HYPERFINE STRUCTURE AND PREDISSOCIATION OF THE B STATE OF BROMINE By James L. Booth B. Sc. (Physics) McGill University, 1983. M. Sc. (Physics) University of British Columbia, 1985. A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF PHYSICS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA March 1994 © James L. Booth, 1994 In presenting this thesis in partial fulfilment of the requirements for degree at the University of British Columbia, I agree that the Library freely available for reference and study. I further agree that permission copying of this thesis for scholarly purposes may be granted by the department or by his or her representatives. It is understood an advanced shall make it for extensive head of my that copying or publication of this thesis for financial gain shall not be allowed without my written permission. (Signature) Department of pri’1 s CS The University of British Columbia Vancouver, Canada Date DE6 (2188) 2- J4 Abstract Investigations have been carried out in bromine of the hyperfine structure of the B H+ and X electronic states and of the predissociation of the B ll+ state by the ‘ll dissociative level. The technique of laser induced fluorescence of a molecular beam was 81 hyperfine spectra were recorded for various B Br used. 79 11 through 17 and v” — X vibrational bands 0, 1, and 2, and for various rotational (v’ f—v”) with v’ = transitions (J’ 2 Br J”) with J’ from 0 to 11 and J” from 0 to 10. As well, the 79 i— 2 hyperfine spectra of the (13’ Br and 81 — = 0”) and (17’ — 2”) bands over the same range of rotational states were measured. The spectra are well described using one X state parameter: the electric quadrupole coupling constant eqQx; and two B state parameters: the electric quadrupole coupling constant eqQB, and the nuclear spin-rotation constant Br) 79 Csr. The results show that eqQB( = (177.0 ± 0.6) MHz for v’ = 11 and increases by approximately 0.5 MHz per vibrational quantum up to (180.6 ± 1.4) MHz for v’ Br) T9 Similarly the ground state electric quadrupole coupling constant, eqQx( 17. (808.1 ± 1.4) MHz for v” = 0 and increases by about 1 MHz per vibrational quantum to (811.4 ± 1.4) MHz for v” = 2. The hyperfine data also provided a check on the accuracy of some of the published rovibronic constants’ for each isotopomer. In order to reproduce the observed relative spacings of the transitions for all three isotopomers, the published 00 have to be modified ; this can be done by decreasing the published T term values, , 1 by (177 ± 8) MHz and (326 ± 8) MHz, respectively. Br 8 r 2 and 79 Br 0 for 81 T values of 0 The phase shift technique was applied to the study of the predissociation of the v’ 13 B = electronic state of bromine. The lifetimes of individual hyperfine levels were ‘S. Gerstenkorn and P. Luc, J. Phys. France 50, 1417 (1989). 11 measured for the rotational states J’ = 0 — 7 (except for J’ 2) for each isotopomer of bromine. Revised values are given for the radiative decay rate Frad, the gyroscopic predissociation parameter C, and the magnetic dipole predissociation parameter a. The first observation of electric quadrupole predissociation is reported and is characterized by a new molecular parameter, b. 111 Table of Contents Abstract List of Tables vii List of Figures xiv Acknowledgements xv Introduction 1 1.1 6 1 2 Thesis Organization General Theoretical Considerations 2.1 2.2 The Labelling of Electronic Wavefunctions 10 2.1.1 11 Electronic State Labels Hund’s Coupling Cases 13 2.2.1 Hund’s Case (a) 13 2.2.2 Hund’s Case (c) 2.3 Electronic symmetries 2.4 Nuclear Spin 2.5 Hyperfine Interactions • . . • . . 15 16 17 • . • . . Calculation of Hyperfine Energies • . . 2.6.1 Hyperfine Interactions Between the Electrons and the Nuclei • • . 2.6.2 Hyperfine Interactions Between the Nuclei • • . 2.5.1 2.6 8 Hyperfine Symmetries iv 20 21 24 25 29 2.7 3 Transition Intensities 32 2.7.1 Allowed Transitions 33 2.7.2 Fluorescence 35 Predissociation 42 Theory 44 3.1 3.1.1 Gyroscopic predissociation 45 3.1.2 Hyperfine predissociation 46 3.1.3 Special Cases 54 4 General Experimental Considerations 59 5 Hyperfine Structure of Bromine 66 6 5.1 Experimental Details 68 5.2 The Model 74 5.3 Results and Discussion 77 5.4 Observation of Near Coincident Transitions 84 Natural Predissociation of the B state of Bromine 92 6.1 Saturation Tests 6.2 Polarization Effects 103 6.3 Relative Intensity Measurements 105 6.4 The Lifetime Plan 108 6.5 The Phase Shift Method 114 6.6 Full Theory 119 6.7 Experimental Arrangement 125 6.7.1 6.8 96 Fitting the Hyperfine Lifetimes Discussion 127 136 V 6.9 7 A Closer look at the Molecular Parameters 145 6.9.1 Franck—Condon Factors 145 6.9.2 The Cyroscopic Predissociation Parameter 148 6.9.3 Quadrupole terms 149 Conclusions 151 7.1 153 Future Work Bibliography 155 Appendices 157 A Derivation of Predissociation Rates 157 A.1 Hyperfine Predissociation Rates 161 A.2 Gyroscopic Predissociation Rates 173 A.3 Hyperfine 175 — Gyroscopic Interference terms A.4 Putting it all Together 176 B Hyperfine Data 180 C Phase Shift Data 193 vi List of Tables 2.1 The selection rules for the magnetic dipole, 1 (H ( x), and 1 H ( 1, 2)) and the electric quadrupole 2 (H ( x), and 2 H ( 1,2)) hyperfine Hamiltonians 2.2 The calculated fluorescence signals with two different laser polarsizations for the R(O) and P(4) hyperfine transitions of 79 2 Br 3.1 41 The selection rules for three Hamiltonians: the gyroscopic, HG, the hyper fine magnetic dipole, H’(x), and hyperfine electric quadrupole, 2 H ( x).. 3.2 57 A list of the rovibronic transitions whose hyperflne spectra were recorded in this study 5.2 48 The predissociation rates for several the odd—J’ (para-bromine) F hyper fine levels including up to the electric quadrupole terms 5.1 31 73 The electric quadrupole parameter for the (13’ — 0”) and (17’ — 2”) bands 2 and 81 Br of 79 2 Br 77 5.3 The B state eqQ parameters for each isotopomer 78 5.4 The X state eqQ parameters for each isotopomer 79 5.5 A comparison of the observed electric quadrupole coupling constants, eqQ( Br), 79 for different vibrational levels of the B ll+ state 5.6 A comparison of observed electric quadrupole coupling constants for dif ferent vibrational levels of the X 5.7 79 1 state of 79 2 Br 80 Br) values for the B state vibrational levels recorded as mea 79 The eqQ( sured and calculated from Equation 5.12 vii 81 5.8 The eqQ( Br) values for the X state vibrational levels recorded as ob 79 served and calcuated from Equation 5.13 5.9 81 The observed B state 79 eqQ( values from Reference [6] and the corre Br) sponding values calculated from Equation 5.14 82 5.10 The electric quadrupole parameters reported by Ref. [5] for the B state of bromine 83 5.11 The observed hyperfine free separations, LS’, of the transitions listed com pared with the theoretical values based upon the rovibronic constants of Reference [7] 86 5.12 The observed hyperfine free frequency separations, /.v, for some P(J) R(J+3) lines of the (17’ — 2”) band of 79 2 Br , 2 B 8 r , and 79 81 Br . — . 87 5.13 The difference between the observed and calculated values of a 88 5.14 The observed and calculated values of 3 88 5.15 Observed hyperfine free frequency differences, v, between rovibronic transitions for different isotopomers as compared with the calculated dif ferences based upon Reference [71 89 5.16 The values of the B state term values reported by Reference [7], (B), 00 T and the values deduced from this work, 0 T ( B) 6.1 91 The Franck—Condon factors for various B—X vibrational transitions for 2 as given by Reference [38] and the calculated saturation laser powers Br 79 based on Equation 6.6 6.2 The intensity ratios of the B ( F” = 2 —* F’ = 99 — 1) to (F” X R(O) (13’ = 0 —* F’ — = 0”) hyperfine transitions a:b 1) for 79 2 with the laser Br polarized vertically (z-direction) and horizontally (y-direction) yin 105 6.3 The intensities, S, of the two 81 2 B—X (5’— 1”) P(l) hyperfine transitions Br as a function of the laser beam 6.4 = — 0”) hyperfine J 130 Prad, and the predissociation constants, C, a, and b 132 The results of fitting the 79 81 lifetime data are compared with the Br predictions based on the results for 79 2 and 81 Br 2 Br 6.7 133 The results of the global eight parameter fit to lifetimes for 2 81 , Br 79 81 Br and 79 6.8 107 The results of fitting the 79 2 and 81 Br 2 lifetime data separately for the ra Br diative decay rate, 6.6 molecular beam interaction position, X The total decay rates for various para bromine B—X (13’ states having F 6.5 — 135 The ratio of the parameters B :A and B 2 :A as compared to the ratios of 3 magnetic dipole moments and electric quadrupole moments for 79 Br and Br 81 6.9 135 The results of the global six parameter fit to lifetimes for 79 2 Br , and 79 81 Br 136 6.10 A comparison of the molecular parameters derived for the B ll+ v’ level of 79 2 Br , 2 Br 81 , 6.11 •The values for C and = 13 and 79 81 with three different models Br “rad 137 for this work compared with References [13,16] (Clyne and co-workers) and [14] (Peeters et al.) 141 6.12 The predissociation parameters reported by References [3,4] compared to the results of this work 144 6.13 The ratios of the vibrational overlap integrals and Franck—Condon factors (FCF) between the B ll+ and ‘H electronic states for the different isotopomers of bromine 145 ix 6.14 The ratios of the vibrational overlap integrals times reduced mass Z(X, 79) and 2 Z ( X, 79) between the B ll+ and ‘ll electronic states for the dif ferent isotopomers of bromine 146 6.15 The ratios of the vibrational overlap integrals and FCF’s for 79 2 and Br 2 as deduced from (I) the global fit results, and (II) the J’ Br 81 = 0 hyperfine level decay rates 147 6.16 The the overlap integral, (a, b) and FCF between the B electronic states for v’ = and [f 13 148 6.17 The electronic matrix elements between the B ll+ and 1fl matrix ele ments involved in the gyroscopic predissociation term: a comparison be tween the current results and those of Ref. [13] as corrected here 149 6.18 The effective quadrupole coupling constant between the ‘H and B states compared with the electric quadrupole coupling constant of the B state of bromine 2 (17’ Br B.1 The 79 — 150 2”) P(1) observed and calculated hyperfine transition frequencies for separated data sets (part I) B.2 The 79 2 (17’ Br — 2”) P(1) observed and calculated hyperfine transition frequencies for separated data sets (part II) B.3 The 79 2 (17’ Br — — — 184 2”) P(1) observed and calculated hyperfine transition frequencies for separated data sets (part I) B.5 The 81 2 (17’ Br 183 2”) P(1) observed and calculated hyperfine transition frequencies for separated data sets (part III) B.4 The 81 2 (17’ Br 182 185 2”) P(1) observed and calculated hyperfine transition frequencies for separated data sets (part II) x 186 B.6 The 79 81 (17’ Br — 2”) P(1) observed and calculated hyperfine transition frequencies for separated data sets B.7 A summary of the 79 2 (17’ Br B.8 A summary of the 81 2 (17’ Br — — 2”) P(1) data results 188 2”) P(1) data results 188 B.9 A summary of the 79 81 (17’ Br 2 (13’ Br B.l0 A summary of the 79 187 — 2”) P(1) data results 189 0”) P(2) data results 190 B.1l A summary of the 81 2 (13’— 0”) P(2) data results Br 191 81 (13’ Br B.12 A summary of the 79 192 C.1 — — 0”) P(2) data results The deduced lifetimes of the v’=13, J’ = 1, F’ = 1_ hyperfine state using a modulation frequency of 18.48 kHz 196 C.2 The deduced lifetimes of the v’=13, J’ 1, F’ = 1_ hyperfine state using a modulation frequency of 25.49 kHz 197 C.3 The deduced lifetimes of the v’=13, J’ 1, F’ = 1_ hyperfine state using a modulation frequency of 35.97 kHz 198 C.4 Coefficients, Ri’(x,y) for calculating of 1’ for 79 2 Br 199 2 Br C.5 Coefficients Rit(x, y) for calculating 1’ for 81 200 C.6 Coefficients Rij’(x, y) for calculating of 1’ for 79 81 (part I) Br 201 C.7 Coefficients Riji(x, y) for calculating r for 79 81 (part II) Br 202 C.8 The observed and calculated inverse lifetimes, I’, for 79 2 Br 204 C.9 The observed and calculated inverse lifetimes, 1’, for 81 2 Br 205 C.10 The observed and calculated inverse lifetimes, F, for 79 81 Br 206 C.11 The observed and calculated lifetimes, r, for 79 2 Br . 207 C.12 The observed and calculated lifetimes, T, for 81 2 Br . C.13 The observed and calculated lifetimes, T, for 79 81 Br xi 208 209 List of Figures 2.1 A schematic diagram of two bound electronic state energy potentials. 2.2 Hund’s coupling case (a) 14 2.3 Hund’s coupling case (c) 15 2.4 The effect of interchanging nuclear spin and spatial coordinates 18 2.5 The laser—detector—emitter system with the laser beam propagating along 9 the X axis, the molecular beam traveling along the Y axis and the detector situated along the Z axis 34 2.6 The laser—detector—emitter system using a different coordinate system 37 2.7 2 B—X (13’ Br The simulated 79 — 0”) R(0) hyperfine fluorescence spectra with the laser polarized along the Z-axis (Sz dotted line) and along the Y-axis (Sy solid line) 40 3.1 Crossing bound and dissociative potentials 43 3.2 The hyperfine structure and spectrum characteristic of para-bromine. 4.1 The molecular beam machine 5.1 A schematic of the apparatus used for studying the hyperfine structure of . 63 bromine 5.2 70 A schematic of the optical collection system used in the hyperfine structure studies 6.1 56 71 A two polarizer (P1, P2) arrangement for maintaining the direction of polarization of a laser beam while varying the beam’s intensity xii 97 6.2 The polarizer—Pockels cell (PC) arrangement for varying the intensity of a laser beam while maintaining the direction of polarization of the trans mitted light 6.3 The observed laser power dependence of the signal intensities of two 79 2 Br B—X (13’ F’ 6.4 98 = — 0”) P(4) hyperfine transitions: (a) the LF 3_ transition and (b) the F J F” = 4 — F’ The observed signal intensities of the 79 2 B—X (16’ Br = — = zJ F” = 4 4 transition. 100 . 1”) P(4) (a) and (b) hyperfine transitions as a function of laser power 6.5 The observed signal intensities of the 79 2 B—X (17’ Br 101 — 2”) P(4) (a) and (b) hyperfine transitions as a function of laser power 6.6 The observed 79 2 B—X (13’ Br — 102 0”) R(0) hyperfine spectrum with the laser polarized vertically (z-direction) and horizontally (y-direction) 104 6.7 Illustration of the design principle used for the final optical stack 109 6.8 A pair of identical lenses separated by 2 focal lengths, 2f, produces the image of an object 4f away from the object independent of changes in the 6.9 position of the object with respect to the lenses 110 The new optical arrangement 111 6.10 The response of the optical collection system across the viewing area. . . 112 6.11 The two types of measurements used to deduce the molecular beam veloc ity. 115 2 (13’—O”) P(1) hyperfine spectra using the initial optical Br 6.12 The observed 79 collection system (A) and the redesigned system (B) 116 6.13 A schematic diagram of the laser beam and optical collection region with respect to the molecular beam 119 6.14 The apparent lifetime as a function of the viewing length along the molec ular beam that is imaged 123 XII’ 6.15 The apparent lifetime as a function of the viewing length and molecular beam velocity for a state with an 8ps lifetime 124 6.16 A schematic of the apparatus used for measuring the lifetimes of individual hyperfine levels 126 6.17 The hyperfine spectrum for the B—X (13’ — 0”) P(4) transition of 79 2 Br observed with laser modulated at 25.5 kHz C,1 A simulated hyperfine absorption spectrum for 79 2 B—X (13’ Br transition. 129 — 0”) P(2) 194 xiv Acknowledgements This thesis, like most, began by asking one question and ended up answering another, quite different one. The success of this work is, in no small way, due to the many excellent people I had guiding, urging, helping, and, when necessary, cajoling me along the way. I would like to express my sincere thanks to my supervisor, Dr. F. W. Dalby, for his 99% crazy and 1% inspired suggestions, for his encouragement and guidance. It was a pleasure and privilege working with him. I am also very grateful to Dr. I. Ozier for his constant help and patience throughout this work. His suggestions during the research and the writing of this thesis (however popular or unpopular at the time!) proved invaluable. I am indebted to Dr. A. G. Adam for introducing me to the molecular beam machine and teaching me the necessary skills to operate the apparatus. I would also like to thank Dr. A. J. Merer for the loan of equipment and lab space necessary for this research and for the very helpful discussions. My thanks as well to Dr. L. Whitehead for the conversations about the “tricks of the trade” when designing optical systems. The refinements to the optical collection system used in this work were a direct consequence of our discussions. I would also like to thank (proto-Dr.) A. Chanda with whom I have shared the lab for several years. His encouragement and help were much appreciated. I would like to thank the large number of students and researchers who made the lab much more enjoyable. In particular, Dr. W. Ho, K. Mah, R. Abraham, J. Spencer, P. Tang, Q. Deng, J. Zhou, S. Wang, and C. Boone have all contributed to making the working environment lively and enjoyable. Finally, I would like to thank P. LeCerf for constant encouragement and support. xv Chapter 1 Introduction The study of the excited electronic states of the diatomic halogens has been an active area of research over the past two decades. In particular, work on molecular iodine has been extremely rewarding. Researchers have used iodine as a test molecule for developing new spectroscopic techniques, for observing predicted but previously unobserved effects, for refining spectroscopic theory, and, in the process, have discovered new, unexpected phenomena. Molecular bromine has been somewhat less studied owing to its weaker absorption and stronger predissociation. However, bromine has several features that make it more attractive to the researcher. First, atomic bromine occurs in two isotopic forms, 79 Br and Br, in almost equal abundance. The molecular species, 79 81 2 Br , 81 Br 79 , and 81 2 Br (called isotopomers) appear in the ratio 1:2:1, presenting an opportunity to study dif ferent molecular systems with very similar properties. Second, the nuclear spin of both the isotopes of bromine is 3/2 as compared to 5/2 for iodine. Therefore the hyperfine structure of molecular bromine is less complicated and insights into the dynamics of the excited molecules may be easier to glean. Finally, the radiative lifetime of the B ll+ electronic state of bromine was reported to be at least three to four times longer than that of the B state of iodine. With the longer lifetime, it is easier to observe inter actions of the B l1+ state with other electronic levels. Understanding these interactions of electronic levels, especially the interactions of dissociative and bound electronic states (e.g. predissociation) has some important implications. As was observed for molecular 1 Chapter 1. Introduction 2 iodine, predissociation occurs through several different physical mechanisms, one of which arises through hyperfine interactions. This type of predissociation affects the different hyperflne levels of the molecules by different amounts. Therefore, if one understands the mechanisms that lead to predissociation then one could, perhaps, manipulate the predissociation or inverse-predissociation, so as to prepare molecules of bromine in spe cific states for further experimentation. In addition, for atoms in optical traps, their recombination into molecules represents one way in which the atoms can leak out. With an understanding of the recombination and predissociation processes one may be able to manipulate the conditions in the trap so as to hold the atoms for a longer period of time. This thesis presents the results of work on the hyperfine structure of the X and B ll+ electronic states of bromine and on the predissociation of the B state by the dissociative 1 fJ state. The hyperfine structure of the X and B states (both are = 0+ levels) is determined primarily by electric quadrupole interactions (characterized by the parameters eqQx and eqQB. respectively) plus a small contribution from the B state nuclear spin — rotation interaction (characterized by the parameter, Csr). For all three isotopomers, hyperflne spectra were recorded of the (13’—O”) band for the rotational transitions P(1) — R(3) and R(l0). For the (17’ P(l) — — — P(7), excluding the P(6), and R(0) 0”), (14’ — 1”), (15’ — P(5), R(0) 2”) band, hyperfine transitions were measured for the 81 , the hyperfine spectra of R(0) Br 79 (12’ — — 1”), and (16’ — R(10), excluding the R(9). In addition, for R(2) and P(1) were measured for the (11’ —0”), — 1”) bands. Of particular importance were the J (so-called cross over) transitions which allowed one to separate the hyperfine parameters for each electronic state. The results indicated that, for both the X and B electronic states, the ratio of the molecular electric quadrupole coupling constants, 2 Br 79 eqQ[ 2 Br 81 ]/eqQ[ 1’ agreed with the ratio of the nuclear quadrupole moments, Br), was (177.0 ± 0.6) MHz for 79 Q(79)/Q(81). For the B state, the parameter, eqQB( Chapter 1. Introduction v’ = 3 11, and increased by about 0.5 MHz per vibrational quantum up to (180.6 ± 1.4) MHz for v’ = 17. The X state had eqQx( Br) 79 = (808.1 ± 1.4) MHz increasing by about 1 MHz per vibrational quantum up to (811.5 ± 1.9) MHz for v” = 2. These results were in excellent agreement with the findings of previous workers [1, 2, 3, 4, 5, 6]. The study of the hyperfine structure of each isotopomer of bromine provided a second type of information. Because the B—X spectrum of bromine is dense, many near coin cidences of transitions from different levels within an isotopomer and from levels from different isotopomers were observed. A study of the Doppler—limited B—X spectrum of 2 had been carried out earlier by Gerstenkorn and Luc [7]. Analysis of the data Br 79 provided the rovibronic (i.e. rotational, vibrational, and electronic) constants for the B and X states of 79 2 Br . Using isotopic relations, the authors of Reference [7] also calcu lated the corresponding parameters for 81 2 and T9 Br Br. By measuring the frequency 81 Br separations of the nearly coincident transitions in the current work, the accuracy of the molecular parameters was tested. The results of the hyperfine studies presented here indicated that the rovibronic parameters of Reference [7] reproduce the the observed frequency separations of transitions arising from levels of the same isotopomer to ± 5 MHz. The observed frequency separations of nearly coincident transitions arising from levels belonging to different isotopomers led to the conclusion that the term values for the different isotopomers, T , 0 0 reported by Gerstenkorn and Luc [7] had to be adjusted. Because the reported 79 2 constants were derived directly from observed data while Br the constants reported for 81 2 and 79 Br 81 were deduced using isotopic relations, 0 Br T 0 for 79 2 was kept fixed and T Br 00 for 81 2 and 79 Br 81 were adjusted. The new term Br values are (177 ± 8) MHz and (386 ± 8) MHz lower than those reported by Gerstenkorn and Luc [7] for 79 81 and 81 Br 2 Br , respectively. With these new constants, the fre quency separations between spectral features arising from different isotopomers should be accurate to ±9 MHz (±0.0003 cm’). Chapter 1. Introduction 4 The fluorescence from many of the rovibrational levels of the B ll+ state of bromine is diminished owing to the phenomenon of predissociation. Predissociation occurs when a bound and unbound electronic state are coupled allowing some of the molecules in the bound state to fall apart into constituent atoms. The decay rate, 1’, of a level subject to predissociation is the sum of the radiative decay rate, ’rad, 1 and the predissociation rate, P (neglecting other decay channels such as collisions and stimulated emission). The strength of the predissociation rate depends upon the types of coupling and is strongest in the region where the two electronic potentials cross. The theory of the predissociation of the B ll+ state of molecular iodine by a disso ciative ‘ll state has been worked out in detail [8]. In this theory, the predissociation arises primarily from terms in the rotational Hamiltonian (gyroscopic terms), from inter actions between the electrons and nuclei in the hyperfine Hamiltonian (hyperfine terms), and from the interference of these two types of coupling. The gyroscopic terms couple all of the hyperfine levels associated with a specific rovibronic state to the continuum by the same amount. By contrast, the hyperfine and interference terms lead to predissociation rates which vary from hyperfine level to hyperfine level. The hyperfine operators arise from magnetic dipole interactions, electric quadrupole interaction, magnetic octupole in teractions and so on. Previous studies have determined that only the gyroscopic terms (characterized by the parameter Cv), and the magnetic dipole terms (characterized by the parameter a) are necessary to explain the observed predissociation rate of the B state of iodine [8, 9, 10, 11, 12]. A variety of techniques has been used to determine the predissociation of a state: (i) direct lifetime measurements of rotational levels (this yields an accurate determination of C) [13, 14]; (ii) direct lifetime measurements of hyperfine levels (to give “rad, C..,, and a) [12]; (iii) observations of the linewidths of hyperfine transitions (giving “rad, C, and a) [11]; (iv) measurements of relative intensities of hyperfine spectral features (producing Chapter 1. Introduction 5 C/jT and av/VT) [3, 4, 9]. The predissociation of the B state of bromine has been investigated by several different researchers [3, 4, 13, 14] using techniques (i) and (iv). The reported values of the parameters, ’racl, 1 C.., and a show a great deal of variation from research group to research group, and in the case of References [3] and [4], from one rotational state to another. In particular, the radiative lifetime has been reported to lie between 1.9 and 15.7 s [3, 13, 14, 15, 16]. In this thesis, a different experimental approach has been taken to studying the pre dissociation of the B state of bromine. A technique known as the phase shift method was successfully applied to the determination of the lifetimes of individual hyperfine levels. In this method, bromine molecules in a molecular beam were excited by a narrow band tunable laser. The laser’s intensity was modulated at an angular frequency approaching the total decay rate of the level being studied. The resulting laser induced fluorescence signal was shifted in phase with respect to the laser’s intensity modulation signal. The amount of phase shift provides a measure of the total decay rate, F, of the state. For ex ample, a phase shift of 450 is observed when F equals the angular modulation frequency. This is the first time that the phase shift method has been applied to the measurement of lifetimes of individual hyperfine levels. The decay rates of the v’ = = 13 B state hyperfine levels of J’ = 0 through 7 (excluding J’ 2) were measured for each isotopomer of bromine. The previous predissociation theory including only the magnetic dipole hyperfine terms proved to be inadequate to describe the data. In this thesis the theory has been expanded to include the electric quadrupole terms. This led to the introduction of a new molecular parameter, b, corresponding to electric quadrupole predissociation. Revised values of Frau, C..,, and a are reported along with the values of b for each isotopomer. For the first time, a determination of a quadrupole coupling constant between a bound and unbound electronic state has been ni ade. Chapter 1. Introduction 1.1 6 Thesis Organization This thesis is organized into 7 chapters. The theoretical concepts are covered in Chapters 2 and 3 while the experimental details and results are found in Chapters 4 through 6. Chapter 7 contains a summary of the findings and some suggestions for future work. Chapter 2 presents an overview of the basic ideas of the spectra of diatomic molecules including the Born—Oppenheimer approximation, symmetry considerations and labelling of energy states, a brief treatment of the rovibronic energy levels, and a description of hyperfine interactions and energy levels. Chapter 2 ends with a discussion of the fluores cence intensities of hyperfine transitions and their dependence upon the laser polarization as compared to absorption line intensities. Chapter 3 introduces the topic of predissociation and presents the theory of natural predissociation of Vigué et al. [8] The theory has been expanded to include heteronuclear molecules. The chapter includes a brief discussion of which hyperfine states are the most sensitive to the various predissociation effects. Chapter 4 discusses the molecular beam machine and laser system common to both the hyperfine structure studies and the predissociation work. Chapter 5 contains the specific optical collection scheme and frequency calibration system used along with the results and discussion of the hyperfine structure of the B and X states of bromine. It concludes with a discussion of the accuracy of the rovibronic constants of Reference [7] and suggests that small corrections should be added to the B state term values to reconcile the calculated and observed transition frequencies. Chapter 6 details the study of the natural predissociation of the v’=13 level of the B ll÷ electronic state of bromine. The various saturation and laser polarization consider ations required for relative intensity measurements are pointed out as are the anomalous results that were originally observed. These were remedied by a redesign of the optical Chapter 1. Introduction 7 collection system. Measurements of the lifetimes of individual hyperfine levels were per formed using the phase shift method. The theory of the phase shift method is presented and the final results for the radiative decay rate, and for the gyroscopic and hyperfine predissociation parameters are given and discussed. Chapter 2 General Theoretical Considerations In this chapter, the main theoretical considerations for unravelling the rovibronic and hy perfine spectra of diatomic molecules will be presented. For more complete treatments of the subject the readers are referred to [17, 18], two of the excellent texts on spectroscopy. The diatomic molecule may be conceptualized as a dumb-bell; the nuclei are separated by a distance RN and surrounded by an electron cloud. The nuclei rotate and vibrate while interacting with the electrons. The energy levels of the molecules are determined from the time-independent Schrödinger equation, HI’) = (2.1) EIi,b) where H is the Hamiltonian describing the system, ) represents the eigenfunction cor responding to eigenenergy E. The Hamiltonian of the system can be written as H where T and N T = Te+TjJ+Vee(re)+VNe(re,RN)+VNJ\T(RN) (2.2) are the kinetic energies of the electrons and nuclei respectively; Vee(re) represents the electron—electron interactions; Vrv(r, RN) represents the electron—nuclear interactions; and VNN(RN) represents the internuclear interactions. (re and RN represent the electron and nuclear coordinates, respectively.) In order to simplify the solution of Equation 2.1, one uses the Born-Oppenheimer approximation (BOA). The BOA relies on the fact that the electrons are much lighter than the nuclei and move much more rapidly. Therefore the orbital frequencies of the electrons are much higher than the characteristic 8 Chapter 2. General Theoretical considerations 9 E R Figure 2.1: A schematic diagram of two bound electronic state energy potentials. The separation of the electronic states, Ee is much larger than the separation of the vibra tional levels, zE which is, in turn, larger than the rotational state separations, Er. frequencies of the rotation and vibration of the nuclei. The wavefunction is factored as I) = (2.3) RL’e)Itt’N) The electronic motion is deduced from the Schrödinger equation keeping the nuclei fixed and treating the internuclear distance, RN, as a parameter. [‘Te + Vee + VNejIbe) = Ue(RN)I’çbe) (2.4) Ue(RN) is then put back into the the nuclear Schrödinger equation. [T + VNN + Ue(RN)lkbN) = EkhN) (2.5) Ue(RN) and VNN provide a background potential in which the nuclei vibrate and rotate. Two typical potentials are illustrated in Figure 2.1. To solve Equation 2.5, the nuclear wavefunction is factored into vibrational, rotational, and nuclear spin functions as l1I)N) = Iv)IhI’r)IXN) (2.6) Chapter 2. 10 General Theoretical Considerations The total energy of the molecule, excluding nuclear spin interactions and terms that couple different electronic states, is resolved as: E = (2.7) T + G(v) + F(J) where T is the electronic term energy, G(v) is the vibrational energy of the state labelled by the quantum number v, and F(J) is the rotational energy of the state labelled by the quantum number J. The subscript v in the rotational energy term takes into account the coupling between the vibrational and rotational motions of the nuclei. In general [17], 1 G(v) 12 We(V+)WeXe(T+) 13 +Weye(V+) 14 +wz(v+) +,.. (2.8) F(J) = BVJ(J + 1) — DV(J(J + l))2 + HV(J(J + i)) (2.9) with = e(V+)2+ 7 Be_e(V+)+ (2.10) (2.11) Typically, Te >> We >> Be. That is, the separation between different electronic states is much greater than the separation of vibrational states which is, in turn, larger than the separation of neighbouring rotational states. This is shown schematically in Figure 2.1. The energy of an electronic-vibrational-rotational level is known as the rovibronic energy. 2.1 The Labelling of Electronic Wavefunctions An infinite number of electronic potentials are possible depending upon the configurations of the valence electrons. These potentials may be bound or repulsive; the latter are Chapter 2. General Theoretical Considerations 11 called dissociative and molecules excited into these potentials break apart into constituent atoms. When several molecular electronic states are known, the letters X, A, B, h, c, ... ... and a, are used to label them. The state lowest in energy is traditionally called X. In itself, this labelling is insufficient to distinguish all the states and determine the allowed transitions from one electronic state to another. In order to do better, one must consider the symmetry properties of the electronic energy levels. 