results (e.g. wrong sign or error larger than the parameter itself), and those results are therefore not reported. The fi-doubling constant for V — 19 to 29 are listed in Table 5-3. The R and P lines used in the determination of these constants are in Table A - l of Appendix A, while the Q lines used are listed in Table A-2. The values determined for B^ from the fits including the Q-transitions agreed with the results of the fits to R and P data only. A major perturbation in V = 27 of the A state (discussed in detail later) did not permit determination of the ^-doubling constant for that vibration. There was little variation in the ^-doubling constants measured in this experiment. This was somewhat surprising, considering the trend of the values measured previously by Coxon [1], also shown in Table 5-3. 5 RESULTS AND ANALYSIS 102 Figure 5-8: Difference between the energies of f parity levels (E-^(J), determined from Q lines) and e parity levels (E e(J), determined from R and P lines) as a function of J(J+1). The f parity levels are lower in energy than the e parity levels. Q-type doubling for v = 19 in the A state 0.00-(cm -0.05-3 , -0.10- ( 5 ' 1 2 ) where r_ and r + are the inner and outer classical turning points, respectively; v is the vibrational quantum number; and G(v) is the energy of the vibrational state. Of course, G(v) = U(r_) = U(r+). Although v must be an integer, it will be treated in Eq 5-12 as a continuous variable. Le Roy and Bernstein [55] developed an expression from Eq 5-12 to describe the spacing of vibrational levels near the dissociation limit. The derivation uses the limiting form of the potential, where only the leading term from Eq 5-11 is kept, the term with n = 5 in the case of the A state of Br2- By differentiating both sides of Eq 5-12 with respect to G(v), an analytical expression describing the variation of vibrational energy as a function of v, valid only for vibrations near the dissociation limit, is derived to be [55]: K'n represents a collection of constants, defined (and tabulated for the various possible values of n) in [56]. The expression in Eq 5-13 can be integrated with respect to v. This yields the following expression: G(v) = D 2 n n - 2 (5-14) LVM(Cn)' where = n^YJn, and v/j is an integration constant that represents the effective (usually non-integer) value of the vibrational quantum number at the dissociation limit. To extend Eq 5-14 beyond the near-dissociation region, toward lower v, an empirical function can be introduced that goes to 1 as v —> v/j>, to account for deviation from the limiting form in Eq 5 RESULTS AND ANALYSIS 108 5-14. The function is chosen in such a manner that its leading terms will simulate the effects of C6 and Cg. Following [57], the chosen form for this empirical function is a Pade approximant, the ratio of two polynomials: G{v) = D — H ^ - T ( v D - v ) 2n n - 2 [ C ( L ) 1 k(M)J with and L v W n ) = C(L) = l+pi(vD -v)2 + p2(vD -v)3 + ... +pL(vD - v ) L + 1 , (5-15) C(M) = 1 + qi(vD - v)2 + q2(vD - v)3 + ... + qM(vD - v ) M + l . The expression in Eq 5-15 replaces the expansion in (v+5) from Eq 2-6. Since Eq 5-15 incor-porates the limiting behaviour of the vibrational spacings, it reproduces the data with many fewer parameters than is required for the expansion in Eq 2-6. By trying different combinations of L and M , it was determined that 9 parameters (L = 7 and M = 2) gave the best fit of the vibrational energies to Eq 5-15. There were actually 12 parameters varied in the fit, since D, vp and C5 were left as adjustable parameters. By comparison, with 15 parameters in the expansion from Eq 2-6, the value of x 2 is about 50 times larger than that for the fit to the near-dissociation expression. In addition, the predictions for the band-origins for v = 1 through 6 (which were not in the fit, since there was no high-precision data for these vibrations) were quite obviously not reliable when the fit to the expansion in (v+5) was used. The near-dissociation approach, on the other hand, yields very reasonable behaviour through this region, possibly due to the use of Pade approximants, since they often give more reliable extrapolations than would straight polynomials. The expected variation of B^ near the dissociation limit can also be estimated from the limiting form of the potential [56]: Bv = _®'n _a_ [vD -v]^s. (5-16) The constant, Q^, is defined and tabulated in [56]. As was done with the expression for the vibrational energies, an empirical function is introduced into Eq 5-16 that goes to 1 as v —> v/j. This gives an expression that can replace the expansion in (v+|) for the B-values (Eq 2-12). Again, because the expression incorporates the correct limiting behaviour of the B-values near the dissociation limit (the region where the expansion in (v+5) has the most difficulty), fewer parameters are required to fit the data. The empirical function typically suggested for the B-values [58] is the exponential of a polynomial: exp ^ 2 S*(VD ~ v)^j • (5-17) 5 RESULTS AND ANALYSIS 109 This form was found to give a poor extrapolation of the B-values for v = 1 through 6, the region for which there is no data. The B-value came out to be larger for v = 1 than for v = 0, a result which is unphysical. A Pade approximant form was therefore chosen for the B-values rather than the more widely used exponential function in Eq 5-17. The expression to which the B-value were fit was: Bv = ^ [VD -V]^i 0 ) " - 2 ( -1.8-1 CO -2.0 A 10 15 20 25 30 35 40 The data points involving v = 34 and up show significant scatter, which could be the result of perturbations, since many electronic states come together near the dissociation limit (see e.g. the potential energy diagram for I2 in [53]). However, it is more likely a matter of increasing correlations in the fits for vibrations near the dissociation limit. There are fewer bound levels for higher vibrations, and so the parameters cannot be determined as well from the fit. Taking the results for v = 35 and 36 from the fits that do not include H^ (i.e. the values listed in Table 5-2), the derivative of the B-values as a function of v behave much more smoothly, as seen in Figure 5-11. Excluding H„ from the fit for v = 34 yields results that deviate significantly from the curve in Figure 5-11. It is only for v above 34 that leaving H^ out of the fit makes the curve smoother. The B-values that went into making the plot in Figure 5-11 were fit to the expression in Eq 5-18, and the corresponding vibrational energies were fit to Eq 5-15. Once these fitted functions are determined, from which the values of G(v) and B^ can be calculated for any (not necessarily integer) value of v, it becomes a straight-forward process to calculate the potential for the state. The most common approach for determining diatomic potential curves from spectroscopic parameters is known as the R K R (Rydberg-Klein-Rees) method. In the R K R method, the inner and outer classical turning points of the potential (r_ and r + , respectively) are determined through intermediate quantities, £v and g„, according to [60] 2fv = r+(v) - r_(t>), and 2gv = — - . (5-19) 5 RESULTS AND ANALYSIS 112 Figure 5-11: First differences for B-values. The B-values were determined from least-squares fits that included distortion constants up to Kv for all vibrations except v = 35, 36 and 37, for which the fits included distortion constants only up to D„. -0.8-1 I -1.0-cm -1.2-CO I O -1.4---1.6-CO • > -1.8-CO -2.0--2.2-10 15 — i — 20 25 30 35 40 The quantities f„ and g„ are defined by the integrals: ^ \] 8TT2CH JVml _ J8n2ciJ, fv i y/G(v) - G(v>) =dv'; -.dv': (5-20) (5-21) y/G(v)-G{v>) where it has been assumed that the vibrational energies and B-values are expressed in c m - 1 . The value of vmin in Eq 5-20 and Eq 5-21 is taken to be equal to - \ , the expected value from the quantization condition (Eq 5-12). The inner and outer turning points (r_ and r + , respectively) are then determined from the following equation: J ' ^ 1 * (5-22) r± + f2±f-Calculation of the two integrals in Eq 5-20 and Eq 5-21 are carried out following the procedure described by Tellinghuisen [60]. A FORTRAN program was written to implement this procedure. The integration routine divides the integration region into successively smaller intervals (i.e. divides it in half, and then into four intervals, then eight intervals, etc.) until the calculated integral changes by less than one part in 106 on successive divisions. The integral in a given interval is calculated using a four-point quadrature. The contribution from the uppermost interval, where there is an 5 RESULTS AND ANALYSIS 113 integrable singularity, is evaluated using Gauss-Mehler quadrature with the weights provided by Tellinghuisen [60]. Al l other intervals are evaluated using the Gauss-Legendre formula [52]. Once r_ and r + are known for a particular value of v, the potential at r_ and r + is given by U(r_) = U(r+) = G(v), where G(v) is calculated from Eq 5-15. One of the major benefits of calculating a potential energy curve is that it can be used to improve spectroscopic constants. The values of Bv and G(v) in Table 5-2 suffer from correlations with the centrifugal distortion constants D„ and H„. It has long been known, however, that the centrifugal distortion constants are not independent physical parameters, but can be calculated from the potential. Therefore, an improved set of values for B„ and G(v) can be determined by fixing the centrifugal distortion constants to the values calculated from the potential and refitting the experimental data. The potential energy curve for the A state was calculated, but there was anomalous behaviour in the repulsive inner branch. As shown in Figure 5-12, the calculated potential flattened off above v = 33 instead of having its negative slope increase in magnitude. Figure 5-12: Uppermost portion of the inner, repulsive wall of the calculated R K R potential. 15900-1 15820 -I , . , , , , , , , 2.409 2.410 0 2.411 2.412 2.413 r.(A) It is common to observe anomalies on the repulsive inner branch of an R K R potential. The R K R method is known to generate very accurate representations of the lower part of the potential, but numerical instabilities and errors may appear for high, near-dissociation levels [61]. The most common method for dealing with the anomalous behaviour involves smoothing the potential by 5 RESULTS AND ANALYSIS 114 using an analytical function, such as where m is usually assumed to be an integer. To select an appropriate value for m in Eq 5-23, a plot is made of l n ( ^ ) versus ln(r_); see Figure 5-13. The plot is well-behaved between v = 13 and v = 25, roughly a straight line with a slope of about -14, which would make m equal to 13. In the subsequent iterative processes, the end result gave a value closer to 14. Setting the value of m to 14 at the very beginning gave a faster and more stable convergence to the iterative process, and so m was fixed at 14 for the remainder of the analysis. Figure 5-13: Plot to find the form of the smoothing function and the vibration above which smoothing should be performed. 10.0-9.8-TJ 9.6-"O 9.4 9.2-9.0-v = 25 0.880 0.885 ln(r.) v = 13 0.890 With the form of the smoothing function established, the potential is re-calculated, using a finer grid this time. As a preliminary step, the inner and outer turning points are evaluated from Eq 5-22 for v from -5 to 38 in steps of 0.005. The inner turning points at v = 24 and 25 were used to determine the constants a and b in Eq 5-23, where the exponent m was set to 14. Al l inner turning points above v = 25 were calculated from the resulting expression. The difference between the inner and outer turning points, defined by 2iv in Eq 5-19 and Eq 5-20, involve only G(v), not B,;. The band-origin is expected to be affected less than B-values by perturbations or parameter correlations with distortion constants, and so the difference calculated from the expression in Eq 5-19 should be more reliable than the calculated sum. Thus, when the 5 RESULTS AND ANALYSIS 115 inner turning point is calculated from Eq 5-23, the outer turning point is adjusted to maintain the calculated difference between inner and outer turning points. In subsequent calculations, a potential with points equally spaced in r (rather than equally spaced in v) is required. A cubic spline interpolation routine [52] was used to generate 16,000 points equidistant in r. It was also necessary to extrapolate the inner branch upwards. The routine for finding the vibrational wavefunctions (described in more detail later) involves setting the wavefunction to zero at the first point in the potential, i.e. the point with the smallest value of r_. The potential at the first point should therefore be quite high in energy for this approximation to be valid. The inner branch was extrapolated upwards 200 points with Eq 5-23. It was not deemed necessary to extrapolate the outer branch, since the calculation of distortion constants were not sensitive to the behaviour of the potential in this region, whereas the calculations were very sensitive to changes in the inner, repulsive wall of the potential. With the potential constructed, the next step is to find the vibrational energies and wave-functions by solving Eq 2-4, the radial part of Schrodinger's equation, for its eigenvalues and eigenvectors. The solutions to this equation can be determined by solving the following differential equation: d 2 ^ 2 l*W«t) =. iU(r) ~ E) \Vvib,rot) • (5-24) This equation is solved numerically using the procedure outlined by Cooley [62]. A FORTRAN routine was written to implement this procedure. Once the wavefunctions are known (from the solutions of Eq 5-24), the centrifugal distortion constants can be calculated. Treating the rotational Hamiltonian (Eq 2-8) as a perturbation, D„[J(J+1)-1] 2 is the second order correction to the energy from perturbation theory, H„[J(J+1)-1] 3 is the third order correction, etc. The numerical procedure for calculating the distortion constants from perturbation theory is given in [63]. A FORTRAN program was written to implement this procedure. Distortion constants up to seventh order (D„, H„, Lv, Mv, N„ and 0„) were evaluated, and the data was refit with the distortion constants held fixed to these calculated values. Note that the original program only calculated distortion constants up to M„; the subroutine used to calculate the higher-order terms was obtained from [64], and this subroutine was employed for the remainder of the analysis. The fits with the distortion constants fixed yielded new values of Bv and G(v), and these new values were then fit to the expressions in Eq 5-18 and Eq 5-15, respectively. It is a standard procedure to iterate this process: the potential is recalculated from the new set of fitted constants; new distortion constants are evaluated from the recalculated potential; the distortion constants are held fixed at their calculated values as the experimental data is refit for 5 RESULTS AND ANALYSIS 116 new values of Bv and G(v); the new values of B„ and G(v) are fit to the expressions in Eq 5-18 and Eq 5-15, and the process begins again with recalculation of the potential. This is continued until convergence is achieved. Using only the Klein integrals in Eq 5-20 and Eq 5-21 to determine the potential, the procedure would not converge. On the second iteration, the results deteriorated (i.e. agreement with the experimental data got worse). Subsequent iterations gave increasingly worse results, finally con-verging to a point for which agreement with the experimental data was significantly worse than was the case for the first iteration: the value of %2 was more than a factor of two larger, and there were oscillatory trends to the residuals in latter iterations that were not present in the first iteration. The parameter x 2 is a measure of the quality of a least-squares fit, i.e. how well the fitted results reproduce the experimental data in comparison to the uncertainties (see [52]). Many things were attempted to try to remedy this situation, such as limiting the data to various subsets, e.g. fitting only data from selected vibrations or fitting only low-J data (J of 20 or less). Changing the value of m used in the smoothing function in Eq 5-23 altered how well the first iteration reproduced the data, but did nothing to improve subsequent iterations. The problem was the inherent inaccuracies involved in the R K R method. For the A state of B t 2 , it would seem that there is a breakdown of the W K B approximation, and the experimental data therefore do not satisfy the quantization condition in Eq 5-12. It is a standard procedure to correct the R K R potential to improve the agreement of the data with results determined from the potential. Usually this involves taking higher order terms in the quantization condition. The most commonly applied of these corrections is known as the Kaiser correction [65]. It was shown [66] that the R K R potential has an error in energy equal to: Y M = i ( B « - ^ « ) + gg + ^ | , (5-25) where the parameters (B e , u>e, etc.) are from the expressions in Eq 2-6 and Eq 2-12. The Kaiser correction accounts for this error in energy by setting v T O j n in Eq 5-20 and Eq 5-21 to -The parameters on the right hand side of Eq 5-25 are not readily available when working with near-dissociation expansions, but they can be easily estimated. The Kaiser correction was calculated and included in the analysis, but it served to make the results worse instead of better. This is not surprising. Firstly, because of the lack of data at low-v, the values determined for the parameters used in Eq 5-25 are likely to be very inaccurate, much too inaccurate to make the correction scheme viable. Secondly, according to [61], the Kaiser correction greatly improves the results for low-v, but it can make the results worse for high-v. This agrees with the discussion in 5 RESULTS AND ANALYSIS 117 [67], where it was suggested that the correction should probably be a slowly varying function of v and J. An alternate correction scheme was therefore necessary. The ideal situation would be to deter-mine the quantum mechanical potential, e.g. by the inverted perturbation approach method [68] or by using a modified Lennard- Jones analytical potential [69]. A closer representation of the true potential is ensured in these approaches by constraining the potential curve to satisfy the boundary conditions of the Hamiltonian. The determination of such a potential is still in progress. The chosen correction for the R K R potential is taken from [61], a procedure that involves simple linear corrections applied to both G(v) and B„. G(v) and B„ are quantities that can be calculated from the potential. The eigenvalues of the potential are equal to G(v), and the wavefunctions can be used to calculate the B-value (from the expectation value of 4?). When the W K B approximation breaks down, i.e. when the experimental energies do not satisfy the quantization condition in Eq 5-12, the values of G(v) and Bv calculated from the potential will be different from the values that went into calculating the potential. The correction scheme makes use of this difference. The potential calculated from the experimental energies is not a good representation of the quantum mechanical potential, because of the limitations inherent in using the quantization condi-tion. The calculated potential is also not a good representation of the target R K R potential because the experimental data does not satisfy the quantization condition. Most correction schemes involve taking higher-order terms in the quantization condition. The correction scheme in [61] goes in the opposite direction in a sense, in that it finds the energies and the B-values that satisfy the first-order quantization condition and determines the potential curve from those. The correction procedure thus involves determining an initial R K R potential from the experi-mental data. Differences between the values of G(v) calculated from Eq 5-15 and the eigenvalues of the potential are used to determine a correction function ( 0.00 -0.05 uncorrected corrected 10 •••• 1 — 20 V 40 Figure 5-15: Difference between the B-values used to calculate the R K R potential and the B-values determined from the potential, in both the corrected and uncor-rected cases. E 1°i o -i—« o a? o-i • uncorrected • corrected • r v , ; . fi • • • • • • m 10 20 V 30 40 5.6 Summary of Fitting Procedure A condensed description of the procedure used for the fit is given to summarize the method. First, the vibrational energies were fit to Eq 5-15. For v = 7 through 12, the data used was from Coxon [1], and for v = 0, the data was from Hwang et al [3]. These data points were adjusted to be 5 RESULTS AND ANALYSIS 120 consistent with the approach used in the current work, as described previously. The data for v = 13 through 37 were the band-origins presented in Table 5-2. For subsequent calculations and least-squares fits, the values of the parameters D, v/j and C5 were fixed to the values resulting from this initial fit to the vibrational energies alone. A least-squares fit of the B-values to Eq 5-18 was performed. The data used was from the same sources as the data for G(v), and the B-values from other authors were adjusted as described previously. The potential was calculated, making use of the correction functions in Eq 5-26 and Eq 5-27. During the calculation of the potential, the inner branch smoothed using Eq 5-23. The values of a and b in Eq 5-23 were determined from the inner branch of the potential at v = 24 and 25, and the value of m was set to 14. The distortion constants were calculated from the potential, and the experimental data were re-fit with the distortion constants held fixed at these calculated values. The experimental data consisted of 2396 transitions (compiled in Table A - l in Appendix A) spanning the range v = 13 through 37, with the exclusion of v = 27 (which was omitted because of a perturbation). The ultimate data set was limited to transitions involving J < 80, an exclusion of 22 transitions from among the vibrations v = 16 through 19. There was some indication that the absence of neglected higher-order centrifugal distortion constants (i.e. eighth order and higher) were affecting these lines. Neglected higher-order terms can be empirically accounted for by the inclusion of an additional parameter [70] from which an effective value for the next higher-order term can be calculated. It seemed more appropriate, however, to drop this small number of data points rather than introduce an extra parameter. There were also a small number of transitions below J = 80 culled from the data set before the final fit. Transitions that did not fit well were re-examined in the spectrum, and if the signal had the appearance of suffering from interference effects with a nearby line (as described previously), or there was a blending not previously identified, the transition was removed from the data set. If a line did not fit well but there was no evidence in the spectrum of possible sources of measurement error, the transition was not removed. Approximately 40 data points were deleted in this process. The new set of G(v) and B„ values were refit to the near-dissociation expansions, the potential was recalculated, and the process was iterated until convergence. The resulting "improved" set of vibrational energies were fit to Eq 5-15 to get new values for D, vp, and C5. Note that the data for v < 13 were the same as in the beginning; only vibrational energies corresponding to transitions observed in the present work changed as a result of the iterative procedure described above. With the new values for D, V£>, and C5, the entire iterative procedure was repeated. Changes in D, vr>, 5 RESULTS AND ANALYSIS 121 and C5 (as determined from a least-squares fit to the vibrational energies alone) were insignificant after the fifth re-determination of these constants. 5.7 Results of Global Analysis The results for the global analysis are presented in Table 5-5. The standard deviation from the fit was 0.00417 c m - 1 , and the average error was -0.00039cm-1. In the fit, 1882 of the 2396 data points (over 78 percent) agreed with the calculated values to within their uncertainty. Following [70], no error estimates are given for the empirical parameters involved in the fit. The statistical uncertainties in the nde (near-dissociation expansion) parameters describing the vibrational energies (pi to P7 and qi and q2 in Table 5-5) were all the order of half a percent. The statistical uncertainties in the parameters describing the B-values (si to ss and t i to t3 in Table 5-5) were in the range of five to ten percent. Table 5-5: The parameters resulting from the global analysis of the A state. D a 15,894.6185(±0.007) cm" 1 41.54442(±0.013) c 5 61,694.23(±860) c m - 1 ! 5 vibration rotation Pi -5.229 140 314 374 x 10" -5 Sl 0.234 067 027 562 P2 -7.921 997 605 45 x 10~ 4 S2 -0.024 594 679 506 P3 8.013 936 275 879 x 10" -5 S3 4.245 363 438 022 x 10" -6 P4 -3.592 547 045 428 x 10' -6 s4 9.356 420 132 142 x 10" -5 P5 8.659 563 846 083 x 10" -8 S5 -5.537 523 511 222 x 10 -6 P6 -1.085 874 845 554 x 10 -9 S6 1.554 583 611 539 x 10" -7 P7 5.565 708 234 849 x 10~ 12 S7 -2.201 529 711 688 x 10 - 9 sg 1.259 604 164 432 x 10" 11 qi -3.323 856 459 600 x 10 -3 8.545 614 371 858 x 10' -5 t l 0.480 322 553 059 t 2 -0.037 793 572 657 t 3 7.422 901 352 29 x 10" 4 "Energy of the dissociation limit relative to v.= 0, J = 0 in the ground state. 5 RESULTS AND ANALYSIS 122 The distortion constants calculated from the potential are given in Table 5-6. Comparing them to the fitted values listed in Table 5-2, the values for D„ in the two tables are fairly consistent, with the ones in Table 5-6 typically larger in magnitude, while the values of H,; calculated from the potential are roughly a factor of two smaller in magnitude than the fitted values. The experimental data used in the global fit is given in Table A - l in Appendix A. The calculated frequencies in Table A - l were calculated using the parameters in Table 5-5 and with the centrifugal distortion constants fixed to the values in Table 5-6. The discrepancies for v = 27 in Table A - l result from a perturbation that will be discussed in more detail later. The standard deviation of 0.00417 c m - 1 from the global analysis does not include the discrepancies in v = 27, since data from this vibration was not included in the fit. Comparing the entries for v = 27 in Table 5-2 and Table 5-7, the perturbation has the effect of shifting the apparent band-origin upwards in frequency by about 0.023 c m - 1 . The error introduced in the fitted B-value by the perturbation is 0.000058 c m - 1 , the order of 0.2%. There are also some indications in Table A - l of other, less significant perturbations than the one in v — 27. There are a few cases where all transitions involving a particular rotational level in the A state are off by roughly the same amount (e.g. v = 31, J = 43). These are local perturbations, and are consistent with the observations of isolated extra lines in the spectrum. These local perturbations will be discussed in detail later. Short stretches of systematic deviations of the residuals in Table A - l for J in the high teens to low twenties could be due to the evolution of the lineshape discussed previously. Table 5-7 presents the values of G(v) and Bv calculated from the parameters in Table 5-5. These should be better representations of the true values for these parameters than the values listed in Table 5-2, since contamination effects from missing or poorly-fit centrifugal distortion constants have been removed. Note, however, that there may still be effects present due to perturbations from other electronic states. The inner and outer classical turning points from the suggested potential are also listed. Note that the smoothing function in Eq 5-23 is still in use during calculation of the turning points. Comparing the values for G(v) and Bv in Table 5-2 and Table 5-7, the changes incurred by fixing the distortion constants to their "mechanical" values are significant, often by an amount the order of ten times larger than the statistical errors given in Table 5-2. The need for calculating the distortion constants from the potential is obvious, both for getting good values for the parameters (G(v) and B„) and for an accurate evaluation of the potential. 5 RESULTS AND ANALYSIS 123 Table 5-6: Centrifugal distortion constants calculated from RKR potential. (All values in cm - 1) V 107D„ H.„ Lv M„ N„ 0 0.35735 -8.2108 x 10-14 -4.059 x 10-19 -1.155 x 10" -24 -3.637 x io--30 -3.536 x 10" -35 1 0.37837 -6.7321 x 10-14 -4.140 x 10-L9 -2.876 x 10" -24 -2.165 x 10" -29 -2.073 x 10" -34 2 0.40306 -7.1235 x 10" L4 -5.131 x 10" L9 -5.223 x 10" -24 -4.258 x io--29 -3.982 x 10" -34 3 0.43261 -9.2548 x 10-L4 -6.991 x 10" 19 -8.086 x 10" -24 -7.514 x io--29 -7.413 x 10" -34 4 0.46810 -1.2925 x 10" L3 -9.995 x 10" 19 -1.185 x 10" -23 -1.299 x 10" -28 -1.429 x 10" -33 5 0.51056 -1.7982 x 10" 13 -1.464 x 10-L8 -1.750 x 10" -23 -2.244 x 10" -28 -2.818 x 10" -33 6 0.56115 -2.4418 x 10-13 -2.173 x 10-18 -2.679 x 10" -23 -3.931 x 10" -28 -5.674 x 10" -33 7 0.62128 -3.2458 x 10-13 -3.252 x 10-18 -4.284 x 10' -23 -7.032 x 10" -28 -1.166 x 10" -32 8 0.69296 -4.2643 x 10-13 -4.902 x 10-18 -7.120 x 10" -23 -1.286 x 10' -27 -2.426 x 10" -32 9 0.77907 -5.5909 x 10-13 -7.422 x 10" 18 -1.210 x 10" -22 -2.386 x 10" -27 -5.042 x 10" -32 10 .0.88337 -7.3578 x 10-13 -1.122 x 10-17 -2.057 x 10" -22 -4.407 x 10" -27 -1.026 x 10" -31 11 1.0099 -9.7076 x 10-13 -1.672 x 10-L7 -3.402 x 10" -22 -7.861 x 10" -27 -1.989 x 10" -31 12 1.1612 -1.2713 x 10-12 -2.407 x 10-17 -5.313 x 10" -22 -1.306 x 10" -26 -3.529 x 10--31 13 1.3355 -1.6248 x 10-L2 -3.270 x 10-17 -7.603 x 10" -22 -1.949 x 10" -26 -5.506 x 10--31 14 1.5237 -1.9891 x 10-12 -4.103 x 10" L7 -9.759 x 10" -22 -2.554 x 10' -26 -7.427 x 10--31 15 1.7094 -2.3044 x 10-L2 -4.729 x 10-L7 -1.134 x 10" -21 -3.048 x 10" -26 -9.359 x 10" -31 16 1.8756 -2.5316 x 10" L2 -5.132 x 10-L7 -1.274 x 10" -21 -3.728 x 10" -26 -1.304 x 10" -30 17 2.0131 -2.6879 x 10" L2 -5.556 x 10" 17 -1.546 x 10' -21 -5.406 x 10" -26 -2.214 x 10" -30 18 2.1249 -2.8450 x 10-12 -6.413 x 10 - 17 -2.105 x 10" -21 -8.405 x 10" -26 -3.734 x 10" -30 19 2.2234 -3.0855 x 10" 12 -7.967 x 10" 17 -3.031 x 10" -21 -1.330 x 10" -25 -6.139 x 10" -30 20 2.3239 -3.4689 x 10-12 -1.036 x 10-16 -4.355 x 10" -21 -2.047 x 10" -25 -1.030 x 10" -29 21 2.4391 -4.0254 x 10-12 -1.369 x 10-16 -6.234 x 10" -21 -3.159 x 10" -25 -1.796 x 10" -29 22 2.5778 -4.7766 x 10-12 -1.827 x 10-16 -9.188 x 10" -21 -5.160 x 10" -25 -3.050 x 10" -29 23 2.7469 -5.7714 x 10" 12 -2.468 x 10" 16 -1.358 x 10" -20 -8.600 x 10" -25 -6.087 x 10" -29 24 2.9530 -7.0760 x 10-12 -3.404 x 10-16 -2.102 x 10' -20 -1.459 x io--24 -1.101 x 10" -28 25 3.2041 -8.8111 x l O - 12 -4.772 x 10" 16 -3.313 x 10" -20 -2.651 x 10" -24 -2.274 x 10" -28 26 3.5104 -1.1147 x 10-11 -6.859 x 10" 16 -5.366 x 10" -20 -4.836 x 10" -24 -4.886 x 10" -28 27 3.8847 -1.4325 x 10-11 -1.010 x 10" 15 -9.078 x 10" -20 -9.278 x 10" -24 -1.048 x 10" -27 28 4.3439 -1.8724 x 10" 11 -1.521 x 10-15 -1.587 x 10' -19 -1.909 x 10" -23 -2.501 x 10--27 29 4.9083 -2.4923 x 10-11 -2.367 x 10" 15 -2.899 x 10" -19 -4.113 x io--23 -6.384 x 10" -27 30 5.6054 -3.3852 x 10-11 -3.815 x 10-15 -5.581 x 10" -19 -9.538 x 10" -23 -1.790 x 10' -26 31 6.4713 -4.7093 x 10" 11 -6.430 x 10-15 -1.145 x 10" -18 -2.381 x 10" -22 -5.496 x 10' -26 32 7.5570 -6.7409 x 10" 11 -1.141 x 10-14 -2.545 x 10" -18 -6.615 x io--22 -1.901 x 10" -25 33 8.9377 -1.0007 x 10-10 -2.162 x 10-14 -6.195 x 10" -18 -2.075 x 10" -21 -7.677 x 10--25 34 10.729 -1.5560 x 10-10 -4.448 x 10-14 -1.694 x 10" -17 -7.556 x 10" -21 -3.727 x 10" -24 35 13.120. -2.5709 x 10-10 -1.017 x 10" 13 -5.379 x 10--17 -3.335 x 10" -20 -2.289 x 10" -23 36 16.445 -4.6102 x 10-10 -2.673 x 10-13 -2.079 x 10--16 -1.897 x 10" -19 -1.920 x 10" -22 37 21.348 -9.2725 x 10-10 -8.533 x 10-13 -1.056 x 10--15 -1.536 x 10' -18 -2.487 x 10" -21 38 29.235 -2.2143 x 10" 9 -3.637 x 10-12 -8.059 x 10" -15 -2.105 x 10" -17 -6.097 x 10" -20 39 42.856 -5.4929 x 10" 9 -7.349 x 10-12 3.712 x 10" -14 4.476 x 10" -16 2.416 x 10" -18 5 RESULTS AND ANALYSIS 124 Table 5-7: Turning points for the R K R potential, along with the values of G(v) and resulting from the global analysis. v G(v) (cm-1) B„ (cm"1) r_ (A) r + (A) 0 13,821.8979 0.0581508 2.6348 2.7859 l a 13,968.3995 0.0573689 2.5922 2.8580 2 a 14,109.7245 0.0566464 2.5651 2.9134 3 a 14,246.0828 0.0559384 2.5441 2.9624 4" 14,377.4353 0.0552111 2.5268 3.0086 14,503.5895 0.0544395 2.5120 3.0536 6 a 14,624.2708 0.0536060 2.4991 3.0988 7 14,739.1726 0.0526984 2.4879 3.1450 8 14,847.9867 0.0517082 2.4781 3.1930 9 14,950.4189 0.0506291 2.4694 3.2435 10 15,046.1944 0.0494553 2.4619 3.2974 11 15,135.0645 0.0481813 2.4552 3.3553 12 15,216.8269 0.0468033 2.4493 3.4181 13 15,291.3703 0.0453232 2.4441 3.4869 14 15,358.7421 0.0437546 2.4396 3.5622 15 15,419.2093 0.0421270 2.4356 3.6446 16 15,473.2734 0.0404822 2.4322 3.7337 17 15,521.6135 0.0388626 2.4291 3.8288 18 15,564.9771 0.0372977 2.4264 3.9290 19 15,604.0705 0.0357986 2.4241 4.0334 20 15,639.4894 0.0343610 2.4220 4.1416 21 15,671.6984 0.0329723 2.4201 4.2537 22 15,701.0441 0.0316169 2.4184 4.3703 23 15,727.7806 0.0302796 2.4168 4.4922 24 15,752.0952 0.0289467 2.4154 4.6207 25 15,774.1280 0.0276054 2.4142 4.7574 26 15,793.9864 0.0262443 2.4131 4.9041 27 15,811.7558 0.0248529 2.4121 5.0634 28 15,827.5086 0.0234214 2.4112 5.2384 29 15,841.3113 0.0219412 2.4105 5.4333 30 15,853.2320 0.0204051 2.4098 5.6535 31 15,863.3470 0.0188074 2.4093 5.9063 32 15,871.7464 0.0171447 2.4088 6.2017 33 15,878.5394 0.0154166 2.4085 6.5541 34 15,883.8568 0.0136258 2.4082 6.9843 35 15,887.8522 0.0117790 2.4080 7.5249 36 15,890.7006 0.0098877 2.4078 8.2290 37 15,892.5940 0.0079686 2.4077 9.1913 38 a 15,893.7349 0.0060448 2.4076 10.6013 39 a 15,894.3268 0.0041474 2.4076 12.9107 40 a 15,894.5635 0.0023207 2.4076 17.6059 41 a 15,894.6168 0.0006571 2.4076 38.2218 ano data used for this vibration 5 RESULTS AND ANALYSIS 125 The first differences in the B-values as a function of v for the results in Table 5-7 is plotted in Figure 5-16. The data below v = 13 are the values calculated from the parameters in Table 5-5. The data points for v = 13 and up are from vibration-by-vibration least-squares fits to the experimental data with the distortion constants fixed to their "mechanical" values. Figure 5-16: First difference of B-values. The B-values come from least-squares fits with the distortion constants fixed to the values determined from the potential in the global analysis. -0.8-"f" -1.0-E o r - . -1.4-CQ CQ * -1.6--1.8--2.0-10 20 V 30 40 This plot should be compared the one in Figure 5-11. The perturbation at v = 27 is still indicated, but the curve in Figure 5-16 varies much more smoothly for the vibrations near the dissociation limit than was observed for the original vibration-by-vibration results. There is still some fluctuation, to be expected with less data for these vibrations. Note that the data for 1 < v < 6 are extrapolations and are not expected to be as accurate, but the predictions for the B-values of these vibrations should be good to within about .5 to 1%, based on agreement with manual extrapolations. The turning over of the curve at low-v in Figure 5-16 is something not observed in electronic states similar to the A state of Br2 for which low-v data has been experimentally measured. Because of this, the actual B-values for 1 < v < 6 are almost certainly larger than the values listed in Table 5-7. The accuracy of the band-origins listed in Table 5-7 for 1 < v < 6 is uncertain, but it is expected that these calculated values should be good to within about 0.5 c m - 1 . The minimum of the (corrected) R K R potential occurs at r e = 2.7026 A , which is slightly lower than the value given in [3] (2.704 A ) . The resulting value of B e (= 8ir2\.uri) is 0.05849 c m - 1 . 