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Magnetic rotation study of the A3[pi]1u-X1[epsilon]g+ system of 79Br2 Boone, Christopher D. 1999

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Magnetic Rotation Study of the A ^ i ^ - X ^ j j " system of 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 F U L F I L M E N T OF T H E REQUIREMENTS FOR T H E D E G R E E OF DOCTOR OF PHILOSOPHY in T H E FACULTY OF G R A D U A T E STUDIES (Department of Physics) We accept this thesis as conforming to the required standard  T H E UNIVERSITY OF BRITISH COLUMBIA July 1999 © Christopher D. Boone, 1999  7 9  Br2  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 I I i - X £ + system of B r 2 has been studied using magnetic rotation spectroscopy. A 3  1  7 9  u  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 cm  - 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 , YD (effective vibration at dissociation) = 41.544 ± 0.013 and C5 (a parameter - 1  describing the long-range form of the potential) = 61,700 ± 900 c m A . From the potential curve - 1  5  generated in the global analysis, the following spectroscopic parameters were determined for the A state: r (equilibrium distance) = 2.7026 A , B (equilibrium rotational constant) = 0.05849 e  e  c m , D (binding energy) = 2147.82 c m , and T (energy relative to the minimum of the X - 1  - 1  e  e  state potential) = 13,909.14 c m . From an analysis of the long-range potential of the A state, - 1  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 ; this differed significantly from the value obtained in the global analysis, a - 1  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 ' ! ] ^ electronic state. Using 3  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 2.1  Angular Momenta  2.2  Energy  2.3  2.4  2.5  2.6  6 6 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  Symmetry  17  2.3.1  Molecular Symmetry and cr  18  2.3.2  Homonuclear Molecules  19  v  Perturbations  20  2.4.1  The Rotational Hamiltonian  21  2.4.2  Uncoupling Phenomena  23  2.4.3  A-doubling  23  Interaction with Magnetic Fields  25  2.5.1  27  Splitting of Levels in a Magnetic Field  Interaction with Electromagnetic Radiation iii  29  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  6  . .  133  5.11 Building the A ' State  151  5.12 Hyperfine Effects  158  5.13 Doppler-free Magnetic Rotation  161  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  32  Selection rules for transitions (excluding hyperfine effects)  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 B r 2 A I I i  98  3  7 9  u  5-3 fi-doubling constants (All values in c m )  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  -1  5-6 Centrifugal distortion constants calculated from R K R potential. (All values in c m ) 123 -1  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 c m ) . . 131 -1  5-10 A selected set of transitions in the perturbed state v = 27 (All values in c m ) . . . 139 -1  5-11 Results of fitting the extra series (All values in c m )  142  5-12 Extra lines observed in the spectrum  144  -1  5-13 Preliminary test of extrapolation procedure in the A state. (All values in c m ) -1  . . 155  5-14 Extrapolation from dissociation in the A ' state. (All values in c m )  157  A - l Data used in the global analysis (all units in c m )  171  -1  -1  A-2 Q-lines used in the analysis of il-type doubling (all units in c m )  214  -1  A-3 Unblended lines from quasi-bound levels above the dissociation limit. These were not used in analysis. All units in c m  220  - 1  A-4 Transitions in the extra series associated with v = 27 of the A state (all units in cm ).226 -1  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 rotational 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<f> 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 A n R(0) transition in the presence of an external magnetic field. The transition frequency in the absence of an external magnetic field is u  49  Q  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. v is the transition Q  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. v is the transition frequency in the absence of external Q  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 A m j = -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. 3-6 Relative transition probabilities for transitions within a P(4) and within a Q(3) line.  58 60  3-7 Calculated frequency shift and intensity perturbation contributions to the first har-  3-8  monic signal in Aa  71  Calculated first harmonic signal from A a  71  3-9 Calculated frequency shift and intensity perturbation contributions to the first harmonic signal in A n  72'  3-10 Calculated first harmonic signal from A n  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 perturbation 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 electronic state mixing affecting the lineshapes. The signal shown was taken in second harmonic  78  4- 1 Potential energy curves for selected valence states of 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  7 9  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(J1) 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 viii  94  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. 5-8  . . . 101  Difference between the energies of f parity levels (E-^(J), determined from Q lines) and e parity levels (E (J), determined from R and P lines) as a function of J(J+1). e  The f parity levels are lower in energy than the e parity levels 5-9  102  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  Ill  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 uncorrected cases  119  5-15 Difference between the B-values used to calculate the R K R potential and the Bvalues 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 B r 2 7 9  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 B r 2 as a function of v. . . 154 7 9  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  Introduction  1  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 I I i electronic state of Br2 using standard spectroscopic techniques such 3  u  as absorption or fluorescence is hampered by the presence of the B i l + electronic state. Transi3  0  U  tions from the ground X E^" electronic state to the B state are generally much stronger with these X  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 Br2The current foray into the exciting world of magnetic rotation spectroscopy began when this author was handed a magnetic rotation spectrum of B r 2 , originally taken by Dr. Alak Chanda. In 7 9  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 I I i - X E + system. These high-J signals, together with 3  1  u  the suppression of signals from the overlapping B n + - X E ^ system, opened up the possibility of 3  1  0  U  investigating the A state of B r 2 in more detail than has ever been achieved before. 7 9  1.1  Background  Previous knowledge of the A I I i state of Br2 comes mainly from an absorption study by Coxon 3  u  [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 I I + - X E + system of I2 was inves3  1  0  U  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 I I i state of I2. The treatment of magnetic rotation theory of the time was only valid for frequen3  u  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 arise from the known [7] interaction of the B state with a Ui 1  2  u  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 . In [8], it was not clear how the 2  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 . The 2  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 c m ) magnetic rotation study of diatomic inter-halogens -1  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 spectroscopy 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 spectroscopy [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 rotation spectroscopy has also been proposed as a technique for measuring the concentrations of NO and N 0 in the atmosphere [25] and [26]. 2  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 general. 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 Br2), 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 cm  - 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 I I i state of Br2 made it worth 3  u  the extra effort.  1 1.2  INTRODUCTION  5  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 magnetic 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 B r 2 , and the frequency 7 9  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 I i i state is used to perform 3  u  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 ' I l 2 state of Br23  u  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  2  THEORETICAL  6  CONSIDERATIONS  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, B r 2 , tends more 7 9  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. A n 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 spacefixed 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  \m \. L  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. A n 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 interaction 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 J (= L + S) is defined. The a  component of J along the internuclear axis is ftft, and the sum of this component with R to give a  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.  R Q  = Q t i k  2  GENERAL  2.2  THEORETICAL  CONSIDERATIONS  11  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„) = E  n  \m )  (2-1)  n  where H is the Hamiltonian operator, E is a particular eigenvalue and |vl/ ) represents the waven  n  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 "BornOppenheimer wavefunctions,"  ty ^, 30  can be written as the product of two wavefunctions: *  B O  )  = |tt ) \y , ), vib rot  e  (2-2)  where |$ ) is an electronic wavefunction, and \ty ib,rot) represents the part of the wavefunction e  V  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  2.2.2  12  CONSIDERATIONS  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  symbol  S  n  2  3 .... A  $ ....  The designation for an electronic state has the form: symbol(A)  {e.g. H  2S+1  3  n  3 U  H,  So)-  1  2  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 Ui l  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: H = H + H ib . Q  e  V  (2-3)  iTOt  Consider first the effect of H acting on the electronic part of the wavefunction in Eq 2-2. Keep 0  in mind that H. ib,rot does not operate on this part of the wavefunction. V  H |* ) - He |*e> = (T + U (r)) |* ) = E |* ) . 0  e  e  e  e  e  e  T is the electron's kinetic energy, r denotes internuclear separation, and U (r) is the potential e  e  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  13  CONSWERATIONS  The approximate solutions to Eq 2-1 using the BO basis set and zeroth-order Hamiltonian are determined by solving [T + T + [V (r) + U {r)}] \^ , ) = E\^ , ot) • N  e  N  e  vib  rot  vib r  (2-4)  The term Vjv(r) + U (r) describes a background potential in which the molecule can vibrate and e  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  e  I  i  I  i  I  i  L  Internuclear separation r (A)  T in Figure 2-4 is the electronic kinetic energy. At low r (small internuclear separation), the e  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 , the equilibrium internuclear separation. The e  binding energy, D , is the difference in energy between the dissociation limit and the minimum of e  the potential.  2  GENERAL  2.2.4  THEORETICAL  CONSIDERATIONS  14  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) = hiu (v + ^),  (2-5)  osc  where ui  osc  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-r )\ e  where k is the spring constant and r is the equilibrium length of the spring. This potential is e  symmetric about r . In the diatomic molecule, there is a strong repulsion at small r, while at e  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) = u (v + ^) - wex (v + ^) +u y (v 2  e  e  e  e  + ^) + u z (v + i ) + 3  4  e  e  (2-6)  where u —• hco in the harmonic limit. to x is a positive quantity and is much smaller than u> . e  osc  e  e  e  The coefficients of higher order terms (co ye, uj z ,...) could be either positive or negative and are e  e  e  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 = \M , ot  (2-7)  2  GENERAL  THEORETICAL  CONSIDERATIONS  15  where I is the moment of inertia about the axis of rotation, and u t is the angular frequency of ro  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 = fir , where fi is the reduced mass of the two 2  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> t = hy/R(R + 1)), leads to the rotational Hamiltonian: ro  H ot — , T  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 E  = F(J) = B [j(J + 1) - ft ] ,  (2-9)  2  rot  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: ^  8n ficr ' 2  ^  2  ^  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 (fir ). This leads to a decrease in the rotational 2  energy. This is accounted for by the introduction of "distortion terms" into Eq 2-9: F {J) V  = B [ j ( J + 1) - ft ] -D 2  v  v  [ j ( J + 1) -  ft ] +#„ 2 2  [ j ( J + 1) - ft ] +L„ [ j ( J + 1) 2 3  ft 4 :  (2-11) where D^, H^, L , etc. are called centrifugal distortion constants. D is a positive quantity, but v  v  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 vibrational dependence: B — Bv  a (v + ^)+ (3 (v + ^) + {v +  +  2  e  e  e  le  (2-12)  where a is a positive quantity, and higher order terms can be either positive or negative. e  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 =T +  G(v)+F (J).  e  v  Typically, the electronic energy, T , is the largest contribution and the rotational energy, F„(J), is e  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: [T (r) +T + ^ N  e  2  [J (J + 1) -  + V (r) + U (r)} | ^ , ( r ) ) = E | ^ , ( r ) > . N  e  6  r o t  T  6  rot  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 [J(J + l)-n }  k  2  21;  8n ficr 2  + V (r) + U (r) N  e  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 treating 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. All 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 conservation 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 > , i.e.  (2-13)  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 I i commutes with the full S  Hamiltonian for the molecule, and so its eigenvalues (+1 or —1) must be rigorously good quantum numbers. 2.3.1  Molecular Symmetry and a  v  Rather than working out the eigenvalues of I i from Eq 2-13 in the space-fixed frame, it turns S  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 a is used to denote v  such a reflection. This operation is equivalent to I ,[33], i.e. they have the same eigenvalues: +1 s  for + parity states and —1 for — parity states. To consider reflection of only electronic coordinates (somewhat confusingly, a is always used v  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: a  v  |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)  Ae  = +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 a  v  (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  - | - A , 5 , - E ; J,-SI), n  (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 <r„|A,S,E; J,Q) = (-1) - + ^ J  S  X  |—A, S, - £ ; J, —Q).  (2-15)  2  GENERAL  THEORETICAL  CONSIDERATIONS  19  Other than for a £ state (A = 0, E = 0, ft = 0), it is evident that the wavefunction in Eq x  2-15 is not an eigenfunction of a . The operation of a on the basis function changes the function v  v  itself, and so parity is not a good quantum number in this basis set. However, a basis set for which parity is a good quantum number can be constructed by taking the following symmetric and antisymmetric combinations: 2 S + 1  A , J, ± ) = ^  [|AS£; J, ft) ± ( - 1 ) ~ J  n  s+X  * \-AS - E; J, -ft)] ,  (2-16)  where the + label in the basis function goes with the upper sign in the combination and the — label goes with the lower sign. Then A ,J,+)  2S+1  0~  v  tl  = + \ A ,J,+) iS+1  a  and a |  2 5 + 1  v  A , J,-) n  =- |  2 S + 1  A , J, f i  The wavefunction for which the eigenvalue of a is +1 is defined as a + parity state, and the v  wavefunction with eigenvalue —1 is a — parity state. Instead of labelling the doubly degenerate levels in states with A ^ 0 as having either +A or —A, the two degenerate levels are referred to as a parity doublet, one with symmetric (+ parity) and the other with antisymmetric (— parity) combinations of +A and —A. An alternative, equivalent labelling scheme uses e and f parity labels, rather than + and — parity [35]. For molecules with an even number of electrons, the e/f symmetrized basis functions are 2 5 + 1  An, J / ) = ^  [|A5E; J, ft) ± (-l)- + * s  x  \-AS - E; J, -ft)] ,  (2-17)  where the e parity level in the basis function goes with the upper sign in the combination, and the f parity level goes with the lower sign. For molecules with an odd number of electrons (not dealt with in this thesis), the phase factor would be ( - 1 )  _ s + A e +  ^ instead of (-l)~ . s+Xs  One advantage  of using e/f parity labels instead of +/- parity comes from the fact that the awkward (-1) factor J  is removed from the basis function. This is more convenient than +/- parity for many applications [35]. 2.3.2  Homonuclear Molecules  When the two nuclei in a diatomic molecule are identical, there are additional symmetries that must be considered. There is a plane of symmetry halfway between the two nuclei, and if the electronic coordinates are reflected through this plane (the only symmetry operation that will be considered here; further consequences of this symmetry will be given in Section 2.10, in relation to hyperfine effects), the electronic part of the wavefunction can either change sign or remain unchanged. If  2  GENERAL  THEORETICAL  CONSIDERATIONS  20  the wavefunction remains unchanged from this reflection, a subscript g (from the German word "gerade" for even) is added to the label for the electronic state. If the wavefunction changes sign, the state label is given the subscript u (from "ungerade" - odd). For example: 2.4  3  IIi , IIi ,.... 3  1x  g  Perturbations Table 2-1: Selection rules for perturbation (excluding hyperfine effects) A A = 0, ±1 An = o, ±1  Electronic  AS = 0; A S = 0, ±1 u <->• u; g <-> g E+ <-» £+; £ " «-» E " AJ = 0 Rotational e <-> e; f <-> f There are standard techniques for the treatment of perturbations (see e.g. [29]). The Hamiltonian operator is separated into the sum of a zeroth-order Hamiltonian (H„) and an interaction Hamiltonian: H = H + H', Q  where H has the following properties: 0  BO  Ho  =E  n  and  (**° H ^ ° ) = 0 ( m ^ n ) . 0  (2-18)  E is, of course, the (zeroth-order) energy of the level labelled by "n." n  For diatomic molecules, operators that provide a coupling between nuclear rotation and electronic 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.  *'  = -sin([3)^  0  Tn  + 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 (<IC 1), and the mixing coefficients were assumed to be real. The labelling scheme is no longer exact, since it was based on quantum numbers associated with the pure states, but the labels are retained for convenience. The state with the larger portion of *b is labelled by the quantum numbers from state n, and the one with more ^i^f character is BO  labelled by the quantum numbers from state m. There are constraints on whether or not two states can perturb each other. These constraints, known as selection rules, are listed in Table 2-1. These selection rules are specific to Hund's case (a). For the additional possible perturbations in Hund's case (c), from the coupling of different electronic states by spin-orbit effects, see [30]. When the energy separation is large compared to the mixing matrix element (^> ° H' B  l  the energy shift in a state n (6E ) is given by second-order perturbation theory to be n  ;BOV n / Ego - E£0 BO  6E = n  2  H'  (2-19)  The perturbation provides a mutual repulsion: the state with higher energy is shifted upwards in energy by an amount 6E , and the lower state gets shifted down by the same amount. n  2.4.1  The Rotational Hamiltonian  The rotational Hamiltonian in Eq 2-8 can be written more explicitly as 1  Hrot =  ;(R  X  +R ), y  where R = 0 because R is perpendicular to the internuclear axis, the z-axis in the molecule-fixed z  frame. In Hund's case (a), since J = R + L + S, this becomes 1  H rot  2fj,r  2  (Jx  L  Sx) ~\~ (Jy  x  Ly  Sy)  Using ladder operators, symmetric and antisymmetric combinations of the x and y components of angular momenta, J-i- = J ± ij , x  y  L± = L ± iL , x  y  and S± = S ± iS , x  y  2 GENERAL  THEORETICAL  CONSIDERATIONS  22  the rotational Hamiltonian can be written in a more convenient form: 1 {J -J ) + {L -L ) 2yur  Hrot — •  2  2  2  + {S -S ) + (L S-+L_S )-(J+L^+J-L )-(J S-+J-S+)\  2  2  z  2  2  +  +  +  +  . (2-20)  The operators J , L , and S represent J • J, L • L, and S • S, respectively. The matrix elements 2  2  2  associated with these three operators are diagonal, and give the square of the magnitude of the vectors: J | J, ft; S, £, A) = J( J + l)h | J, ft; S, £, A) , 2  2  L | J , f t ; S , £ , A ) = L ( L + l ) f t | J , f t ; S , £ , A ) , and 2  2  S | J, ft; S, £, A) = S(S + l)h | J, ft; S, £, A ) . 