2.1.1 Electronic State Labels The diatomic molecule, here visualized as a dumb-bell, has cylindrical symmetry; i.e. there is a symmetry axis along the line joining the two nuclei. As pointed out in Ref erences [17, 18], the electric fields in the molecule are symmetric about the internuclear axis. This leads to a precession of the total orbital angular momentum of the electrons, L , about the internuclear axis. Thus, in general, L, is not conserved but its projection along the internuclear axis, L , (with eigenvalue ML) is. The states with the same absolute value of ML have the same energy. Using the convention of Reference [18] the label A with A = ML is used to classify the different electronic states. (As is evident, states 0 are doubly degenerate.) By analogy with atomic spectroscopy the electronic states are labelled according to: Chapter 2. General Theoretical Considerations Atomic 12 Molecular L Label Al Label 0 S 0 E 1 P 1 II 2 D 2 3 F 3 4 G 4 1’ Similarly, the total electronic spin angular momentum, S , is the vector sum of the spins of the individual electrons. The corresponding quantum number, S, is integral or half—integral depending upon the whether the total number of electrons in the molecule is even or odd [17]. S is coupled to the internuclear axis by the magnetic field resulting from the orbital motion of the electrons. S precesses about the internuclear axis and its projection along the internuclear axis, S, (with eigenvalue M) is conserved. By convention E = M, which may take on 2S+l different values. The quantity, 1, is defined as = lA+l (2.12) In general, the different electronic states are labelled with the electron spin multiplicity, 2S+l, indicated as a superscript to the left of the tAt label and 2 S +l is added as a subscript. Al (2.13) This labelling is sufficient for the most commonly encountered singlet, doublet, and triplet states. However, for quartet or higher spin multiplicity electronic states the label may be insufficient to distinguish all of the different components. For example, consider a 4l state; here At + E f = 5/2, 3/2, 1/2, and —1/2 are the four distinct components. Chapter 2. General Theoretical Considerations 13 Using Equation 2.13 to denote the states leads to the same label for the 1/2 and —1/2 components, even though they are distinct. Therefore, in such cases it is preferable to list the levels as, S+lIAI 2 2.2 (2.14) Hund’s Coupling Cases In addition to the electron orbital, L , and spin, S , angular momenta, diatomic molecules possess the nuclear end over end rotation, R , and nuclear spin, I , angular momenta. Neglecting the nuclear spin for the moment, there are various different schemes for cou pling L S and R together to form J , the total angular momentum exclusive of nuclear spin. In general, J Usually, R , L , and S = R+L+S (2.15) are not conserved. One describes the molecules in terms of the eigenvalues (represented by quantum numbers) of conserved quantities. The most appropriate angular momentum quantum numbers with which to describe the molecular system depend upon the manner in which the various momenta are coupled together. The different types of idealized couplings were first treated by Hund and are called Hund’s coupling cases [17, 18]. The specific case applied to a given molecule depends upon the relative strengths of the couplings of the various angular momenta to the internuclear axis and to each other. For the purposes of this thesis only cases (a) and (c) will be briefly presented. 2.2.1 Hund’s Case (a) Hund’s coupling case (a) is characterized by both L and S being strongly coupled to the internuclear axis. Thus, the operators L and S (the projections of electron orbital and Chaptr 2. General Theoretical Considerations 14 3 R z S L Figure 2.2: Hund’s coupling case (a). The electron orbital angular momentum, L , and spin, S , are strongly coupled to the internuclear axis (Z axis) making their projections, A and E, respectively, good quantum numbers. 1 (see text) is then coupled to the rotation of the nuclei, R , to form J spin angular momenta on the internuclear axis, respectively) have well-defined quantum numbers A and E. In this description the mixing of different electronic levels is ignored; a vector Z is defined whose magnitude is IA + El and whose direction lies along the internuclear axis. (See Figure 2.2.) IZ and R are coupled together to form J. The energy levels are labelled by the quantum numbers J, A, E, and Il. The basis functions are written as fryASEJ1) (2.16) (Here -y is used as a label for the rest of the quantum numbers that have been omitted.) In this case the description of the electronic levels given in Equation 2.14 is the appropriate one. Chapter 2. General Theoretical Considerations 15 3 R z L S Figure 2.3: Hund’s coupling case (c). The electron orbital angular momentum, L , and spin, S are strongly coupled together to form Ja A and E are no longer good quantum numbers but is. and R couple to form J . 2.2.2 Hund’s Case (c) For heavy molecules, like bromine or iodine, L and S are more strongly coupled to each other than to the internuclear axis. Thus, L and S form the resultant, Ja which precesses around the internuclear axis with a well defined projection along the internuclear axis. (See Figure 2.3.) As in case (a), the coupling between different electronic states is neglected and the projection of a is called . 1 is coupled to R to give J . Here, A and are no longer defined so that the energy levels are labelled by the quantum numbers J arid and the basis functions are written, J1 7 12) (2.17) For molecules which follow case (c) coupling, (such as bromine and iodine), the electronic states are given labels such as, O, O, l, where the integer is the value of ft Chapter 2. General Theoretical Considerations 16 Electronic symmetries 2.3 For diatomic molecules, any plane passing through the two nuclei is a plane of symmetry. For 1 = 0 states, a reflection of the electrons through this plane of symmetry multiplies the wavefunction by a phase factor ). This operation will be called o here and it is easily demonstrated that must be either 1 or —1. oR1’) This allows the labelling + or — (2.18) = for these levels. (e.g. etc.) Parity, P, is the operation in which the spatial coordinates of all of the electrons and nuclei are reflected through the origin of the coordinate system. (re RN — —RN). The parity of each level, b) 1 FI Here ..\p .‘ —re and is well defined. = )pRb) (2.19) = ±1. Finally, when the nuclei have the same charge (e.g. H 2 or HD), the molecular charge distribution has a center of symmetry half way between the two nuclei [17]. The reflection of all of the electrons through the center of symmetry is denoted here by the operator Rug. Under this operation the electronic wavefunction again either changes sign or remains unchanged. Rugb) = )iugb) where (2.20) = ±1. The states with ) = —1 are labelled u and those with ) = 1 are labelled g. 1-’-f- 3rr ‘o’ 3rr Lllu, . . It should be noted that for heteronuclear molecules such as 79 81 , when the Br rotation and vibration of the nuclei are taken into account the u and g electronic levels Chapter 2. General Theoretical Considerations 17 are mixed [19, 20, 21]. (For homonuclear molecules the u—g symmetry remains.) The mixing is usually weak so the u—g label will be retained here. A final word about nomenclature: for both diatomic iodine and bromine the electronic states have been described in the literature according to the description 2.21. That is the X and B states are labelled ‘E and respectively. Being relatively heavy molecules, the B states of iodine and of bromine are better described using a Hund’s case (c) description, B 0. In this thesis the prevalent Hund’s case (a) description of the B state will be used but it should be understood that only the quantum number Il is relevant. (i.e. A and E are not defined.) 2.4 Nuclear Spin Each of the nuclei in a diatomic molecule may have nuclear spin, ix, where x (= 1 or 2) labels the nucleus being referred to. A first approximation to the energy levels neglects any coupling between the nuclear spin and electronic motion. The total nuclear spin of a given state is the vector sum of the nuclear spins of the two nuclei, I I = (2.22) The various nuclear spin states are represented in the coupled representation as IXN) = = 1 ) 2 I(iii ) IM 1 ,m m 2 (2.23) C(ii, 1 ,m m 2 i , ; IM 2 ) m 2 )Iiimi; i 1 (2.24) where C(i, 1 ,m m 2 i , ; 1M 2 ) is the Clebsch-Gordan coupling coefficient [22j between the 1 two representations. It is more convenient to use the symmetric 3-j symbols [22] which are simply related to the Clebsch-Gordan coefficients. I(ii ) 2 IMz> = ( ,2 1 m m _il_MI 2 l)i J +i ( \ ‘ 2 1 m m 2 —M 1 ) m 2 Iiimi; i ) Chapter 2. General Theoretical Considerations 18 R(12) •®— ®• X12 4, OO—OO Figure 2.4: The effect of interchanging nuclear spin and spatial coordinates. R(12) is equivalent to the operation X12 followed by Rug and the parity operation, P. The two nuclei are represented by the large black and a white circles, each with its own nuclear spins (arrow). The small black and white dots represent two different electrons. (2.25) When the nuclear spin interactions are neglected, the energy levels of different total nuclear spin states are degenerate. The wavefunction of a homonuclear molecule either remains the same or changes sign when the (identical) nuclei are interchanged. That is, R(12)I) (2.26) = where R(1 ,2) denotes the operator which interchanges the nuclear spatial and spin coor dinates. As illustrated in Figure 2.4, R(12) is equivalent to R(12) = P Rug . X2 (2.27) 19 Chapter 2. General Theoretical Considerations In Equation 2.27 P is the parity operation, R 9 is the reflection of the electronic coor dinates through the center of symmetry of the molecule, and X12 is the operator which interchanges the nuclear spin coordinates. Therefore to deduce the effect of R(12) on the wavefunction of a homonuclear molecule one needs to know the effect of these three operations on the wavefunction. The results of the operations P and R 9 on a wavefunction were defined previously in Equations 2.19 and 2.20, respectively. From Reference [22] I 1 I 2 2 m m — 1 M I ) It follows, then, that the effect of = 2 (_1)t12+’ I l 1 —M 2 m m 1 X12 (2.28) ) on the nuclear spin wavefunction described in Equation 2.25 is, (_1)t12+ = (2.29) ) 1 2 )IM ( Thus, R(12)I&) = p . (2.30) xiI’> (2.31) (2.32) = For nuclei, like bromine and iodine, with half integer nuclear spin, the wavefunction changes sign under R(12). This allows one to rewrite Equation 2.32 as — b) (2.33) = From Equation 2.33 and the fact that = (i)J for the 2 = 0 electronic states of interest here, if follows that = \ug(_1)12+J (2.34) Chapter 2. General Theoretical Considerations 20 Equation 2.34 can be simplified knowing that 1 i ( =i ) is half integral. This gives, 2 1 Consequently, in the X Aug(1)’ (2.35) electronic state, for the even rotational states J only the even nuclear spin states I are populated. For the odd J states only odd I levels are populated. The inverse is true for the B ll+ electronic levels. This correlation between the nuclear spin and the rotational state leads to a further label for homonuclear diatomic molecules. The rotational states with the higher nuclear spin degeneracy are called ortho levels while the rotational levels with the lower degeneracy are called para-levels. To give E+ an example, consider 79 2 in which each nucleus has nuclear spin 3/2. In the X 1 Br electronic state the para-levels are the even J levels coupled to the I = 0 and 2 nuclear spin states, and the ortho-levels are the odd J states which are coupled to the I = 1 and 3 nuclear spin states. For heteronuclear molecules there are no such ortho-para considerations and each rotational level has the full complement of nuclear spin states (e.g. for 79 81 I = 0, Br 1, 2, and 3). 2.5 Hyperfine Interactions The hyperfine effects arise from the interaction of the nuclear multipole moments with the electric and magnetic fields produced by the electrons and other nuclei in the molecule. For a diatomic molecule where both nuclei have the same nuclear spin, i 1 = 2 i = i the two nuclear spins are coupled together to form the total nuclear spin, I In turn, the . total nuclear spin is coupled to J to form the total angular momentum, F = I+J (2.36) Chapter 2. General Theoretical Considerations 21 The hyperfine interactions result in the splitting of the rotational states into several levels each labelled by a quantum number F. The separations of the hyperfine levels are usually much smaller than the separation of the rotational states. 2.5.1 Hyperfine Symmetries Symmetry considerations allow one to deduce some of the properties of the hyperfine Hainiltonian, Hhf, without knowing its explicit form. To obtain the hyperfine energies, one calculates the matrix elements (2.37) in the uncoupled I and J basis and then diagonalizes the resulting matrix to obtain the eigenvalues, EF, and eigenstates, ‘II’F). The hyperfine Hamiltonian must have positive parity. That is, Hhf (Ft is the hermitian conjugate of P.) = (PHhfP) (2.38) Equation 2.38 implies that the only non-zero hyperfine interactions are those electric and magnetic terms which have positive parity; namely the electric 2k-pole and magnetic (2k+1)-pole terms [18]. The hyperfine Hamiltonian for a diatomic molecule can be expressed as the sum of the interactions between nucleus I and the electrons, Hhf(1), between nucleus 2 and the electrons, Hhf(2), and between nucleus 1 and nucleus 2, H,q(l, 2). Hhf = )-f-Hhf(l, 2 Hhf(1)+Hhf( ) (2.39) The behaviour of Hhf(1) and Hhf( ) is different from that of Hhf(l, 2). The first two, 2 because they act on both the electronic and nuclear coordinates, can couple different electronic states, while Hhf(l, 2), which acts only on the nuclear coordinates, is diagonal Chapter 2. General Theoretical Considerations 22 in the molecular electronic state. When calculating the hyperfine structure of the rovi bronic levels within an electronic state, both the nuclear—electronic and nuclear—nuclear interaction must be taken into account. However, hyperfine predissociation, the coupling of a bound and a dissociative electronic state can only be accomplished through Hhf(l) and H,f(2). The electron—nuclear hyperfine interactions are considerably larger than the nuclear—nuclear terms [23, 24]. For homonuclear molecules, upon interchange of the nuclear spatial and spin coordi nates, R(12)Hhf (1 )R(12) = Hhf (2) Remember that the nuclear interchange operator, R(12) = Rug ) 2 Hhf( = R(12)Hhf(l)R(12)t = Rug ( 2 ) t 9 R PH,q(l )4 )P xi = 2 RugX12Hhf( R g l)X (2.40) P. Hence, (2.41) From Equations 2.31 and 2.41 it follows that the matrix element of the hyperfine Hamil tonian, H,q(l), between the states (‘I”j’I11hf(2)I’dI’ I) k’I’) and I?J’I) obeys the condition 1 (_ + ”Ag)ug(’Ik’I’IHhf(l)I’1) l) I) (2.42) The labels I and I’ are the total nuclear spin quantum numbers associated with the states I’’ I) and Iib’I’), respectively. Similarly for the nuclear—nuclear interactions, R(12)Hhf(l,2)R(12)t = Hhf(l, ) 2 (2.43) The H,q(l, 2) terms involve only the nuclear coordinates and therefore must be invariant with respect to l?ug. i.e. RugHhf(12)Rg = Hhf(l,2) (2.44) Chapter 2. General Theoretical Considerations 23 Similarly, it follows from the definition of R(l, 2), Equation 2.38, and Equation 2.44, that 2 X12Hhf(1,2)X, = (2.45) Hhf(l,2) This leads to the following condition on the matrix elements: (/“I’Hhf(l,2)IL’ I) I> = (2.46) As a consequence of Equation 2.42, electronic u and g states can be coupled via ‘u I’IHhf(l) + Hhf(2)b g I) = (i + (_l)I+h’+1) (b’u I’IHhf(l)Ik g I) (2.47) Thus Hhf (1) and Hhf (2) can couple u and g electronic states if I + I’ is odd. This means that ortho u states are mixed with para g states and vice-versa. On the other hand, the ortho-para character of the states is preserved when a g (or u) state interacts with another g (or u) state. As mentioned in Section 2.3, for 12 = 0 electronic states, o, a reflection of the electrons in a plane containing the nuclei is a symmetry operation. States, b, 12 have A = 0, ),,), that ±1 are labelled 12 = 0, )) = = The = (2.48) symmetry operator can be applied to the hyperfine Hamiltonian matrix elements between two 12 = 0 states. (bo±tHhfI±) = (O±I(oZaV)Hhf(oo)I±) = Av(o±I(ovHhfo)1I4±) (2.49) Chapter 2. General Theoretical Considerations 24 Equation 2.49 indicates that of the hyperfine matrix elements that couple two states, = 0 the only ones which can be nonzero are those which satisfy: Hhf = )Q’V (v (2.50) Hhf The o operation changes the sign of the the magnetic dipole operators which couple two = 0 electronic states while the electric quadrupole operators are unchanged. Therefore, the magnetic dipole terms can oniy couple 0+ terms couple only 0+ 2.6 :.‘ 0+ and 0— .‘-‘ +-* 0— levels and the electric quadrupole 0— electronic states [25]. Calculation of Hyperfine Energies A nucleus with spin i 1 can take on (2i 1 + 1) different orientations in an external field. Therefore to describe completely the energy level structure of the nucleus one needs only (2i 1 + 1) labels or properties [18]. One label is taken up by the total charge of the nucleus. Z. leaving 2i 1 labels. These are the electric and magnetic multipoles of the nucleus. Therefore the hyperfine Hamiltonian can be expressed as the sum of the multipole interactions, Hhf = HMD+HEQ+HMO+•• (2.51) where HMD represents the magnetic dipole interactions, HEQ represents the electric quadrupole interactions, HMO represents the magnetic octupole interactions, and so on. Although a nucleus having spin i 1 can have up to 221 multipoles, the resolution of most experimental data in visible spectroscopy is insufficient to observe anything but the magnetic dipole and electric quadrupole effects. Consequently these are the only two hyperfine interactions that will be discussed. Chapter 2. 2.6.1 General Theoretical Considerations 25 Hyperfine Interactions Between the Electrons and the Nuclei The interactions between each nucleus and the electrons will be discussed first. The magnetic dipole terms are the interaction between the magnetic dipole moment of the nucleus, im(a)and the magnetic fields produced by the electrons, H HMD(X) E = H (2.52) The hyperfine magnetic dipole Hamiltonian is made up of three terms [23, 25). JIMD(X) (Here x (= = HLJ(x) + Hsj(x) + (2.53) HFI(x) 1 or 2) is a label for the nucleus.) The first term, the nuclear spin—orbit term, is the interaction of the nuclear spin with the electron orbital angular momentum, le(X). (To simplifly the discussion x HLJ(l) = 1 is used.) —2gIuBuN = l () (2.54) r’ (In the equations presented here, all units are in MKS. The conventions of Reference [23, 25] are adopted; namely, the Bohr magneton is negative, is positive, /tN > 0, and the charge of the electron is moment of the nucleus is ILm(1) = thlfLNil, e, 1LB <0, the nuclear magneton where e < 0. The magnetic where gjl is the nuclear g-factor of nucleus 1.) Here, ne = re — 1 r (2.55) where r is the position of the electron and ri is the position of nucleus 1. As well, hle(1) = ne X (2.56) Pe The second term in Equation 2.53 is the nuclear spin — electron spin interaction, a tensor dipole—dipole coupling between the nuclear spin and the electron spins, 51 H (l) = —/gsg1!LBILN e Se (IAo” 3 (se .nie)(ii .rie)(ii se)(rierie) \‘±71 ne (2.57) Chapter 2. General Theoretical Considerations 26 3 is the electron spin g-factor.) The third is the Fermi contact term, (g HFJ(1) = —gsgiILB[tN () ii 1 se6(r e ) (2.58) Following References [23, 25], each of the magnetic dipole interactions has the same form and can be re-expressed as HMD(1) [(Here, je(1) = (_1) ) Vq(je(1)) 1 Q(i (2.59) le(1) + se(1).] Q(ii) and 1 V_ q (je(1)) are spherical tensor operators of rank 1 constructed from i 1 and je(1), respectively. The electric quadrupole contributions to Hhf(x) are from the interactions of the quadrupole moment of the each nucleus, Q(x), with the gradient of the electric field of the electrons, VE. For nucleus 1, HEQ(1) = (_)q Q(ii) Vq(je(1)) (2.60) ) and Vq(1) are spherical tensor operators of rank 2. They arise from the multipole 1 Q(i expansion of the Coulomb potential. It can be shown that [23, 25] Q(l) The sum over er(O, q) = (2.61) p represents the sum over the charge distribution in nucleus 1. r, O, and are the spherical coordinates of proton, p, of nucleus 1. Similarly V(1) = OT1e Here °e and &e (Cq 2 Oie, 1i€) (2.62) are the angular coordinates of the vector ne. In Equations 2.61 and 2.62 C(O,) = y2() (2.3) Chapter 2. and the 27 General Theoretical Considerations }2(&, ) are spherical harmonics. In summary, both the magnetic dipole and electric quadrupole interactions between the electrons and the nuclei can be expressed as Hk(x) = (_l) Q(i) Vq(je(X)) (2.64) where k = I is the magnetic dipole interaction and k = 2 is the electric quadrupole interaction. The matrix elements of the electron—nuclear hyperfine interactions can be worked out in terms of the coupled basis set. Mf(x, F’, ft’, F, l) JI)cVFM) (7’(J’I’)cz’V’F’M’IHif(X)I ( 7 = (2.65) These can be expressed as the product of a reduced matrix element of intrinsic molecular parameters, fk(x, fZ’, v’, , v), and calculable rotational and nuclear factors. The matrix elements have been worked out previously for homonuclear iodine [23, 25] and and may 2 Br be applied to 79 M(l, F’, , , 2 Br 81 , 2 = 3/2. and 79 81 using i Br 1 =i = 6F,F,6M,M,(_1) F, Q) 1+2+F+l+I+.J+lNfk(1, S2’, v’, 12, v) (2.66) where Nk = ,/(2I+1)(2J’+1)(2J+1)(2J’+l) I xl j’ k J —12’ 12 12) I ( 1 2 i i k I 1’ i 1 ‘I I J F I’ J’ k J I (2.67) J The total matrix element for the electron-nuclei interactions can then be written: = Mjf(l,F,12’,12)+Mf(2,F,12’,12) = F,F’ SM,MI (—1) 8 x (fk(1, k+ii +i 2 +F+I+I’+J+J’ Nc 12’, v’, 12, v) + fug(1) “fk(2, 12’, v’, 12, v)) (2.68) Chapter 2. General Theoretical Considerations 28 Here, = ( if both states are u or g 1 —1 if one state is u and the other is g (The quantum number F’ has been dropped owing to the delta function in Equation 2.66.) (1,f’,v’,12,v) and f 1 (1j’,v’,1Z,v) have been given by Ref 2 The explicit forms for f [23] in terms of Hund’s case (a) coupling. Since the magnetic terms do not directly contribute to the hyperfine structure of the X 2 Br and B H÷ states of 79 , and 79 81 they are expressed in a condensed form, Br f(1, fZ’, v’, Il, v) (—1)’B(1, .9’, .9, A’, A, l’, 1, v’, v)ii(ii + 1 1)(2i + 1) (2.69) = B(l..S’,S.A’,A,fl’j,v’,v) is the sum of three different reduced matrix elements corre sponding to the nuclear spin-electron orbital interaction, the nuclear spin-electron spin interaction and the Fermi contact term defined in Equations 2.54 through 2.58. For the electric quadrupole term (1.Q’,v’j,v) 2 f = (_1)t’ [(2 + 1)] (2.70) Here, Q(1) = (i’i’I (3z — r)Iiiii) (2.71) This is the electric quadrupole moment of nucleus 1. (As in Equation 2.61, the sum over p represents the summation over the protons in nucleus 1.) 2 = (‘v’II e C(Oie,&e)IIfV) (2.72) q is the electric field gradient produced by the electrons at nucleus 1. (Remember that e is the charge of the electron so that e < 0.) The constant eqçQ(x) is referred to as eqQ(x). (Note that the e in eqQ(x) is not the electron charge but e.) Chapter 2. General Theoretical Considerations 2.6.2 29 Hyperfine Interactions Between the Nuclei The two nuclei interact magnetically and electrically in a manner analogous to the elec trons and the nuclei. The magnetic interactions occur in two forms. First, the nuclear magnetic moment of nucleus 1 is coupled to the magnetic field produced by nucleus 2 as the molecule rotates in analogy with the HLI term in the previous section. In this case, ‘e is replaced by R, the rotational angular momentum of the nuclei about the center of mass of the molecule. R = J—(L+S) (2.73) This gives [23], HJR(1,2) (..._I.(L+s)) = (2.74) where 1 R 2 is the internuclear separation. Second, there is a nuclear dipole—dipole inter action analogous to the electron spin—nuclear spin interaction of Equation 2.75. Hi1 — i2 — 2 (/LON 1 i (i 12 ) 2 (R . — 12 R R . 2 3(ij )(i ) 1 2 . gi1gi2p (2.75) The electric quadrupole terms are the interactions of the quadrupole moment of nu cleus 1 with the gradient of the electric field produced by the charge distribution of nucleus 2 and vice versa. HEQ(l,2) = (_1) [Q(1)Vq(1,2)+Q(2)Vq(2,1)] (2.76) Q(x) was defined in Equation 2.61 and V ( 2 1,2) = C(O12,c5l2) 43 (2.77) Chapter 2. General Theoretical Gonsiderations 30 The matrix elements for HEQ(1,2), Mf(l, 2)(F, 1’, Q) = (7’(J’I’)IZ’F’M’ IHEQ(l, 2) 1 (JI)I2FM) 7 (2.78) are very similar to those of the corresponding electron—nuclear interaction, M(1, 2)(F, 12’, 12) = +F+I+I’+f 2 (5F,F,6M,M,6c2,c(lYtI +i [(2i + 3)(2z+2)(2i + 1)] ( J’ 2 J ‘ J 2 i i 1 2 (Zj I’ (2I + l)(21’ + 1)(2J + l)(2J’ +1) f iiJ I +”Q(2)) 1 (Q1 + (_fl F I’ 2 J I (12v’14 R 1 3112V) (2.79) (The definition of Q(x) was given in Equation 2.71.) The matrix elements M,f(F, 12,12) in Equation 2.68 and M, (l, 2)(F, 12,12) have exactly same dependence on the quantum 1 numl)ers, J, J’, I, I’, and F. The magnetic term of the most interest in this work is the first part of HIR(1, 2), namely H j 1 (l,2) (4gj 2)i9.J_ — This has matrix elements, Msr(F,12’,12) IJ’6JJ’FF’6MM1(F(F+ = () l) (2.80) J(J+1) —1(1+ 1)) (2.81) where C,. = (4gj — 2)p_(c2v’I_I12v) (2.82) Other electronic states interact with the electronic states being studied. As is pointed out by Broyer et al [23), many of these interactions, treated using second order pertur bation theory, lead to energy shifts which have the same dependencies on the quantum Chapter 2. General Theoretical Consderations Hamiltonian H’(l,2) (1,2) 2 H (x) 1 H (x) 2 H 1 2 <1 2 1 2 1 2 31 Ij u/g 1 2 < 1 2 u(g) 4-* u(g) u(g) -÷ u(g) no rule no rule ± ± ± ± 44-* ± ± *+—* ± Table 2.1: The selection rules for the magnetic dipole, (H’(x), and H’(l, 2)) and the electric quadrupole (H (x), and H 2 (1, 2)) hyperfine Hamiltonians. 2 numbers F, J, J’, I, I’ as do the shifts produced by the matrix elements, Mf(x, F, 1’, ) and Msr(F, , 1). These two types of shifts are not easily separated. The net effect is that the measured hyperfine parameters are mixtures of arising from the two different effects. In particular, the second order terms which behave the same way as Msr(F, ft’, f) are important. It can be shown [23] that the nuclear—spin rotation contribution of H (1, 2) 1 in Equation 2.80 is negligible. The observed nuclear spin—rotation effects in the B ll+ state of bromine are due primarily to the second order interactions between the B state and other electronic states [6, 23]. The selection rules for the various hyperfine Hamiltonians are summarized in Ta ble 2.1. As noted previously, for the homonuclear bromine isotopomers, only the I and 1±2 nuclear states interact in the X and in the B electronic states. In addition, for 2 = 0 electronic states, only the J and J±2, ±4, ±6,... levels are coupled together. To deduce the hyperfine energies of the X and B 11+ states of bromine, one begins by working out the matrix representation, M, of each term of the Hamiltonian in terms of the basis set, I (JIWvFM). These include the rovibronic Mr.,, the nuclear 7 spin—rotation Msr, and hyperfine electric quadrupole MEQ(1) + MEQ( ), 2 ) + MEQ(l, 2 Chapter 2. General Theoretical Considerations 32 matrices. It should be noted that for 79 81 the nuclear spin—rotation interaction Br is really characterized by two constants, one for each nucleus. However, for the spectra observed in this research the nuclear spin—rotation contribution to the energy is extremely small and, therefore, this interaction will be treated in the same manner for both the homonuclear and heteronuclear molecules. Then, M = Mrv+Msr +MEQ(1) + MEQ( ) + MEQ(1, 2) 2 (2.83) In practice, the final electric quadrupole term is much smaller than MEQ(fl and 2 MEQ( ) . It also has the same explicit dependence on the quantum numbers. Therefore it is absorbed into them to give, M = Mrv+Msr +MEQ(l) + MEQ( ) 2 (2.84) The matrix, M, is diagonalized for each hyperfine level to produce the hyperfine energy, E(F,e), and the corresponding eigenfunction, IFEM) (JI)1lvFM) 7 j 1 = a’ (2.85) JI The label is adopted to remind the reader that more than one hyperfine level may have the same quantum number F. 2.7 Transition Intensities In this work spectra were taken in a molecular beam apparatus using laser induced fluorescence (LIF). That is, the molecules were excited with a tunable laser and then relaxed back down to the ground state by emitting a photon. A fraction of these photons was collected and detected with a photomultiplier. It is important to appreciate that Chapter 2. General Theoretical Considerations 33 absorption and fluorescence measurements are not the same. In the absence of external electric or magnetic fields absorption measurements are independent of the polarization of the laser field exciting the molecules. Fluorescence, on the other hand, is sensitive to the polarization of the laser. In addition, fluorescence measurements also depend on the positioning of the detector, and the amount of solid angle collected. A simple way of seeing this is to imagine the detector and emitter arranged as shown in Figure 2.5, and to treat the molecules as classical dipoles. A classical dipole oscillating in the Z direction will radiate primarily in the X-Y plane. Therefore, if a molecule, sitting at the origin, is excited by a laser beam polarized in the Z direction, one expects to observe relatively little fluorescence at the detector. Conversely, if the molecule is excited by a laser polarized in the Y direction, the dipole radiates primarily in the X-Z plane and one expects to observe a strong signal. Thus, the fluorescence signal measured should depend upon the laser polarization. Furthermore, in the limit that all of the fluorescent light is collected (i.e. one observes over 4ir steradians) the signal observed must be independent of the laser polarization just as for the absorption signal. In the quantum mechanical treatment of the problem, such polarization effects persist and are important especially when one observes transitions between low J levels. If one is interested in making measurements of relative intensities of LIF spectral features then the laser polarization, the detector geometry, and the amount of solid angle observed must be taken into account. 2.7.1 Allowed Transitions The laser induced transitions studied here were the electric dipole allowed ones. The selection rules for an electric dipole transition are, [17] Chapter 2. General Theoretical Considerations 34 Detector z F ‘Laser Figure 2.5: With the coordinate system shown above, the laser beam propagates along the X axis, the molecular beam travels along the Y axis and the detector is situated along the Z axis. Chapter 2. General Theoretical Considerations 35 0 0 0— u*-g u(g) u(g) LJ O,±1 ZJ Ofor =0€-*Q=0 s=0 The + and — levels of the X = 2.7.2 refer to o, symmetry here. The transitions studied were between hyperfine and B ll-- electronic states of bromine. This adds the selection rules 0,±l and zI = 0. Fluorescence Consider the geometry shown in Figure 2.5, in which the molecular beam is travelling along the Y direction, the laser radiation propagates along the X axis, and the detector is on the Z axis. (This coordinate system is chosen so as to simplify the calculations.) With this arrangement the laser may be polarized either in the Y or Z directions. To be detected, the molecules must emit photons which travel into the detector, namely in the Z direction. Therefore the emitted light observed must be polarized in the X—Y plane only, Before discussing the details of the calculations, it is instructive to consider a simple system composed only of the rotational states J” = 0 in the X state and J’ = 1 in the B state with no hyperfine structure. With the laser polarized in the Z direction the only allowed excitation transition is J” = 0 M” = 0) —* = 1 M’ = 0). In order to be detected, the emitted light must be polarized in the X—Y plane. This implies a = ±1 transition. However, this would mean making the transition IJ’ = 1 M’ = Chapter 2. General Theoretical Considerations 0) —f jJ” = 0 M” 36 ±1) which is ruled out. Therefore, no fluorescent light is observed. = When the laser light is polarized along the Y direction, the excitation transitions o M” = 0) —* = 1 M’ = ±1) occur. As before, to be detected, the emitted light must be polarized in the X-Y plane implying decay transitions ZM 1 M’ = ±1) — IJ” = IJ” = 0 M” = = ±1. Namely, 0) and fluorescent light should be detected. In this simplified example, one recovers the classical result that the LIF signal depends upon the polarization of the laser light. Of course, the existence of hyperfine structure and other rotational levels for the molecules to decay to reduces the polarization sensitivity but it is not completely eliminated. This example offers another useful insight. Consider the coordinate system rotated as shown in Figure 2.6, with the laser beam propagating along the Z’-axis, the detector on the Y’-axis, and the molecular beam travelling along the X’-axis. The results of the physical measurements must be independent of the coordinate system used. Therefore, when the laser is polarized along the detector axis (which was the Z-axis of Figure 2.5 and is now the Y’-axis) one should observe exactly the same result as above, namely no signal. With the new coordinate system the excitation transitions are 100) —* = ±1. i.e. 11±1). To be observed, the emitted light must be polarized along the X’ or Z’ axes, giving M (M = = 0, ±1 decay transitions. The emitted light polarized in the Z’ direction 0 or i ± 1) direction (M = — 0 ± 1)) is ruled out, while the emitted light polarized in the X’ ±1) is not. This would seem to imply that one expects to observe some fluorescence with Y’-polarized light, in contradiction with the previous calculation. (See Figure 2.6.) In reality there is no difficulty because the observed signal is the square of the sum of the amplitudes of the transitions [100) — Ill) — 100)1 and [100) —*11—1) —* 00)]. When the laser is polarized along the Y’-axis (parallel to the detector) these amplitudes cancel out and they add up when the laser is polarized along the X’-axis (perpendicular to the detector), in exact agreement with the results of the (X,Y,Z) coordinate system. Chap er 2. General Theoretical Considerations 37 M -1 Laser J=o 0 z, Figure 2.6: The laser—detector—emitter system using a different coordinate system. The molecular beam is travelling along the X’ axis. With this arrangement, the transitions = ±1 from the ground state to upper states and back down, shown for the X state J”=O and B state J’=l system, are allowed for both laser polarization directions. Chapter 2. General Theoretical Considerations 38 With the (X’,Y’,Z’) system, one has an excellent demonstration of the 2 slit problem in which the slits here correspond to the M’ difference between the two paths, [00) — = ±1 levels of the upper state and the phase ill) —. 