5 RESULTS AND ANALYSIS 126 The energy at the bottom of the corrected potential (relative to v = 0, J = 0 in the X state) is 13,746.76 c m - 1 . This makes T e 13,909.14 cm" - 1, in reasonably good agreement with [3] (13910 cm - 1 ) . The value of D e (using the energy of the dissociation limit determined in the following section) is 2,147.82 c m - 1 . The potential in Table 5-7 should be a significant improvement over the one in [3]. The current work uses high-resolution data extending right up to within 2 c m - 1 of the dissociation limit. The data used above v = 24 is more precise and more numerous (several hundred compared to nine) than the data used in [3], and there were no gaps in the high-v region in the present work. The sparsity of high-v data in [3] forced the use of a hybrid Dunham/near-dissociation approach, where there results were apparently quite sensitive to the choice of transition point between the two representations. That may have been a product of not employing a correction for the R K R potential, since it was observed in the current work that, in the absence of a correction, the results were strongly dependent upon the details of the function used to smooth and extrapolate the inner branch. Several parameters had to be fixed in [3] in order to perform the analysis, namely D and C5 for the near-dissociation expansion part and ae from the Dunham expansion part. The missing data at low-v was a major complication to the analysis in [3], but in the current work, the lack of low-v data posed less of a problem. The data in the current work allowed precise determination of the parameters D and C5 (although they may have been effective values to some extent, as will be discussed shortly). Another improvement of the current analysis over the one performed in [3] is the fact that the experimental data was used to refine the potential. This was not an option in [3], since their own experimental data was only for v = 0. Eventually, it is hoped to determine a potential more closely resembling the true potential, but for now, the current work represents the most comprehensive and complete description of the A state and its potential. 5.8 Long-Range Behaviour of the Potential Theoretical values have been published previously for the parameters in Eq 5-11 for the A state (and other valence states) of BT2 [71]. These theoretical values are: C 5 = 3.2 x 104 c m " 1 ! 5 C 6 = 6.37 x 105 cm'1 A6 C 8 = 1.55 x Wcm^A8 5 RESULTS AND ANALYSIS 127 The value determined for C5 from the global analysis, listed in Table 5-5, is almost a factor of two larger than the theoretical value. There have been no other experimental determinations of C5 for this state, and therefore no corroboration or repudiation of the theoretical value from an independent source, but it is assumed that the large discrepancy between the experimental and theoretical values is due to the fact that the value in Table 5-5 is actually an effective value. This is very useful in determining the potential, but its physical meaning is somewhat ambiguous. From the theoretical values for C5 and C6, the contribution from the term in Eq 5-11 is larger than the contribution from the C5 term for r less than about 20 A . Notice that the outer turning point for v = 37, the highest vibration used in the analysis, was around 9.2 A(see Table 5-7). So, for the entire span of the data analyzed, the contribution from the C6 term was larger than the contribution from the C5 term. This could certainly affect the values determined for D, V£> and C5 from the global analysis. Excluding v = 37 from the least-squares fit to the vibrational energies yields values of D and C5 that are smaller (D = 15,894.586 c m - 1 , C5 = 5.8 x 104 c m - 1 A 5 ) . Excluding both v = 36 and 37 makes them smaller still (D = 15,894.525 c m - 1 , C 5 = 5.4 x 104 c m - 1 A 5 ) . The results are not as stable as one would like to the choice of the highest vibration to include in the analysis. It seems to be an indication of minor troubles reconciling the effects of the dominant C6 term with the chosen model. In view of these difficulties, it is instructive to consider alternate methods to determine values for D and C5. The data used to determine the parameters in Eq 5-11 must be limited to the region where the perturbation theory originally used to derive this equation is actually valid. Different criteria have been proposed for the constraints to impose on the data. Only data for which the outer turning point is greater than some limit (r^m) will be used. This limiting value shall be calculated from the inequality given in [51], which is part of what is known as the Kreek, Pan and Meeth criterium. Using the theoretical values for C6 and Cg, xum is 6.976 A . Only data for v = 34 (for which r + is 6.9843 A—see Table 5-7) or greater can be used to determine the parameters. Since the data extends up to v = 37, this gives only four data points, not many considering that the number of parameters involved also equals four. In performing least-squares fits of the outer turning points to the long-range form of the potential in Eq 5-11, it was decided to fix the value of the dissociation energy. The results for various fixed values of the dissociation limit are presented in Table 5-8. The quoted errors are the statistical la errors from the fit. (5-28) 5 RESULTS AND ANALYSIS 128 Table 5-8: Results of fitting turning points in the long-range portion of the potential. D a (cm"1) 10 5C 5 (cm- 1 A 5 ) 10 6C 6 (cm" 1 A 6 ) 10 8C 8 (cm- 1 A 8 ) 15,894.6185 .56(3) .50(4) .178(8) 15,894.59 .41(3) .64(4) .156(8) 15,894.58 .36(3) .69(4) .149(8) 15,894.57 .32(3) .73(4) .142(8) 15,894.546 .19(3) .85(4) .123(8) Theoretical .32 .637 .155 "Dissociation energy fixed to this value for the least-squares fit The first row in Table 5-8 lists the results obtained when D is fixed to the value from Table 5-5. The fitted values for C5 , C6 and Cs differ significantly from their theoretical values. In the next three rows, D is scanned in steps of 0.01 c m - 1 through the most promising region. In the third row of Table 5-8, the fitted values of the parameters C5, C6 and Cs were close to their theoretical values, almost but not quite within the (probably underestimated) statistical errors. From the data in Table 5-8, taking the dissociation energy somewhere in the range 15894.57 to 15894.59 cm" 1 gives the best agreement with the theoretical values. Because of the sensitivity of the fitted parameters in Table 5-5 to the upper limit of v used in the data set, it is assumed that the value for D listed in Table 5-5 is an effective value, and the more accurate value for D is taken from Table 5-8, based on the energy that gives the best correspondence with the theoretical values for the parameters C5. C6 and Cs- The suggested value for the dissociation energy of the A state (relative to v = 0, J = 0 in the ground state), based on the reasoning outlined above, is 15,894.58 ±0.01 c m - 1 . The entry of D = 15,894.546 c m - 1 in Table 5-8 is the most precise previous measurement of the dissociation limit for the A state of 7 9 B r 2 [51]. In that paper, information on the potential at high-v in the B 3II 0+ U state of 7 9 B r 2 was used to determine the ( 2 P i + 2 P3) dissociation energy, the energy of the dissociation limit for the B state. This energy was determined in [51] to be 19,579.692 ±0.008 c m - 1 , although the validity of the quoted uncertainty has been called into question [72]. The energy of the ( 2Pa + 2 Ps) dissociation limit, the limit corresponding to the A state, is then calculated by subtracting the known energy difference between the 2 P i and 2P3 states of atomic 2 2 bromine. According to [51], this energy separation, shown in Figure 5-17, was 3,685.146 ±0.002 c m - 1 . 5 RESULTS AND ANALYSIS 129 Figure 5-17: Energy separation between the lowest two dissociation limits for 7 9 B r 2 . 23 r There is a slight discrepancy (a difference just over the 2cr level) between the dissociation energy determined in [51], 15,894.546 ± 0.010 c m - 1 and the value quoted in the current work, 15,894:58 ± 0 . 0 1 c m - 1 . As mentioned previously, though, the uncertainty on the value determined in [51] is likely to be overly optimistic [72]. The analysis in [51] involved fitting the turning points in the long-range portion of the potential point-by-point to the form in Eq 5-11, and the uncertainties in the fitted parameters were artificially reduced by generating a large number of points in the potential. Table 5-8 shows that fixing the dissociation energy to the value from [51] yields parameters significantly different from their theoretical values. Despite the paucity of qualified data (i.e. greater than r^m) in the A state, the upper limit of the data extends very close to the dissociation limit (approximately 2 cm - 1 ) and should therefore be reliable. As a comparison, the data used in [51] went up to 5.3 c m - 1 from the dissociation limit of the B state, although there were more data points used in the B state than there were in the A state (7 as opposed to 4). Using the known energy difference between the lowest two dissociation limits of 7 9 B r 2 and the dissociation energy determined for the A state, the suggested value for the dissociation energy of the B 3 n 0 + U state of 7 9 B r 2 , relative to v = 0, J = 0 in the X state, is 19,579.726 ±0.012 c m - 1 , an increase of 0.034 c m - 1 from the value quoted in [51]. However, it should be noted that the fit in [51] went up to Cio, whereas the least-squares fits performed here go only up to Cg. To investigate 5 RESULTS AND ANALYSIS 13,0 this further, the data from [51] was taken and fit to Eq 5-11 with the expansion going up to Cio, and the same dissociation energy was obtained as was reported in [51]. The data was then refit to a model that only went up to Cs, and the new value obtained for the dissociation energy from this second fit was larger than the result from the first fit by an amount 0.031 c m - 1 . The second value was almost exactly equal to the value suggested by the current results from the A state. This brings up the possibility that neglecting Cio in the fits performed for the A state data might have had a significant effect on the results. There is insufficient data to extend the analysis in the A state to Cio- Regardless, the result from analysis of the A state seem quite consistent with the results from [51], serving as a test of the results from the B state. There was another analysis of the B state of Br2 that used data up to v = 55, within 1 c m - 1 of the dissociation limit [73]. The dissociation limit of the B state for 7 9 B r 2 was found in [73] to be 19,579.76 ±0.01 c m - 1 . The dissociation limit of the A state, using the expected energy separation given in [51], is then predicted to be 15,894.614 ±0.012 c m - 1 . This is in almost exact agreement with the value in Table 5-5 determined from the global analysis. The value suggested for the dissociation limit of the A state based on the results of the current work, 15,894.58 ± 0.01 c m - 1 , is halfway between the values determined from [51] and [73]. The reason for the minor discrepancy with the result in [73] is not clear. The dissociation limit was determined in [73] by using a near-dissociation analysis, and is consistent with the results of using near-dissociation expansions for the A state in the current work. The remarkably good agreement is assumed to have resulted from random chance, since the value determined for D in the current work was very sensitive to the upper limit in v used for the data set, as discussed previously. It would, of course, be preferable to determine the dissociation energy without reliance upon theoretical values. For the A state of B r 2 , there are three parameters with theoretical values to which the results may be compared, and so it is not as ill-posed an approach as comparing to a single theoretical value. The possibility of errors in the otherwise untested theoretical values for the A state cannot be discounted, however, and an independent experimental measure of these parameters would therefore be in order. A method proposed in [72] makes use of the B-values (not just the energies) in the determination of the parameters in Eq 5-11, thereby doubling the number of data points available. It is hoped that the increase in data points will enable a determination of the parameters describing the long-range potential, as well as an improved determination of the dissociation energy. This analysis is in progress. With more available data (i.e. more vibrational levels meet the criterium of r > r^m) in the B state of B r 2 , a more precise value for the dissociation 5 RESULTS AND ANALYSIS 131 energy should be determined from the B state data following the procedure in [72], which can then be fixed during the determination of actual experimental values for C5, C6 and Cs for the A state. It is hoped that some of the weaker bands very near the dissociation may be identified to provide more data for this analysis. Currently, a few transitions in the 38'-2" band have been tentatively assigned, and there are many weak, unidentified peaks in the spectrum where 39'-2" is expected to fall, but these peaks have yet to be sorted out. 5.9 Lines Above the Dissociation Limit Table 5-9: Deviation of quasi-bound levels from expectation for v7 = 24 (All values in cm 1) Observed Calculated" Obs-Calc Observed Calculated" Obs-Calc R(72) 15,146.6568 15,146.6765 -0.0197 P(74) 15122.7245 15122.7455 -0.0210 ' R(73) blended — — P(75) 15114.1713 15114.1921 -0.0208 R(74) 15,130.0477 15,130.0804 -0.0327 P(76) blended — — R(75) 15,121.5378 15,121.5755 -0.0377 P(77) 15,096.6359 15,096.6735 -0.0376 R(76) 15,112.8877 15,112.9304 -0.0428 P(78) blended — — R(77) 15,104.0858 15,104.1436 -0.0578 P(79) 15,078.5414 15,078.5947 -0.0533 R(78) 15,095.1504 15,095.2130 -0.0626 P(80) 15,069.2742 15,069.3407 -0.0665 R(79) 15,086.0476 15,086.1367 -0.0891 P(81) blended — — R(80) 15,076.8077 15,076.9125 -0.1048 P(82) blended — — R(81) 15,067.4006 15,067.5382 -0.1376 P(83) 15,040.5592 15,040.6965 -0.1373 R(82) 15,057.8446 15,058.0115 -0.1669 P(84) blended — — R(83) 15,048.1209 15,048.3296 -0.2087 P(85) 15,020.6288 15,020.8419 -0.2131 R(84) 15,038.2328 15,038.4900 -0.2572 P(86) blended — — a Calculated from the least-squares fit to levels below the dissociation limit Table 5-2 lists the value of J for the highest rotational state below the dissociation limit, called j£>. Note that for v' > 23, the data used in the fit for the rotational constants (for the e parity levels) extends all the way up to the final bound level. Transitions to quasi-bound levels above the dissociation limit were also measured, but these levels were not included in the least-squares fits because they initially seemed to be inconsistent with the data below the dissociation limit. As an example, the data above the dissociation limit in the 24'-l" band is shown in Table 5-9. There is a significant difference in the frequencies and the frequencies calculated from fitting the 5 RESULTS AND ANALYSIS 132 data below the dissociation limit. The assignments of the lines are not in question. The intensity remains strong, and the "drift" away from the predicted values is gradual. Combination differences can also be used to ensure con-sistency. Recall that R(J-l) and P(J+1) transitions end on the same upper state, so a discrepancy in the upper state would have the same effect for both transitions. The discrepancies for transitions ending on the same upper state listed in Table 5-9 agree within their errors (typically 0.0035 cm - 1 ) . The same behaviour—levels above the dissociation limit being lower in energy than expected—was observed for 23 < V < 36. One possible explanation for this behaviour is that higher order distortion constants in Eq 2-11 are required. Least-squares fits were carried out, including data from both above and below the dissociation limit, with higher order terms. Adding Lv< (the term proportional to [J(J+1)-1]4) was not sufficient; the data above the dissociation limit still did not fit with the data below. Including Mv< (the fifth order term) in the least-squares fits gave positive values for Lv>. Getting positive values of Lv> was inconsistent with preliminary calculations done for the potential of the A state, and this result was therefore considered to be invalid. It was therefore initially concluded that there was a strong perturbation from a dissociative state shifting the levels above the dissociation limit. After the global analysis described previously was performed, however, the data above the dissociation limit did turn out to be consistent with neglected higher order distortion constants. The global analysis used distortion constants up to 0„< (proportional to [J(J+1)-1]7). The data above the dissociation limit still did not fit with the data below, but it comes closer than before. If the discrepancy were due entirely to neglect of the next higher order distortion constant (oc [J(J+1)-1]8), then v(calc) - v{exp) = -PV>[J(J + 1) - l ] 8 , where j/(calc) represents the frequency of a particular transition calculated from the results of the global fit, and z/(exp) represents the experimentally measured transition frequency. Taking the natural log of both sides of the equation yields ln( u(calc) - i/(exp) ) = ln(-Pv>) + 8ln( [J(J + 1) - 1] ). Figure 5-18 shows the plot of ln( i/(calc)-i/(exp) ) versus ln( [J(J+l)-l] ). The slope of this plot is 8.48, close to the theoretical value of 8. The fact that it is larger probably means that even higher orders are required (i.e. the term proportional to [J(J+1)-1]9). The intercept gives an estimate for P24': -5.3 x 1 0 - 3 4 . This is within about a factor of 10 of the value that might be expected for this parameter, based on the trend of the distortion constants for v = 24 in the A state shown in Table 5 RESULTS AND ANALYSIS 133 5-6. It is clear that the data above the dissociation limit is consistent with the data below, but only if the analysis goes to very high order in the distortion constants. Figure 5-18: Discrepancy of data above the dissociation limit from the results of the global analysis. Data above dissociation, v = 24 -1.0--1.5--2.0-O £5 -2.5-> T3 -3.0--3.5--4.0-8.5 8.6 8.7 8.8 8.9 in(J(J+1)-1) A list of all of the unblended lines from transitions to levels above the dissociation limit is given in Table A-3 in Appendix A. This data was not used in any of the analysis. 5.10 E x t r a Lines Perturbations can lead to extra lines in the spectrum. These are signals from transitions that are normally forbidden, but become allowed as a result of a perturbation. There are no allowed electric dipole transitions between the ground state and the A/3n.2u state (since Afi = 2), but a small probability of transition from X to A ' is introduced as a result of the A ' state interacting with the A state. The interaction between the A and A ' states has a significant effect on the magnetic rotation signal, since this interaction gives rise to the strong signals from high-J transitions in the X - A system. It is expected that there would be some reciprocal effects. There could be observable magnetic rotation signals from the A ' - X system as a result of the perturbation. There were some isolated extra lines in the spectrum that could be from A' <— X transitions. These extra lines will be discussed in more detail shortly, but they represent extreme cases (large mixing due to small energy denominator), and were the exception rather than the rule. In the second harmonic spectrum, all but twenty or thirty weak lines have received at least a tentative identification, and so 5 RESULTS AND ANALYSIS 134 perturbation-induced magnetic rotation signals in the A ' - X system must be relatively weak, since they are not readily apparent. The perturbation between the A and the A' states affects both the transition moment and the g-factor. This is a very important point for understanding the effects of perturbations in magnetic rotation spectroscopy. In absorption spectroscopy, only the effects on the transition moment play a role. Therefore, one must be careful not to think in terms of concepts specific to absorption spectroscopy when considering the effects of perturbations in magnetic rotation spectroscopy, since that will often lead to erroneous conclusions. The admixture of the A state introduced into the A ' state (and vise-versa) as a result of the perturbation is small, typically the order of a fraction of a percent, except for those relatively uncommon instances where there is a resonant perturbation. The probability of transition from X to A ' is therefore very small for most of the levels in the A' . The contribution to the g-factor resulting from the perturbation (described in Eq 5-6) is also quite small, the order of a few nuclear magnetons. However, the increasing number of magnetic sublevels as J increases can lead to a large magnetic rotation signal at high-J from this small magnetic moment, even for a relatively small transition moment. The magnetic sublevels in the A' state will also split as a result of this perturbation, and so the magnetic rotation signal for A' <— X transitions is expected to increase at high-J as well, but the transition moment is just so small that the signals were generally too weak to be observed with the sensitivity used in the experiment. Strong signals for A' <— X transitions only appeared where there is a resonant interaction between the A and A ' states. Quite possibly the most interesting feature of the 7 9 B r 2 spectrum was an entire series of extra lines. For transitions involving 2 < J < 32 in the 27'-2" band of the A - X system, extra lines were observed. Several examples from the R branch are shown in Figure 5-19. Each plot spans 0.1 c m - 1 . The signals in Figure 5-19 labelled as the main lines (i.e. arising from the A - X 27'-2" transitions) were the lines that fit better with the data above J = 32 in the 27'-2" band. The extra lines were on the low-frequency side of the main lines up to J = 30 in the A state. There was a crossing at J = 31 in the A state, and the extra lines were on the high-frequency side for J = 32 in the A state. For 27'-l" R(32), there was a small, unassigned peak on the high-frequency side that could belong to the extra series. However, all other transitions to vA = 27, J = 33 (i.e. 11'-1" R(32) and P(34) and 27'-l" P(34)) were blended, and therefore there was no confirmation for an extra line associated with J = 33. For J above this, no further lines were measured for the extra series, presumably because the energy separation became too large for the perturbing state to borrow enough transition moment to give a signal strong enough to observe. The frequency 5 RESULTS AND ANALYSIS Figure 5-19: A selection of extra lines associated with transitions in v = 27 of the A state. For R(l), it is not clear which line is "main" and which is "extra." 27'-2i" R(4) extra 27'-2" R(1) 15167.55 15167.60 15167.65 15166.70 15166.75 5 RESULTS AND ANALYSIS 136 spacing between the main and extra lines remained roughly constant (approximately 0.05 cm - 1 ) over a wide range of J, and then decreased near the crossing. The strengths of the main and extra lines were almost equal at low-J, but the amplitude of the extra line relative to the main line decreased with increasing J, and then increased again as the separation between main and extra lines got smaller. To ensure the lines were not some sort of experimental artifact (e.g. from two different modes in the laser), the frequency region containing the 27'-1" low-J lines was also measured. There were again extra lines, and no other lines in the spectrum showed doubling. Figure 5-20 shows the frequency difference between main and extra lines as a function of J. The results were averaged over all transitions to a particular J state (for R and P lines, but not for the weaker Q lines) from lower states in v" = 1 and v" = 2. Figure 5-20: Separation between the lines assigned as main and extra lines in v = 27 of the A state. UJ 0.06 0.04 0.02 0.00 c 'ro -0.02 • E LU -0.04- ^ 1 0 5 10 15 20 25 30 35 Figure 5-21 shows the strength of the extra line relative to the main line as a function of J in the A state. A value of 1 would indicate that the two lines were of equal strength. A rough estimate of relative heights was used rather than the more rigorous approach of finding the area under the lines. The results were averaged over data from all R and P lines for a particular J state, including data from both v" = 1 and 2. The point at J = 31 is somewhat speculative, based upon one partially blended transition. Combination relations show that the extra lines must have the same J-numbering as the main lines and must originate in the same lower state. Other clues available to identify the state giving 5 RESULTS AND ANALYSIS 137 Figure 5-21: The relative intensity of the line assigned as the extra line as compared to the line assigned as the main line in v = 27 of the A state. The data point for J = 31 is speculative; see text. J rise to the extra lines are the fact that there are no apparent extra lines associated with transitions to J = 1 in the A state, and the fact that there are extra lines associated with Q-transitions (which end on f parity levels) as well as R- and P-transitions (which end on e parity levels). Some of the signals from Q-transitions actually have some structure to them, as shown in Figure 5-22. Al l three peaks appear to be associated with the line. With the R and P lines, there is always a clearly resolved doublet (the main line and the extra line). The Q lines are much weaker and could only be followed up to J = 13, but the few unblended Q lines that were strong enough to observe had structure like the examples shown in Figure 5-22. The separation between the outer peaks is approximately the same as the separation between the doublets in the R and P lines. The reason for the structure in the Q lines is not clear, but it may-have something to do with hyperfine structure, since the e and f parity levels have different values of eQq e//, as described by Eq 2-65. None of the models considered to explain the structure in Figure 5-22 was successful, and so the role of hyperfine effects remains in question. Another potentially useful piece of information comes from the perturbation-induced shifts in the transition frequencies. Data for V = 27 were not used in the global analysis, but the results of the global analysis can be used to predict where the transitions for this vibration should occur. Table 5-10 compares the predicted and observed frequencies for a selected set of transitions involving 5 RESULTS AND ANALYSIS Figure 5-22: Structure in Q lines in v = 27 of the A state. 138 27'-1"Q(5) predicted frequency 15487.40 ' 15487.45 ' 15487.50 v' = 27. The difference between the observed an calculated values is an estimate of the shift in the v = 27 levels resulting from the perturbation. This is not expected to be perfectly accurate, but it should give a rough idea of the effects of the perturbation. From Table 5-10, at low-J, the predicted position of the A - X transition is almost exactly halfway between the main and extra lines. If the prediction is accurate, this is an indication that the two levels would have been almost degenerate in the absence of the perturbation and are therefore almost completely mixed. As J increases, the discrepancy between the calculated and observed frequencies for the transitions decreases. Again assuming that the predictions are accurate, this would indicate that the shift resulting from the perturbation is getting smaller, either because the interaction between the two states is decreasing or because the energy denominator is getting larger (or possibly both). The bound electronic states in the energy region that are candidates for the perturbing state are the B 3II 0+ U state, the X 1 S+ state and the A / 3 n .2 U state. There are also several repulsive states that go to the same dissociation limit as the A state, and some of these states might have a shallow minimum at large r in their potential energy curves, the result of van der Waals forces between the two nuclei. It is possible that these van der Waal states could extend down far enough in energy to get a resonant perturbation with v = 27 of the A state. As discussed in the section on f2-type doubling, for example, the 3 I I 0 - U state is expected to be weakly bound and probably extends farther downwards in energy than v = 27. 5 RESULTS AND ANALYSIS 139 Table 5-10: A selected set of transitions in the perturbed state v = 27 (All values in cm - 1 ) Transition Observed" Calculated6 Obs-Calc Splittingc 27-1 R( 3) 15,488.1276 15,488.0983 0.0293 0.0508 27-1 R( 5) 15,487.1992 15,487.1753 0.0239 0.0514 27-1 R( 7) 15,485.8143 15,485.7974 0.0169 0.0428 27-1 R(10) 15,482.8942 15,482.8761 0.0181 0.0432 27-1 R ( l l ) 15,481.6905 15,481.6742 0.0163 0.0465 27-1 R(12) 15,480.3735 15,480.3581 0.0154 0.0487 27-1 R(13) 15,478.9458 15,478.9276 0.0182 0.0489 27-1 R(14) 15,477.3931 15,477.3826 0.0105 0.0495 27-1 R(15) 15,475.7375 15,475.7229 0.0146 0.0536 27-1 R(18) 15,470.0675 15,470.0545 0.0130 0.0578 27-1 R(19) 15,467.9459 15,467.9347 0.0112 0.0591 27-1 R(21) 15,463.3570 15,463.3484 0.0086 0.0524 27-1 R(22) 15,460.8876 15,460.8815 0.0061 0.0500 27-1 R(23) 15,458.3007 15,458.2985 0.0022 0.0472 27-1 R(24) 15,455.6029 15,455.5991 0.0038 0.0471 27-1 R(25) 15,452.7855 15,452.7830 0.0025 0.0458 27-1 R(26) 15,449.8558 15,449.8501 0.0057 0.0420 27-1 R(27) 15,446.8072 15,446.8000 0.0072 0.0387 27-1 R(29) 15,440.3560 15,440.3470 0.0090 0.0336 "Frequency of what was assigned as the main line in the doublet ^Calculated from the global analysis parameters in Table 5-5 Experimentally measured frequency separation between main and extra lines Consider first the bound states. The energy of the B state is well known, and it does not have a resonant interaction with v = 27 of the A state. The vibrational structure of the X state in this energy region is not known. However, the X state, since it is a X S state, could not possibly give rise to extra lines associated with Q lines as well as with R and P lines. It could gives rise to extra lines with Q lines only, or with R and P lines only, but not both. Since the perturbation is known (from combination relations) to be A J = 0, the X state would give extra lines for the R and P lines, but not for the Q lines. The fact that there are extra lines in all three branches excludes the 5 RESULTS AND ANALYSIS 140 X state as a candidate. That leaves the A ' state as the only viable candidate for the identity of the perturbing state from among the bound states. Now consider the dissociative states. The 3U.0-U state and the other $1 = 0 states cannot be the perturber for reasons similar to why the X state cannot; the perturbation would give extra lines for only one set of parity levels and not the other. Of the remaining dissociative states, the 1 I I i u state (see Figure 4-1) is the most likely candidate. The perturbing state must have a B-value almost identical to that of v = 27 in the A state. This would seem to exclude the possibility of the 1 I I i u state, or any other van der Waals state for that matter, since such states are bound only at large r and should therefore have a smaller B-value than that of v = 27 in the A state. Another telling argument against the 1 I I i u state is the fact that there is no apparent doubling for R(0), P(2) or Q(l), the transitions involving J = 1 in the upper state, but there is doubling for transitions involving J = 2 and up. Because the J-numbering of the main and extra lines has to be the same, this implies that the perturber has a minimum J of 2, and therefore must have Cl = 2. That would make the A / 3 i l 2 U state the only acceptable candidate. However, this assignment appears to be inconsistent with the observed variation of the relative intensities of extra and main lines, shown in Figure 5-21, and with the observed frequency separa-tions between main and extra lines, shown in Figure 5-20. The interaction of an Cl — 1 state with an fi = 2 state (a so-called heterogeneous perturbation) is expected to increase with increasing J, since the rotational Hamiltonian in 2-20 has matrix elements between the two states that are roughly proportional to J. In particular, the small frequency separation between the main and extra lines in R(29) and R(31) are difficult to reconcile with an increasing interaction. However, this can be modelled by taking both the S-uncoupling Hamiltonian and the hyperfine Hamiltonian coupling the A and A' states. The magnetic hyperfine part of the hyperfine Hamiltonian has matrix elements between the two states that are roughly constant as a function of J. The matrix elements of the S-uncoupling Hamiltonian between the two states increase roughly linearly with J, which is why the interaction is expected to increase with J. However, if the two contributions have opposite signs, there can be some cancellation of the two interactions. To fit the observations, though, the magnetic hyperfine matrix elements need to be a factor of 40 or 50 larger than expected (or the matrix elements for the S-uncoupling a factor of 50 smaller) in order to have the partial cancellation occur near J = 30. A model that included the hyperfine Hamiltonian, the S-uncoupling operator, and another operator whose matrix elements increased with J (and were opposite in sign to the matrix elements for the S-uncoupling operator) would fit the observations very well, but it is not immediately evident what that third operator could be. 5 RESULTS AND ANALYSIS 141 This discrepancy has yet to be worked out, but the most likely alternative would be that a weakly bound state was resonant with v = 27 and had the same B-value. This would be inconsistent with all of the weakly bound states previously observed in the halogens and inter-halogens (see e.g. [74] and [75]), and it is very unlikely that it could occur. Another possibility is that an interaction greatly increases the hyperfine effects within the A state, and the doubling of the lines is actually the result of a very large internal splitting of the hyperfine levels in the A state. However, there is no simple explanation as to why this would persist over such a large range of J. A model where only the hyperfine Hamiltonian couples the A state to the perturbing state gives reasonably good agreement with the observations, but again, the B-value in the perturbing state must be almost identical to that in v = 27 of the A state, and a weakly bound state is not likely to satisfy this criterium. The final option that will be pointed out is that different electronic states could perturb the different parity levels, e.g. the X state interacts with the e parity levels and the 3 n 0 - u state interacts with the f parity levels. Without knowing the details of either of these two states in the energy region, though, this must be considered to be pure speculation. As an interesting aside, a bound = 2, g symmetry state (e.g. the 3n2 5 state) would satisfy all of the necessary criteria to explain the observations. Since it is a g parity state, only the hyperfine Hamiltonian couples this state to the A state. The presence of such a state would be difficult to detect experimentally, and so the fact that the state had never been previously observed would not preclude its existence, but the theory describing the electronic states of the halogens is well established (see e.g. [53]), and these theoretical treatments predict that the X , A, A ' and B states are the only bound states in the energy region. Some work obviously remains to be done on explaining the observations, but this could be most easily solved by a multiphoton experiment. Transitions to ion-pair states from the low-lying valence states have a strong Af2 = 0 selectivity. If the perturbing state is the A' state, then there should be strong transitions to the D' state from these interacting levels. The D' state is an f2 = 2 ion-pair state at higher energy than the A and A ' states. Using the perturbed levels in the A state as intermediates, the D' state could be studied in a multiphoton experiment from the ground state, something that has never been done before. At the very least, it would give information on the value of Q in the perturbing state. However, such an experiment is beyond the scope of the current work. The series of extra lines were fit to the expression in Eq 5-8, as before with the ground state constants held fixed to the accurate values from [51]. The data used in the fit is given in Table A-4 in Appendix A. The parameters determined from the least-squares analysis are listed in Table 5-11 5 RESULTS AND ANALYSIS 142 Table 5-11: Results of fitting the extra series (All values in cm - 1 ) E a = 15811.8092(9) B = 0.024754(6) D = 3.04(6) x 10- 7 ° Energy relative to v = 0, J = 0 in the X state Using the constants [76] from a fit to the A' state data, the predicted B-value for v = 28 of the A' state is B2&(A') (predicted) = 0.024763 cm'1, a value which is remarkably close to the fitted value in Table 5-11. The agreement is actually better than the agreement between the fitted and predicted B-values for v = 27 in the A state. It was this correspondence between the calculated and observed B-values that first led to serious consideration of the A ' state as a candidate for the perturbing state. Because of the lack of viable alternatives, the A' state remains the leading candidate, but it was later found that the (one might say) uncanny agreement between the predicted and measured B-values was most likely accidental. The first difference of the B value as a function of v has a distinctive shape for the A and A ' states of the diatomic halogens and inter-halogens. Figure 5-23(a) and 5-23(b) show the plots for the A and A' states, respectively, of I 2 . Figure 5-23(c) shows the result for the A state of Br2- Note that each of these three curves either levels off or increases after an initial decrease (ignoring the anomalous behaviour at low v for the A state of Br2) . Looking at figure 5-23(d), the plot for the lowest 21 levels of the A ' state of B r 2 , it is quite likely the derivative will level off at v = 21 or just above. The prediction for v = 28 assumes the derivative would continue downward at a constant slope. In actuality, the vibration quantum number of the extra series would be greater than 28 (assuming the extra series resulted from the A' state), but exactly how much greater is difficult to say with any degree of certainty. Working under the assumption that the extra series comes from the A' state, based on the argument that it is the only electronic state with Cl ^ 0 in the energy region that would have a large enough B-value, some analysis can be performed on the A ' state. The vibrational and rotational structure of the A ' state has been studied through measurement of transitions in the D ' -A ' system [77]. Even though the A' state is fairly well-characterised up to v = 21, the energy of the state relative to the ground state is not known accurately. An estimate of the energy of the A' state was determined in [77] by extrapolating the vibrational energies up to the dissociation limit. 