2  2  Note that the matrix elements shown here for L and S (and later on for L , S , L± and S±) are 2  2  2  2  defined in pure Hund's case (a), and not in Hund's case (c). The operators J , L and S also have only diagonal matrix elements: 2  z  z  J | J, Q; S, E, A) = Qh \ J,ft;5, E, A ) , z  L | J , Q ; 5 , E , A ) = Aft|J,f2;5,E,A), and 2  5 |J,ft;5,E,A) = Eft|J,ft;5,E,A). 2  The ladder operators have no diagonal elements. The rotational energy is given by the diagonal matrix elements of the operator in Eq 2-20: E  = B [j(J + 1) - 0? + S(S + 1) - E + L ( L + 1) - A ] , 2  rot  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 E  = B [J(J + l ) - f t ] , 2  rot  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  2.4.2 L  +  THEORETICAL  CONSIDERATIONS  23  Uncoupling Phenomena  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  +  J± | J, ft; 5, E, A) =  is a lowering operator (decreases ft by one unit): +  - f t ( f t T l ) 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}  En, ,j - -^s±y,j v  v, J  -h  v', j)I  2  [J(J + 1) - ft(ft + 1)] [L(L + 1)]  (2-22)  Ej[,v,J ~ E-z± ',j tV  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, ,J ~ E^±yj]  is roughly constant as a function of  v  J. With these conditions, the energy shift is AE(v, J})=  q [J(J + 1) v  + 1)] « q [J(J + 1) - ft ], 2  v  (2-23)  where q is called the A-doubling (or ft-doubling) constant. The purpose of the approximation in v  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: BP  £rturbed  (2-25)  = B + q. v  v  Often, A-doubled states are incorrectly represented as being symmetrically split about the mean'. position, with the apparent B-values for the two different sets of parity levels written as B +^ v  and  B -%jf. This is an expedient approximation when it is not known which set of levels is perturbed, v  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 fi = —euivr,  (2-26)  2 GENERAL  THEORETICAL  CONSIDERATIONS  26  where the magnetic moment is directed opposite the angular momentum vector, since it is a negatively charged particle. The classical angular momentum of this electron is L = mvu r  = 2nmvr , 2  near  (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  """Obs;)*  (2  -  31)  where m is the mass of the proton, and ^ is the ratio of charge to mass number (ratio of number p  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 perturbations 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 ) of a magnetic dipole (p) in a magnetic field (H) is defined as m  V  m  =  -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: H  Zee  where H ^  e e  = -p.H=^-(g L L  + g S) • H, s  (2-33)  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 conserved. 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 A+^)^ M  + 1 ) % / J (  7  + 1 )  |g|  = mAH\. WS1  (2-34)  For convenience, an effective rotational g-factor, gj, has been introduced. ( A + g E)n 9L  9 J  =  s  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  field off  field  J  on  momentum along the field direction need be considered. Treating this situation quantum mechanically, the magnetic field can cause perturbations off-diagonal in J by ± 1 (and diagonal in mj). A l 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) = E e^- -^, ?  0  and H(r, t) = H e^-^. 0  (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  rl  2TT  For simplicity, the fields are taken to be plane waves travelling the +Z direction in the spacefixed frame: E = E e - -^ i(  0  kZ  and H = H e - ~^. i(  0  kZ  (2-37)  The direction and magnitude of E M radiation's fields are important considerations for interaction with matter. The polarization of an E M wave is defined according to the electricfieldin 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 lefthanded 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 E =  E [x-iy}e ( -« t>*\ i kZ  R  t+  oR  and the equation for LHCP light with arbitrary phase (f>L is E =E [x L  +  oL  iy]e ( -" ^\ i  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 — <f>L, the electric field for a wave with an arbitrary elliptical  polarization can be written E= (ER + EL) = [x (EoLe-^  + E e ^)  + iy ( f ^ e " ^ - J 5 e ^ ) ] e^-^+V.  i  oR  ofl  (2-38)  Setting one of the amplitudes (E R or E L) to zero obviously gives circular polarization. When 0  0  the two amplitudes are equal, Eq 2-38 becomes E = 2E  Q  x  c o s  /A<M \ 2 )  +  y  „ . /A(f>\ \~Y) s  m  i(kZ-ut+4>)^  e  ^2-39)  This is linearly polarized light, with its plane of polarization oriented along the axis shown in Figure 2-8, at an angle ^ page.  from the -l-X-axis. Note that the +Z-axis in Figure 2-8 is directly out of the  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 H l tric = ~PE • E(f, t)  (2-40)  e ec  The electric dipole moment, /Ug, is defined by = YLwi,  (2-41)  i  where qi is the charge of the i  th  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*-**) = E e-  iujt  a  [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 (e " w 1) [34]. This is known as the lk  r  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) = ^ where E  m  m  e ' ^  and ip (t) =  ipne ^ , 1  n  111  and E are the respective energies of the two states. The matrix element for an electric n  dipole transition, in Eq 2-43, has no time-dependence for a frequency of E M radiation v = m- n _ E  E  When this condition is satisfied, the matrix element in Eq 2-43 reduces to 4>m ~PE • E 1p.  (2-44)  0  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 , which is equivalent to saying that the molecule can n  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. fi represents the component of JIB along a particular axis. See e.g. [28]. a  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) AA =  0, ±1  Aft = 0, ±1  Electronic  AS =  0; A E  =  £ + *-+ £ + ; X T  0 S-  u <-> g AJ =  Rotational  0, ±1  ±1 (ft = o -> n = o)  AJ =  + <-» AJ = AJ =  0 =^ e <-> f  ±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 H i 3  «— £ + transitions being studied in the current work would not/be x  u  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 a . However, the electric dipole moment in v  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 witha 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:  (v  m  \-jt  E  • E \ipn) = 0  (cf i r  )  ib n  (vc'c\-fi  E  • E \cv*) 0  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: dx va—pr = —Fix) w —Dx, 2  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 = -eE e-^ , at at t  0  z  (2-46)  where u — 2iru. The solution to this equation is —eE e~ . ° .  iult  S=  2  2  (2-47)  v  where 7 = £ and UJ = 2  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: Ne  2  P=  (  2  °  E„e~  iujt  2  •  y  The polarization, as derived from Maxwell's equations, should have the form [37], P = e E, oX  (2-48)  (2-49)  2 GENERAL  THEORETICAL  CONSIDERATIONS  where e is the permittivity constant, and %  36  the susceptibility. The complex index of refraction,  l s  0  n, is defined by the relation X = n - 1. 2  A comparison of Eq 2-48 and Eq 2-49 leads to Ne e m(u) — LO — iju)' 2  2  n =1 + 1  1  2  2  0  In a gaseous medium at sufficiently low pressures, the index of refraction is close to unity: n - 1 = (n + l)(n - 1) « 2(n - 1), 2  which gives Ne  2  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 =  where  A is the wavelength in vacuum. The wavenumber k = ^ = | ^ n = k n, where k =  If the  n  0  0  D  complex index of refraction is separated into real and imaginary parts, n = n + tK, 0  the electric field component of E M radiation propagating in the Z-direction from Eq 2-37 becomes E = E e ° -^ i{k  = E e Zre ( ° ° - \  nZ  :z2  0  i k  0  n  z UJt  (2-51)  The real part of the index of refraction, n , describes the dispersion of the medium (the variation 0  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\  c  n  no  0  1  a  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) =  I e- , aZ  0  where I is the intensity at Z = 0. From Eq 2-51, a is a  a  =  -Xo-  2 GENERAL  THEORETICAL  CONSIDERATIONS  37  Expressions for n and K come from evaluating the real and imaginary parts of Eq 2-50: a  Ne u -u n =1+ 2e m {UJ — u> ) + 7 o> ' 2  2  2  0  a  2  2  2  2  2  0  Ne AC = 2e m (to 2  70; 2  0  — UJ ) 2  +  2  ^ UJ 2  '  2  For CO u> , these expressions reduce to 0  _ U o  =  Ne u) — UJ 4e nuv (u, -u)* + qy 2  0  l  +  "  ]  0  0  (2  0  _ Ne 7 ~ 8e muj (u> - u) + (^) '  52)  2  K  2  0  0  2  0  ^" ^ 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 complex index of refraction.  2.9 2.9.1  Widths of Spectral Lines 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 , the uncertainty principle imposes n  a limit on the ability to measure the state's energy, E [37]: n  8E ^-. n  (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 = E - E , the uncertainty in the frequency for the transition is n  m  SE + SE, 7 ' 71 h  6v «  n  Using Eq 2-54, the resulting expression for 7 becomes: 2n8v = 6u = 7 = ( — + — ) . 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 7 = -rad 771  coll -j-rad 'm n  T  1  + T;•coll ' n  (2-55)  2  GENERAL  where T  RAD  THEORETICAL  CONSIDERATIONS  is the mean radiative lifetime, and T  39  is the mean nonradiative lifetime, governed by  CO11  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 (»-»o)\2 e  I{u) = I e a  K  ">°vp ' ,  (2-56)  where the most probable velocity, v is defined by !2kT VM p  F  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 u  0  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 10 ) than the natural (radiative) 6  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+i ,  (2-58)  2  where i\ is the spin of nucleus 1 and i is the spin of nucleus 2. 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 Br2, and i i = 12 = § for I2. Particles with halfintegral 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 X comes from the electronic symmetry operation that led to u/g labelling for the electronic ug  states of homonuclear molecules. This parameter is equal to 0 for g electronic states and 1 for u electronic states. The term (-1) ~ + ^ is just the effect of cr described in Eq 2-15. J  s  v  Since h = 12 = § for B r 2 , the possible values for the quantum number I (as calculated from 7 9  Eq 2-58) are 0, 1, 2 and 3. In a £ + state, for example, for even J states, only the even I levels (0 1  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 E+ X  state in B r 2 , odd-J states are ortho and even-J states are para. 7 9  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): H  hf  = H (l) hf  + H (2) +  H (l,2).  hf  hf  The electron-nuclear terms are considerably larger than the nuclear-nuclear terms [39], and contributions 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' \H {l)\u,l) hf  (g,I'\H (l)\g,I) hf  (u,I' \H (l)\g,I) hf  = (-l)  (u,I  I+r  1  \H (2)\u,I), hf  = (-l) '(g,I'\H (2)\g,I)  and  I+I  hf  = (-l/+''  + 1  (u, I' \H (2)\g, hf  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 i to give I. To illustrate the various magnetic dipole hyperfine 2  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)| ^ 0), there 2  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)| ?i • S. 2  (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 HiL^)  1 .£ = '2g g fXBm- ^--  (2-62)  L  L  N  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. A n 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:  b= b  - -,  FC  (2-64)  87T  °FC = "g 9s9NL B N |*(0)| , and -  L  L  2  r  3  /3cos e-l\ 2  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. ( ) denotes an average over the electron spins contributing to s  the interaction with i\, and ( ) denotes an average over the orbital angular momenta of electrons e  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 i • J is slightly more 2  complicated. Also recall that only certain combinations of i\ and i are allowed for a homonuclear 2  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. A n 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 irreducible 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.  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 : <Zo = 2 ( A | T ( V £ ) 2  0  The component of the electric field gradient perpendicular to the internuclear axis is given by q2, which is defined in terms of matrix elements off-diagonal in A by 2 [40]: 92  = -2v 6/A + 2 r (V£) A /  2  2  A third component, qi, is defined in terms of matrix elements of T (\7E), which are off-diagonai 2  in A by 1. This term does not enter into the current work, and is therefore not presented here, but the definition may be found in [40]. The different electric quadrupole parameters (eQqo, eQqi and eQq2) are determined from experimental data. For most electronic states, the diagonal matrix element of HEQ (the electric quadrupole part of Hhf) is proportional to eQqo. The terms eQqi and eQq2, on the other hand, are usually associated with perturbations to other electronic states. However, for the special case of a Hi  electronic state, the term involving eQq2 can also contribute to the diagonal matrix  2S+l  elements of Hgg. Recall that an electronic state with A ^ 0 consists of doubly degenerate states, one with e parity and the other with f parity. For a H i state, the state of interest for the current 3  work, the symmetrized wavefunction describing an e parity level (from Eq 2-17) is 1 I i i , e) = -j= [|A = 1, S = 1, £ = 0; ft = 1) - |A = - 1 , S = 1, £ = 0; ft = -1)], v2 1  and the wavefunction for an f parity level is !  rii,/) =  [|A = 1,5 = 1,E = 0;ft = 1) + |A = - 1 , 5 = 1,E = 0;ft = -1)].  Since the term in H # Q associated with eQq2 has matrix elements between A = +1 and A = -1 (off-diagonal in A by 2), there is an extra contribution to the diagonal matrix element of HEQ, but the contribution is opposite in sign for e levels than it is for f levels: U J\H (q )\ 1  EQ  2  Il ,f)  3  l  = -( Il ,e\H (q )\ 3  1  EQ  2  Il e  3  u  2  GENERAL  THEORETICAL  CONSIDERATIONS  45  Therefore, the contribution of eQq2 subtracts from the eQqo contribution for e parity levels, while it adds for the f parity levels. This can be accounted for [43] by defining different effective eQq values for the two parity levels QQeff( ) = eQq  e  f  e  0  6 J(J + 1)  ± eQq ,  (2-65)  2  where the upper sign on the right hand side of the equation corresponds to an f parity level, and the lower sign corresponds to an e parity level. The electric quadrupole contribution to the hyperfine energy is proportional to eQq //(^). e  An interesting consequence of this difference in the effective value of eQq is that e parity levels and f parity levels have different hyperfine energies, and this contributes to the splitting between the two states. As an analogy to the normal A-doubling (described in Section 2.4), this effect is referred to as a hyperfine A-type doubling (or hyperfine ft-type doubling in Hund's case (c) coupling). It should be stressed that this hyperfine A-type doubling does not result from a perturbation between the e and f parity levels (such an interaction requires an electric field), but rather comes from different diagonal matrix elements of H#Q for e and f parity levels. Contributions from higher order magnetic and electric multipoles of the nucleus are small compared to magnetic dipole and electric quadrupole effects, and are therefore not considered here. Because J couples with / to give F, J is no longer a good quantum number, and the list of selection rules given in Table 2-1 is no longer complete. In addition to the selection rules for perturbations listed in Table 2-1, magnetic dipole effects allow A J = ± 1 , and A A = ±2. Additional selection rules for electric quadrupole effects are A J = ± 1 and ±2, A A = ±2 and Aft = ±2. The +/- parity is still conserved for all perturbations, and every A J = ± 1 perturbation has 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 A I = 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  / ( 2 F ' + l)(2F" + l)(2J'+l)(2J" + l ) ( - l ) ( J"  1  J'\ (  F"  { -ft" ft"-ft' ft'J ^ -m"  F  / + J  ' - " ' "- ^- "+ +  7  + F  + F  m  n  1  (v"\fi \v )  1 F' \ [I J' F'\ m" -m' m' ) { 1 F" J"} " F  F  F  1  a  ( 9  ,  a ] 66)  "  (2  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 (m  F  - m' ), can be 0, +1 or —1. In the current work, it is the Amp = ± 1 transitions that are F  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: (2-67)  ip" = aipi + bip2 and ip' = atp + Pipy, x  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-E \i> ) = \aa(^i ,S  f2-E ip ^+ap(i)  2  0  0  x  ±  p,-E ipy}+ba(tp 0  2  fl-E  a  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 = /-<-B9F FH. M  This equation has the same form as E q 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]: F{F + l) + J{J + 1) 2F(F + l)  r  9F =  -1(1+1)"  9J-  (2-69)  Note that this g-factor is for the matrix elements of the Zeeman Hamiltonian diagonal in F, J and I.  3 MAGNETIC  3  ROTATION  47  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 transition, 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 A m j = +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 A m j = ± 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  48  ROTATION  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 Br2, 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 L H C P light (ax) and the absorption coefficient for R H C P light {an), (a - OR) ^ 0. L  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 E L and E JI in Eq 2-38 will have changed, Q  0  thereby changing the degree of ellipticity of the polarization. For example, MCD changes linearly polarized light (E L = E R) into elliptically polarized light (E L ^ E R). 0  0  D  D  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: A n R(0) transition in the presence of an external magnetic field. The transition frequency in the absence of an external magnetic field is v  a  Am,  =+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 A m j = ± 1 transitions are possible; the A m j = 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 approximation, 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 A m j = 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 direction, 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, L H C P 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 counterintuitive 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 A m j = -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 A m j = +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. v is Q  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  7^ OCR.  L  A  By the definition of the absorption coefficient a, the amplitude of the electric field decreases as E e~2 ,  where E is the amplitude of the electric field at Z = 0. The amplitudes of the electric  z  0  0  fields for two polarizations are therefore given by: E (Z) = E e~^  and E {Z) = E e~^ .  