100>] and [100> —* 1 — 1) —* 00)], is determined by the relative orientations of the detector and laser beam polarization. The fluorescent light signal detected is proportional to the product of the probability of absorption and the probability of detection. Sobs cc P(absorption)P(detection) cc 1(’li’ EiaserlL”)(,L”Ii.t where Etaser is the electric field of the laser, and (2.86) (2.87) 2 emittedR1’’)l is the unit vector along the emitted direction of polarization of the emitted photons. For bromine the states under study are the eigenfunctions = (2.88) crczj ( 7 J1WvFM) .1,I It is easiest to solve the above expressions when the electric dipole moments and electric fields are expressed in spherical tensor notation [22]. For example, fLy I’z = (2.89) = (2.90) = (2.91) /Jq Consider the signal, S(z), observed by a detector situated along the Z-axis from a molecule sitting at the origin excited by a laser polarized along the Z-axis. S(z) cc 2 1fLo1 2 1fLo1 F”,M”,M I xl F 1 F’ { T(F, F’, J, J’, I, M) T(F’, F”, J’, J”, I”, M) all J,all I \ I 1±1 —M0M) / F’ 1 F” \ I 1+1 _M1M”) F’ I F” —M—1M” (2.92) Chapter 2. 39 General Theoretical Considerations where P1 J’ ‘ I,U i i’ 1L1) i ,jg’ Li.L1 F* F’ j ‘F+F’+J+J’+I+Mf r i’ VI.L,.L JifJ’I’k’-) — — x./(2F + 1)(2F’ + 1)(2J + 1)(2J’ + 1) (I J (1 F’ F1(J H J’ J \ f 1 —1 I —ct’ ) (2.93) Consider now the signal, $z(y), observed with the detector along the Z-axis produced by a molecule at the origin excited by a laser polarized along the Y-axis. S(y) F”,M”,M I xl ( +1 F { T(F,F’,J,J’,I,M) .T(F’,F”,J’,J”,I”,M) all J,all I 1 F’’\ I 1 M) \—M S. F’ \ 1 F” 1 M”) 1 F 1+1 F’ —1 M’ 2 / I F’ 1+1 —M 1 —1 F” \ 1 I M”) (2.94) J E+ and B H+ states By solving the above equations for transitions between the X 1 of bromine, one observes that the polarization effects are most noticeable for hyperfine transitions of the lower rotational levels. The values of S(z) and S(y) for the R(O) and P(4) transitions are given in Table 2.2. The simulated R(O) hyperflne fluorescence spectra for both laser polarizations are shown in Figure 2.7. The above fluorescence expressions may be compared with the absorption intensity expression, Sa&s c l 2 IuoI > T(F, F’, J, J’, 1)12 (295) JI Equations 2.92 and 2.94 are the correct expressions when the fluorescent light is collected over a small amount of solid angle. When a large solid angle is observed the solutions must be integrated over the solid angle with the effect of decreasing the signal sensitivity to the laser polarization. Chapter 2. General Theoretical Considerations 40 3.0 -‘ 2.0 .— CI, 1.0 0.0 -80.0 -60.0 -40.0 -20.0 0.0 20.0 40.0 0 (MHz) v—v 60.0 80.0 100.0 Figure 2.7: The simulated 79 2 B—X (13’ 0”) R(O) hyperfine fluorescence spectra with Br the laser polarized along the Z-axis (Sz dotted line) and along the Y-axis (Sy solid line). The spectra are normalized to line a. The peaks labelled a and b are the transitions, F” 2—* F’ lL and F” 0—i F’ 1, respectively. Those labelled c and d are F” 2—* F’ 1, and F” 0—* F’ 1, respectively. For clarity the Sz spectrum is displaced by 20 MHz and, in both spectra, the central peak heights are divided by a factor of 5. — = = = = = = = = Chapter 2. General Theoretical Considerations Transition 2 F” F’ Br 79 41 E 1 (MHz) (z) 2 S (arb.) (y) 5 S (arb.) S ( 2 y)/S(z) (arb.) Sabs R(0) 2 0 .2 2 2 0 l.. 1_ 2 3 1 1 —54.8 —45.3 2.2 2.2 58.8 67.7 0.517 0.518 1.667 2.333 0.483 0.482 0.192 0.071 0.719 1.126 0.176 0.068 0.194 0.256 1.121 1.511 0.178 0.230 1.01 3.61 1.56 1.34 1.01 3.38 P(4) 4_ 4_ 2 3 2 2 3 5 3 5 6 2 3 4÷ 4÷ 4 3_ 3 3 2 1 2 4 4 5 5 3_ 3_ 3+ 4 —206.3 —160.1 —49.0 —46.5 —2.8 —2.6 —0.4 0,1 0.1 0.3 1.2 43.9 46.3 159.9 206.5 0.349 3.630 0.014 0.347 0.476 1.714 2.381 4.400 0.036 0.489 5.778 0.018 0.348 3.625 0.358 0.050 2.731 0.0011 0.054 0.087 1.452 1.691 3.116 0.003 0.076 4.308 0.0016 0.054 2.719 0.051 0.046 2.844 0.0010 0.057 0.124 1.503 1.798 3.168 0.002 0.064 4.412 0.0014 0.057 2.834 0.057 0.92 1.04 0.91 1.06 1.42 1.04 1.06 1.02 0.59 0.84 1.02 0.88 1.06 1.04 1.12 Table 2.2: The calculated fluorescence signals with two different laser polarsizations for the R(0) and P(4) hyperfine transitions of 79 2 The situation considered is that the Br detector is on the Z-axis, and the laser radiation propagates along the X-axis. S(z) is the signal calculated with the laser polarized along the Z-axis and S(y) is that with the laser polarized along the Y-axis. The calculated absorption signals are given under the Sabs column. The most striking variations in the fluorescence signals occur for low J hyperfine transitions. . Chapter 3 Predisso ciation In general, diatomic molecules have an infinite number of excited electronic levels, some of which are bound while others are unbound. Many of the electronic states of iodine arising from different electron configurations have been worked out by Mulliken [26]. A similar type of electronic structure is expected and observed for bromine. Many of the bound and unbound potentials intersect as illustrated in Figure 3.1. In such cases there is the possibility that the two states can interact in the region of intersection. This implies that when a molecule is excited from the ground electronic into a bound excited state which is coupled to a repulsive state, there is a significant probability that the molecule will pass from the bound excited electronic state into the unbound state resulting in the decomposition of the molecule into its constituent atoms. This phenomenon is known as predissociation. There are various ways to couple the bound and unbound electronic states. These fall into two categories; field-induced predissociation which requires an external electric or magnetic field, and natural predissociation. Both types of predissociation have been previously observed [8,9, 10, 27, 28, 29, 30, 31, 32]. In particular the study of the B H+ state of iodine has been instrumental to the understanding of excited state dynamics of diatomic molecules. Two types of natural predissociation were studied in this work; gyroscopic and hy perfine predissociation. Gyroscopic predissociation arises from part of the rotational Hamiltonian which couples the rotation of the molecule to the electronic orbital and spin 42 Chapter 3. Predissociation 43 bound level E unbound level R Figure 3.1: Crossing bound and dissociative potentials. 44 Chapter 3. Predissociation angular momenta. The hyperfine Hamiltonians Hhf(1) and Hhf( ) can couple different 2 electronic states, as was pointed out in the previous chapter. An important property of hyperfine predissociation is that the different hyperfine levels of each rotational state will display different amounts of predissociation. By contrast, gyroscopic predissociation acts equally on all of the hyperfine levels of a given rotational state. 3.1 Theory 81 has Br 2 , Br Br 81 , and 79 2 In this thesis, the natural predissociation of the B H+ of 79 been studied. The coupling is primarily to the dissociative ‘ll state which crosses the B state between the v’ = 4 and 5 vibrational levels [13]. The theory for gyroscopic and hyperfine predissociation has been worked out in detail by Vigué et al. [8] and applied to the B 3ll electronic state of iodine which also interacts with a repulsive 1fl level. The remainder of this chapter will present the theoretical results for gyroscopic and hyperfine 2 Br predissociation following Vigué et al. as applied to 79 , 2 Br 81 , and 79 Br. 81 Br The predissociation rate, T’,, is given by Fermi’s Golden Rule [8, 33, 34], (3.1) = where ‘) is the continuum wavefunction, Rt’b) is the bound wavefunction, and V is the Hamiltonian which couples them. The total decay rate for the level is the sum of the radiative decay rate, rrad, the predissociation rate, I’, the collisional decay rate, 1’011, and the stimulated emission rate, I’ = Psat. r + r + “coil + “sat (3.2) In a molecular beam the collisional decay rate is extremely low and is neglected in the rest of the discussion. The stimulated emission rate, Psat, can be made very small by reducing Chapter 3. Predissociation 45 the intensity of the laser beam which interacts with the molecules and is, therefore, neglected too. As previously mentioned, there are two primary types of interaction terms which enter the Hamiltonian for natural predissociation: the gyroscopic and hyperfine Hamiltonians, V = HG + Hhf. The fact that more than one mechanism can lead to predissociation introduces the possibility of observing interference effects between HG and Hhf. “p = (3.3) ’ T ’ hf+Fint G+ where P 1 are the interference terms. When Vigué et al. studied the predissociation of the B H+ state of iodine in a magnetic field, it was the existence of these interference terms that allowed the first observation of hyperfine predissociation [27, 28]. 3.1.1 Gyroscopic predissociation The gyroscopic term is part of the rotational Hamiltonian which has matrix elements that are off-diagonal in the electronic state. HG = = Here 2 2R {J+(L_ +S_)+J_(L÷ +S+)+2Jz(Lz+Sz)} (3.4) is the reduced mass of the molecule and R is the internuclear distance. The first two parts of Equation 3.4 which have the L± and S operators are responsible for coupling different electronic states. Rewriting the Hamiltonian in terms of spherical tensors, and dropping the diagonal parts, one obtains: HG (S)] + T 1 (J) [T(L) + T(S)] 1 +T = } (3.5) As per Chapter 2, the eigenfunctions of the bound hyperfine levels are written IFeM) (3.6) = JI Chapter 3. Predissociation 46 The continuum wavefunctions are expressed = ’(J’I’)1’E’F’M’) 7 (3 7) 6(E’ (3.8) and normalized as (J4(E’)Ib(E)) — E) The predissociation rate between the bound and continuum levels is (J’1’)c’E’FM)I (3.9) 1 7 I IEi(7(JI)vFM [HG +EHk(X)] 2 = J’I’’ JI k (The sum over J’, I’, Q’ has been inserted to take into account that the bound level may couple to more than one dissociative state.) To evaluate the predissociation rate it is useful to work out matrix elements of the form, Mx(1, Il’) ( ( 1 7 J’I’)1’E’F’M’) JI)1lvFMIHxI = (3.10) where H is the gyroscopic Hamiltonian or a hyperfine Hamiltonian. The matrix element of the gyroscopic Hamiltonian is [8] Mc;(2, 1) = FF,6MM,6II,8JJ,(1)J1 ( 1 J —1 3.1.2 J(J + 1)(2J + 1) 2 h (c2vI_. [T’(L) + T.(S)] I Il’) l’E’) (3.11) Hyperfine predissociation As indicated in Chapter 2, the hyperfine Hamiltonian can be written as, Hhf = Hhf(1)+Hhf(2)+Hhf(1,2) (3.12) Chapter 3. Predissociation 47 Only the hyperfine interactions between the electrons and the nuclei, Hhf(x), are able to couple different electronic states. (Here x 1 or 2 is a label for each nucleus.) Each of the first two hyperfine terms is made up of a sum of different rank interactions starting with the magnetic dipole, then the electric quadrupole, and so on. Hhf(x) EHk(x) = (3.13) The expression for the matrix element, 1 (1, 1, 12’) M (7(JI)f2vFMIHf(1)fry’(J’I’)fl’E’F’M’) = (3.14) is [8} M(1, 12, 12’) = 6F,F,6M,w(_1)k+t1+2+F+I+I’’fk(1, 12, v, 12’, E’) + 1)(21’ + l)(2J + 1)(2J’ + 1) (J k I — 1 i 2 J’’\1i I11FI’J’ cZ’)lk J 1k 1 i I (3.15) M(2. 12,12’) is obtained by multiplying 1 M, ( 1, 12, 12’) by the factor Eug(1 , 2 [fk( 12, V, 12’, E’)/fk(1, 12, V, 12’, E’)] (3.16) where 1 ( = 1 if both states are u or g (3.17) —1 if one state is u and the other is g and fk(l, 12, v, 12’, v’) is defined in Equations 2.69 and 2.70. The predissociation rate given in Equation 3.9 can be rewritten, = = 2 TI(bIHG+EH !c(x)Rbc)I I [HG + EHk(x)] [HG + H()] I&) (3.18) Chapter 3. Predissociation Hamiltonian 48 ic2i I’JI III 1 <1 <2 H H’(x) (x) 2 H 0 0 < 1 < 2 < 1 2 u/g u(g) ÷-+ u(g) no rule no rule ± ± ± -+ -* ± T ± Table 3.1: The selection rules for three Hamiltonians: the gyroscopic, HG, the hyperfine magnetic dipole, H (x), and hyperfine electric quadrupole, H (x). 2 The first sum is over all of the possible continuum levels that the Hamiltonians may couple to. Schematically, then, the predissociation rate can be seen to be the sum of several terms: FCC k,k’ {F(1) + -I- FkG(l) + ) 2 FGk( (3.19) + Fka(2)} k The first term, FCC, is the pure gyroscopic predissociation. The terms, F(x, y), are the pure hyperfine predissociations of order k. (e.g. Fii(x, y) is the pure hyperfine mag netic dipole predissociation.) The interference terms between hyperfine Hamiltonians of different rank, y) (k k’, and x, y = 1 or 2), and the gyroscopic—hyperfine interfer ences. FGk(x), exist only when different Hamiltonians couple the bound state to the same continuum state. Table 3.1 summarizes the selection rules for three Hamiltonians: the gyroscopic, HG, the hyperfine magnetic dipole, H’(x), and hyperfine electric quadrupole, (x). (The reader is reminded that although the B electronic state for each isotopomer 2 H of bromine is referred to as B H+ it should be thought of as an = 0 state.) One will observe interference terms between the gyroscopic and hyperfine Hamiltonians when the B state is coupled to an = 1, dissociative electronic state. Following Vigué et 1. [8], it is easiest to evaluate the predissociation rate using the Chapter 3. Predissociation 49 tensorial operator method. The method will be outlined here and given in more detail in Appendix A. Equivalent tensorial operators are derived for the gyroscopic and hyperfine Harniltonians. This is accomplished by, first, introducing the projection operator, Pçjj, which restricts the action of the Hamiltonian to the sublevels flvJI>. Second, one defines two irreducible operator basis sets, 9’T and p 1 U. The former operates on all of the coordinates except for the nuclear spins and the latter acts only on the nuclear spins. 9’Tk has non-vanishing matrix elements only between the states KJ and IJ’’) and has non-zero matrix elements only between the states ((i )I and I(i 2 1i i ) I’). 12 The matrix element of the scalar product, q (_ ) 1 q 95’T iiiUq is [8, 22] (—J)’ j3Tç 7 iiiUq( ’ (J’I’)f’F’M’) F (J ( k I’ J’ ITk J I J’’)(I = I I’) (3.20) The equivalent tensorial form of the Hamiltonian, H, is expanded in terms of these new irreducible operator basis sets as follows: PcjjHx Fo’E’J’J’ = Fk(, Q’, , E’, J, J’, I, I’) (_ ) 1 q ‘Tqk ii’(/q (3.21) The factor, Fk(1, 1’, v, E’, J, J’, I, I’), will be abbreviated as Fk. The equivalent gyro scopic operator becomes FOVJIHGFO’E’J’I’ = V(, 1’,v, E’, J, J’) 5 (3.22) By comparing with the matrix element in Equation 3.11, one has V(M’,v,E’,J,J’) = (_l)J1( J 2 + 1)J(J+ (QvIR2 [To(L) + 1) ( T(S)j I’E’) 1 J —s c’ (3.23) Chapter 3. Predissociation 50 Hc(x), Similarly for the hyperfine Hamiltonian, PiVJIHk(x)P,E,J,I, = Wk(X, f, ft’, v, E’, J, J’, I, I’) > )q 1 (_ ‘T 1Ih1q (3.24) with Wk(X, , 12’, v, E’, J, J’, I, I’) = (_1)k+u1+i2I+J’fk(x, 12, v, 12’, E’)/(2k + 1) [(21 + l)(21’ + 1)(2J + 1)(2J’ + 1)11/2 IJ I The coefficients fk(x, fk(x, J’’\Iii H 12’ ) ( k I’ k —12 —12 i i (3.25) J 12, v, 12’, E’) are defined as, 12, v, 12’, E’) (3.26) = as per Equations 2.69 and 2.70. For the homonuclear molecules, fk(i, ll,V, 12’, E’) = , 2 fugfk( 12,v, 12’, E’) (3.27) and Wk( 1, 12, 12’, v, E’, J, J’, I, I’) = wk(2, 12, 12’, v, E’, J, J’, I, I’) 9 (—1)’” (3.28) The predissociation rates are generally of the form, I’ = -- (?,bbIHI9t,f,c)(?,cIHkI?I,b) (3.29) If the projection operators are introduced, this becomes F (-y(J”I”)IZvFMI [pvJ,1p1Hkp1E1J,pj 1 (J’I’)12’E’F’M’) 7 = 3111111 x( (J’I’)fZ’E’F’M9 1 7 ,E,JII,HkIpclvJIl 11 {p (JI)12vFM) 7 (3.30) Chapter 3. Predissociation 51 With the help of Equation 3.21 this reduces to, F > = (J”I”)1vFMI Fk 7 ( x( (J’I’)’E’F’M’I Fk’ 1 7 IIT q Iiiiiuq] ] 1 u 111 (_1)’ (1) 11 ( 7 ( I”)vFMj J F Fk’ = (_1) (J’I’)12’E’F’M’) 1 7 I ( 7 JI)vFM) IIT q J,pcv x( ’(J’I’)f’E’F’M’I 7 (—1 )‘ U’, 7 11 (I JI)fvFM) (3.31) The evaluation of the matrix elements of the irreducible tensor operator basis sets is presented in Appendix A. The results indicate that the predissociation rates can be written as the product of a molecular parameter which is to be determined experimentally, and some rotational and nuclear factors to be denoted Rxy(a, b). (The labels X and Y refer to the type of predissociation term; a and b label the nuclei.) The Rxy(a, b) are sums of products of 3-, 6-, and 9-j symbols. (For example see Equation 3.36.) Specializing to the states, B 11+ 1 (i = = ( = O) and 1 11n (‘ = lu), one can define the molecular parameters, i) a(x) = VOvH [To(L) +T(S)] huE’) (3.32) fi(x,O,v,1u,E’)/[i(i+ 1)(2i+ 1)] (3.33) and b(x) = f ( 2 x,O,v,1u,E’) [(2 + 1)(2i+2)(2i + 3)] (3.34) C is the gyroscopic predissociation parameter, a(x) is the hyperfine magnetic dipole (k = 1) predissociation parameter for nucleus x, and b(x) is the hyperfine electric Chapter 3. Predissociation quadrupole (k = 52 2) predissociation parameter for nucleus x. b(x) is a new parame ter that is introduced in this thesis. The rotational and nuclear factors are now considered. For the purely gyroscopic term, RGG 1 J(J+1)6jj’j (3.35) For the interference terms between the gyroscopic and hyperfine Hamiltonian of order k, RGk(x) = J,I,Ju ,I,I Xk(il) x (—1 )k+2+F+I+P’+J”+f1+1 /J”(J” + 1)(2J” + 1) xW(J, J”, 1,1”) gk(J J”, 1,1”) I x k (3.36) I” i J The factor arising from the product of two hyperfine matrix elements which both couple nucleus x to the continuum is R J 1 (i(x,x) = > J”, 1,1”) Xk(i) Xkl(i) x (2K + 1) Zfr (J, J”, 1,1”) K Ir x 11hz , K I” ix J ( K k j k’ (337) J The factor from the product of two hyperfine matrix elements which couple different nuclei (x y) to the dissociative electronic state is R1d’(x, ) = a’ja,j,,,,(_1)F+f2’+I”T,V(J, J”, I, I”) Xk(i) Xk’(i) J,I,Ju,I,j x x (_1)’(2K + 1) Z’(J, J”, 1,1”) k K k’ Zx I” Z (3.38) 53 Chapter 3. Predissociation in these equations (i) 1 X = (2ix+1)(2ix+2)(2ir+3) 2i(2i, 1) — 2(2) = $k(JJ!fJJf!) = = (3.41) \/(21+1)(21”+ l)(2J+l)(2J”+ 1) I I j” — Z (J,J”,I,I”) (3.40) — — T’V(J,J”,I,I”) (3.39) [i(i + 1)(2i + 1)] I ( j” i f’ — F I” J” I I J \ K J I II k J” k J —1’ k’ Q —1 K 0 I I I I ) \ I F I” J” k j J I J” K ci —ci 0 (3.42) The predissociation rate for a level with wavefunction IFfM) = (3.43) (JI)civFM) 7 Ecr’iI JI expanded up to the electric quadrupole hyperfine terms (k F(Fv) = = 2) is 11 + 2a(1)a(2)R R i(1,2) 1 CRccj + [a(1) + a(2)] (1,1) (1) 02 + 2/C[a(1) + a,(2)jRai(1) + 2V’C[b(1) + b(2)]R 1,i) a(2)b(2)]Ri ( + 2[a(1)b(1) + 2 (3.44) (1,1) 2 2 + 2b(1)b(2)R (1,2) 22 + [b(1) + b(2)] R 81 Br Equation 3.44 must be used to describe the predissociation rate of 79 2 by realizing that a(1) Br 2 and 81 Br the expression can be simplified for 79 = . However a(2) and Chapter 3. Predissociation b(l) = 54 b(2) for these molecules. This gives F(Fv) = CRGG + 2a[Rii(1,1) + (1,2)] 11 R + 4v’avCvRrn(1) + 2 4vbVCVRG ( 1) (1, 1) 2 + 4abRi (1, 1) + (1, 22 22 2)] R + 2b [R 3.1.3 (3.45) Special Cases For the homonuclear molecules, 79 2 and 8 Br 2 ’ Br , the predissociation rates of some of the B ll+ state hyperfine levels are sensitive to only one or to a specific combination of the predissociation parameters. In particular, the hyperfine levels associated with the J’ state and the two F’ = = 0 J’ hyperfine levels associated with the odd J’ rotational states (para-states) are discussed below. Previous workers [3,4, 8, 11, 14] have expanded the predissociation rate only up to the magnetic dipole hyperfine terms. This was sufficient to fit the observed predissociation rates for the hyperfine levels of the B ll+ state of iodine and reported to be sufficient to describe the predissociation of the B of bromine as well. When this simplification is made, a new selection rule appears, I = I”. (This arises because the selection rules for hyperfine Hamiltonians that couple two u electronic states provide iM the magnetic dipole selection rule is F(F,€,v) = = 0,2,4,... while < 1.) EIcrii2{CJ2_v/avCvI.J j2 3 3(1 2 J) + J 1 J 2 (2J—l)(2J+3) . + — . J ‘3 46 The usual definitions apply, = Z(Z + 1) (3.47) 55 Chapter 3. Predissociation = (F2_J2_I2) 1 Recall that for homonuclear bromine, each nucleus has spin i (3.48) = 2 i = 3/2. The para states (the odd—J levels in the B electronic state) are those with total nuclear spin I = 0 and 2. This produces a very simple and symmetric hyperfine pattern. The two F = J byperfine states are shifted equally to either side of the hyperfine—free energy. (The F J hyperfine states lie close to the hyperfine—free energy.) The level shifted to higher energy will be labelled F and the level shifted to lower energy will be called F. (See Figure 3.2.) To a good approximation the two F = J states can be described by the eigenfunctions, in the limit that b — F M) = J 0)JM) [ ( 7 IF M) = J 0)JM) + [ ( 7 - I-x(J 2)JM)] (3.49) 1 ( 7 J 2)JM)j (3.50) 0 (i.e. no hyperfine electric quadrupole predissociation) the predis sociation rates for these hyperfine levels will be denoted F”(F,v). Using Equation 3.46 and eigenfunctioris 3.49 and 3.50 one has: TMD(F,V) = CJ(J+1)+/a,Cv + 6+ 6 3(3) — 2 (3)—6J(J+l) (2J—i)(2J+3) 351 Both the F and F hyperfine levels have the same predissociation rate. When the electric quadrupole predissociation is included, = 0 and 2, and the expressions for the predissociation rates are not so simple as the above equation. One important deduction is that the predissociation rates for the F and F hyperfine levels are no longer equal: Tp(F,v) = F9F,v)+ i3b+e{/C +a}b (3.52) Chapter 3. Predissociation 56 V I I : 3, I I I I F#J’ Fr 3” F*J” Figure 3.2: The hyperfine structure and spectrum characteristic of para-bromine (even J in the X and odd J in the B ll+ ). The FJ levels are nearly degenerate and lie close to the hyperfine—free energy with the F levels lying symmetrically to high and low energy. The strong F = J transitions produce a central peak composed of several nearly coincident transitions, a low frequency F” = FE —* F’ = F transition, and a high frequency F” = F F’ = F peak. — 57 Chapter 3. Predissociation F V l 2C + l.5/aC + l.35a 2.l8(JC + a)b + 2.6b 2C + l.5iJaC + l.35a + 2.2l(/C + a)b + 2.6b 33 l2C + 1.5/aC + O.8l8a 1.79(v’C + + 2.1b l2C + l.5’aC + O.8l5a + 1.79(-/C + a)b + 2.lb — — 3 55 5 3OC + l.5/aC + O.775a 1.75(v’C + a)b + 2.Ob 3OC + l.5v’aC + O.777a + l.75(/C + a)b + 2.Ob 7 7 56C + l.5/aC + O,762a l.74(v’C + a)b + 2.Ob 56C + l.5/aC + O.765a + 1.74(/C + a)b + 2.Ob — — Table 3.2: The predissociation rates for several the odd—J’ (para-bromine) F hyperfine levels including up to the electric quadrupole terms. As is evident, the difference in the rates for F and F. would vanish if b = 0. where j3 and € are numerical factors. One has the remarkable result that the difference in the predissociation rates of the F and F hyperfine levels is directly proportional to the hyperfine electric quadrupole predissociation constant, b. The expressions for F(F, v) for a few para hyperfine levels are given in Table 3.2. These have been calculated from the eigenfunctions which included the contributions of the fry(J, I)JM) and I = 1 ( 7 J ± 2, I)JM), 0 and 2, basis states. The hornonuclear P(l) hyperfine transitions present another important test case. The upper rotational state in these transitions is the J’ states, F’ = = 0 level with corresponding hyperfine 1 and 3. To a good approximation, 1) = (0,l)IM’) 7 (3.53) III’ F’ = 3) = fry(0,3)1M’) (3.54) /‘ F’ = Chapter 3. Predissociation 58 These levels suffer no gyroscopic or gyroscopic—hyperfine interference predissociation. They have predissociation rates, Fp(F’=l,v) = a+2.4b (3.55) l’(F’3,v) = 4a+2.4b (3.56) “rad + I’p(F’,v) (3.57) and total decay rates 1’(F’,v) = As is evident, the difference in decay rates between these two hyperfine levels is directly proportional to the hyperfine magnetic dipole predissociation parameter, a. Notice that if the electric quadrupole predissociation is not negligible then the parameter derived from relative intensity measurements of the P(l) hyperfine spectral lines will be 1’ rad a 2 -‘4b r and not a ’T rad as reported by other authors [3, 4]. Chapter 4 General Experimental Considerations The experimental work for this thesis divided naturally into two parts. The first was a and B H-i. electronic states of bromine. study of the hyperflne structure of the X The second part was a measurement of the natural predissociation of the B H+ state of bromine by a dissociative ‘Thu state. Each type of experiment had its own unique experimental requirements but they also had many common aspects. The measurements of the hyperfine structure of bromine were considerably simpler than the predissociation measurements in which saturation effects and optical collection efficiency were of great importance. This chapter will describe the experimental features common to both studies while the specifics of the hyperfine and the predissociation measurements will be covered in Chapters 5 and 6, respectively. Both of the experimental studies involved direct observations of the individual hyper fine transitions between the B and X electronic states. These transitions occurred in the visible spectral region around 5700 A. The technique for observing the transitions was laser induced fluorescence (LIF). This technique relies on a laser with a tunable frequency. When the laser is brought into resonance with a molecular transition, the molecules ab sorb light and jump into an excited state. They de-excite by falling apart, by colliding with another object, or by relaxing back down to a lower level by emitting a photon. For bromine, the probability of dropping back into the original state is very low. This means that the emitted (fluorescent) light is shifted, usually to the red, of the excitation source. This allows the detector to be conveniently “blinded” to the scattered laser light by using 59 Chapter 4. General Experimental Considerations 60 an appropriate red—pass filter which blocks out the laser light but allows light of longer wavelength (red shifted) through. The transitions are, therefore, recorded against a dark background giving the LIF technique a very high sensitivity. A further enhancement to the sensitivity is obtained by modulating either the light or the molecular source at a frequency, f. This guarantees that the signal of interest is also modulated at the same frequency. By using a lock-in amplifier, a device which selectively amplifies input signals at the frequency chosen by the experimenter, (the modulation frequency, f, of the laser light or molecular source, here), much more of the residual background may be filtered out of the detector signal. The requirements of the excitation light source are determined by the type of exper iment planned and the system being studied. Here both studies required that measure ments be performed on individual hyperfine levels of bromine. The natural width of the B state bromine levels is of the order of 0.1 MHz and the separations of the hyperfine levels vary from a few to tens of MHz. Therefore the light source could not exceed 10 MHz in width in order to resolve at least some of the hyperfine transitions. A Coher ent CR699-2l dye laser system operating with Rhodamine 6G dye and pumped by the 5145 A line from a Coherent Innova 420 argon ion laser was used. The dye laser had a line width of about 2 MHz and with this dye could operate in the range 560 nm 600 nrn with a power \ 200 mW and a maximum at about 570 nm of 800 mW. The system is able to sweep continuously through a frequency window of between 10 and 30000 MHz in a time of 2.5 seconds to 10 minutes. The studies conducted here are difficult to perform in a static cell for several reasons. At 20°C bromine has a vapour pressure of about 170 Torr and a Doppler width of roughly 400 MHz. At this temperature the maximum in the rotational state population occurs at J = 35. This presents two difficulties; first, the hyperfine structure of the transitions are hidden under the Doppler width and second, the most interesting states Chapter 4. General Experimental Considerations 61 to study are those with low J which are weakly populated. To measure the electric quadrupole constants of both the X and B states separately one must be able to observe the F # J transitions. These decrease in intensity at least as fast as 1/J 2 compared to the ZF = J transitions [18]. Therefore, for high rotational states, only the differences in the hyperfine parameters of the B and X electronic states can be determined from the spectra. As for the predissociation measurements, the gyroscopic term and gyroscopic — magnetic dipole interference terms increase as J 2 and J, respectively, while the other terms remain constant or decrease slowly with J. To best measure the hyperfine predissociation parameters, one must study the hyperfine levels of the low lying J states. Furthermore, in a sealed gas cell the collisions between bromine molecules quench the fluorescence and affect the lifetimes of the upper states. These difficulties are eliminated by studying bromine in a molecular beam. A molecu lar beam is produced by sending molecules through a small nozzle into a vacuum chamber. The molecules, especially when mixed with a noble carrier gas, collisionally cool in the nozzle concentrating the molecules into low lying rotational and vibrational states. After passing through the nozzle, the molecules may be further collimated. The molecules in such a beam are free of collisions and travel primarily in one direction with only a small residual velocity perpendicular to the molecular beam axis. Now, if a laser beam intersects the molecular beam at 900, the Doppler broadening, which is caused by the velocity distribution of the molecules along the axis of the laser, is greatly reduced. The amount of residual Doppler width depends upon the degree of collimation of the molecu lar beam and the angle between the laser and molecular beam axes. In our experiments the Doppler shift was reduced from about 400 MHz down to 4 MHz or better. The molecular beam machine, shown schematically in Figure 4.1, consisted of an injection chamber and a high vacuum section separated by a gate valve, GV. The injection chamber was evacuated with a roots blower (Edwards Mechanical Booster Pump model Chapter 4. General Experimental Considerations 62 EH500), RB, backed by a mechanical pump (Edwards mechanical pump model E2M40), M2, to a pressure of about I mTorr. (The cold trap, Cl, was used to trap excess bromine.) The high vacuum side consisted of two cylinders joined together; the first will be referred to as the main chamber where all the measurements were made and the second will be called the auxiliary chamber. The main chamber was evacuated with a 4.5 inch diameter diffusion pump (Norton diffusion pump model 0161), Dl, backed by a roughing pump (Duo Seal model 1402), M3. The auxiliary chamber was evacuated with a 6 inch diameter diffusion pump (CVC model 88), D2, backed by a mechanical pump (Duo Seal model 1402), M4. (Both diffusion pumps were preceded by cold traps, C2 and C3 in Figure 4.1.) The high vacuum side of the molecular beam machine achieved a pressure of 4 x 10—6 Torr with no gas load. Both chambers had flanges attached to the top and the two sides perpendicular to the molecular beam axis as shown in Figure 4.1. The auxiliary chamber also had an end flange closing it off. The inner surfaces of both chambers were coated with graphite to minimize scattered laser light. The main chamber was an aluminum cylinder 12 inches high with a diameter of roughly 18 inches. The flanges perpendicular to the molecular beam axis had various entrance and exit windows for the laser radiation. The laser beam interacted with the molecules in the region labelled A during the predissociation study and in the region la belled B during the hyperfine structure work. The photomultiplier (RCA model 3l034A), housed in a cooling jacket (Products for Research model TE1O4RF) which brought it to a temperature of roughly —30° C, sat on top of the main chamber over either the A or B region. An optical stack holding a series of lenses to collect the LIF was mounted inside the vacuum chamber and the light was fed through a port on the top flange to the active area of the photomultiplier. ( The exact arrangement of the lenses was different for the hyperfine and predissociation studies and will be presented in detail in Chapters 5 and 6.) The mount for the cooling jacket — photomultiplier housing was designed so that a Chapter 4. General Experimental Considerations Figure 4.1: The molecular beam machine. (See text for explanation of symbols.) 