5 RESULTS AND ANALYSIS 143 Figure 5-23: First differences in B-values for various states in I2 and B r 2 . a o o PQ a o O PQ PQ -2.5--3.0 -3.& -4.0 -4.5 -5.0 -5.5--6.0 -6.5--0.8--1.0--1.2 -1.4--1.6--1.8--2.0 (a) A 3 ^ , , O f I; .. .. 10 20 30 40 50 V (c) _ A 3 n l u of Br 2 10 20 30 40 V B o © PQ PQ a o co O PQ PQ > -7--0.4 -0.6--0.8--1.0--1.2 -1.4 -1.6-(b) A ' 3 n 2 u of i . v . 10 20 30 40 50 60 V (d) A ' 3 n 2 u of Br 2 5 10 15 20 25 V The dissociation energy—the same as that for the X state and the A state (see Figure 4-1)—is known very accurately, and so the extrapolated energy to dissociation gave an estimate for the absolute energy of the state. This extrapolation was done in a reasonable manner, using near-dissociation techniques, but it was a long extrapolation, about 430 c m - 1 up to the estimated dissociation energy of 2,835 c m - 1 . The error assigned to the extrapolation was ±100 c m - 1 . The goal in this part of the current work is to improve the estimate of the energy of the A' state. In the discussions that follow, it will be implicitly assumed that the extra series arises from the A ' state. The extra series is (unfortunately) well above the data measured previously for the A ' state. With no clear indication of the vibrational numbering of the extra series, nothing definitive can be said on the basis of the extra series alone. More information is required to connect the extra series to the data previously measured for the A ' state. Therefore, an effort was made to identify 5 RESULTS AND ANALYSIS 144 more extra lines in the spectrum. The results are listed in Table 5-12. The sign of the separation indicates whether the line assigned as the extra line was above or below the line assigned as the main line. Table 5-12: Extra lines observed in the spectrum. Main line level0. Ebextra (cm-1) separationc (cm x) v = 16, J = 44 15,552.6430 -0.0225 v = 20, J = 72 15,813.0440 -0.0270 v = 23, J = 16 15,735.9397 0.0251 v = 24, J = 18 15,761.9040 0.0410 v = 24, J = 58 15,847.3340 -0.0200 v = 30, J = 43 15,889.5035 0.0245 v = 31, J = 24 15,874.3817 -0.0250 v = 31, J = 42 15,894.8206 -0.0170 "Perturbed level in the A state, i.e. the upper level in the main line transition. ^Energy of perturbing level, i.e. the upper level in the extra line transition. This energy is given relative to X state v = 0, J = 0 Emain ~ E e x j r a In v = 16 of the A state, there were extra lines associated with transitions to J ' = 44. Two transitions to this level were observed in the spectrum, 16'-1" R(43) and 16'-0" R(43). The two P-lines in the spectrum were blended, and the Q lines would be much too weak to observe for J this high. Figure 5-24 shows the 16'-1" R(43) line. The predicted position of the transition is shown, and the feature assigned as the extra line is indicated. It is unlikely that a weakly bound state would extend so far down in energy, and so the perturber is either the X state or the A ' state. If it is the A ' state, it would have to be from a A J ^ 0 (i.e. a hyperfine) interaction, since the S-uncoupling operator would cause a large energy separation at this value of J. In v = 20 of the A state, a short series of perturbation-allowed transitions occurs at high-J. There is a splitting of the lines for transitions to J ' = 70, 71 and 72. Figure 5-25 shows an example for J ' = 70 (20'-l" P(71)) and J ' = 72 (20'-l" R(71)). There were extra lines observed for 20'-l" R(69), R(70), R(71), P(71), and P(73). The 20'-l" P(72) line was obscured by blending. Taking the weaker lines as the extra lines, the perturbing level is at a lower energy than the corresponding A state level at J = 70 and is higher in energy at J — 72. 5 RESULTS AND ANALYSIS Figure 5-24: Example of an extra line observed with the v = 16 R(43) line. 145 — i 1 1 . i 15075.10 15075.12 15075.14 Figure 5-25: Examples of the extra lines observed in v = 20. Shown are 20'-l" R(71) and P(71). predicted frequency predicted position 15073.10 15073.15 15073.20 15064.08 15064.12 15064.16 In v = 23 of the A state, extra lines were assigned for transitions to J ' = 16. 23'-l" R(15) and 23'-2" P(17) are shown in Figure 5-26. Extra lines were also observed for 23-2" R(15) and 23-1" P(17), but as can be seen in Figure 5-26, there was no apparent shift in the main line, as there should be from a perturbation, but it could the predicted frequencies that are in error. In this same vibration, there might be extra lines associated with transitions to J ' = 24 and/or 25 as well, and the low-J lines for this vibration showed extra structure that may have extra lines as well. Most low-J lines showed structure due to hyperfine effects, though, as will be discussed later, and it is difficult to distinguish extra lines from hyperfine structure. 5 RESULTS AND ANALYSIS 146 Figure 5-26: Examples of extra lines observed in v = 23. Shown are 23'-l" R(15) and 23'-2" P(17). predicted f requency R ( 1 5 ) predicted frequency 15393.18 15393.22 15393.26 15066.92 15066.96 15067.00 In v = 24 of the A state, four extra lines were observed for transitions to J ' = 18. Figure 5-27 shows the experimental traces for 24'-2" R(17) and P(19) from the A - X system, both of which end on the J = 18 level. For each case, the predicted position of the transition is shown, and falls close to the midpoint between the two observed lines. Figure 5-27: Examples of extra lines observed in v = 24. Shown are 24'-2" R(17) and P(19). extra? extra? R(17) predicted frequency P(19) predicted frequency 15092.90 15092.95 15086.85 15086.90 15086.95 The R(17) and P(19) transitions in 24'-l" look similar to those in Figure 5-27. The J = 19 level in the A state might be slightly perturbed as well, but no extra lines were confirmed. The shift in the v = 24, J = 18 level was significant, equal to one-half the separation between the main and extra lines, an indication that the two interacting levels are almost completely mixed. The 5 RESULTS AND ANALYSIS 147 question of the identity of the perturbing state again arises. The Q(18) transitions from v" = 1 and 2 were both blended, and so gave no evidence either way as to whether the X state could be the perturber. The weakly bound states and the A ' states are also candidates. In v = 24 of the A state, there were also extra lines associated with transitions to J ' = 58. Figure 5-28 shows 24'-l" R(57) and P(59), the two lines observed in the spectrum from transitions ending on this level. The 24'-l" R(58) line also showed some evidence of a perturbation. The 24'-l" P(60) line was obscured by blending, and therefore provided no confirmation of this. There may have also been effects in J = 57, but blending make it difficult to say for certain. Figure 5-28: Extra lines associated with high-J lines in v = 24. Shown are 24'-l" R(57) and P(59). 15254.52 15254.56 15254.60 15235.46 15235.48 15235.50 The extra lines with J = 18 and J = 58 in v = 24 could be due to a single perturbing vibrational level. With the hyperfine Hamiltonian, rotational levels differing by ± 1 or ± 2 in J can interact, and thereby give rise to extra lines. Therefore, one of the extra lines could be from a A J = 0 interaction and the other from A J = 1, for example, or A J = 0 and A J = 2, etc. Alternatively, the perturbations could be from different vibrational states, or even different electronic states, e.g. one perturbation from the X state and the other perturbation from the A ' state. In v — 30 of the A state, a set of extra lines occurs for transitions involving J ' = 43. Al l four transitions in the spectrum, 30'-l" R(42) and P(44) and 30'-2" R(42) and P(44), show a shift from the predicted frequency and have an associated extra line. Figure 5-29 shows the lines from v" = 1. The extra line is assumed to be the weaker line. Clearly, each line shifted as a result of the perturbation. Again there is no way to determine the identity of the perturber. For v = 31 in the A state, there are extra lines for transitions to J ' = 24. Figure 5-30 shows 31'-2" R(23) and P(25), the only two lines for transitions to this level that were observed in the 5 RESULTS AND ANALYSIS 148 Figure 5-29: Examples of extra lines observed in v = 30. Shown are 30'-l" R(42) and P(44). predicted frequency predicted frequency 15418.98 15419.02 15419.06 15404.80 15404.85 15404.90 spectrum. The extra lines are assumed to be the weaker of the two transitions. Figure 5-30: Extra lines in v = 31. Shown are 31'-2" R(23) and P(25). predicted frequency predicted frequency 15185.34 15185.36 15185.38 15185.40 15177.36 15177.38 15177.40 15177.42 For v = 31 in the A state, there are also extra lines associated with transitions to J' = 42 (which is just above the dissociation limit). Extra lines were observed for 31'—1" R(41) and P(43) and 31'-2" R(41) and P(43). Figure 5-31 shows the lines from v" = 1. Again the extra lines are assumed to be the weaker lines. There may have also been extra lines associated with transitions to J ' = 43, but blending makes it difficult to say for sure. In several of the cases shown above, the predicted position of the main line transitions falls halfway between the main and extra lines. This is usually an indication that the two perturbing levels are completely mixed, but the extra lines are much weaker than the main lines, e.g. (v = 30, 5 RESULTS AND ANALYSIS 149 Figure 5-31: Examples of extra lines observed with high-J transitions in v = 31. Shown are 31'-1" R(41) and P(43). predicted frequency predicted frequency 15431.15 15431.20 15417.28 15417.30 15417.32 15417.34 J = 43) in Figure 5-29 and (v = 31, J = 24) in Figure 5-30. This could be due to small errors in the predicted frequencies, or it could be different g-factors in the two states. If the Zeeman Hamiltonian couples the A state and the perturbing state, strange things can happen with the g-factors. For example, the contribution to the g-factor will be opposite in sign for the two states, and so for one state the magnetic activity adds to that from rotation of the nuclear framework (described by Eq 2-31), and for the other state it is opposite in sign to JAR. Recall that the goal of this study is to gather information on the A ' state, to search for supple-mental data (on top of the extra series) to assist in connecting the new observations to the data previously measured in the A ' state. There is no hard evidence as to the identity of the perturber for any of the extra lines listed in Table 5-12. Deducing which ones (if any) belong to the A' state thus becomes a matter of trial and error, finding which ones fit best together and agree with the expectations for the A ' state. Many different models were tried for the A ' state, with different energies for the state and different spacings of the vibrational and rotational levels in the region of the extra lines. The models were then adjusted to be consistent with causing various combinations of the extra lines. Without going into details, the results were judged on the basis of a combination of plots of d ^ -(which is expected to follow the shape shown in Figure 5-23), plots of d fjffl (which should also follow a characteristic shape) and near-dissociation criteria. Several of the models fell within the expectations for the A ' state, and none of the models was vastly superior to the others. There are several thousand permutations of quantum numbers for the extra lines, with interactions of A J from -2 to +2 and the different possible vibrational numbering of the levels. It is felt that this 5 RESULTS AND ANALYSIS 150 analysis is worth pursuing further, in order to determine an accurate value for the energy of the A' state. To proceed with a relatively simple analysis, consider the extra lines associated with transitions to v' = 24, J ' = 18 of the A state. The signals from transitions to this level are down by a factor of about 1.5 to 2 from lower-J lines (e.g. to v' = 24, J ' = 17 of the A state) in this band. This would favour the A ' state over the X state as the perturber. The A ' state would steal transition moment in the perturbation, but would increase the magnetic moment in the A state, since the magnetic moment for the A ' state is much larger (about a factor of six times larger) than that for the A state. The X state, on the other hand, would steal both transition moment and magnetic moment and would give nothing in return, and the magnetic rotation signal would be expected to decrease by a much larger factor (and recall that the signal is expected to decrease with J even in the absence of perturbations). The weakly bound states are also candidates, if they extend this far down in energy (~140 cm - 1 ) , but the A' state is considered to be most likely. From the modelling of the A' state described previously, all of the models that were accepted as possibilities had the B-value in the (v e xtm-3) vibration in the A ' state (where vextra is the vibrational numbering in the extra series which accompanies vA = 27) within 10% of the B-value for the v = 24 vibration in the A state. The vibrational numbering of the level interacting with v — 24 in the A state must be 3 less than the vibrational numbering of the level interacting with v = 27 in the A state (assuming both interactions come from the A' state). Trying models with v^/ different by 2 or by 4 in the two perturbing levels does not give reasonable results. Therefore, it will be assumed that the interacting levels (i.e. v = 24 in the A state and the perturbing level in the A ' state) have B-values that differ by less than 10%. It will also be assumed that the interaction in v = 24, J = 18 of the A state is a A J = 0 interaction. With a band origin for the A ' state level roughly 0.5 c m - 1 above the band origin for v = 24 in the A state, a B-value of 0.0277 c m - 1 , about 4% smaller than the B-value in the A state, and the distortion constants, D„ and H„, for the A ' state set equal to the known values for v = 24 of the A state, there was a A J = +1 interaction predicted for J = 58 (J = 59 in the A ' state with J = 58 in the A state), which would explain the extra line there. It also predicts a A J = +1 perturbation for J = 59 in the A state. As discussed previously, there was some indication that v = 24, J = 59 of the A state was in fact perturbed. Similarly, if the band origin for the A ' state were just below that for the A state (again by about 0.5 cm - 1 ) and the A ' state had a B-value that was 4% larger, there would be A J = -1 interaction at J = 58 (J = 57 in the A ' state with J = 58 in the A state). This model could also be adjusted 5 RESULTS AND ANALYSIS 151 to provide a perturbation in J = 59 of the A state. In both cases (with the band origin above or below the band-origin for v = 24 in the A state), the choice of distortion constants did not have a significant effect on the value determined for the band origin. Doubling the value of D„ changed the value determined for the band origin by only ~0.02 c m - 1 . It is the band origin that shall be used in the analysis that follows. The theoretical value of C5 for the A' state is 1.39 x 105 c m - 1 A 5 , larger than the effective value for C5 for the A state, given in Table 5-5. Although Eq 5-14 and Eq 5-16 are not valid outside the near-dissociation region, the trends they show should extend beyond that region. According to these equations, a larger value for C5 would mean a smaller B-value and a smaller energy spacing between vibrations. The model with the band origin for the A ' state above that of the A state is therefore more likely. With models for which the B-value in the A ' state was about 10% larger than that in the A state, A J = 2 interactions were possible for J = 58 of the A state, but there were also A J = 1 interactions predicted for J in the mid-30s, and these were not observed. The model with the A ' state B-value 4% smaller than that in the A state is therefore considered to be the best possibility. 5.11 Building the A' State With two band origins for the A ' state, one for the extra series and one for the level perturbing v = 24 in the A state, an improved estimate (assuming that all of the assignments are correct) for the energy of the A ' state can be determined. In [77], Eq 5-13 was used to estimate the energy of the A' state: The vibrational energies were extrapolated to the dissociation limit, where ^ in Eq 5-13 goes to zero. Since the energy of the dissociation limit is known, this gave an estimate of the A' state's energy. ( 2 n dGJv^) " + 2 8 5 a f u n c t i ° n °f G(v) (which shall be referred to as a "LeRoy-Bernstein" plot) is expected to give a straight line in the near-dissociation region. Consider, for example, Figure 5-32, which shows such a plot for the A state of B r 2 , with n fixed at its theoretical value of 5, and C5 taken from Table 5-5, the value determined from the global analysis. Notice that the curve levels off slightly above v = 16 or 17, and then steepens again around v = 27, to go linearly from v = 32 or 33 up to the dissociation limit. Figure 5-33 shows the data near the dissociation limit on an expanded scale. Recall that v = 34 and above are in the near-dissociation region, where Eq 5-13 is truly valid, in good agreement with this plot. An accurate estimate of the dissociation limit would be obtained by extrapolating ^ in Eq 5-13 to zero using data for v > 33. In the A state of B r 2 , for example, the extrapolation of the plot 5 RESULTS AND ANALYSIS Figure 5-32: LeRoy-Bernstein plot for the A state of B r 2 . 152 700 600 5^ 500H -O 400H > 300 200-100-0 v= 16 v = 24 15000 15200 15400 15600 15800 16000 G(v) Figure 5-33: Expanded view of the LeRoy-Bernstein plot for the A state of Br2 for data near the dissociation limit. 70-60-o 50-S~ "O 40-> 30-20-— • 10-0-v= 33 15780 15800 15820 15840 15860 15880 15900 G(v) in Figure 5-32, using data for 34 < v < 37, yields a value for the dissociation energy of 15,894.776 c m - 1 . This is somewhat higher than the value determined for the dissociation energy previously because there is a slight downward curve to the data for 34 < v < 37, as can be seen by careful inspection of Figure 5-33. This is a result of the comparable contributions from the C5 term and the C6 term to the long-range potential in 5-11 for the A state of Br2- If the C5 term were dominating 5 RESULTS AND ANALYSIS 153 in this region, the data for 34 < v < 37 in Figure 5-33 would follow a straight line rather than curve. Because of the S-shape of the curve in Figure 5-32, extrapolating from data for v < 34 actually yields reasonably accurate results. For example, if there had only been data with v < 24 in the A state, extrapolating the curve in Figure 5-32 blindly—i.e. assuming the slope of the curve remained constant from the chosen point all the way up to the dissociation limit—from v = 24 to the dissociation limit would overestimate the dissociation energy by about 22 c m - 1 . Extrapolating blindly from v = 16 would underestimate the dissociation energy by approximately 41 c m - 1 . A blind extrapolation from v = 21 in the A ' state was the method used in [77] to determine the dissociation energy. Note that v = 16 in the A state is about 421 c m - 1 below the dissociation limit, close to the estimate in [77] of 430 c m - 1 for v = 21 of the A ' state. Since blind extrapolation from v = 16 in the A state comes within 41 c m - 1 of the correct answer, the error for the extrapolation in the A' state performed in [77] is probably well within their quoted uncertainty of ±100 c m - 1 . The question is, can this uncertainty be improved upon by using the new data, i.e. the two band origins at higher v. To begin with, it is appropriate to use the A state as a test case, to investigate the accuracy of any procedures used. The v = 16 level in the A state will be used as the primary test case, for two reasons. First because the energy of v = 21 in the A' state is estimated to be close in energy to v = 16 in the A state. Secondly, it was noted in [77] that the slope of the curve analogous to Figure 5-32 for the A ' state was close to the expected value of the slope for the near-dissociation region. On this basis, the blind extrapolation was performed to the dissociation limit, assuming that the slope would remain constant through the intervening region (i.e. from VA1 = 21 to the dissociation limit). There was no choice but to make such an assumption, of course. However, it is not likely accurate, as can be seen by examining the trend of the slope of the LeRoy-Bernstein plot for the A state. Figure 5-34 shows the slope of the curve in Figure 5-32 as a function of vibrational quantum number v. It is expected that the slopes for the LeRoy-Bernstein plot of the A' state would follow a similar trend. The important thing to notice from Figure 5-34 is that the slope for v = 16 is close to that for v = 37, i.e. the slope of the LeRoy-Bernstein plot at v = 16 is close to the slope in the near-dissociation region, similar to what appears to be the situation for v = 21 in the A' state. As a preliminary step, a test will be performed using data up to v = 16 in the A state, along with the energies for v = 27 and v = 24 from the A state. This data set is roughly equivalent to the data set available for the A' state. Taking the energies for v = 27 and v = 24, a derivative can 5 RESULTS AND ANALYSIS 154 Figure 5-34: The slope of the LeRoy-Bernstein plot for the A state of Br 2 as a function of v. -0.6--0.7-Q. -0.9-— v = 1 6 v = 3 7 — --1.0-10 - 1 1 1 ' 1 — 15 20 25 — i — 30 35 40 be constructed as: ^ (A) « I [G(vA = 27) -G(vA = 24)]. (5-29) av 6 If (ff(A)J 7 were to be plotted in Figure 5-32 versus Gave = ^[G(vA=27) + G(vA=24)], the average of the two energies, the resulting data point would be fairly consistent with the other data in the plot (i.e. would come close to falling on the curve established by the data points shown in Figure 5-32). Taking two data points, one from v = 16 (the data point plotted in Figure 5-32 for this vibration) and a second one from the concatenation of v = 24 and v — 27, a linear equation is determined 10 10 for ( ^ ) 7 as a function of G(v) (i.e. f ^ J 7 = mG(v) + b, where m and b are constants determined from the two data points). The value of G(v) in this linear equation where 7 goes to zero represents the dissociation energy, D. Using this approach, the estimated value for D comes out to 15,890.3 c m - 1 , off by 4.3 c m - 1 , as compared to the 41 c m - 1 discrepancy from the blind extrapolation. It is evident that the extra data point generated from the two energies (for v = 24 and v = 27 in the A state) greatly improves the extrapolation to the dissociation limit. However, the procedure described above made use of the known energy of v = 16 in the A state, G(vyi=16). In the A' state, G(v^/=21) is not known; it is the quantity being sought. It is thus necessary to invert the procedure, to extrapolate from the known dissociation limit towards lower 5 RESULTS AND ANALYSIS 155 v. This is possible because the values of (ff) 7 in Figure 5-32 do not depend on the energy of the level, only on the change in energy as a function of v. In other words, if the vibrational spacings are known (as is the case for the A ' state), then ^ is known even if the absolute energy is not. This inverted procedure will be tested first in the A state. Again, two data points are used to determine a linear equation of (ff) 7 as a function of G(v). One of the data points comes from the derivative in Eq 5-29, and the second data point comes from the dissociation limit itself, where 10 10 (c£u ) 7 = 0- The value of G(v) where ( f ^ ) 7 is equal to the known value for the vibration in question represents the estimate for the energy of the state. For example, once the linear equation, 10 (ifu ) 7 = m G ( v ) + b, is determined from the two data points, the estimate for G(v^==16) is the 10 value of G(v) that makes J 7 equal to 275.64 cm 7 the known value for this vibration. In the A state, the linear equation is described by the parameters 3 10 m =-0.6357 cm" 7 , b = 10,104 crrT~. The estimated energy of v = 16 from the extrapolation procedure is 15,461 cm 1 , as compared to the actual value of 15,473.2780 c m - 1 , a difference of about 12.3 c m - 1 . Table 5-13: Preliminary test of extrapolation procedure in the A state. (All values in cm - 1 ) V G(v) " Qextrap^y^ b G E X T MP(v)-G(v) 6G c foQextrap d SGextrap_6G 14 15,358.7421 15,298.1 -60.6 67.4 96.6 29.2 15 15,419.2093 15,385.3 -33.9 60.5 87.2 26.7 16 15,473.2734 15,461.0 -12.3 54.1 75.6 21.5 17 15,521.6135 15,524.4 2.8 48.3 63.4 15.1 18 15,564.9771 15,576.6 11.6 43.4 52.1 8.7 19 15,604.0705 15,619.3 15.2 39.1 42.7 3.6 20 15,639.4894 15,654.8 15.3 35.4 35.5 0.1 21 15,671.6984 15,684.8 13.1 32.2 30.1 -2.1 "Actual energy for the given vibration, from Table 5-7 6The extrapolated energy for the given vibration cG(v) - G(v-l), the difference in measured energies between neighbouring vibrations d G e x t r a p ( v ) - G e x t r a p (v-1) , the difference in extrapolated energies between neighbouring vibrations To investigate the accuracy of this procedure, the energy for several vibrational levels was estimated. The results are shown in Table 5-13. In the A' state, the separations between vibrational 5 RESULTS AND ANALYSIS 156 levels are well determined, even though the absolute energies of the levels themselves are not known. This will be used as a diagnostic tool for the analysis of the A' state data. Table 5-13 therefore includes the differences between the extrapolated energies for neighbouring vibrations. For example, <5Gex for v = 21 is the predicted energy separation between v = 21 and v = 20 in the A state, based on the extrapolated energies for the two vibrations. Comparing the predicted separations to the actual separations in Table 5-13, it is evident that the extrapolation procedure reaches its optimum accuracy when the predicted and observed energy separations between vibrational levels are in good agreement. Note that the small difference of 2.8 c m - 1 for v = 17 is accidental; 15 c m - 1 is probably a better indication of the accuracy of the extrapolation technique, and this achieved when the predicted separation between vibrational levels is in best agreement with the experimentally observed separations. Now that the procedure has been established, the data for the A ' state can be analyzed. The plot for the A' state analogous to Figure 5-32 is given in Figure 5-35. The data for the plot is calculated from the parameters in [76]. The values plotted for G(v) are relative to the bottom of the A ' state's potential. Figure 5-35: LeRoy-Bernstein plot for the A ' state of Br2-> T3 1600 1400 1200-1 .1000 800 600-| O 200-0-500 1000 1500 2000 2500 G(v) The energies of two levels are required to construct a derivative like the one in Eq 5-29. One energy comes from the results of the least-squares fit to the extra series, given in Table 5-11. The second energy comes from energy estimated for the level perturbing v = 24 of the A state. This 5 RESULTS AND ANALYSIS 157 was 15,752.58 c m - 1 . As mentioned previously, the difference in vibrational numbering between the two levels must be 3. Following the same procedure as was used in the A state, the linear equation for the A ' state is defined by the parameters: 3 10 m = -0.6311cm -r, b = 10,030cm~T. As before, the estimate for G(vJ4/=21) is given by the value of G(v) that makes ( ^ ) 7 equal to the known value for v = 21 in the A' state (297.89 cm"^). The resulting estimate for the energy of v = 21 is 15,422.6 c m - 1 . On this basis of this estimate, the suggested energy of the A' state, the energy of the bottom of the A ' state potential relative to v = 0, J = 0 in the X state, is T0je(A') = 13,020.5 cm-1. (5-30) The energy relative to the bottom of the potential in the X state is Tete(A') = 13,187 cm-1. This is a difference 43 c m - 1 from the value estimated by [77], well within their quoted uncertainty. The energy of the D' state, based on their measured energy difference between the A' and D' states, is Te,e(L>') = 48,890 cm'1. To determine a confidence level, predicted differences between neighbouring vibrational levels in the A ' state are compared to the actual differences. The results are presented in Table 5-14. Table 5-14: Extrapolation from dissociation in the A ' state. (All values in cm - 1 ) V 6G a fiQextrap b 6Gextrap_6Q 18 72.0 82.6 10.6 19 66.3 75.8 9.5 20 60.9 67.4 6.5 21 56.1 57.1 1.0 aG(v) - G(v-l), the difference in measured energies between neighbouring vibrations ftGea;imp(v) - Gextrap(v-1), difference in extrapolated energies between neighbouring vibrations It would appear from the results in Table 5-14, that the extrapolation has reached its optimal region by v = 21. The extrapolated energy difference between v = 20 and 21 is very close to the actual energy difference. When this was the case in the A state, the extrapolated energies agreed to 5 RESULTS AND ANALYSIS 158 within roughly 15 c m - 1 . To work on the conservative side, the uncertainty on the A ' state energy given in Eq 5-30 is set at ±20 c m - 1 . As an indication of the sensitivity of the procedure, a decrease of 1 c m - 1 in the energy used for the level perturbing v = 24 in the A state led to an increase of 9.1 c m - 1 in the estimate for the energy of the A ' state. It is hoped that this estimate of the A' state's energy, and possible further refinements using the extra lines listed in Table 5-12, can serve as a stepping stone for analysis of spectra involving the A ' state. A multiphoton experiment using the interacting levels associated with the extra lines observed in the current work is a promising way to measure accurately the energy of the A' state. 5.12 Hyperfine Effects The signals for many low-J transitions in the spectrum exhibited significant structure. It was observed for a wide range of vibrations. Figure 5-36 shows a collection of traces for overlapping R(0) + R(l) lines for several different vibrations in the A state. The structure looks as though it could be due to extra lines from perturbations, but it occurs so consistently and over such a broad range of vibrations, that it seems more likely to be the result of large internal splitting in the A state itself, i.e. the result of hyperfine structure. The fact that the structure is only observed for low-J is consistent with the expected decrease in hyperfine effects as J increases. See Eq 2-63. A computer program was written in a combination of C and ASSEMBLY to allow a manual fit of the hyperfine parameters. The program displayed a maximum of four different experimental traces on the screen at one time, along with the calculated signal, which could include contributions from up to two transitions. This program gave the ability to view the effect of changing the hyperfine parameters on several calculated lines at once. The value of eQqo in the X state was fixed to the accurate value determined in [38]. The hyperfine parameters in the A state, namely eQqo, eQq2 and C s r (the spin-rotation constant, equal to [aA + (b+c)£]f2 from Eq 2-63) were varied by hand to try to reproduce the spectrum. A matrix was constructed for each possible value of quantum number F. The matrix included all possible interactions (excluding perturbations from different electronic states) within A J of ±2 for the state of interest. For each J, there were two possible values of I, namely 0 and 2 if the level was a para state or 1 and 3 if the level was an ortho state. This gave a matrix of size 10 by 10. The matrix elements were worked out in the symmetrized basis set, and careful track was kept of the sign of the off-diagonal matrix elements involving eQq2. The matrix was diagonalized to find the eigenvalues (which represent the energies of the hyperfine levels) and the eigenvectors (which 5 RESULTS AND ANALYSIS 159 Figure 5-36: R(0) + R(l) spectra for several vibrations in the A state. 