z  L  R  An average field is defined as E (Z) = L{ )+ R{ ) E  (3-1)  z  oL  Z  E  oR  ^a n ( j  Z  a  difference between the two fields is  defined as AE(Z) = E r,(Z) - E#(Z). Starting from Eq 2-38, E L and E R in this equation are J  0  Q  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)  E {Z) + a  -  * m )  AE{Z)\ e~*2 2 )  -  i  *+ AE{Z)\  E (Z)  i(kZ-ut+<j>)  a  a  which reduces to (A(j>\ _ . (A(j)\ ; E{Z) = \2E {Z) cos \ — ) x+sin\^—J y^ +iAE{Z) V2  -sin{  a  A(j)  x+cos(^^  i(kZ-ujt+(p)  e  )']}•  (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 • If — <t>\I y, x + sin A  cos  (2  (3-3)  and a component along the axis defined by the unit vector . /A(j>\ . [A^ -sin I — jx + cos I — ] y,  (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 — R, the component of electric field along PA, EPA, is A  E  = i(AE(Z))e - -" ® i{ kZ  PA  t+  = ie-% \E e z  oL  E e 4 oR  i{kZ-wt+<t>)  0  (3-5)  3 MAGNETIC  ROTATION  52  To allow for some ellipticity in the original polarization, a parameter d is introduced such that e  2d  = g * . That is, e E  = e- E  D  D  OL  OR  E  = E . Eq 3-5 then simplifies to c  = -2ie-% E sinh z  PA  0  [^Z  + d) ' ( ^ - ^ + < « e  ( 3  .  6 )  In the case where the initial polarization is linear (E L = E ), the parameter d is zero. Note that Eq 0  OR  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: (n  oL  - n ) ^ 0. oR  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. v is the transition frequency in the D  absence of external fields.  In Figure 3-3, the higher frequency curve is associated with the A m j = +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 n between LHCP and RHCP light at the frequency indicated by the Q  dotted line. A difference in n means the two polarizations travel at different velocities through the Q  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 n for the two polarizations is n = <>L+ aR n  G  n  a  a n  d the difference is A n =  oL — n-oR- Starting again from Eq 2-38 and making the substitution k = 2™a.^ the electric field in  n  the E M wave after travelling a distance Z through the medium is given by -*  • Ad> „•  f r  7r  l  l  • A<£  An v  E(Z) = { [E e- 2e —  +E e  Z  l  oL  oR  E  o L  e-^e  7r  A  x +  2 —5; e  i E  ^  z  - E  e^e-^ ] z  o R  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 (E L = E Q  O R  = E ). 0  Eq 3-7 then reduces to E(Z) = 2E cos  (3-8)  0  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 — ^ Z, E  k  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 6 from PA ( | - 0 radians from IA). Starting again from the 0  o  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  , , Aa cosh [ —Z 4  E{Z) = 2 £ e " 0  \ f + d)sin[9 / V  7r A n „ . —Z ] A  n  0  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 9 from PA). a  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 9 radians from PA, using the expression Q  for the electric field in Eq 3-9, is I(Z) = pE e 2  -aZ  • /Aa, cosh' Z + d) sin  IT An \ . , o / A a „, —Z^+sinh^—Z +  9,  2  \ / TT A n d ) ^ - — : 9  where p is a constant of proportionality, into which all constant factors not explicitly included have been collected. This equation reduces to I(Z) =  p^-e-* cosh 2 —Z 4 z  +d  7r A n A  cos 2  (3-10)  0  The series expansions (cos# « 1 — ^ + ... and cosh# RS 1 +01 2!|T + •••) can be used to simplify this. The first order terms from Eq 3-10 are I(Z) =  2  7r (An) A ' 2  I(Z) = pEle  -ccZ  IT  pE e-  aZ  2  2  29,  IT  An z  An  e  \  2  +  0+  /Aa„  2  ^ z  2 +  ,  ^z  N 2  d  2  d  +  (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 , 6 and d are small (<< 1); this 0  assumption appears to be very well justified for the experimental conditions used in the current work. The terms d and 9 are "background" terms—they do not depend on properties of the 2  2  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 « !  (3-12)  2  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 Nje fij 2  E  Ni& fij  uijj - UJ  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 n =1 + Q  \-nN(J",m"j)  u -u 0  (3-13)  transitions  and Air  ° transitions  6e h 0  (aj -u)* + qy  \{v ,J ,mj,il \fj, \v , J ,mj,ll )| a  (3-14)  0  where the sum extends over all transitions possible for the polarization in question, e.g. The sum for II L would be over all transitions driven by LHCP light, all A m j — +1 transitions. The 0  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 u (2n times the transition 0  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,m ) H  J  = ^e-W ">'*fi, Z  (3-15)  J  p  where  is the total number of molecules per unit volume in the sample, Z is the partition P  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'i ') = (v",r,m'S = m'jTl,tf'\l**\v', J',m'j,n') J  (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 :  ^(J,J  + 1) =  Amj = - 1 :  y/(J + mj)(J + mj + l) 4(J + 1)V(2J + 3)(2J + 1)  (3-17)  y/{J-mj)(J-mj + l) S(J,Q", J+l,ft') 4(J + 1)^(2J + 3)(2J + 1)  (3-18)  P-transitions (J —> 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)  Amj = -1  ^ ( J , J -1) = +  y  M  J  V  '  K  (  J  +  {  '  J  '  U  / ! ^ ± ! ! ^ g ( J , f t " ,  4JV(2J +  1 ) ( 2 J - 1)  v  J  J-i,ft')  '  '  '  ;  Q-transitions (J —> J)  A m j = +1:  ^  (J, J) = + ^ -  A 1 Amj = -1:  — / t x\ , V{J -mj){J + mj + Tj „ , fj, (J,J) = + 4 j ( j + i) G(J,Sl ,J,Sl)  +  7;^^  +  1 }  Q(J, n", J,  ft')  mj  (3-19)  (3-20)  Of particular importance for the current work, note that the sign is the same for A m j = +1 and A m j = -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 ft' = ft"+l = 12+1 ft' = ft" = ft ft' = ft"-i = n-i  j' = J "  -y/(J + Q + l)(J + Sl + 2) 2y/(J + Q + l){J-n  j' = j"-i = j-i  +n +  -y/(J-Q)(J-Q-l)  2ft  + l)  V ( J - n + i ) ( j - n + 2)  =J  V(J +  ft)(J-ft  -2y/(J-n)(J + i)  yj  + Q.)  + 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  58  ROTATION  Figure 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 A m j = -1 transitions are driven by RHCP light.  R(2) /  \  /1  A  1\  1\  I\  +3 +2 +1 0 -1 -2 -3  A m , = +1  -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.  (b)  (a)  lMmj(J,J+l)| as a function of mj, for Amj= -1 transitions  Mmj(J,J+l)| as a function of m for Amj= +1 transitions r  +  + ^  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 (|£^ R|) were equal. The plot in Figure 3-5(a) should be compared to.the 0  corresponding results for A m j = -1, plotted in Figure 3-5 (b). 2  Notice that fi (J, mj  J + 1)  _  2  for A m j = +1 transitions is equal to l^^_ ^(J, J + 1) for A m j mj  = -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 = H cos(2nfHt) 0  = H cos(u>Ht), 0  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 signalto-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 firstharmonic 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 co in Eq 3-13 and Eq 3-14 resulting from the 0  shifts of the magnetic sublevels can be expressed as . , ,, AE(m'j) 1 Au (m'j) = ^ = - VBgjmj h 0  cos{u>Ht) = T]jm'jCos(ujHt),  (3-21)  3 MAGNETIC  ROTATION  61  where the parameter 77j (= \^B9J H ) has been introduced for convenience. 0  Using Eq 3-14 and keeping in mind that LHCP light drives A m j = +1 transitions and RHCP light drives A m j = -1, an expression can be written for magnetic circular dichroism: A a = ax - a  R  =  47r 7riVj» X Qe h Amj=+1 0  -  7  (w - to + rijM cos(u> t))  + (2)2  2  0  0  +  H  (3-22)  E I ^ V ' ) I % ,,,,,^ Am - l ( W q ~ W + TljM-COsiiVHt))' + (^) a  2  J=  where M  +  represents the value of m' (the value of mj in the upper state) in a A m j = +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 denominator of Eq 3-22. To remedy this problem, the first term in the denominator is rewritten: (u - to + r]jM±cos(oj t))  = (u - u) + 2{OJ - u)njM±cos{uj t)  2  0  2  H  0  H  0  + ^ j ( M ± ) (1 + cos{2u t)), 2  H  where use was made of the trigonometric relation cos (a;#t) = 1(1 + cos(2cjfjt)). 2  If the maximum splitting of the magnetic sublevels is small compared to the H W H M (^), the series expansion  « l - x + x -..., | x | < l can be used to bring terms involving cosines into the 2  numerator:  7  ^_  (Uo - OJ + rjjMcosiiJHt)) + (§)2 ~ ( 2{u - ui)r]jMcos(uJHt) + 2  W q  1 _ ) 2 l 2 2 + (2)2 \rjjM cos(2u) t) w  +  v  jM  2  Q  H  ( a ; - ) 2 + l 7 ? M 2 + (2)2 0  4(w - u) (r]jMcos(uj t)) 2  0  2  H  + 2(u - u)n M' cos{uj t)cos{2u t) z  0  [  +  2  W  K  _  w  i  J  )  2  H  1 ^ 2  +  M  2  +  H  ( i )  2]  +  \rjJM cos (2oj t) 4  2  H  2  (3-23) for M equal to either M or M _ . The \r] M 2  +  2  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 Dopplerlimited 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  Aa  fs  « -———X oe h 0  [(u> -u;) + (2) ] 2  a  1 2  ,n  2  [(  (  Am  + A  m  j  =  +  1  J =  -l  J(3K-W) -(|) )  2  x  2  0  £  W o  _ )2 w  +  2  (2)2]<»  l^ (J",J')| (M ) 2  2  +  +  \Amj=+l  J2  \»M_(J",J')\ (M-) ) Amj=-1 2  +...| ,  2  (3-24)  /  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 / must be substituted into Eq 3-11. S  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 identities cos (u>Ht) — 3 [cos(3o;#t) + 3cos(u;tft)], and cos(cj^i)cos(2a;^i) = ^[cos(u;#£) + cos(3wf/£)], 3  and so on, are useful. In Eq 3-11, once A a / is substituted from Eq 3-24, there will be contribuS  tions only to odd harmonics (first, third,...) from the A a term. By the same token, the A a term, 2  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 S „(Aa,.)  =  F  - p E l e - ^ Z i ^ " A ie h 0  a  pEy-z^Zd 7  [(  W o  i  W  _ ,)2 a  Z i ^ +  (2)2]  E\^(J",f)fM , +  A  m  j  =  +  (3-25)  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 / ) , depends on the intensity of the incident radiation (|.E | ), 2  S  0  the distance travelled through the medium (according to Ze~ , roughly proportional to Z for weak aZ  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 . Using n from 0  c  Eq 3-13, the general expression for A n is An = n  -  oL  ITNJII  n  3e h  oR  E  Q  \VM (J  Amj=+1  ,J)\  +  { U o  _u  UJ  0  - A mE, = - i  u>„ — U njM+cos(u> t)) + mi 2  +  H  — L)  \»MAJ",J')f K - w + ruM-cos(u t)y  Following the same procedure that was used to determine A a /  (3-26)  + (2)2  H  S l  the frequency-shift contribution  to Eq 3-26 is [(3r) -(u; -u;) ] r)jcos(cv t) 2  2  0  A n •fa'  3e h  H  [(u; -a,)2 + ( 2 ) 2 ]  0  2  M+  0  [ K - a ; ) - 3(2)2] 2  +^n cos(2u; ji) 2  2  £  \^ (J",J')\ M Amj=+1  2  {  U  q  _  u  +  | ^ J / , / ) |  2  K  Amj=-1  )  ^2 2 , ^ 2 1 3  [(w -a;) + (2)2]  /  0  E  |^ (J",J')| (M ) 2  2  +  E  \^ _{J",J')\\MA Amj=-1  +  ^Amj=+l  (3-27)  +....  M  /  Again breaking the summations over A m j = +1 and A m j = -1 into a consideration of complementary 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 A n in Eq 3-11, the leading contribution to the first harmonic signal from Eq 3-27 is S (An ) FH  = - Ele-« W*-^Z  =  z  fs  p  47rATj» (CJ -UJ) - (2)2 A 3e h [ K - a ; ) 2 + ( 2 ) 2 ] 2 E ^ 7T  pE e- 9 Z 2  aZ  o  Vj  0  2  0  0  A m  0  '\^+(J",J')\ M+.  (3-28)  2  + i  None of the other terms in Eq 3-11 give a contribution to the first harmonic signal. The terms in ( A a ) and (An) only have even harmonics, just as the terms in Aa and A n have 2  2  only odd harmonics. Taking the first order term for A a / from Eq 3-24 and the first order term S  for Anf from Eq 3-27, the leading frequency shift contribution to the second harmonic magnetic s  rotation signal, SSH, is SsH((&a ) ) 2  fs  p E ^ W j Z  ^ * ^ A 96^2  2  + S H((An ) )  =  2  S  fs  1  7  _ )2  2  [(u}  UJo  +  q 2y )  £  \^ (J",J')\ M  i Amj=+1  2  M+  +  ,  (3-29)  3 MAGNETIC  64  ROTATION  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 Z e~ , 2  aZ  is proportional to the square of the amplitude of the magnetic field, and does not  depend on the "background" parameters 9 and d. a  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: J,mj,Cl)'  « \J,mj,Cl) +  (J, mj, Cl \H ee\ J, mj, Cl) p  Z  EJ-EJ  J,mj,Clp) +  J + l,mj,Cl ) + p  \J-l,mj,Clp),  (3-30)  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.  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 A m j = -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 , m j , f t \Hzee\ J,mj,Q) = p  x  / ( J + mj + l ) ( J - m j + l ) 2(J + 1 V ( 2 J + 3)(2J + 1) G(J + 1,ftp,J, ft) fi cos(cj t) >  B  (ft  H  \g S + g L\  p  s  L  ft)  (3-31)  Perturbation of J with J:  (J,mj,Q,p\H ee\ J,mj,ft) = Z  mj G(J,ftp,J, ft) n cos(uj t) |i? | (ftp \g S + gLL\ft) (3-32) 2 J ( J + 1)B  H  0  s  Perturbation of J with J - 1: (J - l,mj,ftp \H e\ J, raj, CI) = Ze  y/(J + mj)(J - mj) ftp,J, ft) jjL cos{Lo t) H \ (ftp gsS + g L 2 J / ( 2 J + 1 ) ( 2 J - 1 ) "Q(J " - 1, - > - T o - > - v r ^ ~ - v ~ n ~ j |««| yv\**~ V  B  H  0  L  N  (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 (Q gsS + gjjj ft) must be left as a parameter to p  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). A n 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: ,  ±  ZeeK J))  H  m  ,\  (J ±l,mj,n\H e\  J,mj,Q)  Ze  =  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 3-30 leads to a perturbed transition probability:  MM ( V)| J  ±  2  *  + (Ht (M ))^ (J",J'  \VM (J",J') ±  E  ±  (^  + 1) +  ±  E E  (M ))^ (J", ±  ±  J' - 1)  Expanding this gives: MM ( "> ±  J  J')\ = | ^ M ( ^ " > ^ ) | + | ( ^ ( M ) ) ± ( j " , J' + l ) | + | ( i ^ ( M ) ) / 4 ( J " , J' - 1 ) 2  2  2  ±  + 2 (H+ (M )) ee  ±  e e  ±  M  ±  e e  / 4 ( J " , J ' ) / 4 ( J " , J' + l) + 2 {H~ {M )) ±  ±  ±  ± ( J " , J ' ) / 4 ( A J' ~ 1)  M  ±  ±  ( f T j ( M ) ) a4 (J", J' + l ) / 4 ( J " , J' ~ 1)  + 2 (H+ (M )) EE  ZEE  ±  ±  ee  ±  ±  ±  (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 ^ ( M ) ) ± ( J , J)| + 2  e e  ±  M  ±  2(i^ee(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 wavefunction. 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,", J ) 1  ±  replaced by the expression in Eq 3-34. Keeping in mind the fact that (H^ {Mj)^ ee  , are  is proportional  to cos(wj^t), the signals for the various harmonics are determined as they were before: the first  3 MAGNETIC  67  ROTATION  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 / , there are only odd harmonics for the intensity perturbation S  contribution to A a , A a ^ (where ip stands for intensity perturbation); all of the terms that would p  give rise to even harmonic signals cancel out in the subtraction performed in Eq 3-22. The leading contribution to A a , is: p  JLfiv  £^ 1  ^ ^ T ^ h (u X 5e n {u> - coy + 0  0  0  A t 7  [F(M ,J' +  +  l,J',J")+HM ,J'-l,J',J")), +  + 1  (3-36) where the fact that the contribution from the A m j = -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  _ g  >  HM+\J  W  ,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: S (^ ) FH  = pE^Zd^-^f-. A  ip  x  -7^2 3e h (u -co) + (f) 2  c  a  2  0  [^"(Af+^'+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 A n . Introducing the perturbed transition moment in Eq 3-34 into the expression for A n in Eq 3-26, the resulting first order term is AnNj" a  n  i  x  p  *  (Up - LP)  3e h ( - u / ) 2 + (2)2 Q  J2  W o  [F{M+,J' + 1,J',J")+F(M ,J'  -1,J',J")}  +  ,  (3-38)  The intensity perturbation contribution to A n 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 X 3e h (u) - cv) + {^y z  0  Y  0  0  [J (M ,J'+l,J',J")+F(M ,J'-l,J',J")],  (3-39)  7  +  +  Amj=+1  The intensity perturbation contributions to the terms in A a and A n in Eq 3-11 have only even 2  2  harmonics, as was the case for the frequency-shift contributions. With the inclusion of intensity  3 MAGNETIC  68  ROTATION  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 „ ( ( A a ) ) + SsM(An) ) = E 2  5  2  2  P  aZ  e  Z  2  7T  2  — ( A a + Aa f / S  + -rj ( A n + A n )  ip  / s  ip  (3-40)  where Actf is taken from the leading term in Eq 3-24, Acti is defined in Eq 3-36, A n / is the s  p  S  leading term in Eq 3-27 and An^ is defined in Eq 3-38. p  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 external 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): ^ ( M , J ' , J ' , J " ) , j ( M , J ' + l , J ' , J " ) r  r  ex  +  and .Fea^M+jJ'-ljJ'jJ"), where the subscript "ex" stands for external.  ex  +  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 i  f  Q  4 l n ( 2 ) _ ( * o — / )  O  F(uj,J)e  2  **£duj',  (3-41)  Jo 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/ ) from s  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> ) gets replaced by the variable of integration (UJ'). The general procedure for including the 0  Doppler effect is to replace the frequency-dependent factor in a particular expression by an integral  3 MAGNETIC  69  ROTATION  of that frequency-dependent factor of the form given in 3-41. For Sjr.ff(An/ ) from Eq 3-28, the s  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  = W^TWY  M  The effect of Doppler broadening on second harmonic signals can be calculated from the above expressions. Ss#((Aa/ ) ) involves the square of the lineshape in Eq 3-42, for example. The 2  s  procedure for calculating Ss//((Aa/ ) ) is to find the integral in Eq 3-41 using the lineshape in Eq 2  s  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) + S H(&a ) F  + S H(An )  ip  F  + S (An ).  fs  F H  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 9 , the offset of the polarizer from PA. In a  practise, it is difficult to measure either d or 6 , which makes calculating the lineshape a challenge. 0  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 9 is much larger Q  than d, the first harmonic magnetic rotation signal will be predominantly from An. Conversely, if 9 is set to zero, the signal comes entirely from A a . To simplify the lineshape, it is best to choose a  one of these two regimes (9 — 0 or 0 3> d). a  o  Consider first the case for 9 — 0. The contributions from A n vanish, and the first harmonic D  signal is given by SFH = % ( A a / ) + S>w ( A a ) . s  ip  YU  3 MAGNETIC ROTATION  The expression for the frequency shift contribution is given in Eq 3-25, and the intensity perturbation 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 0 is equal to zero, the parameter d is also set to 1. O  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 c m ) . To change this into the appropriate units for the -1  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 / and Acti . The larger the linewidth is, the more significant the S  p  contribution from Acti becomes relative to the contribution from Aotf . For the experiments in P  s  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 c m / a t m , which -1  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 0 = 0 are shown in Figure 3-7. o  The signal from Aon looks like a Lorentzian, as the lineshape in Eq 3-37 suggests. The lineshape p  for the signal from Actf looks like a dispersion, which is just the derivative of a Lorentzian. This is s  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 Aai  P  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 0 = 0. O  MAGNETIC  71  ROTATION  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 9 is much larger than d, the first harmonic magnetic rotation signal is 0  SFH ~ S (&n ) FH  fs  + S (An ). FH  ip  (3-44)  3 MAGNETIC  72  ROTATION  The expression for the frequency shift contribution is given in Eq 3-28, and the intensity perturbation contribution is in Eq 3-39. The constant C in Eq 3-43 is again set arbitrarily to 1. The parameter 6 is also set to 1, and d is set to zero. Using the same molecular constants as was used 0  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 A n .  S  _4 I  i  -10000  .  F  i  (e  H  i  0  »  i  -5000  d)  1  0  i  5000  .  1  .  10000  co (Mrads/sec) The lineshape for the signal from Anj is dispersion, while the A n / signal has a second derivaP  S  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 9 3> d is a  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 - and oc An ) and the cross terms (oc A a / A a i and oc A n / A n ; ) . 2  p  p  s  P  s  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 A n .  S  _4 I  F H  i  -10000  i  (A n ) + S fs  i  i  1  -5000  F H  1  (A n ) ip  1  0  1  1  5000  •  10000  co (Mrads/sec)  Figure 3-11: Calculated frequency shift and intensity perturbation contributions to the second harmonic signal.  Second harmonic signal S (( fs) ) SH(( fe) ) Aa  2  + S  An  2  SH  i  -5000  .  '  i  0  5000  i  1 —  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  74  ROTATION  Figure 3-12: Calculated second harmonic signal, including both frequency shift and intensity perturbation contributions.  