63 Chapter 4. General Experimental Considerations 64 colored glass filter could be placed in front of the photomultiplier to cut out the scattered laser light. The gas, typically a mixture of 20 % bromine in argon, was introduced through a pyrex nozzle, NI, ranging from 20 to 40 microns in diameter. After expanding through the nozzle into the injection chamber, the gas passed through a 1 mm diameter skimmer, Sk, to help collimate it. The beam then entered the high vacuum side. A pair of adjustable stainless steel slits, Si, mounted vertically, were placed about 10 cm downstream of the skimmer. These were followed by an adjustable iris 1 cm further downstream. By varying the slit width the amount of residual Doppler width in the fluorescence signals could be reduced. Before evacuating the beam machine, the skimmer, iris, and optical stack were aligned optically. The end flange of the auxiliary chamber was removed and replaced by a cardboard alignment plate which had a small hole cut in its center. A HeNe laser beam was then sent through the centers of the alignment plate, the iris, and the skimmer — defining the molecular beam axis. The optical stack was then positioned so that the HeNe laser beam passed through the center of the optical collection region. The stack was then locked in place. The nozzle was mounted so that its distance from the skimmer as well as its position in the plane perpendicular to the molecular beam propagation direction could be varied. Initial alignment was performed by placing the nozzle about 3 mm away from the skimmer and injecting a pure argon beam into the high vacuum chamber. The pressure inside the main chamber was monitored with an ion gauge and the nozzle’s position in the perpendicular plane was adjusted until the pressure was maximized. The optimal distance between the nozzle and skimmer was determined by introducing a bromine beam and monitoring its LIF. The best signal to noise was usually obtained when the nozzle skimmer distance was from 1.5 to 2.0 cm. — Chapter 4. General Experimental Considerations 65 The gas delivery system consisted of a 500 ml stainless steel lecture bottle holding about 100 ml of liquid bromine connected by a needle valve (Whitey model SS 22RS4), N, to a piece of copper tubing feeding into the nozzle. A 0 — 760 Torr pressure gauge (Matheson model 63-5601), P, was attached to the copper tubing to allow a measurement of the gas pressure behind the nozzle. The gas system was designed so that the argon could be fed directly into the copper tubing or into the bromine cylinder. The bromine cylinder plug was designed so that the argon could be be fed into the inlet and a mixture of bromine and argon was carried through the outlet. The cylinder was charged up with argon and then left for at least an hour before use to allow the two gases to mix. The system could be operated on a single charge for roughly 5 hours. The entire delivery system could be evacuated with a roughing pump, Ml. The pressure of the argon and bromine mixture behind the nozzle was measured with the pressure gauge and controlled with the needle valve. The best signal to noise depended upon the nozzle diameter, the backing pressure, and the pumping speed of the injection system. Using a 20 micron diameter nozzle the best results were achieved with a backing pressure of roughly 300 Torr. The resulting rotational temperature of the molecular beam was about 6 K. Chapter 5 Hyperfine Structure of Bromine The first part of this thesis involved the measurement of sub—Doppler spectra of the B H+ — X ‘E rovibronic transitions of each isotopomer of bromine. Bromine has several features that make it particularly attractive to study. Bromine occurs naturally in two isotopes, 79 Br and 81 Br, in almost equal abundance, providing three isotopomers of molecular bromine 79 2 81 Br 2 and 79 Br 81 in the ratio 1:1:2. Each isotope has the Br , same nuclear spin, i = , 3/2, but quite different nuclear magnetic and electric quadrupole moments [35]. As well, bromine is a heavy molecule so that the rovibronic spectra of each isotopomer are very dense. Therefore, when studying molecular bromine in natural abundance, one observes many near coincidences of transitions arising from different isotoporners. This gives the experimenter the ability to compare sub—Doppler spectra and hyperfine structure of three distinct, yet very similar molecular systems. Recently, an atlas of the rovibronic B—X transitions of 79 2 was prepared [36]. The B ll+ and X Br rovibrational constants were deduced from these measurements and were published along with the rovibrational parameters of the other two isotopomers, 81 2 Br , and 79 81 Br derived from those for 79 2 [7]. The near coincidences of bromine transitions provides Br an opportunity to investigate the accuracy of these published constants. There are two main techniques of observing sub—Doppler spectra. The first is to use saturation spectroscopy on the molecules in a sealed glass cell. Saturation techniques are very sensitive but, as discussed, they generally yield information about the high J states. The second technique, measurement of LIF of a molecular beam, provides 66 Chapter 5. Hyperfine Structure of Bromine 67 information mainly about low—lying rotational states. Previous authors [1, 2, 3, 4, 5, 6] have reported measurements of the hyperfine structure of the B and X electronic states of bromine. Half of the studies [1, 5, 6] used saturation spectroscopy techniques, and the others [2, 3, 4] measured the LIF of a molecular beam of bromine. The observed spectra have been adequately described by the molecular constants, eqQx, eqQB, (the molecular electric quadrupole coupling constants of the X and B electronic states respectively) and Csr (the nuclear spin-rotation constant of the B state). Since both the B and X levels are 12 = 0 states, the predominant hyperfine splitting of the rovibronic levels is due to the electric quadrupole interaction given by Equation 2.84. The hyperfine splitting patterns are very similar in the two electronic states with their magnitudes being set by eqQx and eqQ. Transitions between levels with F = J only yield information about eqQ = (eqQB — eqQx). In order to deduce eqQB and eqQx separately one must observe some of the zF LSJ (so-called crossover) transitions. As mentioned in Chapter 4, the crossover transitions decrease in intensity at least as fast as J 2 compared to the LF = J peaks. Thus, the hyperfine spectra of the low rotational states are of particular importance. The trade-off is that the nuclear spin-rotation interaction is very small for low lying rotational states and increases in magnitude with J. Thus, molecular beam studies are suited to the study of the hyperfine parameters eqQx and eqQB while the saturation spectroscopy techniques are useful for measuring eqQ and Csr. In this chapter the experimental details and results of the hyperfine structure study are presented and compared with the findings of previous authors. Chapter 5. Hyperfine Structure of Bromine 5.1 - 68 Experimental Details The hyperfine spectra of various B—X rovibronic transitions of bromine were recorded using the molecular beam machine and laser system described in Chapter 4. The exper imental apparatus is shown schematically in Figure 5.1. The argon—ion (AL) pumped dye laser (DL) beam was steered through the molecular beam machine using mirrors Ml, M2, and M3. The laser induced fluorescence (LIF) was collected with the optical system and imaged on the photomultiplier (PMT). The signals were fed into a lock-in amplifier (LIA) tuned to the frequency of the mechanical chopper which modulated the molecular beam. Part of the laser beam was sent into two reference etalons, Eti and Et2, using beam splitters Sl and S2 and mirror M4. Eti was pressure and temperature stabilized with a 750 MHz free spectral range (FSR) and Et2 was a passive 150 MHz FSR etalon. The etalon transmission peaks along with the LIA signal were recorded on a jWax computer. The optical collection system, shown schematically in Figure 5.2, had a ff0.88 con denser lens (Melles Criot aspheric glass condenser number 01 LAG 007, focal length 26.5 mm, diameter 30 mm), Ll, one focal length above the laser—molecule interaction region. In addition a spherical mirror (Melles Griot concave spherical reflector number 01 MCG 007, focal length 25.0 mm, diameter 30.0 mm), Ml, was located one focal length below the interaction region to enhance the collection efficiency of the laser induced fluores cence (LIF). This light collected was then fed into a spatial filter consisting of two ff0.88 condenser lenses (L2 and L3) held 2f apart with a plate having a 2mm hole in it half way between them. This allowed the on axis light to travel up the optical stack and blocked much of the scattered (off-axis) light. The spatially filtered light was then focussed by a 110 mm focal length lens (Melles Griot symmetric convex glass lens number 01 LDX 183, Chapter 5. Hyperfine Structure of Bromine 69 focal length H0.0 mm, diameter 35.0 mm), L4, onto the active area of the photomulti plier. (L4 also acted as the window from inside to outside the vacuum chamber.) The system was designed so that a red pass filter could be inserted above L4 to reduce the stray laser light detected. To further cut down the noise, the molecular beam rather than the laser beam was modulated with a mechanical chopper inside the vacuum chamber. When one modulates the laser beam, the scattered light is also modulated allowing any scattered light which reaches the photomultiplier to be detected by the lock in amplifier, LIA. However, in the absence of absorption, the molecular beam does not give off any visible light so that it presents a dark background to the detector. Crucial to this work was the calibration system, designed and built by Adam et al. [37], which supplied the frequency scale for the observed hyperfine spectra. The system consists of two confocal Fabry-Perot etalons. The first etalon, Eti, has a free spectral range (FSR) of 750 MHz and is housed in a temperature controlled vacuum chamber. It is locked to a transmission peak of a polarization stabilized HeNe laser and was calibrated to allow absolute frequency measurements. The second is a passive etalon, Et2, with a FSR of 150 MHz. Part of the laser beam used to probe the molecules was split off and sent into these two etalons (using beam splitters Si and S2 and mirror M4 as per Figure 5.1). As the laser was scanned over the frequency region of interest, the etalon transmission peaks were recorded along with the spectral features providing frequency markers to be used for analyzing the data. The designers have determined that this system provides absolute frequency measurements accurate to ±25 MHz and relative frequency measurements accurate to slightly better than ±1.0 MHz [37]. Unfortunately, during these studies, the stabilized HeNe laser used to lock the cal ibration system was under repair. The 750 MHz etalon (FSR = (750.83 ± 0.30) MHz) was no longer fixed to a HeNe transmission peak but was still pressure and temperature regulated. This, of course, ruled out absolute frequency measurements but had almost Chapter 5. Hyperfine Stricture of Bromine 70 jiVax M4 M2 F M3 Ml Br 2 i-Ar Figure 5.1: A schematic of the apparatus used for studying the hyperfine structure of bromine. (See text for explanation of symbols.) Chapter 5. Hyperfine Structure of Bromine 71 4T Filter IA L3 L2 Li Br + 2 Ar Laser Ml Figure 5.2: A schematic of the optical collection system used in the hyperfine structure studies. 72 Chapter 5. Hyperfine Structure of Bromine no effect on relative frequency measurements. For example, an error in the free spectral range of 1 MHz yielded only an error of 1 MHz in the relative frequency separations of peaks which were 750 MHz apart. To put this in perspective, recall that the FSR of a confocal Fabry-Perot etalon is given by: FSR = —s-4nd (5.1) where c is the speed of light, n is the index of refraction of the material between the mirrors, and d is the mirror separation. For an etalon built with invar spacers, one °C. For a change in 10 / may overestimate the thermal expansion coefficient, c, as 5 x 5 temperature T, the change in FSR is, = —cT (5.2) To obtain a 1 MHz change in FSR would require a temperature change of 26 °C. Certainly a more realistic change in temperature for an etalon which is not in a temperature controlled container would be zT = 5 °C giving a change in FSR of only 0.19 MHz. Similar considerations for a catastrophic failure of the vacuum chamber housing the etalon (ZIP = 1 atmosphere) show that the FSR would only change by 0.24 MHz. Therefore, one is confident that the measurements were reliable to at least 1.0 MHz for a 750 MHz frequency separation (0.13%). The data were recorded on a microVAX (VAX) minicomputer which took 2048 channels per scan. The 150 MHz markers were used to interpolate between the 750 MHz peaks allowing the frequency separations between spectral lines to be deduced. The list of B—X rovibronic hyperfine spectra recorded is given in Table 5.1 The hyper fine spectrum of each transition studied was recorded at least 3, and usually 4 times. The results are contained in 379 data sets comprising over 3000 hyperfine spectral features. Chapter 5. Hyperfine Structure of Bromine Isotopomer 2 Br 79 (v’—v”) (11’ (13’ — — 0”) 0”) 73 Rotational Transitions R(18) R(O) R(3), R(l0) P(1) P(5), P(15) P(17) R(11), R(12), R(14) P(7), P(17) R(0) R(8), R(10), R(15) P(1) P(5), P(7) , P(12) P(6) — — (14’ — 1”) (17’ —2”) — — — (19’ 81 Br 79 (11’ (12’ (13’ — — — — 2”) 0”) 0”) 0”) R(0) R(2), P(1) R(0) R(2), P(1) R(0) R(3), R(10) P(1) P(5) R(0) R(2), R(8), R(11) P(1), P(5), P(7) R(0) R(2), P(1) R(0) R(2), P(1) R(0) R(8), R(10) P(1) P(5) — — — — (14’ (15’ (16’ (17’ — — — — 1”) 1”) 1”) 2”) — — — — — 2 Br 81 (13’ (17’ — — 0”) 2”) R(0) R(3), R(10) P(1)—P(5) R(0) R(8), R(10) P(1) —P(5) — — Table 5.1: A list of the rovibronic transitions whose hyperfine spectra were recorded in this study. Chapter 5. Hyperfine Structure of Bromine 5.2 74 The Model The B ll+ and X electronic states are both 1 0 levels. For this work, it is sufficient to include only the electric quadrupole Hamiltonian and the nuclear spin— rotation Hamiltonian to adequately describe the observed hyperfine spectra. Esr = [F(F+1)—J(J+1)—I(I+l)j (5.3) To the accuracy needed here, the nuclear spin—rotation Hamiltonian can be treated as diagonal in I, J, and F. (In addition, even though one has one constant, Csr, for each nucleus for 79 81 Br , characterizing the energy by a single, effective constant was suffi cient.) The electric quadrupole terms couple different nuclear spin states, rotational states, the J and J±2, ±4, < ..., 2. For zMI 2, and = 0 levels, one has the added restriction that only states are mixed. All of the even J levels are connected to one another as are all the odd J levels. This implies that one must set up and digaonalize an N x N matrix which includes all of the rotational and nuclear spin states. In practice it is sufficient to truncate this matrix to include only the J and J±2 rotational states and the appropriate nuclear spin states (see Chapter 2) to fit the observed hyperfine spectra. It is convenient to begin by describing the hyperfine states with a coupled basis set, 7 ( J, I)vFM), and work out the matrix elements for the hyperfine electric quadrupole Hamiltonian. Then one adds the nuclear spin—rotation terms and, finally, the rovibronic energies for the states. Schematically, for each electronic rotational—vibrational hyper fine level, one must diagonalize a matrix, M, which is the sum of a (diagonal) rovibronic energy matrix, Mrv, a (diagonal) nuclear spin—rotation matrix, Msr, and symmetric elec tric quadrupole matrices, MQ and MQ. (The superscripts 1 and 2 refer to the electric quadrupole contributions from nucleus 1 and 2 respectively.) M = Mrv + Csr Msr + eqQ(l) . M+ eqQ(2) . (5.4) Chapter 5. Hyperfine Structure of Bromine 75 For the homonuclear molecules Equation 5.4 may be simplified by recognizing that eqQ(l) = eqQ(2) and combining the two matrices into MEQ. Diagonalization gives the eigenenergies, E(F,), and eigenvectors, labelled here as IFM) = Ec’Jj7(J,I)1lvFM) (5.5) JI The label is used to emphasize that several hyperfine states can have the same F quantum number but different eigenenergies and wavefunctions. As with any energy calculation, one has an arbitrary zero point. In this work, where only relative frequencies were measured, it was convenient to subtract off the energy of the hyperfine-free rovibrational level, = M = — . from M. I (5.6) + Csr’ Msr + eqQ(l) . M+ eqQ(2) . M (5.7) where I is the identity matrix. This maiipulation does not change the eigenfunctions but translates the eigenenergies by an amount For each electronic state, the new energies are, E(F, (Of course, EI,JI,B ) is the B — = E(F, ) X (v’, J’) (5.8) — +— (v”, J”) rovibronic energy difference.) The calculated hyperflne contribution to the transition between the X and the B F”c”M) level IF’f’M> level is v(F’, F”) = E(F’, E’) — E(F”, f”) (5.9) The absolute transition frequencies are obtained by adding to these values the appropriate rovibronic energy differences, (E,J,B — The observed frequencies were not absolute frequency measurements so that one had the freedom to arbitrarily choose a zero point for the experimental frequency scale. The Chapter 5. Hyperfine Structure of Bromine 76 difference between this arbitrary zero and the center of the hyperfine free rovibronic transition frequency, 6, was an adjustable parameter applied to the data. lIobg(F, F”) = 6 + v(F’, F”) (5.10) where v(F’, F”) is the calculated transition frequency and depends upon the hyperfine parameters, eqQx, eqQB, Cs,., and the known rovibronic constants for the X and B electronic states [7]. The nuclear spin — rotation constant, Csr, was measured for the vibrational levels, 16 < v’ <28, of the B state of 79 2 in Reference [6]. This parameter Br increases with vibrational state, and for B state vibrational levels studied here is less than 0.100 MHz. Therefore its overall contribution to the energy of a hyperfine level for J’ = 5 is less than 1.5 MHz and smaller for hyperfine levels of lower J’ states, compared to electric quadrupole hyperfine contributions of tens to hundreds of MHz. Therefore, the data were fit for eqQx, eqQB, and 6 holding Ce,. fixed at the value appropriate for the vibrational state, v’, of the transition under study. (The values of Cs,. were interpolated from the 79 2 data presented in Reference [6]. The values used for 81 Br 2 were scaled Br Br). The values used for 79 Br)/g( 81 by the ratio of nuclear magnetic dipole moments, g( 2 and 79 Br 81 were taken as the average of the 79 Br 79 81 constants.) Br The heteronuclear data could, in theory, be used to deduce the parameters eqQ( Br) 79 Br) for both electronic states. However, these were highly correlated so that 81 and eqQ( the restriction, eqQ( Br) 79 Br) 81 eqQ( — - Q( Br) 79 Br) 81 Q( — - 119707 U 3 L5 11 ) was imposed [35]. The heteronuclear data were fit for eqQ( Br) (for both electronic 79 states) and 6. Each hyperfine spectrum was recorded 4 to 5 times and the results for each vibrational band were averaged. To test the validity of constraint 5.11 on the heteronuclear data hyperfine spectra, the (13’ — 0”) and (17’ — 2”) bands were recorded for each isotopomer. The deduced Chapter 5. Hyperfine Structure of Bromine v 77 v” 13 = 0 eqQ( Br) 79 (MHz) 178.1 ± 1.0 808.3 ± 7.0 eqQ( Br) 81 (MHz) 148.3 ± 1.0 676.2 ± 5.6 1.201 ± 0.015 1.195 ± 0.020 v’ v” 17 = 2 180.7 ± 1.8 810.9 ± 1.8 150.0 ± 2.1 675.1 ± 3.5 1.205 ± 0.029 1.201 ± 0.009 v’ = Ratio Table 5.2: The electric quadrupole parameter for the (13’ 0”) and (17’ 2”) bands of 2 and 81 Br 79 The ratio of eqQ for the different homonuclear isotopomers agrees, 2 Br within experimental uncertainty, with the ratio of nuclear electric quadrupole moments of the two isotopes. — — . hyperfine electric quadrupole parameters for 79 2 and 81 Br 2 are given in Table 5.2. Br The measured ratios of the electric quadrupole moments between the two isotopes equal Br) to within the experimental uncertainty. This implies that the gradient 81 Br)/Q( 79 Q( of the electric field, q, at the nuclei is the same for both 79 Br in 79 2 and 81 Br Br in 2 Br 81 5.3 Results and Discussion For each data set, the unblended transitions were selected, weighted by their relative heights, and then subjected to a nonlinear least squares fit for the parameters, eqQB, eqQx, and 5. Having recorded over 3000 transitions in 379 data sets, it is not practical to list completely the observed and calculated transition frequencies. Instead, Appendix B shows, iii (i’caic), and their differences tabular form, the raw data 0 (v ) 3 b , the calculated hyperfine transition frequencies (‘obs — caic) referred to as (obs—caic), for the B—X (17’ — 2”) P (1) hyperflne spectra of each isotopomer. In addition a summary of the average values of 8 b 0 V and their 5 b 0 (v — is given. These data, with a root-mean-square (rms) Chapter 5. Hyperfine Structure of Bromine v’ 79 81 Br Br) 79 eqQ( (MHz) 11 12 13 14 15 16 17 177.0±0.6 177.8 ± 3.2 178.8 ± 0.5 179.8±1.0 180.1±0.4 179.9±0,5 180.6 ± 1.4 78 79 2 Br Br) 79 eqQ( (MHz) 81 2 Br Br) 81 eqQ( (MHz) — — — — 178.1 ± 1.0 148.3 ± 1.0 — — — — — 180.7 ± 1.8 — 150.0 ± 2.1 Table 5.3: The B state eqQ parameters for each isotopomer. obs—caic of about 0.7 MHz, are typical of most of the data recorded and well within the experimental uncertainty. The exceptions are the (13’ — 0”) band data which had an experimental uncertainty that was larger than the rest. The B—X (13’ — 0”) P(2) hyperfine data are also summarized in Appendix B demonstrating the slightly worse rms (obs—calc) of 1.2 MHz. The deduced electric quadrupole coupling constants for the different vibrational levels of the B and X state studied are given in Tables 5.3 and 5.4, respectively. These are compared with the results reported by previous authors [2, 3, 4, 5, 6] in Tables 5.5 and 5.6. As can be seen, the values presented in this thesis are in excellent agreement with those of previous workers and, with the exception of results reported by Katzenellenbogen Br) parameters. 79 and Prior [5], have comparable or better error limits on the B state eqQ( The molecular electric quadrupole coupling constants increase with vibrational quantum in both electronic states. Performing a linear least squares fit on the molecular constants Chapter 5. Hyperfine Structure of Bromine 79 v” 79 B 8 Br r Br) 79 eqQ( (MHz) 79 2 Br Br) 79 eqQ( (MHz) 81 2 Br Br) 81 eqQ( (MHz) 0 1 2 808.1 ± 1.4 810.8±0.7 811.5 ± 1.9 808.3 ± 7.0 676.1 ± 5.6 — — 810.9 ± 1.8 675.1 ± 3.5 Table 5,4: The X state eqQ parameters for each isotopomer. v’ 11 Reference eqQ( Br) 79 (MHz) [3] [4] this work (176.5 ± 1.0) (176.97 ±0.62) (177.0 ± 0.6) 12 this work (177.9 ± 3.2) 13 [2] [4] this work (179.1 ± 1.6) (177.12 ± 0.98) (178.8 ± 0.5) [2] this work (178.7±1.6) (179.8 ± 1.0) 14 v’ Reference eqQ( Br) T9 (MHz) 15 this work (180.1 ± 0.4) 16 [2] [4] [6] this work (178.5 (180.05 (180.9 (179.9 17 [6] [5] this work (179.7 ± 2.2) (180.20 ± 0.03) (180.6 ± 1.4) — ± ± ± ± 1.2) 0.69) 2.7) 0.5) Table 5.5: A comparison of the observed electric quadrupole coupling constants, Br), for different vibrational levels of the B H+ state. 79 eqQ( Chapter 5. Hyperfine Structure of Bromine 80 X ‘Et state eqQ( Br) (MHz) 79 Reference v” 0 [3] [41 this work (808.7 ± 0.9) (809.32 ± 0.59) (808.1 ± 1.4) (809.91 ± 0.82) (810.8 ± 0.7) (810.15 ± 0.62) (811.5 ± 1.4) [2] — 1 (810.0 ± 0.6) 2 (810.0 ± 1.1) — — Table 5.6: A comparison of observed electric quadrupole coupling constants for different state of 79 vibrational levels of the X 2 Br determined here gives, Br) 79 B state : eqQ( Br) 79 X state: eqQ( = (170.82 ± 1.91) + (0.598 ± 0.134)v’ (MHz) (5.12) (808.76 ± 1.14) + (1.70 ± 0.99)v” (MHz) (5.13) The calculated values are shown in Tables 5.7 and 5.8. Liu et al. [6] have reported the B state electric quadrupole coupling constants for the vibrational levels, 16 v’ 28, using polarization spectroscopy. Because this technique investigates the high rotational states, the eqQ( Br) values reported in Reference [6] for the B state were deduced 79 Br) parameters of 79 from the measurements of eqQ and the values of the X state eqQ( Reference [4]. (See Table 5.9.) Fitting these data in the same manner yields: Br) 79 B state : eqQ( = (172.06 ± 3.52) + (0.484 ± 0.158)v’ (MFIz) (5.14) again, in excellent agreement with this work. The authors of Reference [5] report uncertainties in the B state electric quadrupole coupling constants about 30 times lower than those of all other authors. Like Liu et al. [6], Chapter 5. Hyperfine Structure of Bromine v’ 11 12 13 14 15 16 17 eqQ( b 0 Br) 79 (MHz) 177.0 177.9 178.8 179.8 180.1 179.9 180.6 ± ± ± ± ± ± ± 0.6 3.2 0.5 1.0 0.4 0.5 1.4 eqQ( Br)caic 79 (MHz) 177.40 178.00 178.60 179.20 179.80 180.39 180.99 81 obs caic (MHz) — —0.40 —0.10 0.20 0.60 0.30 —0.49 —0.39 Table 5.7: The eqQ( Br) values for the B state vibrational levels recorded as measured 79 and calculated from Equation 5.12. v’ eqQ( b 0 Br) 79 (MHz) eqQ( Br) 79 (MHz) 11 12 13 808.1 ± 1.4 810.8 ± 0.7 811.5 ± 1.4 808.76 810.47 812.17 ohs caic (MHz) — —0.66 0.33 —0.67 Table 5.8: The eqQ( Br) values for the X state vibrational levels recorded as observed 79 and calcuated from Equation 5.13. Chapter 5. Hyperfine Structure of Bromine 82 v’ eqQ( b 0 Br) 79 (MHz) eqQ( Br)caic 79 (MHz) 16 17 18 19 20 21 22 23 24 25 26 27 28 180.9 ± 2.7 179.7 ± 2.2 180.0 ± 1.9 180.6 ± 1.8 181.5 ± 1.7 183.0 ± 1.8 183.5±1.8 184.6 ±2.0 182.2 ± 2.5 183.8 ± 1.8 185.3 ± 1.9 185.2 ± 2.0 184.3 ± 2.3 179.80 180.28 180.76 181.25 181.73 182,22 182.70 183.18 183.67 184.15 184.63 185.12 185.60 ohs caic (MHz) — 1.10 —0.58 —0.76 —0.65 —0.23 0.78 0.80 1.42 —1.47 —0.35 0.67 0.08 —1.30 Table 5.9: The observed B state eqQ( Br) values from Reference [6] and the correspond 79 ing values calculated from Equation 5.14. Cthapter 5. Hyperfine Structure of Bromine Isotopomer 2 Br 79 2 Br 81 81 Br 79 v’ 17 18 17 18 17 18 eqQ( Br) T9 (MHz) 180.20 ± 0.03 185.17 ± 0.07 — — 179.05 ± 0.07 181.02 ± 0.06 83 eqQ( Br) 81 (MHz) — — (kHz) 90.18 ± 0.09 98.30 ± 0.08 151.39 ± 0.04 150.19±0.12 91.77 ± 0.08 103.07±0.08 151.29 ± 0.07 151.48 ± 0.06 91.93 ± 0.10 101.35 ± 0.09 Table 5.10: The electric quadrupole parameters reported by Ref. [5] for the B state of bromine. the technique employed by Katzenellenbogen and Prior [5] was polarization spectroscopy on a sealed cell of bromine. The hyperfine structure of the states J’ 30 to 100 of the v’ = 17 and 18 vibrational levels were used to deduce, for each isotopomer, the parameters eqQB, Csr. and F, the half—width at half maximum (HWHM) of the hyperfine peaks. The HWHM values will be discussed in more detail in Chapter 6.) Csr and ( eqQB are summarized in Table 5.10. The work of Reference [5] should only be sensitive to eqQ and not eqQB directly. To obtain the reported values, Katzenellenbogen et al. fixed the X Br) = 810.0 MHz and eqQ(8lBr) = 676.7 MHz. The former was 79 state parameters eqQ( chosen from Reference [2] and the latter, it must be assumed, was deduced from the ratio Br). The uncertainty in the value of eqQx used by the authors was the order 81 Br)/Q( 79 Q( of 0.8 MHz. No account was taken of this uncertainty to the errors given in Reference [5] and quoted here in Table 5.10. The variation in the electric quadrupole coupling constant from v’ = 17 to v’ = 18 is markedly different from isotopomer to isotopomer considering Br) is considerably 81 Br)/eqQ( 79 the quoted accuracy. As well, the B state ratio eqQ( Chapter 5. Hyperfine Structure of Bromine 84 different from the ratio assumed for the ground state. Thus, the reported uncertainties seem optimistic. (In passing, it should be noted that the rovibronic transition frequencies for the B—X (17’ — 2”) and (18’ — 2”) hyperfine spectra shown in Reference [5] are both roughly 4.8 cm too high. These are probably calculated values and point to a trivial error in the term value for the X state v” = 2 level used in the calculation.) In general, the observation of the hyperfine spectra corresponding to the low—lying rotational B—X transitions were particularly important for obtaining both the signs and the magnitudes of the electric quadrupole parameters. For example, the P(1) hyperfine spectrum for each isotopomer is almost entirely due to the hyperfine energy pattern of the X J=l level and, therefore, a direct measure of eqQ of the ground state. For each isotopomer this pattern is quite asymmetric so that the sign of the ground state electric quadrupole coupling constant, eqQx, may be determined along with the magnitude. The observed spectra lead to the conclusion that the sign of eqQx is positive as indicated in Table 5.4. From this observation, and measurements of other hyperfine splitting patterns one may further deduce that the upper state quadrupole coupling constant, eqQB, is also positive as reported in Table 5.3. (It should be noted that, in eqQ, the symbol, e, is treated as positive.) 5.4 Observation of Near Coincident Transitions The B—X spectrum of the three species of bromine is extremely dense. Even with th rotational and vibrational cooling inherent in the molecular beam there are many acci dental near coincidences of rovibronic transitions. For each isotopomer, a rotational band head was observed between the R(l) and R(2) transitions. This allowed the recording of either the R(O) and R(2) or the R(1) and R(2) hyperfine structures in a single laser scan. By fitting the observed hyperflne spectra for each rovibronic transitions separately Chapter 5. Hyperfine Structure 3f Bromine 85 one obtained, in addition to the molecular hyperfine constants, values for 6 for each tran sition. For transitions recorded in the same scan these 6 could be used to deduce the hyperfine free frequency separation, zv, of the two rovibronic transitions. That is, (6 — 62) = = [E0(1) E°(l) — (scale zero)] — [E0(2) — (scale zero)] E°(2) where E°(i) is the (hyperfine free) rovibronic transition energy of peak i (i (5.15) = 1, 2). The measurements of Lv in the R branch band head region for the different vibrational levels studied are shown in Table 5.11. The entries in the column labelled v(calc) are the corresponding values deduced from the constants published by Gerstenkorn and Luc [7]. In their work, Gerstenkorn and Luc measured the B—X absorption of 79 2 using Fourier Br transform spectroscopy, recording 80 000 rovibronic transitions in the region 11 600 cm 1 to 19 577 cm . The data were fit with 39 Dunham coefficients and one empirical scaling 1 factor to take into account the centrifugal distortion effects higher in order than the dectic term. The equivalent constants for 81 2 and Br Br 79 were deduced from isotopic 81 relations. These authors claim an uncertainty in the absolute transition frequencies of ±0.0016 cm 1 (±48 MHz). The agreement shown Table 5.11 indicate that the predicted spacings of the rovibronic transitions within a given isotopomer (for low rotational states) are accurate to at about ±5 MHz. The study of the (17’ — 2”) band of each type of molecular bromine was particu larly useful. For this band the molecular constants are such that the P(J) and R(J+3) transitions are nearly coincident allowing them to be recorded within one data set. (See Table 5.12.) The frequency difference between these rovibronic transitions, including Chapter 5. Hyperfine Structure of Bromine Isotopomer 2 Br 79 2 Br 81 Vibrational Band (13’ (17’ (13’ — — — 0”) 2”) 0”) (17’— 2”) 79 B 8 Br r (11’ (12’ (13’ (14’ (15’ (16’ (17’ — — — — — — — 0”) 0”) 0”) 1”) 1”) 1”) 2”) Rotational Transitions R(1) R(2) R(1) R(2) R(1) R(2) R(1) R(2) R(1) R(2) R(1) R(2) R(1) R(2) R(1) R(2) R(1) R(4) R(1) — — — — — — — — — — — — — — — — — — — 86 I1obs 1t’caIc (MHz) (MHz) obs-caic (MHz) R(2) R(0) R(0) 538.0 ± 4.0 46.6 ± 1.4 1054.2 ± 1.2 538.2 48.2 1052.1 —0.1 —1.6 1.9 R(0) R(2) R(0) 754.1 ± 1.0 514.5 ± 2.1 6.2± 1.6 757.1 504.2 2.0 —3.0 10.3 4.2 R(2) R(0) R(2) R(0) R(2) R(0) R(2) R(0) R(2) R(0) R(2) P(1) R(2) 276.2 ± 0.8 947.3 ± 2.3 396.2 ± 1.2 756.7 ± 5.1 516.3 ± 0.8 593.8 ± 4.6 614.7 ± 1.3 377.0 ± 1.4 740.4 ± 1.4 150.8 ± 3.0 873.0 ± 2.2 456.2 ± 1.4 21.0 ± 0.4 278.2 954.6 397.8 749.2 521.0 594.4 610.1 376.3 741.0 151.5 876.0 452.4 22.8 —2.0 —7.3 —1.6 7.5 —4.7 —0.6 4.6 0.7 —0.6 —0.7 —3.0 3.8 —1.8 Table 5.11: The observed hyperfine free separations, v, of the transitions listed com pared with the theoretical values based upon the rovibronic constants of Reference [7]. Chapter 5. Hyperfine Structure of Bromine Transitions P(l)—R(4) P(2)—R(5) P(3)—R(6) P(4)—R(7) P(5)—R(8) P(7)—R(l0) 2 Br 79 (144.2 (192.6 (243.0 (294.3 (346.0 (451.6 87 81 Br 79 ± 0.6) ± 1.2) ± 0.7) ± 1.3) ± 1.6) ± 0.9) 2 Br 81 (68.1 ± 0.1) (92.1 ± 0.4) (115.4 ± 0.8) (140.8 ± 0.6) (170.2 ± 0.9) (223.6 ± 0.3) —(6.2 ± —(6.6 ± —(8.7 ± —(7.2 ± —(7.5 ± —(1.8 ± 1.6) 0.6) 0.2) 1.0) 0.6) 0.6) Table 5.12: The observed hyperfine free frequency separations, zzJ, for some P(J) R(J+3) lines of the (17’ 2”) band of 79 2 , 81 Br 2 , and 79 Br 81 Br — — terms up to those involving the quartic distortion constant (Dy) satisfies the relation ship: (5.16) where a = 7 + 6B’ [(—10B 7 ’) + (120D 2 = 7 (20D — — 24Dç)] (5.17) 12D) (5.18) Tables 5.13 and 5.14 contain the results of a least squares fit for a and for each isotopomer and compares the results with the values of Reference [7]. The observed and calculated values of 9 are in good agreement. However, there is a systematic difference between the observed and calculated values of a of —(0.304 ± 0.017) MHz — well outside of the experimental uncertainties. The higher order centrifugal distortion terms (sextic and above) only contribute corrections of the order of tens of Hz to the calculated value of a so that the discrepancy appears to be significant. The determination made here of Chapter 5. Hyperfine Structure of Bromine Isotopomer 2 Br 79 79 ’ 8 Br Br 2 Br 81 88 (MHz) cr. (MHz) ohs caic (MHz) 47.872 ± 0.134 22.447 ± 0.035 —(2.376 ± 0.061) 48.124 22.758 —2.060 —0.252 —0.311 —0.316 b 0 r — Table 5.13: The difference between the observed and calculated values of cr. Isotopomer 2 Br 79 79 81 Br 2 Br 81 / o 3 bs / c 3 alc (MHz) (MHz) ohs calc (MHz) 0.0288 ± 0.0023 0.0298 ± 0.0006 0.0268 ± 0.0013 0.0304 0.0294 0.0284 —0.0016 0.0004 —0.0016 — Table 5.14: The observed and calculated values of . this linear combination of molecular parameters may prove useful in the future in refining the values of the rotational and distortion constants. Finally, the observations of frequency spacings between transitions arising from dif ferent bromine isotopomers provided an unusual opportunity to measure the relative positioning of the electronic potentials of 79 2 Br , 81 Br 79 , and 81 2 Br . A list of the observed frequency differences is given in Table 5.15 along with the predicted values. The results are divided into three groups for clarity. The first consists of 12 79 2 Br 81 observations. The second consists of 4 79 Br 79 81 Br third consists of a single 79 2 Br — — —* 2 observations, and the Br 81 2 observation. Clearly the experimental frequency Br 81 separations are systematically larger than the calculated values. This leads to the con clusion that the term values are slightly in error. For example, the 79 2 Br —* 81 Br 79 Chapter 5. Hyperfine Structure of Bromine Transitions (II’ (11’ (13’ (13’ (13’ (17’ (17’ (14’ (14’ (14’ (14’ (19’ (14’ (14’ (14’ (14’ (14’ — — — — — — — — — — — — — — — — — z.’