22'-2" R(0) + R(1) 23'-2" R(0) + R(1) 15056.85 15056.90 15056.95 15057.00 1 5083.60 1 5083.65 1 5083.70 24'--2" R(0) + R(1) 2 5 ' - 1 " R(0) + R(1) 15107.90 15107.95 15108.00 15108.05 15450.90 15450.95 15451.00 15451.05 give the mixing of the wavefunctions resulting from the various interactions). The relative intensities of the various possible transitions were calculated from Eq 2-66 using the procedure described in Eq 2-68. The g-factor for a given level was a combination of contributions from the (maximum of) 10 different interacting hyperfine levels. To use a simple example to illustrate the procedure, if a transition is between the two superposition states defined in Eq 2-67, and only the upper state involved in the transition is magnetically active, the effective g-factor for the upper state, gF^{^'), is calculated as ge/fW) = a2gF(1,x)+P2gFU>y), where gF^xory) was calculated from Eq 2-69. The value of gj in this equation was taken from Eq 2-35, i.e. included only the intrinsic g-factors of the levels and not the contribution from the 5 RESULTS AND ANALYSIS 160 interaction between electronic states (which would take gj from Eq 5-6). The magnetic rotation signal was then calculated from Eq 3-40, but only including the frequency shift contributions. The intensity perturbation contributions to the magnetic rotation signal were not included in the calculation. The signal at a given frequency must be summed over all hyperfine transitions in the vicinity. The linewidth was left as an adjustable parameter, and it could be set to Lorentzian for calculating a Doppler-free spectrum, or it could calculate the Doppler-limited spectrum by convolving the lineshape with a Gaussian distribution, like the example shown in Eq 3-41. The ability to calculate an absorption spectrum was also included in the program. Using this option, the program was tested on the A state of IC1, using data from [78] and [79]. The calculations used parameters determined by the analysis of other workers, and the agreement of the calculations from this program with the experimental data was very good. Unfortunately, the agreement was not as good for Br2 . Figure 5-37 shows the experimental and calculated signals for several transitions in 21'-1". The first picture shows the overlapped R(0) + R(l) lines; the second picture shows the overlapped R(2) + Q(l) lines; and the third picture shows the P(3) line. The A state hyperfine parameters used in the calculation were eQqo = -200 MHz, eQq2 = 500 MHz and C s r = 650 MHz. The values for Csr and eQqo are relatively close to the corresponding values for these parameters in the Br atom: 884.810 MHz for the spin-rotation constant, and -384.878 MHz for eQq0 [80]. Figure 5-37: Experimental and calculated spectra for selected low-J lines showing structure in the 21'-1" band of A - X . 2 1 ' - 1 " R ( 0 ) + R ( 1 ) 2 1 ' - 1 " R ( 2 ) + Q(1) 2 1 ' - 1 " P(3) ' e X P ? n m e n t a l c a l c u l a t e d - ^ experimental fA \— calculated calculated < experimental 0.1 cm" 1 0.1 cm" 1 0.1 cm" 1 The fit is not perfect for the first two, but appears to be getting close. The fit is quite poor for the P line, however. With different sets of hyperfine constants, it was possible to get reasonable agreement for different sets of lines, but agreement for all lines simultaneously was not achieved. The problem could be the fact that there are things missing from the calculation. The intensity perturbation contributions to the magnetic rotation signal were not calculated, but this contribution 5 RESULTS AND ANALYSIS 161 is considerably more complicated than the case where only the interaction between neighbouring J states was considered. Now neighbouring F states must also be considered. Including these contributions would be no small undertaking, but with the small energy spacings between hyperfine levels, the contribution is almost certainly very important. Another possible complication comes from perturbations to other electronic states. There is some indication that the A ' state might be close in energy at low-J for several vibrations of the A state. Since the energy of the A ' state levels (or of other electronic states that might perturb the A state, for that matter) is unknown, the effects of the perturbation on the hyperfine levels is difficult to take into account. 5.13 Doppler-free Magnetic Rotation Of course, the analysis would be much simpler if Doppler broadening of the lines were not obscuring the hyperfine structure. To that end, a Doppler-free technique was developed for magnetic rotation spectroscopy. It was later discovered that a very similar technique had been reported previously [23], and so only minimal details will be included on the technique in the current work. The setup for Doppler-free magnetic rotation is shown in Figure 4-3. It is a pump-probe technique, and relies on the hole-burning phenomenon associated with saturation spectroscopy. If the pump beam is intense enough and the transition moment is large enough, a portion of the ground state population can be depleted, while an anomalous peak is created in the upper state population. Because there are fewer molecules in the ground state and significant population in the upper state, the medium becomes partially saturated to transitions at that frequency, i.e. the absorption of light from the probe beam at that frequency by the medium is reduced compared to what it would be without the saturation induced by the pump beam. If the linewidth of the laser is narrow enough and there is no power broadening (or other broadening effects), then the frequency profile of the hole burned in the ground state population should reflect the absorption lineshape for the transition, with a width given by the homogeneous linewidth (natural width plus collision broadening). The magnetic rotation signal depends on the population in the lower state, Nj» in e.g. Eq 3-25 and Eq 3-28. The probe beam propagates in the opposite direction to the pump beam, and the absorption of the probe beam is affected by the saturation effects created by the pump beam. In particular, for the small portion of molecules that see the pump and probe beams as being at the same frequency (i.e. molecules moving transverse to the propagation directions of the beams), the absorption of the probe beam is strongly attenuated. Modulating the intensity of the pump beam 5 RESULTS AND ANALYSIS 162 (e.g. by chopping the beam) then allows the measurement of a Doppler-free signal. If the pump beam is chopped at frequency fi and the magnetic field is modulated at frequency i2, the first harmonic magnetic rotation signal, SFH, has a component that varies as SFH OC Njncos{2'nfit)\H\cos(2'Kf2t). From the trigonometric relation cos(A)cos(B) = i (cos(A + B) + cos(A - B)), measuring with phase-sensitive detection at fi + i2 or |fi— i2\ yields a signal with a linewidth (in the absence of broadening effects) equal to the homogeneous linewidth, because the hole burning in Nj» reflects this linewidth. The setup shown in Figure 4-3 should be more sensitive than the setup in [23]. The experiment in [23] used a circularly polarized pump beam in order to make the technique a variation on polar-ization spectroscopy (see e.g. [37] for an explanation of polarization spectroscopy). Consequently, there had to be a non-zero angle between the direction of propagation of the pump beam and the direction of propagation of the probe beam. This was necessary to get the circularly polarized beam past the polarizer (if it went through the polarizer, it would be linearly polarized) and presumably to avoid the pump beam being reflected back into the dye laser and interfering with the lasing process (a fairly common problem with pump/probe techniques). For the setup in Figure 4-3, if the polarizers are completely crossed, the pump and probe beams can be perfectly counterpropa-gating (or at least as close as the experimenter can manage to make them) without any adverse effects on the lasing process. This allows for a longer overlap region between the two beams and gives a stronger signal. For instance, one of the benefits of the technique presented in [23] was the fact that it got rid of signals from B - X in \ 2 that obscured the spectrum of IBr. The Doppler-free technique in the current work was tested on signals from B - X in I2 (see Figure 5-38), and typically gave better signal-to-noise than the examples for IBr that were shown in [23]. An example trace is shown in Figure 5-38 for 20'-l" R(70) in B - X of I2. The signal was taken with the polarizers decrossed slightly, and the lines have the characteristic second harmonic shape. The smaller peaks in the trace come from 25'-3" R(70) in B - X of I2. For the trace measured in Figure 5-38, the intensity of the pump beam was 45 mW and the intensity of the probe beam was approximately 10 mW. The time constant was set to 100 ms, and the length of the scan was 10 minutes. The linewidth of the signals is approximately 5 MHz. The signal-to-noise is comparable to what can be achieved with polarization spectroscopy for this transition, and probably better 5 RESULTS AND ANALYSIS 163 than what could be achieved with intermodulated fluorescence. See e.g. [37] for an explanation of intermodulated fluorescence. The best signals in I 2 were with the diameter of the beams 1.5 to 2 mm, and the intensities of the pump and probe beams roughly equal. However, the B - X transitions in I2 are relatively easy to saturate, and the best setup for weaker transitions would likely require a stronger pump beam than probe beam. Figure 5-38: Doppler free magnetic rotation for 20'-l" R(70) in the B - X system One interesting aspect observed during the testing of the method in I2 came from the observation of low-J lines. These lines were much weaker than the high-J lines, since the only contribution to the magnetic rotation signal in the B - X system of I2 comes from the electronic mixing contribution to gj, the term that increases with J. There is no intrinsic magnetic moment that gives rise to a strong signal at low-J, as there was in the A - X system. A trace for 20'-l" R(6) in the B - X system of I2 is shown in Figure 5-39. The signal for this transition was much weaker than the one shown in Figure 5-38. With the same intensities in the pump and probe beams as for the measurement for 20'-l" R(70), the time constant was set to 3 seconds and the measurement was taken in three segments, with a 10 minute scan for each segment. It is difficult to assess the linewidths of the of I 2 . 1.1 GHz 5 RESULTS AND ANALYSIS 164 signals. The trace in Figure 5-39 spans 1 GHz (~ 0.033 cm - 1 ) . Figure 5-39: Doppler free magnetic rotation for 20'-l" R(6) in the B - X system of I 2 . As was the case with all the low-J lines observed in I2, there appears to be different lineshapes associated with the different hyperfine transitions, and the phase seems to be positive for some lines and negative for other lines. This could be a result of different signs for g^ in the various levels, resulting from the different combinations of g/ and gj*. There is potentially a great deal of information in Figure 5-39, but this was not pursued further. The Doppler-free technique was then used for Br 2 . The transition moment for the A - X system of Br 2 are very small, and so transitions in this system are difficult to saturate. It was hoped that the large magnetic effects in the A state would overcome this deficiency, but no Doppler-free measurements were obtained for the A - X system of Br 2 . The major problems included backscatter of the pump beam, and the fact that the measured signals were very weak (even in Doppler-limited) when the pressure of the Br 2 was decreased to the point where collision effects were not expected to prevent the measurement of the Doppler-free signal. With the sensitivities used on the detection equipment during the attempts for an A - X Doppler-free signal, noise from sources such as pick-up and fluctuations in laser intensity were likely larger than the signal trying to be measured. It is felt that the measurements are possible, perhaps with a larger magnetic field and the use of measurement techniques that would reduce the effects of laser noise, e.g. the F M spectroscopy technique described in [22]. 6 FUTURE WORK AND CONCLUSIONS 165 6 Future Work and Conclusions Great strides have been made toward a better understanding of the A 3 LTi u state of B r 2 , but it should be evident that more needs to be done. A more detailed study of the fi-type doubling in the A state would be in order. This could give some information on the 3 n 0 - u and 3 £ j _ u electronic states. It might be interesting trying an experiment with the magnetic field perpendicular to the direction of propagation of the laser, since Q-transitions would give the strong signals instead of the R- and P-transitions. This would give the information necessary for analysing the f2-type doubling. Firm identification of signals from transitions to v^ = 38 and up would give an improved description of the long-range potential, important for getting a better understanding of the interactions between atoms. It is hoped to determine a potential that is a closer approximation of the true quantum mechanical potential for the state. The energies of the levels could then be determined directly from the potential, and this would avoid the need to calculate distortion constants to such high order. On a more long-range scale, magnetic rotation spectroscopy could be used to investigate other diatomic halogens and inter-halogens, as well as other molecules. In particular, the higher vibra-tional levels in the A state of I2 could be studied with this technique. The paramagnetic B state in inter-halogens such as BrCl would be an ideal experiment for the technique. Transitions to the lower vibrations in the A state of Br 2 from the lowest vibrations in the X state are very weak, but it would be possible to investigate these vibrations with magnetic rotation spectroscopy by using the more sensitive first harmonic measurements. This would fill in a significant hole in the experimental measurements of the A state of Br 2 . The Doppler-free technique would be well-suited to studying the hyperfine structure of states such as the A 3 I I i u state in I 2 , or the A 3LTi states in IC1 and IBr. The hyperfine structure of these states (except for IC1) has not been well-studied, and these states could also provide accurate frequency standards, because of their small natural linewidths. With sufficient enhancements to the sensitivity, it should also be possible to measure the hyperfine structure in the A3Hiu state of B r 2 , a challenge, perhaps, but an attainable goal. The A ' state could be characterized in more detail, by analyzing the extra lines observed in the current work and/or performing multiphoton experiments using the extra lines as intermediates. It should also be possible to deduce some information on the A ' state by a quantitative analysis of the intensities of high-J lines. This would require careful measurement of the intensities, and would probably have to proceed by taking relative intensities of high-J lines to low-J lines in the 6 FUTURE WORK AND CONCLUSIONS 166 same scan to avoid systematic effects. This would also require a good model for the intensities of the low-J lines. Finally, it would be interesting to investigate a technique analogous to magnetic rotation spec-troscopy that used an electric field instead of a magnetic field. 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An i n d i c a t o r i s included i n the f i n a l column, meant as a guide for the eye, which gives a rough i n d i c a t i o n of how well the calculated frequencies agree with the observed frequencies. No stars are printed i f the cal c u l a t e d frequencies agrees with the observed frequency to within the assigned uncertainty. A single star indicates that the discrepancy i s greater than the uncertainty, but less than 1.3 times the uncertainty. Two stars indicates that the discrepancy i s greater than 1.3 times the uncertainty and less than 1.6 times the uncertainty. Three stars indicates a discrepancy greater than 1.6 times the uncertainty and less than twice the uncertainty. Four stars indicates that the discrepancy i s twice the uncertainty or greater. The d i r e c t i o n of the stars r e l a t i v e to the bar indicates whether the discrepancy i s po s i t i v e or negative. Note that the data for v = 27 i n Table A - l was not ac t u a l l y used i n analysis; i t i s included for observation purposes only. The t r a n s i t i o n s are labeled i n the t r a d i t i o n a l manner. A t r a n s i t i o n i s labeled according to the value of the quantum number J i n the state of lower energy. In addition, v i b r a t i o n a l quantum numbers (v) i n the two states are used to l a b e l a t r a n s i t i o n . Indicating quantum numbers for the higher energy state with a sing l e prime and quantum numbers for the lower energy state with double primes, the la b e l i n g scheme for a t r a n s i t i o n i s v ' ~ v " R ( J " ) , v ' — v " P ( J " ) , or v'--v' ' Q ( J " ) • A t r a n s i t i o n where J i s larger by one i n the higher energy state i s labeled an R t r a n s i t i o n . When J i s lower by one unit i n the higher energy state, i t i s a P t r a n s i t i o n . When J does not change i n the t r a n s i t i o n , i t i s labeled a Q t r a n s i t i o n . T a b l e A - l : Data used i n the global analysis ( a l l units i n cm ) . T r a n s i t i o n Observed C a l c u l a t e d Uncert. Obs-Cal 13—0 R( 8) 13 — 0 R( 9) 13 — 0 R(10) 13 — 0 R(12) 13 — 0 R(13) 13 — 0 R(14) 13 — 0 R(15) 13 — 0 R(17) 13 — 0 R(19) 13 — 0 R(23) 13 — 0 R(27) 13 — 0 R(50) 13 — 0 R(58) 13 — 0 R(59) 15289 4953 15289 5029 0 0150 -0 0076 15288 9424 15288 9338 0 0075 0 0086 15288 2991 15288 2913 0 0150 0 0078 15286 7899 15286 7861 0 0075 0 0038 15285 9345 15285 9232 0 0100 0 0113 15284 9881 15284 9868 0 0075 0 0013 15283 9828 15283 9768 0 0100 0 0060 15281 7409 15281 7358 0 0150 0 0051 15279 2022 15279 2000 0 0075 0 0022 15273 2484 15273 2417 0 0150 0 0067 15266 0986 15266 0980 0 0100 0 0006 15201 7246 15201 7203 0 0150 0 0043 15169 8411 15169 8380 0 0100 0 0031 15165 5045 15165 5004 0 0100 0 0041 Appendix 13- -0 R 62) 15152 0134 15152 0127 0 0100 0 0007 13- -0 R 70) 15112 5271 15112 5221 0 0055 0 0050 13- -0 R 72) 15101 8367 15101 8373 0 0055 -0 0006 13- -0 P 8) 15287 9601 15287 9625 0 0100 -0 0024 13- -0 P 11) 15285 4914 15285 4922 0 0100 -0 0008 13- -0 P 12) 15284 5216 15284 5220 0 0100 -0 0004 13- -0 P 13) 15283 4726 15283 4784 0 0075 -0 0058 13- -0 P 15) 15281 1665 15281 1707 0 0065 -0 0042 13- -0 P 17) 15278 5751 15278 5689 0 0075 0 0062 13- -0 P 18) 15277 1582 15277 1577 0 0075 0 0005 13- -0 P 21) 15272 4805 15272 4817 0 0075 -0 0012 13- -0 P 31) 15252 0892 15252 0872 0 0150 0 0020 13- -0 P 36) 15239 0879 15239 0996 0 0150 -0 0117 13- -0 P (52) 15184 8711 15184 8619 0 0150 0 0092 13- -0 P (53) 15180 8288 15180 8224 0 0150 0 0064 14- -0 R ( 4) 15358 3704 15358 3719 0 0100 -0 0015 14- -0 R 5) 15358 0697 15358 0774 0 0075 -0 0077 14- -0 R 6) 15357 7007 15357 7064 0 0075 -0 0057 14- -0 R 7) 15357 2625 15357 2589 0 0100 0 0036 14- -0 R 8) 15356 7410 15356 7349 0 0150 0 0061 14- -0 R 10) 15355 4614 15355 4573 0 0150 0 0041 14- -0 R 12) 15353 8734 15353 8733 0 0150 0 0001 14- -0 R 13) 15352 9708 15352 9663 0 0150 0 0045 14- -0 R 14) 15351 9825 15351 9826 0 0100 -0 0001 14- -0 R 15) 15350 9334 15350 9221 0 0150 0 0113 14- -0 R 16) 15349 7863 15349 7847 0 0150 0 0016 14- -0 R 47) 15275 9711 15275 9718 0 0150 -0 0007 14- -0 R 49) 15268 6086 15268 6010 0 0075 0 0076 14- -0 R 52) 15256 9381 15256 9378 0 0150 0 0003 14- -0 R 54) 15248 7518 15248 7553 0 0055 -0 0035 14- -0 R 60) 15222 2307 15222 2343 0 0100 -0 0036 14- -0 R 64) 15202 8883 15202 8893 0 0045 -0 0010 14- -0 R 65) 15197 8440 15197 8425 0 0075 0 0015 14- -0 R 68) 15182 1842 15182 1921 0 0055 -0 0079 14- -0 R 70) 15171 3302 15171 3303 0 0100 -0 0001 14- -0 P 5) 15357 1014 15357 1150 0 0100 -0 0136 14- -0 P 6) 15356 5622 15356 5691 0 0100 -0 0069 14- -0 P 8) 15355 2561 15355 2480 0 0150 0 0081 14- -0 P 11) 15352 6878 15352 6928 0 0100 -0 0050 14- -0 P 14) 15349 4401 15349 4485 0 0150 -0 0084 14- -0 P 17) 15345 5173 15345 5142 0 0150 0 0031 14- -0 P 44) 15278 7438 15278 7483 0 0100 -0 0045 14- -0 P 67) 15176 0894 15176 0937 0 0100 -0 0043 14- -0 P 69) 15165 0979 15165 0944 0 0075 0 0035 14- -0 P 73) 15142 0743 15142 0690 0 0150 0 0053 14- -1 R 3) 15035 4325 15035 4434 0 0065 -0 0109 14- -1 R 4) 15035 2249 15035 2278 0 0035 -0 0029 14- -1 R 5) 15034 9306 15034 9365 0 0045 -0 0059 14- -1 R 6) 15034 5659 15034 5694 0 0055 -0 0035 14- -1 R 8) 15033 6058 15033 6076 0 0030 -0 0018 14- -1 R 9) 15033 0100 15033 0129 0 0035 -0 0029 14- -1 R 10) 15032 3404 15032 3422 0 0030 -0 0018 14- -1 R 11) 15031 5990 15031 5956 0 0055 0 0034 14- -1 R 13) 15029 8758 15029 8744 0 0045 0 0014 14- -1 R 15) 15027 8505 15027 8489 0 0070 0 0016 14- -1 R 16) 15026 7153 15026 7218 0 0065 -0 0065 14- -1 R 18) 15024 2409 15024 2390 0 0040 0 0019 14- -1 R 21) 15019 9488 15019 9419 0 0060 0 0069 14- -1 R 25) 15013 1483 15013 1401 0 0050 0 0082 14- -1 P 6) 15033 4347 15033 4321 0 0040 0 0026 14- -1 P 10) 15030 5033 15030 5059 0 0035 -0 0026 14- -1 P 12) 15028 5850 15028 5877 0 0060 -0 0027 Appendix 14- -1 p 13) 15027 5147 15027 5147 0 0055 0 0000 14- -1 p 14) 15026 3676 15026 3657 0 0050 0 0019 14- -1 p (17) 15022 4587 15022 4623 0 0040 -0 0036 14- -1 p 19) 15019 4754 15019 4793 0 0050 -0 0039 14- -1 p (21) 15016 1840 15016 1912 0 0040 -0 0072 14- -1 p (22) 15014 4301 15014 4326 0 0040 -0 0025 14- -1 p (23) 15012 5985 15012 5975 0 0055 0 0010 14- -1 p (25) 15008 6996 15008 6975 0 0050 0 0021 14- -1 p (27) 15004 4959 15004 4908 0 0065 0 0051 14- -1 p (29) 14999 9710 14999 9767 0 0065 -0 0057 15- -0 R 5) 15418 4751 15418 4778 0 0100 -0 0027 15- -0 R 6) 15418 0814 15418 0839 0 0065 -0 0025 15- -0 R ( 7) 15417 6095 15417 6104 0 0065 -0 0009 15- -0 R ( 9) 15416 4270 15416 4239 0 0065 0 0031 15- -0 R (10) 15415 7155 15415 7109 0 0100 0 0046 15- -0 R 11) 15414 9216 15414 9181 0 0065 0 0035 15- -0 R 13) 15413 0977 15413 0925 0 0065 0 0052 15- -0 R 14) 15412 0653 15412 0597 0 0065 0 0056 15- -0 R 18) 15407 1306 15407 127 0 0 0150 0 0036 15- -0 R 19) 15405 6938 15405 6931 0 0065 0 0007 15- -0 R 20) 15404 1789 15404 1788 0 0055 0 0001 15- -0 R 23) 15399 1678 15399 1526 0 0150 0 0152 15- -0 R 31) 15382 1943 15382 1885 0 0100 0 0058 15- -0 R 41) 15353 6263 15353 6252 0 0150 0 0011 15- -0 R 48) 15328 6919 15328 6917 0 0065 0 0002 15- -0 R 50) 15320 8058 15320 8101 0 0150 -0 0043 15- -0 R 51) 15316 7424 15316 7419 0 0055 0 0005 15- -0 R 53) 15308 3511 15308 3497 0 0100 0 0014 15- -0 R 57) 15290 5296 15290 5348 0 0065 -0 0052 15- -0 R 59) 15281 1021 15281 1079 0 0100 -0 0058 15- -0 R 60) 15276 2655 15276 2636 0 0100 0 0019 15- -0 R 61) 15271 3199 15271 3316 0 0100 -0 0117 15- -0 R 62) 15266 3162 15266 3117 0 0100 0 0045 15- -0 R 64) 15256 0062 15256 0068 0 0100 -0 0006 15- -0 R 65) 15250 7195 15250 7211 0 0100 -0 0016 15- -0 R 66) 15245 3487 15245 3462 0 0100 0 0025 15- -0 R 69) 15228 6823 15228 6823 0 0100 0 0000 15- -0 R 70) 15222 9509 15222 9467 0 0075 0 0042 15- -0 R 72) 15211 2081 15211 2017 0 0100 0 0064 15- -0 R 73) 15205 1876 15205 1915 0 0100 -0 0039 15- -0 P 6) 15416 9878 15416 9890 0 0045 -0 0012 15- -0 P 7) 15416 3447 15416 3472 0 0055 -0 0025 15- -0 P 8) 15415 6223 15415 6256 0 0065 -0 0033 15- -0 P 9) 15414 8162 15414 8243 0 0100 -0 0081 15- -0 P 11) 15412 9805 15412 9823 0 0100 -0 0018 15- -0 P 14) 15409 6182 15409 6205 0 0100 -0 0023 15- -0 P 15) 15408 3405 15408 3400 0 0100 0 0005 15- -0 P 17) 15405 5343 15405 5390 0 0075 -0 0047 15- -0 P 20) 15400 7315 15400 7363 0 0150 -0 0048 15- -0 P 21) 15398 9786 15398 9748 0 0150 0 0038 15- -0 P 25) 15391 1226 15391 1242 0 0150 -0 0016 15- -0 P 26) 15388 9600 15388 9599 0 0150 0 0001 15- -0 P 30) 15379 4856 15379 4935 0 0150 -0 0079 15- -0 P 58) 15276 2998 15276 3019 0 0065 -0 0021 15- -0 P 60) 15266 4034 15266 3966 0 0065 0 0068 15- -0 P 71) 15205 6221 15205 6318 0 0100 -0 0097 15- -0 P 72) 15199 5633 15199 5709 0 0100 -0 0076 15- -1 R 3) 15095 8700 15095 8796 0 0065 -0 0096 15- -1 R 4) 15095 6455 15095 6478 0 0045 -0 0023 15- -1 R 5) 15095 3362 15095 3369 0 0055 -0 0007 15- -1 R 7) 15094 4801 15094 4779 0 0075 0 0022 15- -1 R 8) 15093 9278 15093 9297 0 0035 -0 0019 Appendix 15- -1 R (10) 15092 5958 15092 5959 0 .0035 -0 0001 15- -1 R (11) 15091 8105 15091 8101 0 .0055 0 0004 15- -1 R (15) 15087 8749 15087 8736 0 0055 0 0013 15- -1 R (16) 15086 6905 15086 6908 0 0035 -0 0003 15- -1 R (18) 15084 0868 15084 0866 0 0050 0 0002 15- -1 R (21) 15079 5841 15079 5823 0 0060 0 0018 15- -1 R (22) 15077 9284 15077 9211 0 0045 0 0073 15- -1 R (24) 15074 3646 15074 3586 0 0040 0 0060 15- -1 R (25) 15072 4621 15072 4570 0 0050 0 0051 15- -1 R (27) 15068 4190 15068 4129 0 0060 0 0061 15- -1 R (28) 15066 2761 15066 2702 0 0070 0 0059 15- -1 R (33) 15054 3427 15054 3444 0 0070 -0 0017 15- -1 R (35) 15048 9988 15049 0062 0 0070 -0 0074 15- -1 R (36) 15046 2139 15046 2148 0 0045 -0 0009 15- -1 R (41) 15031 0241 15031 0296 0 0060 -0 0055 15- -1 R (46) 15013 7917 15013 7814 0 0070 0 0103 15- -1 R (48) 15006 3061 15006 2992 0 0050 0 0069 15- -1 R (49) 15002 4337 15002 4324 0 0055 0 0013 15- -1 R (50) 14998 4764 14998 4815 0 0050 -0 0051 15- -1 R (52) 14990 3292 14990 3265 0 0040 0 0027 15- -1 P ( 8) 15092 4980 15092 4983 0 0035 -0 0003 15- -1 P ( 9) 15091 7011 15091 7027 0 0060 -0 0016 15- -1 P (10) 15090 8260 15090 8281 0 0045 -0 0021 15- -1 P (11) 15089 8752 15089 8743 0 0055 0 0009 15- -1 P (12) 15088 8374 15088 8413 0 0065 -0 0039 15- -1 P (14) 15086 5350 15086 5376 0 0035 -0 0026 15- -1 P 15) 15085 2653 15085 2668 0 0055 -0 0015 15- -1 P (16) 15083 9157 15083 9166 0 0035 -0 0009 15- -1 P 18) 15080 9804 15080 9779 0 0035 0 0025 15- -1 P 19) 15079 3945 15079 3893 0 0055 0 0052 15- -1 P 21) 15075 9725 15075 9731 0 0070 -0 0006 15- -1 P 23) 15072 2398 15072 2379 0 0055 0 0019 15- -1 P 28) 15061 4954 15061 4998 0 0065 -0 0044 15- -1 P 46) 15006 0907 15006 0897 0 0045 0 0010 15- -1 P 49) 14994 2639 14994 2660 0 0065 -0 0021 15- -1 P 50) 14990 1561 14990 1578 0 0045 -0 0017 16- -0 R 3) 15473 0610 15473 0591 0 0065 0 0019 16- -0 R 12) 15467 8115 15467 8111 0 0065 0 0004 16- -0 R 14) 15465 7371 15465 7297 0 0055 0 0074 16- -0 R 16) 15463 3173 15463 3146 0 0055 0 0027 16- -0 R 18) 15460 5691 15460 5653 0 0045 0 0038 16- -0 R 33) 15429 2238 15429 2180 0 0100 0 0058 16- -0 R 35) 15423 6001 15423 5956 0 0100 0 0045 16- -0 R 39) 15411 3212 15411 3205 0 0075 0 0007 16- -0 R 40) 15408 0365 15408 0361 0 0075 0 0004 16- -0 R 42) 15401 2065 15401 2071 0 0065 -0 0006 16- -0 R 46) 15386 5016 15386 5030 0 0150 -0 0014 16- -0 R 48) 15378 6240 15378 6246 0 0065 -0 0006 16- -0 R 49) 15374 5547 15374 5529 0 0050 0 0018 16- -0 R 50) 15370 3927 15370 3927 0 0100 0 0000 16- -0 R 51) 15366 1468 15366 1436 0 0100 0 0032 16- -0 R 53) 15357 3782 15357 3777 0 0075 0 0005 16- -0 R 54) 15352 8625 15352 8605 0 0075 0 0020 16- -0 R 58) 15333 8884 15333 8892 0 0055 -0 0008 16- -0 R 59) 15328 9224 15328 9193 0 0065 0 0031 16- -0 R 61) 15318 7085 15318 7048 0 0065 0 0037 16- -0 R 64) 15302 6927 15302 6912 0 0055 0 0015 16- -0 R 65) 15297 1673 15297 1674 0 0040 -0 0001 16- -0 R 66) 15291 5461 15291. 5500 0 0100 -0 0039 16- -0 R 67) 15285 8346 15285 8386 0 0075 -0 . 0040 16- -0 R 71) 15262 0418 15262. 0450 0 0045 -0 . 0032 16- -0 R 72) 15255. 8579 15255. 8575 0. 0045 0. 0004 Appendix 16- -0 R (73) 15249 5736 15249 5734 0 0045 0 0002 16- -0 R (74) 15243 1945 15243 1922 0 0100 0 0023 16- -0 R (76) 15230 1382 15230 1369 0 0055 0 0013 16- -0 R (77) 15223 4577 15223 4616 0 0050 -0 0039 16- -0 P ( 8) • 15469 5967 15469 5992 0 0055 -0 0025 16- -0 P ( 9) 15468 7655 15468 7715 0 0055 -0 0060 16- -0 P (10) 15467 8628 15467 8608 0 0075 0 0020 16- -0 P (11) 15466 8690 15466 8669 0 0055 0 0021 16- -0 P (14) 15463 3829 15463 3863 0 0065 -0 0034 16- -0 P (16) 15460 6500 15460 6496 0 0075 0 0004 16- -0 P (18) 15457 5806 15457 5792 0 0100 0 0014 16- -0 P (20) 15454 1701 15454 1746 0 0150 -0 0045 16- -0 P (21) 15452 3460 15452 3468 0 0100 -0 0008 16- -0 P (22) 15450 4352 15450 4353 0 0100 -0 0001 16- -0 P (24) 15446 3522 15446 3605 0 0075 -0 0083 16- -0 P (67) 15275 4175 15275 4186 0 0065 -0 0011 16- -0 P (69) 15263 4421 15263 4388 0 0055 0 0033 16- -0 P (71) 15251 0692 15251 0817 0 0100 -0 0125 16- -0 P (74) 15231 8247 15231 8314 0 0100 -0 0067 16- -0 P (76) 15218 5180 15218 5162 0 0055 0 0018 16- -1 R ( 3) 15149 9104 15149 9124 0 0100 -0 0020 16- -1 R ( 4) 15149 6668 15149 6642 0 0100 0 0026 16- -1 R ( 5) 15149 3317 15149 3336 0 0100 -0 0019 16- -1 R (10) 15146 4463 15146 4442 0 0100 0 0021 16- -1 R (12) 15144 7143 15144 7109 0 0100 0 0034 16- -1 R (14) 15142 6541 15142 6469 0 0100 0 0072 16- -1 R (15) 15141 4991 15141 4907 0 0050 0 0084 16- -1 R 17) 15138 9358 15138 9299 0 0035 0 0059 16- -1 R 19) 15136 0426 15136 0370 0 0035 0 0056 16- -1 R 21) 15132 8213 15132 8116 0 0150 0 0097 16- -1 R 22) 15131 0766 15131 0739 0 0040 0 0027 16- -1 R 25) 15125 3679 15125 3599 0 0075 0 0080 16- -1 R 26) 15123 2936 15123 2879 0 0045 0 0057 16- -1 R 27) 15121 1373 15121 1321 0 0100 0 0052 16- -1 R 29) 15116 5683 15116 5683 0 0065 0 0000 16- -1 R 31) 15111 6659 15111 6677 0 0045 -0 0018 16- -1 R 33) 15106 4304 15106 4290 0 0035 0 0014 16- -1 R 34) 15103 6829 15103 6825 0 0075 0 0004 16- -1 R 35) 15100 8508 15100 8511 0 0040 -0 0003 16- -1 R 36) 15097 9341 15097 9346 0 0040 -0 0005 16- -1 R 38) 15091 8477 15091 8456 0 0050 0 0021 16- -1 R 39) 15088 6742 15088 6727 0 0045 0 0015 16- -1 R 41) 15082 0692 15082 0694 0 0045 -0 0002 16- -1 R 42) 15078 6408 15078 6386 0 0050 0 0022 16- -1 R 43) 15075 1196 15075 1214 0 0100 -0 0018. 16- -1 R 44) 15071 5181 15071 517 6 0 0035 0 0005 16- -1 R 45) 15067 8256 15067 8270 0 0035 -0 0014 16- -1 R 46) 15064 0494 15064 0493 0 0035 0 0001 16- -1 R 47) 15060 1830 15060 1845 0 0035 -0 0015 16- -1 R 49) 15052 1980 15052 1921 0 0075 0 0059 16- -1 R 50) 15048 0658 15048 0641 0 0035 0 0017 16- -1 R 51) 15043 8464 15043 8479 0 0040 -0 0015 16- -1 R 53) 15035 1522 15035 1498 0 0035 0 0024 16- -1 R 54) 15030 6700 15030 6674 0 0035 0 0026 16- -1 R 58) 15011 8491 15011 8420 0 0045 0 0071 16- -1 R 59) 15006 9071 15006 9102 0 0050 -0 0031 16- -1 R 60) 15001 8913 15001 8876 0 0035 0 0037 16- -1 R 62) 14991 5691 14991 5686 0 0035 0. 0005 16- -1 P 9) 15145. 6511 15145. 6500 0 0075 0. 0011 16- -1 P 10) 15144. 7440 15144. 7457 0 0050 -0 . 0017 16- -1 P 13) 15141. 5392 15141. 5379 0 0055 0. 0013 16- -1 P 14) 15140. 3012 15140. 3035 0. 0150 -0 . 0023 Appendix 16- -1 p (15) 15138 .9885 15138 . 9864 0 .0045 0 .0021 16- -1 p (16) 15137 .5877 15137 .5867 0 .0040 0 .0010 16- -1 p (17) 15136 .1064 15136 .1042 0 .0045 0 .0022 16- -1 p (18) 15134 .5354 15134 .5388 0 .0050 -0 .0034 16- -1 p (19) 15132 .8899 15132 8906 0 0035 -0 .0007 16- -1 p (27) 15116 .7095 15116 7107 0 0035 -0 .0012 16- -1 p (28) 15114 .3142 15114 .3126 0 0045 0 .0016 16- -1 p (29) 15111 8289 15111 8306 0 0045 -0 0017 16- -1 p (44) 15064 4447 15064 4495 0 0040 -0 .0048 16- -1 p (46) 15056 .6747 15056 6775 0 0040 -0 0028 16- -1 p (48) 15048 5576 15048 5585 0 0035 -0 0009 16- -1 p (49) 15044 3680 15044 3683 0 0050 -0 0003 16- -1 p (50) 15040 0892 15040 0907 0 0045 -0 0015 16- -1 p (52) 15031 2731 15031 2722 0 0035 0 0009 16- -1 p (54) 15022 0957 15022 1011 0 0100 -0 0054 16- -1 p (56) 15012 5718 15012 5753 0 0075 -0 0035 16- -1 p (57) 15007 6729 15007 6787 0 0075 -0 0058 16- -1 p (59) 14997 6171 14997 6166 0 0045 0 0005 17- -0 R (28) 15488 6976 15488 7033 0 0065 -0 0057 17- -0 R (29) 15486 2573 15486 2620 0 0100 -0 0047 17- -0 R (30) 15483 7240 15483 7324 0 0100 . -0 0084 17- -0 R (32) 15478 4069 15478 4079 0 0055 -0 0010 17- -0 R (33) 15475 6045 15475 6126 0 0100 -0 0081 17- -0 R (35) 15469 7506 15469 7553 0 0100 -0 0047 17- -0 R 39) 15456 9692 15456 9684 0 0065 0 0008 17- -0 R 40) 15453 5477 15453 5473 0 0055 0 0004 17- -0 R 43) 15442 7437 15442 7419 0 0055 0 0018 17- -0 R 45) 15435 0844 15435 0846 0 0055 -0 0002 17- -0 R 46) 15431 1168 15431 1191 0 0065 -0 0023 17- -0 R 48) 15422 9167 15422 9133 0 0045 0 0034 17- -0 R 53) 15400 7857 15400 7835 0 0055 0 0022 17- -0 R 57) 15381 3937 15381 3995 0 0100 -0 0058 17- -0 R 60) 15365 8700 15365 8682 0 0055 0 0018 17- -0 R 61) 15360 4986 15360 5001 0 0100 -0 0015 17- -0 R 63) 15349 4751 15349 4752 0 0075 -0 0001 17- -0 R 65) 15338 0641 15338 0627 0 0055 0 0014 17- -0 R 66) 15332 2116 15332 2102 0 0045 0 0014 17- -0 R 68) 15320 2112 15320 2103 0 0050 0 0009 17- -0 R 69) 15314 0654 15314 0621 0 0050 0 0033 17- -0 R 70) 15307 8117 15307 8145 0 0100 -0 0028 17- -0 R 72) 15295 0148 15295 0190 0 0150 -0 0042 17- -0 R 74) 15281 8197 15281 8199 0 0100 -0 0002 17- -0 R 75) 15275 0663 15275 0676 0 0055 -0 0013 17- -0 R 76) 15268 2124 15268 2129 0 0035 -0 0005 17- -0 R 78) 15254 1968 15254 1938 0 0100 0 0030 17- -0 P 28) 15484 3170 15484 3109 0 0100 0 0061 17- -0 P 34) 15467 4295 15467 4332 0 0100 -0 0037 17- -0 P 40) 15447 3590 15447 3622 0 0100 -0 0032 17- -0 P 42) 15439 9599 15439 9559 0. 0075 0 0040 17- -0 P 44) 15432 1866 15432 1891 0. 0075 -0 . 0025 17- -0 P 53) 15392 7285 15392. 7289 0. 0150 -0 . 0004 17- -0 P 54) 15387. 8815 15387. 8840 0. 0075 -0 . 0025 17- -0 P 73) 15277 7549 15277. 7660 0. 0100 -0 . 0111 17- -0 P 75) 15264. 1191 15264. 1191 0. 0100 0. 0000 17- -0 P 77) 15250. 0596 15250. 0667 0. 0100 -0 . 0071 17- -1 R 3) 15198. 2167 15198. 2217 0. 0045 -0 . 0050 17- -1 R 4) 15197. 9593 15197 . 9572 0. 0065 0. 0021 17- -1 R 5) 15197. 6054 15197. 6072 0. 0045 -0 . 0018 17- -1 R 7) 15196. 6595 15196. 6502 0. 0100 0. 0093 17- -1 R( 8) 15196. 0423 15196. 0431 0. 0045 -0 . 0008 17- -1 R( 10) 15194. 5704 15194. 5719 0. 0035 -0 . 0015 17- -1 R( 12) 15192. 7541 15192. 7573 0. 0065 -0 . 0032 Appendix 177 17 — 1 R (13) 15191 .7216 15191 .7212 0 .0040 0 .0004 | 17 — 1 R (14) 15190 .5984 15190 .5990 0 .0035 -0 .0006 | 17 — 1 R (15) 15189 .3864 15189 .3909 0 .0075 -0 .0045 | 17 — 1 R (16) 15188 .0960 15188 .0965 0 .0035 -0 .0005 | 17 — 1 R (17) 15186 .7224 15186 .7160 0 .0055 0 .0064 | 17 — 1 R (18) 15185 .2497 15185 .2492 0 .0035 0 .0005 | 17 — 1 R (20) 15182 .0569 15182 .0564 0 .0035 0 .0005 | 17 — 1 R (21) 15180 .3285 15180 .3302 0 .0035 -0 .0017 | 17 — 1 R (23) 15176 . 6223 15176 . 6178 0 0045 0 .0045 | 17 — 1 R (24) 15174 . 6330 15174 . 6313 0 0045 0 .0017 | 17 — 1 R (27) 15168 .1520 15168 .1495 0 0050 0 .0025 | 17 — 1 R (28) 15165 .8126 15165 .8144 0 0045 -0 0018 | 17 — 1 R (29) 15163 .3826 15163 3918 0 0045 -0 0092****| 17 — 1 R (31) 15158 .2823 15158 2836 0 0055 -0 0013 | 17 — 1 R (33) 15152 8187 15152 8236 0 0055 -0 0049 | 17 — 1 R (35) 15147 0065 15147 0108 0 0040 -0 0043 * | 17 — 1 R (36) 15143 9675 15143 9715 0 0035 -0 0040 * | 17 — 1 R (37) 15140 8420 15140 8436 0 0035 -0 0016 | 17 — 1 R (38) 15137 6258 15137 6266 0 0035 -0 0008 | 17 — 1 R (39) 15134 3181 15134 3206 0 0035 -0 0025 | 17 — 1 R (40) 15130 9253 15130 9252 0 0035 0 0001 | 17--1 R (42) 15123 8626 15123 8657 0 0075 -0 0031 | 17 — 1 R (43) 15120 2016 15120 2011 0 0100 0 0005 | 17 — 1 R (44) 15116 4462 15116 4463 0 0035 -0 0001 | 17 — 1 R 46) 15108 6632 15108 6654 0 0035 -0 0022 | 17 — 1 R 50) 15092 0122 15092 0108 0 0035 0 0014 | 17 — 1 R 52) 15083 1301 15083 1331 0 0075 -0 0030 | 17 — 1 R 54) 15073 8852 15073 8854 0 0035 -0 0002 | 17 — 1 R 57) 15059 3136 15059 3149 0 0035 -0 0013 | 17 — 1 R 59) 15049 1276 15049 1315 0 0040 -0 0039 | 17 — 1 R 60) 15043 8987 15043 8978 0 0035 0 0009 | 17 — 1 R 61) 15038 5688 15038 5691 0 0035 -0 0003 | 17 — 1 R 64) 15022 0099 15022 0087 0 0045 0 0012 | 17 — 1 R 65) 15016 2988 15016 2958 0 0075 0 0030 | 17 — 1 R 66) 15010 4881 15010 4859 0 0075 0 0022 | 17 — 1 R 67) 15004 5814 15004 5785 0 0045 0 0029 | 17 — 1 R 68) 14998 5755 14998 5732 0 0045 0 0023 | 17 — 1 R 69) 14992 4613 14992 4696 0 0055 -0 0083 * * * | 17 — 1 P 5) 15196 7498 15196 7525 0 0035 -0 0027 | 17 — 1 P 6) 15196 1616 15196 1615 0 0045 0 0001 | 17 — 1 P 7) 15195 4812 15195 4850 0 0035 -0 0038 * | 17 — 1 P 8) 15194 7222 15194 7228 0 0035 -0 0006 | 17 — 1 P 9) 15193 8753 15193 8750 0 0045 0 0003 | 17 — 1 P 10) 15192 9426 15192 9415 0 0050 0 0011 | 17 — 1 P 12) 15190 8181 15190 8173 0 0035 0 0008 | 17 — 1 P 13) 15189 6145 15189 6266 0. 