S ((Aa +Aa ) ) + S ((An +An ) ) 2  SH  3.5 3.0 2.5  c rs  2.0  •e  1.5  fs  sH  ip  fs  ip  centre of^ gravity  TO 1.0 0.5 0.0 -0.5  -10000  -5000  0  5000  10000  co (Mrads/sec) 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 6 2> d for a P(5) line is shown in Figure 3-13. This should be compared to Figure 3-10. 0  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  75  ROTATION  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 6 » d is shown in Figure 3-14. a  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 symmetric. The phase is the same as that of an R line.  Calculated Q(4) (e » d) 0  .15  I  i -10000  .  i - 6 0 0 0  i  i 0  .  i 5 0 0 0  .  1  1  10000  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  76  ROTATION  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 A m j = +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 measurements. 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 , the direction of the 2  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 (mj) = r)jmjcos(LO t) + ^0™j 0  H  +  ^(jmjCos(2u t), H  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 ([(u — to) +• (^) ] ) in expressions such as 2  2  n  0  Eq 3-25 is replaced by  [LJ  —  0  to + \C,jm j) 2  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 \ri jrn ), but everything else is the same. 2  2  J  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  77  ROTATION  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 and A n terms 2  2  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. A n 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  15218.00  i  i  1  15218.05  wavenumber units  1  —  15218.10  4  EXPERIMENT  4  Experiment  79  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 B r 2 79  79  B r Electronic states 2  0  Internuclear separation r (A) As mentioned previously, when using absorption spectroscopy to study Br2, signals for transitions 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  80  EXPERIMENT  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; C H O P Mechanical chopper; PE-Photo-emitter; PD-Photo-detector; 12-Iodine reference cell; PMT-Photomultiplier tube; PO-Polarizer; L-Lens; MR-Magnetic rotation setup, consisting of a cell filled with 7 9  Br2  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 . The frequency spacing between - 1  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  81  EXPERIMENT Figure 4-2:  Experimental setup for Doppler-limited magnetic rotation spec-  troscopy.  CHOP  •••«PE  Etalon  PMT PO  BS  MR  PO L El 0 f  M  mVax 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~ . Strain on the cell windows caused significant transmission even when 5  4  82  EXPERIMENT  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 P M T 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 B r 2 [47]. The solenoid consisted of a single layer of 18-gauge wire wrapped around a 6 cm 7 9  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 L C R 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 Br2. 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 B r 2 was measured from 15,027 to 15,488.5 c m . The - 1  7 9  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 . Measurements for this portion of the thesis were taken in - 1  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 consistency check for adjoining scans.  - 1  of unique data on each scan), to provide a  4  83  EXPERIMENT  4.1  Frequency Measurements  A computer program was written in a combination of C and ASSEMBLY for the purpose of determining 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 . It was found that there were often serious discrepancies for lines in the overlapping - 1  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 cm . - 1  For the small percentage of scans that contained only one I2 reference line, the fringe  4  84  EXPERIMENT  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 frequencies 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 . The pressure and temperature of the etalon were not controlled, - 1  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 acomplete, continuous, calibrated form rather than in 1 c m , uncalibrated chunks). Using this new - 1  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 cm  - 1  4.2  for a few cases. 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  85  EXPERIMENT Figure 4-3: Experimental setup for Doppler-free magnetic rotation spectroscopy.  ^  Dye  M  CHOP  o|a«PI  Etalon B S 2 /  E]PO  P  M  T  MR  pom  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  86  EXPERIMENT  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.  87  5 RESULTS AND ANALYSIS  5  Results and Analysis  The magnetic rotation spectrum of B r 2 was much more congested than originally expected, and 7 9  many of the strong signals in the spectrum were initially unassigned, as mentioned in the Introduction. 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), v (j) = v n , + B'J{J - 1) - B"J(J + 1), and P  v  v  »Q{J) = "v»,v> + B'J(J + 1) - B"J(J + 1), where z v > ' 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  = 2{B' - B")(J + 1) + 2B',  m  and the second differences (the differences of these differences) are A (J R  + 1) - A ( J ) = 2(B' - B"), fl  (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 experimental 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  88  5 RESULTS AND ANALYSIS  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 ) A (cm" ) second difference (cm ) x  1  1  R  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 v (j)-"p(j) R  = 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: v {j-i) ~ vp(j+i) R  = 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 (X S^") state, B " is set to the value determined by a Fourier transform 1  89  5 RESULTS AND ANALYSIS  study of the B n + -X E+ system [51] The vibrational states 0 < v" < 2 were the only possibilities 3  1  0  U  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.  J+l A  R(J-1) P(J)  A  J J-l  P(J+1)  R(J) 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 Jnumbering 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 n~i 3  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 spectrum, 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.  90  5 RESULTS AND ANALYSIS 5.1  Cause of High-J Signals  Both the S-uncoupling operator {j^J±S^)  and the Zeeman Hamiltonian (ii )  couple the A n i 3  Zee  u  state to the A n . 2 state as well as the A state to the B n + state. Second-order perturbation /3  3  U  0  U  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\H  =  >  1  where H  s u  + H \A')\ E -E ,  \(A\H \A')\ E -E ,  2  Zee  su  A  (A\H \A') (A'\H \A) E -E ,  2  =  Zee  A  A  Zee  A  ,  SU  A  \(A\H \A')\ E -E , ' (5-4) 2  SU  +  A  A  A  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 c m ) becomes (in the Hund's case (a) coupling -1  scheme): (A' AA\H \A>) (A'\H \A) _ (A\g S-\A') E -E , E^E^, Zee  SU  A  ~  ^^{A,v\S-\A'J)(A',r/\$S \A,v) ^ E ,,-E ,  (J  +  2  9  S  A  A  '  v l  ( j + 2)(J - 1)  w w r  A  , =  A)  s  2  ,  +  J  (  j  T  j+  1 )  2)(J-1) , 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 should be replaced by (A \gsS- + gLL-\A'),  (A\gsS-\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! (A'  2^ 'i (S+ + L+)  8n  cr  ^ i S+ Aj  &1T  cr  would therefore be replaced by  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.  91  5 RESULTS AND ANALYSIS  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 E q 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: [(g E+g A)n} s  9 J  =  L  ^  eff  J(J + 1)  ~  2  9  8  a f e (A,v \S-1 A', v>) (A', v> \^S  +  \  E , ,j A v  - E , ,j A v  | A, t,) j (  +  j  2)(  J ( J + 1)  _  1 }  '  ( 5  "  6 )  where (gs£+g.£,A)r2 = 1 for a Hund's case (a) n i electronic state, but must be left as an effective 3  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 B HQ+ 3  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 H i state, the two terms in Eq 5-6 are opposite in sign, and the g-factor actually 3  u  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,  92  5 RESULTS AND ANALYSIS  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 n + - X S ^ " system of molecular iodine [4] arise 3  1  0  u  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 H i state, as well as to a dissociative H\ 3  l  u  u  state.  See Figure 4-1 (which is for Br2, 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 n i state) give rise to 1  u  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 H i state is in the same 1  u  energy region as the B state. It has been shown [7] that there is an interaction between the B state and the n i x  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 H\ l  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, 0 , was negative in the experiment. o  negative value for this parameter.  The calculated lineshapes that follow therefore assumed a 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  93  5 RESULTS AND ANALYSIS 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 contributions 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):  ^jfJTT)- ' 0  0 0 1 5  -  <-> 57  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  94  5 RESULTS AND ANALYSIS 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)  Calculated P(21)  1.0[  Calculated P(54)  0.5  0.0  -0.*  -14 -10000  -5000  0  5000  -10000  10000  -5000  0  5000  -10000  10000  -5000  0  5000  10000  co (Mrads/sec)  to (Mrads/sec)  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 9J  1 J ( J + 1)  0.001,  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.  Re-calculated P(54) 4k  -10000  -5000  0  5000  co (Mrads/sec)  10000  95  5 RESULTS AND ANALYSIS  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 Br2, 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  96  ANALYSIS  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 , with a typical - 1  discrepancy of about 0.0075 c m . As is evident in Figure 5-5, the apparent shift is smallest for - 1  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 B r 2 in this frequency region, calculated from Eq 2-57, is about 450 MHz 7 9  (~0.015 c m ) , but the observed spectral widths are about twice that, approximately 0.03 c m , -1  - 1  because hyperfine structure broadens the lines. A typical signal-to-noise ratio for the second harmonic 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 . The skew in the sig- 1  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 , the standard error of the global analysis performed in [50]. - 1  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, except 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  5.3  - 1  or more, as discussed previously.  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 H i 3  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') = v (i/, v") + B ,K' - D ,K'  2  0  V  V  + H ,K'  3  V  - B „K" + D >,K!  12  V  V  - H »K" , 3  V  (5-8)  97  5 RESULTS AND ANALYSIS 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 (v ). The results of these fits are shown in Table 5-2. The Q-transitions 7  (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 E + state. The L-uncoupling operator does not couple these two states X  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 +q . e  v  v  (5-9)  5 RESULTS AND ANALYSIS Table 5-2: Vibration-by-vibration results for the e parity levels of B r 2 A Ui 3  7 9  E(v')  b  10 T> ,  -10 ILy  7  c  V  U  c  #  d  a u  T  e  3  f D  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 c m ) of V," J ' = 0" in the A state relative to v" = 0, J " = 0 in the X state  c  Constants (in c m ) 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  -1  -1  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 c m ) , and the normal fi-type doubling must be larger than this separation in order for q„ to -1  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 frequency 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 . The minimum uncertainty was set to - 1  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  Br2  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. v (v',v")) for the e 0  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 (J), determined from R and P lines) as a e  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  (cm  0.00-0.05-  3,  -0.10-  <u LU -0.15-  LU  -0.20-0.250  1000  2000  3000  4000  5000  J(J+1)  Coxon's data, from an absorption experiment, consisted primarily of Q-transitions. The data from the current work consists primarily of R- and P-transitions. Although no fi-doubling constants were measured for v' < 1 9 in the current work, estimates for the ^-doubling constants can be determined by making comparisons to Coxon's results. The B-values from the current work in Table 5-3 are taken from Table 5-7, the results of a global analysis on the A state to be described shortly. The accuracy of Coxon's results is estimated to be  ±0.000005  c m , while the results from - 1  the current work should be better by at least a factor of five or more. The comparison between the results of the current work and Coxon's results agrees very well with the ^-doubling constants measured by Coxon for 1 3 < v < 16. The discrepancies for these vibrations are all less than the uncertainty in Coxon's B-values  (0.000005  c m ) . Above v' = -1  16,  Coxon did not measure enough high-J data to include ELy in his least-squares fits, which is likely why v' = 1 7 and 18 do not agree. The value of the ^-doubling constant for v' = 1 9 is consistent with the extrapolation from Coxon's results. For vibrations above this, however, q levels off to an almost constant value. v  To explain the reason for this, consider the potentials shown in Figure 5-9. The two dissociative states ( S Q _ and II _ ) are taken to be similar to those for I2 shown in [53], but Br2 should be 3  3  u  0  U  qualitatively the same, as discussed previously.  5 RESULTS AND ANALYSIS  103  Table 5-3: Q-doubling constants (All values in cm ) 1  v'  10 q„/ 5  a  #  6  I  C  10V  B,  d  f  e  IO (B^,-B£) 5  11  -  -  -  2.4  12  -  -  -  2.8  13  -  -  -  3.2  .045294  .045323  2.9  14  -  -  -  3.6  .043721  .043755  3.4  15  -  -  -  4.1  .042090  .042127  3.7  16  -  -  -  4.4  .040439  .040482  4.3  17  -  -  -  4.8  .03884  .038863  2.3  18  -  -  -  5.2  .03726  .037300  4.0  19  5.66(5)  35  65  20  5.59(8)  43  52  21  5.63(14)  30  46  22  5.86(12)  19  53  23  5.90(9)  46  54  24  5.84(7)  50  59  25  5.60(11)  22  65  26  5.59(9)  36  62  27  9  -  -  28  5.55(19)  24  40  29  5.75(17)  22  43  Q, doubling constant measured in the current work.  a  b  The number of Q lines (associated with the f parity levels in the A state) used in the analysis.  c  The maximum J for the Q lines used in the analysis. Q doubling constant measured by Coxon [1]  d  e  B-value for f parity levels, as measured by Coxon [1]. B-value for e parity levels, measured in the current work. See text.  9  No fit for q„ was possible in this case. See text.  The first thing to notice is that there are two E-type states that would interact with the f parity levels and leave the e parity levels unperturbed: the I l o - state and the S Q _ state. There is, 3  3  u  u  however, only one S-type state that perturbs the e parity levels in the A state of B r 2 , the B n + „ 7 9  3  0  5 RESULTS AND ANALYSIS  104  Figure 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.  state. The extra perturbation for the f levels could certainly explain why they are lower in energy. The drastic differences between the potential energy curves also means that the matrix element of 4j between the two levels (which enters into Eq 2-24) will vary significantly as a function of v. This might explain the unusual levelling-off of the Cl doubling constant. The energy of the n 3  0  u  state is not known, since transitions from the ground  state are  forbidden. However, according to Mullikan [53], in I2, case (c) coupling depresses the n - state 3  0  u  to lower energy than the B n + state. The same should be true for Br2, which is also described 3  0  U  by case (c) coupling. The energy of the B state is well known [51]. The v = 0 level in the B state is close to being coincident with the v = 28 level in the A state. For the n - state to be below 3  0  u  the B state, it must dip down into the A state. In other words, the n - state is expected to be 3  0  u  weakly bound. This was not depicted in Figure 5-9, since the location of the n - state has not 3  0  u  been verified experimentally, even in I2. The £ Q _ state is also most likely weakly bound as well, 3  u  but its binding energy is also unknown. Doing a quantitative estimate of expected variation of the £2-type doubling would obviously be challenging, without knowing the potential curves for the two states interacting with the f parity levels. It might be possible to deduce some information on these two states from the experimental data, however. More data than was measured in the current work would be required for such an  5 RESULTS AND ANALYSIS  105  analysis. 5.5  Global Analysis  This work represents the first complete set of high-precision measurements of the levels close to the dissociation limit for the A state of Br2. If perfect calibration were possible, the absolute limit of precision would be determined by the effective linewidth of the laser. Including frequency jitter, this was about 2 MHz, or ~0.00007 c m ) . However, the accuracy of the I2 reference lines and the -1  problems associated with lineshapes makes the actual precision closer to 0.0035 c m . The current - 1  state of knowledge for the A state is summarized in Table 5-4. Table 5-4: Current state of knowledge for A state. Previous  New  V  ref  #°  0  [3]  ?  0.1  -  7-12  [1]  531  0.02  -  13-24  [1]  707  0.02  25-29  -  -  30-35  [2]  10  36-37  -  -  ±(cm ) - 1  •  ±(cm )  6  - 1  -  -  1600  0.0035  780  0.0035  0.2  209  0.0035  -  44  0.0035  -  b  a  The number of unblended lines used in analysis (except in [2]—see text)  b  The estimated precision of frequency measurements  The only previous measurements above v = 24 were 10 transitions measured to a precision of ±0.2 c m  - 1  during the course of a multiphoton experiment [2]. There were actually only three  frequencies measured, and these were assigned to 10 overlapping transitions from different isotopemers. The data in the current work represents a vast improvement in the knowledge of these states, both in precision and in quantity. The data here is also an improvement in the precision and quantity of Coxon's data [1] (±0.02 c m ) over the range of vibrations from v = 16 through -1  v = 24. For v = 15, the number of transitions measured in Coxon's work was comparable to the number measured here, although the current work has greater precision. For v = 13 and 14, Coxon measured more transitions. Information for v = 7 through 12 is taken solely from Coxon's work, since no transitions to these vibrations were measured in the current study. Recently, data for v = 0 in the A state became available from the study of the /5-A system of Br2 [3]. The (3 electronic  5 RESULTS AND ANALYSIS  106  state is an ion-pair state (i.e. its dissociation product is two ions, rather than two neutral atoms) at higher energy than the A state. With high-precision data that now spans the entire range of the A state, from v = 0 right up to very close to the dissociation limit, an improved global description of the A state and its potential is possible. The most comprehensive analysis prior to the current work was performed in [3], using their own data for v = 0, Coxon's data for v = 7 through 24, and the handful of high v data from Brand et al for v = 30 through 35. In [3], the band-origins and B-values measured in [2] were adjusted to include the effects of distortion constants. These adjusted values agreed with the results measured here (in Table 5-2) to within 0.14 c m  - 1  for the band-origins and to within 0.00025 c m  - 1  for the B-values. The Brand data was given lower weight in the analysis performed in [3]. The extent and precision of the high-v data in the current work allows an enormous improvement in the determination of the potential in this region. The improved precision for the range of vibrations in the present work that were also measured by Coxon should improve the determination of the potential there as well. Traditionally, the band-origins and B-values are fit to the expressions in Eq 2-6 and Eq 2-12, respectively. With data that spans the entire range of the electronic state, this approach becomes very inefficient, i.e. an excessive number of parameters are required to reproduce the data to within the experimental uncertainties. An alternative approach for a global representation of the" band-origins and B-values involves what is known as near-dissociation theory. When the internuclear separation is large, where processes such as electron exchange can be neglected, interactions between the two nuclei can be described in terms of interactions between separated atoms. The theory of such interactions is well-established (see e.g. [54]). The potential, U(r), associated with the interaction has the form:  m  where D is the dissociation limit, and the coefficients C  m  are constants that depend on the details  of the interaction between the two atoms. Which terms to include in the sum depends on the dissociation products of the molecule. The A state of Br2 dissociates into two atoms, each of which is in a 4p configuration (equivalent to a single electron in a p orbital). 