(obs) (MHz) O”) R(18) —(11’ O”) R(18) —(11’ O”) P(15) —(13’ O”) P(16) —(13’ O”) P(17) —(13’ 2”) R(15) —(17’ 2”) P(12) —(17’ i”) i”) 79 i”) 79 i”) 89 P(7) —(17’ R(11) —(17’ R(12) —(17’ R(14) —(17’ 1hI)7981 1l)7981 1l)7981 — (176.1 ± 2.8) (182.0 ± 4.4) 0”)’’ R(3) O”)’’ P(4) 0”)’’ R(10) (1038.6 ± 5.2) (511.0 ± 1.4) (1219.5 ± 1.6) 854.7 330.8 1031.6 (183.9 ± 5.2) (180.2 ± 1.4) (187.9 ± 1.6) 211)79.81 (299.6 ± 0.6) (1057.8 ± 2.3) 117.8 875.0 (181.8 ± 0.6) (182.8 ± 2.3) (1071.1 (693.1 (971.9 (829.8 1.9) 1.6) 4.0) 2.2) 897.8 525.4 807.7 662.7 (173.3 (167.7 (164.2 (167.1 (459.4 ± 2.2) 283.9 (175.5 ± 2.2) R(4) R(4) 2)7981 R(3) R(5) 2h/)7981 R(7) 2lI)7981 R(10) 2l1)7981 — — — — 0hI)7981 R(8) —(17’ P(7) —(17’ P(5) —(17’ R(11) —(17’ i”) P(17) —(17’ 251.8 —902.1 211)7981 — — — — — — (427.9 ± 2.8) —(720.1 ± 4.4) — — 2”) P(6) —(13’ 11l)7981 — obs caic (MHz) O”)’’ R(0) 01)7981 R(2) — — — v(ca1c) (MHz) P(2) R(1) P(4) 2l)81 R(3) 2’4’)81 R(8) 2h1)81 211)81 — 2l)81 R(10) ± ± ± ± —(106.7 ± (337.8 ± —(458.9 ± (1086.1 ± ± 2.8) ± 1.6) ± 4.0) ± 2.2) 3.5) 0.5) 3.0) 2.2) —317.4 134.4 —669.0 874.4 (210.7 (203.4 (210.1 (211.7 (126.1 ± 2.1) —260.0 (386.1 ± 2.1) ± ± ± ± 3.5) 0.5) 3.0) 2.2) Table 5.15: Observed hyperfine free frequency differences, zj, between rovibronic tran sitions for different isotopomers as compared with the calculated differences based upon Reference [7]. Chapter 5. Hyperfine Structure of Bromine 90 data show that the discrepancy between the observed and calculated values is about 180 MHz. (The value of zv for (14’ 1”) — — (17’ — 2”) data is an exception. The fact that the z’ in this case shows a discrepancy of about 167 MHz may be evidence for some vibrationally dependent correction or may be a reflection of the uncertainty in the con stants of Reference [7]. More data are required to resolve this question.) The 79 81 Br —k 2 data display a 210 MHz correction. That is, Br 81 T 8 ’ 79 0 1 0 79,81 T 0 0 = (177 ± 8)MHz (5.19) _81 T 0 0 = (209 ± 4)MHz (5.20) Adding Equations 5.19 and 5.20 leads to the prediction, 79 0 T 0 81 T 0 0 = (386 ± 9)MHz, (5.21) in remarkable agreement with the observation of (386.1 ± 2.1) MHz given in Table 5.15. Because Gerstenkorn and Luc directly observed the 79 2 spectra and subsequently Br calculated the molecular constants for the other two isotopomers, it is reasonable to assume this shift is due to small terms that were neglected in the constants deduced for 2 and 79 Br 81 81 Br . To correct these, one needs only to shift the term values, 0 T 8 ’ 79 1 0 by —(177±9)MHz and T 81 0 0 by —(386±9) MHz. The original (T ) and corrected (T 00 ) 0 term values are shown in Table 5.16. The absolute values reported in the table are still only accurate to ±0.0016 cm (±48 MHz), but the relative positions should be accurate to about ±0.0003 cm (±9 MHz). Chapter 5. Hyperfine Structure of Bromine 79 2 Br 81 Br 79 2 Br 81 91 ‘P (D 100t)) 100 ( crn) ) 1 (crn 15823.4297 15823.9199 15824.4146 15823.4297 15823.9140 15824.4018 00 and T Table 5.16: The values of the B state term values reported by Reference [7], (B), (B). 0 the values deduced from this work, T Chapter 6 Natural Predissociation of the B state of Bromine In the presence of predissociation the lifetime of a state decreases from its nonpredis sociated lifetime. This is easy to understand as, in addition to radiation and collisions, the molecule now has another decay channel available. As a result the molecule will spend less time in the excited state. Two of the consequences of predissociation are that the level broadens and the fluorescence yield of the radiative transition is decreased owing to the molecules which fall apart without emitting a photon. These suggest differ ent experimental techniques for studying the predissociation of excited state levels: (i) measurements of transition linewidths, (ii) direct lifetime measurements of the predis sociated state, and (iii) relative intensity measurements of spectral lines. The approach taken depends upon the properties of both the molecular system under study and the type of predissociation being investigated. In this thesis the gyroscopic and hyperfine predissociation of molecular bromine was studied so that a technique that is sensitive to the properties of the individual hyperflne levels was required. The method of making measurements of the linewidths of different hyperfine transi tions has been performed on iodine [11] but the technique is very difficult to set up and use. Most of the levels of interest have sub-MHz linewidths whereas the linewidths of the lasers available for experiments are larger than 1 MHz. The authors of study [11] relied on a 2 photon resonant scattering technique in which the forward scattered light is recorded. The technique employed two frequency stabilized lasers, whose beams were made to pass colinearly through a long absorption cell. Direct lifetime measurements 92 Chapter 6. Natural Predissociation of the B state of Bromine 93 have previously been performed on the rotational states of bromine [13, 14] and, in one case, on individual hyperfine levels of the B H+ state of iodine [12]. The studies used a laser beam that was pulsed on and off and the fluorescence decay of the excited molecules was recorded. The lifetime, r, is related to the radiative, predissociation, and collisional decay rates by, = Frad+T’p+Tcoll (6.1) = Frad+CJ(J+1)+1mt+rcoll (6.2) where the CJ(J+l) term is the pure gyroscopic predissociation rate, and p,mnt represents the sum of the hyperfine and interference predissociation terms. One can obtain accurate values of the gyroscopic predissociation parameter, C, by measuring the lifetimes of a series of rotational levels. The measurement for each rotational state must be performed at various different pressures to enable the researcher to extrapolate the results to zero pressure, thereby eliminating the collisional effects. By plotting the extrapolated inverse lifetimes as a function J(J+1) one obtains C from the slope. Theoretically, the intercept can provide the radiative decay rate and the predissociation parameters of the hyperfine levels underlying the rotational transitions. However, because the intercept is very small, and because one must correctly account for the different lifetimes of the hyperfine states, it is not possible to obtain very accurate values of “rad from the intercept of such a plot. To measure the decay rates of individual hyperflne levels using this laser pulsed technique is much more difficult. Such experiments are best performed on a molecular beam and require a laser which has a very narrow linewidth to avoid exciting several hyperfine transitions at the same time. The simplest and most widely used technique [3, 4, 9] to measure the hyperfine pre dissociation relies on the fact that hyperflne predissociation couples different hyperfine states to the continuum by different amounts. This implies that the fluorescence yield Chapter 6. Natural Predissociation of the B state of Bromine 94 will be different for different hyperfine levels of a given rotational state. Thus, the hyper fine fluorescence spectra will display anomalies in the relative intensities of the spectral lines. The intensity of a peak may be fit to the model S(x) o So(x) ’1 rad l’rad + Pp (6.3) where 0 $ ( x) is the intensity of the transition in the limit that P = 0. The predissociation rate, P, has been described in Chapter 3. It is very difficult to measure the absolute intensity of a given transition as this requires a very detailed knowledge of the detector efficiency, the laser beam profile, and so on. It is much easier to measure the relative intensities of two transitions labelled here as a and b. S(a) S(b) — — So(a) T’rad+Pp(b) S ( 0 b) Praa+Pp(a) (64) One of the problems with this technique is that it only produces the ratios of the predis sociation parameters to the square root of the radiative decay rate, namely, aV/\/, and b//T. The main advantage of the relative intensity measurement technique is that it is comparatively easy to set up and use, requiring almost exactly the same experimental apparatus as was used for observing the hyperfine structure of bromine, There are some serious drawbacks to this technique as well. First, the amount of time required to collect the data for the hyperfine structure of a single rotational transition was of the order of ten minutes. Thus, any variations in the intensities of either the laser beam or the molecular beam over this time introduce errors into the relative intensity measurements. Second, as was discussed in Chapter 2, the peak intensities of the different hyperfine transitions are sensitive to the polarization of the laser beam, to the positioning of the detector, and to the amount of solid angle observed by the optical system. Third, care had to be taken to ensure that the transitions were not saturated. A final, more Chapter 6. Natural Predissociation of the B state of Bromine 95 subtle point is that the experiment had to be designed so as to collect the fluorescent light with equal efficiency from the shortest to the longest-lived states being observed. If this is not done then many of the molecules excited into long-lived states will move out of the detection region without radiating while most of the molecules excited into short-lived states will decay inside the detection area. The net result is to enhance the relative strength of the states with short-lifetimes as compared to those with long lifetimes. This experimentally introduced bias will be reflected in the predissociation parameters obtained. The radiative lifetime of the B state of bromine has been reported by two different authors [13, 14]. Clyne et al. [13] determined Trad by measuring the lifetimes of rotational states of 79 2 Br to J = , 2 ’ 8 Br , and 79 81 for various vibrational states and extrapolating Br 0. The lifetimes reported varied from 5 to 16 Its. A second study of the B state of 81 2 by Peeters et al. [14] using a similar direct lifetime measurement technique Br found Trad to be from 2 to 5 s. The values of Trad reported by Clyne et al. displayed a large variation with vibrational state and no attempt was made to take into account the hyperfine predissociation. In contrast, the work of Peeters et al. [14] attempted such a correction. Consequently, the latter work was accepted and it was initially assumed here that the radiative lifetime of the B state of bromine was of the order of 3 ps. Since the predissociation decreases the lifetime of a state, Trad = F, is the longest lifetime of any state. The optical stack described in Chapter 5 could image a circle of roughly 3 mm in diameter where the laser and molecular beams intersected. Assuming that the average velocity of the molecules in the beam was 400 m/s, this meant that a molecule with a 3 lifetime travelled, on average, 1.2 mm from the point of excitation to the point of emitting a photon. With the existing optics, one could observe about 2.5 decay lengths or longer, sufficient for measurements with accuracy better than 10%. With this in mind, work was started on measuring the predissociation of the B -‘- Chapter 6. Natural Predissociation of the B state of Bromine state of 79 2 Br , , 96 and 79 81 employing the relative intensity technique. Several Br previous studies [3, 4, 14] had already characterized the hyperfine predissociation of bromine by the model developed by Vigué et al. [8] for iodine. This model included only up to the magnetic hyperfine predissociation terms. The results reported for bromine by References [3, 4] displayed some anomalous variation of the parameters with rotational state. This difficulty was thought to be not very serious and that the relative intensity measurements could be used to refine the previous results. 6.1 Saturation Tests The first step in performing the predissociation measurements was to ensure that the laser was not saturating the transitions under study. The signal from a specific hyperfine transition was recorded several times as the laser intensity was varied. The signal intensity as a function of the laser intensity could be plotted. From this plot the point at which the laser began to saturate the transition could be estimated as the point where the plot deviated significantly from a straight line. In practice this was difficult because the observed LIF signals were sensitive to the variations in the position of the laser beam. Initially two consecutive polarizers were placed in the laser beam path with the second polarizer’s transmission axis held fixed (as shown in Figure 6.1) aligned with the laser’s initial axis of polarization. This was to ensure that the polarization of the laser beam interacting with the molecules remained constant to eliminate the possibility that changes in the LIF were due to variations of polarization of the laser beam. The intensity of the laser beam passing through the two polarizers is, I(/3) = Iocos C 4 6) (6.5) where /9 is the angle between polarizer (P1) and fixed polarizer (P2) axes. With this arrangement the intensity of the laser beam interacting with the molecular beam could be Chapter 6. Natural Predissociation of the B state of Bromine 97 P2 P1 Laser Figure 6.1: A two polarizer (P1, P2) arrangement for maintaining the direction of polar ization of a laser beam while varying the beam’s intensity. continuously varied. Unfortunately, as P1’s axis was changed the direction of propagation of the laser beam also changed. The beam could be displaced by about one beam diameter over a 2 meter path length ruling out this arrangement. A method of rotating the polarization of the laser beam without displacing the beam’s position was required. The solution was to place a Pockels cell between two fixed po larizers as shown in Figure 6.2. A Pockels cell is an isotropic material which becomes birefringent when a voltage is applied across it. The result is that the axis of polarization of the light can be rotated as a function of the voltage. This provides a straightforward technique for varying the laser’s intensity while maintaining the polarization of the laser light passing through the polarizer — Pockels cell arrangement. With this technique no displacement of the laser, to better than 0.1 beam diameter, was observed over a 6 meter Chapter 6. Natural Predissociation of the B state of Bromine P1 RD 98 P2 Laser Q P2 ‘P1 Laser Figure 6.2: The polarizer—Pockels cell (PC) arrangement for varying the intensity of a laser beam while maintaining the direction of polarization of the transmitted light. Chapter 6. Natural Predissociation of the B state of Bromize ( v’ (13’ (16’ (17’ — v”) Franck—Condon Factor (F — — — 99 J) 60 mW 3 mW 1 mW 0.00041 0.00799 0.02405 0”) 1”) 2”) = 2 as Br Table 6.1: The Franck—Condon factors for various B—X vibrational transitions for 79 Equation and the calculated saturation laser powers based on 6.6. given by Reference [38] path. The intensities of the B—X P(4), F” transitions for the (13’ — 0”), (16’ 4 to F’ = 1”), and (17’ — 4 and the F” — = 4 to F’ = 3. hyperfine 2 were Br 2”) vibrational bands of 79 recorded as a function of laser power. The results are shown in Figures 6.3 to 6.5. For transitions from the same lower state to different upper states, the stronger the transition, the easier it is to saturate. Therefore, the F” saturation threshold than the F” = 4 to F’ = 4 to F’ = 4 peaks should show a higher 3_ peaks, exactly as observed. The vibrational band to vibrational band saturation dependence is primarily deter mined by the Franck—Condon factors (FCF) for the transitions. The larger the FCF, the stronger the transition, and the easier it is to saturate. The FCF’s for these three bands, quoted from [38] are given in Table 6.1. Again observed transitions follow the behaviour expected with the (13’ — 0”) band being the hardest to saturate and the (17’ — 2”) the easiest. The laser power required to saturate transitions from different vibrational bands may be estimated from the data in Figures 6.3 to 6.5 and the FCF’s. i.e. Psat(v’ — v”) = FCF(13’ 0” FCF(v’ _Vl)t( — 0”) (6.6) Chapter 6. Natural Predissociation of the B state of Bromine 100 300.0 ‘ 200.0 I •— /) 100.0 0.0 0.0 50.0 100.0 150.0 200.0 250.0 300.0 Laser Power (mW) 2 Br Figure 6.3: The observed laser power dependence of the signal intensities of two 79 J F” = 4 —* F’ = 3_ transition B—X (13’ — 0”) P(4) hyperfine transitions: (a) the zF = 1 and (b) the LF LSJ F” = 4 —* F’ = 4 transition. Transition (b) shows a linear response, i.e. no saturation, for laser powers of up to 250 mW while the stronger transition (a) displays nonlinear behaviour at an input power of about 60 mW. Chapter 6. Natural Predissociation of the B state of Bromine 101 500.0 400.0 300.0 .— 200.0 C,) 100.0 0.0 0.0 50.0 100.0 150.0 200.0 250.0 300.0 Laser Power (mW) 2 B—X (16’ 1”) P(4) (a) and (b) Br Figure 6.4: The observed signal intensities of the 79 hyperfine transitions as a function of laser power. These both display nonlinear behaviour beginning at very low laser power, P < 15mW. — Chapter 6. Natural Predissociation of the B state of Bromine 102 150.0 ‘‘ 100.0 •— ) 1 Cl 50.0 0.00.0 50.0 100.0 150.0 200.0 250.0 300.0 Laser Power (mW) Figure 6.5: The observed signal intensities of the 79 2 B—X (17’ Br 2”) P(4) (a) and (b) hyperfine transitions as a function of laser power. These transitions are more easily saturated than their (16’ 1”) and (13’ 0”) band counterparts. — — — Chapter 6. Natural Predissociation of the B state of Bromine 103 A final note is made here that, although the laser intensity should be quoted in power per unit area, the saturation intensity has been written as a power. This is done because the area of the laser beam when it intersects the molecular beam is constant (roughly 6 ) and it is more convenient to measure the laser power than the intensity before it 2 mm enters the molecular beam machine. 6.2 Polarization Effects As discussed in Chapter 2, the LIF signal depends upon the polarization of the laser and the positioning of the detector. The LIF signals of the low J hyperfine levels are the most sensitive to the polarization of the laser radiation. To test this effect, a similar arrangement as was used for the saturation tests was employed (Figure 6.2). The second polarizer, P2, was removed and, by using the Pockels cell, the polarization of the laser could be rotated from vertical (z direction) to horizontal (y direction) without changing the direction of propagation of the laser beam. The 79 2 B—X (13’ Br — 0”) R(0) transition was chosen as suitable for study because the hyperfine spectrum provides pairs of resolved peaks, labelled a, b and c, d in Figure 6.6, which start in separate lower hyperfine states and end in the same upper hyperfine state. Thus any change in relative intensities for a:b or c:d will be due solely to polarization effects and not to any differences in predissociation of the upper hyperfine states. The observed and predicted relative intensities are given in Table 6.2. The discrepancy between the theoretical and experimentally observed ratios is due to the fact that the theoretical values were calculated assuming that the detector collected light only over a very small solid angle. These measurements of the polarization depen dence of the LIF may be used to calculate the amount of solid angle observed. This, in turn, must be used to deduce the expected relative intensities of the different hyperfine Chapter 6. Natural Predissociation of the B state of Bromine sy sz 1÷ II 104 ÷5 C d v (MHz) v (MHz) Figure 6.6: The observed 79 2 B—X (13’ — 0”) R(0) hyperfine spectrum with the laser Br polarized vertically (z-direction) and horizontally (y-direction). The relative intensities of peak a ( F” = 2 —* F’ = 1) compared to peak b ( F” = 0 —f F’ = 1) and peak c (F” = 2 — F’ = 1) compared to peak d ( F” = 0 —* F’ = 1) show a strong polarization dependence. Chapter 6. Natural Predissociation of the B state of Bromine (a:b)theory b 0 (a:b) z polarization sz y polarization sy 2.70 1.26 ± 0.15 0.76 0.96 ± 0.10 105 Table 6.2: The intensity ratios of the B — X R(0) (13’ — 0”) hyperfine transitions a:b F” = 2 —* F’ = 1) to (F” = 0 —* F’ = 1) for 79 2 with the laser polarized vertically Br (z-direction) and horizontally (y-direction). transitions in the absence of predissociation. Without taking this correction into account, the values obtained for the predissociation parameters obtained from relative intensity measurements will be in error. Relative Intensity Measurements 6.3 A series of measurements of the relative intensities of the hyperfine transitions of 79 2 Br 2 Br 81 , Br was made for various upper electronic state vibrational levels. The 81 11r and 79 relative intensities for the levels in v’ = 11, 12, and 13 appeared to be in agreement with the results previously published [3, 4]. However, our investigation of the v’ J’ = 0 hyperfine levels with F’ = = 5, 1 and 3, showed inconsistent results from day to day and between isotopomers. To make sense of this observation one must realize that the v’ = 5 level is strongly predissociated [13] so that the (J’ = 0) hyperfine levels have very different lifetimes. One can picture the apparatus as follows. The molecules in the beam travel in the positive x direction with a constant velocity, V. Let x = —S be defined as the position of the molecular source. The optical system collects LIF photons emitted between the 106 Chapter 6. Natural Predissociation of the B state of Bromine positions x = —B and x = B; x = 0 lies at the center of this region. The laser radiation intersects the molecular beam at 90° at position X. If one studies two levels, one with a (long) lifetime, TL, and the other with a (short) lifetime ‘rs, then the observed relative intensities of the two transitions will depend upon the velocity of the molecular beam, V, the length of the detection region, (2B), the position of the laser beam—molecular beam interaction, X, and the two lifetimes, 2B = 5 mm, TL = 10 ps, and Ts = TL and rs. As a concrete example, let V = 500 m/s, 1 ns. This implies that the molecules excited into the long-lived state travel 5 mm before emitting a photon (on average) and those excited into the short-lived state travel only 0.5 tm before decaying (on average). The consequences of this are illustrated by the following three experiments. (i) Suppose that the interaction point is at the center of the detection region (X = 0). In this case one will observe virtually all of the fluorescence from the shortlived level, rs, and about 40% of the fluorescence from the long-lived level, TL. (i.e. Almost all of the short-lived molecules decay within the detection region while only about 40% of the long-lived decay in the same region.) Thus one records a spectrum with two peaks. (ii) The laser radiation and molecular beam intersect 5 mm before the detection region (i.e. X = —B—S mm). Here, almost none of the molecules excited into the short- lived level will be observed while about 24% of the molecules excited into the long-lived level are seen. One observes only one spectral line and could conclude that the absence of the transition from the short-lived level is evidence for strong predissociation of this level. (iii) Finally, the laser beam is arranged so as to intersect the molecules 1 mm behind B ( X = the TS B—I mm). With this arrangement one observes almost all of the LIF from level and almost no LIF from the TL level. Again one records a spectrum with Chapter 6. Natural Predissociation of the B state of Bromine X (mm) —1.54 0.00 1.24 2.46 S(F’=3) (arb.) 3.32 4.77 3.94 2.15 ± ± ± ± S(F’=l) (arb.) 0.13 0.32 0.21 0.13 4.42 4.53 2.56 1.90 ± ± ± ± 0.53 0.38 0.24 0.26 107 S(F’=3)/S(F’=1) 0.76 1.06 1.52 1.14 ± ± ± ± 0.09 0.08 0.11 0.12 a/T’rb (2.154 (0.866 (0.363 (0.726 ± ± ± ± 0.255) 0.065) 0.026) 0.076) Table 6.3: The intensities, S, of the two 81 2 B—X (5’ — 1”) P(1) hyperfine transitions Br as a function of the laser beam molecular beam interaction position, X. S(F’ 3) arises from the transition F” = 4 —* F’ = 3, and S(F’ = 1) arises from the transition F” = 1 —f 1. — only one peak. This time one would conclude that the long-lived level is completely predissociated. The point of this discussion is that if the optical system does not collect the LIF from the long- and short-lived states under study with equal efficiency, then the predissociation parameters deduced from the measurements of the relative intensities of spectral features will be in error. In light of the anomalous findings for the (v’ = 5 J’ = 0) hyperfine levels of bromine the optical system used for the relative intensity measurements was tested. A flat glass plate was placed in the path of the laser beam so that the laser beam — molecular beam interaction position, X, could be varied simply by rotating the plate. The intensities of the two hyperfine components were studied as a function of X in the P(l) rotational transition of the (5’ and (F” = 1 —* — F’ 1”) band. The transitions selected were (F” = = 4 —* F’ = 3) 1). The former had a much shorter lifetime than the latter. The results are summarized in Table 6.3. As is evident, the signal from the longer-lived F’ = 1 hyperfine level decreased rapidly as the laser was moved away from the source. Chapter 6. Natural Predissociation of the B state of Bromine The signal from the shorter-lived F’ = 108 3 state dropped less rapidly when the laser beam was moved away from the source and also decreased when the laser was translated closer to the molecular beam source with respect to the center of the optical detection region. These observations are easily understood in the light of experiments (i), (ii), and (iii) described above. The final column in Table 6.3 lists the parameter, 2 — l’rb — “rad a2 + 2.4b 67 deduced from the different measurements. (Note that previous authors [3, 4, 14] have assumed b = 0.) These findings clearly indicated that the optical collection system was inadequate and had to be re-designed. As well, they pointed to the possibility that the radiative lifetime of the B state of bromine was much longer than the value of about 3its initially assumed. 6.4 The Lifetime Plan The active area of the photomultiplier was a rectangle (22 x 7) mm 2 arranged so that the long axis coincided with the molecular beam axis. With an optical system of unity magnification one could, at most, detect light from a 22 mm length along the molecular beam. For molecules travelling at 500 m/s, this distance represents 3 decay lengths for a molecule excited into a 14.5 s lifetime level. The new optical design relied upon the lens arrangement shown in Figure 6.7. With a lens (Li) placed 2 focal lengths (2f) in front of an object (0), an inverted, unity magnification image (Ii) is formed 2f behind the lens. A second lens (L2) placed 4f behind the first forms another unity magnification, inverted image of II 2f behind the second lens (12). This process can be continued with more lenses to transport a unity magnification image of the original object along a distance D = 4Nf (where N is a positive integer). This arrangement can be refined by adding a lens identical to LI 2f away from it, as shown in Figure 6.8. One of the advantages Chapter 6. Natural Predissociation of the B state of Bromine 109 I 0 Li Ii L2 12 Figure 6.7: Illustration of the design principle used for the final optical stack. of using pairs of lenses separated by 2f is that the image formed is 4f away from the original object even if the object position is considerably different from 2f away from the first lens. In addition the second lens helps to collect more light and correct some of the aberrations of the first lens giving a brighter, sharper image. Three such lens pairs (Melles Griot aspheric glass condensers number 01 LAG 007, focal length 26.5 mm, diameter 30 mm) were used to collect the fluorescent light and pipe it up to the active surface of the photoinultiplier. The final optical arrangement is shown in Figure 6.9. The 110 mm focal length lens was kept out of convenience as this acted as a the vacuum seal for the molecular beam apparatus. Direct measurements of the final optical arrangement found that the new stack imaged an area of (18 x 7) mm . 2 Next the input and exit laser windows were enlarged from 3 mm to roughly 1.2 cm to allow the laser beam to interact with the molecular beam at various positions, X. (As in the previous discussion, X = 0 represents the center of the optical collection region, X < 0 is closer to the source of the molecules, and X > 0 is farther downstream.) This arrangement allowed one to make relative intensity measurements as a function of the number of decay lengths observed in the optical collection region, and a determination Chapter 6. Natural Predissociation of the B state of Bromine 110 2f Figure 6.8: A pair of identical lenses separated by 2 focal lengths, 2f, produces the image of an object 4f away from the object independent of changes in the position of the object with respect to the lenses. of the response of the optical system. The response across the detection region, shown in Figure 6.10, was measured by recording the strength of the B—X (13’ F” = 11 —* F’ = — 0”) R(9) 12 hyperfine transition as a function of X. The optical stack’s collection region was —9 mm X 9mm. By making X 0 the laser intersected the molecular < beam closer to the molecular beam source, and one had a longer LIF collection length. Conversely, making X > 0 the optical collection length was shortened. With the new design the laser beam could be translated from X approx —8 mm to X observing this relatively short-lived transition, r 2.5 mm. By 1is, one did not have to worry about losing signal due to molecules decaying outside the viewing region over the range of X tested. The response of the optical stack was assumed to be symmetric about X = 0 and the data was fit with to a quadratic, R(X) = 0.96 — 0.000239X — 2 0.00207X (6.8) This large window also allowed a determination of the velocity of the molecular beam. Chapter 6. Natural Predissociation of the B state of Bromine 111 PMT L7 7 L6cç Filter IA” L3 Li s’ ÷Ar 2 Br Laser Mi Figure 6.9: The new optical arrangement. Three pairs of identical lens doublets L1—L2, L3—L4, and L6—L7, were used to move the LIF onto the photomultiplier (PMT) with unity magnification. The 110 mm focal length lens, L5, was used as a window out of the vacuum chamber and slightly demagnified the image. This arrangement allowed the the . 2 detection of an area of (18x7) mm Chapter 6. Natural Predissociation of the B state of Bromine 112 1.10 I 1.00 Cfj 0.90 C 0.80 0.70 -10.0 — -5.0 0.0 5.0 10.0 X (mm) Figure 6.10: The response of the optical collection system across the viewing area. Chapter 6. Natural Predissociation of the B state of Bromine 113 By recording a hyperfine spectrum with the laser beam reflected back through the molec ular beam as shown in Figure 6.11 (a), one observed two overlapping spectra shifted in frequency by amounts, v. Iv = — ii 2 L1i1a = — 6a (6.9) —sina C V• sin(/3 — — — z) — o) (6.10) Lii I) = (sina—sin(3—a)) (2a—$) (6.11) Similarly one observes two overlapping spectra with the arrangement in Figure 6.11 (b) shifted by, — — ii C Lv — = s]n(cr +7) (6.12) v. ——s1n(c+7) (6.13) - ii I) = 2 sin(Q +7) (2c’+2j’) (6.14) By measuring these two frequency shifts one obtains the molecular beam velocity, v, according to: C v = — ii 6 b a 27+/9 (6.15) Using a nozzle of diameter (21.8 ± 3.6)m, a skimmer of 1 mm diameter and a mixture of 20% bromine in argon at a backing pressure of roughly 300 Torr, the velocity of the molecular beam was measured to be 2 VAr+Br = (420 ± 60)m/s Chapter 6. Natural Predissociation of the B state of Bromine 114 For a pure bromine beam using the same nozzle and skimmer at a backing pressure of 180 Torr, the velocity was, 2 vBr = (350 ± 80)m/s Using the new lens configuration, very different LIF spectra were observed. Fig ure 6.12 shows the hyperfine spectrum of the B—X (13’ — 2 as Br 0”) P(1) transition of 79 observed with the old optical system (A) and the new system (B). As can be seen, the relative intensities of the spectral lines arising from the long-lived F’ = 1 levels were much larger with the re-designed optical system. This was clear evidence that the previous LIF optical collection arrangement was inadequate. This re-design worked well but had three drawbacks: first, the results of the relative intensity measurements had to be corrected for the variations in collection efficiency across the optical viewing region; second, the new optical stack collected roughly 4 times less light compared to the original design; third, the new stack had more components leading to more reflection losses. The signal to noise with the new system dropped by about a factor of 8 to 10 compared to the previous arrangement. To compensate, the slits collimating the molecular beam were widened to increase the size of the molecular beam, regaining a factor of 2 in signal to noise. This led to an increase in the residual Doppler widths of the observed hyperfine lines from 3—4 MHz to about 8 MHz. In the end, the overall signal to noise was decreased by a factor of 4 to 5. 