0100 -0 0121 * * | 17 — 1 P 15) 15186 9812 15186 9874 0. 0055 -0 0062 * | 17 — 1 P 16) 15185 5382 15185. 5389 0. 0035 -0 . 0007 | 17 — 1 P 17) 15184. 0026 15184. 0043 0. 0035 -0 . 0017 | 17 — 1 P 19) 15180. 6833 15180. 6768 0. 0045 0. 0065 | 17--1 P 23) 15172. 9898 15172. 9858 0. 0040 0. 0040 | 17 — 1 P 24) 15170. 8486 15170. 8467 0. 0065 0. 0019 | 17 — 1 P 27) 15163. 9092 15163. 9086 0. 0050 0. 0006 | 17 — 1 P 29) 15158. 8446 15158. 8481 0. 0075 -0 . 0035 | 17 — 1 P 30) 15156. 1820 15156. 1869 0. 0045 - 0 . 0049 * | 17 — 1 P ( 31) 15153. 4263 15153. 4382 0. 0055 -0 . 0119****| 17 — 1 P ( 33) 15147 . 6684 15147 . 6780 0. 0045 -0 . 0096****| 17--1 P ( 34) 15144. 6634 15144. 6661 0. 0055 -0 . 0027 | 17--1 P ( 36) 15138. 3748 15138. 3781 0. 0055 -0 . 0033 | 17 — 1 P ( 37) 15135. 0916 15135. 1016 0. 0040 -0 . 0100****| 17 — 1 P ( 38) 15131. 7316 15131. 7365 0. 0100 -0 . 0049 | 17 — 1 P ( 40) 15124. 7397 15124. 7401 0. 0035 -0 . 0004 | Appendix 17- -1 p (42) 15117 .3829 15117 .3874 0 .0035 -0 .0045 17- -1 p (43) 15113 .5792 15113 .5769 0 .0075 0 .0023 17- -1 p (45) 15105 . 6798 15105 . 6865 0 .0045 -0 .0067 17- -1 p (47) 15097 .4312 15097 . 4357 0 .0045 -0 .0045 17- -1 p (49) 15088 8123 15088 .8225 0 .0075 -0 .0102 17- -1 p (50) 15084 3794 15084 .3794 0 .0035 0 .0000 17- -1 p (51) 15079 8418 15079 .8450 0 .0045 -0 .0032 17- -1 p (53) 15070 5019 15070 .5010 0 .0055 0 0009 17- -1 p (57) 15050 7046 15050 .7050 0 .0040 -0 0004 17- -1 p (58) 15045 5257 15045 .5234 0 .0055 0 0023 17- -1 p (59) 15040 2456 15040 .2482 0 .0055 -0 0026 17- -1 p (60) 15034 8795 15034 .8789 0 .0035 0 0006 17- -1 p (61) 15029 4176 15029 .4153 0 .0045 0 0023 17- -1 p (64) 15012 4554 15012 . 4551 0 0045 0 0003 17- -1 p (65) 15006 6149 15006 6107 0 0100 0 0042 17- -1 p (67) 14994 6328 14994 6329 0 0050 -0 0001 17- -1 p (68) 14988 4944 14988 4988 0 0100 -0 0044 18- -0 R (42) 15486 7967 15486 7974 0 0065 -0 0007 18- -0 R (43) 15482 9606 15482 9634 0 0035 -0 0028 18- -0 R (44) 15479 0362 15479 0352 0 0055 0 0010 18- -0 R (45) 15475 0102 15475 0124 0 0055 -0 0022 18- -0 R (47) 15466 6877 15466 6825 0 0065 0 0052 18- -0 R (49) 15457 9726 15457 9715 0 0065 0 0011 18- -0 R (50) 15453 4740 15453 4725 0 0045 0 0015 18- -0 R (51) 15448 8827 15448 8773 0 0055 0 0054 18- -0 R (59) 15408 6163 15408 6189 0 0055 -0 0026 18- -0 R (60) 15403 1451 15403 1444 0 0055 0 0007 18- -0 R (61) 15397 5697 15397 5704 0 0100 -0 0007 18- -0 R (62) 15391 8984 15391 8965 0 0055 0 0019 18- -0 R 64) 15380 2487 15380 2475 0 0100 0 0012 18- -0 R 72) 15329 5726 15329 5691 0 0075 0 0035 18- -0 R 74) 15315 8596 15315 8594 0 0055 0 0002 18- -0 R 75) 15308 8423 15308 8456 0 0065 -0 0033 18- -0 R 77) 15294 4960 15294 4971 0 0065 -0 0011 18- -0 P 40) 15488 2551 15488 2583 0 0075 -0 0032 18- -0 P 44) 15472 5496 15472 5523 0 0045 -0 0027 18- -0 P 45) 15468 3850 15468 3915 0 0075 -0 0065 18- -0 P 48) 15455 3386 15455 3428 0 0065 -0 0042 18- -0 P 55) 15421 5600 15421 5600 0 0065 0 0000 18- -0 P 57) 15411 0525 15411 0397 0 0150 0 0128 18- -0 P 59) 15400 1273 15400 1288 0 0100 -0 0015 18- -0 P 65) 15365 0263 15365 0248 0 0065 0 0015 18- -0 P 68) 15346 1187 15346 1196 0 0065 -0 0009 18- -0 P 74) 15305 5432 15305 5431 0 0065 0 0001 18- -0 P 77) 15283 8414 15283 8447 0 0100 -0 0033 18- -0 P 79) 15268 8379 15268 8467 0 0075 -0 0088 18- -0 P 80) 15261 1885 15261 1861 0 0055 0 0024 18- -0 P 81) 15253 4113 15253 4170 0 0100 -0 0057 18- -1 R 3) 15241 5524 15241 5556 0 0075 -0 0032 18- -1 R 4) 15241 2774 15241 2754 0 0050 0. 0020 18- -1 R 6) 15240. 4526 15240 4490 0 0045 0. 0036 18- -1 R 7) 15239 8966 15239 9026 0 0075 -0 . 0060 18- -1 R 10) 15237 7310 15237 7303 0 0040 0. 0007 18- -1 R 11) 15236. 8290 15236 8283 0 0075 0. 0007 18- -1 R 12) 15235. 8335 15235 8373 0 0055 -0 . 0038 18- -1 R 13) 15234. 7608 15234 7572 0. 0085 0. 0036 18- -1 R 14) 15233. 5898 15233. 5880 0. 0035 0. 0018 18- -1 R 15) 15232. 3298 15232. 3295 0. 0050 0. 0003 18- -1 R 16) 15230. 9841 15230. 9818 0. 0040 0. 0023 18- -1 R 17) 15229. 5477 15229. 5447 0. 0045 0. 0030 18- -1 R 18) 15228. 0208 15228. 0181 0. 0040 0. 0027 18- -1 R 19) 15226. 4062 15226. 4020 0. 0040 0. 0042 Appendix 179 18- -1 R (20) 15224 .7013 15224 . 6962 0 .0040 0 0051 18- -1 R (21) 15222 .9101 15222 .9007 0 0045 0 0094 18- -1 R (24) 15216 .9830 15216 .9745 0 0045 0 0085 18- -1 R (26) 15212 .5749 15212 .5730 0 0035 0 0019 18- -1 R (27) 15210 .2326 15210 .2366 0 0035 -0 0040 * 1 18- -1 R (28) 15207 .7991 15207 .8096 0 0060 -0 0105 * * * 1 18- -1 R (29) 15205 2912 15205 2918 0 0035 -0 0006 18- -1 R (30) 15202 6776 15202 6832 0 0035 -0 0056 • * * | 18- -1 R (34) 15191 3365 15191 3367 0 0030 -0 0002 18- -1 R (36) 15185 1187 15185 1139 0 0040 0 0048 18- -1 R (37) 15181 8684 15181 8645 0 0100 0 0039 18- -1 R (38) 15178 5173 15178 5228 0 0065 -0 0055 18- -1 R (39) 15175 0830 15175 0886 0 0045 -0 0056 18- -1 R (40) 15171 5570 15171 5617 0 0040 -0 0047 * 1 18- -1 R (41) 15167 9393 15167 9419 0 0055 -0 0026 18- -1 R (42) 15164 2259 15164 2289 0 0035 -0 0030 18- -1 R (44) 15156 5202 15156 5228 0 0045 -0 0026 18- -1 R (45) 15152 5272 15152 5291 0 0040 -0 0019 18- -1 R (48) 15139 9816 15139 9824 0 0035 -0 0008 18- -1 R (49) 15135 6087 15135 6107 0 0045 -0 0020 18- -1 R (50) 15131 1453 15131 1439 0 0055 0 0014 18- -1 R (52) 15121 9232 15121 9237 0 0030 -0 0005 18- -1 R (53) 15117 1681 15117 1696 0 0040 -0 0015 18- -1 R (55) 15107 3708 15107 3721 0 0045 -0 0013 18- -1 R 56) 15102 3283 15102 3279 0 0035 0 0004 18- -1 R (57) 15097 1821 15097 1864 0 0045 -0 0043 18- -1 R (58) 15091 9483 15091 9472 0 0065 0 0011 18- -1 R (60) 15081 1720 15081 1741 0 0065 -0 0021 18- -1 R 62) 15070 0047 15070 0056 0 0045 -0 0009 18- -1 R 64) 15058 4388 15058 4386 0 0035 0 0002 18- -1 R 65) 15052 5047 15052 5045 0 0035 0 0002 18- -1 R 69) 15027 7451 15027 7546 0 0100 -0 0095 18- -1 R 70) 15021 3114 15021 3113 0 0045 0 0001 18- -1 R 71) 15014 7627 15014 7647 0 0055 -0 0020 18- -1 R 72) 15008 1115 15008 1142 0 0045 -0 0027 18- -1 R 73) 15001 3545 15001 3593 0 0050 -0 0048 18- -1 R 74) 14994 5042 14994 4995 • 0 0050 0 0047 18- -1 R 75) 14987 5335 14987 5342 0 0055 -0 0007 18- -1 P 5) 15240 0830 15240 0863 0 0055 -0 0033 18- -1 P 6) 15239 4796 15239 4797 0 0045 -0 0001 18- -1 P 8) 15237 9985 15238 0003 0 0040 -0 0018 18- -1 P 9) 15237 1259 15237 1274 0 0040 -0 0015 18- -1 P 12) 15233 9756 15233 9757 0 0045 -0 0001 18- -1 P 13) 15232 7494 15232 7473 0 0050 0 0021 18- -1 P 14) 15231 4282 15231 4299 0 0045 -0 0017 18- -1 P 15) 15230 0233 15230 0234 0 0050 -o 0001 18- -1 P 17) 15226 9414 15226 9430 0 0100 -0 0016 18- -1 P 18) 15225 2665 15225 2689 0 0035 -0 0024 18- -1 P 19) 15223 5065 15223 5054 0 0050 0 0011 18- -1 P 20) 15221 6515 15221 6525 0 0045 -0 0010 18- -1 P 21) 15219 7083 15219 7101 0 0035 -0 0018 18- -1 P 22) 15217 6753 15217 6780 0 0040 -0 0027 18- -1 P 23) 15215 5623 15215 5562 0 0040 0. 0061 18- -1 P 25) 15211 0434 15211 0431 0 0035 0. 0003 18- -1 P 29) 15200 9228 15200 9351 0. 0045 -0 . 0123****| 18- -1 P 32) 15192 3943 15192 4036 0. 0045 -0 . 0093****| 18- -1 P 38) 15172 8721 15172 8789 0. 0045 -0 . 0068 * * * 1 18- -1 P 39) 15169 2963 15169 3037 0. 0045 -0 . 0074 * * * j 18- -1 P 41) 15161 8685 15161. 8764 0. 0040 -0 . 0079 * * * t 18- -1 P 48) 15132 9524 15132. 9504 0. 0045 0. 0020 18- -1 P 49) 15128 4394 15128. 4429 0. 0055 -0 . 0035 18- -1 P 51) 15119. 1424 15119. 1440 0. 0045 -0 . 0016 Appendix 18 — 1 P (52) 15114 3532 15114 3520 0 0030 0 0012 18 — 1 P (54) 15104 4800 15104 4816 0 0035 -0 0016 18 — 1 P (55) 15099 4001 15099 4025 0 0055 -0 0024 18 — 1 P (58) 15083 5843 15083 5860 0 0075 -0 0017 18 — 1 P (60) 15072 5587 15072 5557 0 0075 0 0030 18 — 1 P (63) 15055 2812 15055 2741 0 0100 0 0071 18 — 1 P (66) 15037 0983 15037 1000 0 0065 -0 0017 18 — 1 P (67) 15030 8405 15030 8416 0 0045 -0 0011 18 — 1 P (68) 15024 4902 15024 4825 0 0050 0 0077 18 — 1 P (69) 15018 0200 15018 0220 0 0050 -0 0020 18 — 1 P (70) 15011 4609 15011 4597 0 0055 0 0012 18 — 1 P (71) 15004 8028 15004 7952 0 0060 0 0076 18 — 1 P (72) 14998 0205 14998 0280 0 0055 -0 0075 ** 18 — 1 P (73) 14991 1579 14991 1574 0 0065 0 0005 19 — 0 R (50) 15488 5147 15488 5171 0 0075 -0 0024 19 — 0 R (51) 15483 7624 15483 7598 0 0040 0 0026 19 — 0 R (55) 15463 7328 15463 7292 0 0075 0 0036 19 — 0 R (56) 15458 4741 15458 4696 0 0055 0 0045 19 — 0 R (57) 15453 1164 15453 1084 0 0055 0 0080 19 — 0 R (58) 15447 6489 15447 6452 0 0045 0 0037 19 — 0 R (59) 15442 0797 15442 0798 0 0045 -0 0001 19 — 0 R (60) 15436 4150 15436 4117 0 0045 0 0033 19 — 0 R (61) 15430 6434 15430 6405 0 0045 0 0029 19 — 0 R (64) 15412 7060 15412 7039 0 0045 0 0021 19—0 R (67) 15393 8227 15393 8234 0 0065 -0 0007 19—0 R 69) 15380 7059 15380 7054 0 0100 0 0005 19 — 0 R 71) 15367 1563 15367 1577 0 0100 -0 0014 19 — 0 R 72) 15360 2185 15360 2213 0 0150 -0 0028 19 — 0 R 73) 15353 1720 15353 1758 0 0075 -0 0038 19 — 0 R 74) 15346 0161 15346 0206 0 0075 -0 0045 19 — 0 R 76) 15331 3780 15331 3783 0 0065 -0 0003 19 — 0 P 49) 15486 3227 15486 3146 0 0075 0 0081 19--0 P 50) 15481 5298 15481 5282 0 0040 0 0016 19--0 P 51) 15476 6497 15476 6431 0 0040 0 0066 19 — 0 P 52) 15471 6621 15471 6588 0 0045 0 0033 19 — 0 P 53) 15466 5708 15466 5750 0 0045 -0 0042 19 — 0 P 55) 15456 1039 15456 1080 0 0045 -0 0041 19 — 0 P 56) 15450 7279 15450 7240 0 0055 0 0039 19 — 0 P 57) 15445 2360 15445 2393 0 0055 -0 0033 19 — 0 P 68) 15378 1519 15378 1483 0 0100 0 0036 19 — 0 P 75) 15328 8210 15328 8248 0 0100 -0 0038 19 — 1 R 4) 15280 3255 15280 3254 0 0065 0 0001 19 — 1 R 6) 15279 4612 15279 4599 0 0050 0 0013 19 — 1 R 7) 15278 8891 15278 8895 0 0045 -0 0004. 19—1 R 8) 15278 2291 15278 2273 0 0035 0 0018 19 — 1 R 10) 15276 6279 15276 6271 0 0035 0 0008 19 — 1 R 11) 15275 6904 15275 6891 0 0045 0 0013 19 — 1 R 13) 15273 5395 15273 5369 0 0040 0 0026 19 — 1 R 15) 15271 0193 15271 0160 0 0040 0 0033 19 — 1 R 16) 15269 6201 15269 6171 0 0035 0 0030 19 — 1 R 17) 15268 1300 15268 1258 0 0040 0 0042 19 — 1 R 18) 15266 5438 15266 5419 0 0035 0 0019 19 — 1 R 20) 15263 0972 15263 0964 0 0035 0 0008 19 — 1 R 21) 15261 2414 15261 2345 0 0050 0 0069 19 — 1 R 22) 15259 2768 15259 2797 0 0045 -0 0029 19—1 R 23) 15257 2369 15257 2318 0 0065 0 0051 19—1 R 27) 15248 1012 15248 1076 0 0035 -0 0064 *** 19 — 1 R 28) 15245 5930 15245 5927 0 0035 0 0003 19 — 1 R 30) 15240 2729 15240 2811 0 0035 -0 0082**** 19 — 1 R 31) 15237 4786 15237 4842 0 0035 -0 0056 *** 19 — 1 R 33) 15231 6038 15231 6071 0 0065 -0 0033 19 — 1 R 35) 15225 3518 15225 3513 0. 0065 0. 0005 Appendix 19- -1 R (37) 15218 7172 15218 7153 0 .0065 0 .0019 19- -1 R (38) 15215 2549 15215 2541 0 .0030 0 0008 19- -1 R (40) 15208 0478 15208 0446 0 .0100 0 .0032 19- -1 R (41) 15204 2953 15204 2957 0 .0040 -0 0004 19- -1 R (42) 15200 4535 15200 4505 0 .0055 0 0030 19- -1 R (43) 15196 4995 15196 5086 0 .0100 -0 0091 19- -1 R (44) 15192 4725 15192 4699 0 .0035 0 0026 19- -1 R (46) 15184 1018 15184 1009 0 .0075 0 0009 19- -1 R (48) 15175 3431 15175 3412 0 .0035 0 0019 19- -1 R (49) 15170 8121 15170 8141 0 .0040 -0 0020 19- -1 R (50) 15166 1863 15166 1886 0 0065 -0 0023 19- -1 R (52) 15156 6433 15156 6406 0 0055 0 0027 19- -1 R (53) 15151 7171 15151 7176 0 0040 -0 0005 19- -1 R (54) 15146 6960 15146 6947 0 0100 0 0013 19- -1 R (55) 15141 5829 15141 5717 0 0100 0 0112 19- -1 R (56) 15136 3492 15136 3482 0 0030 0 0010 19- -1 R (57) 15131 0272 15131 0237 0 0035 0 0035 19- -1 R (59) 15120 0728 15120 0707 0 0035 0 0021 19- -1 R (61) 15108 7090 15108 7095 0 0040 -0 0005 19- -1 R (62) 15102 8768 15102 8748 0 0035 0 0020 19- -1 R (63) 15096 9398 15096 9368 0 0065 0 0030 19- -1 R (65) 15084 7549 15084 7490 0 0075 0 0059 19- -1 R (67) 15072 1417 15072 1423 0 0035 -0 0006 19- -1 R (68) 15065 6851 15065 6807 0 0055 0 0044 19- -1 R (71) 15045 6537 15045 6563 0 0035 -0 0026 19- -1 R (73) 15031 7659 15031 7681 0 0035 -0 0022 19- -1 R (75) 15017 4434 15017 4436 0 0050 -0 0002 19- -1 R (77) 15002 6823 15002 6775 0 0050 0 0048 19- -1 R (78) 14995 1264 14995 1272 0 0050 -0 0008 19- -1 P 5) 15279 1459 15279 1513 0 0050 -0 0054 19- -1 P 6) 15278 5290 15278 5297 0 0035 -0 0007 19- -1 P 7) 15277 8124 15277 8163 0 0035 -0 0039 * 19- -1 P 8) 15277 0130 15277 0112 0 0075 0 0018 19- -1 P 10) 15275 1245 15275 1257 0 0030 -0 0012 19- -1 P 11) 15274 0448 15274 0451 0 0040 -0 0003 19- -1 P 14) 15270 2509 15270 2517 0 0030 -0 0008 19- -1 P 15) 15268 8048 15268 8031 0 0035 0 0017 19- -1 P 17) 15265 6309 15265 6294 0 0035 0 0015 19- -1 P 18) 15263 9036 15263 9042 0 0035 -0 0006 19- -1 P 20) 15260 1759 15260 1764 0 0075 -0 0005 19- -1 P 21) 15258 1761 15258 1736 0 0045 0 0025 19- -1 P 22) 15256 0772 15256 0782 0 0035 -0 0010 19- -1 P 23) 15253 8897 15253 8901 0 0045 -0 0004 19- -1 P 24) 15251 6099 15251 6090 0 0045 0 0009 19- -1 P 25) 15249 2345 15249 2349 0 0060 -0 0004 19- -1 P 27) 15244 2013 15244 2073 0 0035 -0 0060 * * * 19- -1 P 30) 15235 9605 15235 9651 0 0035 -0 0046 * * 19- -1 P 31) 15233 0258 15233 0303 0 0045 -0 0045 19- -1 P 33) 15226 8718 15226 8786 0 0065 -0 0068 * 19- -1 P 34) 15223 6520 15223 6614 0 0045 -0 0094**** 19- -1 P 35) 15220 3424 15220 3497 0 0045 -0 0073 * • * 19- -1 P 37) 15213 4388 15213 4421 0 0040 -0 0033 19- -1 P 39) 15206 1425 15206. 1545 0 0075 -0 . 0120 * * * 19- -1 P 40) 15202. 3657 15202. 3676 0 0035 -0 0019 19- -1 P 42) 15194. 5068 15194. 5067 0 0035 0. 0001 19- -1 P 43) 15190. 4228 15190. 4322 0 0035 -0 . 0094**** 19- -1 P 44) 15186. 2606 15186. 2614 0 0065 -0 . 0008 19- -1 P 45) 15181. 9782 15181. 9941 0. 0100 -0 . 0159 *** 19- -1 P 46) 15177. 6369 15177. 6299 0. 0100 0. 0070 19- -1 P 47) 15173. 1644 15173. 1686 0. 0045 -0 . 0042 19- -1 P 49) 15163. 9548 15163. 9538 0. 0065 0. 0010 19- -1 P 50) 15159. 2008 15159. 1997 0. 0035 0. 0011 Appendix 182 19 — 1 P (51) 15154 3453 15154 3474 0 0040 -0 0021 19 — 1 P (54) 15139 2011 15139 1985 0 0035 0 0026 | 19 — 1 P (55) 15133 9458 15133 9504 0 0045 -0 0046 * 1 19 — 1 P (56) 15128 6036. 15128 6026 0 0035 0 0010 | 19 — 1 P (57) 15123 1526 15123 1546 0 0045 -0 0020 | 19 — 1 P (58) 15117 6050 15117 6063 0 0035 -0 0013 | 19 — 1 P (59) 15111 9615 15111 9570 0 0065 0 0045 | 19 — 1 P (60) 15106 1990 15106 2066 0 0075 -0 0076 * 1 19 — 1 P (61) 15100 3503 15100 3546 0 0100 -0 0043 | 19—1 P (62) 15094 3957 15094 4006 0 0065 -0 0049 | 19—1 P (63) 15088 3458 15088 3442 0 0040 0 0016 | 19—1 P (64) 15082 1872 15082 1850 0 0035 0 0022 | 19—1 P (66) 15069 5526 15069 5564 0 0075 -0 0038 | 19 — 1 P (69) 15049 8306 15049 8311 0 0035 -0 0005 | 19--1 P (70) 15043 0446 15043 0453 0 0085 -0 0007 | 19 — 1 P (71) 15036 1501 15036 1534 0 0035 -0 0033 19 — 1 P (72) 15029 1545 15029 1549 0 0035 -0 0004 | 20 — 0 R (58) 15477 8337 15477 8290 0 0030 0 0047 i * * * 20 — 0 R (60) 15466 2288 15466 2245 0 0055 0 0043 | 20 — 0 R (61) 15460 2616 15460 2625 0 0065 -0 0009 | 20 — 0 R (62) 15454 2001 15454 1932 0 0065 0 0069 1 * 20 — 0 R (63) 15448 0151 15448 0163 0 0065 -0 0012 | 20 — 0 P 56) 15481 6085 15481 6075 0 0050 0 0010 | 20 — 0 P 61) 15452 2772 15452 2857 0 0075 -0 0085 * 1 20 — 0 P 63) 15439 8126 15439 8165 0 0075 -0 0039 | 20 — 0 P 65) 15426 9091 15426 9188 0 0075 -0 0097 20 — 0 P 66) 15420 3043 15420 3081 0 0075 -0 0038 | 20 — 1 R 3) 15316 0055 15316 0121 0 0065 -0 0066 * 1 20 — 1 R 4) 15315 7111 15315 7026 0 0035 0 0085 1 * * * 20 — 1 R 5) 15315 2990 15315 2985 0 0045 0 0005 | 20 — 1 R 6) 15314 8011 15314 7997 0 0035 0 0014 | 20 — 1 R 7) 15314 2087 15314 2063 0 0045 0 0024 | 20 — 1 R 8) 15313 5169 15313 5182 0 0035 -0 0013 | 20 — 1 R 9) 15312 7378 15312 7353 0 0040 0 0025 | 20 — 1 R 11) 15310 8870 15310 8850 0 0040 0 0020 | 20 — 1 R 14) 15307 3984 15307 3973 0 0035 0 0011 | 20 — 1 R 15) 15306 0535 15306 0445 0 0055 0 0090 1 * * * 20 — 1 R 16) 15304 6000 15304 5966 0 0035 0 0034 | 20 — 1 R 17) 15303 0594 15303 0533 0 0045 0 0061 20 — 1 R 18) 15301 4157 15301 4145 0 0035 0 0012 | 20 — 1 R 19) 15299 6849 15299 6803 0 0040 0 0046 1 *' 20 — 1 R 20) 15297 8575 15297 8504 0 0040 0 0071 1 * * * 20 — 1 R 21) 15295 9314 15295 9248 0 0045 0 0066 1 * * 20 — 1 R 22) 15293 9076 15293 9033 0 0050 0 0043 | 20 — 1 R 23) 15291 7862 15291 7859 0 0065 0 0003 | 20 — 1 R 25) 15287 2656 15287 2626 0 0075 0 0030 | 20 — 1 R 27) 15282 3524 15282 3538 0 0045 -0 0014 | 20 — 1 R 30) 15274 2629 15274 2651 0 0035 -0 0022 | 20 — 1 R 31) 15271 3699 15271 3747 0 0035 -0 0048 20 — 1 R 32) 15268 3856 15268 3870 0 0035 -0 0014 | 20 — 1 R 33) 15265 2978 15265 3018 0 0035 -0 0040 * 1 20 — 1 R 35) 15258 8337 15258 8380 0 0035 -0 0043 * * 1 20 — 1 R 36) 15255 4586 15255 4590 0 0035 -0 0004 | 20 — 1 R 37) 15251 9803 15251 9817 0 0040 -0 0014 | 20 — 1 R 39) 15244 7306 15244 7311 0 0040 -0 0005 | 20 — 1 R 40) 15240 9587 15240 9574 0 0100 0 0013 | 20 — 1 R 41) 15237 0819 15237 0845 0 0075 -0 0026 | 20 — 1 R 42) 15233 1128 15233 1120 0 0045 0 0008 20 — 1 R 43) 15229 0442 15229 0398 0 0055 0 0044 | 20 — 1 R 44) 15224 8684 15224 8675 0 0040 0 0009 | 20 — 1 R 45) 15220 5949 15220 5950 0 0035 -0 0001 | 20 — 1 R 46) 15216 2245 15216 2218 0 0035 0 0027 | Appendix 20 — 1 R (47) 15211 .7466 15211 .7478 0 0065 -0 .0012 | 20 — 1 R (48) 15207 .1687 15207 .1725 0 0065 -0 .0038 | 20 — 1 R (50) 15197 .7201 15197 .7173 0 0040 0 .0028 | 20 — 1 R (51) 15192 8382 15192 .8366 0 0035 0 .0016 | 20 — 1 R (53) 15182 7595 15182 7674 0 0100 -0 .0079 | 20 — 1 R (54) 15177 5805 15177 5782 0 0035 0 .0023 | 20 — 1 R (55) 15172 .2876 15172 2854 0 0055 0 .0022 | 20 — 1 R (56) 15166 8910 15166 8887 0 0055 0 0023 | 20 — 1 R (57) 15161 3876 15161 3876 0 0035 0 0000 | 20 — 1 R (58) 15155 7842 15155 7818 0 0035 0 0024 | 20 — 1 R (60) 15144 2562 15144 2542 0 0050 0 0020 | 20 — 1 R (62) 15132 3052 15132 3023 0 0035 0 0029 | 20 — 1 R (64) 15119 9254 15119 9224 0 0035 0 0030 | 20 — 1 R (69) 15087 0697 15087 0732 0 0085 -0 0035 | 20 — 1 R (70) 15080 1750 15080 1733 0 0075 0 0017 | 20 — 1 R (71) 15073 1533 15073 1619 0 0055 -0 0086 * * * | 20 — 1 R (72) 15066 0364 15066 0385 0 0065 -0 0021 | 20 — 1 R (73) 15058 7966 15058 8024 0 0035 -0 0058 * * * | 20 — 1 R (74) 15051 4519 15051 4529 0 0065 -0 0010 | 20 — 1 R (77) 15028 7088 15028 7167 0 0050 -0 0079 * * * | 20 — 1 R 79) 15012 9770 15012 9784 0 0050 -0 0014 | 20 — 1 P ( 5) 15314 5386 15314 5428 0 0065 -0 0042 | 20 — 1 P ( 6) 15313 9067 15313 9068 0 0045 -0 0001 | 20 — 1 P 7) 15313 1741 15313 1763 0 0045 -0 0022 | 20 — 1 P 8) 15312 3483 15312 3510 0 0045 -0 0027 | 20 — 1 P 9) 15311 4301 15311 4311 0 0045 -0 0010 | 20 — 1 P 10) 15310 4156 15310 4165 0 0035 -0 0009 | 20 — 1 P 11) 15309 3079 15309 3072 0 0040 0 0007 | 20 — 1 P 12) 15308 1024 15308 1030 0 0035 -0 0006 | 20 — 1 P 13) 15306 8040 15306 8040 0 0040 0 0000 | 20 — 1 P 14) 15305 4117 15305 4100 0 0035 0 0017 | 20 — 1 P 15) 15303 9225 15303 9211 0 0035 0 0014 | 20 — 1 P 16) 15302 3378 15302 3371 0 0035 0 0007 | 20 — 1 P 17) 15300 6590 15300 6580 0 0040 0 0010 | 20 — 1 P 18) 15298 8842 15298 8836 0 0035 0 0006 | 20 — 1 P 21) 15292 9933 15292 9884 0 0045 0 0049 | 20 — 1 P 22) 15290 8310 15290 8322 0 0045 -0 0012 | 20 — 1 P 23) 15288 5816 15288 5804 0 0045 0 0012 | 20 — 1 P 24) 15286 2346 15286 2327 0 0045 0 0019 | 20 — 1 P 25) 15283 7868 15283 7890 0 0045 -0 0022 | 20 — 1 P 26) 15281 2500 15281 2493 0 0050 0 0007 | 20 — 1 P 29) 15273 0459 15273 0523 0 0035 -0 0064 * * * | 20 — 1 P 30) 15270 1206 15270 1269 0 0100 -0 0063 | 20 — 1 P 31) 15267 0937 15267 1047 0 0040 -0 0110**** | 20 — 1 P 32) 15263 9814 15263 9855 0 0035 -0 0041 * | 20 — 1 P 34) 15257 4515 15257 4555 0 0045 -0 0040 | 20 — 1 P 35) 15254 0417 15254 0444 0 0035 -0 0027 | 20 — 1 P 37) 15246 9253 15246 9288 0 0035 -0 0035 * | 20 — 1 P 38) 15243 2190 15243 2240 0 0035 -0 0050 * * | 20 — 1 P 39) 15239 4146 15239 4209 0 0055 -0 0063 * | 20 — 1 P 40) 15235 5211 15235 5193 0 0045 0 0018 | 20 — 1 P 41) 15231 5207 15231 5189 0 0055 0 0018 | 20--1 P 42) 15227 4162 15227 4196 0 0035 -0 0034 | 20 — 1 P 43) 15223 2158 15223 2210 0. 0055 -0 0052 | 20 — 1 P 44) 15218 9272 15218 9230 0. 0040 0 0042 | 20 — 1 P 45) 15214 5239 15214. 5252 0. 0035 -0 . 0013 | 20 — 1 P 46) 15210 0288 15210. 0275 0. 0035 0. 0013 | 20 — 1 P 47) 15205. 4265 15205. 4295 0. 0050 -0 . 0030 | 20--1 P 48) 15200. 7307 15200. 7309 0. 0100 -0 . 0002 | 20 — 1 P 49) 15195. 9313 15195. 9316 0. 0035 -0 . 0003 | 20 — 1 P 50) 15191. 0354 15191. 0310 0. 0035 0. 0044 | 20 — 1 P 51) 15186. 0312 15186. 0291 0. 0045 0. 0021 | Appendix 20- -1 p (52) 15180 9258 15180 .9254 0 0035 0 0004 20- -1 p (54) 15170 4147 15170 4113 0 0100 0 0034 20- -1 p (57) 15153 8678 15153 8684 0 0035 -0 0006 20- -1 p (58) 15148 1462 15148 1468 0 0035 -0 0006 20- -1 p (59) 15142 3206 15142 3209 0 0035 -0 0003 20- -1 p (60) 15136 3900 15136 3904 0 0040 -0 0004 20- -1 p (61) 15130 3572 15130 3547 0 0035 0 0025 20- -1 p (62) 15124 2155 15124 2134 0 0035 0 0021 20- -1 p (64) 15111 6142 15111 6125 0 0035 0 0017 20- -1 p (65) 15105 1518 15105 1519 0 0040 -0 0001 20- -1 p (66) 15098 5884 15098 5838 0 0040 0 0046 20- -1 p (68) 15085 1171 15085 1234 0 0065 -0 0063 20- -1 p (69) 15078 2283 15078 2299 0 0035 -0 0016 20- -1 p (70) 15071 2297 15071 2269 0 0055 0 0028 20- -1 p (71) 15064 1181 15064 1138 0 0065 0 0043 20- -1 p (74) 15042 1058 15042 1075 0 0045 -0 0017 20- -1 p (75) 15034 5421 15034 5477 0 0100 -0 0056 20- -1 p (78) 15011 1828 15011 1853 0 0055 -0 0025 20- -1 p (79) 15003 1605 15003 1678 0 0050 -0 0073 20- -2 R ( 4) 14994 7272 14994 7 346 0 0050 -0 0074 20- -2 R ( 5) 14994 3306 14994 3337 0 0065 -0 0031 20- -2 R ( 7) 14993 2485 14993 2499 0 0065 -0 0014 20- -2 R (10) 14990 9230 14990 9187 0 0045 0 0043 20- -2 P ( 6) 14992 9461 14992 9459 0 0050 0 0002 20- -2 P ( 7) 14992 2223 14992 2199 0 0050 0 0024 20- -2 P ( 8) 14991 3954 14991 3998 0 0045 -0 0044 20- -2 P ( 9) 14990 4873 14990 4858 0 0055 0 0015 20- -2 P (10) 14989 4801 14989 4776 0 0045 0 0025 20- -2 P (11) 14988 3729 14988 3754 0 0100 -0 0025 21- -0 R (68) 15440 6499 15440 6459 0 0055 0 0040 21- -0 R (69) 15433 5962 15433 5931 0 0055 0 0031 21- -0 R (72) 15411 7364 15411 7374 0 0045 -0 0010 21- -0 R (76) 15380 9403 15380 9414 0 0100 -0 0011 21- -0 R (78) 15364 8109 15364 8192 0 0100 -0 0083 21- -0 R (79) 15356 5620 15356 5738 0 0100 -0 0118 21- -0 P (60) 15485 4807 15485 4795 0 0055 0 0012 21- -0 P 61) 15479 2276 15479 2242 0 0045 0 0034 21- -0 P 62) 15472 8628 15472 8591 0 0065 0 0037 21- -0 P 68) 15432 3281 15432 3328 0 0100 -0 0047 21- -0 P 69) 15425 1835 15425 1839 0 0100 -0 0004 21- -0 P 70) 15417 9145 15417 9207 0 0065 -0 0062 21- -0 P 71) 15410 5299 15410 5426 0 0075 -0 0127 21- -1 R 4) 15347 8734 15347 8713 0 0055 0 0021 21- -1 R 6) 15346 9343 15346 9323 0 0045 0 0020 21- -1 R 7) 15346 3151 15346 3166 0 0045 -0 0015 21- -1 R 8) 15345 6029 15345 6035 0 0035 -0 0006 21- -1 R 10) 15343 8855 15343 8844 0 0050 0 0011 21- -1 R 11) 15342 8814 15342 8784 0 0045 0 0030 21- -1 R 12) 15341 7779 15341 7747 0 0040 0 0032 21- -1 R 13) 15340 5758 15340 5731 0 0045 0 0027 21- -1 R 14) 15339 2760 15339 2737 0 0045 0 0023 21- -1 R 15) 15337 8785 15337 8763 0 0045 0. 0022 21- -1 R 16) 15336 3829 15336 3809 0 0035 0. 0020 21- -1 R 17) 15334 7930 15334 7873 0 0040 0. 0057 21- -1 R 18) 15333 0992 15333 0955 0 0045 0. 0037 21- -1 R 20) 15329 4203 15329 4167 0. 0045 0. 0036 21- -1 R 21) 15327 4322 15327 4294 0. 0055 0. 0028 21- -1 R 22) 15325 3469 15325 3435 0. 0065 0. 0034 21- -1 R 23) 15323. 1608 15323. 1588 0. 0055 0. 0020 21- -1 R 24) 15320. 8753 15320. 8750 0. 0065 0. 0003 21- -1 R 25) 15318. 4874 15318. 4922 0. 0065 -0 . 0048 21- -1 R 27) 15313. 4221 15313. 4286 0. 0040 -0 . 0065 Appendix 21 — 1 R (28) 15310 7436 15310 7475 0 0035 -0 0039 * 21 — 1 R (29) 15307 9596 15307 9667 0 0040 -0 0071 *** 21 — 1 R (30) 15305 0843 15305 0859 0 0040 -0 0016 21 — 1 R (31) 15302 1004 15302 1051 0 0040 -0 0047 * 21 — 1 R (32) 15299 0209 15299 0239 0 0035 -0 0030 21 — 1 R (33) 15295 8331 15295 8423 0 0035 -0 0092**** 21 — 1 R (34) 15292 5587 15292 5600 0 0035 -0 0013 21 — 1 R (35) 15289 1730 15289 1767 0 0040 -0 0037 21 — 1 R (36) 15285 6901 15285 6924 0 0035 -0 0023 21 — 1 R (38) 15278 4196 15278 4194 0 0035 0 0002 21 — 1 R (39) 15274 6287 15274 6303 0 0100 -0 0016 21 — 1 R (40) 15270 7395 15270 7391 0 0035 0 0004 21 — 1 R (41) 15266 7432 15266 7456 0 0055 -0 0024 21 — 1 R (42) 15262 6506 15262 6494 0 0035 0 0012 21 — 1 R (44) 15254 1491 15254 1482 0 0035 0 0009 21 — 1 R (46) 15245 2357 15245 2332 0 0035 0 0025 21 — 1 R (48) 15235 9062 15235 9018 0 0065 0 0044 21 — 1 R (49) 15231 0829 15231 0792 0 0065 0 0037 21 — 1 R (50) 15226 1562 15226 1514 0 0035 0 0048 21 — 1 R 51) 15221 1203 15221 1182 0 0050 0 0021 21 — 1 R 53) 15210 7376 15210 7340 0 0035 0 0036 21 — 1 R 54) 15205 3873 15205 3822 0 0065 0 0051 21 — 1 R 55) 15199 9247 15199 9233 0 0045 0 0014 21 — 1 R 56) 15194 3612 15194 3571 0 0035 0 0041 21 — 1 R 58) 15182 9024 15182 9005 0 0035 0 0019 21 — 1 R 59) 15177 0104 15177 0093 0 0075 0 0011 21 — 1 R 60) 15171 0108 15171 0089 0 0035 0 0019 21 — 1 R 61) 15164 9037 15164 8987 0 0045 0 0050 21 — 1 R 62) 15158 6808 15158 6782 0 0035 0 0026 21 — 1 R 63) 15152 3424 15152 3469 0 0060 -0 0045 21 — 1 R 65) 15139 3496 15139 3498 0 0035 -0 0002 21 — 1 R 66) 15132 6829 15132 6828 0 0035 0 0001 21 — 1 R 67) 15125 9033 15125 9027 0 0035 0 0006 21 — 1 R 68) 15119 0125 15119 0088 0 0075 0 0037 21 — 1 R 69) 15111 9984 15112 0005 0 0045 -0 0021 21 — 1 R 70) 15104 8745 15104 8772 0 0045 -0 0027 21 — 1 R 74) 15075 2102 15075 2186 0 0050 -0 0084 *** 21 — 1 R 78) 15043 6497 15043 6571 0 0035 -0 0074**** 21 — 1 P 5) 15346 7231 15346 7254 0 0055 -0 0023 21 — 1 P 6) 15346 0760 15346 0756 0 0035 0 0004 21 — 1 P 8) 15344 4828 15344 4836 0 0045 -0 0008 21 — 1 P 9) 15343 5396 15343 5415 0 0045 -0 0019 21 — 1 P 10) 15342 5016 15342 5018 0 0040 -0 0002 | 21 — 1 P 11) 15341 3645 15341 364 6 0 0045 -0 0001 | 21 — 1 P 13) 15338 7997 15338 7974 0 0040 0 0023 | 21 — 1 P 14) 15337 3669 15337 3673 0 0035 -0 0004 | 21 — 1 P 15) 15335 8440 15335 8393 0 0040 0 0047 21 — 1 P 16) 15334 2119 15334 2135 0 0035 -0 0016 | 21 — 1 P 17) 15332 4920 15332 4898 0 0040 0 0022 | 21 — 1 P 19) 15328 7497 15328 7481 0 0040 0 0016 | 21 — 1 P 20) 15326 7296 15326 7299 0 0035 -0 0003 | 21 — 1 P 22) 15322 3980 15322 3985 0 0065 -0 0005 | 21 — 1 P 23) 15320 0872 15320 0850 0 0055 0 0022 21 — 1 P 24) 15317 6734 15317 6729 0 0055 0 0005 | 21 — 1 P 25) 15315 1645 15315 1619 0 0065 0 0026 | 21 — 1 P 26) 15312 5515 15312 5520 0 0065 -0 0005 | 21 — 1 P 28) 15307 0278 15307 0348 0 0075 -0 0070 | 21--1 P 29) 15304 1183 15304 1271 0 0065 -0 0088 * * | 21 — 1 P 30) 15301 1144 15301 1200 0 0055 -0 0056 * | 21 — 1 P 31) 15298 0038 15298 0131 0 0055 -0 0093 * * * | 21 — 1 P 32) 15294 7969 15294 8063 0. 0055 -0 0094 * * * | 21 — 1 P 33) 15291 4896 15291 4995 0. 0065 -0 . 0099 * * * | Appendix 186 21 — 1 P (34) 15288 0866 15288 0924 0 0055 -0 0058 * 21 — 1 P (35) 15284 5762 15284 5848 0 0045 -0 0086 * * * 21 — 1 P (37) 15277 2630 15277 2675 0 0035 -0 0045 21 — 1 P (38) 15273 4563 15273 4574 0 0035 -0 0011 21 — 1 P (39) 15269 5404 15269 5459 0 0100 -0 0055 21 — 1 P (40) 15265 5308 15265 5329 0 0035 -0 0021 21 — 1 P (41) 15261 4151 15261 4181 0 0035 -0 0030 21 — 1 P (42) 15257 2002 15257 2012 0 0040 -0 0010 21 — 1 P (44) 15248 4605 15248 4604 0 0040 0 0001 21 — 1 P (45) 15243 9353 15243 9359 0 0035 -0 0006 21 — 1 P (47) 15234 5801 15234 5771 0 0065 0 0030 21 — 1 P (48) 15229 7428 15229 7423 0 0035 0 0005 21 — 1 P (49) 15224 8039 15224 8035 0 0065 0 0004 21--1 P (50) 15219 7625 15219 7603 0 0035 0 0022 21 — 1 P (51) 15214 6192 15214 6124 0 0065 0 0068 * 21 — 1 P (52) 15209 3641 15209 3595 0 0035 0 0046 * * 21 — 1 P (53) 15204 0078 15204 0012 0 0065 0 0066 * 21 — 1 P (54) 15198 5488 15198 5371 0 0100 0 0117 * 21 — 1 P (55) 15192 9642 15192 9668 0 0100 -0 0026 21 — 1 P (56) 15187 2926 15187 2900 0 0035 0 0026 21 — 1 P (57) 15181 5094 15181 5063 0 0035 0 0031 21 — 1 P (59) 15169 6194 15169 6163 0 0035 0 0031 21 — 1 P (62) 15150 9700 15150 9681 0 0035 0 0019 21 — 1 P (63) 15144 5346 15144 5334 0 0035 0 0012 21 — 1 P (64) 15137 9902 15137 9884 0 0040 0 0018 21 — 1 P (66) 15124 5687 15124 5658 0 0035 0 0029 21 — 1 P (67) 15117 6859 15117 6869 0 0035 -0 0010 21 — 1 P (68) 15110 6995 15110 6957 0 0035 0 0038 * 21 — 1 P 69) 15103 5909 15103 5914 0 0035 -0 0005 21 — 1 P 72) 15081 5942 15081 5939 0 0035 0 0003 21 — 1 P 73) 15074 0252 15074 0309 0 0050 -0 0057 * 21 — 1 P 75) 15058 5467 15058 5548 0 0045 -0 0081 * * * 21 — 1 P 76) 15050 6355 15050 6402 0 0035 -0 0047 * * 21 — 1 P 78) 15034 4438 15034 4537 0 0100 -0 0099 21 — 1 P 80) 15017 7866 15017 7849 0 0050 0 0017 21 — 2 R 4) 15026 9085 15026 9033 0 0040 0 0052 * * 21 — 2 R 5) 15026 4909 15026 4857 0 0045 0 0052 * 21—2 R 6) 15025 9756 15025 9714 0 0040 0 0042 * 21 — 2 R 7) 15025 3634 15025 3603 0 0040 0 0031 21—2 R 8) 15024 6561 15024 6523 0 0075 0 0038 21—2 R 10) 15022 9391 15022 9455 0 0075 -0 0064 21—2 R 11) 15021 9495 15021 9466 0 0040 0 0029 21—2 R 12) 15020 8526 15020 8507 0 0040 0 0019 21—2 R 13) 15019 6534 15019 6575 0 0045 -0 0041 21—2 R 14) 15018 3733 15018 3672 0 0040 0 0061 * * * 21—2 R 15) 15016 9875 15016 9795 0 0045 0 0080 * * * 21—2 R 17) 15013 9162 15013 9119 0 0040 0 0043 * 21—2 R 18) 15012 2375 15012 2317 0 0040 0 0058 * * 21—2 R 22) 15004 5323 15004 5329 0 0075 -0 0006 21 — 2 R 23) 15002 3578 15002 3630 0 0055 -0 0052 21—2 R 25) 14997 7300 14997 7282 0 0050 0 0018 21 — 2 R 27) 14992 7018 14992 6989 0 0045 0 0029 21—2 R 28) 14990 0388 14990 0359 0 0045 0 0029 21 — 2 P 5) 15025 7569 15025 7606 0 0050 -0 0037 21 — 2 P 6) 15025 1143 15025 1146 0 0050 -0 0003 21 — 2 P 8) 15023 5409 15023 5324 0 0075 0 0085 * 21 — 2 P 9) 15022 5998 15022 5961 0 0040 0 0037 21 — 2 P 10) 15021 5576 15021 5629 0 0040 -0 0053 * * 21—2 P 11) 15020 4400 15020 4329 0 0040 0 0071 *** 21—2 P 12) 15019 2080 15019 2059 0 0050 0 0021 21—2 P 13) 15017 8909 15017 8819 0 0055 0 0090 * * * 21—2 P 14) 15016 4628 15016 4608 0 0040 0 0020 Appendix 21—2 P (15) 15014 9458 15014 9425 0 0045 0 0033 21 — 2 P (16) 15013 3208 15013 3271 0 0040 -0 0063 * * * 1 21 — 2 P (18) 15009 7990 15009 8042 0 0040 -0 0052 • * 1 21 — 2 P (19) 15007 8955 15007 8966 0 0040 -0 0011 21 — 2 P (20) 15005 8906 15005 8914 0 0035 -0 0008 21 — 2 P (22) 15001 5819 15001 5878 0 0050 -0 0059 * I 21—2 P (23) 14999 2923 14999 2893 0 0045 0 0030 21 — 2 P (27) 14989 1115 14989 1133 0 0045 -0 0018 22 — 0 R (68) 15463 0056 15463 0023 0 0065 0 0033 22 — 0 R (69) 15455 7249 15455 7283 0 0055 -0 0034 22 — 0 R (71) 15440 8220 15440 8181 0 0055 0 0039 22 — 0 R (73) 15425 4256 15425 4195 0 0065 0 0061 22--0 R (75) 15409 5177 15409 5251 0 0065 -0 0074 * 1 22--0 P (65) 15476 4966 15476 4956 0 0045 0 0010 22--0 P (69) 15447 7517 15447 7570 0 0055 -0 0053 22 — 0 P (70) 15440 2805 15440 2772 0 0065 0 0033 22 — 0 P (71) 15432 6694 15432 6778 0 0065 -0 0084 ** t 22 — 0 P (72) 15424 9516 15424 9581 0 0100 -0 0065 22 — 0 P (74) 15409 1472 15409 1543 0 0150 -0 0071 22 — 0 P (75) 15401 0659 15401 0685 0 0100 -0 0026 22 — 1 R ( 3) 15377 5069 15377 5146 0 0075 -0 0077 * 1 22 — 1 R ( 4) 15377 1784 15377 1776 0 0045 0 0008 22 — 1 R ( 5) 15376 7364 15376 7406 0 0055 -0 0042 22 — 1 R ( 6) 15376 2018 15376 2034 0 0035 -0 0016 22 — 1 R ( 7) 15375 5643 15375 5660 0 0040 -0 0017 22 — 1 R ( 8) 15374 8274 15374 8284 0 0045 -0 0010 22 — 1 R 9) 15373 9907 15373 9905 0 0055 0 0002 22 — 1 R 10) 15373 0503 15373 0522 0 0035 -0 0019 22 — 1 R 11) 15372 0149 15372 0136 0 0045 . 0 0013 22 — 1 R 12) 15370 8726 15370 8745 0 0035 -0 0019 22 — 1 R 14) 15368 2936 15368 2 946 0 0035 -0 0010 22 — 1 R 16) 15365 3149 15365 3118 0 0045 0 0031 22 — 1 R 18) 15361 9256 15361 9254 0 0045 0 0002 22 — 1 R 20) 15358 1345 15358 1344 0 0065 0 0001 22 — 1 R 21) 15356 0922 15356 0870 0 0065 0 0052 22 — 1 R 22) 15353 9416 15353 9380 0 0075 0 0036 22 — 1 R 23) 15351 6765 15351 6874 0 0075 -0 0109 * * 1 22 — 1 R 24) 15349 3342 15349 3350 0 0065 -0 0008 22 — 1 R 25) 15346 8812 15346 8806 0 0065 0 0006 22 — 1 R 26) 15344 3201 15344 3242 0 0065 -0 0041 22 — 1 R 27) 15341 6587 15341 6655 0 0055 -0 0068 * * 1 22 — 1 R 30) 15333 0694 15333 0740 0 0055 -0 0046 22 — 1 R 31) 15329 9992 15330 0044 0 0055 -0 0052 22 — 1 R 33) 15323 5577 15323 5554 0 0045 0 0023 22 — 1 R 34) 15320 1743 15320 1755 0 0035 -0 0012 22 — 1 R 35) 15316 6895 15316 6918 0 0040 -0 0023 22 — 1 R 36) 15313 1033 15313 1040 0 0035 -0 0007 22 — 1 R 37) 15309 4092 15309 4119 0 0045 -0 0027 22 — 1 R 38) 15305 6148 15305 6151 0 0035 -0 0003 22 — 1 R 39) 15301 7126 15301 7135 0 0040 -0 0009 22 — 1 R 40) 15297 7095 15297 7067 0 0035 0 0028 22 — 1 R 41) 15293 5919 15293 5946 0 0040 -0 0027 22 — 1 R 42) 15289 3742 15289 3767 0 0035 -0 0025 22 — 1 R 43) 15285 0535 15285 0528 0 0035 0 0007 22 — 1 R 45) 15276 0882 15276 0857 0 0100 0 0025 22 — 1 R 46) 15271 4441 15271 4419 0 0035 0 0022 22 — 1 R 47) 15266 6905 15266 6907 0 0040 -0 0002 22 — 1 R 48) 15261 8340 15261 8319 0 0035 0 0021 22 — 1 R 49) 15256 8671 15256 8650 0 0040 0 0021 22 — 1 R 50) 15251 7924 15251 7897 0 0035 0 0027 22 — 1 R 52) 15241 3153 15241 3122 0 0040 0 0031 22 — 1 R 54) 15230 3989 15230 3960 0 0035 0 0029 Appendix 22 — 1 R (55) 15224 7662 15224 7723 0 0100 -0 0061 22 — 1 R (56) 15219 0450 15219 0376 0 0050 0 0074 22 — 1 R (57) 15213 1962 15213 1913 0 0035 0 0049 22 — 1 R (58) 15207 2396 15207 2331 0 0040 0 0065 22 — 1 R (59) 15201 1646 15201 1623 0 0040 0 0023 22 — 1 R (62) 15182 2732 15182 2693 0 0035 0 0039 22 — 1 R (64) 15169 1050 15169 1009 0 0035 0 0041 22 — 1 R (65) 15162 3420 15162 3428 0 0035 -0 0008 22 — 1 R (68) 15141 3635 15141 3652 0 0075 -0 0017 22 — 1 R (69) 15134 1355 15134 1357 0 0065 -0 0002 22 — 1 R (70) 15126 7864 15126 7865 0 0035 -0 0001 22 — 1 R (71) 15119 3159 15119 3167 0 0035 -0 0008 22 — 1 R (72) 15111 7240 15111 7254 0 0035 -0 0014 22--1 R (73) 15104 0079 15104 0118 0 0050 -0 0039 22 — 1 R (74) 15096 1745 15096 1749 0 0035 -0 0004 22 — 1 R (75) 15088 2078 15088 2137 0 0035 -0 0059 *** 22 — 1 P ( 5) 15376 0408 15376 0453 0 0055 -0 0045 22 — 1 P ( 6) 15375 3811 15375 3819 0 0040 -0 0008 22 — 1 P ( 8) 15373 7533 15373 7546 0 0035 -0 0013 22 — 1 P ( 9) 15372 7862 15372 7908 0 0040 -0 0046 * 22 — 1 P (10) 15371 7246 15371 7267 0 0035 -0 0021 22 — 1 P (11) 15370 5595 15370 5624 0 0040 -0 0029 22 — 1 P (12) 15369 2953 15369 2977 0 0035 -0 0024 22 — 1 P (13) 15367 9356 15367 9327 0 0065 0 0029 22 — 1 P (14) 15366 4658 15366 4671 0 0035 -0 0013 22 — 1 P (15) 15364 9026 15364 9011 0 0035 0 0015 22 — 1 P (16) 15363 2331 15363 2344 0 0040 -0 0013 22 — 1 P (17) 15361 4742 15361 4671 0 0055 0 0071 22 — 1 P (18) 15359 5995 15359 5989 0 0035 0 0006 22 — 1 P (19) 15357 6338 15357 6299 0 0055 0 0039 22 — 1 P (20) 15355 5595 15355 5598 0 0040 -0 0003 22 — 1 P (21) 15353 3874 15353 3887 0 0045 -0 0013 22 — 1 P (22) 15351 1164 15351 1163 0 0045 0 0001 22 — 1 P (24) . 