5  For the A state of Br2, the long-range potential has the form: lr)  u  =  D  - ^ - ^ - ^ s - ^ i o -  ••••  (-) 5  n  This expression comes from an expansion of the interaction energy determined by treating the Coulomb interaction between the two atoms with perturbation theory. Odd powers of r  _ 1  come  5 RESULTS AND ANALYSIS  107  from the first-order terms for the Coulombic interaction. The C5 term, for example, represents the quadrupole-quadrupole interaction between the two atoms. C i would obviously represent the case where the diatomic molecule dissociated into two ions. It vanishes because the A state products are neutral. C3 would arise from identical atoms in electronic states whose total angular momentum differed by one [55]. It vanishes because the A state products do not meet this condition. Even powers of r  come from second-order effects in the perturbation theory treatment. C6, for example,  - 1  arises from the induced-dipole—induced-dipole interaction. The classical turning points for a potential energy curve, U(r), are usually calculated from the W K B quantum condition [56]: V  where r_ and r  +  l  =  ^lr  +  "J  G{V)  ~  >  U(r)dr  ( 5  '  1 2 )  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' represents a collection of constants, defined (and tabulated for the various possible values of n) n  in [56]. The expression in Eq 5-13 can be integrated with respect to v. This yields the following expression: G(v) where  = ^YJ , n  n  =D  2n n-2  LVM(Cn)'  (5-14)  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: 2n  G{v)  n-2  =D—  H  ^- (v -v) T  D  LvWn)=  [  C ( L )  1  (5-15)  k(M)J  with C(L) = l+pi(v  -v)  + p (v  2  D  2  -v)  + ... +p (v  3  D  L  -v)  D  L + 1  ,  and C(M) = 1 + (v qi  - v) + q (v  - v) + ... + q (v  2  D  2  3  D  M  D  -  v)  M + l  .  The expression in Eq 5-15 replaces the expansion in (v+5) from Eq 2-6. Since Eq 5-15 incorporates 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 is about 50 times larger than that for the fit to the near-dissociation expression. In 2  addition, the predictions for the band-origins for v = 1 through 6 (which were not in thefit,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]: B =  _®'  n  v  _a_ [v -v]^s. D  (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 tofitthe data. The empirical function typically suggested for the B-values [58] is the exponential of a polynomial: exp  ^2 *( D ~ v)^j • S  V  (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 =  (5-18)  ^ [VD -V]^i 0 ) " - (<?n)"2  2  where £(L) = 1 + si(v -v) + s (v D  2  - v) + ... + s (v 2  D  L  - v) , L  D  and £(M) = l + ti(v -v)+ D  t (v - v) + ... + t (y 2  2  D  M  D  - v) M  Note that the polynomials contain a linear term, unlike those in Eq 5-15. The extrapolation through the range of missing data is much smoother with the expression in Eq 5-18 than when the exponential form in Eq 5-17 is used. The optimum configuration (from the point of view of compactness and accuracy) for fitting the B-values to Eq 5-18 is L = 8 and M = 3. With the expansion in (v+^) from Eq 2-12, even 15 or 16 parameters is insufficient tofitthe data well. The band-origins from Table 5-2 werefitto the expression in Eq 5-15. There were 12 parameters in thefit,the values for D, C5 and vp, plus the 9 empirical parameters. Coxon's data needed to be adjusted slightly to be included in the fit. In Coxon's analysis, K' in Eq 5-8 was taken to be J(J+1) rather than [J(J-fl)-l]. As a test, some bands were refit using Coxon's form for K', and the only fitted parameter that had a significantly different value than was obtained in the original fit was the band-origin. The fitted band-origin using K' = J(J+1) was smaller than that determined with K = [J(J+1)-1], the difference being equal to the B-value, as would be expected, since the 1  main difference between the two approaches is the extra constant term in the rotational energy (B„[J(J+1)-1] versus B„J(J+1)). In Coxon's analysis, the constant term from the rotational energy was absorbed into the band-origin, making the fitted value smaller by an amount B„. The differing expressions for centrifugal distortion terms (i.e. involving D„, H„ and higher order) for the two approaches changes the fitted values of G(v) and B„ by amounts that are negligible in comparison to the uncertainties, and the effects from these can therefore be ignored. For consistency with the data from the current work, Coxon's data in [1] was adjusted by adding the measured B-values to the vibrational energies. The band-origin for v = 0 in [3] was adjusted in  5 RESULTS AND ANALYSIS  110  the same manner, since the same form as Coxon's was used for the rotational energy («' = J(J+1)), apparently for the purpose of including data from older works (such as Coxon's) in the analysis. The effect of other terms from the rotational Hamiltonian ( H ) that are absorbed into the rot  band-origin (see Section 2.4) were not taken into account. A trial fit was tried where the bandorigins were adjusted according to the contribution from the rotational Hamiltonian (assuming case (a) coupling), but little difference in the quality of the fit was observed. The actual values for the contributions are not accurately known (since Br2 is closer to case (c) coupling than to case (a)), and so, to avoid complicating matters needlessly, these effects were neglected in subsequent analysis. As mentioned in [56] and observed with the data for the current work (although not explicitly proven here), the approximations inherent in the derivation of Eq 5-16 make the equation less reliable than Eq 5-14. As a result, the values of C5 and v/j in Eq 5-18 are fixed to the values determined from this fit to the vibrational energies. The B-values from Table 5-2 are then fit to Eq 5-18. Coxon's data once again needed to be adjusted before being included in the fit. In the previous section, it was determined that the e parity levels should give a closer representation of the true B-values than the f levels. The B-values reported by Coxon [59] are for the average of the e and f levels. Coxon's measured fi-doubling constants were used to calculate the B-values associated with the e parity levels, the values required for the current analysis. Below v = 11, for which Coxon did not measure fi-doubling, an extrapolation of his values was used. Since the (^-doubling was seen to decrease with decreasing v (see Table 5-3), this should introduce minimal error. The Q-doubling would likely be almost negligible at v = 0, and so the B-value from [3] for this vibration was not adjusted. In Table 5-2, the fits for v = 35 through 37 only went up to D„, (second order in [J(J+1)~ 1]) while the fits for all of the other vibrations went up to H^ (third order in [J(J+1)-1]). This inconsistency might be expected to lead to problems in fitting the band-origins and B-values to Eq 5-15 and Eq 5-18. However, fitting v = 35 and 36 with an H„ included leads to worse problems than leaving H„ out. Figure 5-10 shows B ^ - B ^ - i (essentially ^ ) as a function of v. The data for vibrations less than 13 are from Coxon [1], with the adjustments described previously. Note that data points in Figure 5-10 that involve v = 27 deviate from the trend of the surrounding data. There is a perturbation in v = 27 (which will be discussed in more detail later) that affects the fitted B-value for that vibration. As a result, v — 27 was excluded from the global fit, because its fitted constants are known to be unreliable.  5 RESULTS AND ANALYSIS  111  Figure 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.  -0.8 ^  -1-0-  CO  1  o  -1.4 "5 -1.6-1  CO  ' > -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, £ and g„, according to [60] v  2f = r (v) - r_(t>), and 2g = — v  +  v  -.  (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 K for all vibrations except v  v = 35, 36 and 37, for which the fits included distortion constants only up to D„.  -0.8-1  -1.0-  cm  I  -1.2-  CO  I  O  -1.4-  CO •  > CO  -1.6-1.8-2.0-2.2-  10  15  —i— 20  25  30  35  40  The quantities f„ and g„ are defined by the integrals:  ^  J  J8n ciJ,  f  2  _  i =dv'; y/G(v) - G(v>) Vml  \] 8TT CH 2  (5-20)  v  y/G(v)-G{v>)  -.dv':  (5-21)  where it has been assumed that the vibrational energies and B-values are expressed in c m . The - 1  value of v i m  n  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: r±  J  +'  ^  * f ±f-  1  2  (5-22)  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 10 on successive divisions. The integral in a given interval is calculated 6  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]. A l 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 B and G(v) in Table 5-2 suffer from correlations v  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.  10.09.8TJ 9.6-  v = 25  "O  v = 13  9.4 9.29.0-  0.880  0.885  0.890  ln(r.) 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. All inner turning points above v = 25 were calculated from the resulting expression. The difference between the inner and outer turning points, defined by 2i in Eq 5-19 and Eq v  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 wavefunctions 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 l * W « t ) =. i ( ) 2  U  r  ~ ) \Vvib,rot) •  (5-24)  E  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] is the second order correction to the energy from perturbation theory, H„[J(J+1)-1] 2  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„, L , M , N„ and 0„) were evaluated, v  v  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 B and v  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 B and G(v); the new values of B„ and G(v) are fit to the expressions in Eq 5-18 and v  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 E q 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 converging to a point for which agreement with the experimental data was significantly worse than was the case for the first iteration: the value of % was more than a factor of two larger, and there were 2  oscillatory trends to the residuals in latter iterations that were not present in the first iteration. The parameter x is a measure of the quality of a least-squares fit, i.e. how well the fitted results 2  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 , u> , etc.) are from the expressions in Eq 2-6 and Eq 2-12. The Kaiser e  e  correction accounts for this error in energy by setting v j in Eq 5-20 and Eq 5-21 to TO  n  -  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 determine 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 B calculated from the potential will be different from the values that v  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 condition. 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 experimental 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 (<SG(v)) for G(v). The correction function is chosen such that G(v) + <5G(v) satisfy the quantization condition in Eq 5-12. Following [61], this correction is a linear function of v, set to some value at v = 0 and constrained to go to zero at the dissociation limit. Similarly, a linear correction function  (<5B„)  is determined for B„ by comparing  the B-values determined from the potential to the experimental B-values. This procedure was iterated until the best agreement between input and output values of G(v) and B^ for the potential was achieved. The two correction functions (in c m ) for the A state of -1  6G(v) = 0.2 -  0.2 41.5 v:  5  Br2  were: (5-26)  6B = - 9 x 10~ + 5 x 1 0 V V  7 9  -  (5-27)  5 RESULTS AND ANALYSIS  118  Note that <5G(v) goes to zero at the dissociation limit, where v = vp ( « 41.5—see Table 5-5). Further calculations of the R K R potential involve these correction functions. It must be stressed that the energies and B-values determined from this potential are intended to be consistent with the input values of these quantities, i.e. the first-order, corrected values. They will not, in general, agree with the experimental values. This is less gratifying than working with a quantum mechanical potential, where the values determined from the potential should correspond to the experimental data, but it is the price that must be paid for the convenience of using R K R potentials. Note that this does not represent a correction to G(v) and B„ themselves; it is an adjustment used to determine a more accurate R K R potential. It is also possible that the correction functions play some additional role to compensate for problems near the bottom of the potential. Since there are no data at low-v, the results will not be as accurate in this region. The integrals in Eq 5-20 and Eq 5-21 must be integrated through this region, and the solution for the wavefunctions involves the potential in this region. This might have provided some contribution to the difficulty of getting the iterative process to converge. It is not clear how significant that contribution might have been, though, or if it was a factor at all. Figure 5-14 and Figure 5-15 illustrate the improvement in the internal consistency of the potential as a result of including the corrections. The plots show the discrepancy between input values (i.e. values of G(v) and B„ used to calculate the potential) and output values determined from the potential. The curves labelled as "uncorrected" are for the results for when no correction scheme is employed, and the curves labelled "corrected" are the results for when the correction functions in Eq 5-26 and Eq 5-27 are used. For both G(v) and B , there is little effect at low-v from inclusion of the correction functions. v  In the region for which there is data for the current work, the improvement is striking, particularly for the B-values. The results for the vibrational energy has the appearance of perhaps requiring a quadratic correction function, but with perturbations and possible incompatibilities between the data from the current work and data for v < 13 taken from other sources, the reliability of such an approach would be questionable. Some difficulties seem to be indicated for the extrapolation of the energy through the region of missing data at low-v. With the inclusion of these correction functions, the iterative procedure described previously improved on successive iterations and converged to a very reasonable fit, unlike the original situation without the correction, where the procedure diverged. Also, the final results were no longer as sensitive to the choice of m in Eq 5-23.  5 RESULTS AND ANALYSIS  119  Figure 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 uncorrected cases.  0.25 §  0.20-  13  0.15-  uncorrected corrected  S~ o.io0.05-  1  0.00  >  -0.05  •••• 1—  10  40  20  V  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 uncorrected cases.  E °i o 1  •  uncorrected  •  corrected  -i—«  o a? o-i  • r v , ; fi  .  ••••••  m 10  20  30  40  V  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 , and the average error was -0.00039cm . In the fit, 1882 of the 2396 data - 1  -1  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 q in Table 5-5) were all the order of half a percent. The statistical uncertainties 2  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) c m "  1  41.54442(±0.013)  c  61,694.23(±860) c m - ! 1  5  vibration  5  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  s  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  tl  0.480 322 553 059  t  2  -0.037 793 572 657  t  3  qi  4  -3.323 856 459 600 x 10 - 3 8.545 614 371 858 x 10'-5  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 . The error introduced - 1  in the fitted B-value by the perturbation is 0.000058 c m , the order of 0.2%. - 1  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 B calculated from the parameters in Table 5-5. These v  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 B in Table 5-2 and Table 5-7, the changes incurred by v  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 V 10 D„ H.