6.5 The Phase Shift Method Direct measurements of the lifetimes of individual hyperfine levels is the best way to determine not only the predissociation parameters but also the radiative lifetime of the Chapter 6. Natural Predissociation of the B state of Bromine 6a II 115 b 6 I I I 0 0 —135 v (MHz) laser : I I I —135 ‘ (MHz) laser /Yads is Figure 6.11: The two types of measurements used to deduce the molecular beam velocity. (a) The laser beam crosses the molecular beam (moving at an angle a to the y-axis shown), is reflected back at an angle /3, and crosses the molecular beam again, resulting in the two overlapping hyperfine spectra shown. (b) The laser beam crosses the molecular beam at an angle —y to the y- axis and is reflected back on itself give the two overlapping spectra shown. Chapter 6. Natural Predissociation of the B state of Bromine 116 (B) (A) 1* Ij 31* 1 3* 1 3* I II H I III I I I 1 1 I —200 I I 0 200 v (MHz) I —200 v 0 200 (MHz) 2 (13’—O”) P(1) hyperfine spectra using the initial optical Br Figure 6.12: The observed 79 collection system (A) and the redesigned system (B). The long-lived F’ = 1 levels show a dramatic increase in intensity relative to the short-lived F’ = 3 levels with the longer viewing length indicating that the initial collection system was inadequate. The peaks labelled with both a 1 and 3 are blended lines with the * indicating the stronger of the two forming the feature. Chapter 6. Natural Predissociation of the B state of Bromine 117 electronic state being studied. Unfortunately, with the available equipment it was not possible to rapidly switch the laser beam off and on to directly monitor the LIF decay. However, there was an alternative technique which could be applied here: the phase shift method [39]. In this technique the laser beam’s intensity is modulated at angular frequency. fi, and the resulting LIF lags behind the excitation by phase, tan = , —fir according to (6.16) where r is the lifetime of the state being studied. To understand the technique one can conceptualize the molecular system as consisting of the levels. n , the lower and upper states respectively. Molecules are excited out 1 0 and n of n 0 with a modulated resonant laser beam into level n 1 with lifetime T. For simplicity the laser intensity, ‘laser, will be described by Ilaser = Io(1+sinflt) (6.17) where fi is the angular modulation frequency. The excited molecules then decay at rate I’ = r into other levels. Assuming that the population of the lower level, n , is not 0 appreciably altered by the laser field (i.e. weak laser intensity) one has, 1 dn = nollaser = nolo(1+sinflt)—T’ni — “fli (6.18) Choosing the trial solution, ni(t) = Msinflt+Ncosflt+O (6.19) one obtains, ni (t) = 1’ 1 [fl2 0 n + sin fit — nojo ci 1 f2 cos fit + 1 0 n —i-- (6.20) Chapter 6. Natural Predissociation of the B state of Bromine 118 This solution is of the form, ni(t) = 0 sin(Qt + q) + 0 N = Nocosqsin1t+Nosinqcos1t+O (6.21) By inspection, the phase shift is given by tan4 (where r = = — = —fT (6.22) 1/f) in agreement with Equation 6.16. It. is important to realize the main advantage of this type of approach. This method is a ratiometric technique. That is, one needs only to measure the ratio of the coefficients of the cos t and the sin 2t terms to deduce the lifetime of a given state. This is achieved by detecting the LIF with a lock in amplifier. The in-phase (IP) component corresponds to the part of the signal oscillating in phase with the excitation (the sin ft part above), and the in-quadrature (IQ) component is the part of the signal 900 out of phase with the laser modulation (the cos2t term). This gives, tan = = —1r (6.23) Because one is not comparing the intensities of one hyperfine peak to the next, the results of this technique are no longer sensitive to temporal variations in the laser and molecular beam intensities. As well, the polarization, and detector efficiency variations over the field of view cancel out. In short, this method is insensitive to many of the systematic errors inherent in other techniques. Chapter 6. Natural Predissociation of the B state of Bromine 119 Detector Window I I I I — — aser 1h 0 L W Figure 6.13: A schematic diagram of the laser beam and optical collection region with respect to the molecular beam. 6.6 Full Theory The arrangement of the laser beam and optical collection region with respect to the molecular beam is described schematically in Figure 6.13. The molecular beam is shown with molecules travelling at velocity, v, along the x-axis. The laser beam intersects the molecular beam at 900 and is idealized as being of length L, as having a constant electric field amplitude over this length, and is modulated according to ‘laser I(1 +asin1t) The optical system collects photons over the length 0 x (6.24) W. The probability, P(XD, t), that. a photon is emitted at a position x = xD to xD + dx within the observation region, Chapter 6. Natural Predissociation of the B state of Bromine 120 and at a time t, is given by the probability, Pe(Xo, to), that molecule was excited at a positioii x 0 at an earlier time, to = t (XD — — xo)/v , times the probability, Pd(xD, xo), that the molecule remains in the excited state from the position where it was excited to position XD, where it decays. The probability that the molecule was excited at time to and position x 0 is proportional to the laser intensity, ‘laser; ) cx I(1 0 0 Pe(Xo,t +asinct ) . (6.25) The probability that the molecule remains in the excited state is proportional to a simple exponential decay; Pd(xD,xo) cx ( exp F(xD—xo)” dxD (6.26) — Therefore, using this model, the amount of signal collected by the detector between x = XD to xD + dxc, at time, t, dS(t,xD) is, (_fxD exp xo)) (1 + asin 1to)} = (6.27) where iV is a normalization factor. The total signal observed at time t is, S(t) JdS(t;xD) (6.28) These integrations must be performed for decays in two separate regions; region (I) is the area outside the laser beam, and region (II) is inside the laser beam. For region (I), the integral in Equation 6.27 is over 0 L <xD S ( 1 t) xc, L and the integral in Equation 6.28 is over W. The result is = (exp (—PT) +(p2f2)2 — exp (—PU)) (exp (—PT) ) 1 {F(r,c,T ) 1 {G(r,c,T — — ) 2 F(P,f,T ) 2 G(r,c,T — — — 1) ) + (P 3 F(P,,T 2 — i2)} ) +2rc}] 3 G(P,f,T (6.29) Chapter 6. Natural Predissociation of the B state of Bromine 121 where 1 T = W/v (6.30) 2 T = L/v (6.31) 3 T = (W — L)/v (6.32) and F(P, Q, T) = exp(—PT) [(P2 — ) cos fT 2 ç 2P[ sin lIT ] 1 (6.33) G(F, lI, T ) 1 = ) 1 exp(—T’T [(P2 ) sin lIT 2 lI 1 + 2FlI cos lIT ] 1 (6.34) 0(1’, lI, T ) 0(1’, fz, T 1 ) — G(1’, lI,T 2 ) + 21’lI 3 F(P, lI, T ) F(P, lI, T 1 ) F(F, lI, T 2 ) + (P2 3 (6.35) — — By inspection one has — tan — — — In the limiting case where W — — cc and T 2 —* 0, tan bi —* —lI/P as expected. For region (II) the integral in Equation 6.27 is performed over 0 integral in Equation 6.28 is performed over 0 Sij(t) = .N 2 T XD xo xD and the L. The solution for region (II) is, (1 —exp 2 (—PT ) ) f2 asinlIt F(P,lI,T ) 2 +lI (P ) 2 acos lIt ) 2 G(P,lI,T l1 IT 2 + (P2 — (P2 + 2 lI j ) ) 2 + lI 1 P T 2+ (P — 2 V) +lI (P ) 2 2PlI (P2 + lI ) 2 (6.36) and tan ji = — Here, again, tan qjj — (P + lI 2 lIT ) + G(P,cl,T 2 ) — 2PlI 2 (P + lI PT 2 ) + F(P, lI, T 2 ) — (P2 — lI 2 ) 2 —lI/P as T 2 — cc. The total signal observed is the sum of 1 S ( t) and 1 Sj ( t), (6.37) Chapter 6. Natural Predissociation of the B state of Bromine st’ T 2 exp (—PT ) (1 exp (—PT 1 )) 2 P asinft ) 1 F(P,f,T ) 3 F(P,Q,T PT 2+ 1 +(P22)l ) 22 (P +Q (P ) 2 acosZt ) 1 G(f,1,T ) 3 G(F,f,T T 2+ (P 1 2) 2 — — 122 — — (f2+2)l — (p2+2) (6. 38 ) with an — — Provided PT 1 = (r + c1 2 cT ) + G(PM,T 2 ) 3 ) G(P,Q,T 1 ( PT + ) 2 ) 1 ) 3 —F(P, +F(Pj, P 1 T 1,T PW/v and PT 3 — = [P(W — (639) L)]/v are large compared to 1, the F(F, Q, T ) 1 and G(P, f, T ) terms in the expression for tan 1 are negligible and one obtains the usual result. tan4 = — = —ar. (6.40) (6.41) Equation 6.41 holds provided one collects a large enough fraction of the LIF and the LIF is due to a single transition. The behaviour of the expression for (tan q)/fZ (referred to here as the apparent lifetime) as a function of the viewing length, W, is shown in Figure 6.14 for states with various different lifetimes. In Figure 6.14 it is assumed that the molecules in the molecular beam are travelling at 420 rn/s and the laser beam— molecule interaction length, L, is 2 mm. From the plot one observes that under such conditions, a viewing length of 16 mm is sufficient to measure lifetimes of up to 12 1 us to within 8%. The behaviour of the apparent lifetime with molecular beam velocity is shown in Figure 6.15. As expected, the reliability of the measurement of longer lifetimes decreases as the velocity increases. For a level with an 8s lifetime, the accuracy is degraded to ±5% when the velocity changes from 420 rn/s to 500 rn/s. Another advantage of using the phase shift method was the insensitivity of the results to the variations in the efficiency of the detector over the field of view and to the exact Chapter 6. Natural Predissociation of the B state of Bromine 123 15.0 a CID b I 10.0 . d a) t 0.0 • . 0.0 a.. 10.0 5.0 • ..1... 15.0 a.. • 20.0 .1.... 25.0 30.0 Viewing Length (mm) Figure 6.14: The apparent lifetime as a function of the viewing length along the molecular beam that is imaged. Curves a f are for states with true lifetimes of l2its, lOits, 8ts, 5is, 2s, and 1ts, respectively. The above calculation (with v = 420 rn/s and 1 = 2ir(10 kHz)) indicates that a viewing window of 16 mm will lead to an 8% error in the lifetime of a 12s state, a 5% error for a lOgs state, and a 3% error for an 8its state. — Chapter 6. Natural Predissociation of the B state of Bromine 124 10.0 9.0 Cl) a S a • b C 50 I. 3.0 a:v=300m/s b: v=400m/s C: v=500m/s 2.0 1.0 •••• 0.0 • . • . .1... • .1.... 20.0 25.0 • 0.0 5.0 10.0 15.0 30.0 Viewing Length (mm) Figure 6.15: The apparent lifetime as a function of the viewing length and molecular beam velocity for a state with an 8ps lifetime. Chapter 6. Natural Predissociation of the B state of Bromine 125 spatial profile of the laser beam. Both effects were investigated by numerically integrating the equations for the observed signal. 6.7 Experimental Arrangement The new optical stack was placed in the beam machine as shown in Figure 6.16. The laser was modulated using the Pockels cell (Lasermetrics LM4-A), PC, placed between two crossed polarizers. The first polarizer, P1, had its axis aligned vertically while the second, P2, passed light with horizontal polarization. The Pockels cell sinusoidal modu lation voltage was supplied by tunable frequency function generator (IEC F33 Function Generator). SG, fed into a power amplifier (Techron model 7560) followed by a 32:1 step up transformer, T. With this arrangement the modulation frequency could be varied from about 5 kHz up to over 40 kHz. The laser beam power entering the molecular beam machine was kept below 45 mW for studying the B—X (13’ — 0”) transitions of 79 2 Br , 2 Br 81 , and 79 81 to avoid Br saturation. The beam passed through the Pockels cell, PC, and entered the molecular beam machine approximately 6 — 7 mm upstream of the center of the optical stack. This provided an optical collection length of at least 15 mm for the LIF. The light collected passed through a Corning red pass filter (CS2-62) to reduce the scattered laser light before entering the photomultiplier. The signal from the photomultiplier was fed into a lock-in amplifier (EG&G model 5204), LIA, which measured both the in-phase (IP) and in-quadrature (IQ) responses of the input signal. The IP and IQ outputs were recorded on a two channel chart recorder (Phillips 8252A PM), CR. Tests were made on the lock-in to ensure that the zeroes, gains, and phase measurements were consistent between the IP and IQ channels. The zero values and gains of each channel of the chart recorder were checked as well. Chapter 6. Natural Predissociation of the B state of Bromine 126 SG T M2 Ml Br + 2 Ar Figure 6.16: A schematic of the apparatus used for measuring the lifetimes of individual hyperfine levels. (See text for description of symbols.) Chapter 6. Natural Predissociation of the B state of Bromine 127 After selecting a modulation frequency, !2, the lock-in amplifier’s phase setting, pLIA, was chosen so that the hyperfine signals observed on the IP and IQ channels were roughly equal. The hyperfine transitions of interest were recorded between 5 and 8 times. The reference phase of the laser beam, thref, was measured by closing off the molecular beam, removing the red-pass filter, and observing the scattered laser light. The phase shift of any given hyperfine signal could be deduced from = arctan ({) + LIA — &ef (6.42) Each series of hyperfine transition measurements was repeated for at least two and usually three different modulation frequencies. By knowing the phase shift and the modulation frequency, the lifetimes of the individual B state hyperfine levels involved in the observed transitions were deduced from Equation 6.41. The hyperfine transitions for the (13’ P(1) through P(9) excluding the P(3) of 79 2 Br , 2 B 8 r , — 0”) and 79 81 were recorded. Br A complete list of the hyperfine levels and lifetimes used in the analysis is given in Appendix C. 6.7.1 Fitting the Hyperfine Lifetimes The inverse lifetime for a hyperfine level subject to both gyroscopic and hyperfine pre dissociation is given by, r = r’rad+rp (6.43) (assuming no saturation or collisions). The predissociation rate is given by = CRaG + 2a.[Rii(1,1) + (1,2)] 11 R + 4V’aCVRGI(1) + 4vbVCvRG (1) 2 (1,1) 12 + 4abR (1, 1) + (l, 22 22 2)] R + 2b [R (6.44) Chapter 6. Natural Predissociation of the B state of Bromine 128 for 79 2 arid 8 Br 2 ’ Br and by 1, = CRGG + [a(l) + a(2)jRii(1,l) + 2a(1)a(2)Rii(1,2) +2v”C[a(1) + a(2)]RGl(1) + 2v’C[b(1) + b(2)]RG (1) 2 + 2[a(1)b(l) + av(2)bv(2)]R (1,1) 12 22 + 2b(1)b(2)R R + [b(l) + b(2)j (1,1) (1,2) 22 for 79 81 Br (6.45) In the above expressions, the R(x, y) are numerical factors, described in Chapter 3, which depend upon the hyperfine state and the type of predissocia tion. (A complete list of the RAB(X, y) used in the analysis are given in Appendix C.) C, a(x), and b(x), respectively, are the gyroscopic, hyperfine magnetic dipole and hyperfine electric quadrupole predissociation parameters to be determined from the data. x = 1, 2 in Equations 6.44 and 6.45 label the different nuclei in the heteronuclear isotoporner. Previous workers [3, 4, 14] have assumed that for bromine, like iodine, the electric quadrupole predissociation terms were too small to observe. The para-bromine F = J hyperfine states provide a direct experimental measure of the relative importance of the electric quadrupole predissociation parameter. As was discussed in Chapter 3, the odd—J (para) levels for the B electronic state of 79 2 and 81 Br 2 are coupled to the I Br = 0 and 2 nuclear spin states. The resulting hyperfine splitting is very simple and symmetric with the two F = J levels displaced equally above (F) and below (F) the hyperfine free energy. The F J hyperfine levels lie close to the hyperfine free energy. The predissociation rates for the two F F(F,v) = = J levels, labelled F are, FMD(F,V) + 9b ± E{vC + a}b (6.46) Here T’D(F, v), is the predissociation rate up to the magnetic dipole terms in the limit that b = 0. In this limit, F, is the same for both the F and F. hyperflne levels. Any Chapter 6. Natural Predissociation of the B state of Bromine 129 (4.48 ± 0.40)ILs (3.65 ± O.26)#s FL F Ir • ‘P. . . I I I -160 0 160 f (MHz) Figure 6.17: The hyperfine spectrum for the B—X (13’ 0”) P(4) transition of 79 2 Br observed with laser modulated at 25.5 kHz. The relative heights of the two hyperfine peaks labelled by F and F differ in the in-phase (IP) and in-quadrature (IQ) traces. The difference in lifetime is entirely due to electric quadrupole predissociation effects. The • indicate weak transitions with F J. — Chapter 6. Natural Predissociation of the B state of Bromine 81D D 79 r 2 F (10 s’) 5 2.20 ± 0.14 1.58 ± 0.09 0.60 ± 0.07 2.74 ± 0.20 2.23 ± 0.20 0.50 ± 0.06 4.28 ± 0.59 3.60 ± 0.30 0.57 ± 0.20 6.10 ± 0.93 5.38 ± 0.50 0.83 + 0.90 P(2) 1 l) P(4) 3 — 33 P(6) 5 5 P(8) 7 77 (7 —7) 130 ir 2 ) 1 s 5 F (10 2.54 ± 0.29 1.92 ± 0.20 0.54 ± 0.12 3.07 ± 0.14 2.56 ± 0.12 0.51 ± 0.08 4.42 ± 0.36 3.83 ± 0.32 0.90 ± 0.30 6.06 ± 0.57 5.78 ± 0.58 0.27 ± 0.27 Table 6.4: The total decay rates for various para bromine B—X (13’ 0”) hyperfine states having F = J. The difference in the rates labelled z(F F) (determined directly from the data, not by subtracting the average values for two decay rates) vanishes in the limit b —*0. — — observed difference in the lifetimes of these levels arises solely from interactions involving electric quadrupole (or perhaps higher order) predissociation. The best place to look for evidence of electric quadrupole effects is in the hyperfine structure of the low J levels because the gyroscopic predissociation rate increases as J(J+1) and, at high J, could mask the electric quadrupole contributions. Figure 6.17 shows the in-phase (IP) and in2 (13’ Br quadrature (IQ) signals recorded for the 79 — 0”) P(4) hyperfine spectrum. The fact that the ratio of the heights of the F and F peaks is different in the IP trace from that in the IQ trace is direct evidence that the two levels have different lifetimes and, therefore, for the existence of electric quadrupole predissociation. The results of lifetime measurements of the F = J hyperfine levels of the B—X (13’ — 0”) P(2), P(4), P(6), and 2 and 81 Br 2 are given in Table 6.4. Br P(8) for 79 These data show clear evidence that the electric quadrupole parameters are important Chapter 6. Natural Predissociation of the B state of Bromine for the B state of bromine. In particular, the difference in the decay rates, 131 — FL), for the levels 1 and 3 are well determined, having an experimental error of about 10%. It should be pointed out that the differences in decay rates reported in Table 6.4 were determined directly from the raw data and not by subtracting the average values of the decay rates from one another. As explained in Appendix C, out of a total of nearly 200 hyperfine transitions studied for the three isotopomers (each recorded between three and five times for at least two and usually three different modulation frequencies), 73 were chosen as suitable for analysis. Those with a signal to noise ratio of less than 10 were rejected. As well, the blended lines in the spectra, for which Equation 6.16 does not hold, were not used. This process selected 29 clean measurements from a total of 50 for the 79 2 data and another 20 mea Br surements out of 50 for the 812 data. The heteronuclear data, having more hyperfine levels associated with each rotational level, suffered a higher rate of rejection owing to blended transitions. Out of 100 hyperfine transitions, 33 were chosen from the 79 81 Br data set. Appendix C contains the complete subset of hyperfine transitions used in the predissociation analysis of this thesis. It lists the observed and calculated decay rates, F, along with the differences (obs caic). In addition, the observed and calculated lifetimes, - r = F, with their (obs — calc) are also tabulated. The homonuclear molecules, 79 2 and 81 Br 2 Br for the parameters, Frad, , were fit individually to model 6.44 Ci,, a, and b. The results of the individual fits for both homonuclear isotopomers are given in Table 6.5. The 79 81 data were somewhat more difficult to fit because the parameters, a(79) Br with a(81) and b(79) with b(81) were highly correlated. The solution was to recast the model in terms of new parameters, .4± and B±, defined as, .4± = (a(79) ± a(81)) (6.47) Chapter 6. Natural Predissociation of the B state of Bromine M IN X 2 i-i (10 s 4 ’) ) 2 (s’/ a ) 2 (s’/ 1 r ad 132 L ) 2 (s’/ 2 Br 79 20 9.2 (8.26 ± 0.57) (87.1 ± 1.1) (203.1 ± 6.9) (38.2 ± 6.8) 2 B 8 r 20 11.6 (7.94 ± 0.54) (89.0 ± 0.8) (239.3 ± 3.8) (39.2 ± 6.4) Table 6.5: The results of fitting the 79 2 and 81 Br 2 lifetime data separately for the Br radiative decay rate, frad, and the predissociation constants, C, a, and b, N is the number of hyperfine lifetimes used in the fit and x 2 is a measure of the goodness of fit of the data. In theory x should less be than or equal to the number of data points minus 2 the number of parameters being fit. 13± = (b(79)±b(81)) (6.48) Equation 6.45 can then be written, = (1,l) + (1,2)] 11 CRGG + A[R 11 + A[R R (1,1) 11 -- — (l,2)] 1 R 1 4CA+RGl(1) + 4Cl3÷R (1) 02 +2[A+B÷ + 12 A_B_]R (l,l) (l,1) + R 2 (1,2)] 2 2 + B. 22 + B [R [R — (1,2)] (1,1) 22 R (6.49) There was still a large correlation between A÷ and 4L. and between B÷ and B_ ultimately making it impossible to obtain meaningful results for all four of these parameters. The magnitudes of the parameters A_ and 13.... were of the order of their uncertainties. Con sequently the lifetimes were fit with both ..&.. and B_ held fixed at zero. The results are given in Table 6.6 along with the prediction for the parameters based upon the pa rameters derived from 79 2 and 81 Br 2 Br . As can be seen, the agreement between the measured and predicted parameters is very good although the uncertainty is relatively large for B. (Notice that the value of T ’ rad deduced for each isotopomer is the same Chapter 6. Natural Predissociation of the B state of Bromine 1T “ 81 Br 79 33 X 2 1 13.3 P 133 A rad (10 s 4 ’) ) 2 (s’/ (s_h/2) ) 2 (s_li (8.98 ± 0.84) (86.9 ± 2.4) (309.6 ± 11.3) (58.2 ± 11.7) (8.10 ± 1.05) (88.0 ± 1.4) (312.8 ± 11.2) (54.7 ± 13.2) Table 6.6: The results of fitting the T9 81 lifetime data are compared with the pre Br dictions based on the results for 79 2 and 81 Br 2 , labelled P. The constants A_ and B_ Br were held fixed at 0. within the uncertainty. As expected. T’rad does not vary much between isotopomers.) A closer examination of the definition of the parameters shows that, a(x) Q T(L)T(S) cl’E’) = (a,b) = fi(xj,v,1l’,E’)//ii(ii + 1)(2i 1 + 1) g(x)(cvTcIE!) = f ( 2 x,,v,’,E’) b?(x) = Q(x) (6.50) (6.51) +2)(21— 1 +3)] 1 1 + 1)(2i [(2i cz’E’) (6.52) where the superscript (a,b) with each parameter refers to the particular isotopomer being studied. e.g. C’ 79 is the gyroscopic predissociation parameter for 79 2 Br . Each of the parameters may be re-expressed as the product of a known molecular property, o t(a,b) the reduced mass of isotopomer (a,b) (6.53) x g(x) the magnetic dipole moment of nucleus x (6.54) cc Q(x) the electric quadrupole moment of nucleus x (6.55) Chapter 6. Natural Predissociation of the B state of Bromine 134 and a matrix element between the bound and continuum state. To a good approximation this matrix element may be written as the product of a measured parameter times a vibrational overlap integral for isotopomer (a,b), o-(a, b) (the square root of the Franck — (ondon factor). Namely, pa,b — Using definitions 6.56 — (-10 ‘v — ( )cTa ‘ a(x) = g(x)ao(a,b) (6.57) b(x) = Q(x)bo(a,b) (6.58) 6.58, one can fit the lifetime data from all 3 isotopomers simul taneously. The first fit was performed using Frad and the following seven parameters: 1 B = 2 B = g(79)ao(79,79) (6.60) 2 A = g(8l)acr(79,79) (6.61) 3 B = Q(79)b(79,79) (6.62) 3 A = Q(8l)br(79,79) (6.63) 4 B = o(81,81)/o(79,79) (6.64) 5 B = u(79,81)/u(79,79) (6.65) (7979)(7979) (6.59) From these, one is able to deduce all of the necessary molecular constants for the predis sociation expressions for each isotopomer. (e.g. b (81) 8 ’ 9 1 = in this form has the built in self-consistency checks: B /A 2 5 . 3 A . B ) Casting the model = g(79)/g(81) and B /A 3 = Q(79)/Q(81). The global fit results are given in Table 6.7. The ratios of the molecular parameters B , /A 2 , and , 2 /A given in Table 6.8, are in excellent agreement with the 3 B known ratios [35] of the magnetic dipole and electric quadrupole moments of the 79 Br and 81 Br nuclei. Having verified the validity of modelling the parameters as per Equa Chapter 6. Natural Predissociation of the B state of Bromine Parameter Value s 4 (8.46 ± 0.36)10 “rad B 1 2 B 2 A 3 B 3 A 4 B 5 B 135 = = = = = = = [C/(79, 79)] cr(79, 79) g(79)a(79)cr(79, 79) g(81)a(81)cr(79, 79) Q(79)b(79)o-(79, 79) Q(81)b(81)u(79, 79) o(81,81)/o-(79,79) (79,81)/o(79,79) (86.5 ± 0.8)s_h/2 (203.7 (226.8 (38.6 (34.8 (1.045 (1.029 2 ± 5.2)sI ± 3.9)s_h/2 12 ± 4.9)s’ 2 ± 4.7)s/ ± 0.010) ± 0.011) Table 6.7: The results of the global eight parameter fit to lifetimes for 79 2 Br 81 and 79 2 Br 81 The fit to 73 hyperfine lifetimes, gave a reduced x Br 2 = 0.85 (See Equation B.1). , Observed /A 2 B /A B 3 (0.898 ± 0.086) (1.106 ± 0.298) Expected g(79)/g(81) Q(79)/Q(81) 0.928 1.197 Table 6.8: The ratio of the parameters 2 :A and 3 B :A as compared to the ratios B of magnetic dipole moments and electric quadrupole moments for 79 Br and 81 Br. The nuclear moments were taken from Reference [35]. Chapter 6. Natural Predissociation of the B state of Bromine Parameter Value “rad 1 B B 2 3 B 4 B 5 B = = = = = 136 79) g(79)a(79)a(79, 79) Q(79)b(79)o(79, 79) (81, 81)/(79, 79) cr(79, 81)/a(79, 79) L(7979)°(’ (8.40 (86.5 (208.3 (40.5 (1.052 (1.030 ± 0.34) x 10 s 4 0.8)s_h/2 ± 2 ± 3.1)s’/ 4.0)S_h/2 ± ± 0.008) ± 0.011) Table 6.9: The results of the global six parameter fit to lifetimes for 79 2 Br 81 and 79 2 Br 81 The fit to 73 hyperfine lifetimes gave a x Br 2 = 56.5, and a reduced 2 = 0.84. x , tions 6.56 . — 6.58, the data for all three isotopomers were fit again, this time eliminating parameters A 2 and A , and holding the ratios of 3 and bb(79)/b(81) to 0.928 and 1.197, respectively. The results of this final six parameter fit are shown in Table 6.9. The predissociation parameters for each isotopomer of bromine may be derived from the appropriate combinations of global fit parameters for either the 8 parameter model or the 6 parameter model. These derived molecular parameters are compared with the molecular parameters deduced from the individual isotopomer fits in Table 6.10. The values deduced by the different models are in excellent agreement. 6.8 Discussion The results of these predissociation measurements yielded some interesting surprises. First, the radiative lifetime of the B ll+ state of bromine was determined to be Trad p-I rad Trad = (11.9 ± 0.5)its (6.66) 137 Chapter 6. Natural Predissociation of the B state of Bromine 2 Br 79 Frad C 1 3 13 a 13 b Combined Fit 6 parameter 8 parameter I Individual Fit 4 Parameter s’ 4 (8.40 ± 0.34)10 12 (86.5 ± 0.8)s’ (208.3 ± 3.1)s/ 2 2 (40.5 ± 4.0)s’/ s’ 4 (8.46 ± 0.36)10 12 (86.5 ± 0.8)s’ 2 (203.7 ± 5.2)s/ (38.6 ± 4.7)s_h/2 s’ 4 (8.26 ± 0.57)10 2 (87.1 ± 1.1)s/ 2 (203.1 ± 6.9)s/ (38.2 ± 6.8)s’/ 2 1 s 4 (8.40 ± 0.34)10 (88.7 ± 1.5)s_h1’2 (.236.1 ± 5.4)s’/ 2 12 (35.6 ± 3.8)s’ 1 s 4 (8.46 ± 0.36)10 2 (88.2 ± 1.7)s/ 12 (237.0 ± 6.3)s’ 2 (36.4 ± 5.3)sI (7.94 ± 0.54)10 1 s 4 2 (89.0 ± 0.8)s/ (239.3 ± 3.8)s_h/2 (39.2 ± 6.4)s_h/2 s 4 (8.40 ± 0.34)10 2 (88.0 ± 1.8)s’/ 5.7)s_h/2 (222.9 ± 2 (38.3 ± 4.2)sI s 4 (8.46 + 0.36)10 2 (89.0 ± l.8)s’/ 7.1)s_h/2 (221.4 ± 2 (37.8 ± 5.4)s/ s’ 4 (8.98 ± 0.84)10 2 (86.9 ± 2.4)s’/ (218.9 ± 8.0)s/ 2 (41.2 ± 8.2)s_h/2 81o 2 ur ’rad 1 C 1 3 13 a 13 b 81 Br 79 Frad 13 C Table 6.10: A comparison of the molecular parameters derived for the B ll+ v’ = 13 level 81 , with three different models: (i) by fitting the data from Br 2 , and 79 Br of 79 2 , 81 Br together with 6 parameters; (ii) by fitting the data from all three three isotopomers all isotopomers together with 8 parameters; (iii) by fitting the data from each isotopomer individually. Chapter 6. Natural Predissociation of the B state of Bromine 138 certainly in agreement with the rather broad range of values reported by Clyne and co workers [13, 15, 16] (particularly for the value reported in Reference [16] for the v’ = 2 level namely, (12.4 ± O.2)s) but much longer than the values of Peeters et al. [14]. (See Table 6.11.) This finding played a crucial role in designing an experiment to observe the predissociation of bromine. The second surprise was the detection of the first example of electric quadrupole predissociation in any molecule. Previous workers have assumed that the b parameter was negligible, omitting it from their analysis. In fact, both Koffend et al. [3] and Siese et al. [4] who investigated the predissociation using the relative intensity technique, did not bother to measure any para-bromine (odd J’) transitions assuming that the two strong, resolved hyperfine transitions in the para-bromine spectra would have exactly the same predissociation rates and, consequently, the same relative intensities. Part of the effect of neglecting the electric quadrupole predissociation is that these terms are absorbed into the grad parameter. The variation with J of the electric quadrupole terms will be reflected in variations with J in l’raa(expt). It will also produce a slight variation with J of av/1’rad(expt) and Cv/\JI’rad(expt). (A larger variation in the parameters with J is expected if the experimental apparatus is inadequately designed as discussed below.) For (J’=O v’=13) in the B ii+ state of 79 2 , the electric quadrupole Br predissociation term, 2.4b, is roughly 5% of “rad Finally, the values for the ratios of the predissociation parameters to the radiative decay rate for bromine reported in References [3, 4] are much lower than the results observed here while the gyroscopic predissociation parameters reported in References [13, 14] are in satisfactory agreement with the values reported here. (One other study [5, 40] of the hyperfine structure of the v’ = 17 and 18 levels of each isotopomer of bromine reported a value of C that was an order of magnitude lower than that found in this work. This result will be addressed a little later.) As discussed, the rotational state lifetime measurements only yield precise values of Chapter 6. Natural Predissaciation of the B state of Bromine 139 the gyroscopic predissociation parameter, C, from the slope of a plot of the zero pressure inverse lifetime versus J(J+1). The intercept is a combination of the radiative decay rate and the hyperfine and hyperfine — gyroscopic interference predissociation rates averaged over the hyperfine levels associated with the J = 0 state. In iodine these measurements provided direct evidence for hyperfine predissociation as the ortho and para rotational decay rates showed the same slope as a function of J(J+1) but different intercepts [25]. No such behaviour has been observed for ortho and para rotational states of the B fl+ electronic state of bromine owing to the much larger value of C for bromine. Clyne ci. al. [13) carried out a systematic study of the gyroscopic predissociation parameter (which they called k ( = C in this thesis)) for the vibrational bands 4 v’ 24 for each isotopomer of bromine. The intercepts were erroneously interpreted as the radiative decay rate of the B state of bromine. This, coupled with the relatively small values of the intercept, yielded measurements of the radiative lifetime of the different vibrational bands that varied by a factor of 3, from about 5 us up to about 16 p5. Clyne and co workers [16] also measured the radiative lifetime of the B state v’ = subject to any strong predissociation. The value reported for the J’ was Trad = 2 level which is not = 29 rotational state (12.4 ± 0.2)ps, in excellent agreement with the findings of this thesis. It must be pointed out, however, that lifetime measurements for many v’ = 2 rotational states were reported in Reference [16] having values between 9.5 and 12.6 ps. For reasons that are not made clear in the publication, Clyne and co-workers [16] chose to use only the data from the v’ = 2, J’ = 29 level to deduce Tr. Therefore, it is felt that the reported uncertainty of ±0.2 ps is underestimated. A second group, Peeters et al. [14], also carried out rotational state lifetime measure ments on the B state of 81 2 for the levels 13 Br v’ 16. This time the data were fit to two models, one which ignored the hyperfine predissociation and the other taking it into account (up to the magnetic dipole terms). The results obtained for C, varied by up to Chapter 6. Natural Predissociation of the B state of Bromine 140 10% depending upon the method of data analysis, but are in agreement with the values of Clyne and co-workers. The radiative decay rates quoted also vary from vibrational level to vibrational level and are roughly a factor of 6 larger than the radiative decay rate obtained in this study. As can be seen from Table 6.11, the agreement between the C values reported is satisfactory but not excellent. A final study by Katzenellenbogen and co-workers [5, 40] should be noted in which the hyperfine structure of the B state v’ = 17 and 18 levels of each isotopomer of bromine was investigated using polarization spectroscopy. The hyperfine data were fit for the B state electric quadrupole parameter, eqQ, the nuclear spin—rotation constant, and the linewidth of the hyperfine spectral lines observed, y. In addition, each of the parameters was allowed to vary with J as 0 = J 1 + 0 (J+1) 0 where 0 (6.67) eqQ, Csr, or -y. Katzenellenbogen et al. assumed that, 7 = -yo+CJ(J+1) (6.68) The value of C was taken as the slope of a plot of y versus J(J+1). For the v’ = 18 level of 81 2 their result was (392 ± 10) Hz [40]. This value was later revised to (410.5 ± 0.9) Br Hz [5], and values for the other isotopomers were also reported for the v’ = 17 and 18 vibrational levels. The authors of Reference [40] attribute the discrepancy between their value and those of Reference [13] as the result an error in the experimental technique of Clyne and co-workers. However, Katzenellenbogen and co-workers failed to appreciate several key points. First, the linewidth used to fit their hyperfine spectra, 7, is the half width at half-maximum (HWHM) in Hz and is related to the inverse lifetime in, 1’, in s by r = 27r7 (6.69) Chapter 6. Natural Predissociation of the B state of Bromine 2 Br 81 [16] [13] v’ k = iO s—i 2 4 5 7 11 14 — (4.4 ± 0.2) (25±1) (2.1 ±0.2) (7.4 ± 0.8) (6.0 ± 0.2) (4.8 ± 0.6) (3.8 ± 0.2) 19 20 23 24 [14] 13 14 15 16 this work 13 — (2.8 ± 0.2) (a) (b) (c) (a) (b) (c) (a) (b) (c) (a) (b) (c) “rad 5 s_i Q (0.806 ± 0.013) 141 Tr4 LS (12.4 ± 0.2) — — (1.96 (1.35 (0.88 (1.64 (0.64 (0.98 + 0.58, —0.35) + 0.27, —0.18) + 0.15, —0.12) + 0.67, —0.38) + 0.10, —0.08) + 0.14, —0.12) (5.1 + 1.5, —0.9) (7.4 + 1.5, —1.0) (11.3 + 2.0, —1.5) (6.1 + 2.5, —1.4) (15.7 + 2.5, —1.9) (10.2 + 1.5, —1.2) (6.97 ± 0.10) (8.48 ± 0.50) (8.17 ± 0.61) (6.68 ± 0.42) (8.94 ± 0.79) (8.91 ± 0.82) (6.366 ± 0.076) (6.71 ± 0.16) (6.48 ± 0.30) (6.06 ± 0.11) (6.28 ± 0.20) (5.80 ± 0.45) (5.24 ± 0.83) (5.8 ± 3.7) (6.2 ± 4.4) (4.5 ± 2.7) —(2.6 ± 5.9) —(4.2 ± 6.1) (2.26 ± 0.96) (5.1 ± 2.2) (4.9 ± 4.4) (2.4 ± 1.5) (6.7 ± 3.7) (7.4 ± 8.4) (1.91 ± 0.30) (1.7± 1.1) (1.6 ± 1.1) (2.2 ± 1.3) —(3.8 ± 8.7) —(2.4 ± 3.5) (4.4 ± 1.9) (1.96 ± 0.85) (2.0 ± 1.8) (4.2 ± 2.6) (1.49 ± 0.82) (1.4 ± 1.5) (7.87 ± 0.27) (0.840 ± 0.034) (11.9 ± 0.5) Table 6.11: The values for C and FraA for this work compared with References [13,16] (Clyne and co-workers) and [14] (Peeters et a!.). Peeters et al. fit the data in three ways: (a) using a purely gyroscopic predissociation model; (b) using a predissociation model that included gyroscopic and magnetic dipole predissociation; (c) fitting to a purely gyroscopic predissociation model and restricting the data to 3r observation window. The agreement among the three research groups is satisfactory but not excellent. Chapter 6. Natural Predissociation of the B state of Bromine The authors of References [5, 40] used a gyroscopic predissociation rate of C 142 = 5.5 kHz from Reference [13] and a radiative decay rate of 0.89 MHz to calculate, for the J’ = 72 hyperfine peaks, that 7 = Frad+CJ(J+1) = lO H z + 5.5(72)(73)(10 )Hz 3 0.89 6 = 29.8MHz (6.70) In the light of Equation 6.69 this should be changed to 7 much closer to the value of = = (29.8MHz)/(27r) = 4,74MHz (6.71) 5.9 MHz which they reported. Second, the fact is not addressed that hyperfine predissociation affects different hyperfine levels in different ways, leading to different HWHM for the different features. Third, polarization spectroscopy is a saturation technique, but no discussion of the effect of saturation on the observed HWHM is made. Finally, the linewidth of the laser used in the study is not disclosed. Thus, several of the conclusions of References [5, 40] are suspect. Two other studies [3, 4] have been carried out measuring the hyperfine predissoci ation using the relative intensity of hyperfine spectral lines technique. Both of these studies yielded predissociation parameters that displayed anomalous variations with the rotational state studied. This strong J dependence of the predissociation parameters is probably due to an experimental bias introduced by the optical systems used. As men tioned, if the observation region does not collect the LIF from long- and short-lived states with the same efficiency then the results are biased towards the short-lived states. The gyroscopic predissociation rate and the gyroscopic—magnetic dipole predissociation rate increase with the rotational state as J 2 and J, respectively, while the other predissociation Chapter 6. Natural Predissociation of the B state of Bromine 143 terms remain the same or decrease slowly with rotational quantum number. Consequently the lifetimes of the individual hyperfine states generally decrease with increasing J. The optical detection system can be inadequate for recording hyperfine lifetimes of low ro tational states but adequate for measuring high J hyperfine levels. Thus, the deduced predissociation parameters will vary from rotational level to rotational level reflecting the changing amount of experimental bias introduced by the apparatus. A second consequence of oversampling the short-lived levels with respect to the longlived ones is to increase the relative intensity of the more highly predissociated (short lived) states. When one relies on relative intensity measurements to deduce the pre dissociation parameters, as was done in References [3, 4], then an oversampling of the short-lived levels leads to an underestimate of the parameters, C//T and a//T. The data reported by Koffend et al. [3] and Siese et al. [4] display both of these charac teristics supporting an experimental bias hypothesis. (See Table 6.12.) The lifetime data obtained here for the B ll+ v’ = 13 for 79 2 , Br Br 81 , and 79 2 81 Br are well described by the predissociation model given in Chapter 3. The existence of the interference terms between the gyroscopic and hyperfine terms verifies that the electronic state predominantly responsible for the predissociation must be the 1fl state. That is, in order to observe interference terms arising from the gyroscopic and hyperfine Hamil tonians, both these Hamiltonians must be able to couple the bound level to the same continuum level. The selection rules for the gyroscopic Hamiltonian, HG, (Table 3.1) allows the coupling u state to a lu state. 