15346 2680 15346 2673 0 0065 0 0007 22 — 1 P (25) 15343 6799 15343 6905 0 0075 -0 0106 ** 22 — 1 P (26) 15341 0016 15341 0119 0 0075 -0 0103 ** 22 — 1 P 27) 15338 2176 15338 2314 0 0065 -0 0138**** 22 — 1 P 30) 15329 2716 15329 2768 0 0065 -0 0052 22 — 1 P (32) 15322 7872 15322 7944 0 0045 -0 0072 *** 22 — 1 P (33) 15319 3911 15319 3988 0 0040 -0 0077 *** 22 — 1 P (34) 15315 8972 15315 9001 0 0035 -0 0029 22 — 1 P (36) 15308 5883 15308 5922 0 0045 -0 0039 22 — 1 P (37) 15304 7754 15304 7826 0 0035 -0 0072**** 22 — 1 P (38) 15300 8655 15300 8690 0 0035 -0 0035 * 22 — 1 P (39) 15296 8445 15296 8511 0 0045 -0 0066 ** 22 — 1 P (40) 15292 7277 15292 7286 0 0035 -0 0009 22 — 1 P 41) 15288 4937 15288 5013 0 0040 -0 0076 *** 22 — 1 P (42) 15284 1694 15284 1689 0 0035 0 0005 22 — 1 P 44) 15275 1879 15275 1876 0 0035 0 0003 22 — 1 P 45) 15270 5369 15270 5382 0 0035 -0 0013 22 — 1 P 47) 15260 9187 15260 9202 0 0040 -0 0015 22 — 1 P 48) 15255 9520 15255 9510 0 0035 0 0010 22 — 1 P 49) 15250 8731 15250 8745 0 0040 -0 0014 22 — 1 P (51) 15240 3977 15240 3983 0 0040 -0 0006 | 22 — 1 P 52) 15235 0006 15234 9978 0 0035 0 0028 | 22 — 1 P 53) 15229 4920 15229 4885 0 0045 0 0035 | 22 — 1 P 54) 15223 8723 15223 8700 0 0035 0 0023 22 — 1 P 55) 15218 1456 15218 1420 0 0100 0 0036 | 22 — 1 P 56) 15212 3127 15212 3038 0 0100 0 0089 | 22 — 1 P 57) 15206 3575 15206 3552 0 0035 0 0023 | 22 — 1 P 58) 15200 2985 15200 2957 0 0035 0 0028 22 — 1 P 59) 15194 1276 15194 1246 0 0035 0 0030 | Appendix 189 22 — 1 P 61) 15181 4450 15181 4462 0 0040 -0 0012 | 22 — 1 P 62) 15174 9474 15174 9378 0 0100 0 0096 | 22 — 1 P 63) 15168 3131 15168 3157 0 0035 -0 0026 | 22 — 1 P (67) 15140 6810 15140 6800 0 0035 0 0010 | 22 — 1 P (68) 15133 4759 15133 4809 0 0100 -0 0050 | 22 — 1 P (72) 15103 5036 15103 5032 0 0035 0 0004 | 22 — 1 P (74) 15087 7929 15087 7944 0 0035 -0 0015. | 22 — 1 P (75) 15079 7478 15079 7570 0 0100 -0 0092 | 22 — 1 P (76) 15071 5959 15071 5965 0 0035 -0 0006 | 22 — 1 P (77) 15063 3021 15063 3117 0 0035 -0 0096****| 22—2 R ( 3) 15056 5410 15056 5440 0 0055 -0 0030 | 22—2 R ( 6) 15055 2413 15055 2425 0 0040 -0 0012 | 22—2 R ( 7) 15054 6079 15054 6096 0 0035 -0 0017 | 22—2 R ( 8) 15053 8767 15053 8772 0 0035 ^0 0005 | 22 — 2 R ( 9) 15053 0440 15053 0451 0 0150 -0 0011 | 22 — 2 R (10) 15052 1118 15052 1134 0 0035 -0 0016 | 22—2 R (11) 15051 0802 15051 0819 0 0040 -0 0017 | 22—2 R (12) 15049 9496 15049 9505 0 0035 -0 0009 | 22—2 R (13) 15048 7228 15048 7193 0 0035 0 0035 | 22—2 R 14) 15047 3861 15047 3881 0 0035 -0 0020 | 22 — 2 R 16) 15044 4277 15044 4254 0 0035 0 0023 | 22 — 2 R 17) 15042 7980 15042 7937 0 0035 0 0043 | 22 — 2 R 18) 15041 0651 15041 0616 0 0035 0 0035 | 22 — 2 R 19) 15039 2327 15039 2291 0 0035 0 0036 | 22—2 R 21) 15035 2713 15035 2621 0 0100 0 0092 | 22—2 R 24) 15028 5499 15028 5548 0 0065 -0 0049 | 22—2 R 27) 15020 9285 15020 9358 0 0055 -0 0073 * * | 22—2 R 30) 15012 4011 15012 4006 0 0035 0 0005 | 22 — 2 R 32) 15006 2043 15006 1991 0 0035 0 0052 | 22—2 P 7) 15053 6583 15053 6620 0 0035 -0 0037 * | 22—2 P 8) 15052 8002 15052 8035 0 0035 -0 0033 | 22 — 2 P 9) 15051 8433 15051 8454 0 0035 -0 0021 | 22 — 2 P 11) 15049 6318 15049 6306 0 0035 0 0012 | 22 — 2 P 12) 15048 3706 15048 3737 0 0035 -0 0031 | 22 — 2 P 14) 15045 5568 15045 5606 0 0045 -0 0038 | 22 — 2 P 16) 15042 3467 15042 3480 0 0035 -0 0013 | 22—2 P 17) 15040 5939 15040 5916 0 0040 0 0023 | 22—2 P 19) 15036 7795 15036 7784 0 0035 0 0011 | 22 — 2 P 20) 15034 7227 15034 7213 0 0035 0 0014 | 22 — 2 P 21) 15032 5643 15032 5638 0 0035 0 0005 | 22—2 P 22) 15030 3051 15030 3056 0 0035 -0 0005 | 22—2 P 23) 15027 9466 15027 9468 0 0035 -0 0002 | 22 — 2 P 26) 15020 2563 15020 2647 0 0060 -0 0084 * * | 22—2 P 29) 15011 6660 15011 6712 0 0055 -0 0052 | 22—2 P 35) 14991 7240 14991 7315 0 0060 -0 0075 * * | 22—2 P 36) 14988 0446 14988 0491 0 0045 -0 0045 | 23 — 0 R 71) 15459 8290 15459 8390 0 0045 -0 0100****| 23 — 0 R 73) 15443 9392 15443 9461 0 0040 -0 0069 * * * | 23 — 0 P 68) 15475 2831 15475 2780 0 0055 0 0051 | 23 — 1 R 3) 15404 2206 15404 2257 0 0075 -0 0051 | 23 — 1 R 4) . 15403 8733 15403 8754 0 0055 -0 0021 | 23 — 1 R 5) 15403 4327 15403 4222 0 0055 0 0105 | 23 — 1 R 6) 15402 8652 15402 8663 0 0035 -0 0011 | 23 — 1 R 8) 15401 4426 15401 4457 0 0035 -0 0031 | 23 — 1 R 9) 15400 5826 15400 5810 0 0040 0 0016 | 23 — 1 R 11) 15398 5431 15398 5425 0 0040 0 0006 | 23 — 1 R 12) 15397 3672 15397 3685 0 0035 -0 0013 | 23 — 1 R 13) 15396 0992 15396 0912 0 0040 0 0080 | 23 — 1 R 14) 15394 7090 15394 7106 0 0035 -0 0016 | 23 — 1 R 15) 15393 2259 15393 2265 0 0035 -0 0006 | 23 — 1 R 19) 15386 2560 15386 2537 0 0040 0 0023 | 23 — 1 R 20) 15384 2512 15384 2508 0 0040 0 0004 | Appendix 190 23- -1 R (21) 15382 1447 15382 1438 0 0045 0 0009 23- -1 R (23) 15377 6217 15377 6166 0 0045 0 0051 23- -1 R (27) 15367 3001 15367 3058 0 0055 -0 0057 23- -1 R (28) 15364 4616 15364 4653 0 0045 -0 0037 23- -1 R (30) 15358 4625 15358 4677 0 0035 -0 0052 * * 23- -1 R (31) 15355 3068 15355 3101 0 0035 -0 0033 23- -1 R (32) 15352 0443 15352 0463 0 0035 -0 0020 23- -1 R (34) 15345 1979 15345 1995 0 0035 -0 0016 23- -1 R (35) 15341 6100 15341 6160 0 0040 -0 0060 * * 23- -1 R (36) 15337 9243 15337 9253 0 0035 -0 0010 23- -1 R (37) 15334 1266 15334 1273 0 0040 -0 0007 23- -1 R (38) 15330 2239 15330 2216 0 0040 0 0023 23- -1 R (39) 15326 2060 15326 2079 0 0040 -0 0019 23- -1 R (40) 15322 0883 15322 0861 0 0035 0 0022 23- -1 R (41) 15317 8565 15317 8556 0 0040 0 0009 23- -1 R (42) 15313 5169 15313 5163 0 0035 0 0006 23- -1 R (43) 15309 0673 15309 0678 0 0040 -0 0005 23- -1 R (44) 15304 5126 15304 5098 0 0035 0 0028 23- -1 R (45) 15299 8404 15299 8418 0 0040 -0 0014 23- -1 R (46) 15295 0678 15295 0636 0 0040 0 0042 23- -1 R (47) 15290 1749 15290 1747 0 0040 0 0002 23- -1 R (48) 15285 1795 15285 1747 0 0035 0 0048 23- -1 R (49) 15280 0650 15280 0633 0 0035 0 0017 23- -1 R (50) 15274 8403 15274 8399 0 0035 0 0004 23- -1 R (51) 15269 5116 15269 5042 0 0075 0 0074 23- -1 R (52) 15264 0590 15264 0557 0 0035 0 0033 23- -1 R (53) 15258 4963 15258 4940 0 0055 0 0023 23- -1 R (54) 15252 8217 15252 8184 0 0035 0 0033 23- -1 R (55) 15247 0297 15247 0286 0 0035 0 0011 23- -1 R (56) 15241 1276 15241 1239 0 0035 0 0037 23- -1 R (57) 15235 1006 15235 1039 0 0065 -0 0033 23- -1 R (58) 15228 9723 15228 9678 0 0035 0 0045 23- -1 R 59) 15222 7078 15222 7153 0 0100 -0 0075 23- -1 R 60) 15216 3529 15216 3455 0 0055 0 0074 23- -1 R 61) 15209 8578 15209 8580 0 0055 -0 0002 23- -1 R 63) 15196 5277 15196 5266 0 0065 0 0011 23- -1 R 64) 15189 6869 15189 6814 0 0035 0 0055 23- -1 R 65) 15182 7152 15182 7154 0 0035 -0 0002 23- -1 R 67) 15168 4166 15168 4182 0 0035 -0 0016 23- -1 R 68) 15161 0804 15161 0852 0 0075 -0 0048 23- -1 R 70) 15146 0439 15146 0458 0 0035 -0 0019 23- -1 R 72) 15130 5020 15130 5021 0 0035 -0 0001 23- -1 R 73) 15122 5290 15122 5384 0 0035 -0 0094**** 23- -1 P 5) 15402 7492 15402 7564 0 0065 -0 0072 * 23- -1 P 7) 15401 3091 15401 3000 0 0045 0 0091 23- -1 P 8) 15400 4176 15400 4176 0 0035 0 0000 23- -1 P 9) 15399 4316 15399 4323 0 0040 -0 0007 23- -1 P 10) 15398 3429 15398 3441 0 0035 -0 0012 23- -1 P 11) 15397 1533 15397 1530 0 0035 0 0003 23- -1 P 12) 15395 8610 15395 8588 0 0040 0 0022 23- -1 P 14) 15392 9595 15392 9611 0 0035 -0 0016 23- -1 P 15) 15391 3569 15391 3574 0 0040 -0 0005 23- -1 P 16) 15389 6466 15389 6504 0 0035 -0 0038 * 23- -1 P 17) 15387 8367 15387 8399 0 0040 -0 0032 23- -1 P 18) 15385 9271 15385 9260 0 0035 0 0011 23- -1 P 19) 15383 9110 15383 9084 0 0040 0 0026 23- -1 P 21) 15379 5615 15379 5619 0 0040 -0 0004 23- -1 P 27) 15364 0171 15364 0219 0 0045 -0 0048 23- -1 P 28) 15361 0586 15361 0657 0 0040 -0 0071 *** 23- -1 P 29) 15357 9967 15358 0044 0 0045 -0 0077 * * * 23- -1 P 31) 15351 5582 15351. 5658 0 0040 -0 . 0076 * * * 23- -1 P 33) 15344 6971 15344. 7045 0 0040 -0 . 0074 * * * Appendix 23 — 1 P (34) 15341 1114 15341 1148 0 0035 - 0 0034 23 — 1 P (36) 15333 6146 15333 6162 0 0035 - 0 0016 23 — 1 P (38) 15325 6880 15325 6903 0 0035 - 0 0023 23 — 1 P (39) 15321 5641 15321 5665 0 0040 - 0 0024 23 — 1 P (40) 15317 3345 15317 3351 0 0035 - 0 0006 23 — 1 P (41) 15312 9923 15312 9957 0 0040 - 0 0034 23 — 1 P (42) 15308 5483 15308 5482 0 0045 0 0001 23 — 1 P (43) 15303 9916 15303 9922 0 0040 - 0 0006 23 — 1 P (45) 15294 5509 15294 5533 0 0040 - 0 0024 23 — 1 P (46) 15289 6682 15289 6697 0 0035 - 0 0015 23 — 1 P (49) 15274 3581 15274 3585 0 0035 - 0 0004 23 — 1 P (50) 15269 0359 15269 0332 0 0035 0 0027 23 — 1 P (51) 15263 5972 15263 5965 0 0035 0 0007 2 3 - - 1 P (52) 15258 0505 15258 0480 0 0035 0 0025 23 — 1 P (53) 15252 3887 15252 3872 0 0035 0 0015 23 — 1 P (55) 15240 7287 15240 7268 0 0035 0 0019 23 — 1 P 56) 15234 7297 15234 7263 0 0055 0 0034 23 — 1 P 57) 15228 6146 15228 6115 0 0035 0 0031 23 — 1 P 58) 15222 3867 15222 3820 0 0035 0 0047 23 — 1 P 59) 15216 0415 15216 0371 0 0065 0 0044 23 — 1 P 61) 15202 9977 15202 9992 0 0035 - 0 0015 23 — 1 P 63) 15189 4949 15189 4927 0 0035 0 0022 23 — 1 P 64) 15182 5640 15182 5621 0 0035 0 0019 23 — 1 P 65) 15175 5068 15175 5124 0 0045 - o 0056 * * 23 — 1 P 66) 15168 3504 15168 3428 0 0055 0 0076 23 — 1 P 67) 15161 0569 15161 0525 0 0075 0 0044 23 — 1 P 69) 15146 1053 15146 1069 0 0035 - 0 0016 23 — 1 P 71) 15130 6653 15130 6687 0 0035 - 0 0034 23 — 1 P 72) 15122 7609 15122 7625 0 0035 - 0 0016 2 3 - - 1 P 73) 15114 7221 15114 7303 0 0065 - 0 0082 * * 2 3 - - 1 P 74) 15106 5641 15106 5711 0 0100 - 0 0070 2 3 - - 1 P 75) 15098 2814 15098 2837 0 0065 - 0 0023 23 — 1 P 77) 15081 3194 15081 3196 0 0100 - 0 0002 2 3 — 2 R 3) 15083 2483 15083 2551 0 0055 - 0 0068 ** 23 — 2 R 4) 15082 9050 15082 9073 0 0040 - 0 0023 23 — 2 R 9) 15079 6369 15079 6357 0 0035 0 0012 2 3 — 2 R 10) 15078 6724 15078 6744 0 0040 - 0 0020 2 3 — 2 R 12) 15076 4445 15076 4445 0 0035 0 0000 23 — 2 R 13) 15075 1764 15075 1756 0 0055 0 0008 23 — 2 R 15) 15072 3352 15072 3297 0 0055 0 0055 23 — 2 R 16) 15070 7559 15070 7525 0 0035 0 0034 2 3 — 2 R 17) 15069 0718 15069 0722 0 0035 - 0 0004 2 3 — 2 R 19) 15065 4049 15065 4023 0 0035 0 0026 2 3 — 2 R 20) 15063 4122 15063 4123 0 0035 - 0 0001 2 3 — 2 R 21) 15061 3178 15061 3188 0 0035 - 0 0010 2 3 — 2 R 28) 15043 7486 15043 7538 0 0035 - 0 0052 * * 2 3 — 2 R 29) 15040 8359 15040 8266 0 0055 0 0093 2 3 — 2 R 30) 15037 7887 15037 7943 0 0055 - 0 0056 * 23 — 2 R 31) 15034 6537 15034 6568 0 0035 - 0 0031 23 — 2 R 32) 15031 4067 15031 4138 0 0035 - 0 0 0 7 1 * * * * 23 — 2 R 33) 15028 0611 15028 0651 0 0035 - 0 0040 * 23 — 2 R 35) 15021 0457 15021 0495 0 0055 - 0 0038 23 — 2 R 36) 15017 3768 15017 3822 0 0050 - 0 0054 * 23 — 2 R 38) 15009 7243 15009 7271 0 0060 - 0 0028 23 — 2 R 40) 15001 6405 15001 6428 0 0045 - 0 0023 23 — 2 R 42) 14993 1305 14993 1268 0 0045 0 0037 23 — 2 P 5) 15081 7894 15081 7917 0 0075 - 0 0023 23 — 2 P 6) 15081 1192 15081 1187 0 0040 0 0005 23 — 2 P 7) 15080 3483 15080 3436 0 0100 0 0047 23 — 2 P 8) 15079 4660 15079 4664 0 0035 - 0 0004 23 — 2 P 9) 15078 4784 15078 4869 0 0055 - 0 0085 * * * 23 — 2 P 10) 15077 4046 15077 4052 0 0035 - 0 0006 Appendix 192 23 — 2 P 11) 15076 2199 15076 2212 0 0065 -0 0013 | 23 — 2 P 12) 15074 9337 15074 9348 0 0035 -0 0011 | 23 — 2 P 13) 15073 5456 15073 5459 0 0035 -0 0003 | 23—2 P (14) 15072 0517 15072 0546 0 0035 -0 0029 | 23—2 P 15) 15070 4608 15070 4606 0 0035 0 0002 | 23 — 2 P 16) 15068 7592 15068 7639 0 0035 -0 0047 * * | 23—2 P 17) 15066 9605 15066 9645 0 0050 -0 0040 | 23—2 P 20) 15060 9470 '15060 9485 0 0040 -0 0015 | 23—2 P (21) 15058 7362 15058 7369 0 0035 -0 0007 | 23—2 P 22) 15056 4198 15056 4220 0 0035 -0 0022 | 23—2 P 23) 15054 0089 15054 0036 0 0065 0 0053 | 23 — 2 P 24) 15051 4813 15051 4815 0 0050 -0 0002 | 23 — 2 P 25) 15048 8527 15048 8557 0 0040 -0 0030 | 23—2 P (27) 15043 2830 15043 2922 0 0065 -0 0092 * * | 23—2 P 30) 15034 1567 15034 1645 0 0100 -0 0078 | 23—2 P 33) 15024 0842 15024 0933 0 0045 -0 0091****| 23—2 P (34) 15020 5310 15020 5257 0 0040 0 0053 | 23—2 P (36) 15013 0756 15013 0731 0 0060 0 0025 | 23—2 P (37) 15009 1784 15009 1876 0 0045 -0 0092****| 23—2 P (39) 15001 0904 15001 0973 0 0050 -0 0069 * * | 23—2 P 40) 14996 8841 14996 8918 0 0065 -0 0077 * | 24 — 0 R 70) 15484 2759 15484 2775 0 0035 -0 0016 | 24 — 1 R 4) 15428 1441 15428 1513 0 0045 -0 0072 * * * | 24 — 1 R 5) 15427 6771 15427 6822 0 0055 -0 0051 | 24 — 1 R 7) 15426 4248 15426 4274 0 0050 -0 0026 | 24 — 1 R 8) 15425 6428 15425 6416 0 0040 0 0012 | 24 — 1 R 10) 15423 7522 15423 7530 0 0035 -0 0008 | 24 — 1 R 11) 15422 6521 15422 6500 0 0040 0 0021 | 24 — 1 R 12) 15421 4408 15421 4411 0 0035 -0 0003 | 24 — 1 R 13) 15420 1286 15420 1263 0 0040 0 0023 | 24 — 1 R 14) 15418 7044 15418 7054 0 0035 -0 0010 | 24--1 R 16) 15415 5436 15415 5450 0 0035 -0 0014 | 24 — 1 R 21) 15405 7830 15405 7798 0 0040 0 0032 | 24 — 1 R 22) 15403 5057 15403 5061 0 0055 -0 0004 | 24 — 1 R 23) 15401 1221 15401 1251 0 0040 -0 0030 | 24 — 1 R 24) 15398 6359 15398 6367 0 0065 -0 0008 | 24 — 1 R 25) 15396 0313 15396 0407 0 0055 -0 0094 * * * | 24 — 1 R 26) 15393 3353 15393 3369 0 0035 -0 0016 | 24 — 1 R 28) 15387 6023 15387 6051 0 0045 -0 0028 | 24 — 1 R 29) 15384 5705 15384 5767 0 0035 -0 0062 * * * | 24 — 1 R 30) 15381 4379 15381 4397 0 0035 -0 0018 | 24 — 1 R 31) 15378 1886 15378 1938 0 0035 -0 0052 * * | 24 — 1 R 33) 15371 3698 15371 3744 0 0035 -0 0046 * * | 24 — 1 R 34) 15367 7995 15367 8004 0 0035 -0 0009 | 24 — 1 R 35) 15364 1138 15364 1165 0 0040 -0 0027 | 24 — 1 R 36) 15360 3220 15360 3224 0 0035 -0 0004 | 24 — 1 R 37) 15356 4164 15356 4178 0 0035 -0 0014 | 24 — 1 R 39) 15348 2767 15348 2760 0 0035 0 0007 | 24 — 1 R 40) 15344 0405 15344 0380 0 0035 0 0025 | 24 — 1 R 41) 15339 6896 15339 6883 0 0035 0 0013 | 24 — 1 R 42) 15335 2294 15335 2264 0 0035 0 0030 | 24 — 1 R 44) 15325 9680 15325 9647 0 0035 0 0033 | 24--1 R 45) 15321 1664 15321 1640 0 0035 0 0024 | 24 — 1 R 46) 15316 2530 15316 2497 0 0055 0 0033 | 24 — 1 R 47) 15311 2248 15311 2211 0 0045 0 0037 | 24 — 1 R 48) 15306 0817 15306 0780 0 0035 0 0037 | 24 — 1 R 49) 15300 8217 15300 8197 0 0040 0 0020 | 24--1 R 50) 15295 4476 15295 4459 0 0040 0 0017 | 24 — 1 R 51) 15289 9581 15289 9560 0 0075 0 0021 | 24 — 1 R 54) 15272 7918 15272 7843 0 0045 0 0075 | 24 — 1 R 55) 15266 8338 15266 8245 0 0050 0 0093 | 24 — 1 R 57) 15254 5532 15254 5475 0 0050 0 0057 | Appendix 193 24 — 1 R (58) 15248 2268 15248 2289 0 0075 -0 0021 | 24 — 1 R (59) 15241 7831 15241 7893 0 0075 -0 0062 | 24 — 1 R (60) 15235 2266 15235 2279 0 0040 -0 0013 | 24 — 1 R (62) 15221 7405 15221 7370 0 0035 0 0035 | 24 — 1 R (63) 15214 8022 15214 8057 0 0075 -0 0035 | 24 — 1 R (64) 15207 7473 15207 7494 0 0065 -0 0021 | 24 — 1 R (65) 15200 5645 15200 5671 0 0050 -0 0026 | 24 — 1 R (66) 15193 2588 15193 2579 0 0100 0 0009 | 24 — 1 R (67) 15185 8176 15185 8207 0 0055 -0 0031 | 24 — 1 R (68) 15178 2517 15178 2544 0 0075 -0 0027 | 24 — 1 R (69) 15170 5487 15170 5580 0 0100 -0 0093 | 24--1 R (70) 15162 7287 15162 7302 0 0035 -0 0015 | 24 — 1 P ( 6) 15426 3557 15426 3556 0 0035 0 0001 | 24 — 1 P ( 7) 15425 5569 15425 5599 0 0045 -0 0030 | 24 — 1 P ( 8) 15424 6587 15424 6588 0 0035 -0 0001 | 24 — 1 P ( 9) 15423 6527 15423 6522 0 0035 0 0005 | 24 — 1 P (10) 15422 5401 15422 5399 0 0035 0 0002 | 24 — 1 P (13) 15418 5698 15418 5690 0 0035 0 0008 | 24 — 1 P (14) 15417 0370 15417 0337 0 0045 0 0033 | 24 — 1 P (15) 15415 3932 15415 3925 0 0040 0 0007 | 24 — 1 P (16) 15413 6423 15413 6452 0 0065 -0 0029 | 24 — 1 P (17) 15411 7940 15411 7918 0 0040 0 0022 | 24 — 1 P (18) 15409 8299 15409 8321 0 0065 -0 0022 | 24 — 1 P (20) 15405 5950 15405 5935 0 0035 0 0015 | 24 — 1 P (21) 15403 3146 15403 3143 0 0050 0 0003 | 24 — 1 P (23) 15398 4377 15398 4354 0 0065 0 0023 . | 24 — 1 P (27) 15387 3823 15387 3915 0 0065 -0 0092 * * | 24 — 1 P (28) 15384 3519 15384 3616 0 0050 -0 0097 * * * | 24 — 1 P (29) 15381 2156 15381 2237 0 0035 -0 0081****| 24 — 1 P (32) 15371 1528 15371 1601 0 0035 -0 0073****| 24 — 1 P (33) 15367 5852 15367 5882 0 0040 -0 0030 | 24 — 1 P (34) 15363 9047 15363 9073 0 0040 -0 0026 | 24 — 1 P 36) 15356 2131 15356 2171 0 0035 -0 0040 * | 24 — 1 P 37) 15352 2032 15352 2073 0 0035 -0 0041 * | 24 — 1 P 38) 15348 0840 15348 0874 0 0035 -0 0034 | 24 — 1 P 39) 15343 8551 15343 8571 0 0055 -0 0020 | 24 — 1 P 40) 15339 5160 15339 5159 0 0035 0 0001 | 24 — 1 P 41) 15335 0643 15335 0638 0 0035 0 0005 | 24--1 P 42) 15330 4992 15330 5002 0 0035 -0 0010 | 24 — 1 P 43) 15325 8242 15325 8248 0 0035 -0 0006 | 24 — 1 P 44) 15321 0395 15321 0374 0 0035 0 0021 | 24 — 1 P 45) 15316 1318 15316 1374 0 0040 -0 0056 * * | 24 — 1 P 46) 15311 1257 15311 1246 0 0035 0 0011 | 24 — 1 P 47) 15305 9991 15305 9986 0 0035 0 0005 | 24 — 1 P 48) 15300 7601 15300 7588 0 0035 0 0013 | 24 — 1 P 49) 15295 4062 15295 4049 0 0035 0 0013 | 24 — 1 P 53) 15272 8416 15272 8389 0 0065 0 0027 | 24 — 1 P 54) 15266 9104 15266 9073 0 0035 0 0031 | 2.4 — 1 P 55) 15260 8602 15260 8586 0 0040 0 0016 . | 24 — 1 P 56) 15254 6980 15254 6922 0 0035 0 0058 | 24 — 1 P 58) 15242 0125 15242 0039 0 0065 0 0086 | 24 — 1 P 59) 15235 4730 15235 4808 0 0050 -0 0078 * * * | 24 — 1 P 60) 15228 8420 15228 8374 0 0065 0 0046 | 24 — 1 P 61) 15222 0703 15222 0731 0 0075 -0 0028 | 24 — 1 P 63) 15208 1706 15208 1788 0 0100 -0 0082 | 24 — 1 P 64) 15201 0538 15201 0472 0 0035 0 0066 | 24 — 1 P 65) 15193 7830 15193 7915 0 0065 -0 0085 * * | 24 — 1 P 68) 15171 2615 15171 2708 0 0100 -0 0093 | 24 — 1 P 71) 15147 5854 15147 5986 0 0055 -0 0132****| 24 — 1 P 72) 15139 4445 15139 4469 0 0030 -0 0024 | 24—2 R 5) 15106 7122 15106 7174 0 0065 -0 . 0052 | 24—2 R 6) 15106 1466 15106 1466 0 0065 0. 0000 | Appendix 24 — 2 R 7) 15105 4717 15105 4710 0 0075 0 0007 | 24—2 R ( 8) 15104 6894 15104 6904 0 0035 -0 0010 | 24—2 R ( 9) 15103 8075 15103 8048 0 0035 0 0027 | 24—2 R (10) 15102 8069 15102 8141 0 0100 -0 0072 | 24—2 R (13) 15099 2092 15099 2107 0 0035 -0 0015 | 24—2 R (14) 15097 7985 15097 7989 0 0035 -0 0004 | 24—2 R (16) 15094 6600 15094 6586 0 0035 0 0014 | 24—2 R (18) 15091 1063 15091 0953 0 0075 0 0110 | 24—2 R (19) 15089 1602 15089 1547 0 0045 0 0055 | 24—2 R (20) 15087 0950 15087 1079 0 0075 -0 0129 * * * | 24—2 R (22) 15082 6946 15082 6954 0 0100 -0 0008 | 24—2 R (24) 15077 8524 15077 8565 0 0035 -0 0041 * | 24—2 R 25) 15075 2733 15075 2767 0 0035 -0 0034 | 24—2 R (26) 15072 5859 15072 5897 0 0065 -0 0038 | 24—2 R (27) 15069 7917 15069 7954 0 0035 -0 0037 * | 24—2 R (30) 15060 7703 15060 7664 0 0065 0 0039 | 24—2 R (31) 15057 5322 15057 5405 0 0035 -0 0083****| 24 — 2 R 32) 15054 2048 15054 2063 0 0035 -0 0015 | 24 — 2 R 33) 15050 7598 15050 7633 0 0075 -0 0035 | 24 — 2 R 34) 15047 2096 15047 2113 0 0035 -0 0017 | 24 — 2 R 35) 15043 5478 15043 5501 0 0035 -0 0023 | 24 — 2 R 36) 15039 7784 15039 7793 0 0035 -0 0009 | 24 — 2 R 37) 15035 9002 15035 8987 0 0035 0 0015 | 24 — 2 R 38) 15031 9085 15031 9079 0 0035 0 0006 | 24—2 R 39) 15027 8091 15027 8067 0 0045 0 0024 | 24—2 R 40) 15023 5979 15023 5947 0 0055 0 0032 | 24 — 2 R 41) 15019 2726 15019 2716 0 0050 0 0010 | 24 — 2 R 42) 15014 8403 15014 8369 0 0055 0 0034 | 24—2 R 43) 15010 2893 15010 2904 0 0050 -0 0011 | 24—2 R 44) 15005 6275 15005 6316 0 0055 -0 0041 | 24—2 R 45) 15000 8563 15000 8601 0 0065 -0 0038 | 24—2 R 46) 14995 9748 14995 9755 0 0040 -0 0007 | 24—2 R 47) 14990 9892 14990 9775 0 0075 0 0117 | 24—2 P 5) 15106 0759 15106 0809 0 0040 -0 0050 * * | 24—2 P 7) 15104 6064 15104 6036 0 0040 0 0028 | 24—2 P 8) 15103 7073 15103 7076 0 0035 -0 0003 | 24—2 P 9) 15102 7100 15102 7068 0 0040 0 0032 | 24—2 P 11) 15100 3835 15100 3903 0 0055 -0 0068 * * | 24 — 2 P 12) 15099 0734 15099 0744 0 0035 -0 0010 | 24 — 2 P 15) 15094 4919 15094 4957 0 0075 -0 0038 | 24 — 2 P 16) 15092 7611 15092 7588 0 0035 0 0023 | 24 — 2 P 24) 15075 0577 15075 0552 0 0050 0 0025 | 24 — 2 P 27) 15066 6507 15066 6618 0 0035 -0 0111**** | 24—2 P 28) 15063 6447 15063 6500 0 0035 -0 0053 * * * | 24—2 P 29) 15060 5205 15060 5309 0 0035 -0 0104****| 24—2 P 30) 15057 2972 15057 3043 0 0035 -0 0071****| 24 — 2 P 31) 15053 9583 15053 9699 0 0040 -0 0116****| 24 — 2 P 32) 15050 5230 15050 5276 0 0035 -0 0046 * * | 24 — 2 P 34) 15043 3174 15043 3181 0 0065 -0 0007 | 24 — 2 P 37) 15031 6852 15031 6882 0 0035 -0 0030 . | 24 — 2 P 38) 15027 5960 15027 5929 0 0100 0 0031 | 24 — 2 P 39) 15023 3889 15023 3878 0 0050 0 0011 | 24 — 2 P 40) 15019 0743 15019 0726 0 0045 0 0017 | 24 — 2 P 41) 15014 6421 15014 6470 0 0055 -0 0049 | 24 — 2 P 43) 15005 4566 15005 4632 0 0060 -0 0066 * | 24 — 2 P 44) 15000 7004 15000 7043 0 0040 -0 0039 | 24 — 2 P 45) 14995 8304 14995 8335 0 0050 -0 0031 | 24 — 2 P 46) 14990 8581 14990 8505 0 0045 0 0076 | 25 — 1 R 7) 15448 3666 15448 3648 0 0075 0 0018 | 25--1 R 9) 15446 6382 15446 6364 0 0075 0 0018 | 25 — 1 R 10) 15445 6016 15445 6096 0 0075 -0 0080 * | 25 — 1 R 11) 15444 4696 15444 4743 0 0100 -0 0047 | Appendix 195 25 — 1 R 13) 15441 8760 15441 8776 0 0075 -0 0016 25 — 1 R 15) 15438 8485 15438 8458 0 0075 0 0027 25 — 1 R 23) 15422 3506 15422 3451 0 0055 0 0055 25 — 1 R 30) 15402 1259 15402 1168 0 0055 0 0091 25 — 1 R 31) 15398 7801 15398 7814 0 0040 -0 0013 25 — 1 R 32) 15395 3370 15395 3337 0 0035 0 0033 25 — 1 R 33) 15391 7699 15391 7737 0 0040 -0 0038 25 — 1 R 34) 15388 1012 15388 1009 0 0035 0 0003 25 — 1 R 35) 15384 3146 15384 3150 0 0035 -0 0004 25 — 1 R 36) 15380 4166 15380 4157 0 0035 0 0009 25 — 1 R 37) 15376 4014 15376 4027 0 0035 -0 0013 25 — 1 R 38) 15372 2785 15372 2756 0 0035 0 0029 25 — 1 R 39) 15368 0329 15368 0341 0 0035 -0 0012 25--1 R 40) 15363 6776 15363 6778 0 0045 -0 0002 25 — 1 R 41) 15359 2058 15359 2063 0 0035 -0 0005 25 — 1 R 42) 15354 6223 15354 6192 0 0035 0 0031 25 — 1 R 43) 15349 9163 15349 9160 0 0035 0 0003 25--1 R 44) 15345 0991 15345 0964 0 0035 0 0027 25 — 1 R 45) 15340 1607 15340 1598 0 0035 0 0009 25 — 1 R 46) 15335 1091 15335 1058 0 0035 0 0033 25 — 1 R 47) 15329 9362 15329 9339 0 0035 0 0023 25 — 1 R 48) 15324 6485 15324 6435 0 0035 0 0050 25 — 1 R 49) 15319 2371 15319 2341 0 0035 0 0030 25 — 1 R 50) 15313 7096 15313 7052 0 0035 0 0044 25 — 1 R 51) 15308 0571 15308 0561 0 0035 0 0010 25 — 1 R 52) 15302 2898 15302 2862 0 0035 0 0036 25 — 1 R 53) 15296 3976 15296 3949 0 0035 0 0027 25 — 1 R 54) 15290 3841 15290 3815 0 0035 0 0026 25--1 R 55) 15284 2459 15284 2453 0 0035 0 0006 25 — 1 R 56) 15277 9838 15277 9855 0 0030 -0 0017 25--1 R 57) 15271 6046 15271 6013 0 0045 0 0033 25 — 1 R 58) 15265 0945 15265 0920 0 0030 0 0025 25 — 1 R 59) 15258 4535 15258 4566 0 0055 -0 0031 25 — 1 R 60) 15251 6951 15251 6943 0 0035 0 0008 25 — 1 R 61) 15244 8020 15244 8042 0 0035 -0 0022 25 — 1 R 62) 15237 7813 15237 7851 0 0035 -0 0038 * 1 25 — 1 R 63) 15230 6310 15230 6361 0 0035 -0 0051 25 — 1 R (64) 15223 3520 15223 3560 0 0035 -0 0040 * 1 25 — 1 R (65) 15215 9405 15215 9437 0 0055 -0 0032 25 — 1 R (66) 15208 3886 15208 3979 0 0045 -0 0093 k * * * 1 25 — 1 P ( 6) 15448 3522 15448 3494 0 0075 0 0028 25 — 1 P 9) 15445 5904 15445 5896 0 0075 0 0008 25 — 1 P 11) 15443 2105 15443 2083 0 0075 0 0022 25 — 1 P 12) 15441 8645 15441 8551 0 0075 0 0094 25 — 1 P 14) 15438 8305 15438 8229 0 0065 0 0076 25 — 1 P (22) 15422 3309 15422 3372 0 0055 -0 0063 * 1 25 — 1 P (29) 15402 1478 15402 1512 0 0055 -0 0034 25 — 1 P (30) 15398 8209 15398 8246 0 0040 -0 0037 25 — 1 P (31) 15395 3843 15395 3867 0 0040 -0 0024 25 — 1 P (32) 15391 8338 15391 8372 0 0035 -0 0034 25 — 1 P (33) 15388 1716 15388 1758 0 0040 -0 0042 * 1 25 — 1 P (34) 15384 3999 15384 4022 0 0035 -0 0023 25 — 1 P (35) 15380 5123 15380 5162 0 0035 -0 0039 * 1 25 — 1 P (37) 15372 4008 15372 4058 0 0040 -0 0050 * * j 25 — 1 P (38) 15368 1814 15368 1807 0 0035 0 0007 25 — 1 P (39) 15363 8413 15363" 8419 0 0040 -0 0006 25 — 1 P (40) 15359 3887 15359 3891 0 0035 -0 0004 25 — 1 P (41) 15354 8198 15354 8219 0 0035 -0 0021 25 — 1 P (42) 15350 1436 15350 1399 0 0035 0 0037 25 — 1 P (44) 15340 4328 15340 4301 0 0035 0 0027 25 — 1 P (46) 15330 2591 15330 2563 0 0035 0 0028 25 — 1 P (47) 15325 0010 15324 9943 0 0075 0 0067 Appendix 196 25 — 1 P (48) 15319 6156 15319 6149 0 0035 0 0007 25 — 1 P (49) 15314 1188 15314 1177 0 0035 0 0011 | 25 — 1 P (50) 15308 5034 15308 5020 0 0035 0 0014 | 25 — 1 P (51) 15302 7688 15302 7674 0 0035 0 0014 25 — 1 P (52) 15296 9155 15296 9133 0 0035 0 0022 | 25 — 1 P (53) 15290 9377 15290 9390 0 0040 -0 0013 | 25 — 1 P (54) 15284 8509 15284 8441 0 0035 0 0068 1 * + • 25 — 1 P (56) 15272 2897 15272 2894 0 0030 0 0003 | 25 — 1 P 57) 15265 8280 15265 8282 0 0035 -0 0002 | 25 — 1 P (58) 15259 2432 15259 2436 0 0035 -0 0004 | 25 — 1 P (59) 15252 5351 15252 5346 0 0035 0 0005 | 25 — 1 P (61) 15238 7389 15238 7405 0 0055 -0 0016 | 25--1 P (62) 15231 6563 15231 6536 0 0035 0 0027 | 25 — 1 P (63) 15224 4369 15224 4389 0 0040 -0 0020 | 25 — 1 P (64) 15217 0994 15217 0953 0 0055 0 0041 | 25 — 1 P (65) 15209 6198 15209 6219 0 0035 -0 0021 | 25 — 1 P (66) 15202 0124 15202 0174 0 0035 -0 0050 • * | 25 — 1 P (67) 15194 2740 15194 2808 0 0050 -0 0068 * * 1 25—2 R (12) 15122 2990 15122 3063 0 0100 -0 0073 | 25 — 2 R (14) 15119 5063 15119 5097 0 0075 -0 0034 | 25—2 R (15) 15117 9364 15117 9490 0 0075 -0 0126 * * * I 25—2 R (17) 15114 4999 15114 5023 0 0075 -0 0024 | 25—2 R (19) 15110 6172 15110 6209 0 0075 -0 0037 | 25—2 R (33) 15071 1562 15071 1625 0 0035 -0 0063 * * * I 25—2 R (34) 15067 5118 15067 5117 0 0075 0 0001 | 25—2 R (35) 15063 7483 15063 7485 0 0035 -0 0002 | 25—2 R (36) 15059 8717 15059 8726 0 0065 -0 0009 | 25 — 2 R 37) 15055 8814 15055 8835 0 0055 -0 0021 | 25—2 R 38) 15051 7834 15051 7811 0 0035 0 0023 | 25—2 R 39) 15047 5653 15047 5649 0 0035 0 0004 | 25—2 R 40) 15043 2395 15043 2345 0 0065 0 0050 | 25—2 R 41) 15038 7938 15038 7895 0 0035 0 0043 1 ** 25 — 2 R 42) 15034 2319 15034 2296 0 0035 0 0023 | 25—2 R 43) 15029 5562 15029 5544 0 0045 0 0018 | 25 — 2 R 44) 15024 7680 15024 7632 0 0040 0 0048 1 * 25—2 R 45) 15019 8553 15019 8558 0 0050 -0 0005 | 25 — 2 R 46) 15014 8403 15014 8317 0 0045 0 0086 1 * * * 25—2 R 48) 15004 4381 15004 4310 0 0040 0 0071 1 * * * 25—2 R (49) 14999 0586 14999 0534 0 0045 0 0052 1 * 25 — 2 R 50) 14993 5638 14993 5569 0 0065 0 0069 1 * 25 — 2 R 51) 14987 9448 14987 9408 0 0045 0 0040 | 25—2 P 11) 15122 2815 15122 2766 0 0100 0 0049 | 25—2 P (13) 15119 4847 15119 4777 0 0075 0 0070 | 25—2 P (14) 15117 9210 15117 9164 0 0075 0 0046 | 25—2 P (16) 15114 4791 15114 4696 0 0075 0 0095 1 * * 25—2 P (17) 15112 5774 15112 5838 0 0075 -0 0064 | 25—2 P (18) 15110 5988 15110 5896 0 0075 0 0092 25—2 P 27) 15087 7445 15087 7414 0 0075 0 0031 | 25—2 P (33) 15067 5604 15067 5646 ' 0 0040 -0 0042 * 1 25—2 P (34) 15063 8114 15063 8131 0 0035 -0 0017 | 25—2 P (35) 15059 9453 15059 9498 0 0035 -0 0045 * * 1 25—2 P (36) 15055 9730 15055 9744 0 0035 -0 0014 | 25—2 P 37) 15051 8895 15051 8866 0 0045 0 0029 | 25—2 P 38) 15047 6860 15047 6862 0 0035 -0 0002 | 25 — 2 P 39) 15043 3695 15043 3727 0 0040 -0 0032 | 25 — 2 P 40) 15038 9466 15038 9458 0 0035 0 0008 | 25 — 2 P 42) 15029 7526 15029 7504 0 0035 0 0022 | 25 — 2 P 43) 15024 9790 15024 9812 0 0045 -0 0022 | 25 — 2 P 44) 15020 1003 15020 0970 0 0045 0 0033 | 25 — 2 P 45) 15015 1015 15015 0975 0 0045 0 0040 | 25 — 2 P 46) 15009 9845 15009 9822 0 0045 0 0023 | 25—2 P 47) 15004 7516 15004 7507 0 0045 0 0009 | Appendix 197 25—2 P (48) 14999 3979 14999 4024 0 0040 -0 0045 * 25 — 2 P (49) 14993 9445 14993 9369 0 0045 0 0076 * * * 26 — 1 R ( 3) 15470 3602 15470 354 8 0 0065 0 0054 26--1 R ( 5) 15469 4670 15469 4624 0 0040 0 0046 * 26 — 1 R ( 6) 15468 8549 15468 8499 0 0055 0 0050 26 — 1 R ( 8) 15467 2943 15467 2918 0 0035 0 0025 26 — 1 R ( 9) 15466 3512 15466 3461 0 0035 0 0051 * * 26 — 1 R (10) 15465 2909 15465 2891 0 0035 0 0018 26 — 1 R (11) 15464 1231 15464 1209 0 0040 0 0022 26 — 1 R (12) 15462 8453 15462 8413 0 0035 0 0040 * 26 — 1 R (13) 15461 4552 15461 4502 0 0040 0 0050 * * 26 — 1 R (14) 15459 9533 15459 9475 0 0040 0 0058 * * 26 — 1 R (15) 15458 3434 15458 3330 0 0040 0 0104 * * * * 26--1 R 16) 15456 6138 15456 6066 0 0035 0 0072 * * * * 26--1 R (17) 15454 7765 15454 7683 0 0040 0 0082 k-kkk 26 — 1 R (18) 15452 8273 15452 8177 0 0040 0 0096 kkkk 26—1 R 19) 15450 7654 15450 7549 0 0045 0 0105 kkkk 26—1 R 20) 15448 5823 15448 5795 0 0040 0 0028 26—1 R 22) 15443 8947 15443 8906 0 0045 0 0041 26—1 R 23) 15441 3791 15441 3766 0 0040 0 0025 26 — 1 R 24) 15438 7469 15438 7494 0 0035 -0 0025 26 — 1 R 26) 15433 1527 15433 1541 0 0035 -0 0014 26 — 1 R 27) 15430 1794 15430 1857 0 0055 -0 0063 * 26 — 1 R 28) 15427 0997 15427 1029 0 0035 -0 0032 26 — 1 R 29) 15423 9047 15423 9057 0 0035 -0 0010 26 — 1 R 30) 15420 5946 15420 5937 0 0035 0 0009 26 — 1 R 32) 15413 6274 15413 6242 0 0045 0 0032 26 — 1 R 33) 15409 9645 15409 9660 0 0035 -0 0015 26 — 1 R 34) 15406 1951 15406 1918 0 0035 0 0033 26 — 1 R 35) 15402 3022 15402 3012 0 0035 0 0010 26 — 1 R 36) 15398 2970 15398 2938 0 0035 0 0032 26 — 1 R 37) 15394 1724 15394 1693 0 0035 0 0031 26 — 1 R 38) 15389 9274 15389 9273 0 0035 0 0001 26—1 R 39) 15385 5675 15385 5673 0 0035 0 0002 26 — 1 R 40) 15381 0917 15381 0889 0 0035 0 0028 26—1 R 42) 15371 7762 15371 7751 0 0035 0 0011 26—1 R 43) 15366 9397 15366 9387 0 0035 0 0010 26 — 1 R 44) 15361 9831 15361 9820 0 0035 0 0011 26 — 1 R 45) 15356 9060 15356 9045 0 0035 0 0015 26 — 1 R 46) 15351 7088 15351 7054 0 0035 0 0034 26 — 1 R 47) 15346 3856 15346 3844 0 0035 0 0012 26 — 1 R 48) 15340 9442 15340 9406 0 0035 0 0036 * 26 — 1 R 51) 15323 8673 15323 8668 0 0035 0 0005 26 — 1 R 52) 15317 9273 15317 9256 0 0035 0 0017 26—1 R 54) 15305 6699 15305 6636 0 0035 0 0063 * * * 26 — 1 R 56) 15292 8905 15292 8896 0 0035 0 0009 26 — 1 R 57) 15286 3043 15286 3083 0 0035 -0 0040 * 26 — 1 R 59) 15272 7468 15272 7519 0 0035 -0 0051 * * 26 — 1 R 60) 15265 7766 15265 7746 0 0100 0 0020 26—1 R 61) 15258 6492 15258 6629 0 0035 -0 0137**** 26 — 1 R 62) 15251 4088 15251 4155 0 0035 -0 0067 * * * 26 — 1 P 7) 15467 3375 15467 3402 0 0035 -0 0027 26 — 1 P 8) 15466 4014 15466 4012 0 0035 0 0002 26 — 1 P 9) 15465 3516 15465 3512 0 0035 0 0004 26 — 1 P 10) 15464 1892 15464 1901 0 0035 -0 0009 26 — 1 P 11) 15462 9197 15462 9180 0 0035 0 0017 26 — 1 P 12) 15461 5345 15461 5346 0 0035 -0 0001 26 — 1 P 13) 15460 0381 15460 0400 0 0040 -0 0019 26 — 1 P 14) 15458 4339 15458 4339 0 0035 0 0000 26 — 1 P 15) 15456 7168 15456 7164 0 0040 0 0004 26—1 P 16) 15454 8876 15454 8873 0 0035 0 0003 26 — 1 P 17) 15452 9479 15452 9464 0 0035 0. 0015 Appendix 198 26 — 1 P (18) 15450 8925 15450 8937 0 0035 -0 0012 | 26 — 1 P (19) 15448 7306 15448 7290 0 0045 0 0016 1 26 — 1 P (20) 15446 4465 15446 4522 0 0040 -0 0057 * * 1 26 — 1 P (21) 15444 0606 15444 0630 0 0040 -0 0024 | 26 — 1 P (26) 15430 4255 15430 4263 0 0035 -0 0008 | 26 — 1 P (27) 15427 3518 15427 3594 0 0035 -0 0076 Ar * * * 1 26 — 1 P (28) 15424 1816 15424 1788 0 0065 0 0028 | 26 — 1 P (29) 15420 8778 15420 8842 0 0035 -0 0064 * * * 1 26 — 1 P (30) 15417 4735 15417 4754 0 0035 -0 0019 | 26 — 1 P (31) 15413 9484 15413 9521 0 0035 -0 0037 * 1 26 — 1 P (32) 15410 3192 15410 3141 0 0065 0 0051 | 26 — 1 P (33) 15406 5580 15406 5611 0 0035 -0 0031 | 26 — 1 P (34) 15402 6921 15402 6927 0 0035 -0 0006 | 26 — 1 P (35) 15398 7064 15398 7086 0 0040 -0 0022 | 26 — 1 P (36) 15394 6118 15394 6085 0 0035 0 0033 | 26—1 P (37) 15390 3895 15390 3920 0 0035 -0 0025 | 26—1 P (38) 15386 0596 15386 0588 0 0035 0 0008 | 26 — 1 P (39) 15381 6069 15381 6086 0 0035 -0 0017 | 26 — 1 P (40) 15377 0432 15377 0408 0 0045 0 0024 | 26—1 P (42) 15367 5477 15367 5511 0 0035 -0 0034 | 26—1 P (43) 15362 6270 15362 6282 0 0040 -0 0012 | 26—1 P 44) 15357 5868 15357 5861 0 0035 0 0007 | 26 — 1 P 46) 15347 1453 15347 1420 0 0035 0 0033 | 26 — 1 P (47) 15341 7372 15341 7390 0 0035 -0 0018 | 26 — 1 P 48) 15336 2200 15336 2146 0 0035 0 0054 1 * * * 26 — 1 P 49) 15330 5691 15330 5682 0 0035 0 0009 | 26 — 1 P 50) 15324 8021 15324 7992 0 0035 0 0029 | 26 — 1 P 51) 15318 9082 15318 9069 0 0035 0 0013 | 26 — 1 P 52) 15312 8930 15312 8907 0 0035 0 0023 | 26 — 1 P 53) 15306 7496 15306 7498 0 0035 -0 0002 | 26 — 1 P 54) 15300 4835 15300 4835 0 0035 0 0000 | 26 — 1 P 55) 15294 0893 15294 0910 0 0035 -0 0017 | 26 — 1 P 56) 15287 5771 15287 5715 0 0035 0 0056 j *** 26 — 1 P 58) 15274 1476 15274 1477 0 0030 -0 0001 | 26 — 1 P 60) 15260 2003 15260 2047 0 0040 -0 0044 * 1 26 — 1 P 61) 15253 0309 15253 0358 0 0035 -0 0049 * * 1 26--1 P 62) 15245 7363 15245 7338 0 0040 0 0025 26 — 1 P 63) 15238 2892 15238 2976 0 0035 -0 0084**** | 26--1 P 64) 15230 7170 15230 7257 0 0065 -0 0087 26—2 R 8) 15146 3424 15146 3406 0 0035 0 0018 | 26—2 R 9) 15145 4124 15145 4007 0 0030 0 0117 1 26—2 R 11) 15143 1936 15143 1892 0 0035 0 0044 1 * * 26 — 2 R 12) 15141 9267 15141 9173 0 0030 0 0094 1 * * * * 26—2 R 13) 15140 5377 15140 5346 0 0075 0 0031 | 26—2 R 14) 15139 0436 15139 0409 0 0035 0 0027 | 26 — 2 R 15) 15137 4403 15137 4362 0 0055 0 0041 | 26—2 R 16) 15135 7268 15135 7202 0 0035 0 0066 j * * * 26 — 2 R 17) 15133 9008 15133 8928 0 0045 0 0080 1 * * * 26 — 2 R 18) 15131 9571 15131 9540 0 0040 0 0031 | 26 — 2 R 22) 15123 0790 15123 0799 0 0065 -0 0009 | 26—2 R 23) 15120 5772 15120 5809 0 0035 -0 0037 * 1 26 — 2 R 25) 15115 2415 15115 2446 0 0035 -0 0031 | 26—2 R 26) 15112 4054 15112 4069 0 0035 -0 0015 | 26—2 R 28) 15106 3854 15106 3914 0 0065 -0 0060 | 26—2 R 30) 15099 9215 15099 9204 0 0050 0 0011 | 26—2 R 31) 15096 5139 15096 5134 0 0040 0 0005 | 26—2 R 32) 15092 9952 15092 9917 0 0035 0 0035 1 * 26—2 R 33) 15089 3542 15089 3549 0 0040 -0 0007 | 26—2 R 34) 15085 6042 15085 6027 0 0030 0 0015 | 26 — 2 R 35) 15081 7347 15081 7348 0 0035 -0 0001 | 26 — 2 R 36) 15077 7530 15077 7507 0 0040 0 0023 | 26 — 2 R 37) 15073 6474 15073 6502 0 0035 -0 0028 | Appendix 199 26—2 R (38) 15069 4343 15069 4328 0 0035 0 0015 26—2 R (41) 15056 0795 15056 0749 0 0065 0 0046 26—2 R (42) 15051 3896 15051 3856 0 0035 0 0040 * 26—2 R (44) 15041 6517 15041 6489 0 0035 0 0028 26—2 R (45) 15036 6031 15036 6005 0 0035 0 0026 26—2 R (46) 15031 4373 15031 4313 0 0035 0 0060 * * * 26—2 R (47) 15026 1466 15026 1407 0 0055 0 0059 * 26—2 R (49) 15015 1938 15015 1929 0 0040 0 0009 26—2 R (50) 15009 5307 15009 5343 0 0040 -0 0036 26—2 R (51) 15003 7524 15003 7516 0 0045 0 0008 26—2 R (52) 14997 8487 14997 8441 0 0035 0 0046 * * 26 — 2 P ( 7) 15146 3820 15146 3838 0 0055 -0 0018 26—2 P ( 8) 15145 4491 15145 4500 0 0035 -0 0009 26—2 P (10) 15143 2510 15143 2512 0 0035 -0 0002 26—2 P (11) 15141 9911 15141 9862 0 0035 0 0049 * * 26—2 P (15) 15135 8237 15135 8196 0 0035 0 0041 * 26—2 P (17) 15132 0724 15132 0710 0 0035 0 0014 26 — 2 P (19) 15127 8790 15127 8776 0 0040 0 0014 26 — 2 P (22) 15120 7468 15120 7507 0 0035 -0 0039 * 26—2 P (23) 15118 1430 15118 1513 0 0035 -0 0083**** 26—2 P (26) 15109 6737 15109 6791 0 0075 -0 0054 26—2 P (27) 15106 6227 15106 6297 0 0055 -0 0070 ** 26—2 P (28) 15103 4610 15103 4672 0 0035 -0 0062 *** 26—2 P (29) 15100 1856 15100 1914 0 0040 -0 0058 ** 26—2 P (31) 15093 3004 15093. 2989 0 0075 0 0015 26—2 P 32) 15089 6795 15089 6816 0 0035 -0 0021 26—2 P (33) 15085 9478 15085 9499 0 0040 -0 0021 26—2 P (34) 15082 1074 15082 1035 0 0100 0 0039 26—2 P (35) 15078 1415 15078 1421 0 0075 -0 0006 26—2 P 36) 15074 0681 15074 0654 0 0035 0 0027 26—2 P 37) 15069 8701 15069 8729 0 0035 -0 0028 26 — 2 P 38) 15065 5653 15065 5643 0 0035 0 0010 26 — 2 P 39) 15061 1363 15061 1393 0 0065 -0 0030 26 — 2 P 40) 15056 6029 15056 5975 0 0065 0 0054 26 — 2 P 41) 15051 9368 15051 9384 0 0045 -0 0016 26 — 2 P 42) 15047 1638 15047 1615 0 0035 0 0023 26 — 2 P 43) 15042 2661 15042 2666 0 0035 -0 0005 26—2 P 44) 15037 2543 15037 2530 0 0035 0 0013 26—2 P 45) 15032 1224 15032 1202 0 0040 0 0022 26—2 P 46) 15026 8695 15026 8679 0 0040 0 0016 26—2 P 47) 15021 4959 15021 4953 0 0045 0 0006 26—2 P 48) 15016 0042 15016 0021 0 0035 0 0021 26—2 P 50) 15004 6540 15004 6508 0 0035 0 0032 26—2 P 51) 14998 7931 14998 7916 0 0045 0 0015 26—2 P 52) 14992 8136 14992 8091 0 0035 0 0045 * * 27 — 1 R 3) 15488 1276 15488 0978 0 0050 0 0298 * * * * 27 — 1 R 5) 15487 1992 15487 1748 0 0050 0 0244 * * * * 27 — 1 R 6) 15486 5655 15486 5427 0 0035 0 0228 * * * * 27 — 1 R 7) 15485 8143 15485 7968 0 0035 0 0175 * * * * 27 — 1 R 8) 15484 9510 15484 9371 0 0100 0 0139 * * 27 — 1 R 10) 15482 8942 15482 8757 0 0045 0 0185 **** 27 — 1 R 11) 15481 6905 15481 6738 0 0040 0 0167 **** 27 — 1 R 12) 15480 3735 15480 3577 0 0035 0 0158 * * * * 27 — 1 R 13) 15478 9458 15478 9272 0 0035 0 0186 * * * * 27 — 1 R 14) 15477 3931 15477 3822 0 0045 0 0109 * * * * 27 — 1 R 15) 15475 7375 15475 7226 0 0035 0 0149 * * * * 27 — 1 R 18) 15470 0675 15470 0543 0 0065 0 0132 * * * * 27 — 1 R 19) 15467 9459 15467 9345 0 0045 0 0114 * * * * 27 — 1 R 20) 15465 7041 15465. 