„ L M„ 14 19 0 0.35735 -8.2108 x 10-4.059 x 10-1.155 x 10"-24 14 1 0.37837 -6.7321 x 10-4.140 x 10-L9 -2.876 x 10"-24 L4 2 0.40306 -7.1235 x 10" -5.131 x 10"L9 -5.223 x 10"-24 L4 3 0.43261 -9.2548 x 10-6.991 x 10"19 -8.086 x 10"-24 4 0.46810 -1.2925 x 10"L3 -9.995 x 10"19 -1.185 x 10"-23 5 0.51056 -1.7982 x 10"13 -1.464 x 10-L8 -1.750 x 10"-23 6 0.56115 -2.4418 x 10-13 -2.173 x 10-18 -2.679 x 10"-23 7 0.62128 -3.2458 x 10-13 -3.252 x 10-18 -4.284 x 10'-23 8 0.69296 -4.2643 x 10-13 -4.902 x 10-18 -7.120 x 10"-23 9 0.77907 -5.5909 x 10-13 -7.422 x 10"18 -1.210 x 10"-22 10 .0.88337 -7.3578 x 10-13 -1.122 x 10-17 -2.057 x 10"-22 11 1.0099 -9.7076 x 10-13 -1.672 x 10-L7 -3.402 x 10"-22 12 1.1612 -1.2713 x 10-12 -2.407 x 10-17 -5.313 x 10"-22 13 1.3355 -1.6248 x 10-L2 -3.270 x 10-17 -7.603 x 10"-22 14 1.5237 -1.9891 x 10-12 -4.103 x 10"L7 -9.759 x 10"-22 15 1.7094 -2.3044 x 10-L2 -4.729 x 10-L7 -1.134 x 10"-21 16 1.8756 -2.5316 x 10"L2 -5.132 x 10-L7 -1.274 x 10"-21 17 2.0131 -2.6879 x 10"L2 -5.556 x 10"17 -1.546 x 10'-21 18 2.1249 -2.8450 x 10-12 -6.413 x 1 0 17 -2.105 x 10"-21 19 2.2234 -3.0855 x 10"12 -7.967 x 10"17 -3.031 x 10"-21 20 2.3239 -3.4689 x 10-12 -1.036 x 10-16 -4.355 x 10"-21 21 2.4391 -4.0254 x 10-12 -1.369 x 10-16 -6.234 x 10"-21 22 2.5778 -4.7766 x 10-12 -1.827 x 10-16 -9.188 x 10"-21 23 2.7469 -5.7714 x 10"12 -2.468 x 10"16 -1.358 x 10"-20 24 2.9530 -7.0760 x 10-12 -3.404 x 10-16 -2.102 x 10'-20 25 3.2041 -8.8111 x l O 12 -4.772 x 10"16 -3.313 x 10"-20 26 3.5104 -1.1147 x 10-11 -6.859 x 10"16 -5.366 x 10"-20 27 3.8847 -1.4325 x 10-11 -1.010 x 10"15 -9.078 x 10"-20 28 4.3439 -1.8724 x 10"11 -1.521 x 10-15 -1.587 x 10'-19 29 4.9083 -2.4923 x 10-11 -2.367 x 10"15 -2.899 x 10"-19 30 5.6054 -3.3852 x 10-11 -3.815 x 10-15 -5.581 x 10"-19 31 6.4713 -4.7093 x 10"11 -6.430 x 10-15 -1.145 x 10"-18 32 7.5570 -6.7409 x 10"11 -1.141 x 10-14 -2.545 x 10"-18 33 8.9377 -1.0007 x 10-10 -2.162 x 10-14 -6.195 x 10"-18 34 10.729 -1.5560 x 10-10 -4.448 x 10-14 -1.694 x 10"-17 35 13.120. -2.5709 x 10-10 -1.017 x 10"13 -5.379 x 10--17 36 16.445 -4.6102 x 10-10 -2.673 x 10-13 -2.079 x 10--16 37 21.348 -9.2725 x 10-10 -8.533 x 10-13 -1.056 x 10--15 38 29.235 -2.2143 x 10"9 -3.637 x 10-12 -8.059 x 10"-15 39 42.856 -5.4929 x 10"9 -7.349 x 10-12 3.712 x 10"-14 7  v  -  -  potential. (All values in cm ) N„ -3.637 x io--30 -3.536 x 10"-35 -2.165 x 10"-29 -2.073 x 10"-34 -4.258 x io--29 -3.982 x 10"-34 -7.514 x io--29 -7.413 x 10"-34 -1.299 x 10"-28 -1.429 x 10"-33 -2.244 x 10"-28 -2.818 x 10"-33 -3.931 x 10"-28 -5.674 x 10"-33 -7.032 x 10"-28 -1.166 x 10"-32 -1.286 x 10'-27 -2.426 x 10"-32 -2.386 x 10"-27 -5.042 x 10"-32 -4.407 x 10"-27 -1.026 x 10"-31 -7.861 x 10"-27 -1.989 x 10"-31 -1.306 x 10"-26 -3.529 x 10--31 -1.949 x 10"-26 -5.506 x 10--31 -2.554 x 10'-26 -7.427 x 10--31 -3.048 x 10"-26 -9.359 x 10"-31 -3.728 x 10"-26 -1.304 x 10"-30 -5.406 x 10"-26 -2.214 x 10"-30 -8.405 x 10"-26 -3.734 x 10"-30 -1.330 x 10"-25 -6.139 x 10"-30 -2.047 x 10"-25 -1.030 x 10"-29 -3.159 x 10"-25 -1.796 x 10"-29 -5.160 x 10"-25 -3.050 x 10"-29 -8.600 x 10"-25 -6.087 x 10"-29 -1.459 x io--24 -1.101 x 10"-28 -2.651 x 10"-24 -2.274 x 10"-28 -4.836 x 10"-24 -4.886 x 10"-28 -9.278 x 10"-24 -1.048 x 10"-27 -1.909 x 10"-23 -2.501 x 10--27 -4.113 x io--23 -6.384 x 10"-27 -9.538 x 10"-23 -1.790 x 10'-26 -2.381 x 10"-22 -5.496 x 10'-26 -6.615 x io--22 -1.901 x 10"-25 -2.075 x 10"-21 -7.677 x 10--25 -7.556 x 10"-21 -3.727 x 10"-24 -3.335 x 10"-20 -2.289 x 10"-23 -1.897 x 10"-19 -1.920 x 10"-22 -1.536 x 10'-18 -2.487 x 10"-21 -2.105 x 10"-17 -6.097 x 10"-20 4.476 x 10"-16 2.416 x 10"-18 -1  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- ) B„ (cm" ) r_ (A) r (A) 0 13,821.8979 0.0581508 2.6348 2.7859 l 13,968.3995 0.0573689 2.5922 2.8580 2 14,109.7245 0.0566464 2.5651 2.9134 3 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 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 15,046.1944 0.0494553 2.4619 3.2974 10 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 15,358.7421 0.0437546 2.4396 3.5622 14 15 15,419.2093 0.0421270 2.4356 3.6446 15,473.2734 0.0404822 2.4322 3.7337 16 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 15,639.4894 0.0343610 2.4220 4.1416 20 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 15,793.9864 0.0262443 2.4131 4.9041 26 27 15,811.7558 0.0248529 2.4121 5.0634 15,827.5086 0.0234214 2.4112 5.2384 28 15,841.3113 0.0219412 2.4105 5.4333 29 30 15,853.2320 0.0204051 2.4098 5.6535 15,863.3470 0.0188074 2.4093 5.9063 31 32 15,871.7464 0.0171447 2.4088 6.2017 15,878.5394 0.0154166 2.4085 6.5541 33 34 15,883.8568 0.0136258 2.4082 6.9843 15,887.8522 0.0117790 2.4080 7.5249 35 36 15,890.7006 0.0098877 2.4078 8.2290 37 15,892.5940 0.0079686 2.4077 9.1913 15,893.7349 0.0060448 2.4076 10.6013 38 15,894.3268 0.0041474 2.4076 12.9107 39 15,894.5635 0.0023207 2.4076 17.6059 40 41 15,894.6168 0.0006571 2.4076 38.2218 1  1  +  a  a  a  a  a  a  a  a  a  no 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 *. -1.4-1.6-  CQ CQ  -1.8-2.010  V  20  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 = 2.7026 A , which is slightly e  lower than the value given in [3] (2.704 A ) . The resulting value of B (= 2\. i) is 0.05849 c m . - 1  e  8ir  ur  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 . This makes T 13,909.14 cm" , in reasonably good agreement with [3] (13910 - 1  -1  e  c m ) . The value of D (using the energy of the dissociation limit determined in the following -1  e  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 a from the Dunham expansion part. The missing e  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 BT [71]. These theoretical values are: 2  C = 3.2 x 10 c m " ! 4  1  5  5  C = 6.37 x 10 cm' 5  6  C = 1.55 x 8  1  A  Wcm^A  6  8  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 , C5 = 5.8 x 10 c m - 1  4  - 1  A ) . Excluding both v = 36 and 37 makes them smaller still (D = 15,894.525 c m , C = 5.4 x 10 5  - 1  4  5  cm  - 1  A ) . The results are not as stable as one would like to the choice of the highest vibration to 5  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^ ) will be used. This limiting value shall be calculated from m  the inequality given in [51], (5-28) which is part of what is known as the Kreek, Pan and Meeth criterium. Using the theoretical values for C6 and Cg, xu is 6.976 A . Only data for v = 34 (for which r m  +  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 RESULTS AND ANALYSIS  128  Table 5-8: Results of fitting turning points in the long-range portion of the potential. D  (cm" )  a  10 C (cm- A )  1  5  1  10 C (cm" A ) 10 C (cm- A )  5  6  5  1  6  8  1  6  8  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 C 5 , 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 C 5 , 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" gives 1  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 B r 2 [51]. In that paper, information on the potential at 7 9  high-v in the B II + state of B r 2 was used to determine the ( P i + P3) dissociation energy, the 3  0  7 9  U  2  2  energy of the dissociation limit for the B state. This energy was determined in [51] to be 19,579.692 ±0.008 c m , although the validity of the quoted uncertainty has been called into question [72]. - 1  The energy of the ( Pa + P s ) dissociation limit, the limit corresponding to the A state, is then 2  2  calculated by subtracting the known energy difference between the P i and P3 states of atomic 2  2  2  2  bromine. According to [51], this energy separation, shown in Figure 5-17, was 3,685.146 ±0.002 cm . -1  5 RESULTS AND ANALYSIS  129  Figure 5-17: Energy separation between the lowest two dissociation limits for 7 9  Br . 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 . As mentioned previously, though, the uncertainty on the value determined in [51] - 1  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^ ) in the A state, the upper limit of the data extends very close to the dissociation m  limit (approximately 2 c m ) and should therefore be reliable. As a comparison, the data used in -1  [51] went up to 5.3 c m  from the dissociation limit of the B state, although there were more data  - 1  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 B r 2 and the 7 9  dissociation energy determined for the A state, the suggested value for the dissociation energy of the B n 3  state of B r , relative to v = 0, J = 0 in the X state, is 19,579.726 ±0.012 c m , an 7 9  0 + U  increase of 0.034 c m  - 1  2  - 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 . The second - 1  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 B r 2 was found in [73] 7 9  to be 19,579.76 ±0.01 c m . The dissociation limit of the A state, using the expected energy - 1  separation given in [51], is then predicted to be 15,894.614 ±0.012 c m . This is in almost exact - 1  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 , is halfway between the values determined from [51] and [73]. - 1  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 Br2, 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^ ) in the B state of Br2, a more precise value for the dissociation m  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 v = 24 (All values in cm ) 7  a  Observed  Calculated"  Obs-Calc  R(72)  15,146.6568  15,146.6765  -0.0197  R(73)  blended  —  R(74)  15,130.0477  R(75)  1  Observed  Calculated"  P(74)  15122.7245  15122.7455  -0.0210 '  —  P(75)  15114.1713  15114.1921  -0.0208  15,130.0804  -0.0327  P(76)  blended  —  —  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  —  —  Obs-Calc  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 consistency. 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 c m ) . -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 L < (the term proportional to [J(J+1)-1] ) was 4  v  not sufficient; the data above the dissociation limit still did not fit with the data below. Including M < (the fifth order term) in the least-squares fits gave positive values for L >. Getting positive v  v  values of L > was inconsistent with preliminary calculations done for the potential of the A state, v  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] ). The data 7  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] ), then 8  v(calc) - v{exp) = -P >[J(J + 1) - l ] , 8  V  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(-P >) + 8ln( [J(J + 1) - 1] ). v  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] ). The intercept gives an estimate for 9  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 -1.0-1.5-2.0O  £5  > T3  -2.5-3.0-3.5-4.0-  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 n.2u state (since Afi = 2), but a small /3  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 B r 2 spectrum was an entire series of extra 7 9  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 v  A  = 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'-2" R(4)  27'-2" R(1)  i  extra  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 c m ) -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.  0.06 0.04 0.02  UJ  0.00  c 'ro -0.02 •  E LU  -0.04-  ^ 0  1  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. All 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 mayhave something to do with hyperfine structure, since the e and f parity levels have different values of eQq //, as described by Eq 2-65. None of the models considered to explain the structure in e  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  138  Figure 5-22: Structure in Q lines in v = 27 of the A state.  27'-1"Q(5)  15487.40  predicted frequency  '  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 II + state, the X S + state and the A n.2 state. There are also several repulsive states 3  1  0  /3  U  U  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 I I - state is expected to be weakly bound and probably 3  0  U  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 c m ) -1  Transition  Observed"  Calculated  27-1 R( 3)  15,488.1276  27-1 R( 5)  Obs-Calc  Splitting  15,488.0983  0.0293  0.0508  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  6  c  "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 S state, could not possibly give X  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 U. 3  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 state (see Figure 4-1) is the most likely candidate. The perturbing state must have a B-value u  almost identical to that of v = 27 in the A state. This would seem to exclude the possibility of the 1  I I i state, or any other van der Waals state for that matter, since such states are bound only at u  large r and should therefore have a smaller B-value than that of v = 27 in the A state. Another telling argument against the I I i state is the fact that there is no apparent doubling for R(0), P(2) 1  u  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  il2  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 separations 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 n 3  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 n2 state) would satisfy all 3  5  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 c m ) -1  E = 15811.8092(9) a  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 B &(A') (predicted) = 0.024763 2  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 . Figure 5-23(c) shows the result for the A state of Br2- Note 2  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 Br2, 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 Br2.  (a) A ^,, 3  -2.5-  a  o o  Of  (b)  I;  A'3n  B  -3.0 -3.&  ..  -4.0 -4.5  ..  of i .  2 u  o © v.  -5.0  PQ  PQ  -5.5-  -6.0 -6.5-  > 10  20  30  40  50  -7-  PQ  10  20  V  a  o O  -0.8-  _A n  l u  of Br  2  a  -1.0-  o  -1.2  co O  -1.4-  60  -0.6-  A' n  of Br  3  2 u  2  -0.8-1.0-  -1.2  PQ  -1.8-  -1.4  -1.6-  -2.0  PQ  50  (d) -0.4  -1.6-  PQ  40  V  (c) 3  30  10  20  30  40  PQ  10  5  V  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 . The error assigned to the extrapolation was ±100 c m . The goal in this part of - 1  - 1  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 level  0.  E  b  (cm- ) 1  extra  separation (cm ) c  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  = 31, J = 24  15,874.3817  -0.0250  v = 31, J = 42  15,894.8206  -0.0170  v  x  "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  145  Figure 5-24: Example of an extra line observed with the v = 16 R(43) line.  —  i  15075.10  1  1  .  15075.12  i  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  15073.10  15073.15  15073.20  predicted position  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 frequency  predicted frequency  R(15)  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?  P(19)  R(17) predicted frequency  15092.90  15092.95  predicted frequency  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. All 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  148  5 RESULTS AND ANALYSIS Figure 5-29: Examples of extra lines observed in v = 30. Shown are 30'-l" R(42) and P(44).  predicted frequency  15418.98  15419.02  predicted frequency  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  15431.15  15431.20  predicted frequency  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 supplemental 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 fjffl d  (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 c m ) , but the A ' state is considered to be most likely. -1  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 t -3) vibration in the A' state (where v tra is the vibrational ex  m  numbering in the extra series which accompanies v  ex  A  = 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 , about 4% smaller than the B-value in the A state, and the - 1  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 c m ) and the A ' state had a B-value that was 4% larger, there would be A J = -1 interaction -1  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 . It is the band origin that shall be - 1  used in the analysis that follows. The theoretical value of C5 for the A' state is 1.39 x 10 c m A , larger than the effective value 5  - 1  5  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.  ( Jv^) "  2n  dG  + 2  8 5  a  f  u n c  ti°  n  °f G(v) (which  shall be referred to as a "LeRoy-Bernstein" plot) is expected to give a straight line in the neardissociation region. Consider, for example, Figure 5-32, which shows such a plot for the A state of Br2, 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 Br2, for example, the extrapolation of the plot  5 RESULTS AND ANALYSIS  152  Figure 5-32: LeRoy-Bernstein plot for the A state of Br2.  700 600 5^  500H  v = 16  -O 400H  >  300  v = 24  2001000  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.  7060o  S~  50-  "O 40-  > —•  v = 33  302010-  015780 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 . This is somewhat higher than the value determined for the dissociation energy previously - 1  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 . Extrapolating - 1  blindly from v = 16 would underestimate the dissociation energy by approximately 41 c m . A blind - 1  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 the estimate in [77] of 430 c m  - 1  - 1  below the dissociation limit, close to  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 VA = 21 to the dissociation limit). There was no choice but to make such an 1  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  B r as a 2  function of v.  -0.6-  -0.7-  Q.  —v=16  -0.9-  -1.010  - 1  15  1  1  '  v= 37—-  1—  20  25  —i— 30  35  40  be constructed as: ^ (A) « I [G(v = 27) -G(v av 6 A  If (ff(A)J  = 24)].  A  were to be plotted in Figure 5-32 versus G  7  ave  (5-29)  = ^[G(v =27) + G(v =24)], the A  A  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  for ( ^ )  7  10  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 , off by 4.3 c m , as compared to the 41 c m - 1  - 1  - 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)  in Figure 5-32 do not depend on the energy of the  7  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  (c£u )  7  10  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, (ifu )  10 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 10  3  m =-0.6357 cm" 7 , b = 10,104 crrT~. The estimated energy of v = 16 from the extrapolation procedure is 15,461 cm , as compared to 1  the actual value of 15,473.2780 c m , a difference of about 12.3 c m . - 1  - 1  Table 5-13: Preliminary test of extrapolation procedure in the A state. (All values in c m ) -1  V  G(v)  "  Qextrap^y^ G b  EXTM  P(v)-G(v)  6G  c  foQextrap d extrap_ SG  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 6  The extrapolated energy for the given vibration  c  G(v) - G(v-l), the difference in measured energies between neighbouring vibrations  d  G  e x t r a p  (v) - G  extrap  ( 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, <5G for v = 21 is the predicted energy separation between v = 21 and v = 20 in the A state, based ex  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-  1600 1400 1200-1 .1000  > T3  800 600-|  O 2000-  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 . As mentioned previously, the difference in vibrational numbering between the - 1  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: 10  3  m = -0.6311cm r, b = 10,030cm~T. -  As before, the estimate for G(v 4/=21) is given by the value of G(v) that makes ( ^ )  equal to  7  J  the known value for v = 21 in the A ' state (297.89 c m " ^ ) . The resulting estimate for the energy of v = 21 is 15,422.6 c m . On this basis of this estimate, the suggested energy of the A' state, the - 1  energy of the bottom of the A ' state potential relative to v = 0, J = 0 in the X state, is T (A') = 13,020.5 cm- .  (5-30)  1  0je  The energy relative to the bottom of the potential in the X state is T (A') = 13,187 cm- . 1  ete  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 T , (L>') = 48,890 cm' . 1  e e  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 c m ) fiQextrap b extrap_ Q V 6G -1  a  6G  6  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  G(v) - G(v-l), the difference in measured energies between neighbouring vibrations  a  G (v)  ft  ea;imp  -G  (v-1),  extrap  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 . To work on the conservative side, the uncertainty on the A ' state energy - 1  given in Eq 5-30 is set at ±20 c m . As an indication of the sensitivity of the procedure, a decrease - 1  of 1 c m 9.1 c m  - 1  - 1  in the energy used for the level perturbing v = 24 in the A state led to an increase of 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 eQq . The matrix was diagonalized to find 2  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)  15056.85  15056.90  15056.95  23'-2" R(0) + R(1)  15057.00  1 5083.60  24'--2" R(0) + R(1)  15107.90  15107.95  15108.00  1 5083.65  1 5083.70  2 5 ' - 1 " R(0) + R(1)  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, g ^{^'), F  is calculated as g / W) e  f  =  a g (1, )+P g U> ), 2  2  F  x  F  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 C  sr  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 eQq [80]. 0  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) '  e  X  P  ?  n  m  e  n  t  a  l  2 1 ' - 1 " R ( 2 ) + Q(1) calculated-^  2 1 ' - 1 " P(3) experimental A f  \— calculated experimental  <  calculated  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 i , the first harmonic magnetic rotation signal, SFH, has a component that varies as 2  S H OC Njncos{2'nfit)\H\cos(2'Kf t). F  2  From the trigonometric relation cos(A)cos(B) = i (cos(A + B) + cos(A - B)), measuring with phase-sensitive detection at fi + i or |fi— i \ yields a signal with a linewidth (in 2  2  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 polarization 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 counterpropagating (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 \ that obscured the spectrum of IBr. The Doppler-free 2  technique in the current work was tested on signals from B - X in I (see Figure 5-38), and typically 2  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 I . The signal was taken 2  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 I . For the trace measured in 2  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 were with the diameter of the beams 1.5 to 2 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 of I . 