4-* u or g 4-* g and LMI = 1. Thus HG can only couple the O Chapter 6. Natural Predissociation of the B state of Bromine 2 Br 79 v’ Y [3] 11 4 6 10 16 — — Average [4] 11 13 16 this work 13 0 4 6 8 10 12 14 144 a/jT (0.121 (0.096 (0.141 (0.072 (0.096 ± ± ± ± ± (0.177 (0.218 (0.22 (0.187 (0.10 (0.083 ± 0.016) ± 0.019) ± 30%) ± 0.095) ± 30%) ± 0.006) 0 2 4 6 8 10 (0.146 (0.207 (0.180 (0.20 (0.13 ± ± ± ± ± 0 4 6 8 10 12 (0.148 (0.166 (0.264 (0.16 (0.13 0.030) 0.008) 0.074) 0.026) 0.007) (0.294 (0.363 (0.412 (0.274 (0.292 ± ± ± ± ± 0.039) 0.049) 0.060) 0.017) 0.014) (0.487 ± 0.032) (0.372 + 0.035) (0.459 ± 0.025) (0.47 ± 30%) (0.577 ± 0.073) (0.51 ± 30%) (0.447 ± 0.032) 0.024) 0.010) 0.024) 30%) 30%) (0.436 (0.371 (0.416 (0.449 (0.47 (0.49 ± ± ± ± ± ± ± 0.015) ± 0.010) ± 0.052) ± 30%) ± 30%) (0.421 (0.280 (0.434 (0.512 (0.45 (0.38 ± 0.039) ± 0.052) ± 0.038) ± 0.036) ± 30%) ± 30%) (0.298 ± 0.009) 0.058) 0.071) 0.076) 0.057) 30%) 30%) (0.719 ± 0.025) Table 6.12: The predissociation parameters reported by References [3,4] compared to the results of this work. Chapter 6. Natural Predissociation of the B state of Bromine v’ = 13 81 : 79 2 Br 2 Br 81 : 79 Br 79 2 Br 145 o(a, b)/(79, 79) FCF(a,b)/FCF(79,79) (1.052 ± 0.008) (1.030 ± 0.011) (1.107 ± 0.017) (1.061 ± 0.023) Table 6.13: The ratios of the vibrational overlap integrals and Franck—Condon factors (FCF) between the B ll+ and hll electronic states for the different isotopomers of broni ne. 6.9 A Closer look at the Molecular Parameters From the definitions of the molecular parameters, one can compare the results of the studies presented in this thesis to work done by other authors on the Franck—Condon factors, on C, and on the electric quadrupole structure of the B state of bromine. 6.9.1 Franck—Condon Factors The fact that data from all three isotopomers were fitted at once in this work made it possible to determine the relative sizes of the vibrational overlap integrals between the B TI+ and ‘11U electronic states for the different isotopomers. (See Table 6.13.) The results indicate that the overlap integrals, o(a, b), varied smoothly from isotopomer to isotopomer. increasing slightly with the mass. Cyne et al. [13] have deduced the FCF between these two electronic states for 79 81 Br from the vibrational dependence of the gyroscopic predissociation parameter, k = C. These workers did not observe any significant variation of the gyroscopic predissociation parameters from isotopomer to isotopomer (except for the 1 = V 7 level where the predis sociation is at a minimum). This result agrees well with the findings presented here. The gyroscopic predissociation parameter, C, depends upon the vibrational overlap integral Chapter 6. Natural Predissociation of the B state of Bromine v’ 13 81 : 79 2 Br 2 Br 81 : 79 Br 79 2 Br Z(X,79) (X, 79) 2 Z (1.026 ± 0.010) (1.017 ± 0.011) (1.053 ± 0.021) (1.034 ± 0.023) 146 Table 6.14: The ratios of the vibrational overlap integrals times reduced mass Z(X, 79)(= [(X)/t(X)}[(79, 79)/o(79, 79)J) and Z (X, 79) between the B H+ and ‘TIrn 2 electronic states for the different isotopomers of bromine. and the reduced mass. (6.72) Therefore, if the gyroscopic predissociation parameter displays a very small change from one isotopomer to the next, (i.e. with reduced mass) then one must conclude that the vibrational overlap integral from isotopomer to isotopomer must increase with the reduced mass in such a way that the ratio o(a,b)/ii(a,b) remains nearly constant. The. ratio of gyroscopic predissociation parameters, C(X)/C(Y), for isotopomers X and Y is given by the quantity, — (X) p(X) 673 (.) o(Y) It follows that the ratio of the parameters, k(X)/k(Y) (where k(X) = C(X)) reported in Reference [13] should follow Z (X,Y). The values of Z(X,79) and Z 2 (X,79) based upon 2 the results presented in this thesis for the v’ = 13 level of bromine are given in Table 6.14. As can be seen the variation of C from isotopomer to isotopomer is predicted to be less than about 5 %. Thus one concludes that the predissociation parameters, k for each isotopomer should be the same to within the ±5 % in excellent agreement the results of Clyne et al. [13]. Chapter 6. Natural Predissociation of the B state of Bromine v’ = (I) (II) 13 (81,81)/u(79,79) FCF(81,81)/FCF(79,79) (1.052 ± 0.008) (1.138 ± 0.096) (1.107 ± 0.017) (1.296 ± 0.218) 147 Table 6.15: The ratios of the vibrational overlap integrals and FCF’s for 79 2 and 81 Br 2 Br as deduced from (I) the global fit results, and (II) the J’ = 0 hyperfine level decay rates. The inverse lifetimes of the two hyperfine levels of the J’ = 0 state of the homonuclear molecules provide another estimate of the ratios of the FCF’s. One recalls that for the homonuclear molecules (for J’ = r(F’ 0), = 1) = Frad + 2.4b (6.74) = 3) = Fr + 4a + 2.4b (6.75) + The difference in the decay rates is directly proportional to the parameter [a(x)j . (Here 2 the superscript (a,b) refers to the isotopomer and x(=79 or 81) refers to the nucleus.) The magnetic dipole predissociation parameter, a(X), is proportional to the nuclear magnetic moment, g(x), times an overlap integral, o(X). a,,(X) x g(x)cr(X) (6.76) The ratio FCF(81,81)/FCF(79,79) may be directly deduced from a knowledge of the decay rates, I’(F’ = 1) and r(F’ = 2 and 81 Br 3) for 79 2 Br , and the ratio of the nuclear magnetic moments, g(81)/g(79). The ratio of vibrational overlap integrals and FCF’s based upon the decay rates of the J’ = 0 hyperfine levels (see Tables C.8 and C.9) are given in Table 6.15. The estimates from the J’ = 0 hyperfine decay rates agree with the values obtained by fitting all of the lifetime data but have considerably larger errors. Chapter 6. Natural Predissociation of the B state of Bromine v’ = 13 o(a,b) (10_ . 2 /i) FCF(a,b) (10—i cm) 1.622 1.707 1.671a 2.632 2.913 2.793a 2 Br 79 81 2 Br 81 Br 79 148 Table 6.16: The the overlap integral, (a, b) and FCF between the B 11÷ and ‘11 electronic states for v’ = 13. The values for 79 2 and 81 Br 2 were determined here. The Br a values with superscript Br 79 ( B 8 r ) were reported by Clyne et al. [13] Clyne et al. [13] have tabulated the FCF between the B and hlli states for 79 81 Br 1 v’ 25. Because of the normalization of the continuum wavefunctions, E (‘ll l 1 E l ’) I = 6(E — E’) (6.77) the FCF have units of 1/energy (and are quoted in cm by Clyne et al.). Using the FCF between the B and lliu states for the v’ = 13 vibrational level of 79 81 Br reported in Reference [13], the corresponding FCF’s and o-(a, b) have been deduced from the measurements reported here for 79 2 and 81 Br 2 Br 6.9.2 . (See Table 6.16.) The Gyroscopic Predissociation Parameter The gyroscopic predissociation predissociation parameter is defined in this thesis (see equation 3.32) as, C = = (L) + 1 1 T ( S)] 1E) (0vH [T’ _ R 2 t I(vIE)I (L) + T 1 (S)] I1u)12 1 I(oI [T (6.78) (6.79) Chapter 6. Natural Predissociation of the B state of Bromine )Wor(R)I / 2 2 (°I {T(L) + 1 T ( S)j I1n)12 149 1.76 x 10—2 1.79 x 10-2 Table 6.17: The electronic matrix elements between the B ll-i. and ‘11i matrix elements involved in the gyroscopic predissociation term: a comparison between the current results and those of Ref. [13] as corrected here. (R is an average value of the internuclear separation, R.) Here 2 has I(vIE)1 units of . To convert the units to 1 1 .Joules 1/(cm ) , one must divide by (hc), = R c 2 2,u [T ( L +T (S)j Ilu)12 1 I(vIE)I (0 1 (6.80) In the notation of Clyne et al [13], k = C = R c 2 4 2 We( (vE) 2 (6.81) By comparing equations 6.80 and 6.81, it is clear that 2 IWe()I Clyne et al. reported that = 2 IWe(R)1 [V ( 1 L) + 1 T’ ( S)] lu) 12 (6.82) . However, this value is incompatible 3 8.8x10 with the results that are presented in their paper. A quick check shows that the factor of 4 in Equation 6.81 was neglected, making W0r()I /4 = 8.8x10 2 , in excellent agreement 3 with the findings of the current work as shown in Table 6.17. 6.9.3 Quadrupole terms Both the electric quadrupole predissociation parameter and the electric quadrupole cou pling constant of an electronic state have similar types of matrix elements. From Equa tion 3.34, Chapter 6. Natural Predissociation of the B state of Bromine v’ = 13 79 2 Br 2 Br 81 1 . 0 eq Q s(MHz) (MHz) (1729 ± 188) (1445 ± 152) (178.1 ± 1.0) (148.3 ± 1.0) eqQB 150 qo,iJ1B (9.7 ± 1.1) (9.7 ± 1.1) Table 6.18: The effective quadrupole coupling constant between the 1 f] and B H+ states compared with the electric quadrupole coupling constant of the B ll+ state of bromine, b(x) = = (x,,’,v,E’) 2 f +2)(2i + 1)] 3)(2i 1 +1 [(2i ,) 1 eQ,1U( By dividing the overlap integral and the factor of (6.83) (6.84) J7i out of b, one obtains an effective electric quadrupole coupling constant between the B 11-i- and ‘Hiu electronic states. (See Table 6.18.) This finding represents the first measurement of hyperfine electric quadrupole coupling between a bound and a continuum electronic state. In addition, this is the first time a value for eqQ has been obtained between states which differ in by one. Chapter 7 Conclusions The hyperfine structure of the B 11+ and X electronic states of molecular bromine has been investigated along with the natural predissociation of the v’ = 13 level of the B electronic state. The hyperfine spectra of the two electronic states were well described using three pa rameters. One is the X state electric quadrupole coupling constant, eqQx, second is the B state parameter the electric quadrupole coupling constant eqQB, and third is the nuclear spin—rotation constant Csr, of the B ll÷ state. The ratio of electric quadrupole cou pling constants of the different isotopomers agreed with the ratio of nuclear quadrupole moments of the different isotopes, eqQ( Br) 79 Br) 81 eqQ( — - Q( Br) 79 Br) 81 Q( 7 1 for both electronic states. The B state eqQ( Br) was (177.0 ± 0.6)MHz for v’ = 11 79 and increased by roughly 0.5 MHz per vibrational quantum up to (180.6 ± l.4)MHz for = 17. The ground state eqQ(79) values were recorded for the vibrational levels v” = 0 through 2 beginning with eqQ( Br) = (808.1 ± 1.4)MHz and increased by about 1.0 T9 MHz per vibrational quantum. The hyperfine study also determined the hyperfine free frequency differences between various B—X rovibronic transitions providing a test of the accuracy of the molecular con stants reported by Gerstenkorn and Luc [7] for each isotopomer of bromine. In general, the data showed that the constants faithfully reproduced the observed frequency separa tions of rovibronic transitions within a given isotopomer (to within 5 MHz) for the low J 151 Chapter 7. Conclusions states studied (J < 152 20). The study of the (17’ 2”) band had a near coincidence of the — transitions P(J) and R(J+3) for each isotopomer. The observed frequency separations yielded values for the combination of constants, (—lOB’ 17 + 6B) + (120D 7 — 24D) that were systematically (0.304±0.017) MHz lower than the values derived from Reference [7] for all three species of molecular bromine. Finally, the calculated frequency separations between transitions arising from different isotopomers were systematically offset from the observed values. The offsets may be interpreted as a correction to the B state term values of 79 81 and 81 Br 2 reported [7]. The following adjustments to the term values Br are suggested: 79 ( 0 T B) = (B) 0 T 79 0 T 8 ’ 79 ( 0 B) 1 = (B) 8 ’ 79 0 T 1 0 81 ( 0 T B) = (B) 0 T 81 0 (7.2) — — (177 ± 8)MHz (385 ± 8)MHz (7.3) (7.4) (The primed quantities are the suggested new values.) This leads the calculated and observed transition frequency differences to agree to within about 8 MHz. (The absolute frequencies are still no more accurate than the ±48 MHz value claimed by Gerstenkorn and Luc [7].) The study of the predissociation of the v’ = 13 level of the B state of bromine demonstrated that the dissociative ‘ll level was responsible for most, if not all of the observed predissociation. The radiative lifetime of the B state was found to be (11 .9±0.5)ts, in agreement with the range of lifetimes reported by Clyne et al. [13, 15, 16]. However, it is much longer than the values reported by Peeters et al. [14] or assumed in References [3, 4]. Revised values of the gyroscopic (Cv) and hyperfine magnetic dipole (a) predissociation parameters have been determined and, for the first time, a hyperfine electric quadrupole predissociation has been measured. It has been characterized by the new molecular constant b. From this parameter, the first determination of an electric Chapter 7. Conclusions 153 quadrupole coupling constant between a bound and continuum electronic state was made. Finally, this work has shown that the phase shift technique, which is relatively easy to set up and apply, is very well suited to the measurement of the lifetimes of individual hyperfine levels of excited molecular states. 7.1 Future Work A recent study [6] of the vibrational dependence of the electric quadrupole coupling constant of the B state of bromine has been carried out for v’ = 16 through 28. As was pointed out by Vigué et al. [8], the electric quadrupole coupling constant of the an = 0 state can be “contaminated” by interactions with other electronic states. Therefore, it would be useful to continue this survey of the vibrational dependence of the hyperfine structure to lower vibrational levels of the B state of bromine, especially from the v’ = 2 through 7 states. This is the region where the interaction between the l 1 l and B states goes from being very small (v’=2), through a maximum (v’=5), and down to a minimum (v’=7). It would be interesting to see the effect, if any, this dissociative state has on the hyperfine structure of the B state. A second suggestion is to continue a systematic survey of the separations of different rovibronic transitions as deduced from hyperfine measurements. With more data any systematic variations with v and J may be detected and allow even finer corrections to the constants of Gerstenkorn and Luc [7]. The appeal of this is that bromine provides an unusual opportunity to directly measure the relative positions of molecular electronic potentials between different isotopomers. Third, a systematic study of the predissociation of the same vibrational levels men tioned above of the B state of bromine could prove interesting. In particular, the v’ = 5 level should have a much larger predissociation and may display further discrepancies in Chapter 7. Conclusions 154 the lifetimes of hyperfine states both within and between different isotopomers. As well, a systematic survey using the phase shift technique could determine the shape of the 1 H i,. state for each isotopomer. Finally, another candidate for observing hyperfine electric quadrupole predissociation is molecular chlorine. Like bromine, chlorine has a nuclear spin of 3/2 and, therefore, it is possible that a measurement of the lifetimes of the hyperfine levels of para-chiorine could display an effect similar to that observed in bromine. Bibliography [1] R. S. Eng and J. T. LaTourrette, J. Mol. Spectrosc. 52, 269 (1974). [2] N. Bettin, H. lKnöckel, and E. Tiemaun, Chem. Phys. Letters 80, 386 (1981). [3] J. B. Koffend, R. Bacis, S. Churassy, M. L. Gaillard, J. P. Pique, and F. Hartmann, Laser Chem. 1, 185 (1983). [4] M. Siese, E. Tiemann, and U. Wuif, Chem. Phys. Letters 117, 208 (1985). [5) N. Katzenellenbogen and Y. Prior, J. C’hem. Phys. 93, 897 (1990). [6] P. Liu, J. Kieckhafer, and E. Tiemann, J. Mol. Spectrosc. 150, 521 (1991). [7] S. Gerstenkorn and P. Luc, J. Phys. France 50, 1417 (1989). [8] J. Vigué , M. Broyer, and J. C. Lehmann, J. Physique 42, 937 (1981). [9] J. Vigué , M. Broyer, and J. C. Lehmann, J. Physique 42, 949 (1981). [10] J. Vigué , M. Broyer, and J. C. Lehmann, J. Physique 42, 961 (1981). [11] R. Tench and S. Ezekiel, Chem. Phys. Letters 96, 253 (1983). [12] J. P. Pique, R. Bacis, F. Hartmann, N. Sadeghi, and S. Churassy, J. Physique 44, 347 (1983). [13] M. A. A. Clyne, M. C. Heaven, and J. Tellinghuisen, J. Chem. Phys. 76, 5341 (1982). [14] F. Peeters, J. Van Craen, and H. Eisendrath, J. Phys. B 22, 2541 (1989). [15] M. A. A. Clyne and M. C. Heaven, J. Chem. Soc. Faraday Trans. 2 74, 1992 (1978). [16] M. A. A. Clyne, M. C. Heaven, and E. Martinez, J. Chem. Soc. Faraday Trans. 2 76, 405 (1980). [17] G. Herzberg, Molecular Spectra and Molecular Structure 1. Spectra of Diatomic Molecules, D. Van Nostrand Company, New York (1950). [18] C. H. Townes and A. L. Schalow, Microwave Spectroscopy, McGraw Hill, New York (1955). 155 Bibliography 156 [19] J. P. Pique, These Paris (1984). [20] P. R. Bunker, J. Mo?. Spectrosc. 46, 119 (1973). [21] M. Trefler and H. P. Gush, Phys. Rev. Lett. 20, 703 (1968). [22] A. R. Edmonds, Angular Momentum in Quantum Mechanics, Princeton University Press, Princeton, New Jersey (1957). [23] M. Broyer, J. Vigué , and J. C. Lehmann, J. Physique 39, 591 (1978). [24] J. Vigué , These Paris (1978). [25] M. Broyer, These Paris (1977). [26] R. S. Mulliken, J. Chem. Phys., 55, 288 (1971). [27] M. Broyer, J. Vigué , and J. C. Lehmann, Ghem. Phys. Letters 22, 313 (1973). [28] J. Vigué , M. Broyer, and J. C. Lehmann, J. Phys. B 7, L158 (1974). [29] J. Vigué , M. Broyer, and J. C. Lehmann, J. Ghem. Phys. 62, (1975). [30] M. Brayer, J. Vigué , and J. C. Lehmaun, J. Chem. Phys. 63, 5428 (1975). [:31] J. Vigué , M. Broyer, and J. C. Lehmann, J. Phys. B 10, L379 (1977). [32] F. W. Dalby, C. D. P. Levy, and J. Vanderlinde, Chem. Phys. 85, 23 (1984). [:3:3] E. Merzbacher, Quantum Mechanics, John Wiley and Sons, Inc., New York (1961). [34] A. Messiah, Quantum Mechanics, vol.11, John Wiley and Sons, New York (1976). [:35] H. H. Brown, and J. G. King, Phys. Rev. 142, 53 (1966). [36] S. Gerstenkorn, P. Luc, and A. Raynal, Atlas du Spectre d’Absorption de la Molecule de Brome. Vol 1 and II, Laboratoire Aimé Cotton CNRS II, Orsay, France (1985). — [37] A. C. Adam, A. J. Merer, D. M. Steunenberg, M. C. L. Gerry, and I. Ozier, Rev. Sci. Instrum. 60, 1003 (1989). [38] R. F. Barrow, T. C. Clark, J. A. Coxon, and K. K. Yee, J. Mo?. Spectrosc. 51, 428 (1974). [39] W. Demtröder, Laser Spectroscopy, Springer-Verlag, New York (1981). [40] N. Katzenellenbogen, P. S. Stern, and Y. Prior, J. Ghem. Phys. 92, 7718 (1990). [41] M. Rotenberg, R. Bivins, N. Metropolis, and J. K. Wooten, Sj and 6j Symbols, Technology Press, M. I. T. (1959). Appendix A Derivation of Predissociation Rates This appendix contains the derivation of the critical equations used in this thesis to describe the gyroscopic, hyperfine, and interference predissocaition rates. Specifically, the derivation of the rotational and nuclear factors (Equations 3.35 through 3.38) used in Equations 3.44 and 3.45 is given. The approach taken is that of Broyer and co workers [8, 23]. - The predissociation rate of a bound electronic state, I]’b), coupled to a continuum electronic state, I?1’(E’)), is given by Fermi’s Golden Rule [33, 34] Fp I(I Hk(x)] [HG + 2 I(E’))I (A.l) where HG and H’(x) are the gyroscopic Hamiltonian and the nuclear—electron hyper fine Hamiltonians which couple the two states. (k refers to the rank of the multipole characterizing the particular hyperfine interaction and x refers to the nucleus involved in the coupling.) In Equation A.l it is assumed that the continuum wavefunctions are normalized as, = (E — E’) (A.2) To evaluate the predissociation rate the bound and continuum wavefunctions will be expressed as = c ( 7 ( 1 JI)fvFM) JI 157 (A.3) Appendix A. Derivation of Predissociation Rates (L’bI 158 (A.4) = JI, III I’) = (J’I’)I!’E’F’M’) 1 7 (A.5) Note that for the purposes of these calculations the wavefunctions with primed quantum numbers (e.g. J’, I’, ‘) will refer to the continuum states while the wavefunctions with unprimed or doubly primed quantum numbers will refer to the bound electronic levels. By substituting Equations A.3 through A.5 into Equation A.l one obtains: = cr Hk(x) (y(J”I”)12vFMI HG + JIJ”I” J’PO’ (I 1 7 J’1’)1’E’FM) k,x x(-y’(J’I’)12’E’FM HG + HJc’(y) 7 (I JIWvFM) (A.6) k’,y In Equation A.6, a sum over J’, I’, and Q’ has been added to take into account the possibility that, in general, the bound state can couple to several continuum states with the same F quantum number. Before going through the full mathematical details of the calculations, a brief overview of the calculation will be given. (The predissociation calculations will be performed following the method of Broyer and co-workers [8, 23].) The goal is to cast Equation A.6 into a form that is simpler to evaluate. The products of matrix elements that must be evaluated to obtain the predissociation rate from Equation A.6 are of the form C — J,IIo’ (A.7) where HA and HB can each be either the gyroscopic Hamiltonian or a hyperfine Hamil tonian. To proceed, one recalls that the quantity, = - jy’(J’I’)f’E’FM) ( (J’I’)f’E’FMj 1 7 (A.8) Appendix A. Derivation of Predissociation Rates 159 is the projection operator onto the sublevels -y’(J’I’)f’E’FM). Therefore Equation A.7 can be expressed as C = (( 7 J”I”)fZvFM HA Pç’’J’p HB 7 (1 J1WZvFM) (A.9) J’I1’ A projection operator, P = I)(I, has the following properties Pm) If the states I) [34]: = P (A.1O) = 1 (A.11) = In)(nlrn) (A.12) and rn) in A.12 are orthonormal basis states then (nlm) Sn,m. Using these three properties in Equation A.9 one finds C = (( 7 J”I”)fZvFMI 1 j 211 Pç (HAPOIEIJIIIHB) I J V 2 FI (JI)fZvFM) 1 J v 2 O (A.13) and C = (HAFIEIJIJIFcIE’JIIIHB) (7 JI)1vFM) (A.14) This gives the desired result: C = (K 7 J”I”WZvFMI (P”i” HA Pc’E’J’p) (Pc’E’J’I’ HB P ) I-v(JI)flvFM) j 1 J,Ilc, (A.15) Using projection operators, the expression for the predissociation rate given in Equa tion A.6 can be rewritten as _’p 1 = h c’j’ Appendix A. Derivation of Predissociation Rates x( (J”I”)fvFMt 7 x HG PIE’J’p) HG Pj )+ 1 (Pc2IEIJIP 160 (Po’E’JII’ Hk(x) P1E1JIIi)] + Hk’(y) I ( 7 JI)fvFM) k’,y (A.16) One observes that each Hamiltonian, HA, which couples the bound and continuum levels has been replaced by the equivalent tensorial operator, P, P,. (The subscripts 1 1 HA 1 r and are shorthand for the appropriate quantum numbers.) The equivalent tensorial operators for each Hamiltonian will be derived in terms of the irreducible tensor operator basis sets, 9 T 9 1 and 11 U [8, 23]. 9j’T acts on all of the coordinates except for the nuclear spins, and jj’Ue only acts on the nuclear spins. The extra sub- and superscripts used to label the basis sets are to remind the reader that 1 fi Tqk has non-vanishing matrix elements between the states, (JI and IJ’1’) while has non-zero matrix elements between the states, ((iii 1 and i 2 )II and I(i’i 2 )I’). (i 2 are the nuclear spins of nucleus 1 and 2 respectively.) The properties of these operator basis sets will be explored further in the next section. Once the equivalent operators have been derived, the composite operators, > PGG HGPlEIJ1Il) (P’E’JlJ’ HGPJJ) (A.17) J’I’12’ PkG(x) = H1c(x)PIElJlJi) (Pn1E1JIIIHGPVJI) Hk(x) Po’E’J’JI) (P’E’J’I’ Hk’ () PclvJI) (A.18) J,I,c1, Pkk’ (x, y) = J,I o 1 , (A.19) are introduced and evaluated. These allow the predissociation rate to be written = c JIJ”I” (( 7 J”I”)fivFMl PGG + Pk’G(y) k’,y Appendix A. Derivation of Predissociation Rates + Pi(x) P’(x, y) + k,x 161 (JI)fvFM) 7 (A,20) k, k’,y Equation A.20 is of the form = : a’u i 1 ,, ajFj, Foc + Fk’G(y) + JIJ”I” FGk(x) + 1ji(x,y) k,x k’,y k,a k’,y (A.21) The matrix elements, I’AB(a, FAB(a, b) b), are y(J”I”)fvFMPAB(a, 7 b)1 ( JI)I1vFM) = (A.22) where A or B may be G, k, or k’ and the indices a, b refer to nucleus 1 or 2; if A (and/or B) = G, then the index a (and/or b) is redundant. (NOTE: for the rest of this appendix it will be assumed that all of the quantum numbers are integer except for i 1 A.1 2 i = is.) Hyperfine Predissociation Rates The hyperfine Hamiltonians responsible for coupling the bound and continuum electronic states are the interactions between each nucleus and the electrons. As explained in Chapter 2, the electron—nuclear hyperfine Hamiltonians can be expressed as contractions of spherical tensors of rank k. H(x) = (_)q 1’(je(x))Qq(ix) (k is the order of the multipole interaction, or 2), and je(x) = x is the nuclear spin of nucleus x le(x) + Se(x).) The definintions of the 1’(je(x)) and been given in Chapter 2 for the magnetic dipole (k hyperfine interactions. (A.23) = Qk_q(ix) ( = I have 1) and electric quadrupole (k=2) Appendix A. Derivation of Predissociation Rates l62 The first step is to work out the equivalent operators, PoVJIHif(x)PoIVIJ’J’, in terms of the basis sets, 93T and 11 U. To do this, some of the properties of the irreducible tensor basis sets must be presented. The reader is reminded that 9’Tqk operates on all of the coordinates except for nuclear spin and has non-zero matrix elements between the states (JI and IJ’’) U only acts on the nuclear spin coordinates and has non11 )I and 2 vanishing matrix elements between the states ((iii isotopes have the same nuclear spin, i = = = )I’) 2 I(iii (For bromine, both 3/2. Therefore, for the rest of this appendix i will be used.) The matrix elements of the scalar product, Vr TTk iq OO’’rk IIltJ_q JJ”q Ld’) q is given by (see Equation 7.1.6 of Reference [22]) (JI)cZFM (—l ) %T 7 ( = ( F J’ I’ )F+J’+I 1 (k I iii (J JJ Uq I ’(11’)1’f’M) 7 I Tk J!J I I I’) (A.24) From the normalization condition Tr { T (T) } = 1 &,j j’j 6 c6 6 kk 6 6 c’o 1 qq (A.25) one can show that [23] (J1 J’1’) “2k + 1 (A.26) and (IIii1U” Ill’) The equivalent operator M,f(x, F, 2, 1’) = POIIJIHf(X)P12IV’JIII ./2k + 1 (A.27) is derived from the matrix element, = (JI)flvFMHjf(x) 7 7 ( ’(J’I’)’v’FM) = (7(JI)fvFMIPçvJJHjf (x)Po’’j’j’ I ’(J’I’W’v’FM) (A.29) 7 (A.28) Appendix A. Derivation of Predissociation Rates 163 Replacing, PçVJJHff(x)P1V1J1J Wic(X, = )q 1 ci, ci’, v, v’, J, J’, 1,1’) (_ JIT iih1-Tq (A.30) in Equation A.29 and using relations A.24, A.26 and A.27, one obtains, (_l)F’+I(2k + 1)wk(x, ci, ci’) = I F J’ i’ 1k (Here wk(x, I JJ (A.31) ci, 12’, v, v’, J, J’, I, I’) has been abbreviated by wk(x, ci, 12’).) Equations 2.66 and 2.67 give F,12,12’) Mhf(X, + l)(21’ + 1)(2J + 1)(2J’ + 1) I k J I ix I ) ( I’ x J’ ‘\ i 1< —12 12’ k XI — ‘I I J F I’ J’ k (A.32) (where z1l Wk(X. = 12’ — 12.) Comparing Equation A.31 with Equation A.32 one deduces Ii. 12’). Wk(X, 12,12’, v, v’, J, J’, 1,1’) = (_1)k+2ix4hi+fk(x, 12, v, 12’, v’)/(2k + 1) J(2I+l)(2I’+1)(2J+1)(2J’+1) I \ x j k 12 ci J’Iix ix ii H k I’ 12’) J x Now that the equivalent operator, PcVJJHf(x)PclV’J’p, has been established, one would like to construct the composite operators Pkk’(x, y) = > (pVJ,,I,,Hk(x)p,E,J,I,) (P’E’JlIIH’(y)PJJ) Jul11, Using Equation A.30 one obtains, (A.34) Appendix A. Derivation of Predissociation Rates Pkk’(x, y) (wax, = , ‘, 164 )q 1 v, E’, J”, J’, I”, I’) (_ JIT J,I,fl iuui1q) q ‘, , E’, v, J’, J, I’, I) (_1)’ 9T’ 1qUi (_ ) 1 +’ w,(,1l,1) = wkI(Y,cz,c) J’J’1) qq’ < [11:I’ ] 1 u 111 T’] (A.35) 9Tqk 1 1 1 The product of irreducible tensor basis sets, 9 99T can be developed on the , irreducible basis set, 1 9, 9 T, where K takes on the values 9ii9Tq ji’YTq 1k k’I — Tr 1 11 ( } t {i i :T: 1TqI’ T.<) = K (k + k’) [8, 23]. 9IIJT (A.36) The trace of the product of the three irreducible tensor bases is Tr {9 11 ( } t :T: jT’ T) 11 T’) 1 ( 1 I1”1”) (J”fZ”I,,T YTc’ t = (J’T”I,,?,TIJ’1’) = c2 x (J’f’ljTq’IJ1Z) x(JI (,?T)t lJ””> (A.37) To evaluate the trace, one uses the following properties of irreducible tensorial basis sets [22]: First, KJIY,’TIJ’’) = (-i) ( k (JII,ITkIIJ!!> (A.38) q Second, = (_1)’ (“T) (A.39) Appendix A. Derivation of Predissociation Rates 165 Third, (J2I (]T,) IJ”II”) = (_i).’_(_i)Q / x J \—f K J”’ —Q Q”) I (JII’T’IIJ”12”> (A.40) Substituting these into Equation A.37, one obtains, Tr {9 9119 ( 9;T: 9T T)t} 11 (-1)”°”2k +1 0 ( —a” IJ’ —‘ ( q f’ k’J I x(—1)2K + 1 k J) “ j \—1 q’ K J” —Q ft” (A.41) Usiug Equation 6.2.8 from Reference [22] one has, (_l)J_0+Jl_ +J”c2” 12 ( —a” J” 1k’ = I k J’ q S’ k q’ q () \ J’ k’ J —2’ q’ l K’\Ik’ H —Q I I. J” k () —c J K J” —Q i” K1 J J’ (A.42) J This simplifies Equation A.41 giving, Tr ST 1 { 1 ‘]T’ 11 ( } t T) (-1)(2k + 1)(2k’ + 1)(2K +1) 1k’ k XI q K k’ k K 1< —Q) (J” J J’ (A.43) Appendix A. Derivation of Predissociation Rates 166 The product of irreducible tensor basis sets, 9,,9TQk 99T’, can now be expressed as, E(—1)QV(2k + 1)(2k’ + 1)(2K + 1) = KQ 1k’ k \Ik’ K k K JUJTQ X —Q) I q J” J J’ (A.44) J Similarly, IiiIiUq iiiU’qi (—1)’J(2k + 1)(2k’ + 1)(2K + 1) = K’Q’ x Ik’ K’’\Ik’ k I Q’ I I. —q’ —q kK’ (AA5) I” I I’ J Equations A.44 and A.45 can be substituted back into the expression for Pkk’(x, y) (Equation A.35). Pkkl(x, y) !, 1’)wk’(y, 2’, ) J’I’12’ qq’ + 1)(2k’ + 1)(2K + 1) KQ 1k k’ Ki21frpK —QJ IJ J” J’J (k’ k ‘S I q x K’ I (JhIJLQ ‘ (—1)QV(2k + 1)(2k’ + 1)(2K’ + 1) K’Q’ x Ik’ \ By using the relation, q + q’ = k’ k ( qq’ —q’ •—q Q, and q K’1k Q’ ) I. k’ k —Q) k’ 1 I” K’ I’ ji U 1 , (A.46) J Equation 3.7.8 from Edmonds [22], namely, K ‘ q’ k ( q’ q K’ “1 —Q’) — Appendix A. Derivation of Predissociation Rates 167 Equation A.46 can be reduced to: Pkki(x,y) = J’I’fl’ K XWk(X, I IZ, 1’)wk’(y, 1, 1) k k’ K J JI’ JI x E(.-.-i)Q 1 1 J k k’ K I I” I’ (A.48) 9II9T I’IIUQ Q Next, the definitiion of wk(a, from Equation A.33 is substituted into Equa 2) tion A.48. This gives (_l)K+k+k’(_l)2:x+k+P+Jh’(_l)2tx+k’+I+J’fk(X, Pkk’(x, y) !, ‘)fZ’(y, ! J’I’W K (2J’ + 1)(21’ + 1) [(2J” + 1)(2J + 1)(21” + 1)(21 + 1)] k k’ K I I” I’ j” k ><1 I xl —c — x (_i)Q ‘I I i” i k i I’ ( j’ 1 1 J’ II k’ I’ i k’ I )( J 1< —ci’ ci Q’) i k k’ K j j” J’ (A.49) 9II9T Q The case where x Pkk’(x, x) = y is treated first. (_l)K+k+k’(_l)2iz+k+P+J”(_l)2ix+k’+I+J’fk(x, = , ‘)fZ’(x, 12, 12’) J’I’f’ K (2J’ + 1)(21’ H- 1) [(2J” + 1)(2J + 1)(21” + 1)(21 + 1)] k I xl 11 k’ K I I” I’ k J” k J’ ‘\ 1” I’ I II —12 —12 12’ J\ x E(—1) JO,,TQK JII1UQ Q 1 1 , r j k’ I’ i z J ‘\ I k —12’ 12 12) [ j j” .i’ J’ k’ 1< k’ K (A.50) 168 Appendix A. Derivation of Predissociation Rates Using formula 6.2.12 from Reference [22] one obtains k (2I’ + 1)(—i)” k’ 1. K x x ,, k I’ . 3 i K I I” i i i J ‘ (_l)K+k+kl+I+II I ix J ( J ( I jx = k’ I’ i i K k’ i, (A.51) J k It follows, then, that PkkI(x, x) (_1 )k+k’+Jh’+I”+J’fk(x fi, ‘)fZ’(x, = J’f’ K (2J’ + 1) [(2J” + 1)(2J + 1)(21” + 1)(21 + I I K I I” x x x 1KI J k j” \—1 J’ K x k’ j k x ( c,) — x (_i)Q k’ J’ H I J k ) I. —‘ k’ K “ “ 1< (A.52) U’< 1 ,‘ Q A similar formula for the summation over J’ can be obtained from Equation 2.19 of Reference [41] (2J’+1)(—1) I I j” k J’ —c —Lc c’ (_i)J+J / I k \I II j’ —i’ k’ — Equation A.52 can then be written K k’ J c I I J” — k k’ K j j” j’ K 0) = (A.53) Appendix A. Derivation of Predissociation Rates Pkk(x, x) 169 = 0’ K fZ(x)fk’(x) [(2J” + 1)(2J + 1)(21” + 1)(21 + 1)] I I x K I I” ix ix x k’ 1k 1K1i J ( k’ K i x K’\(J XI II —Lc E(—1) UJT k J” K —ci o 1111U < 1 Q (A.54) Q Having deduced Pkk’ (x, x) one now returns to Equation A.49 and considers the case x y. The evaluation of Pkk’(x, y) relies on the property that = fk(x) (—l)’C ugfk(Y) (A.55) where 1 = foru+—>u,g.