6993 0 0040 0 0048 * * 27 — 1 R 21) 15463 3570 15463. 3483 0 0045 0. 0087 • * * 27 — 1 R 22) 15460 8876 15460. 8814 0 0045 0 0062 * * 27 — 1 R 23) 15458 3007 15458 2984 0 0040 0 0023 Appendix 200 27 — 1 R (24) 15455 6029 15455 5991 0 0035 0 0038 * 27 — 1 R (25) 15452 7855 15452 7831 0 0035 0 0024 • 27 — 1 R (26) 15449 8558 15449 8502 0 0055 0 0056 * 27 — 1 R (27) 15446 8072 15446 8002 0 0055 0 0070 * * 27 — 1 R (29) 15440 3560 15440 3474 0 0035 0 0086 * * * * 27 — 1 R (30) 15436 9384 15436 9441 0 0075 -0 0057 27 — 1 R (32) 15429 7742 15429 7817 0 0035 -0 0075 k * * * 27 — 1 R (33) 15426 0158 15426 0220 0 0035 -0 0062 * * * 27 — 1 R (34) 15422 1409 15422 1428 0 0035 -0 0019 27 — 1 R (35) 15418 1414 15418 1436 0 0040 -0 0022 27 — 1 R (36) 15414 0266 15414 0241 0 0035 0 0025 27 — 1 R (38) 15405 4260 15405 4221 0 0035 0 0039 * 27 — 1 R (39) 15400 9429 15400 9387 0 0035 0 0042 * 27 — 1 R (40) 15396 3334 15396 3331 0 0035 0 0003 27--1 R (41) 15391 6050 15391 6046 0 0035 0 0004 27 — 1 R (42) 15386 7539 15386 7527 0 0035 0 0012 27 — 1 R (44) 15376 6784 15376 6764 0 0035 0 0020 27 — 1 R (45) 15371 4488 15371 4507 0 0035 -0 0019 27--1 R (46) 15366 1022 15366 0991 0 0035 0 0031 27 — 1 R (47) 15360 6198 15360 6208 0 0040 -0 0010 27 — 1 R (48) 15355 0162 15355 0150 0 0035 0 0012 27--1 R (49) 15349 2802 15349 2811 0 0035 -0 0009 27 — 1 R (50) 15343 4207 15343 4181 0 0035 0 0026 27 — 1 R 52) 15331 3020 15331 3011 0 0030 0 0009 27 — 1 R 53) 15325 0435 15325 0453 0 0035 -0 0018 27 — 1 R (54) 15318 6571 15318 6564 0 0035 0 0007 27 — 1 R 55) 15312 1293 15312 1335 0 0035 -0 0042 * 27 — 1 R 56) 15305 4731 15305 4752 0 0035 -0 0021 27 — 1 R 57) 15298 6728 15298 6803 0 0035 -0 0075 k * * * 27 — 1 R 58) 15291 7412 15291 7474 0 0035 -0 0062 * * * 27 — 1 P 5) 15486 6486 15486 6285 0 0035 0 0201 * * * * 27 — 1 P 6) 15485 9194 15485 8974 0 0035 0 0220 * * * * 27 — 1 P 7) 15485 0763 15485 0526 0 0075 0 0237 27 — 1 P 8) 15484 1132 15484 0940 0 0040 0 0192 * * * * 27 — 1 P 9) 15483 0454 15483 0217 0 0100 0 0237 * * * * 27 — 1 P 10) 15481 8511 15481 8355 0 0035 0 0156 * * * * 27 — 1 P 12) 15479 1357 15479 1211 0 0035 0 0146 * * * * 27 — 1 P 13) 15477 6080 15477 5928 0 0040 0 0152 * * * * 27 — 1 P 16) 15472 3330 15472 3220 0 0035 0 0110 * * * * 27 — 1 P 18) 15468 2453 15468 2352 0 0035 0 0101 * * * * 27 — 1 P 20) 15463 7004 15463 6887 0 0055 0 0117 * * * * 27 — 1 P 21) 15461 2524 15461 2426 0 0035 0 0098 27 — 1 P 22) 15458 6830 15458 6811 0 0040 0 0019 27 — 1 P 23) 15456 0082 15456 0039 0 0035 0 0043 * * 27 — 1 P 26) 15447 2761 15447 2760 0 0040 0 0001 27 — 1 P 27) 15444 1366 15444 1339 0 0035 0 0027 27 — 1 P 28) 15440 8754 15440 8749 0 0035 0 0005 27 — 1 P 29) 15437 5025 15437 4987 0 0035 0 0038 * 27 — 1 P 30) 15434 0118 15434 0051 0 0035 0 0067 * * * 27 — 1 P 31) 15430 3957 15430 3938 0 0040 0 0019 27 — 1 P 32) 15426 6697 15426 6645 0 0040 0 0052 * * 27 — 1 P 34) 15418 8405 15418 8502 0 0055 -0 0097 • * * 27 — 1 P 35) 15414 7598 15414 7646 0 0035 -0 0048 ** 27 — 1 P 36) 15410 5586 15410 5595 0 0035 -0 0009 27 — 1 P 37) 15406 2316 15406 2344 0 0040 -0 0028 27 — 1 P 38) 15401 7882 15401 7891 0 0035 -0 0009 27 — 1 P 39) 15397 2227 15397 2230 0 0040 -0 0003 27 — 1 P 40) 15392 5372 15392 5356 0 0035 0 0016 27 — 1 P 41) 15387 7271 15387 7265 0 0035 0. 0006 27 — 1 P 42) 15382 7956 15382 7952 0 0035 0. 0004 27 — 1 P 44) 15372 5651 15372 5637 0 0035 0. 0014 27 — 1 P 45) 15367 2642 15367 2623 0 0035 0. 0019 Appendix 201 27- -1 P 46) 15361 8380 15361 8364 0 0035 0 0016 27- -1 P 47) 15356 2847 15356 2852 0 0035 -0 0005 27- -1 P 48) 15350 6082 15350 6082 0 0035 0 0000 27- -1 P 49) 15344 8111 15344 8046 0 0040 0 0065 * * * 27- -1 P 50) 15338 8779 15338 8736 0 0040 0 0043 * 27- -1 P 51) 15332 8151 15332 8144 0 0035 0 0007 27- -1 P 52) 15326 6264 15326 6261 0 0035 0 0003 27- -1 P 53) 15320 3084 15320 3080 0 0035 0 0004 27- -1 P 54) 15313 8601 15313 8590 0 0035 0 0011 27- -1 P 55) 15307 2751 15307 2781 0 0035 -0 0030 27- -1 P 56) 15300 5621 15300 5643 0 0035 -0 0022 27- -1 P 57) 15293 7101 15293 7164 0 0035 -0 0063 *** 27- -1 P 58) 15286 7297 15286 7333 0 0035 -0 0036 * 27- -1 P 60) 15272 3544 15272 3560 0 0100 -0 0016 27- -2 R 4) 15166 7524 15166 7251 0 0030 0 0273 * * * * 27- -2 R 6) 15165 6035 15165 5818 0 0045 0 0217 * * * * 27- -2 R 7) 15164 8547 15164 8405 0 0035 0 0142 27- -2 R 8) 15164 0037 15163 9859 0 0035 0 0178 * * * * 27- -2 R 9) 15163 0340 15163 0180 0 0035 0 0160 * * * * 27- -2 R 11) 15160 7587 15160 7420 0 0035 0 0167 * * * * 27- -2 R 14) 15156 4887 15156 4757 0 0040 0 0130 27- -2 R 15) 15154 8373 15154 8258 0 0035 0 0115 * * * * 27- -2 R 16) 15153 0725 15153 0617 0 0045 0 0108 * * * * 27- -2 R 17) 15151 1956 15151 1833 0 0045 0 0123 27- -2 R 18) 15149 1997 15149 1905 0 0045 0 0092 * * * * 27- -2 R 21) 15142 5317 15142 5234 0 0040 0 0083 * * * * 27- -2 R 22) 15140 0756 15140 0708 0 0035 0 0048 * * 27- -2 R 25) 15132 0257 15132 0191 0 0035 0 0066 * * * 27- -2 R 26) 15129 1084 15129 1030 0 0035 0 0054 * * * 27- -2 R 27) 15126 0801 15126 0705 0 0040 0 0096 * * * * 27- -2 R 28) 15122 9273 15122 9211 0 0035 0 0062 • * * 27- -2 R 29) 15119 6659 15119 6546 0 0035 0 0113 * * * * 27- -2 R 31) 15112 7553 15112 7690 0 0040 -0 0137**** 27- -2 R 32) 15109 1438 15109 1492 0 0045 -0 0054 ** 27- -2 R 33) 15105 4145 15105 4109 0 0035 0 0036 * 27- -2 R (35) 15097 5746 15097 5772 0 0035 -0 0026 27- -2 R 36) 15093 4831 15093 4810 0 0035 0 0021 27- -2 R (37) 15089 2643 15089 2646 0 0035 -0 0003 27- -2 R 38) 15084 9280 15084 9276 0 0035 0 0004 27- -2 R 39) 15080 4747 15080 4695 0 0050 0 0052 * 27- -2 R (40) 15075 8957 15075 8897 0 0040 0 0060 * * 27- -2 R (42) 15066 3670 15066 3632 0 0040 0 0038 27- -2 R (43) 15061 4162 15061 4152 0 0035 0 0010 27- -2 R (44) 15056 3441 15056 3433 0 0035 0 0008 27- -2 R (45) 15051 1483 15051 1468 0 0035 0 0015 27- -2 R (46) 15045 8275 15045 8249 0 0035 0 0026 27- -2 R (47) 15040 3762 15040 3771 0 0050 -0 0009 27- -2 R (48) 15034 8092 15034 8025 0 0055 0 0067 * * 27- -2 R (49) 15029 1002 15029 1003 0 0035 -0 0001 27- -2 R (51) 15017 3094 15017 3098 0 0050 -0 0004 27- -2 R (52) 15011 2255 15011 2196 0 0050 0 0059 * 27- -2 R (54) 14998 6448 14998 6443 0 0055 0 0005 27- -2 P ( 5) 15165 6817 15165 6638 0 0035 0 0179 27- -2 P ( 6) 15164 9555 15164 9365 0 0030 0 0190 * * * * 27- -2 P ( 8) 15163 1583 15163 1428 0 0030 0 0155 * ** * 27- -2 P ( 9) 15162 0956 15162 0763 0 0045 0 0193 ** * * 27- -2 P (10) 15160 9124 15160 8966 0 0030 0 0158 * ** * 27- -2 P (11) 15159 6203 15159 6035 0 0035 0 0168 * ** * 27- -2 P (12) 15158 2112 15158 1971 0 0035 0 0141 * * * * 27- -2 P (13) 15156 6930 15156 6773 0 0035 0 0157 27- -2 P (14) 15155 0554 15155 0438 0 0035 0 0116 • * * * 27- -2 P (15) 15153 3067 15153 2966 0 0035 0 0101 Appendix 202 27 — 2 P (16) 15151 4470 15151 4356 0 0035 0 0114 27—2 P (17) 15149 4754 15149 4606 0 0150 0 0148 27—2 P (18) 15147 3777 15147 3715 0 0075 0 0062 27—2 P (19) 15145 1794 15145 1681 0 0035 0 0113 27—2 P (20) 15142 8583 15142 8502 0 0040 0 0081 27—2 P (21) 15140 4318 15140 4177 0 0060 0 0141 27 — 2 P (24) 15132 4322 15132 4306 0 0075 0 0016 27 — 2 P (26) 15126 5333 15126 5288 0 0040 0 0045 27 — 2 P (29) 15116 8106 15116 8059 0 0045 0 0047 27 — 2 P (31) 15109 7505 15109 7406 0 0035 0 0099 27—2 P (33) 15102 1905 15102 2055 0 0040 -0 0150**** 27 — 2 P (34) 15098 2523 15098 2611 0 0045 -0 0088 *** 27 — 2 P (35) 15094 1896 15094 1981 0 0035 -0 0085**** 27 — 2 P (37) 15085 7117 15085 7153 0 0035 -0 0036 * 27 — 2 P (39) 15076 7537 15076 7537 0 0035 0 0000 27 — 2 P (40) 15072 0945 15072 0923 0 0035 0 0022 27 — 2 P (41) 15067 3104 15067 3098 0 0035 0 0006 27 — 2 P (42) 15062 4073 15062 4057 0 0035 0 0016 27 — 2 P (43) 15057 3795 15057 3795 0 0035 0 0000 27—2 P (44) 15052 2319 15052 2305 0 0035 0 0014 27—2 P (45) 15046 9538 15046 9583 0 0035 -0 0045 ** 27—2 P (46) 15041 5646 15041 5622 0 0030 0 0024 27—2 P (47) 15036 0409 15036 0416 0 0035 -0 0007 27—2 P 48) 15030 3985 15030 3957 0 0030 0 0028 27—2 P (49) 15024 6246 15024 6238 0 0050 0 0008 27—2 P (50) 15018 7300 15018 7252 0 0050 • 0 0048 27—2 P (51) 15012 6940 15012 6991 0 0045 -0 0051 * 27—2 P (52) 15006 5459 15006 5446 0 0035 0 0013 28 — 1 R (17) 15487 3195 15487 3179 0 0045 0 0016 28 — 1 R (18) 15485 2598 15485 2577 0 0035 0 0021 28 — 1 R (19) 15483 0664 15483 0791 0 0100 -0 0127 ** 28 — 1 R 20) 15480 7777 15480 7819 0 0035 -0 0042 ** 28 — 1 R 21) 15478 3603 15478 3659 0 0035 -0 0056 *** 28 — 1 R 22) 15475 8299 15475 8308 0 0035 -0 0009 28 — 1 R 23) 15473 1694 15473 1763 0 0035 -0 0069 *** 28 — 1 R 24) 15470 4020 15470 4022 0 0035 -0 0002 28 — 1 R 25) 15467 5053 15467 5082 0 0035 -0 0029 28 — 1 R 26) 15464 4917 15464 4940 0 0035 -0 0023 28 — 1 R 27) 15461 3553 15461 3592 0 0035 -0 0039 * 28 — 1 R 28) 15458 1027 15458 1036 0 0035 -0 0009 28 — 1 R 29) 15454 7233 15454 7268 0 0035 -0 0035 28 — 1 R 30) 15451 2279 15451 2283 0 0035 -0 0004 28 — 1 R 31) 15447 6068 15447 6078 0 0035 -0 0010 28 — 1 R 32) 15443 8640 15443 8650 0 0035 -0 0010 28 — 1 R 33) 15439 9986 15439 9993 0 0035 -0 0007 28 — 1 R 34) 15436 0099 15436 0102 0 0035 -0 0003 28 — 1 R (35) 15431 8995 15431 8974 0 0035 0 0021 28 — 1 R 37) 15423 3000 15423 2984 0 0035 0 0016 28 — 1 R 38) 15418 8168 15418 8111 0 0035 0 0057 28 — 1 R 39) 15414 1995 15414 1978 0 0075 0 0017 28 — 1 R 40) 15409 4606 15409 4580 0 0035 0 0026 28 — 1 R 43) 15394 4820 15394 4721 0 0040 0 0099 28 — 1 R 44) 15389 2218 15389 2189 0 0035 0 0029 28 — 1 R 45) 15383 8348 15383 8355 0 0035 -0 0007 28 — 1 R 46) 15378 3226 15378 3210 0 0035 0 0016 28 — 1 R 48) 15366 8950 15366 8948 0 0045 0 0002 28 — 1 R 49) 15360 9801 15360 9812 0 0035 -0 0011 28 — 1 R 50) 15354 9331 15354 9324 0 0035 0 0007 28 — 1 R 51) 15348 7473 15348 7473 0 0035 0 0000 28 — 1 R 52) 15342 4231 15342 4246 0 0035 -0 0015 28 — 1 R 53) 15335 9570 15335 9630 0 0035 -0 0060 *** 28 — 1 P 17) 15485 6983 15485 6974 0 0050 0 0009 Appendix 203 28 — 1 P 18) 15483 5418 15483 5470 0 0045 -0 0052 * | 28 — 1 P 19) 15481 2792 15481 2787 0 0035 0 0005 | 28 — 1 P 21) 15476 3823 15476 3872 0 0035 -0 0049 * * | 28 — 1 P 22) 15473 7605 15473 7638 0 0035 -0 0033 | 28 — 1 P 25) 15465 1715 15465 1794 0 0040 -0 0079 * * * | 28 — 1 P 26) 15462 0713 15462 0791 0 0040 -0 0078 * * * | 28 — 1 P 27) 15458 8528 15458 8590 0 0035 -0 0062 * * * | 28 — 1 P 28) 15455 5155 15455 5186 0 0035 -0 0031 | 28 — 1 P 29) 15452 0507 15452 0578 0 0035 -0 0071**** | 28 — 1 P 30) 15448 4743 15448 4761 0 0035 -0 0018 | 28 — 1 P 31) 15444 7713 15444 7732 0 0035 -0 0019 | 28 — 1 P 32) 15440 9462 15440 9487 0 0035 -0 0025 | 28 — 1 P 33) 15436 9990 15437 0022 0 0035 -0 0032 | 28--1 P (34) 15432 9322 15432 9334 0 0035 -0 0012 | 28 — 1 P 36) 15424 4285 15424 4269 0 0035 0 0016 | 28 — 1 P 38) 15415 4273 15415 4253 0 0035 0 0020 | 28 — 1 P 39) 15410 7364 15410 7376 0 0035 -0 0012 | 28 — 1 P (40) 15405 9252 15405 9246 0 0035 0 0006 | 28 — 1 P (41) 15400 9850 15400 9856 0 0040 -0 0006 | 28 — 1 P 42) 15395 9234 15395 9201 0 0035 0 0033 | 28 — 1 P 43) 15390 7247 15390 7274 0 0035 -0 0027 | 28 — 1 P 44) 15385 4080 15385 4068 0 0035 0 0012 | 28 — 1 P 45) 15379 9506 15379 9575 0 0035 -0 0069 * * * | 28 — 1 P 46) 15374 3795 15374 3789 0 0035 0 0006 | 28 — 1 P 47) 15368 6670 15368 6701 0 0040 -0 0031 | 28 — 1 P 48) 15362 8306 15362 8301 0 0035 0 0005 | 28 — 1 P 49) 15356 8576 15356 8582 0 0035 -0 0006 | 28 — 1 P 50) 15350 7535 15350 7533 0 0035 0 0002 | 28 — 1 P 51) 15344 5119 15344 5145 0 0035 -0 0026 | 28 — 1 P 52) 15338 1398 15338 1405 0 0035 -0 0007 | 28 — 1 P 53) 15331 6273 15331 6302 0 0035 -0 0029 | 28 — 1 P 55) 15318 1898 15318 1958 0 0035 -0 0060 * * * | 28—2 R 3) 15182 8463 15182 8528 0 0035 -0 0065 * * * | 28 — 2 R 4) 15182 4374 15182 4363 0 0045 0 0011 | 28 — 2 R 5) 15181 9031 15181 9040 0 0050 -0 0009 | 28 — 2 R 6) 15181 2557 15181 2557 0 0040 0 0000 | 28 — 2 R 8) 15179 6106 15179 6109 0 0035 -0 0003 | 28 — 2 R 9) 15178 6143 15178 6143 0 0045 0 0000 | 28 — 2 R 10) 15177 5022 15177 5013 0 0030 0 0009 | 28 — 2 R 11) 15176 2771 15176 2718 0 0045 0 0053 | 28 — 2 R 13) 15173 4681 15173 4632 0 0050 0 0049 | 28—2 R (14) 15171 8861 15171 8837 0 0040 0 0024 | 28 — 2 R 15) 15170 1906 15170 1872 0 0035 0 0034 | 28—2 R 16) 15168 3757 15168 3735 0 0040 0 0022 | 28 — 2 R 18) 15164 3942 15164 3939 0 0035 0 0003 | 28—2 R 21) 15157 5340 15157 5410 0 0035 -0 0070 * * * | 28—2 R 22) 15155 0206 15155 0201 0 0040 0 0005 | 28 — 2 R 23) 15152 3772 15152 3805 0 0065 -0 0033 | 28 — 2 R 24) 15149 6213 15149 6220 0 0075 -0 0007 | 28—2 R 25) 15146 7350 15146 7442 0 0100 -0 0092 | 28—2 R 27) 15140 6352 15140 6295 0 0035 0 0057 | 28—2 R (29) 15134 0266 15134 0340 0 0075 -0 0074 | 28—2 R 30) 15130 5553 15130 5549 0 0030 0 0004 | 28—2 R 31) 15126 9493 15126 9546 0 0040 -0 0053 * * | 28—2 R 32) 15123 2363 15123 2324 0 0065 0 0039 | 28 — 2 R 33) 15119 3877 15119 3881 0 0035 -0 0004 | 28 — 2 R 34) 15115 4252 15115 4211 0 0045 0 0041 | 28 — 2 R 35) 15111 3305 15111 3310 0 0035 -0 0005 | 28 — 2 R 36) 15107 1186 15107 1172 0 0065 0 0014 | 28 — 2 R 37) 15102 7804 15102 7793 0 0075 0 0011 | 28 — 2 R 38) 15098 3196 15098 3166 0 0035 0 0030 | 28 — 2 R 39) 15093 7301 15093 7286 0 0035 0 0015 | Appendix 204 28—2 R 41) 15084 1757 15084 1741 0 0035 0 0016 | 28—2 R 42) 15079 2084 15079 2063 0 0035 0 0021 | 28—2 R 43) 15074 1112 15074 1105 0 0035 0 0007 ' | 28—2 R 44) 15068 8868 15068 8858 0 0035 0 0010 | 28 — 2 R 45) 15063 5299 15063 5316 0 0035 -0 0017 | 28 — 2 R 46) 15058 0486 15058 0469 0 0035 0 0017 | 28 — 2 R 47) 15052 4310 15052 4308 0 0055 0 0002 | 28 — 2 R 48) 15046 6821 15046 6823 0 0030 -0 0002 | 28 — 2 R 49) 15040 7957 15040 8004 0 0035 -0 0047 * * | 28 — 2 R 50) 15034 7795 15034 7841 0 0045 -0 0046 * | 28 — 2 R 51) 15028 6279 15028 6320 0 0035 -0 0041 * | 28 — 2 R 52) 15022 3418 15022 3431 0 0045 -0 0013 | 28 — 2 R 53) 15015 9105 15015 9158 0 0050 -0 0053 * | 28—2 P 5) 15181 3841 15181 3893 0 0040 -0 0052 * * | 28—2 P ( 6) 15180 6445 15180 6477 0 0035 -0 0032 | 28—2 P ( 8) 15178 8144 15178 8167 0 0035 -0 0023 | 28—2 P 9) 15177 7260 15177 7272 0 0040 -0 0012 | 28 — 2 P 10) 15176 5208 15176 5216 0 0035 -0 0008 | 28 — 2 P 11) 15175 2005 15175 1998 0 0035 0 0007 | 28 — 2 P 12) 15173 7630 15173 7616 0 0050 0 0014 | 28 — 2 P 13) 15172 2069 15172 2070 0 0040 -0 0001 | 28 — 2 P 16) 15166 8473 15166 8435 0 0040 0 0038 | 28—2 P 18) 15162 6806 15162 6832 0 0035 -0 0026 | 28 — 2 P 21) 15155 5644 15155 5623 0 0100 0 0021 | 28—2 P 22) 15152 9458 15152 9531 0 0035 -0 0073**** | 28 — 2 P 24) 15147 3777 15147 3799 0 0100 -0 0022 | 28 — 2 P 25) 15144 4106 15144 4154 0 0100 -0 0048 | 28 — 2 P 27) 15138 1199 15138 1293 0 0035 -0 0094****| 28 — 2 P 30) 15127 7960 15127 8027 0 0035 -0 0067 * * * | 28 — 2 P 31) 15124 1158 15124 1199 0 0045 -0 0041 | 28 — 2 P 32) 15120 3150 15120 3162 0 0030 -0 0012 | 28 — 2 P 33) 15116 3884 15116 3911 0 0040 -0 0027 | 28 — 2 P 34) 15112 3419 15112 3443 0 0075 -0 0024 | 28 — 2 P 35) 15108 1729 15108 1754 0 0035 -0 0025 | 28 — 2 P 36) 15103 8870 15103 8838 0 0075 0 0032 | 28 — 2 P 37) 15099 4674 15099 4691 0 0035 -0 0017 | 28—2 P 39) 15090 2655 15090 2684 0 0065 -0 0029 | 28 — 2 P 40) 15085 4818 15085 4813 0 0040 0 0005 | 28 — 2 P 41) 15080 5679 15080 5689 0 0035 -0 0010 | 28 — 2 P 42) 15075 5373 15075 5306 0 0065 0 0067 | 28 — 2 P 43) 15070 3644 15070 3658 0 0035 -0 0014 | 28 — 2 P 45) 15059 6571 15059 6536 0 0045 0 0035 | 28 — 2 P 46) 15054 1051 15054 1048 0 0040 0 0003 | 28 — 2 P 47) 15048 4256 15048 4264 0 0055 -0 0008 | 28—2 P 48) 15042 6213 15042 6176 0 0040 0 0037 | 28 — 2 P 49) 15036 6760 15036 6775 0 0035 -0 0015 | 28—2 P 50) 15030 6037 15030 6050 0 0035 -0 0013 | 28—2 P 51) 15024 3944 15024 3992 0 0075 -0 0048 | 28 — 2 P (52) 15018 0663 15018 0590 0 0050 0 0073 | 28—2 P 53) 15011 5787 15011 5831 0 0060 -0 0044 | 28—2 P 55) 14998 2135 14998 2194 0 0035 -0 0059 * * * | 29—1 R 24) 15483 2169 15483 2186 0 0035 -0 0017 | 29—1 R (25) 15480 2381 15480 2432 0 0035 -0 0051 * * | 29 — 1 R (26) 15477 1443 15477 1440 0 0040 0 0003 | 29 — 1 R (27) 15473 9127 15473 9207 0 0035 -0 0080****| 29—1 R 28) 15470 5724 15470 5728 0 0035 -0 0004 | 29—1 R 29) 15467 0966 15467 1000 0 0035 -0 0034 | 29 — 1 R 30) 15463 5005 15463 5017 0 0035 -0 0012 | 29 — 1 R 31) 15459 7772 15459 7776 0 0035 -0 0004 | 29 — 1 R 32) 15455 9295 15455 9271 0 0035 0 0024 | 29 — 1 R 33) 15451 9509 15451 9497 0 0035 0 0012 | 29 — 1 R 34) 15447 8450 15447 8449 0 0035 0 0001 | Appendix 205 29—1 R 35) 15443 6096 15443 6120 0 0035 -0 0024 | 29 — 1 R 36) 15439 2526 15439 2505 0 0035 0 0021 | 29 — 1 R 37) 15434 7607 15434 7597 0 0035 0 0010 | 29 — 1 R 38) 15430 1410 15430 1389 0 0035 0 0021 | 29 — 1 R 39) 15425 3863 15425 3874 0 0035 -0 0011 | 29 — 1 R 40) 15420 5072 15420 5044 0 0035 0 0028 | 29 — 1 R 41) 15415 4894 15415 4890 0 0035 0 0004 | 29 — 1 R 42) 15410 3420 15410 3404 0 0040 0 0016 | 29 — 1 R 43) 15405 0575 15405 0577 0 0040 -0 0002 | 29 — 1 R 44) 15399 6397 15399 6398 0 0035 -0 0001 | 29--1 R 45) 15394 0868 15394 0857 0 0035 0 0011 | 29 — 1 R 46) 15388 3952 15388 3942 0 0035 0 0010 | 29 — 1 R 47) 15382 5617 15382 5641 0 0035 -0 0024 | 29 — 1 R 49) 15370 4773 15370 4829 0 0035 -0 0056 * * * | 29--1 R 50) 15364 2279 15364 2288 0 0035 -0 0009 | 29 — 1 P 22) 15486 8723 15486 8715 0 0065 0 0008 | 29--1 P 25) 15478 0679 15478 0738 0 0035 -0 0059 * * * | 29 — 1 P 26) 15474 8922 15474 8956 0 0035 -0 0034 | 29 — 1 P 30) 15460 9426 15460 9453 0 0035 -0 0027 | 29 — 1 P 31) 15457 1423 15457 1464 0 0035 -0 0041 * | 29 — 1 P 32) 15453 2192 15453 2221 0 0045 -0 0029 | 29—1 P 33) 15449 1655 15449 1720 0 0035 -0 0065 * * * | 29—1 P 34) 15444 9949 15444 9956 0 0035 -0 0007 | 29 — 1 P 35) 15440 6886 15440 6923 0 0035 -0 0037 * | 29 — 1 P 36) 15436 2612 15436 2616 0 0035 -0 0004 | 29—1 P 37) 15431 7020 15431 7029 0 0035 -0 0009 | 29—1 P 39) 15422 1997 15422 1990 0 0035 0 0007 | 29—1 P 40) 15417 2503 15417 2524 0 0045 -0 0021 | 29 — 1 P 41) 15412 1743 15412 1752 0 0035 -0 0009 | 29 — 1 P 42) 15406 9674 15406 9665 0 0035 0 0009 | 29 — 1 P 43) 15401 6227 15401 6255 0 0035 -0 0028 | 29 — 1 P 44) 15396 1531 15396 1514 0 0035 0 0017 | 29 — 1 P 46) 15384 8012 15384 7997 0 0035 0 0015 | 29 — 1 P 47) 15378 9158 15378 9202 0 0035 -0 0044 * * | 29—1 P 48) 15372 9058 15372 9033 0 0035 0 0025 | 29 — 1 P 49) 15366 7451 15366 7479 0 0035 -0 0028 | 29 — 1 P 50) 15360 4522 15360 4527 0 0035 -0 0005 | 29 — 1 P 51) 15354 0104 15354 0162 0 0035 -0 0058 * * * | 29—2 R 3) 15196 6211 15196 6273 0 0065 -0 0062 | 29—2 R 4) 15196 1941 15196 1961 0 0040 -0 0020 | 29—2 R 5) 15195 6448 15195 6460 0 0040 -0 0012 | 29 — 2 R 7) 15194 1909 15194 1887 0 0075 0 0022 | 29—2 R 8) 15193 2760 15193 2815 0 0075 -0 0055 | 29 — 2 R 9) 15192 2586 15192 2550 0 0075 0 0036 | 29—2 R 10) 15191 1143 15191 1091 0 0065 0 0052 | 29—2 R 11) 15189 8443 15189 8437 0 0035 0 0006 | 29—2 R 12) 15188 4564 15188 4588 0 0065 -0 0024 | 29—2 R 13) 15186 9550 15186 9540 0 0055 0 0010 | 29—2 R 14) 15185 3306 15185 3293 0 0045 0 0013 | 29—2 R 15) 15183 5883 15183 5845 0 0045 0 0038 | 29—2 R 19) 15175 3958 15175 3998 0 0045 -0 0040 | 29—2 R 20) 15173 0443 15173 0511 0 0040 -0 0068 * * * | 29—2 R 22) 15167 9834 15167 9890 0 0040 -0 0056 * * | 29—2 R 23) 15165 2702 15165 2749 0 0050 -0 0047 | 29 — 2 R 24) 15162 4343 15162 4384 0 0030 -0 0041 * * | 29 — 2 R 25) 15159 4698 15159 4792 0 0065 -0 0094 * * | 29 — 2 R 26) 15156 3947 15156 3968 0 0030 -0 0021 | 29—2 R 27) 15153 1886 15153 1910 0 0035 -0 0024 | 29 — 2 R 30) 15142 8291 15142 8284 0 0045 0 0007 | 29—2 R 32) 15135 2961 15135 2946 0 0035 0 0015 | 29—2 R 34) 15127 2584 15127 2558 0 0030 0 0026 | 29—2 R 35) 15123 0462 15123 0456 0 0035 0 0006 | Appendix 206 29—2 R 36) 15118 7081 15118 7074 0 0065 0 0007 | 29 — 2 R 37) 15114 2425 15114 2406 0 0035 0 0019 . | 29 — 2 R 38) 15109 6481 15109 6444 0 0040 0 0037 | 29 — 2 R 39) 15104 9186 15104 9182 0 0035 0 0004 | 29 — 2 R 40) 15100 0640 15100 0611 0 0035 0 0029 | 29—2 R 41) 15095 0734 15095 0723 0 0035 0 0011 | 29 — 2 R 42) 15089 9530 15089 9509 0 0035 0 0021 | 29 — 2 R 43) 15084 6981 15084 6960 0 0045 0 0021 | 29 — 2 R 44) 15079 3083 15079 3067 0 0035 0 0016 | 29—2 R 45) 15073 7778 15073 7817 0 0055 -0 0039 | 29 — 2 R 46) 15068 1216 15068 1201 0 0035 0 0015 | 29 — 2 R 47) 15062 3190 15062 3205 0 0035 -0 0015 | 29 — 2 R 48) 15056 3815 15056 3816 0 0040 -0 0001 | 29 — 2 R 49) 15050 2961 15050 3021 0 0045 -0 0060 * * | 29 — 2 R 50) 15044 0769 15044 0804 0 0035 -0 0035 * | 29 — 2 P 5) 15195 1607 15195 1639 0 0035 -0 0032 | 29 — 2 P 6) 15194 4095 15194 4075 0 0045 0 0020. | 29 — 2 P 7) 15193 5326 15193 5322 0 0035 0 0004 | 29 — 2 P 8) 15192 5387 15192 5379 0 0035 0 0008 | 29 — 2 P 9) 15191 4258 15191 4246 0 0035 0 0012 | 29 — 2 P 10) 15190 1903 15190 1921 0 0030 -0 0018 | 29—2 P 11) 15188 8406 15188 8405 0 0040 0 0001 | 29—2 P 12) 15187 3688 15187 3694 0 0035 -0 0006 | 29—2 P 13) 15185 7768 15185 7790 0 0065 -0 0022 | 29—2 P 14) 15184 0672 15184 0689 0 0075 -0 0017 | 29—2 P 15) 15182 2375 15182 2390 0 0065 -0 0015 | 29—2 P 16) 15180 2903 15180 2892 0 0035 0 0011 | 29—2 P 18) 15176 0285 15176 0291 0 0035 -0 0006 | 29—2 P 19) 15173 7151 15173 7183 0 0045 -0 0032 | 29—2 P 20) 15171 2796 15171 2869 0 0075 -0 0073 | 29—2 P 21) 15168 7230 15168 7344 0 0065 -0 0114 * * * | 29—2 P 22) 15166 0524 15166 0608 0 0035 -0 0084**** | 29—2 P 23) 15163 2547 15163 2657 0 0045 -0 0110****| 29—2 P 24) 15160 3459 15160 3487 0 0035 -0 0028 | 29—2 P 26) 15154 1418 15154 1484 0 0030 -0 0066****| 29—2 P 27) 15150 8570 15150 8643 0 0035 -0 0073**** | 29—2 P 28) 15147 4537 15147 4571 0 0030 -0 0034 * | 29—2 P 29) 15143 9205 15143 9264 0 0040 -0 0059 * * | 29—2 P 31) 15136 4860 15136 4931 0 0045 -0 0071 * * * | 29 — 2 P 32) 15132 5891 15132 5896 0 0030 -0 0005 | 29—2 P 33) 15128 5570 15128 5609 0 0035 -0 0039 * | 29 — 2 P 34) 15124 4061 15124 4065 0 0035 -0 0004 . | 29—2 P 35) 15120 1235 15120 1258 0 0035 -0 0023 | 29—2 P 36) 15115 7183 15115 7185 0 0035 -0 0002 | 29—2 P 37) 15111 1842 15111 1837 0 0065 0 0005 | 29—2 P 38) 15106 5200 15106 5210 0 0040 -0 0010 | 29—2 P 41) 15091 7586 15091 7585 0 0035 0 0001 | 29—2 P 42) 15086 5866 15086 5770 0 0075 0 0096 | 29—2 P 44) 15075 8217 15075 8183 0 0035 0 0034 | 29—2 P 45) 15070 2342 15070 2392 0 0065 -0 0050 | 29—2 P 46) 15064 5257 15064 5256 0 0035 0 0001 | 29—2 P 47) 15058 6728 15058 6765 0 0035 -0 0037 * | 29—2 P 48) 15052 6931 15052 6908 0 0035 0 0023 | 29—2 P 50) 15040 3120 15040 3044 0 0100 0 0076 | 29—2 P 51) 15033 8958 15033 9009 0 0035 -0 0051 * * | 29—2 P 52) 15027 3552 15027 3553 0 0040 -0 0001 | 30 — 1 R 27) 15484 5430 15484 5442 0 0040 -0 0012 | 30 — 1 R 28) 15481 1019 15481 0991 0 0065 0 0028 | 30 — 1 R 29) 15477 5237 15477 5249 0 0035 -0 0012 | 30 — 1 R 30) 15473 8207 15473 8212 0 0035 -0 0005 | 30 — 1 R 33) 15461 9267 15461 9267 0 0035 0 0000 | 30 — 1 R 34) 15457 7008 15457 6985 0 0035 0 0023 | Appendix 207 30 — 1 R (35) 15453 3377 15453 3375 0 0035 0 0002 30 — 1 R (36) 15448 8432 15448 8429 0 0035 0 0003 | 30 — 1 R (37) 15444 2172 15444 2139 0 0040 0 0033 | 30 — 1 R (38) 15439 4519 15439 4495 0 0035 0 0024 | 30 — 1 R (39) 15434 5501 15434 5487 0 0035 0 0014 | 30 — 1 R (40) 15429 5120 15429 5107 0 0035 0 0013 | 30 — 1 R (41) 15424 3372 15424 3342 0 0055 0 0030 | 30 — 1 R (42) 15419 0284 15419 0181 0 0035 0 0103 i * * * * 30 — 1 R (43) 15413 5624 15413 5611 0 0040 0 0013 | 30 — 1 R (44) 15407 9616 15407 9619 0 0045 -0 0003 | 30 — 1 R (46) 15396 3334 15396 3304 0 0035 0 0030 | 30 — 1 P (26) 15485 7856 15485 7872 0 0100 -0 0016 | 30 — 1 P (27) 15482 3949 15482 4001 0 0040 -0 0052 * * t 30 — 1 P (29) 15475 2358 15475 2427 0 0035 -0 0069 * * * 1 30 — 1 P (30) 15471 4693 15471 4716 0 0035 -0 0023 | 30 — 1 P (31) 15467 5667 15467 5714 0 0035 -0 0047 * * 1 30 — 1 P (32) 15463 5398 15463 5416 0 0035 -0 0018 | 30 — 1 P (33) 15459 3796 15459 3818 0 0035 -0 0022 | 30 — 1 P (34) 15455 0884 15455 0912 0 0035 -0 0028 | 30 — 1 P (35) 15450 6652 15450 6692 0 0035 -0 0040 * 1 30 — 1 P 36) 15446 1170 15446 1152 0 0040 0 0018 | 30 — 1 P 37) 15441 4271 15441 4283 0 0035 -0 0012 | 30 — 1 P 38) 15436 6098 15436 6079 0 0035 0 0019 | 30 — 1 P 39) 15431 6511 15431 6531 0 0035 -0 0020 | 30 — 1 P 40) 15426 5624 15426 5629 0 0035 -0 0005 | 30 — 1 P (42) 15415 9742 15415 9728 0 0035 0 0014 | 30 — 1 P (43) 15410 4692 15410 4707 0 0035 -0 0015 | 30 — 1 P (45) 15399 0457 15399 0466 0 0045 -0 0009 | 30 — 1 P (47) 15387 0479 15387 0533 0 0040 -0 0054 * * I 30 — 1 P 48) 15380 8352 15380 8395 0 0035 -0 0043 * * 1 30 — 2 R 3) 15208 5226 15208 5188 0 0045 0 0038 | 30 — 2 R ( 4) 15208 0717 15208 0722 0 0065 -0 0005 | 30 — 2 R 5) 15207 5042 15207 5036 0 0040 0 0006 | 30—2 R 6) 15206 8154 15206 8129 0 0035 0 0025 | 30—2 R ( 8) 15205 0660 15205 0649 0 0030 0 0011 | 30—2 R 10) 15202 8325 15202 8274 0 0035 0 0051 i * * 30 — 2 R 11) ' 15201 5294 15201 5246 0 0055 0 0048 | 30—2 R 12) 15200 1026 15200 0991 0 0035 0 0035 1 * 30—2 R 14) 15196 8837 15196 8788 0 0035 0 0049 i * * 30—2 R 15) 15195 0913 15195 0836 0 0035 0 0077 j * * * * 30—2 R 16) 15193 1666 15193 1647 0 0040 0 0019 | 30 — 2 R 18) 15188 9504 15188 9552 0 0040 -0 0048 * 1 30—2 R 19) 15186 6639 15186 6639 0 0045 0 0000 | 30—2 R 20) 15184 2442 15184 2479 0 0040 -0 0037 | 30 — 2 R 22) 15179 0344 15179 0404 0 0040 -0 0060 * * * 1 30 — 2 R 23) 15176 2396 15176 2483 0 0055 -0 0087 * * * 1 30—2 R 24) 15173 3272 15173 3301 0 0035 -0 0029 | 30—2 R 26) 15167 1088 15167 1136 0 0055 -0 0048 | 30—2 R 27) 15163 8122 15163 8145 0 0035 -0 0023 | 30—2 R 28) 15160 3865 15160 3875 0 0035 -0 0010 | 30 — 2 R 29) 15156 8296 15156 8322 0 0035 -0 0026 | 30—2 R 30) 15153 1480 15153 1479 0 0035 0 0001 | 30 — 2 R 34) 15137 1112 15137 1094 0 0035 0 0018 | 30 — 2 R 35) 15132 7716 15132 7711 0 0035 0 0005 | 30—2 R (36) 15128 3014 15128 2998 0 0050 0 0016 | 30—2 R 37) 15123 6974 15123 6947 0 0040 0 0027 | 30 — 2 R 38) 15118 9572 15118 9549 0 0035 0 0023 | 30 — 2 R 39) 15114 0805 15114 0795 0 0035 0 0010 | 30 — 2 R 40) 15109 0683 15109 0674 0 0035 0 0009 | 30 — 2 R 41) 15103 9161 15103 9175 0 0035 -0 0014 | 30 — 2 R 42) 15098 6369 15098 6286 0 0040 0 0083 t * * * * 30 — 2 R 43) 15093 2012 15093 1995 0 0055 0 0017 | Appendix 208 30—2 R (44) 15087 6301 15087 6287 0 0050 0 0014 30—2 R (46) 15076 0553 15076 0563 0 0035 -0 0010 30—2 P ( 5) 15207 0548 15207 0554 0 0040 -0 0006 30—2 P ( 6) 15206 2867 15206 2836 0 0030 0 0031 30—2 P ( 8) 15204 3786 15204 3739 0 0075 0 0047 30—2 P ( 9) 15203 2319 15203 2359 0 0055 -0 0040 30 — 2 P (10) 15201 9765 15201 9756 0 0050 0 0009 30 — 2 P (11) 15200 5938 15200 5929 0 0050 0 0009 30 — 2 P (12) 15199 0895 15199 0877 0 0030 0 0018 30 — 2 P (13) 15197 4603 15197 4599 0 0045 0 0004 30 — 2 P (16) 15191 8364 15191 8387 0 0040 -0 0023 30—2 P (17) 15189 7200 15189 7184 0 0035 0 0016 30 — 2 P (18) 15187 4710 15187 4745 0 0040 -0 0035 30—2 P (20) 15182 6064 15182 6149 0 0050 -0 0085 * * * 30 — 2 P (21) 15179 9902 15179 9986 0 0045 -0 0084 * * * 30--2 P (22) 15177 2501 15177 2576 0 0040 -0 0075 * * * 30--2 P (23) 15174 3839 15174 3916 0 0040 -0 0077 * * * 30 — 2 P (24) 15171 3940 15171 4002 0 0040 -0 0062 *** 30 — 2 P (25) 15168 2716 15168 2832 0 0045 -0 0116**** 30 — 2 P (26) 15165 0359 15165 0400 0 0075 -0 0041 30—2 P (27) 15161 6637 15161 6704 0 0040 -0 0067 * * * 30—2 P (28) 15158 1684 15158 1739 0 0040 -0 0055 * * 30—2 P (29) 15154 5452 15154 5500 0 0035 -0 0048 * * 30—2 P (30) 15150 7939 15150 7982 0 0040 -0 0043 * 30—2 P (31) 15146 9134 15146 9181 0 0035 -0 0047 * * 30—2 P (32) 15142 9088 15142 9091 0 0030 -0 0003 30 — 2 P (33) 15138 7693 15138 7706 0 0035 -0 0013 30—2 P (34) 15134 5000 15134 5021 0 0065 -0 0021 30—2 P (35) 15130 1014 15130 1028 0 0035 -0 0014 30—2 P (36) 15125 5782 15125 5721 0 0075 0 0061 30 — 2 P (37) 15120 9091 15120 9092 0 0040 -0 0001 30—2 P (38) 15116 1137 15116 1134 0 0040 0 0003 30—2 P (40) 15106 1220 15106 1196 0 0055 0 0024 30—2 P (41) 15100 9192 15100 9198 0 0035 -0 0006 30—2 P (42) 15095 5856 15095 5833 0 0045 0 0023 30—2 P (43) 15090 1064 15090 1091 0 0035 -0 0027 30—2 P (44) 15084 5044 15084 4960 0 0040 0 0084 30—2 P (45) 15078 7399 15078 7426 0 0035 -0 0027 30—2 P (46) 15072 8489 15072 8477 0 0035 0 0012 30—2 P (47) 15066 8048 15066 8097 0 0035 -0 0049 31 — 1 R (29) 15486 0664 15486 0678 0 0065 -0 0014 31 — 1 R (30) 15482 2540 15482 2517 0 0035 0 0023 31 — 1 R (31) 15478 2999 15478 3005 0 0035 -0 0006 31 — 1 R (34) 15465 6317 15465 6291 0 0035 0 0026 31 — 1 R (35) 15461 1299 15461 1296 0 0040 0 0003 31 — 1 R (36) 15456 4950 15456 4905 0 0035 0 0045 31 — 1 R (37) 15451 7114 15451 7108 0 0035 0 0006 31 — 1 R (38) 15446 7897 15446 7892 0 0045 0 0005 31 — 1 R (39) 15441 7266 15441 7244 0 0040 0 0022 31 — 1 R (40) 15436 5195 15436 5148 0 0035 0 0047 31 — 1 P (30) 15480 1227 15480 1222 0 0035 0 0005 31 — 1 P (31) 15476 1123 15476 1142 0 0035 -0 0019 31 — 1 P (32) 15471 9750 15471 9721 0 0035 0 0029 31 — 1 P (33) 15467 6972 15467 6949 0 0040 0 0023 31 — 1 P 34) 15463 2851 15463 2821 0 0035 0 0030 31 — 1 P 35) 15458 7315 15458 7327 0 0035 -0 0012 31 — 1 P 36) 15454 0491 15454 0457 0 0035 0 0034 31 — 1 P 37) 15449 2196 15449 2204 0 0035 -0 0008 31 — 1 P 38) 15444 2577 15444 2555 0 0035 0 0022 31 — 1 P 39) 15439 1486 15439 1500 0 0035 -0 0014 31 — 1 P 40) 15433 9025 15433 9027 0 0035 -0 0002 31 — 1 P 42) 15422 9805 15422 9770 0 0035 0 0035 Appendix 209 31—2 R 3) 15218 6056 15218 6034 0 0055 0 0022 31 — 2 R 5) 15217 5577 15217 5529 0 0055 0 0048 31 — 2 R 6) 15216 8440 15216 8397 0 0045 0 0043 31 — 2 R 8) 15215 0481 15215 0370 0 0065 0 0111 * * * 31—2 R 9) 15213 9547 15213 9472 0 0040 0 0075 * * * 31 — 2 R 11) 15211 3950 15211 3898 0 0040 0 0052 * * 31 — 2 R 12) 15209 9261 15209 9219 0 0040 0 0042 * 31 — 2 R 14) 15206 6152 15206 6067 0 0035 0 0085 * * * * 31—2 R (15) 15204 7629 15204 7589 0 0045 0 0040 31—2 R (16) 15202 7850 15202 7839 0 0035 0 0011 31—2 R (19) 15196 0981 15196 0931 0 0075 0 0050 31—2 R (20) 15193 6081 15193 6065 0 0045 0 0016 31—2 R (23) 15185 3629 15185 3719 0 0045 -0 0090**** 31--2 R (26) 15175 9657 15175 9662 0 0035 -0 0005 31--2 R (27) 15172 5649 15172 5683 0 0060 -0 0034 31—2 R (28) 15169 0389 15169 0381 0 0035 0 0008 31—2 R (29) 15165 3742 15165 3750 0 0035 -0 0008 31—2 R (31) 15157 6482 15157 6473 0 0035 0 0009 31—2 R 34) 15145 0436 15145 0400 0 0050 0 0036 31—2 R 35) 15140 5596 15140 5631 0 0075 -0 0035 31 — 2 R 36) 15135 9518 15135 9474 0 0035 0 0044 * * 31 — 2 R 37) 15131 1930 15131 1917 0 0045 0 0013 31 — 2 R 38) 15126 2976 15126 2947 0 0050 0 0029 31—2 R 39) 15121 2573 15121 2551 0 0055 0 0022 31 — 2 R (40) 15116 0784 15116 0715 0 0040 0 0069 * * * 31—2 P 5) 15217 1392 15217 1400 0 0100 -0 0008 31—2 P 7) 15215 4425 15215 4391 0 0065 0 0034 31 — 2 P 8) 15214 4053 15214 4007 0 0040 0 0046 * 31—2 P 9) 15213 2449 15213 2369 0 0035 0 0080 31 — 2 P 10) 15211 9545 15211 9476 0 0030 0 0069 * * * * 31—2 P 11) 15210 5365 15210 5327 0 0040 0 0038 31—2 P 12) 15208 9949 15208 9919 0 0035 0 0030 . 31 — 2 P 14) 15205 5393 15205 5320 0 0040 0 0073 * * * 31—2 P 17) 15199 3942 15199 3937 0 0040 0 0005 31—2 P (20) 15192 1135 15192 1111 0 0035 0 0024 31—2 P 22) 15186 6197 15186 6162 0 0065 0 0035 31—2 P 24) 15180 5999 15180 6061 0 0150 -0 0062 31 — 2 P 26) 15174 0666 15174 0774 0 0040 -0 0108**** 31 — 2 P 27) 15170 6060 15170 6175 0 0075 -0 0115 *** 31 — 2 P 28) 15167 0251 15167 0265 0 0035 -0 0014 31 — 2 P 29) 15163 3018 15163 3038 0 0035 -0 0020 31—2 P 30) 15159 4515 15159 4488 0 0075 0 0027 31—2 P 32) 15151 3399 15151 3395 0 0040 0 0004 31—2 P 34) 15142 6913 15142 6930 0 0065 -0 0017 31 — 2 P 35) 15138 1662 15138 1662 0 0035 0 0000 31 — 2 P 36) 15133 5104 15133 5026 0 0050 0 0078 * * * 31 — 2 P 38) 15123 7635 15123 7610 0 0035 0 0025 31 — 2 P 39) 15118 6888 15118 6808 0 0100 0 0080 31 — 2 P 40) 15113 4613 15113 4594 0 0035 0 0019 31 — 2 P 41) 15108 0914 15108 0954 0 0035 -0 0040 * 31—2 P 42) 15102 5893 15102 5875 0 0040 0 0018 32 — 1 R 31) 15484 7930 15484 7924 0 0035 0 0006 32 — 1 R 32) 15480 5786 15480 5727 0 0055 0 0059 * 32 — 1 R (33) 15476 2099 15476 2101 0 0035 -0 0002 32 — 1 R 34) 15471 7035 15471 7033 0 0035 0 0002 32 — 1 R 35) 15467 0484 15467 0510 0 0035 -0 0026 32 — 1 R 36) 15462 2556 15462 2515 0 0035 0 0041 * 32 — 1 R 37) 15457 3018 15457 3032 0 0035 -0 0014 32 — 1 R 38) 15452 2016 15452 2043 0 0035 -0 0027 32 — 1 P 30) 15486 9790 15486 9777 0 0035 0 0013 32 — 1 P 31) 15482 8540 15482 8541 0 0075 -0 0001 32 — 1 P 32) 15478 5895 15478 5908 0 0035 -0 0013 Appendix 210 32 — 1 P 33) 15474 1852 15474 1868 0 0035 -0 0016 | 32 — 1 P 34) 15469 6410 15469 6412 0 0035 -0 0002 | 32 — 1 P 35) 15464 9485 15464 9527 0 0045 -0 0042 | 32 — 1 P 36) 15460 1198 15460 1200 0 0035 -0 0002 | 32 — 1 P 37) 15455 1415 15455 1418 0 0035 -0 0003 | 32 — 1 P 38) 15450 0208 15450 0165 0 0035 0 0043 | 32 — 1 P 39) 15444 7352 15444 7425 0 0035 -0 0073****| 32 — 1 P 40) 15439 3151 15439 3178 0 0035 -0 0027 | 32—2 R 4) 15226 4968 15226 4919 0 0050 0 0049 | 32 — 2 R 6) 15225 1494 15225 1474 0 0050 0 0020 | 32—2 R 7) 15224 2843 15224 2820 0 0050 0 0023 | 32—2 R 8) 15223 2907 15223 2876 0 0045 0 0031 | 32 — 2 R 9) 15222 1656 15222 1641 0 0045 0 0015 | 32—2 R (10) 15220 9134 15220 9112 0 0035 0 0022 | 32 — 2 R 11) 15219 5348 15219 5289 0 0040 0 0059 | 32 — 2 R 12) 15218 0234 15218 0168 0 0065 0 0066 | 32 — 2 R (15) 15212 7000 15212 6994 0 0045 0 0006 | 32—2 R (16) 15210 6599 15210 6656 0 0040 -0 0057 * * | 32—2 R 19) 15203 7702 15203 7752 0 0035 -0 0050 * * | 32—2 R 20) 15201 2095 15201 2141 0 0065 -0 0046 | 32 — 2 R 22) 15195 6976 15195 6927 0 0080 0 0049 | 32 — 2 R 23) 15192 7339 15192 7314 0 0065 0 0025 | 32 — 2 R 24) 15189 6402 15189 6355 0 0055 0 0047 | 32 — 2 R 26) 15183 0394 15183 0377 0 0065 0 0017 | 32 — 2 R 27) 15179 5305 15179 5343 0 0035 -0 0038 * | 32 — 2 R 28) 15175 8942 15175 8937 0 0035 0 0005 | 32 — 2 R 29) 15172 1133 15172 1149 0 0035 -0 0016 | 32 — 2 R 30) 15168 1985 15168 1970 0 0035 0 0015 | 32 — 2 R 31) 15164 1375 15164 1392 0 0055 -0 0017 | 32 — 2 R 32) 15159 9402 15159 9402 0 0065 0 0000 | 32 — 2 R 34) 15151 1155 15151 1142 0 0055 0 0013 | 32 — 2 R 35) 15146 4875 15146 4845 0 0065 0 0030 | 32 — 2 R 36) 15141 7133 15141 7084 0 0075 0 0049 | 32 — 2 R 37) 15136 7843 15136 7841 0 0035 0 0002 | 32 — 2 R 38) 15131 7110 15131 7098 0 0035 0 0012 | 32 — 2 P 5) 15225 5048 15225 5078 0 0040 -0 0030 | 32 — 2 P 6) 15224 7013 15224 7033 0 0065 -0 0020 | 32 — 2 P 7) 15223 7738 15223 7702 0 0040 0 0036 | 32—2 P 9) 15221 5189 15221 5178 0 0060 0 0011 | 32—2 P 11) 15218 7525 15218 7496 0 0065 0 0029 | 32 — 2 P 12) 15217 1759 15217 1716 0 0035 0 0043 | 32 — 2 P 13) 15215 4669 15215 4641 0 0055 0 0028 | 32 — 2 P 15) 15211 6674 15211 6597 0 0100 0 0077 | 32 — 2 P 17) 15207 3329 15207 3342 0 0055 -0 0013 | 32 — 2 P 20) 15199 8627 15199 8636 0 0035 -0 0009 | 32 — 2 P 23) 15191 1962 15191 2049 0 0035 -0 0087****| 32 — 2 P 24) 15188 0510 15188 0525 0 0060 -0 0015 | 32—2 P 25) 15184 7645 15184 7662 0 0040 -0 0017 | 32—2 P (26) 15181 3389 15181 3455 0 0040 -0 0066 * * * | 32—2 P (27) 15177 7878 15177 7896 0 0040 -0 0018 | 32—2 P (28) 15174 0972 15174 0979 0 0035 -0 0007 | 32—2 P (30) 15166 2999 15166 3043 0 0075 -0 0044 | 32—2 P 31) 15162 2028 15162 2008 0 0065 0 0020 . | 32—2 P (32) 15157 9612 15157 9582 0 0065 0 0030 | 32—2 P (34) 15149 0499 15149 0521 0 0040 -0 0022 | 32—2 P (36) 15139 5755 15139 5769 0 0030 . -o 0014 | 32—2 P (37) 15134 6188 15134 6226 0 0035 -0 0038 * | 32 — 2 P 39) 15124 2704 15124 2732 0 0035 -0 0028 | 32 — 2 P 40) 15118 8746 15118 8745 0 0055 0 0001 | 33 — 1 R 32) 15485 1848 15485 1823 0 0035 0 0025 | 33 — 1 P 32) 15483 4886 15483 488 6 0 0045 0 0000 | 33 — 1 P 34) 15474 2517 15474 2508 0 0035 0 0009 | Appendix 211 33 — 2 R 4) 15233. 2399 15233. 2347 0. 0075 0 0052 | 33 — 2 R 5) 15232. 6095 15232. 6059 0. 0050 0 0036 | 33 — 2 R 6) 15231. 8484 15231. 8449 0. 0035 0 0035 | 33 — 2 R 8) 15229 9334 15229. 9257 0. 0040 0 0077 | 33—2 R 9) 15228. 7673 15228 7670 0. 0040 0 0003 | 33 — 2 R 10) 15227 4723 15227. 4754 0. 