2  1.1 GHz  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  5 RESULTS AND ANALYSIS  164  signals. The trace in Figure 5-39 spans 1 GHz (~ 0.033 c m ) . -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 B r . The transition moment for the A - X system 2  of B r are very small, and so transitions in this system are difficult to saturate. It was hoped 2  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 B r . The major problems included backscatter 2  of the pump beam, and the fact that the measured signals were very weak (even in Doppler-limited) when the pressure of the B r was decreased to the point where collision effects were not expected 2  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  6  165  Future Work and Conclusions  Great strides have been made toward a better understanding of the A L T i state of Br2, but it should 3  u  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  and £ j _ 3  u  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 vibrational 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 B r from the lowest vibrations in the X state are very weak, 2  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 B r . 2  The Doppler-free technique would be well-suited to studying the hyperfine structure of states such as the A I I i 3  state in I , or the A LTi states in IC1 and IBr. The hyperfine structure of 3  u  2  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 A Hi 3  u  state of  Br2, 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 spectroscopy that used an electric field instead of a magnetic field. The major challenge would probably be to devise an experimental setup to give an oscillating, longitudinal electric field (perhaps a series of concentric rings), but the possibilities that such a technique would open up make it worth consideration. It should be clear that magnetic rotation is a very convenient and powerful utility for furthering the studies of the diatomic halogens and inter-halogens. It can probe the details of some of the states that cannot be explored with other spectroscopic techniques. It would have been next to impossible to study the A I I i state of B r to the same level of detail achieved in the current work 3  u  2  through absorption spectroscopy, for example. Using the results of this magnetic rotation study, the A state is now well-characterized almost right up to the dissociation limit.  REFERENCES  167  References [I] J. A. Coxon, J. Mol. Spectrosc. 41, 548 (1972). [2] J. C. D. Brand, A. R. Hoy, S. M . Jaywant, and A. W. Taylor, J. Mol. 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No s t r o n g l y blended l i n e s are i n c l u d e d i n these t a b l e s . An i n d i c a t o r i s i n c l u d e d i n t h e f i n a l column, meant as a guide f o r the eye, which g i v e s a rough i n d i c a t i o n o f how w e l l t h e c a l c u l a t e d f r e q u e n c i e s agree w i t h t h e observed f r e q u e n c i e s . No s t a r s a r e p r i n t e d i f t h e c a l c u l a t e d f r e q u e n c i e s agrees w i t h the observed frequency t o within the assigned uncertainty. A s i n g l e star i n d i c a t e s that the d i s c r e p a n c y i s g r e a t e r than t h e u n c e r t a i n t y , but l e s s than 1.3 times the u n c e r t a i n t y . Two s t a r s i n d i c a t e s t h a t the d i s c r e p a n c y i s g r e a t e r than 1.3 times t h e u n c e r t a i n t y and l e s s than 1.6 times t h e u n c e r t a i n t y . Three s t a r s i n d i c a t e s a d i s c r e p a n c y g r e a t e r than 1.6 times t h e u n c e r t a i n t y and l e s s than twice t h e u n c e r t a i n t y . Four s t a r s i n d i c a t e s t h a t t h e d i s c r e p a n c y i s twice t h e u n c e r t a i n t y o r g r e a t e r . The d i r e c t i o n of t h e s t a r s r e l a t i v e t o the b a r i n d i c a t e s whether the d i s c r e p a n c y i s p o s i t i v e or negative. Note t h a t t h e d a t a f o r v = 27 i n Table A - l was not a c t u a l l y used i n a n a l y s i s ; i t i s i n c l u d e d f o r o b s e r v a t i o n purposes o n l y . The t r a n s i t i o n s a r e l a b e l e d i n t h e 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 l a b e l e d a c c o r d i n g t o t h e v a l u e o f t h e quantum number J i n the s t a t e o f lower energy. In a d d i t i o n , v i b r a t i o n a l quantum numbers (v) i n t h e two s t a t e s a r e used t o l a b e l a t r a n s i t i o n . Indicating quantum numbers f o r t h e h i g h e r energy s t a t e w i t h a s i n g l e prime and quantum numbers f o r the lower energy s t a t e w i t h double primes, the l a b e l i n g scheme f o r 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 l a r g e r by one i n t h e h i g h e r energy s t a t e i s l a b e l e d an R t r a n s i t i o n . When J i s lower by one u n i t i n t h e h i g h e r energy s t a t e , i t i s a P t r a n s i t i o n . When J does not change i n t h e transition, i t i slabeled a Q t r a n s i t i o n .  Table A - l :  Data used i n t h e g l o b a l a n a l y s i s  Transition  Observed  13—0 13 — 0 13 — 0 13 — 0 13 — 0 13 — 0 13 — 0 13 — 0 13 — 0 13 — 0 13 — 0 13 — 0 13 — 0 13 — 0  15289 15288 15288 15286 15285 15284 15283 15281 15279 15273 15266 15201 15169 15165  R( 8) R( 9) R(10) R(12) R(13) R(14) R(15) R(17) R(19) R(23) R(27) R(50) R(58) R(59)  4953 9424 2991 7899 9345 9881 9828 7409 2022 2484 0986 7246 8411 5045  Calculated 15289 15288 15288 15286 15285 15284 15283 15281 15279 15273 15266 15201 15169 15165  5029 9338 2913 7861 9232 9868 9768 7358 2000 2417 0980 7203 8380 5004  ( a l l u n i t s i n cm  Uncert. 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0150 0075 0150 0075 0100 0075 0100 0150 0075 0150 0100 0150 0100 0100  Obs-Cal -0 0 0 0 0 0 0 0 0 0 0 0 0 0  0076 0086 0078 0038 0113 0013 0060 0051 0022 0067 0006 0043 0031 0041  ) .  Appendix 13- -0 13- -0 13- -0 13- -0 13- -0 13- -0 13- -0 13- -0 13- -0 13- -0 13- -0 13- -0 13- -0 13- -0 13- -0 14- -0 14- -0 14- -0 14- -0 14- -0 14- -0 14- -0 14- -0 14- -0 14- -0 14- -0 14- -0 14- -0 14- -0 14- -0 14- -0 14- -0 14- -0 14- -0 14- -0 14- -0 14- -0 14- -0 14- -0 14- -0 14- -0 14- -0 14- -0 14- -0 14- -0 14- -1 14- -1 14- -1 14- -1 14- -1 14- -1 14- -1 14- -1 14- -1 14- -1 14- -1 14- -1 14- -1 14- -1 14- -1 14- -1 14- -1  R R R P P P P P P P P P P P P R R R R R R R R R R R R R R R R R R R R P P P P P P P P P P R R R R R R R R R R R R R R P P P  62) 70) 72) 8) 11) 12) 13) 15) 17) 18) 21) 31) 36) (52) (53) ( 4) 5) 6) 7) 8) 10) 12) 13) 14) 15) 16) 47) 49) 52) 54) 60) 64) 65) 68) 70) 5) 6) 8) 11) 14) 17) 44) 67) 69) 73) 3) 4) 5) 6) 8) 9) 10) 11) 13) 15) 16) 18) 21) 25) 6) 10) 12)  15152 15112 15101 15287 15285 15284 15283 15281 15278 15277 15272 15252 15239 15184 15180 15358 15358 15357 15357 15356 15355 15353 15352 15351 15350 15349 15275 15268 15256 15248 15222 15202 15197 15182 15171 15357 15356 15355 15352 15349 15345 15278 15176 15165 15142 15035 15035 15034 15034 15033 15033 15032 15031 15029 15027 15026 15024 15019 15013 15033 15030 15028  0134 5271 8367 9601 4914 5216 4726 1665 5751 1582 4805 0892 0879 8711 8288 3704 0697 7007 2625 7410 4614 8734 9708 9825 9334 7863 9711 6086 9381 7518 2307 8883 8440 1842 3302 1014 5622 2561 6878 4401 5173 7438 0894 0979 0743 4325 2249 9306 5659 6058 0100 3404 5990 8758 8505 7153 2409 9488 1483 4347 5033 5850  15152 15112 15101 15287 15285 15284 15283 15281 15278 15277 15272 15252 15239 15184 15180 15358 15358 15357 15357 15356 15355 15353 15352 15351 15350 15349 15275 15268 15256 15248 15222 15202 15197 15182 15171 15357 15356 15355 15352 15349 15345 15278 15176 15165 15142 15035 15035 15034 15034 15033 15033 15032 15031 15029 15027 15026 15024 15019 15013 15033 15030 15028  0127 5221 8373 9625 4922 5220 4784 1707 5689 1577 4817 0872 0996 8619 8224 3719 0774 7064 2589 7349 4573 8733 9663 9826 9221 7847 9718 6010 9378 7553 2343 8893 8425 1921 3303 1150 5691 2480 6928 4485 5142 7483 0937 0944 0690 4434 2278 9365 5694 6076 0129 3422 5956 8744 8489 7218 2390 9419 1401 4321 5059 5877  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0100 0055 0055 0100 0100 0100 0075 0065 0075 0075 0075 0150 0150 0150 0150 0100 0075 0075 0100 0150 0150 0150 0150 0100 0150 0150 0150 0075 0150 0055 0100 0045 0075 0055 0100 0100 0100 0150 0100 0150 0150 0100 0100 0075 0150 0065 0035 0045 0055 0030 0035 0030 0055 0045 0070 0065 0040 0060 0050 0040 0035 0060  0 0 -0 -0 -0 -0 -0 -0 0 0 -0 0 -0 0 0 -0 -0 -0 0 0 0 0 0 -0 0 0 -0 0 0 -0 -0 -0 0 -0 -0 -0 -0 0 -0 -0 0 -0 -0 0 0 -0 -0 -0 -0 -0 -0 -0 0 0 0 -0 0 0 0 0 -0 -0  0007 0050 0006 0024 0008 0004 0058 0042 0062 0005 0012 0020 0117 0092 0064 0015 0077 0057 0036 0061 0041 0001 0045 0001 0113 0016 0007 0076 0003 0035 0036 0010 0015 0079 0001 0136 0069 0081 0050 0084 0031 0045 0043 0035 0053 0109 0029 0059 0035 0018 0029 0018 0034 0014 0016 0065 0019 0069 0082 0026 0026 0027  Appendix 14- -1 14- -1 14- -1 14- -1 14- -1 14- -1 14- -1 14- -1 14- -1 14- -1 15- -0 15- -0 15- -0 15- -0 15- -0 15- -0 15- -0 15- -0 15- -0 15- -0 15- -0 15- -0 15- -0 15- -0 15- -0 15- -0 15- -0 15- -0 15- -0 15- -0 15- -0 15- -0 15- -0 15- -0 15- -0 15- -0 15- -0 15- -0 15- -0 15- -0 15- -0 15- -0 15- -0 15- -0 15- -0 15- -0 15- -0 15- -0 15- -0 15- -0 15- -0 15- -0 15- -0 15- -0 15- -0 15- -0 15- -0 15- -1 15- -1 15- -1 15- -1 15- -1  p p p p p p p p p p R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R P P P P P P P P P P P P P P P P P R R R R R  13) 14) (17) 19) (21) (22) (23) (25) (27) (29) 5) 6) ( 7) ( 9) (10) 11) 13) 14) 18) 19) 20) 23) 31) 41) 48) 50) 51) 53) 57) 59) 60) 61) 62) 64) 65) 66) 69) 70) 72) 73) 6) 7) 8) 9) 11) 14) 15) 17) 20) 21) 25) 26) 30) 58) 60) 71) 72) 3) 4) 5) 7) 8)  15027 15026 15022 15019 15016 15014 15012 15008 15004 14999 15418 15418 15417 15416 15415 15414 15413 15412 15407 15405 15404 15399 15382 15353 15328 15320 15316 15308 15290 15281 15276 15271 15266 15256 15250 15245 15228 15222 15211 15205 15416 15416 15415 15414 15412 15409 15408 15405 15400 15398 15391 15388 15379 15276 15266 15205 15199 15095 15095 15095 15094 15093  5147 3676 4587 4754 1840 4301 5985 6996 4959 9710 4751 0814 6095 4270 7155 9216 0977 0653 1306 6938 1789 1678 1943 6263 6919 8058 7424 3511 5296 1021 2655 3199 3162 0062 7195 3487 6823 9509 2081 1876 9878 3447 6223 8162 9805 6182 3405 5343 7315 9786 1226 9600 4856 2998 4034 6221 5633 8700 6455 3362 4801 9278  15027 15026 15022 15019 15016 15014 15012 15008 15004 14999 15418 15418 15417 15416 15415 15414 15413 15412 15407 15405 15404 15399 15382 15353 15328 15320 15316 15308 15290 15281 15276 15271 15266 15256 15250 15245 15228 15222 15211 15205 15416 15416 15415 15414 15412 15409 15408 15405 15400 15398 15391 15388 15379 15276 15266 15205 15199 15095 15095 15095 15094 15093  5147 3657 4623 4793 1912 4326 5975 6975 4908 9767 4778 0839 6104 4239 7109 9181 0925 0597 127 0 6931 1788 1526 1885 6252 6917 8101 7419 3497 5348 1079 2636 3316 3117 0068 7211 3462 6823 9467 2017 1915 9890 3472 6256 8243 9823 6205 3400 5390 7363 9748 1242 9599 4935 3019 3966 6318 5709 8796 6478 3369 4779 9297  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0055 0050 0040 0050 0040 0040 0055 0050 0065 0065 0100 0065 0065 0065 0100 0065 0065 0065 0150 0065 0055 0150 0100 0150 0065 0150 0055 0100 0065 0100 0100 0100 0100 0100 0100 0100 0100 0075 0100 0100 0045 0055 0065 0100 0100 0100 0100 0075 0150 0150 0150 0150 0150 0065 0065 0100 0100 0065 0045 0055 0075 0035  0 0 -0 -0 -0 -0 0 0 0 -0 -0 -0 -0 0 0 0 0 0 0 0 0 0 0 0 0 -0 0 0 -0 -0 0 -0 0 -0 -0 0 0 0 0 -0 -0 -0 -0 -0 -0 -0 0 -0 -0 0 -0 0 -0 -0 0 -0 -0 -0 -0 -0 0 -0  0000 0019 0036 0039 0072 0025 0010 0021 0051 0057 0027 0025 0009 0031 0046 0035 0052 0056 0036 0007 0001 0152 0058 0011 0002 0043 0005 0014 0052 0058 0019 0117 0045 0006 0016 0025 0000 0042 0064 0039 0012 0025 0033 0081 0018 0023 0005 0047 0048 0038 0016 0001 0079 0021 0068 0097 0076 0096 0023 0007 0022 0019  Appendix 15- -1 15- -1 15- -1 15- -1 15- -1 15- -1 15- -1 15- -1 15- -1 15- -1 15- -1 15- -1 15- -1 15- -1 15- -1 15- -1 15- -1 15- -1 15- -1 15- -1 15- -1 15- -1 15- -1 15- -1 15- -1 15- -1 15- -1 15- -1 15- -1 15- -1 15- -1 15- -1 15- -1 15- -1 15- -1 15- -1 16- -0 16- -0 16- -0 16- -0 16- -0 16- -0 16- -0 16- -0 16- -0 16- -0 16- -0 16- -0 16- -0 16- -0 16- -0 16- -0 16- -0 16- -0 16- -0 16- -0 16- -0 16- -0 16- -0 16- -0 16- -0 16- -0  R (10) R (11) R (15) R (16) R (18) R (21) R (22) R (24) R (25) R (27) R (28) R (33) R (35) R (36) R (41) R (46) R (48) R (49) R (50) R (52) P ( 8) P ( 9) P (10) P (11) P (12) P (14) P 15) P (16) P 18) P 19) P 21) P 23) P 28) P 46) P 49) P 50) R 3) R 12) R 14) R 16) R 18) R 33) R 35) R 39) R 40) R 42) R 46) R 48) R 49) R 50) R 51) R 53) R 54) R 58) R 59) R 61) R 64) R 65) R 66) R 67) R 71) R 72)  15092 15091 15087 15086 15084 15079 15077 15074 15072 15068 15066 15054 15048 15046 15031 15013 15006 15002 14998 14990 15092 15091 15090 15089 15088 15086 15085 15083 15080 15079 15075 15072 15061 15006 14994 14990 15473 15467 15465 15463 15460 15429 15423 15411 15408 15401 15386 15378 15374 15370 15366 15357 15352 15333 15328 15318 15302 15297 15291 15285 15262 15255.  5958 8105 8749 6905 0868 5841 9284 3646 4621 4190 2761 3427 9988 2139 0241 7917 3061 4337 4764 3292 4980 7011 8260 8752 8374 5350 2653 9157 9804 3945 9725 2398 4954 0907 2639 1561 0610 8115 7371 3173 5691 2238 6001 3212 0365 2065 5016 6240 5547 3927 1468 3782 8625 8884 9224 7085 6927 1673 5461 8346 0418 8579  15092 15091 15087 15086 15084 15079 15077 15074 15072 15068 15066 15054 15049 15046 15031 15013 15006 15002 14998 14990 15092 15091 15090 15089 15088 15086 15085 15083 15080 15079 15075 15072 15061 15006 14994 14990 15473 15467 15465 15463 15460 15429 15423 15411 15408 15401 15386 15378 15374 15370 15366 15357 15352 15333 15328 15318 15302 15297 15291. 15285 15262. 15255.  5959 8101 8736 6908 0866 5823 9211 3586 4570 4129 2702 3444 0062 2148 0296 7814 2992 4324 4815 3265 4983 7027 8281 8743 8413 5376 2668 9166 9779 3893 9731 2379 4998 0897 2660 1578 0591 8111 7297 3146 5653 2180 5956 3205 0361 2071 5030 6246 5529 3927 1436 3777 8605 8892 9193 7048 6912 1674 5500 8386 0450 8575  0 .0035 0 .0055 0 0055 0 0035 0 0050 0 0060 0 0045 0 0040 0 0050 0 0060 0 0070 0 0070 0 0070 0 0045 0 0060 0 0070 0 0050 0 0055 0 0050 0 0040 0 0035 0 0060 0 0045 0 0055 0 0065 0 0035 0 0055 0 0035 0 0035 0 0055 0 0070 0 0055 0 0065 0 0045 0 0065 0 0045 0 0065 0 0065 0 0055 0 0055 0 0045 0 0100 0 0100 0 0075 0 0075 0 0065 0 0150 0 0065 0 0050 0 0100 0 0100 0 0075 0 0075 0 0055 0 0065 0 0065 0 0055 0 0040 0 0100 0 0075 0 0045 0. 0045  -0 0 0 -0 0 0 0 0 0 0 0 -0 -0 -0 -0 0 0 0 -0 0 -0 -0 -0 0 -0 -0 -0 -0 0 0 -0 0 -0 0 -0 -0 0 0 0 0 0 0 0 0 0 -0 -0 -0 0 0 0 0 0 -0 0 0 0 -0 -0 -0. -0. 0.  0001 0004 0013 0003 0002 0018 0073 0060 0051 0061 0059 0017 0074 0009 0055 0103 0069 0013 0051 0027 0003 0016 0021 0009 0039 0026 0015 0009 0025 0052 0006 0019 0044 0010 0021 0017 0019 0004 0074 0027 0038 0058 0045 0007 0004 0006 0014 0006 0018 0000 0032 0005 0020 0008 0031 0037 0015 0001 0039 0040 0032 0004  Appendix 16- -0 16- -0 16- -0 16- -0 16- -0 16- -0 16- -0 16- -0 16- -0 16- -0 16- -0 16- -0 16- -0 16- -0 16- -0 16- -0 16- -0 16- -0 16- -0 16- -0 16- -1 16- -1 16- -1 16- -1 16- -1 16- -1 16- -1 16- -1 16- -1 16- -1 16- -1 16- -1 16- -1 16- -1 16- -1 16- -1 16- -1 16- -1 16- -1 16- -1 16- -1 16- -1 16- -1 16- -1 16- -1 16- -1 16- -1 16- -1 16- -1 16- -1 16- -1 16- -1 16- -1 16- -1 16- -1 16- -1 16- -1 16- -1 16- -1 16- -1 16- -1 16- -1  R (73) R (74) R (76) R (77) P ( 8) • P ( 9) P (10) P (11) P (14) P (16) P (18) P (20) P (21) P (22) P (24) P (67) P (69) P (71) P (74) P (76) R ( 3) R ( 4) R ( 5) R (10) R (12) R (14) R (15) R 17) R 19) R 21) R 22) R 25) R 26) R 27) R 29) R 31) R 33) R 34) R 35) R 36) R 38) R 39) R 41) R 42) R 43) R 44) R 45) R 46) R 47) R 49) R 50) R 51) R 53) R 54) R 58) R 59) R 60) R 62) P 9) P 10) P 13) P 14)  15249 15243 15230 15223 15469 15468 15467 15466 15463 15460 15457 15454 15452 15450 15446 15275 15263 15251 15231 15218 15149 15149 15149 15146 15144 15142 15141 15138 15136 15132 15131 15125 15123 15121 15116 15111 15106 15103 15100 15097 15091 15088 15082 15078 15075 15071 15067 15064 15060 15052 15048 15043 15035 15030 15011 15006 15001 14991 15145. 15144. 15141. 15140.  5736 1945 1382 4577 5967 7655 8628 8690 3829 6500 5806 1701 3460 4352 3522 4175 4421 0692 8247 5180 9104 6668 3317 4463 7143 6541 4991 9358 0426 8213 0766 3679 2936 1373 5683 6659 4304 6829 8508 9341 8477 6742 0692 6408 1196 5181 8256 0494 1830 1980 0658 8464 1522 6700 8491 9071 8913 5691 6511 7440 5392 3012  15249 15243 15230 15223 15469 15468 15467 15466 15463 15460 15457 15454 15452 15450 15446 15275 15263 15251 15231 15218 15149 15149 15149 15146 15144 15142 15141 15138 15136 15132 15131 15125 15123 15121 15116 15111 15106 15103 15100 15097 15091 15088 15082 15078 15075 15071 15067 15064 15060 15052 15048 15043 15035 15030 15011 15006 15001 14991 15145. 15144. 15141. 15140.  5734 1922 1369 4616 5992 7715 8608 8669 3863 6496 5792 1746 3468 4353 3605 4186 4388 0817 8314 5162 9124 6642 3336 4442 7109 6469 4907 9299 0370 8116 0739 3599 2879 1321 5683 6677 4290 6825 8511 9346 8456 6727 0694 6386 1214 517 6 8270 0493 1845 1921 0641 8479 1498 6674 8420 9102 8876 5686 6500 7457 5379 3035  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.  0045 0100 0055 0050 0055 0055 0075 0055 0065 0075 0100 0150 0100 0100 0075 0065 0055 0100 0100 0055 0100 0100 0100 0100 0100 0100 0050 0035 0035 0150 0040 0075 0045 0100 0065 0045 0035 0075 0040 0040 0050 0045 0045 0050 0100 0035 0035 0035 0035 0075 0035 0040 0035 0035 0045 0050 0035 0035 0075 0050 0055 0150  0 0 0 -0 -0 -0 0 0 -0 0 0 -0 -0 -0 -0 -0 0 -0 -0 0 -0 0 -0 0 0 0 0 0 0 0 0 0 0 0 0 -0 0 0 -0 -0 0 0 -0 0 -0 0 -0 0 -0 0 0 -0 0 0 0 -0 0 0. 0. -0. 0. -0.  0002 0023 0013 0039 0025 0060 0020 0021 0034 0004 0014 0045 0008 0001 0083 0011 0033 0125 0067 0018 0020 0026 0019 0021 0034 0072 0084 0059 0056 0097 0027 0080 0057 0052 0000 0018 0014 0004 0003 0005 0021 0015 0002 0022 0018. 0005 0014 0001 0015 0059 0017 0015 0024 0026 0071 0031 0037 0005 0011 0017 0013 0023  Appendix 16- -1 16- -1 16- -1 16- -1 16- -1 16- -1 16- -1 16- -1 16- -1 16- -1 16- -1 16- -1 16- -1 16- -1 16- -1 16- -1 16- -1 16- -1 17- -0 17- -0 17- -0 17- -0 17- -0 17- -0 17- -0 17- -0 17- -0 17- -0 17- -0 17- -0 17- -0 17- -0 17- -0 17- -0 17- -0 17- -0 17- -0 17- -0 17- -0 17- -0 17- -0 17- -0 17- -0 17- -0 17- -0 17- -0 17- -0 17- -0 17- -0 17- -0 17- -0 17- -0 17- -0 17- -0 17- -0 17- -1 17- -1 17- -1 17- -1 17- -1 17- -1 17- -1  p (15) p (16) p (17) p (18) p (19) p (27) p (28) p (29) p (44) p (46) p (48) p (49) p (50) p (52) p (54) p (56) p (57) p (59) R (28) R (29) R (30) R (32) R (33) R (35) R 39) R 40) R 43) R 45) R 46) R 48) R 53) R 57) R 60) R 61) R 63) R 65) R 66) R 68) R 69) R 70) R 72) R 74) R 75) R 76) R 78) P 28) P 34) P 40) P 42) P 44) P 53) P 54) P 73) P 75) P 77) R 3) R 4) R 5) R 7) R( 8) R( 10) R( 12)  15138 .9885 15137 .5877 15136 .1064 15134 .5354 15132 .8899 15116 .7095 15114 .3142 15111 8289 15064 4447 15056 .6747 15048 5576 15044 3680 15040 0892 15031 2731 15022 0957 15012 5718 15007 6729 14997 6171 15488 6976 15486 2573 15483 7240 15478 4069 15475 6045 15469 7506 15456 9692 15453 5477 15442 7437 15435 0844 15431 1168 15422 9167 15400 7857 15381 3937 15365 8700 15360 4986 15349 4751 15338 0641 15332 2116 15320 2112 15314 0654 15307 8117 15295 0148 15281 8197 15275 0663 15268 2124 15254 1968 15484 3170 15467 4295 15447 3590 15439 9599 15432 1866 15392 7285 15387. 8815 15277 7549 15264. 1191 15250. 0596 15198. 2167 15197. 9593 15197. 6054 15196. 6595 15196. 0423 15194. 5704 15192. 7541  15138 . 9864 15137 .5867 15136 .1042 15134 .5388 15132 8906 15116 7107 15114 .3126 15111 8306 15064 4495 15056 6775 15048 5585 15044 3683 15040 0907 15031 2722 15022 1011 15012 5753 15007 6787 14997 6166 15488 7033 15486 2620 15483 7324 15478 4079 15475 6126 15469 7553 15456 9684 15453 5473 15442 7419 15435 0846 15431 1191 15422 9133 15400 7835 15381 3995 15365 8682 15360 5001 15349 4752 15338 0627 15332 2102 15320 2103 15314 0621 15307 8145 15295 0190 15281 8199 15275 0676 15268 2129 15254 1938 15484 3109 15467 4332 15447 3622 15439 9559 15432 1891 15392. 7289 15387. 8840 15277. 7660 15264. 1191 15250. 0667 15198. 2217 15197 . 9572 15197. 6072 15196. 6502 15196. 0431 15194. 5719 15192. 7573  0 .0045 0 .0040 0 .0045 0 .0050 0 0035 0 0035 0 0045 0 0045 0 0040 0 0040 0 0035 0 0050 0 0045 0 0035 0 0100 0 0075 0 0075 0 0045 0 0065 0 0100 0 0100 . 0 0055 0 0100 0 0100 0 0065 0 0055 0 0055 0 0055 0 0065 0 0045 0 0055 0 0100 0 0055 0 0100 0 0075 0 0055 0 0045 0 0050 0 0050 0 0100 0 0150 0 0100 0 0055 0 0035 0 0100 0 0100 0 0100 0 0100 0. 0075 0. 0075 0. 0150 0. 0075 0. 0100 0. 0100 0. 0100 0. 0045 0. 0065 0. 0045 0. 0100 0. 0045 0. 0035 0. 