0050 -0 0031 | 33—2 R 12) 15224 4946 15224 4923 0 0035 0 0023 | 33 — 2 R 13) 15222 8018 15222 8002 0 0040 0 0016 | 33—2 R 14) 15220 9661 15220 9739 0 0055 -0 0078 * * | 33—2 R 15) 15219 0088 15219 0132 0 0035 -0 0044 * * | 33 — 2 R 17) 15214 6813 15214 6867 0 0055 -0 0054 | 33 — 2 R 22) 15201 4862 15201 4850 0 0035 0 0012 | 33 — 2 R 23) 15198 4294 15198 4310 0 0065 -0 0016 | 33—2 R 24) 15195 2397 15195 2375 0 0035 0 0022 | 33—2 R 26) 15188 4341 15188 4286 0 0065 0 0055 | 33—2 R 27) 15184 8077 15184 8113 0 0040 -0 0036 | 33—2 R 28) 15181 0521 15181 0508 0 0035 0 0013 | 33—2 R 29) 15177 1430 15177 1457 0 0035 -0 0027 | 33—2 R 30) 15173 0961 15173 0949 0 0035 0 0012 | 33—2 R 31) 15168 8965 15168 8968 0 0035 -0 0003 | 33—2 R 32) 15164 5515 15164 5498 0 0030 0 0017 | 33—2 P 5) 15232 2680 15232 2679 0 0045 0 0001 | 33—2 P 6) 15231 4443 15231 4461 0 0065 -0 0018 | 33—2 P 7) 15230 4944 15230 4921 0 0050 0 0023 | 33—2 P 8) 15229 4059 15229 4059 0 0065 0 0000 | 33—2 P 9) 15228 1841 15228 1874 0 0040 -0 0033 | 33 — 2 P 10) 15226 8388 15226 8363 0 0055 0 0025 | 33 — 2 P 11) 15225 3533 15225 3525 0 0100 0 0008 | 33 — 2 P 12) 15223 7353 15223 7358 0 0040 -0 0005 | 33 — 2 P 13) 15221 9893 15221 9858 0 0045 0 0035 | 33—2 P 14) 15220 1027 15220 1024 0 0035 0 0003 | 33—2 P 15) 15218 0791 15218 0852 0 0040 -0 0061 * * * | 33—2 P (17) 15213 6455 15213 6480 0 0085 -0 0025 | 33—2 P 18) 15211 2247 15211 2274 0 0035 -0 0027 | 33—2 P (19) 15208 6662 15208 6715 0 0075 -0 0053 | 33—2 P (21) 15203 1444 15203 1518 0 0040 -0 0074 * * * | 33—2 P (22) 15200 1806 15200 1871 0 0035 -0 0065 * * * | 33—2 P (24) 15193 8457 15193 8448 0 0075 0 0009 | 33—2 P (25) 15190 4654 15190 4659 0 0035 -0 0005 | 33 — 2 P (28) 15179 4917 15179 4889 0 0035 0 0028 | 33 — 2 P (29) 15175 5450 15175 5468 0 0035 -0 0018 | 33 — 2 P (30) 15171 4614 15171 4615 0 0035 -0 0001 | 33—2 P (31) 15167 2293 15167 2317 0 0045 -0 0024 . | 33—2 P (32) 15162 8534 15162 8561 0 0040 -0 0027 | 33—2 P (33) 15158 3326 15158 3333 0 0035 -0 0007 | 33—2 P (34) 15153 6657 15153 6617 0 0065 0 0040 | 34—2 R ( 5) 15237 8541 15237 8496 0 0040 0 0045 | 34—2 R ( 9) 15233 8984 15233 8870 0 0075 0 0114 | 34—2 R (10) 15232 5633 15232 5550 0 0100 0 0083 | 34—2 R (12) 15229 4821 15229 4793 0 0045 0 0028 | 34—2 R (13) 15227 7364 15227 7349 0 0040 0 0015 . | 34—2 R (14) 15225 8476 15225 8522 0 0040 -0 0046 * | 34 — 2 R (15) 15223 8211 15223 8309 0 0100 -0 0098 | 34—2 R (19) 15214 3449 15214 3478 0 0065 -0 0029 | 34—2 R (20) 15211 6259 15211 6244 0 0035 0 0015 | 34—2 R (21) 15208 7554 15208 7584 0 0040 -0 0030 | 34—2 R (22) 15205 7527 15205 7491 0 0035 0 0036 | 34 — 2 R (23) 15202 5907 15202 5954 0 0030 -0 0047 * * * | 34 — 2 R (24) 15199 2986 15199 2962 0 0030 0 0024 | 34 — 2 R (25) 15195 8511 15195 8503 0 0035 0 0008 | 34 — 2 R (27) 15188 5162 15188 5130 0 0030 0 0032 | 34 — 2 P ( 5) 15237 5490 15237 5512 0 0035 -0 0022 | Appendix 212 34 — 2 P 6) 15236 7104 15236 7114 0 0030 -0 0010 | 34 — 2 P 7) 15235 7362 15235 7358 0 0035 0 0004 | 34 — 2 P 8) 15234 6227 15234 6243 0 0045 -0 0016 | 34 — 2 P 9) 15233 3764 15233 3767 0 0040 -0 0003 | 34 — 2 P 10) 15231 9916 15231 9929 0 0040 -0 0013 | 34 — 2 P 11) 15230 4683 15230 4725 0 0075 -0 0042 | 34—2 P 13) 15227 0133 15227 0211 0 0065 -0 0078 * | 34—2 P 14) 15225 0946 15225 0894 0 0075 0 0052 | 34—2 P 16) 15220 8087 15220 8121 0 0040 -0 0034 | 34 — 2 P 17) 15218 4713 15218 4657 0 0040 0 0056 | 34 — 2 P 19) 15213 3495 15213 3548 0 0055 -0 0053 | 34 — 2 P 20) 15210 5892 15210 5891 0 0035 0 0001 | 34 — 2 P 22) 15204 6322 15204 6341 0 0035 -0 0019 | 34 — 2 P 23) 15201 4430 15201 4432 0 0035 -0 0002 | 34 — 2 P 24) 15198 1065 15198 1089 0 0045 -0 0024 | 34 — 2 P 27) 15187 2298 15187 2354 0 0055 -0 0056 * | 35 — 2 R 4) 15242 4322 15242 4417 0 0100 -0 0095 | 35—2 R 7) 15240 0056 15240 0040 0 0040 0 0016 | 35 — 2 R 8) 15238 9112 15238 9113 0 0065 -0 0001 | 35 — 2 R 10) 15236 3047 15236 3041 0 0035 0 0006 | 35 — 2 R 11) 15234 7895 15234 7889 0 0050 0 0006 | 35—2 R 13) 15231 3287 15231 3328 0 0035 -0 0041 * | 35 — 2 R 14) 15229 3916 15229 3910 0 0050 0 0006 | 35 — 2 R 16) 15225 0717 15225 0768 0 0050 -0 0051 * | 35 — 2 R 20) 15214 7069 15214 7040 0 0055 0 0029 | 35—2 P ( 6) 15240 6525 15240 6530 0 0075 -0 0005 | 35—2 P 9) 15237 2396 15237 2398 0 0040 -0 0002 | 35—2 P 12) 15232 5633 15232 5645 0 0055 -0 0012 | 35 — 2 P 14) 15228 7426 15228 7420 0 0075 0 0006 | 35 — 2 P 15) 15226 6065 15226 6178 0 0045 -0 0113****| 35 — 2 P 16) 15224 3405 15224 3509 0 0065 -0 0104 * * * | 35 — 2 P 17) 15221 9308 15221 9407 0 0040 -0 0099****| 35 — 2 P 19) 15216 6800 15216 6876 0 0045 -0 0076 * * * | 35 — 2 P 20) 15213 8464 15213 8431 0 0035 0 0033 | 35 — 2 P 21) 15210 8518 15210 8522 0 0035 -0 0004 | 35 — 2 P 23) 15204 4190 15204 4264 0 0075 -0 0074 | 35 — 2 P 24) 15200 9935 15200 9890 0 0040 0 0045 | 35 — 2 P 25) 15197 3972 15197 3999 0 0050 -0 0027 | 36 — 2 R 3) 15245 7900 15245 7872 0 0035 0 0028 | 36—2 R 5) 15244 5377 15244 5391 0 0035 -0 0014 . | 36—2 R ( 6) 15243 7054 15243 6997 0 0035 0 0057 | 36—2 R 7) 15242 7203 15242 7163 0 0040 0 0040 | 36—2 R ( 8) 15241 5978 15241 5886 0 0075 0 0092 | 36—2 R ( 9) 15240 3073 15240 3161 0 0040 -0 0088**** | 36—2 R (11) 15237 3291 15237 3353 0 0050 -0 0062 * * | 36—2 R (12) 15235 6240 15235 6257 0 0050 -0 0017 | 36—2 R (14) 15231 7658 15231 7649 0 0035 0 0009 | 36—2 R (15) 15229 6125 15229 6121 0 0040 0 0004 | 36—2 R (18) 15222 2582 15222 2509 0 0035 0 0073 | 36—2 R (19) 15219 4990 15219 4917 0 0035 0 0073 | 36—2 P 5) 15244 3259 15244 3238 0 0040 0 0021 | 36—2 P 6) 15243 4425 15243 4463 0 0035 -0 0038 * | 36—2 P (12) 15235 1625 15235 1589 0 0045 0 0036 | 36—2 P (14) 15231 2274 15231 2358 0 0045 -0 0084 * * * | 36—2 P (16) 15226 7218 15226 7248 0 0075 -0 0030 | 36—2 P (17) 15224 2454 15224 2469 0 0035 -0 0015 | 36—2 P (18) 15221 6200 15221 6193 0 0050 0 0007 | 37—2 R ( 1) 15248 3525 15248 3462 0 0055 0 0063 | 37—2 R ( 2) 15248 0627 15248 0686 0 0065 -0 0059 | 37—2 R ( 3) 15247 6466 15247 6440 0 0050 0 0026 | 37—2 R ( 5) 15246 3592 15246 3530 0 0040 0 0062 | 37—2 R ( 6) 15245 4913 15245 4860 0 0100 0 0053 | Appendix 37- -2 R( 7) 15244 4682 15244 4708 0 0040 -0 0026 37- -2 R( 8) 15243 2992 15243 3070 0 0055 -0 0078 37- -2 R(10) 15240 5303 15240 5309 0 0055 -0 0006 37- -2 P( 6) 15245 2839 15245 2836 0 0065 0 0003 37- -2 P( 7) 15244 2429 15244 2392 0 0050 0 0037 37- -2 P(10) 15240 2224 15240 2176 0 0050 0 0048 37- -2 P(12) 15236 7855 15236 7913 0 0100 -0 0058 37- -2 P(13) 15234 8462 15234 8525 0 0075 -0 0063 37- -2 P(15) 15230 5194 15230 5192 0 0100 0 0002 Appendix 214 T a b l e A - 2 : Q-lines used i n the analysis of Omega-type doubling ( a l l units i n cm - 1) . Transition Observed Calculated Uncert. Obs-Cal 19--1 q< 7) 15278. 3097 15278. 3176 0. 0065 -0. 0079 19--1 q 8) 15277. 5815 15277. 5827 0. 0065 -0. 0012 19--1 q 9) 15276. 7500 15276. 7559 0. 0055 -0. 0059 19--1 q 12) 15273. 7210 15273 7233 0 0065 -0. 0023 19--1 q 13) 15272 5248 15272 5281 0 0055 -0 0033 19--1 q 15) 15269 8602 15269 8611 0 0065 -0 0009 19--1 q 18) 15265 1716 15265 1676 0 0065 0 0040 19--1 q 19) 15263 4150 15263 4180 0 0050 -0 0030 19--1 q 20) 15261 5771 15261 5758 0 0050 0 0013 19--1 q 21) 15259 6405 15259 6407 0 0050 -0 0002 19--1 q 22) 15257 6114 15257 6127 0 0050 -0 0013 19--1 q 23) 15255 4940 15255 4918 0 0050 0 0022 19--1 q 24) 15253 2783 15253 2777 0 0050 0 0006 19--1 q 25) 15250 9756 15250 9705 0 0050 0 0051 19--1 q 26) 15248 5782 15248 5698 0 0050 0 0084 19--1 q 27) 15246 0791 15246 0757 0 0050 0 0034 19--1 q 30) 15238 0341 15238 0311 0 0075 0 0030 19--1 q 32) 15232 2029 15232 1980 0 0050 0 0049 19--1 q 33) 15229 1499 15229 1400 0 0050 0 0099 19--1 q 38) 15212 4337 15212 4278 0 0050 0 0059 19--1 q 41) 15201 2647 15201 2558 0 0050 0 0089 19--1 q 43) 15193 3304 15193 3272 0 0055 0 0032 19--1 q 44) 15189 2232 15189 2181 0 0055 0 0051 19--1 q 45) 15185 0188 15185 0120 0 0055 0 0068 19--1 q 48) 15171 8120 15171 8096 0 0055 0 0024 19--1 q 51) 15157 7287 15157 7249 0 0055 0 0038 19--1 q 54) 15142 7456 15142 7498 0 0055 -0 0042 19--1 q 58) 15121 3838 15121 3820 0 0055 0 0018 19--1 q 59) 15115 7818 15115 7875 0 0055 -0 0057 19--1 q 60) 15110 0858 15110 0911 0 0055 -0 0053 19--1 q 61) 15104 2911 15104 2925 0 0055 -0 0014 19--1 q 62) 15098 3901 15098 3913 0 0055 -0 0012 19--1 q 64) 15086 2735 15086 2791 0 0055 -0 0056 19--1 q 65) 15080 0601 15080 0673 0 0055 -0 0072 20--1 q 2) 15316 0280 15316 0227 0 0075 0 0053 20--1 q 4) 15315 3614 15315 3600 0 0075 0 0014 20--1 q 5) 15314 8787 15314 8865 0 0040 -0 0078 20--1 q 6) 15314 3141 15314 3184 0 0045 -0 0043 20--1 q ( 7) 15313 6501 15313 6554 0 0035 -0 0053 20--1 q ( 9) 15312 0401 15312 0451 0 0035 -0 0050 20--1 q (11) 15310 0516 15310 0552 0 0040 -0 0036 20--1 q (1'2) 15308 9177 15308 9178 0 0055 -0 0001 20--1 q (13) 15307 6829 15307 6853 0 0045 -0 0024 20--1 q (14) 15306 3557 15306 3578 0 0055 -0 0021 20--1 q (15) 15304 9327 15304 9350 0 0055 -0 0023 20--1 q (16) 15303 4163 15303 4170 0 0055 -0 0007 20--1 q (17) 15301 8021 15301 8037 0 0055 -0 0016 20--1 q (18) 15300 0958 15300 0949 0 0055 0 0009 20--1 q (19) 15298 2953 15298 2907 0 0055 0 0046 20--1 q (21) 15294 3939 15294 3952 0 0055 -0 0013 20--1 q (22) 15292 3032 15292 3037 0 0055 -0 0005 20--1 q (23) 15290 1171 15290 1164 0 0055 0 0007 20--1 q (24) 15287 .8343 15287 .8330 0 0055 0 0013 20--1 q (25) 15285 4565 15285 .4534 0 0055 0 0031 Appendix 215 20--1 q( 26) 15282. 9794 15282 20--1 q( 28) 15277 . 7379 15277 20--1 q( 29) 15274. 9719 15274 20--1 q( 30) 15272. 1111 15272 20--1 q( 32) 15266. 0941 15266 20--1 q( 33) 15262. 9391 15262 20--1 q( 34) 15259. 6854 15259 20--1 q( 35) 15256. 3381 15256 20--1 q< 37) 15249. 3440 15249 20--1 q 39) 15241. 9512 15241 20--1 q 40) 15238. 1152 15238 20--1 q 42) 15230. 1382 15230 20--1 q 45) 15217. 4137 15217 20--1 q 46) 15212. 9735 15212 20--1 q 47) 15208. 4334 15208 20--1 q 51) 15189. 2539 15189 20--1 q 52) 15184. 2019 15184 20--2 q 3) 14994. 7712 14994 20--2 q 6) 14993 3542 14993 20--2 q 7) 14992 7018 14992 20--2 q 8) 14991 9354 14991 20--2 q 9) 14991 1044 14991 20--2 q 10) 14990 1561 14990 21--1 q 5) 15347 0472 15347 21--1 q 6) 15346 4697 15346 21--1 q 7) 15345 7822 15345 21--1 q 8) 15345 0040 15345 21--1 q 9) 15344 1247 15344 21--1 q 10) 15343 1554 15343 21--1 q 11) 15342 0728 15342 21--1 q 12) 15340 9079 15340 21--1 q (13) 15339 6422 15339 21--1 q (14) 15338 2730 15338 21--1 q (15) 15336 8111 15336 21--1 q (25) 15316 7705 15316 21--1 q (27) 15311 5575 15311 21--1 q (30) 15303 0170 15303 21--1 q (31) 15299 9717 15299 21--1 q (34) 15290 2294 15290 21--1 q (36) 15283 2317 15283 21--1 q (39) 15271 9696 15271 21--1 q (42) 15259 7905 15259 21--1 q (43) 15255 .5313 15255 21--1 q (44) 15251 .1620 15251 21--1 q (46) 15242 .1193 15242 21--2 q ( 4) 15026 .5707 15026 21--2 q ( 7) 15024 . 8259 15024 21--2 q ( 9) 15023 .1868 15023 21--2 q (10) 15022 .2084 15022 21--2 q (11) 15021 . 1530 15021 21--2 q (12) 15019 . 9905 15019 22--1 q (46) 15268 .4682 15268 22--1 q (47) 15263 . 6489 15263 22--1 q (48) 15258 .7258 15258 22--1 q (49) 15253 . 6988 15253 22--1 q (51) 15243 .3269 15243 22--1 q (53) 15232 .5042 15232 22--2 q ( 3) 15056 .2800 15056 22--2 q ( 4) 15055 . 8797 15055 22--2 q ( 5) 15055 .3861 15055 9775 0. 0055 0. 0019 7362 0. 0055 0. 0017 9705 0. 0055 0. 0014 1079 0. 0065 0. 0032 0914 0. 0055 0. 0027 9371 0. 0055 0. 0020 6853 0. 0055 0. 0001 3356 0. 0055 0. 0025 3421 0. 0055 0. 0019 9551 0. 0055 -0. 0039 1134 0. 0055 0. 0018 1324 0. 0055 0. 0058 4135 0. 0055 0. 0002 9735 0. 0055 0. 0000 4327 0. 0055 0. 0007 2568 0. 0055 -0. 0029 2079 0. 0055 -0. 0060 * 7681 0 0065 0 0031 3575 0 0100 -0 0033 6991 0 0055 0 0027 9465 0 0055 -0 0111 * * * * 0997 0 0075 0 0047 1587 0 0050 -0 0026 0558 0 0035 -0 0086 * * * * 4709 0 0040 -0 0012 7885 0 0035 -0 0063 * * * 0084 0 0035 -0 0044 * 1307 0 0035 -0 0060 * * * 1554 0 0100 0 0000 0823 0 0045 -0 0095 * • * * 9114 0 0075 -0 0035 6426 0 0040 -0 0004 2760 . 0 0075 -0 0030 8113 0 0075 -0 0002 7552 0 0150 0 0153 5579 0 0065 -0 0004 0154 0 0055 0 0016. 9682 0 0055 0 0035 2249 0 0075 0 0045 2260 0 0075 0 0057 9682 0 0055 0 0014 7933 0 0055 -0 0028 5301 0 .0075 0 .0012 1638 0 .0075 -0 .0018 1213 0 .0075 -0 .0020 5752 0 .0100 -0 .0045 8321 0 . 0065 -0 .0062 1854 0 .0055 0 .0014 2165 0 .0055 -0 .0081 * * 1505 0 .0065 0 .0025 9874 0 .0065 0 .0031 .4593 0 .0100 0 .0089 . 6478 0 .0055 0 .0011 .7289 0 .0100 -0 .0031 .7021 0 .0055 -0 .0033 .3236 0 .0150 0 .0033 .5091 0 . 0065 -0 .0049 .2928 0 .0065 -0 .0128 * * * .8947 0 .0100 -0 . 0150 * * .3969 0 .0075 -0 .0108 * * Appendix 216 22--2 q 6) 15054 7919 15054 7996 0 0055 -0 0077 * * t 22--2 q 8) 15053 2980 15053 3059 0 0055 -0 0079 ** 1 22--2 q 14) 15046 4277 15046 4300 0 0055 -0 0023 22--2 q 15) 15044 9320 15044 9341 0 0055 -0 0021 22--2 q 18) 15039 8461 15039 8450 0 0100 0 0011 22--2 q 19) 15037 9480 15037 9478 0 0065 0 0002 22--2 q 23) 15029 3648 15029 3523 0 0100 0 0125 22--2 q 25) 15024 4526 15024 4486 0 0100 0 0040 22--2 q 26) 15021 8487 15021 8448 0 0055 0 0039 23--1 q 2) 15404 2883 15404 2935 0 0075 -0 0052 23--1 q 4) 15403 5706 15403 5734 0 0100 -0 0028 23--1 q 5) 15403 0479 15403 0590 0 0100 -0 0111 * 1 23--1 q 8) 15400 8944 15400 8981 0 0055 -0 0037 23--1 q 9) 15399 9660 15399 9718 0 0045 -0 0058 * 1 23--1 q 10) 15398 9403 15398 9423 0 0100 -0 0020 23--1 q 11) 15397 8056 15397. 8096 0 0075 -0 0040 23--1 q 14) 15393 7916 15393 7920 0 0035 -0 0004 23--1 q 15) 15392 2494 15392 2460 0 0045 0 0034 23--1 q 16) 15390 6009 15390 5965 0 0075 0 0044 23--1 q 19) 15385 0291 15385 0255 0 0075 0 0036 23--1 q 20) 15382 9661 15382 9608 0 0075 0 0053 23--1 q 23) 15376 1414 15376 1414 0 0075 0 0000 23--1 q 27) 15365 5959 15365 5850 0 0075 0 0109 23--1 q 29) 15359 6797 15359 6766 0 0150 0 0031 23--1 q 33) 15346 6040 15346 5915 0 0150 0 0125 23--1 q 38) 15327 8476 15327 8388 0 0150 0 0088 23--1 q 39) 15323 7759 15323 7662 0 0150 0 0097 23--1 q 43) 15306 3960 15306 3926 0 0075 0 0034 23--1 q 44) 15301 7751 15301 7768 0 0055 -0 0017 23--1 q 46) 15292 2199 15292 2158 0 0055 0 0041 23--1 q 50) 15271 7614 15271 7656 0 0055 -0 0042 23--1 q 51) 15266 3712 15266 3739 0 0055 -0 0027 23--1 q 53) 15255 2500 15255 2525 0 0075 -0 0025 23--1 q 54) 15249 5192 15249 5218 0 0055 -0 0026 23--2 q 2) 15083 3169 15083 3209 0 0100 -0 0040 23--2 q 3) 15083 0043 15083 0143 0 0040 -0 0100 **** j 23--2 q 4) 15082 5976 15082 6054 0 0040 -0 0078 * * * j 23--2 q 7) 15080 7575 15080 7651 0 0040 -0 0076 * * * 1 23--2 q 8) 15079 9425 15079 9469 0 0035 -0 0044 * 1 23--2 q 9) 15079 0214 15079 0264 0 0055 -0 0050 23--2 q 10) 15078 0030 15078 0034 0 0100 -0 0004 23--2 q 11) 15076 8734 15076 8779 0 0045 -0 0045 23--2 q 12) 15075 6515 15075 6498 0 0100 0 0017 23--2 q 13) 15074 3181 15074 3190 0 0045 -0 0009 23--2 q 14) 15072 8905 15072 8855 0 0065 0 0050 23--2 q 15) 15071 3489 15071 3492 0 0040 -0 0003 23--2 q 17) 15067 9685 15067 9678 0 0040 0 0007 23--2 q 18) 15066 1268 15066 1225 0 0065 0 0043 23--2 q 19) 15064 1765 15064 1741 0 0045 0 0024 23--2 q 20) 15062 1285 15062 1222 0 0045 0 0063 23--2 q 22) 15057 7166 15057 7082 0 0100 0 0084 23--2 q 24) 15052 8797 15052 8792 0 0065 0 0005 23--2 q 27) 15044 8610 15044 8553 0 0100 0 0057 23--2 q 31) 15032 6995 15032 6927 0 0100 0 0068 24--1 q 3) 15428 2767 15428 2839 0 0055 -0 0072 * * j 24--1 q 4) 15427 8542 15427 8618 0 0045 -0 0076 * * * 1 24--1 q 5) 15427 3191 15427 3341 0 0050 -0 0150 * * * * j 24--1 q 7) 15425 9567 15425 9618 0 0035 -0 0051 24--1 q 8) 15425 1118 15425 1171 0 0045 -0 0053 * t 24--1 q 9) 15424 1546 15424 1667 0 0045 -0 0121 * * * * i Appendix 217 24- -1 q 10) 15423. 1024 15423. 1105 0. 0060 - 0 . 0081 24- -1 q I D 15421. 9443 15421. 9484 0 0035 -0 0041 24- -1 q 14) 15417 8149 15417 8264 0 0075 -0 0115 24- -1 q 15) 15416 2408 15416 2401 0 0035 0 0007 24- -1 q 17) 15412 7513 15412 7487 0 0075 0 0026 24- -1 q 19) 15408 8313. 15408 8313 0 0045 0 0000 24- -1 q 23) 15399 7215 15399 7148 0 0100 0 0067 24- -1 q 27) 15388 8930 15388 8811 0 0100 0 0119 24- -1 q 29) 15382 8246 15382 8169 0 0100 0 0077 24- -1 q 30) 15379 6252 15379 6222 0 0075 0 0030 24- -1 q 33) 15369 3886 15369 3850 0 0075 0 0036 24- -1 q 37) 15354 2115 15354 2024 0 0100 0 0091 24- -1 q 39) 15345 9551 15345 9490 0 0150 0 0061 24- -1 q 48) 15303 2587 15303 2615 0 0075 -0 0028 24- -1 q 49) 15297 9465 15297 9502 0 0075 -0 0037 24- -1 q 50) 15292 5221 15292 5237 0 0075 -0 0016 24- -1 q 54) 15269 6515 15269 6541 0 0055 -0 0026 24- -1 q 55) 15263 6426 15263 6431 0 0055 -0 0005 24- -1 q 56) 15257 5087 15257 5137 0 0055 -0 0050 24- -1 q 57) 15251 2634 15251 2650 0 0075 -0 0016 24- -1 q 59) 15238 4116 15238 4075 0 0055 0 0041 24- -2 q 2) 15107 6372 15107 6280 0 0100 0 0092 24- -2 q 3) 15107 3027 15107 3133 0 0045 -0 0106 24- -2 q 4) 15106 8870 15106 8938 0 0045 -0 0068 24 - -2 q 5) 15106 3652 15106 3693 0 0100 -0 0041 24- -2 q 6) 15105 7376 15105 7398 0 0055 -0 0022 24- -2 q 7) 15105 0009 15105 0054 0 0040 -0 0045 24- -2 q 8) 15104 1638 15104 1659 0 0045 -0 0021 24- -2 q 9) 15103 2195 15103 2213 0 0100 -0 0018 24- -2 q 11) 15101 0187 15101 0166 0 0075 0 0021 24- -2 q 12) 15099 7548 15099 7564 0 0055 -0 0016 24- -2 q (13) 15098 3897 15098 3909 0 0075 -0 0012 24- -2 q (14) 15096 9203 15096 9198 0 0150 0 0005 24- -2 q (16) 15093 6625 15093 6611 0 0045 0 0014 24 - -2 q (19) 15087 9842 15087 9799 0 0065 0 0043 24- -2 q (24) 15076 3953 15076 3876 0 0100 0 0077 24- -2 q (28) 15065 1952 15065 1916 0 0075 0 0036 24- -2 q (30) 15058 9548 15058 9488 0 0100 0 0060 24- -2 q (32) 15052 2850 15052 2740 0 0150 0 0110 24- -2 q (35) 15041 4593 15041 4472 0 0100 0 0121 24- -2 q (36) 15037 6183 15037 .6200 0 0075 -0 0017 24- -2 q (37) 15033 6965 15033 6833 0 .0100 0 0132 25- -1 q 24) 15418 3901 15418 3905 0 0075 -0 0004 25 - -1 q 25) 15415 6706 15415 6673 0 0075 0 0033 25 - -1 q 26) 15412 8362 15412 8336 0 0055 0 0026 25 - -1 q 28) 15406 8384 15406 8336 0 0055 0 0048 25 - -1 q 59) 15255 2797 15255 2770 0 0100 0 0027 25 - -1 q 65) 15212 5214 15212 5244 0 0055 -0 0030 25 - -2 q 2) 15129 6485 15129 6573 0 0065 -0 0088 25 - -2 q 3) 15129 3216 15129 3346 0 0100 -0 0130 25 - -2 q 4) 15128 8992 15128 9043 0 0055 -0 0051 25- -2 q 5) 15128 3579 15128 3663 0 0045 -0 0084 25 - -2 q 6) 15127 7176 15127 7207 0 0075 -0 0031 25 - -2 q 11) 15122 8738 15122 8762 0 0045 -0 0024 25 - -2 q 12) 15121 5752 15121 5836 0 0075 -0 0084 25 - -2 q (15) 15117 0550 15117 0568 0 0045 -0 0018 2 5 - -2 q (16) 15115 3307 15115 3313 0 0055 -0 0006 2 5 - -2 q (17) 15113 5024 15113 4972 0 0075 0 0052 25 - -2 q (21) 15105 0838 15105 0737 0 0075 0 0101 25 - -2 q (31) 15076 3487 15076 3460 0 0075 0 0027 25 - -2 q (37) 15053 7812 15053 7808 0 0100 0 0004 Appendix 218 25--2 q( 40) 15040. 9685 15040 25--2 q( 41) 15036. 4883 15036 25--2 q( 42) 15031. 8633 15031 26--1 q( . 3) 15470. 1426 15470 26--1 q< 4) 15469. 6981 15469 26--1 q 5) 15469. 1412 15469 26--1 q 6) 15468. 4778 15468 26--1 q 10) 15464. 7095 15464 26--1 q 12) 15462 1505 15462 26--1 q 13) 15460 7073 15460 26--1 q 17) 15453 8141 15453 26--1 q 18) 15451 8091 15451 26--1 q 20) 15447 4703 15447 26--1 q 22) 15442 6702 15442 26--1 q 25) 15434 6294 15434 26--1 q 32) 15411 8824 15411 26--1 q 33) 15408 1774 15408 26--1 q 34) 15404 3649 15404 26--1 q 37) 15392 1828 15392 26--1 q 39) 15383 4809 15383 26--1 q 57) 15283 4099 15283 26--1 q 59) 15269 7766 15269 26--1 q 61) 15255 6178 15255 26--1 q 62) 15248 3386 15248 26--2 q 2) 15149 4953 15149 26--2 q 3) 15149 1676 15149 26--2 q 4) 15148 7296 15148 26--2 q 5) 15148 1826 15148 26--2 q 6) 15147 5164 15147 26--2 q 8) 15145 8664 15145 26--2 q 11) 15142 5632 15142 26--2 q 12) 15141 2288 15141 26--2 q 13) 15139 7903 15139 26--2 q 14) 15138 2451 15138 26--2 q 15) 15136 5877 15136 26--2 q (21) 15124 3037 15124 26--2 q (25) 15113 8704 15113 26--2 q (37) 15071 6647 15071 26--2 q (43) 15044 2972 15044 28--2 q ( 5) 15181 6159 15181 28--2 q ( 7) 15180 1099 15180 28--2 q ( 8) 15179 .1847 15179 28--2 q ( 9) 15178 .1405 15178 28--2 q (12) 15174 .3149 15174 28--2 q (13) 15172 .8055 15172 28--2 q (15) 15169 .4285 15169 28--2 q (20) 15158 . 9528 15158 28--2 q (22) 15153 . 9448 15153 28--2 q (26) 15142 . 4754 15142 28--2 q (29) 15132 . 6294 15132 28--2 q (32) 15121 . 6925 15121 28--2 q (33) 15117 .8108 15117 28--2 q (34) 15113 .7925 15113 29--1 q (41) 15413 .7155 15413 29--1 q (42) 15408 .5294 15408 29--1 q (43) 15403 .2120 15403 29--2 q ( 2) 15196 .8009 15196 29--2 q ( 3) 15196 .4454 15196 29--2 q ( 4) 15195 . 9772 15195 9752 0. 0100 -0. 0067 4786 0. 0100 0. 0097 8674 0. 0150 -0. 0041 1495 0. 0035 -0. 0069 * * * 7057 0. 0050 -0. 0076 ** 1508 0. 0035 -0. 0096 * * * * 4850 0. 0045 -0. 0072 * * * 7102 0. 0075 -0. 0007 1551 0. 0075 -0. 0046 7103 0. 0045 -0. 0030 8140 0. 0100 0. 0001 8100 0. 0100 -0. 0009 4652 0 0150 0. 0051 6702 0 0100 0 0000 6303 0 0100 -0 0009 8849 0 0075 -0 0025 1761 0 0040 0 0013 3517 0 0100 0 0132 1801 0 0075 0 0027 4802 0 0150 0 0007 4102 0 0055 -0 0003 7776 0 0075 -0 0010 6186 0 0075 -0 0008 3392 0 0100 -0 0006 5097 0 0100 -0 0144 * * 1788 0 0065 -0 0112 * * * 7376 0 0100 -0 0080 1861 0 0040 -0 0035 5241 0 0040 -0 0077 * * * 8688 0 0075 -0 0024 5566 0 0040 0 0066 2311 0 0040 -0 0023 7947 0 0040 -0 0044 * 2474 0 0045 -0 0023 5890 0 0040 -0 0013 2991 0 0100 0 0046 8662 0 0075 0 0042 6610 0 0150 0 0037 2980 0 0150 -0 0008 6229 0 0055 -0 0070 1149 0 0065 -0 0050 1867 0 0065 -0 0020 1423 0 0100 -0 0018 3105 0 0150 0 0044 8001 0 0100 0 0054 4287 0 0065 -0 0002 9488 0 0150 0 0040 9322 0 0100 0 0126 4753 0 0055 0 0001 6277 0 .0065 0 0017 6943 0 .0055 -0 0018 8067 0 .0100 0 0041 7968 0 .0100 -0 .0043 7194 0 .0075 -0 .0039 5360 0 .0075 -0 .0066 2191 0 .0055 -0 0071 * 8103 0 .0065 -0 .0094 4535 0 .0065 -0 .0081 * 9778 0 .0065 -0 .0006 Appendix 29--2 q( 5) 15195 3909 15195 3830 0 0065 0 0079 29--2 q(13) 15186 3323 15186 3321 0 0065 0 0002 29--2 q(16) 15180 9728 15180 9635 0 0100 0 0093 29--2 q(17) 15178 9385 15178 9334 0 0075 0 0051 29--2 q(18) 15176 7895 15176 7828 0 0035 0 0067 29--2 q(19) 15174 5133 15174 5114 0 0065 0 0019 29--2 q(22) 15166 9806 15166 9701 0 0100 0 0105 29--2 q(24) 15161 3399 15161 3340 0 0075 0 0059 29--2 q(26) 15155 2139 15155 2082 0 0065 0 0057 29--2 q(27) 15151 9640 15151 9607 0 0075 0 0033 29--2 q(28) 15148 5971 15148 5898 0 0075 0 0073 29--2 q(31) 15137 7418 15137 7315 0 0075 0 0103 29--2 q(32) 15133 8568 15133 8621 0 0075 -0 0053 29--2 q(34) 15125 7492 15125 7452 0 0065 0 0040 Appendix 220 T a b l e A - 3 : Unblended l i n e s from quasi-bound l e v e l s above the d i s s o c i a t i o n l i m i t . These were not used i n analysis. A l l units i n cm"1. T r a n s i t i o n Observed Uncerta inty 23- -1 r 76) 15097. 8494 0 0035 23- -1 r 78) 15080. 7311 0 0055 23- -1 r 79) 15071. 9580 0 0035 23- -1 r 82) 15044 8174 0 0045 23- -1 r 83) 15035 4926 0 0055 23- -1 r 85) 15016 3644 0 0065 23- -1 r 87) 14996 6086 0 0050 23- -1 P 78) 15072 6313 0 0100 23- -1 P 80) 15054 8472 0 0150 23- -1 P 81) 15045 7620 0 0035 23- -1 P 82) 15036 5360 0 0040 23- -1 P 83) 15027 1571 0 0055 23- -1 P 84) 15017 6634 0 0045 23- -1 P 85) 15007 9834 0 0045 24- -1 r 72) 15146 6568 0 0065 24- -1 r 74) 15130 0477 0 0040 24- -1 r 75) 15121 5378 0 0035 24- -1 r 76) 15112 8877 0 0035 24- -1 r 77) 15104 0858 0 0035 24- -1 r 78) 15095 1505 0 0045 24- -1 r 79) 15086 0476 0 0035 24- -1 r 80) 15076 8077 0 0035 24- -1 r 81) 15067 4006 0 0035 24- -1 r 82) 15057 8446 0 0035 24- -1 r (83) 15048 1209 0 0035 24- -1 r (84) 15038 2328 0 0035 24- -1 r (85) 15028 1615 0 0035 24- -1 r (87) 15007 4639 0 0050 24- -1 r (88) 14996 7617 0 0055 24- -1 P (73) 15131 1453 0 0100 24- -1 P (74) 15122 7245 0 0035 24- -1 P (75) 15114 1713 0 0040 24- -1 P (77) 15096 6359 0 0045 24- -1 P (79) 15078 5414 0 0065 24- -1 P (80) 15069 2737 0 0040 24- -1 P (83) 15040 5592 0 0075 24- -1 P (85) 15020 6288 0 0045 24- -1 P (87) 15000 0202 0 0040 25- -1 r (68) 15192 8996 0 0035 25- -1 r (69) 15184 9255 0 0035 25- -1 r (70) 15176 8280 0 0035 25- -1 r (72) 15160 2002 0 0040 25- -1 r (73) 15151 6629 0 0035 25- -1 r (74) 15142 9842 0 0040 25- -1 r (76) 15125 1457 0 0075 25- -1 r (77) 15116 0049 0 0035 25- -1 r (79) 15097 2033 0 0100 25- -1 r (80) 15087 5465 0 0045 25- -1 r (81) 15077 6992 0 0040 25- -1 P (69) 15178 3893 0 0035 25- -1 P (71) 15161 9765 0 0040 25- -1 P (72) 15153 5389 0 0045 Appendix 221 25- -1 P 73) 15144. 9749 0. 0035 25- -1 P 74) 15136 2660 0. 0035 25- -1 P 76) 15118 4064 0 0035 25- -1 P 77) 15109 2402 0 0035 25- -1 P 79) 15090 4538 0 0035 25- -1 P 80) 15080 8162 0 0045 25- -1 P 81) 15071 0092 0 0035 25- -1 P 82) 15061 0288 0 0045 25- -1 P 83) 15050 8568 0 0035 26- -1 r 63) 15244 0175 0 0035 26- -1 r 64) 15236 4935 0 0035 26- -1 r 65) 15228 8258 0 0045 26- -1 r 66) 15221 0280 0 0035 26- -1 r 67) 15213 0621 0 0035 26- -1 r 69) 15196 7029 0 0035 26- -1 r 70) 15188 2999 0 0075 26- -1 r 73) 15162 1226 0 0075 26- -1 r 74) 15153 0573 0 0055 26- -1 r 75) 15143 8294 0 0055 26- -1 r 76) 15134 4139 0 0045 26- -1 r 77) 15124 7780 0 0035 26- -1 P 65) 15222 9954 0 0055 26- -1 P 66) 15215 1460 0 0045 26- -1 P 68) 15199 0456 0 0040 26- -1 P 69) 15190 7607 0 0040 26- -1 P 70) 15182 3236 0 0035 26- -1 P 71) 15173 7330 0 0100 26- -1 P 73) 15156 1262 0 0040 26- -1 P 76) 15128 4682 0 0100 26- -1 P 77) 15118 9275 0 0100 26- -1 P 78) 15109 1727 0 0035 26- -1 P 79) 15099 2490 0 0075 27- -1 r (61) 15270 0857 0 0035 27- -1 r (62) 15262 5812 0 0040 27- -1 r 63) 15254 9220 0 0035 27- -1 r (64) 15247 1317 0 0035 27- -1 r (65) 15239 1640 0 0035 27- -1 r (66) 15231 0439 0 0040 27- -1 r (68) 15214 3267 0 0045 27- -1 r (69) 15205 7123 0 0040 27- -1 r (71) 15187 9365 0 0035 27- -1 r (72) 15178 7477 0 0035 27- -1 P (61) 15264 9442 0 0040 27- -1 P (62) 15257 4064 0 0045 27- -1 P (65) 15233 9079 0 0100 27- -1 P (66) 15225 7932 0 0035 27- -1 P (67) 15217 5082 0 0040 27- -1 P (68) 15209 0406 0 0035 27- -1 P (70) 15191 6907 0 0100 27- -1 P (72) 15173 6245 0 0055 27- -1 P (73) 15164 3271 0 0035 28- -1 r (54) 15329 3572 0 0035 28- -1 r (55) 15322 6072 0 0035 28- -1 r (56) 15315 7111 0 0100 28- -1 r (57) 15308 6710 0 0055 28- -1 r (58) 15301 4967 0 0035 28- -1 r (59) 15294 1529 0 0035 28- -1 r (60) 15286 6646 0 0040 28- -1 r (61) 15279 0230 0 0050 Appendix 222 28--1 r 62) 15271 2094 0 0035 28--1 r 63) 15263 2261 0 0035 28--1 r 64) 15255 0740 0 0100 28--1 r 65) 15246 7433 0 0045 28--1 r 66) 15238 2192 0 0035 28--1 P 56) 15311 2597 0 0045 28--1 P 57) 15304 1900 0 0035 28--1 P 58) 15296 9770 0 0055 28--1 P 59) 15289 6061 0 0035 28--1 P 61) 15274 4363 0 0035 28--1 P 62) 15266 6226 0 0040 28--1 P 65) 15242 2068 0 0075 28--1 P 66) 15233 7397 0 0100 28--1 P 67) 15225 0717 0 0075 28--1 P 70) 15197 9802 0 0100 28--2 r (54) 15009 3411 0 0055 28--2 r (55) 15002 6264 0 0045 28--2 r (57) 14988 7685 0 0055 28--2 P 56) 14991 3230 0 0035 29--1 r 50) 15364 2279 0 0035 29--1 r (51) 15357 8215 0 0035 29--1 r 52) 15351 2751 0 0035 29--1 r (53) 15344 5753 0 0035 29--1 r (54) 15337 7286 0 0035 29--1 r (55) 15330 7198 0 0035 29--1 r (58) 15308 7581 0 0045 29--1 r (59) 15301 0719 0 0035 29--1 r (60) 15293 2226 0 0035 29--1 P (53) 15340 7029 0 0035 29--1 P (54) 15333 8323 0 0035 29--1 P (55) 15326 8109 0 0075 29--1 P (56) 15319 6441 0 0045 29--1 P (58) 15304 8206 0 0045 29--1 P (59) 15297 1599 0 0035 29--1 P (61) 15281 3565 0 0035 29--1 P (62) 15273 1840 0 0035 29--1 P (63) 15264 8019 0 0035 29--2 r (50) 15044 0769 0 0035 29--2 r (52) 15031 1952 0 0035 29--2 r (53) 15024 5312 0 0045 29--2 r (55) 15010 7539 0 0050 29--2 r (56) 15003 6184 0 0050 29--2 r (57) 14996 3228 0 0045 29--2 P (52) 15027 3552 0 0040 29--2 P (53) 15020 6633 0 0050 29--2 P (54) 15013 8236 0 0045 29--2 P (55) 15006 8289 0 0045 29--2 P (56) 14999 6879 0 0045 29--2 P (57) 14992 4022 0 0055 30--1 r (46) 15396 3334 0 0035 30--1 r (48) 15384 1026 0 0035 30--1 r (50) 15371 2567 0 0040 30--1 r (51) 15364 6051 0 0035 30--1 r (52) 15357 7821 0 0035 30--1 r (53) 15350 7961 0 0035 30--1 P (48) 15380 8352 0 0035 30--1 P (49) 15374 4686 0 0040 30--1 P (50) 15367 9560 0 0050 30--1 P (51) 15361 2892 0 0040 30--1 P (52) 15354 4703 0 0035 Appendix 223 30--1 P 54) 15340 3471 0 0055 30--1 P 55) 15333 0310 0 0040 30--2 r 46) 15076 0553 0 0035 30--2 r 47) 15070 0431 0 0035 30--2 r 48) 15063 8875 0 0040 30--2 r 49) 15057 5771 0 0035 30--2 r 50) 15051 1107 0 0055 30--2 r 51) 15044 4917 0 0035 30--2 r 53) 15030 7503 0 0035 30--2 r 55) 15016 2988 0 0075 30--2 r 56) 15008 7890 0 0050 30--2 P 49) 15054 2859 0 0100 30--2 P 50) 15047 8161 0 0035 30--2 P 51) 15041 1754 0 0035 30--2 P 54) 15020 3379 0 0045 30--2 P 55) 15013 0499 0 0055 30--2 P 57) 14997 9555 0 0045 30--2 P 58) 14990 1051 0 0055 31--1 r 41) 15431 1591 0 0065 31--1 r 44) 15414 1856 0 0075 31--1 r 45) 15408 2191 0 0035 31--1 r 46) 15402 0922 0 0055 31--1 r 47) 15395 8061 0 0035 31--1 r 48) 15389 3461 0 0035 31--1 r 49) 15382 7034 0. 0035 31--1 r 50) 15375 8843 0 0075 31--1 P 43) 15417 2877 0 0045 31--1 P 45) 15405 4783 0 0035 31--1 P 46) 15399 3434 0 0035 31--1 P 47) 15393 0544 0 0035 31--1 P 48) 15386 6010 0 0040 31--1 P 49) 15379 9894 0 0035 31--1 P 50) 15373 2015 0 0035 31--1 P 51) 15366 2358 0 0035 31--1 P 52) 15359 0873 0 0100 31--2 r 41) 15110 7399 0 0075 31--2 r 43) 15099 6339 0 0035 31--2 r 44) 15093 8536 0 0100 31--2 r 45) 15087 9151 0 0035 31--2 r 46) 15081 8256 0 0100 31--2 r 47) 15075 5565 0 0075 31--2 r 48) 15069 1302 0 0075 31--2 r 49) 15062 5222 0 0035 31--2 P 45) 15085 1757 0 0040 31--2 P 46) 15079 0696 0 0035 31--2 P 47) 15072 8067 0 0035 31--2 P 49) 15059 8054 0 0035 31--2 P 50) 15053 0531 0 0075 31--2 P (51) 15046 1197 0 0065 31--2 P 52) 15039 0149 0 0035 32--1 r 37) 15457 3018 0 0035 32--1 r 38) 15452 2016 0 0035 32--1 r 39) 15446 9446 0 0035 32--1 r 40) 15441 5405 0 0040 32--1 r 41) 15435 9617 0 0035 32--1 r 42) 15430 2318 0 0035 32--1 r 43) 15424 3192 0 0055 32--1 r 44) 15418 2358 0 0035 32--1 P 39) 15444 7352 0 0035 32--1 P 40) 15439 3151 0 0035 Appendix 224 32- -1 P 41) 15433 7322 0 0035 32- -1 P 42) 15427 9971 0 0035 32- -1 P 43) 15422 1015 0 0035 32- -1 P 44) 15416 0389 0 0035 32- -1 P 46) 15403 3894 0 0040 32- -2 r 37) 15136 7843 0 0035 32- -2 r 38) 15131 7110 0 0035 32- -2 r 39) 15126 4772 0 0035 32- -2 r 41) 15115 5463 0 0035 32- -2 r 42) 15109 8409 0 0035 32- -2 r 44) 15097 8991 ' 0 0040 32- -2 P 39) 15124 2704 0 0035 32- -2 P 40) 15118 8746 0 0055 32- -2 P 41) 15113 3142 0 0035 32- -2 P 42) 15107 6085 0 0040 32- -2 P 44) 15095 7069 0 0055 32- -2 P 45) 15089 4997 0 0035 32- -2 P 46) 15083 1301 0 0055 33- -1 r 33) 15480 6614 0 0035 33- -1 r 34) 15475 9900 0 0055 33- -1 r 35) 15471 1559 0 0035 33- -1 r 36) 15466 1677 0 0035 33- -1 r 37) 15461 0082 0 0035 33- -1 r 38) 15455 6846 0 0055 33- -1 r 39) 15450 1633 0 0075 33- -1 P 35) 15469 4027 0 0035 33- -1 P 36) 15464 4035 0 0035 33- -1 P 37) 15459 2452 0 0035 33- -1 P 38) 15453 9293 0 0035 33- -1 P 39) 15448 4394 0 0055 33- -1 P 40) 15442 7918 0 0035 33- -1 P 41) 15436 9562 0 0035 33- -2 r 33) 15160 0509 0 0035 33- -2 r 34) 15155 4001 0 0030 33- -2 r 35) 15150 5907 0 0065 33- -2 r 37) 15140 4864 0 0035 33- -2 r 38) 15135 1960 0 0100 33- -2 r (39) 15129 6980 0 0040 33- -2 P 35) 15148 8376 0 0035 33- -2 P 36) 15143 8558 0 0055 33- -2 P (37) 15138 7250 0 0035 33- -2 P (38) 15133 4353 0 0040 33- -2 P (39) 15127 9780 0 0030 33- -2 P (40) 15122 3489 0 0040 33- -2 P 41) 15116 5423 0 0075 34- -1 r 33) 15483 3809 0 0035 34- -1 r 34) 15478 4788 0 0035 34- -1 P (32) 15486 7605 0 0035 34- -1 P (33) 15482 0470 0 0035 34- -1 P (34) 15477 1851 0 0035 34- -2 r (28) 15184 6207 0 0035 34- -2 r (29) 15180 5728 0 0035 34- -2 r (30) 15176 3679 0 0040 34- -2 r (31) 15172 0005 0 0030 34- -2 r (33) 15162 7695 0 0035 34- -2 r (34) 15157 8913 0 0035 34- -2 P (31) 15170 6534 0 0035 34- -2 P (32) 15166 1273 0 0035 34- -2 P (34) 15156 5797 0 0055 34- -2 P (35) 15151 5573 0 0030 Appendix 225 35--2 r (24) 35--2 r(25) 35--2 r (27) 35--2 P(26) 35--2 P(27) 35--2 P(28) 35--2 P(29) 35--2 P(30) 35--2 P(31) 36--2 r (21) 36--2 r (22) 36--2 r (23) 36--2 P(24) 36--2 P(25) 36--2 P(26) 37--2 r (17) 37--2 P(18) 37--2 P.(19) 15201.9521 15198.3737 15190.7380 15193.6549 15189.7634 15185.6985 15181.4712 15177.0858 15172.5244 15213.5097 15210.2790 15206.8563 15202.6330 15198.8776 15194.9130 15226.0007 15222.8488 15219.9781 0.0035 0.0100 0.0075 0.0045 0.0035 0.0050 0.0075 0.0030 0.0065 0.0040 0.0035 0.0040 0.0035 0.0040 0.0050 0.0035 0.0065 0.0040 Appendix 226 T a b l e A - 4 : Transitions i n the extra series associated with v = 27 of the A state ( a l l units i n cm - 1) . v'' Trans. Observed Calculated Uncert. Obs-Cal 1 r ( 1) 15488 5386 15488 5450 0 0075 -0 0064 1 r ( 2) 15488 3794 15488 3670 0 0100 0 0124 1 r ( 3) 15488 0737 15488 0752 0 0100 -0 0015 1 r ( 4) 15487 6794 15487 6696 0 0100 0 0098 1 r ( 5) 15487 1474 15487 1501 0 0035 -0 0027 1 r ( 6) 15486 5232 15486 5168 0 0045 0 0064 1 r ( 7) 15485 7657 15485 7696 0 0055 -0 0039 1 r ( 8) 15484 9131 15484 9083 0 0060 0 0048 1 r ( 9) 15483 9309 15483 9330 0 0150 -0 0021 1 r (11) 15481 6461 15481 6400 0 0045 0 0061 1 r (12) 15480 3279 15480 3221 0 0045 0 0058 1 r (14) 15477 3482 15477 3431 0 0065 0 0051 1 r (15) 15475 6839 15475 6818 0 0040 0 0021 1 r (17) 15472 0163 15472 0151 0 0045 0 0012 1 r (19) 15467 8839 15467 8888 0 0055 -0 0049 1 r (22) 15460 8390 15460 8350 0 0075 0 0040 1 r (23) 15458 2535 15458 2528 0 0055 0 0007 1 r (24) 15455 5552 15455 5547 0 0065 0 0005 1 r (25) 15452 7402 15452 7406 0 0055 -0 0004 1 r (26) 15449 8165 15449 8104 0 0100 0 0061 1 r (29) 15440 3257 15440 3212 0 0045 0 0045 1 P ( 5) 15486 6010 15486 6059 0 0040 -0 0049 1 P ( 6) 15485 8725 15485 8738 0 0035 -0 0013 1 P 7) 15485 0253 15485 0279 0 0030 -0 0026 1 P 9) 15482 9961 15482 9944 0 0035 0 0017 1 P 10) 15481 8055 15481 8067 0 0030 -0 0012 1 P 11) 15480 5052 15480 5049 0 0030 0 0003 1 P 12) 15479 0896 15479 0891 0 0040 0 0005 1 P 13) 15477 5611 15477 5590 0 0065 0 0021 1 P 14) 15475 9127 15475 9147 0 0050 -0 0020 1 P 15) 15474 1566 15474 1560 0 0045 0 0006 1 P 16) 15472 2812 15472 2829 0 0040 -0 0017 1 P 17) 15470 2952 15470 2953 0 0045 -0 0001 1 P 18) 15468 1969 15468 1930 0 0050 0 0039 1 P 19) 15465 9760 15465 9759 0 0035 0 0001 1 P 20) 15463 6426 15463 6439 0 0040 -0 0013 1 P 21) 15461 1959 15461 1969 0 0035 -0 0010 1 P 22) 15458 6322 15458 6347 0 0045 -0 0025 1 P 23) 15455 9588 15455 9573 0 0055 0 0015 1 P 24) 15453 1626 15453 1644 0 0055 -0 0018 1 P 25) 15450 2546 15450 2559 0 0045 -0 0013 1 P 26) 15447 2297 15447 2316 0 0075 -0 0019 1 P 29) 15437 4623 15437 4625 0 0040 -0 0002 1 P 32) 15426 6433 15426 6449 0 0040 -0 0016 2 r 1) 15167 5684 15167 5711 0 0035 -0 0027 2 r 2) 15167 4068 15167 3944 0 0055 0 0124 2 r 4) 15166 7027 15166 7016 0 0035 0 0011 2 r 6) 15165 5605 15165 5559 0 0035 0 0046 2 r 9) 15162 9880 15162 9877 0 0035 0 0003 2 r 10) 15161 9101 15161 9047 0 0040 0 0054 2 r 11) 15160 7102 15160 7082 0 0035 0 0020 2 r 12) 15159 4055 15159 3981 0 0040 0. 0074 Appendix 227 2 r (14) 15156 4384 15156 4366 0 0035 0 0018 2 r (16) 15153 0168 15153 0194 0 0040 -0 0026 2 r (19) 15147 0479 15147 0373 0 0075 0 0106 2 r (20) 15144 8130 15144 8144 0 0045 -0 0014 2 r (21) 15142 4780 15142 4768 0 0035 0 0012 2 r (22) 15140 0221 15140 0244 0 0075 -0 0023 2 r (25) 15131 9808 15131 9766 0 0075 0 0042 2 r (26) 15129 0637 15129 0632 0 0100 0 0005 2 r (27) 15126 0368 15126 0342 0 0065 0 0026 2 r (28) 15122 8907 15122 8893 0 0100 0 0014 2 r (29) 15119 6295 15119 6284 0 0055 0 0011 2 P ( 7) 15164 0708 15164 0715 0 0035 -0 0007 2 P ( 8) 15163 1134 15163 1169 0 0030 -0 0035 2 P ( 9) 15162 0463 15162 0490 0 0035 -0 0027 2 P (10) 15160 8644 15160 8678 0 0035 -0 0034 2 P (11) 15159 5727 15159 5732 0 0030 -0 0005 2 P (14) 15155 0036 15155 0082 0 0045 -0 0046 2 P (15) 15153 2578 15153 2592 0 0040 -0 0014 2 P (16) 15151 3951 15151 3965 0 0035 -0 0014 2 P (18) 15147 3271 15147 3292 0 0040 -0 0021 2 P 19) 15145 1219 15145 1244 0 0040 -0 0025 2 P 21) 15140 3731' 15140 3720 0 0035 0 0011 2 P 22) 15137 8164 15137 8241 0 0050 -0 0077 2 P 24) 15132 3843 15132 3842 0 0045 0 0001 2 P 25) 15129 4871 15129 4919 0 0045 -0 0048 2 P 27) 15123 3600 15123 3617 0 0045 -0 0017 2 P 29) 15116 7735 15116 7697 0 0040 0 0038 2 P 3D 15109 7147 15109 7143 0 0040 0 0004 2 P 32) 15106 0092 15106 0124 0 0040 -0 0032