0065  0 .0021 0 .0010 0 .0022 -0 .0034 -0 .0007 -0 .0012 0 .0016 -0 0017 -0 .0048 -0 0028 -0 0009 -0 0003 -0 0015 0 0009 -0 0054 -0 0035 -0 0058 0 0005 -0 0057 -0 0047 -0 0084 -0 0010 -0 0081 -0 0047 0 0008 0 0004 0 0018 -0 0002 -0 0023 0 0034 0 0022 -0 0058 0 0018 -0 0015 -0 0001 0 0014 0 0014 0 0009 0 0033 -0 0028 -0 0042 -0 0002 -0 0013 -0 0005 0 0030 0 0061 -0 0037 -0 0032 0 0040 - 0 . 0025 - 0 . 0004 - 0 . 0025 - 0 . 0111 0. 0000 - 0 . 0071 - 0 . 0050 0. 0021 - 0 . 0018 0. 0093 - 0 . 0008 - 0 . 0015 - 0 . 0032  Appendix 17 — 1 17 — 1 17 — 1 17 — 1 17 — 1 17 — 1 17 — 1 17 — 1 17 — 1 17 — 1 17 — 1 17 — 1 17 — 1 17 — 1 17 — 1 17 — 1 17 — 1 17 — 1 17 — 1 17 — 1 17 — 1 17--1 17 — 1 17 — 1 17 — 1 17 — 1 17 — 1 17 — 1 17 — 1 17 — 1 17 — 1 17 — 1 17 — 1 17 — 1 17 — 1 17 — 1 17 — 1 17 — 1 17 — 1 17 — 1 17 — 1 17 — 1 17 — 1 17 — 1 17 — 1 17 — 1 17 — 1 17 — 1 17 — 1 17 — 1 17--1 17 — 1 17 — 1 17 — 1 17 — 1 17 — 1 17 — 1 17--1 17--1 17 — 1 17 — 1 17 — 1  R (13) R (14) R (15) R (16) R (17) R (18) R (20) R (21) R (23) R (24) R (27) R (28) R (29) R (31) R (33) R (35) R (36) R (37) R (38) R (39) R (40) R (42) R (43) R (44) R 46) R 50) R 52) R 54) R 57) R 59) R 60) R 61) R 64) R 65) R 66) R 67) R 68) R 69) P 5) P 6) P 7) P 8) P 9) P 10) P 12) P 13) P 15) P 16) P 17) P 19) P 23) P 24) P 27) P 29) P 30) P (31) P (33) P (34) P (36) P (37) P (38) P (40)  177 15191 .7216 15190 .5984 15189 .3864 15188 .0960 15186 .7224 15185 .2497 15182 .0569 15180 .3285 15176 . 6223 15174 . 6330 15168 .1520 15165 .8126 15163 .3826 15158 .2823 15152 8187 15147 0065 15143 9675 15140 8420 15137 6258 15134 3181 15130 9253 15123 8626 15120 2016 15116 4462 15108 6632 15092 0122 15083 1301 15073 8852 15059 3136 15049 1276 15043 8987 15038 5688 15022 0099 15016 2988 15010 4881 15004 5814 14998 5755 14992 4613 15196 7498 15196 1616 15195 4812 15194 7222 15193 8753 15192 9426 15190 8181 15189 6145 15186 9812 15185 5382 15184. 0026 15180. 6833 15172. 9898 15170. 8486 15163. 9092 15158. 8446 15156. 1820 15153. 4263 15147 . 6684 15144. 6634 15138. 3748 15135. 0916 15131. 7316 15124. 7397  15191 .7212 15190 .5990 15189 .3909 15188 .0965 15186 .7160 15185 .2492 15182 .0564 15180 .3302 15176 . 6178 15174 . 6313 15168 .1495 15165 .8144 15163 3918 15158 2836 15152 8236 15147 0108 15143 9715 15140 8436 15137 6266 15134 3206 15130 9252 15123 8657 15120 2011 15116 4463 15108 6654 15092 0108 15083 1331 15073 8854 15059 3149 15049 1315 15043 8978 15038 5691 15022 0087 15016 2958 15010 4859 15004 5785 14998 5732 14992 4696 15196 7525 15196 1615 15195 4850 15194 7228 15193 8750 15192 9415 15190 8173 15189 6266 15186 9874 15185. 5389 15184. 0043 15180. 6768 15172. 9858 15170. 8467 15163. 9086 15158. 8481 15156. 1869 15153. 4382 15147 . 6780 15144. 6661 15138. 3781 15135. 1016 15131. 7365 15124. 7401  0 .0040 0 .0035 0 .0075 0 .0035 0 .0055 0 .0035 0 .0035 0 .0035 0 0045 0 0045 0 0050 0 0045 0 0045 0 0055 0 0055 0 0040 0 0035 0 0035 0 0035 0 0035 0 0035 0 0075 0 0100 0 0035 0 0035 0 0035 0 0075 0 0035 0 0035 0 0040 0 0035 0 0035 0 0045 0 0075 0 0075 0 0045 0 0045 0 0055 0 0035 0 0045 0 0035 0 0035 0 0045 0 0050 0 0035 0. 0100 0. 0055 0. 0035 0. 0035 0. 0045 0. 0040 0. 0065 0. 0050 0. 0075 0. 0045 0. 0055 0. 0045 0. 0055 0. 0055 0. 0040 0. 0100 0. 0035  0 .0004 | -0 .0006 | -0 .0045 | -0 .0005 | 0 .0064 | 0 .0005 | 0 .0005 | | -0 .0017 0 .0045 | 0 .0017 | 0 .0025 | -0 0018 | -0 0092****| -0 0013 | -0 0049 | -0 0043 *| -0 0040 *| -0 0016 | -0 0008 | -0 0025 | 0 0001 | -0 0031 | 0 0005 | -0 0001 | -0 0022 | 0 0014 | -0 0030 | -0 0002 | -0 0013 | -0 0039 | 0 0009 | -0 0003 | 0 0012 | 0 0030 | 0 0022 | 0 0029 | 0 0023 | -0 0083 * * * | | -0 0027 0 0001 | -0 0038 *| -0 0006 | 0 0003 | 0 0011 | 0 0008 | -0 0121 * * | -0 0062 *| - 0 . 0007 | - 0 . 0017 | 0. 0065 | 0. 0040 | 0. 0019 | 0. 0006 | - 0 . 0035 | - 0 . 0049 *| - 0 . 0119****| - 0 . 0096****| - 0 . 0027 | - 0 . 0033 | - 0 . 0100****| - 0 . 0049 | - 0 . 0004 |  Appendix 17- -1 17- -1 17- -1 17- -1 17- -1 17- -1 17- -1 17- -1 17- -1 17- -1 17- -1 17- -1 17- -1 17- -1 17- -1 17- -1 17- -1 18- -0 18- -0 18- -0 18- -0 18- -0 18- -0 18- -0 18- -0 18- -0 18- -0 18- -0 18- -0 18- -0 18- -0 18- -0 18- -0 18- -0 18- -0 18- -0 18- -0 18- -0 18- -0 18- -0 18- -0 18- -0 18- -0 18- -0 18- -0 18- -0 18- -0 18- -0 18- -1 18- -1 18- -1 18- -1 18- -1 18- -1 18- -1 18- -1 18- -1 18- -1 18- -1 18- -1 18- -1 18- -1  p (42) p (43) p (45) p (47) p (49) p (50) p (51) p (53) p (57) p (58) p (59) p (60) p (61) p (64) p (65) p (67) p (68) R (42) R (43) R (44) R (45) R (47) R (49) R (50) R (51) R (59) R (60) R (61) R (62) R 64) R 72) R 74) R 75) R 77) P 40) P 44) P 45) P 48) P 55) P 57) P 59) P 65) P 68) P 74) P 77) P 79) P 80) P 81) R 3) R 4) R 6) R 7) R 10) R 11) R 12) R 13) R 14) R 15) R 16) R 17) R 18) R 19)  15117 .3829 15113 .5792 15105 . 6798 15097 .4312 15088 8123 15084 3794 15079 8418 15070 5019 15050 7046 15045 5257 15040 2456 15034 8795 15029 4176 15012 4554 15006 6149 14994 6328 14988 4944 15486 7967 15482 9606 15479 0362 15475 0102 15466 6877 15457 9726 15453 4740 15448 8827 15408 6163 15403 1451 15397 5697 15391 8984 15380 2487 15329 5726 15315 8596 15308 8423 15294 4960 15488 2551 15472 5496 15468 3850 15455 3386 15421 5600 15411 0525 15400 1273 15365 0263 15346 1187 15305 5432 15283 8414 15268 8379 15261 1885 15253 4113 15241 5524 15241 2774 15240. 4526 15239 8966 15237 7310 15236. 8290 15235. 8335 15234. 7608 15233. 5898 15232. 3298 15230. 9841 15229. 5477 15228. 0208 15226. 4062  15117 .3874 15113 .5769 15105 . 6865 15097 . 4357 15088 .8225 15084 .3794 15079 .8450 15070 .5010 15050 .7050 15045 .5234 15040 .2482 15034 .8789 15029 .4153 15012 . 4551 15006 6107 14994 6329 14988 4988 15486 7974 15482 9634 15479 0352 15475 0124 15466 6825 15457 9715 15453 4725 15448 8773 15408 6189 15403 1444 15397 5704 15391 8965 15380 2475 15329 5691 15315 8594 15308 8456 15294 4971 15488 2583 15472 5523 15468 3915 15455 3428 15421 5600 15411 0397 15400 1288 15365 0248 15346 1196 15305 5431 15283 8447 15268 8467 15261 1861 15253 4170 15241 5556 15241 2754 15240 4490 15239 9026 15237 7303 15236 8283 15235 8373 15234 7572 15233. 5880 15232. 3295 15230. 9818 15229. 5447 15228. 0181 15226. 4020  0 .0035 0 .0075 0 .0045 0 .0045 0 .0075 0 .0035 0 .0045 0 .0055 0 .0040 0 .0055 0 .0055 0 .0035 0 .0045 0 0045 0 0100 0 0050 0 0100 0 0065 0 0035 0 0055 0 0055 0 0065 0 0065 0 0045 0 0055 0 0055 0 0055 0 0100 0 0055 0 0100 0 0075 0 0055 0 0065 0 0065 0 0075 0 0045 0 0075 0 0065 0 0065 0 0150 0 0100 0 0065 0 0065 0 0065 0 0100 0 0075 0 0055 0 0100 0 0075 0 0050 0 0045 0 0075 0 0040 0 0075 0 0055 0. 0085 0. 0035 0. 0050 0. 0040 0. 0045 0. 0040 0. 0040  -0 .0045 0 .0023 -0 .0067 -0 .0045 -0 .0102 0 .0000 -0 .0032 0 0009 -0 0004 0 0023 -0 0026 0 0006 0 0023 0 0003 0 0042 -0 0001 -0 0044 -0 0007 -0 0028 0 0010 -0 0022 0 0052 0 0011 0 0015 0 0054 -0 0026 0 0007 -0 0007 0 0019 0 0012 0 0035 0 0002 -0 0033 -0 0011 -0 0032 -0 0027 -0 0065 -0 0042 0 0000 0 0128 -0 0015 0 0015 -0 0009 0 0001 -0 0033 -0 0088 0 0024 -0 0057 -0 0032 0. 0020 0. 0036 - 0 . 0060 0. 0007 0. 0007 - 0 . 0038 0. 0036 0. 0018 0. 0003 0. 0023 0. 0030 0. 0027 0. 0042  Appendix 18- -1 18- -1 18- -1 18- -1 18- -1 18- -1 18- -1 18- -1 18- -1 18- -1 18- -1 18- -1 18- -1 18- -1 18- -1 18- -1 18- -1 18- -1 18- -1 18- -1 18- -1 18- -1 18- -1 18- -1 18- -1 18- -1 18- -1 18- -1 18- -1 18- -1 18- -1 18- -1 18- -1 18- -1 18- -1 18- -1 18- -1 18- -1 18- -1 18- -1 18- -1 18- -1 18- -1 18- -1 18- -1 18- -1 18- -1 18- -1 18- -1 18- -1 18- -1 18- -1 18- -1 18- -1 18- -1 18- -1 18- -1 18- -1 18- -1 18- -1 18- -1 18- -1  R (20) R (21) R (24) R (26) R (27) R (28) R (29) R (30) R (34) R (36) R (37) R (38) R (39) R (40) R (41) R (42) R (44) R (45) R (48) R (49) R (50) R (52) R (53) R (55) R 56) R (57) R (58) R (60) R 62) R 64) R 65) R 69) R 70) R 71) R 72) R 73) R 74) R 75) P 5) P 6) P 8) P 9) P 12) P 13) P 14) P 15) P 17) P 18) P 19) P 20) P 21) P 22) P 23) P 25) P 29) P 32) P 38) P 39) P 41) P 48) P 49) P 51)  179 15224 .7013 15222 .9101 15216 .9830 15212 .5749 15210 .2326 15207 .7991 15205 2912 15202 6776 15191 3365 15185 1187 15181 8684 15178 5173 15175 0830 15171 5570 15167 9393 15164 2259 15156 5202 15152 5272 15139 9816 15135 6087 15131 1453 15121 9232 15117 1681 15107 3708 15102 3283 15097 1821 15091 9483 15081 1720 15070 0047 15058 4388 15052 5047 15027 7451 15021 3114 15014 7627 15008 1115 15001 3545 14994 5042 14987 5335 15240 0830 15239 4796 15237 9985 15237 1259 15233 9756 15232 7494 15231 4282 15230 0233 15226 9414 15225 2665 15223 5065 15221 6515 15219 7083 15217 6753 15215 5623 15211 0434 15200 9228 15192 3943 15172 8721 15169 2963 15161 8685 15132 9524 15128 4394 15119. 1424  15224 . 6962 15222 .9007 15216 .9745 15212 .5730 15210 .2366 15207 .8096 15205 2918 15202 6832 15191 3367 15185 1139 15181 8645 15178 5228 15175 0886 15171 5617 15167 9419 15164 2289 15156 5228 15152 5291 15139 9824 15135 6107 15131 1439 15121 9237 15117 1696 15107 3721 15102 3279 15097 1864 15091 9472 15081 1741 15070 0056 15058 4386 15052 5045 15027 7546 15021 3113 15014 7647 15008 1142 15001 3593 14994 4995 14987 5342 15240 0863 15239 4797 15238 0003 15237 1274 15233 9757 15232 7473 15231 4299 15230 0234 15226 9430 15225 2689 15223 5054 15221 6525 15219 7101 15217 6780 15215 5562 15211 0431 15200 9351 15192 4036 15172 8789 15169 3037 15161. 8764 15132. 9504 15128. 4429 15119. 1440  0 .0040 0 0045 0 0045 0 0035 0 0035 0 0060 0 0035 0 0035 0 0030 0 0040 0 0100 0 0065 0 0045 0 0040 0 0055 0 0035 0 0045 0 0040 0 0035 0 0045 0 0055 0 0030 0 0040 0 0045 0 0035 0 0045 0 0065 0 0065 0 0045 0 0035 0 0035 0 0100 0 0045 0 0055 0 0045 0 0050 • 0 0050 0 0055 0 0055 0 0045 0 0040 0 0040 0 0045 0 0050 0 0045 0 0050 0 0100 0 0035 0 0050 0 0045 0 0035 0 0040 0 0040 0 0035 0. 0045 0. 0045 0. 0045 0. 0045 0. 0040 0. 0045 0. 0055 0. 0045  0 0 0 0 -0 -0 -0 -0 -0 0 0 -0 -0 -0 -0 -0 -0 -0 -0 -0 0 -0 -0 -0 0 -0 0 -0 -0 0 0 -0 0 -0 -0 -0 0 -0 -0 -0 -0 -0 -0 0 -0 -o -0 -0 0 -0 -0 -0 0. 0. -0. -0. -0. -0. -0. 0. -0. -0.  0051 0094 0085 0019 0040 *1 0105 * * * 1 0006 0056 • * * | 0002 0048 0039 0055 0056 0047 *1 0026 0030 0026 0019 0008 0020 0014 0005 0015 0013 0004 0043 0011 0021 0009 0002 0002 0095 0001 0020 0027 0048 0047 0007 0033 0001 0018 0015 0001 0021 0017 0001 0016 0024 0011 0010 0018 0027 0061 0003 0123****| 0093****| 0068 * * * 1 0074 * * * j 0079 * * * t 0020 0035 0016  Appendix 18 — 1 18 — 1 18 — 1 18 — 1 18 — 1 18 — 1 18 — 1 18 — 1 18 — 1 18 — 1 18 — 1 18 — 1 18 — 1 18 — 1 19 — 0 19 — 0 19 — 0 19 — 0 19 — 0 19 — 0 19 — 0 19 — 0 19 — 0 19 — 0 19—0 19—0 19 — 0 19 — 0 19 — 0 19 — 0 19 — 0 19 — 0 19--0 19--0 19 — 0 19 — 0 19 — 0 19 — 0 19 — 0 19 — 0 19 — 0 19 — 1 19 — 1 19 — 1 19—1 19 — 1 19 — 1 19 — 1 19 — 1 19 — 1 19 — 1 19 — 1 19 — 1 19 — 1 19 — 1 19—1 19—1 19 — 1 19 — 1 19 — 1 19 — 1 19 — 1  P (52) P (54) P (55) P (58) P (60) P (63) P (66) P (67) P (68) P (69) P (70) P (71) P (72) P (73) R (50) R (51) R (55) R (56) R (57) R (58) R (59) R (60) R (61) R (64) R (67) R 69) R 71) R 72) R 73) R 74) R 76) P 49) P 50) P 51) P 52) P 53) P 55) P 56) P 57) P 68) P 75) R 4) R 6) R 7) R 8) R 10) R 11) R 13) R 15) R 16) R 17) R 18) R 20) R 21) R 22) R 23) R 27) R 28) R 30) R 31) R 33) R 35)  15114 15104 15099 15083 15072 15055 15037 15030 15024 15018 15011 15004 14998 14991 15488 15483 15463 15458 15453 15447 15442 15436 15430 15412 15393 15380 15367 15360 15353 15346 15331 15486 15481 15476 15471 15466 15456 15450 15445 15378 15328 15280 15279 15278 15278 15276 15275 15273 15271 15269 15268 15266 15263 15261 15259 15257 15248 15245 15240 15237 15231 15225  3532 4800 4001 5843 5587 2812 0983 8405 4902 0200 4609 8028 0205 1579 5147 7624 7328 4741 1164 6489 0797 4150 6434 7060 8227 7059 1563 2185 1720 0161 3780 3227 5298 6497 6621 5708 1039 7279 2360 1519 8210 3255 4612 8891 2291 6279 6904 5395 0193 6201 1300 5438 0972 2414 2768 2369 1012 5930 2729 4786 6038 3518  15114 15104 15099 15083 15072 15055 15037 15030 15024 15018 15011 15004 14998 14991 15488 15483 15463 15458 15453 15447 15442 15436 15430 15412 15393 15380 15367 15360 15353 15346 15331 15486 15481 15476 15471 15466 15456 15450 15445 15378 15328 15280 15279 15278 15278 15276 15275 15273 15271 15269 15268 15266 15263 15261 15259 15257 15248 15245 15240 15237 15231 15225  3520 4816 4025 5860 5557 2741 1000 8416 4825 0220 4597 7952 0280 1574 5171 7598 7292 4696 1084 6452 0798 4117 6405 7039 8234 7054 1577 2213 1758 0206 3783 3146 5282 6431 6588 5750 1080 7240 2393 1483 8248 3254 4599 8895 2273 6271 6891 5369 0160 6171 1258 5419 0964 2345 2797 2318 1076 5927 2811 4842 6071 3513  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.  0030 0035 0055 0075 0075 0100 0065 0045 0050 0050 0055 0060 0055 0065 0075 0040 0075 0055 0055 0045 0045 0045 0045 0045 0065 0100 0100 0150 0075 0075 0065 0075 0040 0040 0045 0045 0045 0055 0055 0100 0100 0065 0050 0045 0035 0035 0045 0040 0040 0035 0040 0035 0035 0050 0045 0065 0035 0035 0035 0035 0065 0065  0 -0 -0 -0 0 0 -0 -0 0 -0 0 0 -0 0 -0 0 0 0 0 0 -0 0 0 0 -0 0 -0 -0 -0 -0 -0 0 0 0 0 -0 -0 0 -0 0 -0 0 0 -0 0 0 0 0 0 0 0 0 0 0 -0 0 -0 0 -0 -0 -0 0.  0012 0016 0024 0017 0030 0071 0017 0011 0077 0020 0012 0076 0075 * * 0005 0024 0026 0036 0045 0080 0037 0001 0033 0029 0021 0007 0005 0014 0028 0038 0045 0003 0081 0016 0066 0033 0042 0041 0039 0033 0036 0038 0001 0013 0004. 0018 0008 0013 0026 0033 0030 0042 0019 0008 0069 0029 0051 0064 * * * 0003 0082**** 0056 * * * 0033 0005  Appendix 19- -1 19- -1 19- -1 19- -1 19- -1 19- -1 19- -1 19- -1 19- -1 19- -1 19- -1 19- -1 19- -1 19- -1 19- -1 19- -1 19- -1 19- -1 19- -1 19- -1 19- -1 19- -1 19- -1 19- -1 19- -1 19- -1 19- -1 19- -1 19- -1 19- -1 19- -1 19- -1 19- -1 19- -1 19- -1 19- -1 19- -1 19- -1 19- -1 19- -1 19- -1 19- -1 19- -1 19- -1 19- -1 19- -1 19- -1 19- -1 19- -1 19- -1 19- -1 19- -1 19- -1 19- -1 19- -1 19- -1 19- -1 19- -1 19- -1 19- -1 19- -1 19- -1  R (37) R (38) R (40) R (41) R (42) R (43) R (44) R (46) R (48) R (49) R (50) R (52) R (53) R (54) R (55) R (56) R (57) R (59) R (61) R (62) R (63) R (65) R (67) R (68) R (71) R (73) R (75) R (77) R (78) P 5) P 6) P 7) P 8) P 10) P 11) P 14) P 15) P 17) P 18) P 20) P 21) P 22) P 23) P 24) P 25) P 27) P 30) P 31) P 33) P 34) P 35) P 37) P 39) P 40) P 42) P 43) P 44) P 45) P 46) P 47) P 49) P 50)  15218 7172 15215 2549 15208 0478 15204 2953 15200 4535 15196 4995 15192 4725 15184 1018 15175 3431 15170 8121 15166 1863 15156 6433 15151 7171 15146 6960 15141 5829 15136 3492 15131 0272 15120 0728 15108 7090 15102 8768 15096 9398 15084 7549 15072 1417 15065 6851 15045 6537 15031 7659 15017 4434 15002 6823 14995 1264 15279 1459 15278 5290 15277 8124 15277 0130 15275 1245 15274 0448 15270 2509 15268 8048 15265 6309 15263 9036 15260 1759 15258 1761 15256 0772 15253 8897 15251 6099 15249 2345 15244 2013 15235 9605 15233 0258 15226 8718 15223 6520 15220 3424 15213 4388 15206 1425 15202. 3657 15194. 5068 15190. 4228 15186. 2606 15181. 9782 15177. 6369 15173. 1644 15163. 9548 15159. 2008  15218 15215 15208 15204 15200 15196 15192 15184 15175 15170 15166 15156 15151 15146 15141 15136 15131 15120 15108 15102 15096 15084 15072 15065 15045 15031 15017 15002 14995 15279 15278 15277 15277 15275 15274 15270 15268 15265 15263 15260 15258 15256 15253 15251 15249 15244 15235 15233 15226 15223 15220 15213 15206. 15202. 15194. 15190. 15186. 15181. 15177. 15173. 15163. 15159.  7153 2541 0446 2957 4505 5086 4699 1009 3412 8141 1886 6406 7176 6947 5717 3482 0237 0707 7095 8748 9368 7490 1423 6807 6563 7681 4436 6775 1272 1513 5297 8163 0112 1257 0451 2517 8031 6294 9042 1764 1736 0782 8901 6090 2349 2073 9651 0303 8786 6614 3497 4421 1545 3676 5067 4322 2614 9941 6299 1686 9538 1997  0 .0065 0 .0030 0 .0100 0 .0040 0 .0055 0 .0100 0 .0035 0 .0075 0 .0035 0 .0040 0 0065 0 0055 0 0040 0 0100 0 0100 0 0030 0 0035 0 0035 0 0040 0 0035 0 0065 0 0075 0 0035 0 0055 0 0035 0 0035 0 0050 0 0050 0 0050 0 0050 0 0035 0 0035 0 0075 0 0030 0 0040 0 0030 0 0035 0 0035 0 0035 0 0075 0 0045 0 0035 0 0045 0 0045 0 0060 0 0035 0 0035 0 0045 0 0065 0 0045 0 0045 0 0040 0 0075 0 0035 0 0035 0 0035 0 0065 0. 0100 0. 0100 0. 0045 0. 0065 0. 0035  0 .0019 0 0008 0 .0032 -0 0004 0 0030 -0 0091 0 0026 0 0009 0 0019 -0 0020 -0 0023 0 0027 -0 0005 0 0013 0 0112 0 0010 0 0035 0 0021 -0 0005 0 0020 0 0030 0 0059 -0 0006 0 0044 -0 0026 -0 0022 -0 0002 0 0048 -0 0008 -0 0054 -0 0007 -0 0039 * 0 0018 -0 0012 -0 0003 -0 0008 0 0017 0 0015 -0 0006 -0 0005 0 0025 -0 0010 -0 0004 0 0009 -0 0004 -0 0060 * * * -0 0046 ** -0 0045 -0 0068 * -0 0094**** -0 0073 * • * -0 0033 - 0 . 0120 * * * -0 0019 0. 0001 - 0 . 0094**** - 0 . 0008 - 0 . 0159 *** 0. 0070 - 0 . 0042 0. 0010 0. 0011  Appendix 19 — 1 19 — 1 19 — 1 19 — 1 19 — 1 19 — 1 19 — 1 19 — 1 19 — 1 19—1 19—1 19—1 19—1 19 — 1 19--1 19 — 1 19 — 1 20 — 0 20 — 0 20 — 0 20 — 0 20 — 0 20 — 0 20 — 0 20 — 0 20 — 0 20 — 0 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1  P (51) P (54) P (55) P (56) P (57) P (58) P (59) P (60) P (61) P (62) P (63) P (64) P (66) P (69) P (70) P (71) P (72) R (58) R (60) R (61) R (62) R (63) P 56) P 61) P 63) P 65) P 66) R 3) R 4) R 5) R 6) R 7) R 8) R 9) R 11) R 14) R 15) R 16) R 17) R 18) R 19) R 20) R 21) R 22) R 23) R 25) R 27) R 30) R 31) R 32) R 33) R 35) R 36) R 37) R 39) R 40) R 41) R 42) R 43) R 44) R 45) R 46)  182 15154 15139 15133 15128 15123 15117 15111 15106 15100 15094 15088 15082 15069 15049 15043 15036 15029 15477 15466 15460 15454 15448 15481 15452 15439 15426 15420 15316 15315 15315 15314 15314 15313 15312 15310 15307 15306 15304 15303 15301 15299 15297 15295 15293 15291 15287 15282 15274 15271 15268 15265 15258 15255 15251 15244 15240 15237 15233 15229 15224 15220 15216  3453 2011 9458 6036. 1526 6050 9615 1990 3503 3957 3458 1872 5526 8306 0446 1501 1545 8337 2288 2616 2001 0151 6085 2772 8126 9091 3043 0055 7111 2990 8011 2087 5169 7378 8870 3984 0535 6000 0594 4157 6849 8575 9314 9076 7862 2656 3524 2629 3699 3856 2978 8337 4586 9803 7306 9587 0819 1128 0442 8684 5949 2245  15154 15139 15133 15128 15123 15117 15111 15106 15100 15094 15088 15082 15069 15049 15043 15036 15029 15477 15466 15460 15454 15448 15481 15452 15439 15426 15420 15316 15315 15315 15314 15314 15313 15312 15310 15307 15306 15304 15303 15301 15299 15297 15295 15293 15291 15287 15282 15274 15271 15268 15265 15258 15255 15251 15244 15240 15237 15233 15229 15224 15220 15216  3474 1985 9504 6026 1546 6063 9570 2066 3546 4006 3442 1850 5564 8311 0453 1534 1549 8290 2245 2625 1932 0163 6075 2857 8165 9188 3081 0121 7026 2985 7997 2063 5182 7353 8850 3973 0445 5966 0533 4145 6803 8504 9248 9033 7859 2626 3538 2651 3747 3870 3018 8380 4590 9817 7311 9574 0845 1120 0398 8675 5950 2218  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0040 0035 0045 0035 0045 0035 0065 0075 0100 0065 0040 0035 0075 0035 0085 0035 0035 0030 0055 0065 0065 0065 0050 0075 0075 0075 0075 0065 0035 0045 0035 0045 0035 0040 0040 0035 0055 0035 0045 0035 0040 0040 0045 0050 0065 0075 0045 0035 0035 0035 0035 0035 0035 0040 0040 0100 0075 0045 0055 0040 0035 0035  -0 0 -0 0 -0 -0 0 -0 -0 -0 0 0 -0 -0 -0 -0 -0 0 0 -0 0 -0 0 -0 -0 -0 -0 -0 0 0 0 0 -0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0 -0 -0 -0 -0 -0 -0 -0 -0 0 -0 0 0 0 -0 0  0021 0026 0046 0010 0020 0013 0045 0076 0043 0049 0016 0022 0038 0005 0007 0033 0004 0047 0043 0009 0069 0012 0010 0085 0039 0097 0038 0066 0085 0005 0014 0024 0013 0025 0020 0011 0090 0034 0061 0012 0046 0071 0066 0043 0003 0030 0014 0022 0048 0014 0040 0043 0004 0014 0005 0013 0026 0008 0044 0009 0001 0027  |  *1 | | | |  * 1 | | | |  | | | |  i *** | |  1 * | |  *1 | |  * 1 1 *** | | | | | | |  1 *** | |  1 *' 1 * * *  1 ** | | |  | | |  *1 ** 1 | |  | |  | | | | |  Appendix 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1 20--1 20 — 1 20 — 1 20 — 1 20 — 1 20 — 1 20--1 20 — 1 20 — 1 20 — 1  R (47) R (48) R (50) R (51) R (53) R (54) R (55) R (56) R (57) R (58) R (60) R (62) R (64) R (69) R (70) R (71) R (72) R (73) R (74) R (77) R 79) P ( 5) P ( 6) P 7) P 8) P 9) P 10) P 11) P 12) P 13) P 14) P 15) P 16) P 17) P 18) P 21) P 22) P 23) P 24) P 25) P 26) P 29) P 30) P 31) P 32) P 34) P 35) P 37) P 38) P 39) P 40) P 41) P 42) P 43) P 44) P 45) P 46) P 47) P 48) P 49) P 50) P 51)  15211 .7466 15207 .1687 15197 .7201 15192 8382 15182 7595 15177 5805 15172 .2876 15166 8910 15161 3876 15155 7842 15144 2562 15132 3052 15119 9254 15087 0697 15080 1750 15073 1533 15066 0364 15058 7966 15051 4519 15028 7088 15012 9770 15314 5386 15313 9067 15313 1741 15