-*g (—1 foru—g and J The factor g(x)/g(y) for k = Q(x)/Q(y) for k=2 takes into account that the two nuclei may have different nuclear properties such as for 79 81 Br . For k = I (hyperfine magnetic dipole interaction) g(x)/g(y) is the ratio of the nuclear magnetic moments and for k = 2 (electric quadrupole interaction) Q(x)/Q(y) is the ratio of the nuclear quadrupole moments. As is evident from the definition, = 1 for all values of k for homonuclear molecules. Thus, Appendix A. Derivation of Predissociation Rates Pkkl(x, ) 170 E(_l)K+k+kt(_ l)2x+k+I’+J”(l)2ix+k’+J+J’ = J’I’1’ K { ( fk(x, f, c’)f,(y, Q, 1’)(2J’ + 1)(21’ + 1) [(2J” + 1)(2J + 1)(21” + 1)(21 + x x k k’ K I ,, , k J” — — x( }{ ,) i i I” k J’ )QOrpK 1 JIIJLQ — I’ i J f k’ I’ k’ J’ 1 i J ‘ (—a, I”IUQ ){ J k k’ K j j” j’ j Q (A.56) (l)K+k+k’(fj2ix+k+I’+J”(i)2ix+k’+I+J’ Pkk’(x, ) K J’I’O’ (—i)”( f fk(x,Q,S’)f,(x,1M’) [(2J” + 1)(2J + 1)(21” + 1)(21 + 9 k k’ K x(2J’+1)(21’+1){ I I” I’ x ( — — ,) J” xE( Q — k J’ }{ k’ J’ (—a, 1 ) QIZOmK JUJIQ i k “ I’ ){ J i }{ 1 k’ I’ i k k’ K j ju’ j’ = J’I’2’ K 2 (EU fk(x, k k’ K x(2J,+1)(21,+1){ I I” I’ x ( k J” — — x( Q — , }{ J’ ){ Q, Q’)f,(x, !, ft’) [(2J” + 1)(2J + 1)(21” + 1)(21 + 1)] ,) J’ J’ (—a, )1 QIZI2rflK Q IIIIUQ 1 J”J x k k’ J I’ z x k’ 1 j k k’ K j j” j’ J (A.57) Appendix A. Derivation of Predissociation Rates From Equation 6.4.3 in Ref. [22], (2I’+ 1)(_1)2’ k k’ K I I” I’ II J x x ‘, k I’ x x I k K x x k’ I’ J x (A.58) k’ 1” Zx £ When this identity and Equation A.53 are used to simplify Equation A.57 one obtains: k’+J Pkk’(x, i) “+I+c2’ = c’ fk(x, 9 f, 1’)fZ,(x, , ft’) K [(2J” + 1)(2J + 1)(21” + 1)(21 + 1 / \ Ic’ (Ic xl —z2 / £ \ J” K\ O)jl —1 0) K\(J ID I k K k’ x x E(_i) Q 1 JlIIU (A.59) Q Note that in Equation A.59 the following relation has been used: x I x k K k’ Zx I” x = )K+k+k’+I+I”+dtx 1 ( x 1’ Ic K k’ (A.60) x The matrix elements of interest for the hyperfine predissociation, T’kkl(x, y), will now be calculated. For x = y, it can be seen from Equations A.54 and A.24 that the matrix elements are of the form: Appendix A. Derivation of Predissociation Rates f1d’(x, x) 172 (7(J”I”)1ZvFMIPkkl(x, = (x)1 7 JI)IZvFM) (A.61) ( ) 1 F+o’+k+kI [(21 + 1)(21” + 1)(2J + 1)(2J” + [/fk(x’)] I F J” i” xE(2K+1) K (K I I L. Similarly for the x fkJ’(x, y) = ( K I I” ix x x 1 J 1 1 H i K —2 1 J” j II O)j k K — 1 J K k’ k’ k (A.62) y, the hyperfine predissociation matrix elements are, )1 F+1V+Ihl [(21 + 1)(21” + 1)(2J + l)(2J” + 1)j >< g [fk(x’)] I (K K x x F k 1 1 k k’ ) I JJZ—M J” i” I K (J J” II O)\!—cl K 0 I K k’ 1,, 1 (A.63) Zx Expressions A.62 and A.63 may be simplified by defining the quantities W(J, J”, I, I”) = /(2J + 1)(2J” + 1)(21 + 1)(21” + 1) (A.64) and Z/ (J,J”,I,I”) = I ( F J” i” K I J 1 1 I J \ k k’ K —Ls 0 (J II ) \ J” — K 0 (A.65) Appendix A. Derivation of Predissociation Rates 173 Specializing to the B ii+ predissociated by the Fji(x, x) electronic state one obtains = ((J”I”)OFMjPkk’(x, x)l(JI)OFM) = )F+11’+k+k’ 1 (_ V/(J, J”, I, I”) x [\/fk(xOlu)] [\/fZ1(xOu+l:)] k x (2K + 1) Z (J, J”, I, I”) tK I” iJ(K i k’ K (A.66) Pi(x, y) = = ((J”I”)OFMIPkk’(x, y)j(J’I’)OFM) )F+f’+I” T1/(J, 1 (_ J”, I, I”) [fk(xOlu)] [fkI(xOlu)] Zx x E(_flK(2K + 1) Z’(J, J”, 1,1”) k I ‘x K k’ (A.67) x (NOTE: The wavefunctions describing the molecular states are chosen so as to make fk(x A.2 Il, v, 1’, E’) real.) Gyroscopic Predissociation Rates Now the calculation will be performed for the term PGG in Equation A.17. The gyroscopic Hamiltonian, HG = _-J. (L + S) can couple different electronic states through the of these terms between the B 2 state is given by [8] = (A.68) (L + S) terms. The matrix element O electronic state and a dissociative f’ = lu electronic Appendix A. Derivation of Predissociation Rates ( = Qy(J”I”)vFMHGI7’(J’I’)f’, E’, F, M) 174 l)”_01(2Jfl I j” I 0 + 1)J”(J” + 1) 1 J” —1 1 (OvI- [T’(L)+T(s)] huE’) (A.69) The equivalent operator, P1VJIIPIHGPOIE’J’I’, V(Q, v, Q’, E’, J”) = PVJI’IIIHGFIEIJ’II can be written: (A.70) U9IT° where V(, v, ft’, E’, J”) )J”_+1 1 (_ = (2J” + i)J”(J” + 1) Ij” flI 5 Jl4 J,i 5 ‘ I I 1 J” —c ‘ (QVI— (L) + T(S)] h1’E’) (A.71) The gyroscopic predissociation rate is easily evaluated as PGG E’;F M)1 2 = = —(2J” + 1)J”(J” -t- 1)1 h —1 1 I(0vI [T(L) + T(S)] hluE’)1 2 For = 0 and Z’ = 1 the 3—j symbol reduces to I J” 1 J” 0 —1 1 ) = 1 2(2J”+1) Then, = (A.72) [(0vI (L) + T(S)] lu El)] J”(J” + 1) 0 [T’ (A.73) Appendix A. Derivation of Predissociation Rates A.3 Hyperfine — 175 Gyroscopic Interference terms The gyroscopic—hyperfine interference terms in Equation A.18 can be readily derived from Equations A.70 and A.30 PGk (x) HGPiVEIJIII] [PcvE,J,I,Hk(x)PcvJIj J,I,cv = [v(, v, ‘, f f’ p0 1 E’, J ir J”J”oj x [wk(x c’, , E’, V, , 11 J, I”, I) J = V(12, ) wk(X, ‘, 1) q ( ) (—1j\ql2f2’ f’11rnk J”J’q II Uq] k iiiiU.g jiijTq) 012’O (A.75) q From equation A.44 one has: ° °‘ JJ (_flJ+J” — o’Tk T°0 JJ q J(2J” + 1) — °Tq (A.76) J”J Then 2 (_l)J+J+I+2Z J 11 x+k+O+1( PGk(x) + 1)j(2J + 1) xJ”(J”+1)(2I+1)(2P’+1) <‘ >< x ( 1f(x, [Vi j J” 0—1 k ( 1 .,, l)il 0—1 q ){ i” k i’ Ii } JIlJ 2ir = -- j [(oUI-2 [7’(L) + T(S)] Ilu E’)] Is—I) ‘Tç iiiiUq q The matrix element of this operator can then be written: l’Gk(x) ‘ (2k+1) (A.77) (( 7 J”I”)QvFMI {PJ”I”HaP1E1J’I11 x [pElJ,J,Hc(x)PVJI] (j 7 JI)IZvFM) (A.78) Appendix A. Derivation of Predissociation Rates ( = T’Gk(X) l)F+J”+1+I”+2i+k+c2+1 176 (2J” + l)/(2J + 1) xJ”(J” + 1)(21 + l)(21” + 1) [(ovI- [i’L) +T(S)] huE’)] (j XI j” k O —1 II 1) 1 (j” j 1 i” 1< —l)( k 0 I F J I i J(k i” ti” (A.79) To simplify the above equation, one can define the quantity 8 k (JI j, I”, I) I “ I = j” i c, — k I I ) J c I I ) ( F J k I” J” 1 1 J (A.80) This gives: FGk(x) = (_l)F+J <‘ +I+2,+k+0+1 /J”(Jh’ + 1)(2J” + 1) W(J, J”, 1,1”) [(ovI- (L) + T(s)] huE’)] xsk(J/!,J,I,I) (A81) k A.4 i I J Putting it all Together Now that the building blocks (rGG, Fkw(xy), , l’c(x)) have been developed, one recalls that the predissociation rate of a hyperfine level, IFfM), is given by, = c’j JIJ”I” FOG + >k’,y l’k’G(!I) + l’Gk(x) k,x + >k, k’,y Fi’(x, y) (A.82) Appendix A. Derivation of Predissociation Rates The predissociation terms, I’GG, Fkk’(xy), 177 and F Gk(x), can be written as the product of a molecular parameter to be experimentally determined and calculable rotational and nuclear factors. For predissociation between the B H+ (f the dissociative ‘I1 (= = O) electronic state and lu) state the molecular parameters are defined here as, VOVI- [T’(L)+T’(s)] huE’) (A.83) (x) = fi(x,O,v,lu,E’)/[i(i+ 1)(2i+ 1)} (A.84) b(x) = f ( 2 x,O,v, lu,E’) (A.85) [(2j + l)(2i+2)(2i + 3)] Here C is the gyroscopic predissociation parameter, a is the hyperfine magnetic dipole (k 1) predissociation parameter, and b is the hyperfine electric quadrupole (k = 2) predissociation parameter a new molecular parameter introduced here. By including the sums over the coefficients, a, in the rotational and nuclear factors. RaG, RGk(x), and Rj&t(xy), one can rewrite the predissociation rate as, Fp(Fev) = CRGG + + v’Ca,(x)Ri(x) + v”Cb(x)R ( 2 x) + vCa(x)R a 1 (x) v’Cb(x)R ( 2 x) (x,y) + b(x)a(y)R 2 (x,y) 21 + a(x)b(y)Ri (x,y) 22 + a(x)a(y)Rii(x,y) + b(x)b(y)R (A.86) By inspection, the expressions for the rotational and nuclear factors can be deduced. RGG = J(J + 1) (A.87) Appendix A. Derivation of Predissociation Rates 178 )F+J”+I+I”+2ix+k++1 a(— 1 RGk(x) J,,I,, JI xJ”(J” + 1)(2J” + 1) W(J, J”, 1,1”) X Xk(x) Rii(x, x) Sk(Jl, , , I) ,f 1 , ( 1 _i)F+c2’+k+ 111 k’ = Jl,Iu k I x } (A.88) W(J, J”, I, I”) Xk(x) Xk’(x) JI + 1) Z’(J, J”, 1,1”) K Rj(i(x, y) { “ = ,, { I x k x xflK K I” x x k’f (A.89) 1 (_i)F+’f1” l’V(J, J”, I, I”) .Xk(i) .Xk’(x) ,,a’ J,,I,, Ji x E(_1)K (2K + 1) Z’(J, 11 1,1”) J x I Z k K k’ x I x K Here X ( 1 i) = \/x(x + 1)(2i + 1) J (A.90) (A.91) 1 X ( 2 i) — — + 3)(2x + 2)(2i + 1)] 2x(2ix 1) [(2ix (A.92) — The rotational and nuclear terms in Equations A.88, A.89, and A.90 have the following symmetry properties: 1. RGk(x) = Rka(x). 2. RGk(x) = Rak(y). 3. Rii(x,x) = Rk’k(x,x). 4. Rlth’(x,y) = Rji(y,x). Appendix A. Derivation of Predissociation Rates 179 These simplify Equation A.86 to: f(Fv) CRGG + 2/C [a(1) + a(2)] RG1(1) = +2/C [b(1) + b(2)] RG2(1) 11 1) + 2a,(1)a(2)R R + [a(l) + a(2)] (1, (1, 2) 11 +2 [a(1)b(1) + av(2)bv(2)J (1, 12 1) R (1,2) + (1,2)} 12 + [a,(1)b(2) + a(2)b(l)] (R 21 R 22 1) + 2b(1)b(2)R R + [b(1) + b(2)] (1, (1, 2) 22 (A.93) The coefficients 1 R 2 (1, 2) and 2 R 1 (1, 2) are completely negligible for the states of interest in this work. Therefore the predissociation rate for 79 81 can be written: Br 1’(F€v) CRGQ + 2v’C [a(1) -I- a(2)j R(l) = +2ñC [b(1) + b(2)j RG2(1) 11 1) + 2a(1)a(2)R R + [a(l) + a(2)j (1, (1, 2) 11 +2 [a(1)b(1) + a(2)b(2)] (l, 12 1) R + [b(1) +b(2)j (1, 22 2 R (1,1)+2b(1)b 2 (2)R 2) For the hornonuclear molecules, 79 2 and 81 Br 2 , where a(1) Br = a(2) and b(1) (A.94) = the predissociation rate is: 1’(Fv) = CRGG + 4V’CaRoi(1) + 4’ñCbRG (1) 2 (1, 1) + (1, 11 11 2)] R +2a [R +4abRi ( 2 (1, 1) + (1, 22 1, 1) + 2b [R 22 2)] R (A.95) Appendix B Hyperfine Data The study of the hyperfine structure of the B ll-i. and X electronic states involved recording, for each isotopomer of bromine, the hyperfine spectra of the following rovi bronic transitions: (a) for the (13’— 0”) band, P(1) through P(5), R(0) through R(3) and R(l0); (b) for the (17’ — 2”) band, R(0) through R(10) except R(9) and P(1) through P(7) except P(6). In addition, for Br 79 B 8 r R(0) through R(2) and P(1) for the (11’ (16’ — four — , the hyperfine spectra were measured of 0”), (12’ — 0”), (14’ — 1”), (15’ — 1”), and 1”) vibrational bands. Each transition was measured at least three and usually times producing 379 data sets with a total of over 3000 hyperfine transitions for the three isotopomers of bromine. As mentioned in Chapter 5, only the relative separations of the rovibronic hyperfine spectral features observed were recorded. Each data set was analyzed separately for the parameters eqQB, eqQx, and 6, an arbitrary frequency scale offset. The unbiended features were chosen, weighted by their relative heights, and then the observed frequencies were subjected to a nonlinear least squares fit. Each data set provided associated fit parameters and a reduced chi-squared value, X2 (B.1) = Here Vobs is the observed frequency of a given hyperfine transition, hyperfine transition frequency, 8 b 0 a is the calculated is the uncertainty of the observed frequency, N is the total number of transitions included in the fit, and n is the number of parameters varied. Ideally the reduced chisquared value should be 1.0. The final values for the electric quadrupole coupling constants, eqQ, for a given electronic and vibrational state were 180 Appendix B. Hyperfine Data 181 deduced by averaging the values from the appropriate data sets weighted by the reduced chisquared value. Owing to the very large number of transitions recorded it is not practical to tabulate the (observed — calculated) frequency differences for all of the transitions. Instead the observed and calculated frequencies of the B—X (17’ — 2”) P(1) hyperfine data sets for each isotopomer are given in tables B.1 through B.6. These results are typical of the measurements with the exception of the (13’ tainties. The (17’ — — 0”) band which had slightly higher uncer 2”) band results are summarized in Tables B.7 through B.9, which give the reader a feeling for the variation of the observed transition frequencies along with the (observed (13’ — — calculated) residuals. Finally, Tables B.10 to B.12 summarize the 0”) P(2) hyperfine results for each isotopomer to demonstrate the slightly higher uncertainty associated with this band. (The values and uncertainties quoted for the ob served data in the tables labelled AVERAGES are the averages and standard deviations of the frequencies of the individual the data sets.) In the tables in this appendix, the X and B hyperfine states are labelled both by J’ and F’. For many of the rotational levels several hyperfine states with the same value of F’ exist. These are distinguished with the further label a, b, c, and so on in order of ascending energy. (e.g. J’ .1’ = = 4, F’ 4 rotational state having F’ = = 6a labels the lowest energy hyperfine level for the 6.) Appendix B. Hyperfine Data 182 Isotopomer: Band: Line: Data Set F” F’ 79 2 Br (17’ 2”) P(1) — (MHz) vi (MHz) obs—caic (MHz) Vobs br2187.dat 0 3 2a 4 1 2b 1 3 1,3 3 1 1,3 —324.62 —243.32 —132.42 85.18 163.18 297.08 —322.42 —241.60 —132.06 84.10 163.26 296.71 —2.20 —1.72 —0.36 1.08 —0.08 0.37 br2188.dat 0 3 2a 4 1 2b 1 3 1,3 3 1 1,3 —321.73 —241.73 —132.33 83.87 163.67 298.07 —322.41 —241.60 —132.06 84.10 163.26 296.71 0.68 —0.13 —0.27 —0.23 0.41 1.36 br2189.dat 0 3 2a 4 1 2b 1 3 1,3 3 1 1,3 —323.71 —242.81 —131.61 83.49 164.39 298.89 —322.41 —241.59 —132.05 84.11 163.27 296.72 —1.30 —1.22 0.44 —0.62 1.12 2.17 br2190.dat 0 3 2a 4 1 2b 1 3 1,3 3 1 1,3 —321.63 —240.33 —131.63 82.87 161.97 297.77 —322.42 —241.60 —132.06 84.10 163.26 296.71 0.79 1.27 0.43 —1.23 —1.29 1.06 Table B.1: The 79 2 (17’ 2”) P(1) observed and calculated hyperfine transition fre Br quencies for separated data sets (part I). — Appendix B. Hyperfine Data 183 Isotopomer: Band: Line: Data Set F” F’ 79 2 Br (17’ 2”) P(1) — (MHz) v (MHz) obs— caic (MHz) Vobs br2191.dat 0 3 2a 4 1 3 1,3 3 —325.24 —242.74 —134.24 85.86 —322.41 —241.60 —132.06 84.10 2.83 —1.14 —2.18 1.76 br2192.dat 0 3 2a 4 1 2b 1 3 1,3 3 1 1,3 —320.75 —241.95 —132.55 83.85 162.65 298.05 —322.41 —241.60 —132.06 84.10 163.26 296.72 1.66 —0.35 —0.49 —0.25 —0.61 1.33 br2193.dat 0 3 2a 4 1 2b 1 3 1,3 3 1 1,3 —326.49 —242.99 —134.49 84.91 163.41 299.21 —322.42 —241.60 —132.06 84.10 163.26 296.71 —4.07 —1.39 —2.43 0.81 0.15 2.50 br2194.dat 0 3 2a 4 1 2b 1 3 1,3 3 1 1,3 —322.26 —240.86 —131.66 84.04 161.04 295.54 —322.42 —241.60 —132.06 84.10 163.26 296.71 0.16 0.74 0.40 —0.06 —2.22 —1.17 Table B.2: The 79 2 (17’ 2”) P(1) observed and calculated hyperfine transition fre Br quencies for separated data sets (part II). — Appen dix B. Hyperfine Data 184 Isotopomer: Band: Line: Data Set F” F’ 79 2 Br (17’ 2”) P(1) — ‘obs vi (MHz) (MHz) obs— caic (MHz) br2195.dat 0 3 2a 4 1 2b 1 3 1,3 3 1 1,3 —323.00 —240.30 —132.40 84.40 162.10 295.30 —322.41 —241.59 —132.05 84.11 163.27 296.72 —0.59 1.29 —0.35 0.29 —1.17 —1.42 br2196.dat 0 3 2a 4 1 3 1,3 3 1 1,3 —240.78 —133.68 85.72 162.62 294.12 —241.59 —132.05 84.11 163.27 296.72 0.81 —1.63 1.61 —0.65 —2.60 br2197.dat 0 3 2a 4 1 2b 1 3 1,3 3 1 1,3 —322.22 —242.72 —133.32 83.78 165.68 299.18 —322.41 —241.60 —132.06 84.10 163.26 296.72 0.19 —1.12 —1.26 —0.32 2.42 2.46 Table B.3: The 79 2 (17’ Br 2”) P(1) observed and calculated hyperfine transition fre sets (part III). — quencies for separated data Appendix B. Hyperfine Data 185 Isotoporner: Band: Line: Data Set F” F’ 812 (17’ P(1) — 2”) Vobs z’i (MHz) (MHz) ohs— caic (MHz) br2268.dat 3 2a 4 1 2b 3 1,3 3 1 L3 —202.84 —110.14 67.66 138.46 247.26 —202.02 —110.48 69.90 136.26 247.74 —0.82 0.34 —2.24 2.20 —0.48 br2269.dat 0 3 2a 4 1 2b 1 3 1,3 3 1 1,3 —266.59 —202.49 —110.69 68.01 138.61 246.31 —269.57 —202.02 —110.47 69.91 136.26 247.75 2.98 —0.47 —0.22 —1.90 2.35 —1.44 br2270.dat 0 3 2a 4 1 2b 1 3 1,3 3 1 1,3 —269.62 —203.22 —111.22 70.78 138.08 244.68 —269.57 —202.02 —110.48 69.91 136.26 247.75 —0.05 —1.20 —0.74 0.87 1.82 —3.07 br2271.dat 0 3 2a 4 1 2b 1 3 1,3 3 1 1,3 —271.12 —206.32 —110.02 70.18 138.38 248.78 —269.57 —202.02 —110.48 69.90 136.26 247.74 —1.55 —4.30 0.46 0.28 2.12 1.04 Table B.4: The 81 2 (17’ 2”) P(1) observed and calculated hyperfine transition fre Br quencies for separated data sets (part I). — 186 Appendix B. Hyperfine Data Isotopomer: Band: Line: Data Set F” F’ 81 2 Br (17’ 2”) P(1) — 0bs 1 vi (MHz) (MHz) ohs— caic (MHz) br2272.dat 3 2a 4 1 2b 3 1,3 3 1 1,3 —202.81 —111.41 69.49 137.89 247.19 —202.02 —110.48 69.90 136.26 247.74 —0.79 —0.93 —0.41 1.63 —0.55 br2273.dat 0 3 2a 4 1 2b 1 3 1,3 3 1 1,3 —270.46 —201.36 —112.06 68.54 139.64 242.24 —269.57 —202.02 —110.48 69.91 136.26 247.75 —0.89 0.66 —1.58 —1.37 3.38 —5.51 br2275.dat 0 3 2a 4 1 2b 1 3 1,3 3 1 1,3 —269.59 —200.89 —109.69 67.51 139.11 245.31 —269.57 —202.02 —110.48 69.90 136.26 247.74 —0.02 1.13 0.79 —2.39 2.85 —2.43 br2276.dat 0 3 2a 4 1 2b 1 3 1,3 3 1 1,3 —267.69 —202.19 —110.49 70.31 137.21 243.81 —269.57 —202.02 —110.48 69.90 136.26 247.74 1.88 —0.17 —0.01 0.41 0.95 —3.93 Br (17’ 2”) P(1) observed and calculated hyperfine transition fre 8 Table B.5: The 2 quencies for separated data sets (part IT). — Appendix B. Hyperfine Data 187 Isotopomer: Band: Line: Data Set F” F’ Br 8 ’ 79 2 1 (17’ 2”) P(1) Vob vi (MHz) (MHz) ohs— caic (MHz) br2242.dat 0 la 3b 2a 4 lb lc 2c 1 0,2 3 1,2,3 3 1 0,1,2 1,3 —299.13 —235.93 —222.33 —123.83 76.77 148.27 239.17 274.37 —296.23 —235.07 —222.80 —124.24 77.06 148.22 238.61 273.23 —2.90 —0.86 0.47 0.41 —0.29 0.05 0.56 1.14 br2243.dat 0 la 3b 2a 4 lb ic 2c 1 0,2 3 1,2,3 3 1 0,1,2 1,3 —295.43 —234.13 —221.43 —121.83 75.57 146.17 238.67 273.17 —296.23 —235.08 —222.80 —124.24 77.06 148.21 238.61 273.23 0.80 0.95 1.37 2.41 —1.49 —2.04 0.06 —0.06 br2245.dat 0 la 3b 2a 4 lb ic 2c 1 0,2 3 1,2,3 3 1 0,1,2 1,3 —297.55 —236.35 —223.85 —121.05 76.45 146.85 237.85 272.85 —296.23 —235.08 —222.80 —124.24 77.06 148.21 238.61 273.23 —1.32 —1.27 —1.05 3.19 —0.61 —1.36 —0.76 —0.38 Table B.6: The 79 81 (17’ 2”) P(1) observed and calculated hyperfine transition Br frequencies for separated data sets. — Appendix B. I-Iyperfine Data Isotopomer: Band: Line: F” F’ 0 3 2a 4 1 2b 1 3 1,3 3 1 1,3 188 79 2 Br (17’ 2”) P(1) AVERAGES — Vobs ca1c 11 (MHz) (MHz) obs— caic (MHz) —322.41 —241.60 —132.06 84.10 163.26 296.72 —0.90 —0.27 —0.70 0.26 —0.19 0.61 —(323.17 —(241.87 —(132.76 (84.36 (163.07 (297.32 ± ± ± ± ± ± 1.82) 1.13) 1.03) 0.95) 1.32) 1.77) Table B.7: A summary of the Br 79 (17’ 2”) P(1) data results. The root mean square 2 (rms) of the differences (obs caic) is 0.56 MHz, — — Isotopomer: Band: Line: F” F’ 0 3 2a 4 1 2b 1 3 1,3 3 1 1,3 81 2 Br (17’—2”) P(1) AVERAGES ’1 obs ca1c 11 (MHz) (MHz) obs— caic (MHz) —269.57 —202.02 —110.47 69.91 136.26 247.75 0.39 —0.24 —0.25 —0.85 2.16 —0.30 —(269.18 —(202.26 —(110.72 (69.06 (138.42 (247.45 Table B.8: A summary of the 81 2 (17’ Br 0.98 MHz. ± ± ± ± ± ± — 1.71) 0.84) 0.80) 1.29) 0.74) 1.24) 2”) P(1) data results. The (obs — caic) rms is Appendix B. Hyperfine Data Isotopomer: Band: Line: F” F’ 0 la 3b 2a 4 lb ic 2c 1 0,2 3 1,2,3 3 1 0,1,2 1,3 189 79,81 Br 2 (17’ 2”) P(1) AVERAGES — 1 o bs 1 i c (MHz) (MHz) ohs— caic (MHz) —296.23 —235.07 —222.80 —124.24 77.06 148.22 238.61 273.23 —0.53 —0.39 —0.01 1.45 —0.72 —1.12 —0.05 0.23 —(296.76 —(235.46 —(222.81 —(122.79 (76.34 (147.10 (238.56 (273.46 + 1.94) ± 0.96) ± 1.14) + 1.61) ± 0.53) ± 1.07) ± 0.67) ± 0.80) Table B.9: A summary of the 79 81 (17’— 2”) P(1) data results. The (ohs Br is 0.74 MHz. — caic) rms Appendix B. Hyperfine Data Isotopomer: Band: Line: F” F’ 190 79 2 Br (13’ 0”) P(2) AVERAGES — (MHz) 2a 2a 2a 0 1 0 1 2b 2b 2b lb 2 la lb lb la la lb 2 la (MHz) obs— caic (MHz) —272.46 —215.95 —159.69 —67.64 —56.57 45.00 56.07 159.55 216.05 272.19 0.51 0.86 0.85 1.75 2.08 —0.72 —0.82 —1.79 —0.29 —0.36 obs 11 —(271.94 —(215.09 —(158.97 —(65.89 —(54.49 (44.28 (55.26 (157.76 (215.76 (271.84 ± ± ± ± ± ± ± ± ± ± Table B.10: A summary of the 79 2 (13’ Br is 1.17 MHz. 1.59) 0.66) 0.68) 1.66) 1.43) 0.96) 0.87) 0.58) 1.43) 4.28) — 0”) P(2) data results. The (obs — caic) rms Appendix B. Hyperfine Data Isotopomer: Band: Line: F” F’ 2a 2a 2a 0 1 0 1 2b 2b 2b lb 2 la lb lb la la lb 2 la 191 81 2 Br (13’ 0”) P(2) AVERAGES — ’obs 1 vi (MHz) (MHz) ohs— caic (MHz) —227.70 —180.66 —133.90 —55.04 —47.10 38.75 46.69 133.66 180.70 227.45 1.22 0.35 0.74 0.88 1.34 0.87 0.30 —0.92 —1.31 —2.51 —(226.48 ± —(180.31 ± —(133.16 ± —(54.16 ± —(45.76 + (39.62 ± (46.99 ± (132.39 ± (179.39 ± (224.94 ± Table B.11: A summary of the 81 2 (13’ Br is 1.20 MHz. 5.98) 1.19) 1.14) 2.36) 2.50) 1.34) 1.16) 0.79) 0.33) 1.85) — 0”) P(2) data results. The (ohs — caic) rms 192 Appendix B. Hyperfine Data Isotopomer: Band: Line: F” F’ 2a 3c 2a 3c 4b 3c 2a 4b la lb 2d 2d 3a 5 3a 2c 2c 2b lc 2b 2b lc ic 4 2b 2a 4 3b la 3b 0 lc lb 2a 2a 4 3b 2c lc ic 2c 2b 3b 2a 79,81 Br 2 (13’ 0”) P(2) — AVERAGES (MHz) va (MHz) ohs— caic (MHz) —(253.26 ± 3.06) —(231.11 ± 1.40) —(200.86±1.94) —(187.71 ± 1.60) —(176.53±1.30) —(164.53 ± 1.16) —(147.66 ± 1.39) —(109.38 ± 1.34) —(91.00 ± 0.98) —(52.10 ± 1.38) —(32.40 ± 0.88) (27.64 ± 0.56) (78.82 ± 1.36) (92.30 ± 1.48) (100.35 ± 0.76) (132.05±0.98) (139.70 ± 1.00) (169.35±1.69) (198.70±2.43) (220.22 ± 2.62) (270.82 ± 1.67) (286.02±0.86) —251.24 —229.10 —199.13 —185.16 —175.21 —163.45 —146.86 —109.57 —90.23 —52.80 —32.71 27.38 78.27 91.48 99.98 131.33 139.01 167.50 197.01 219.61 269.30 284.79 —2.02 —2.01 —1.73 —2.55 —1.32 —1.08 —0.80 0.19 —0.77 0.70 0.31 0.26 0.55 0.82 0.37 0.72 0.69 1.85 1.69 0.61 1.52 1.26 ’obs 1 81 (13’ Br Table B.12: A summary of the 79 rms is 1.26 MHz. — 0”) P(2) data results. The (obs — caic) Appendix C Phase Shift Data This appendix contains all the lifetime data used in the analysis of the natural predisso ciation of the B ii+ state of bromine. The phase shift method was applied to the hyperfine spectra of the B—X (13’ — 0”) 2 and to the P(1) Br 2 and 81 Br P(l) through P(9) transitions (excluding the P(3)) of 79 81 Br through P(7) and R(9) transitions of 79 the 79 2 P(2) F’ Br = . A sample calculation of the lifetime of L. hyperfine level is given below. The calculated hyperfine spectrum of the P(2) transition is given in Figure C.1. The F” = 2_ — F’ = l_ transition is indicated by the letter A in the figure. The amplitude of the laser radiation was modulated at a frequency, f, and the in-phase (IP) and inquadrature (IQ) signals from a lock-in amplifier were recorded as the wavelength of the laser beam was varied. The phase setting on the lock-in amplifier, LL&, was chosen so as to make IP and IQ almost equal for most of the hyperfine peaks. The reference phase, Qef was determined from the scattered laser light signal. The data for the lifetime measurements of the B ll+ v’ = 13, J’ = 1, F’ = 2 are shown in Br 1... level for 79 Tables C.l to C.3. The phase shift of the observed signal is deduced from, = [LIA + arctan ()] — &ef (C.1) and T = tan 4 2irf tan —_____ = 193 (C.2) Appendix C. Phase Shift Data 194 6.0 5.0 4.0 3.0 tJ 1.0 0.0 -300.0 -150.0 0.0 0 v( MHv z) 150.0 Figure C.l: A simulated hyperfine absorption spectrum for 79 2 B—X (13’ Br transition. The peak labelled A has the F’ = L. level as its upper state. 300.0 — 0”) P(2) Appendix C. Phase Shift Data 195 For example from the second entry in Table C.1, ref = 60.7°, LIA = 70.00°, and the observed IQ/IP phase is 46.36°. Therefore, [70.00 = — 46.36] — 60.70 —37.06° yielding, tan(—37.06°) s’) 3 2K(18.48)(10 6.503 x 10_6s — — — = For each hyperfine spectrum studied the measurements were repeated for at least 2 and usually 3 different modulation frequencies providing a test of the consistency of the data. The variation from modulation frequency to modulation frequency seen for the data shown in Tables C.1, C.2, and C.3 was typical. The average lifetime for this level was (6.313 ± 0.343)ts. Of the nearly 200 hyperfine transitions studied for the different isotopomers, 73 were chosen as suitable for analysis because they had a signal to noise ratio of 10 or better and were not blended with other transitions. (Equation 6.16 does not hold for blended lines.) The coefficients, Ri&i(x, y), used in Equations 6.44 and 6.45 for the calculation of the theoretical decay rates, I’, for each isotopomer are given in Tables C.4 through C.7. For the homonuclear species the use of the tables is straightforward. For example, the expression for the decay rate of the J’ = 0, F’ = 0 hyperfine level may be read directly from the first line of table C.4 as, [‘(F’ = 0) = 0.0C + 0.0003v’Ca(79) + 0.6642a(79) +0.0402 + a(79)] b(79) + 2.4b(79) (C.3) Appendix C. Phase Shift Data g L 5 IA ( degrees) 196 f = 18.48 kHz 4)rei = 60.7° —arctan(IQ/IP) —4) (degrees) (degrees) 60.7 70.0 35.47 46.36 46.56 45.54 48.13 45.84 43.92 45.55 46.13 Table C.l: The deduced lifetimes of the v’=13, J’ modulation frequency of 18.48 kHz. , (bus) 35.47 37.06 37.26 37.26 38.83 36.54 34.62 36.25 36.83 Average For 79 81 Br T 6.137 6.503 6.552 6.313 6.932 6.382 5.945 6.315 6.451 (6.392 ± 0.276) = 1, F’ = 1_ hyperfine state using a the coefficients are split up into Tables C.6 and C.7. These two tables for 81 Br are used in exactly the same manner as the homonuclear tables except for new Br 79 the labels, B = b(79)+b(8l) (C.4) AB = [a(79)b(79)+a(81)b(81)j (C.5) in Table C.7. The decay rate for one of the heteronuclear levels (J’, F’) is given by the sum of all of the entries in Tables C.6 and C.7. For example, for the J’ one has = 0, F’ = 1 level Appendix C. Phase Shift Data Q L 5 IA (degrees) 50.0 49.8 197 f = 25.49 kHz 4ref = 49.8° —arctan(IQ/IP) (degrees) (degrees) r (us) — 46.48 43.84 45,00 47.16 47.04 43.14 47.81 47.54 46.31 43.90 47.54 46.28 43.64 44.80 46.96 46.84 42.94 47.81 47.54 46.31 43.90 47.54 Average Table C.2: The deduced lifetimes of the v’=13, J’ modulation frequency of 25.49 kHz. 6.530 5.955 6.200 6.687 6.660 5.809 6.888 6.823 6.537 6.008 6.823 (6.447 ± 0.387) = 1, F’ = 1... hyperfine state using a Appendix C. Phase Shift Data L1A (degrees) 36.3 30.0 198 f = 35.97 kHz ref = 35.7° —arctan(IQ/JP) 5 —q (degrees) (degrees) 54.54 55.41 55.20 53.92 51.24 49.40 48.87 46.83 51.09 48.41 46.10 49.83 r (us) 53.94 54.81 54.60 53.32 50.64 55.10 54.57 52.53 56.79 54.11 51.80 55.53 Average Table C.3: The deduced lifetimes of the v’=13, J’ modulation frequency of 35.97 kHz. 6.076 6.274 6.226 5.940 5.395 6.342 6.219 5.774 6.759 6.115 5.622 6.445 (6.099 ± 0.373) = 1, F’ = 1_ hyperfine state using a Appendix C. Phase Shift Data J’ F’ C 0 1 3 1_ 1+ 3_ 3÷ 5b 6 7 5_ 5+ 6a 7b 8 9 7_ 7+ 9b 10 11 0.0 0.0 2.0 2.0 12.0 12.0 20.0 20.0 20.0 30.0 30.0 42.0 42.0 42.0 42.0 56.0 56.0 72.0 72.0 72.0 1 3 4 5 6 7 8 199 /Ca(79) a(79) 0.0003 0.0001 1.5018 1.4983 1.4972 1.5029 —0.4800 —5.0000 —12.0000 1.4971 1.5030 1.5624 —2.5155 —9.0000 —18.0000 1.4970 1.5030 —4.5722 —13.0000 —24.0000 0.6642 4.0231 1.3505 1.3495 0.8164 0.8169 2.3039 3.3230 4.9125 0.7747 0.7766 0.8012 2.3071 3.5035 5.0022 0.7623 0.7648 2.3071 3.6079 5.0543 [Cv+a(79)jbv(79) 0.0402 0.0402 —2.1788 2.2029 —1.7887 1.7890 —2.0933 —0.9373 1.2679 —1.7541 1.7541 1.7490 —2.0346 —0.6874 1.3914 —1.7437 1.7438 —1.9830 —0.5427 1.4630 b(79) 2.4000 2.4000 2.6027 2.5972 2.0721 2.0612 1.6918 2.8964 2.2094 2.0303 2.0210 2.0199 1.7733 2.9419 2.1645 2.0179 2.0092 1.8233 2.9629 2.1352 Table C.4: Coefficients, Rii(x, y) for calculating of I’ for 79 2 Br Appendix C. Phase Shift Data J’ F’ C 0 1 3 1_ 1÷ 3_ 3 4a Sb 7 5_ 5+ 6a 7b 8 9 7_ 7+ 9b 10 11 0.0 0.0 2.0 2.0 12.0 12.0 20.0 20.0 20.0 30.0 30.0 42.0 42.0 42.0 42.0 56.0 56.0 72.0 72.0 72.0 1 3 4 5 6 7 8 200 i/C,a(81) a(81) 0.0002 0.0001 1.5018 1.4983 1.4965 1.5035 1.6461 —0.4830 —12.0000 1.4964 1.5036 1.5632 —2.5183 —9.0000 —18.0000 1.4964 1.5036 —4.5750 —13.0000 —24.0000 0.6646 4.0196 1.3505 1.3495 0.8159 0.8174 0.8728 2.3026 4.9120 0.7742 0.7770 0.8018 2.3058 3.5029 5.0019 0.7620 0.7652 2.3057 3.6075 5.0540 [/c + av(81)] b(81) 0.0340 0.0340 —2.1788 2.2029 —1.7887 1.7890 1.7716 —2.0939 1.2666 —1.7541 1.7541 1.7490 —2.0351 —0.6882 1.3905 —1.7438 1.7438 —1.9835 —0.5433 1.4623 Table C.5: Coefficients Rii(x, y) for calculating 1’ for 81 2 Br b(81) 2.4000 2.4000 2.6027 2.5972 2.0727 2.0606 2.0494 1.6917 2.2095 2.0310 2.0203 2.0200 1.7733 2.9418 2.1645 2.0187 2.0085 1.8232 2.9628 2.1352 Appendix C. Phase Shift Data J’ F’ C 0 1 2 3 0 la 2a 2b 3b 4 3c 4b 5 3a 4c 5b 6 3a 5c 7 4a 5a 5d 6c 8 5a 6a 6d 7c 8b 9 lOd llc 13 0.0 0.0 0.0 2.0 2.0 2.0 2.0 2.0 2.0 6.0 6.0 6.0 12.0 12.0 12.0 12.0 20.0 20.0 20.0 30.0 30.0 30.0 30.0 30.0 42.0 42.0 42.0 42.0 42.0 42.0 110.0 110.0 110.0 1 2 3 4 5 6 10 201 V’C[a(79)+a(8l)] a(79)+a(81) 0.0001 0.0000 0.0001 1.0001 0.7840 0.3462 0.0762 0.6206 —1.5000 0.5980 —0.2020 —3.0000 0.5448 0.2183 —0.9712 —4.5000 2.6488 —0.2607 —6.0000 3.4145 0.1670 3.2942 —0.7636 —7.5000 4.1259 —0.0708 3.5434 —1.2737 —3.0057 —9.0000 4.7456 —3.3209 —15.000 1.2550 1.2511 1.2550 0.8520 0.9527 1.1781 1.1865 0.9783 1.3526 0.9807 1.0840 1.3946 1.0101 0.9524 1.1491 1.4179 1.0802 0.9502 1.4328 1.0683 1.0143 1.0336 0.9531 1.4431 1.0588 1.0148 1.0193 0.9570 1.2599 1.4507 0.9986 0.9694 1.4678 aL,(79)a.V(81) —1.8463 —0.5036 1.5106 —0.7039 —0.5202 —0.4135 —1.2086 0.6011 1.8044 0.3746 0.8504 1.9309 —1.1527 0.4064 0.9827 2.0014 —0.8643 0.4053 2.0464 —0.6339 —1.1869 0.8005 0.3995 2.0776 —0.4539 —1.1858 0.7674 0.3925 1.1396 2.1005 0.6835 0.3647 2.1524 Table C.6: Coefficients Rij’(x, y) for calculating of P for 79 81 (part I). Br Appendix C. Phase Shift Data J’ F’ ./CVB 0 1 2 3 0 la 2a 2b 3b 4 3c 4b 5 3a 4c 5b 6 3a 5c 7 4a 5a 5d 6c 8 5a 6a 6d 7c 8b 9 lOd lic 13 0.0198 0.0085 0.0198 —1.3789 —1.1445 —0.8526 0.2011 —1.1167 0.3570 —1.0431 —0.8846 0.5019 —0.9517 —1.0667 —0.7767 0.5826 —0.6693 —1.0542 0.6340 —0.6698 —0.9321 —0.6600 —1.0364 0.6696 —0.6834 —0.9276 —0.7050 —1.0195 —0.6890 0.6957 —0.7773 —0.9726 0.7549 1 2 3 4 5 6 10 202 AB 0.0174 0.0038 0.0174 —1.3789 —1.0298 —0.2491 —0.2199 —0.9413 0.3555 —0.9328 —0.5751 0.5008 —0.8312 —1.0309 —0.3496 0.5817 —0.5881 —1.0386 0.6333 —0.6294 —0.8166 —0.7495 —1.0284 0.6690 —0.6624 —0.8149 —0.7993 —1.0151 0.0341 0.6952 —0.8708 —0.9719 0.7546 b(79)+b(81) b(79)b(81) 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.4000 —1.1999 0.4000 1.2104 0.5985 —0.0501 —1.5459 0.5684 0.3421 —0.4009 0.7465 0.2851 0.0690 —0.3647 0.8174 0.2419 0.3226 —0.3085 0.2094 0.2564 0.0272 0.2276 —0.2624 0.1843 0.2146 0.0195 0.1634 —0.2267 0.8487 0.1645 0.0643 —0.1442 0.1147 Table C.7: Coefficients Ri(x, y) for calculating 1’ for 79 81 (part II). Br Appendix C. Phase Shift Data F(F’ = 0) = 203 0.OC + 0.O001VC [a(79) + a(81)] +1.2550 [a(79) + a(81)] — 1.8463a(79)a(81) +O.0l98VC [b(79) + b(81)] + 0.0174 [a(79)b(79) + a(81)b(81)] +1.000 [b(79) + b(81)] + 0.4000b(79)b(81) (C.6) These expressions were fit to the observed decay rates for the parameters described in Chapter 6. The observed and calculated inverse lifetimes (decay rates) are given in Tables C.8 to C.10 along with the differences, (obs-caic). The calculated values were obtained from a global fit of the data to a six parameter model described in Chapter 6. The parameters were, (C.7) rad Co (C.8) 1 B = 2 B = g(79)ao(79, 79) (C.9) 3 B = Q(79)b(79,79) (C.10) 4 B = o(81,81)/(79,79) (C.11) o(79,81)/u(79,79) (C.12) 5 B (7979)O(7979) The hyperfine states are labelled both by J’ and F’. For many of the rotational levels several hyperfine states with the same value of F’ exist. These are distinguished with the further label a, b, c, and so on in order of ascending energy. (e.g. J’ lowest energy hyperfine level for the J’ = = 4 rotational state having F’ 4, F’ = = 6a is the 6.) As can be seen, the agreement between the observed and calculated values is excellent. For the readers convenience, the lifetimes of the states, r, derived from 1/T, are listed in Tables C.ll through C.13. Appendix C. Phase Shift Data 204 2 B ll+ v’ Br 79 J F’ — 0 1 3 4 5 6 7 8 1 3 1_ 1÷ 3_ 3÷ 5b 6 7 5_ 5+ 6a 7b 8 9 7_ 7 9b 10 11 = 13 l’obs “caic (10 s 5 ’) (10 s 5 ’) (obs caic) s’) 5 (10 1.203 ± 0.077 2.487 ± 0.148 1.584±0.086 2.200 ± 0.140 2.232 ± 0.195 2.729 ± 0.201 2.923 ± 0.093 2.425±0.114 1.551 ± 0.070 3.598 ± 0.302 4.279 ± 0.588 4.776 ± 0.312 4.125±0.272 3.295 ± 0.065 1.759 ± 0.033 5.385 ± 0.505 6.105±0.932 5.734 ± 0.122 4.570±0.142 2.460 ± 0.221 1.173 2.630 1.708 2.292 2.262 2.742 2.953 2.421 1.616 3.587 4.058 4.976 4.083 3.154 1.783 5.517 5.986 5.800 4.435 2.522 0.030 —0.143 —0.124 —0.092 —0.030 —0.013 —0.030 0.004 —0.065 0.011 0.221 —0.200 0.042 0.141 —0.024 —0.132 0.119 —0.066 0.135 —0.062 - Table C.8: The observed and calculated inverse lifetimes, I’, for 79 2 Br in the observed decay rates are the standard deviations of the data. . The uncertainties Appendix C. Phase Shift Data 205 2 B H+ v’ Br 81 J F’ — 0 1 3 4 5 6 7 8 1 3 1_ 1÷ 3_ 3+ 4a 5b 7 5_ 5÷ 6a 7b 8 9 7_ 7+ 9b 10 11 = 13 robs (10 s 5 ’) 1’i (10 s 5 ’) (obs caic) s’) 5 (10 1.202 ± 0.050 3.134+0.055 1.925 ± 0.204 2.542 ± 0.286 2.563 ± 0.123 3.066 ± 0.142 3.928 ± 0.208 3.569 ± 0.185 1.824±0.208 3.830 ± 0.317 4.425 ± 0.360 5.112 ± 0.316 4.130 ± 0.140 3.388 ± 0.125 1.775 ± 0.037 5.780 ± 0.575 6.06 1 ± 0.566 6.180 ± 0.978 4.621 ± 0.049 2.414 ± 0.056 1.245 3.116 1.946 2.507 2.472 2.935 3.632 3.299 1.790 3.862 4.316 5.286 4.429 3.370 1.804 5.891 6.343 6.177 4.613 2.418 —0.043 0.018 —0.021 0.035 0.091 0.131 0.296 0.270 0.034 —0.032 0.109 —0.174 —0.299 0.018 —0.029 —0.111 —0.282 0.003 0.008 —0.004 - 2 Br Table C.9: The observed and calculated inverse lifetimes, 1’, for 81 in the observed decay rates are the standard deviations of the data. . The uncertainties Appendix C. Phase Shift Data 206 81 B ll+ v’ Br 79 J’ F’ = 13 Fobs (10 s 5 ’) o 2 3 4 5 6 10 1 2 3 0 la 2a 2b 3b 4 3c 4b 5 3a 4c 5b 6 3a 5c 7 4a 5a 5d 6c 8 5a 6a 6d 7c 8b 9 lOd lic 13 1.224±0.082 1.933±0.112 2.924±0.352 1.682±0.298 1.895 ± 0.206 1.726 ± 0.200 1.827 ± 0.213 2.367 ± 0.240 2.593 ± 0.211 2.420 ± 0.266 2.394 ± 0.204 2.192±0.132 2.472 ± 0.202 2.888 ± 0.229 2.583 ± 0.192 2.005±0.100 3.762 ± 0.553 3.189±0.059 1.742±0.075 5.402 ± 1.208 3.939 ± 0.399 5.580 ± 1.632 3.897 ± 0.425 1.734 ± 0.072 7.027 ± 1.689 4.805 ± 0.644 7.210 ± 1.669 4.320 ± 0.442 3.571 ± 0.164 1.834±0.056 11.136±1.798 7.675 ± 1.678 3.917±0.305 F (10 s 5 ’) 1.213 1.849 2.879 1.728 1.872 2.058 1.606 2.38 1 2.535 2.561 2.550 2.155 2.297 2.782 2.766 1.866 4.362 3.130 1.702 5.649 3.465 6.240 3.626 1.677 7.039 4.258 7.257 4.272 4.212 1.797 13.074 8.386 3.770 (obs caic) s’) 5 (10 - 0.011 0.084 0.045 —0.046 0.023 —0.332 0.221 —0.014 0.058 —0.141 —0.156 0.037 0.175 0.106 —0.183 0.139 —0.600 0.059 0.040 —0.247 0.474 —0.660 0.271 0.057 —0.012 0.547 —0.047 0.048 —0.641 0.037 —1.938 —0.711 0.147 Table C. 10: The observed and calculated inverse lifetimes, F, for 79 81 The uncer Br tainties in the observed decay rates are the standard deviations of the data. . Appendix C. Phase Shift Data 207 2 B 3fl; Br 79 V J’ Tobs 0 1 3 4 5 6 7 8 F’ 1 3 1_ 1÷ 3_ 3 Sb 6 7 5... 5 6a 7b 8 9 7_ 7+ 9b 10 11 = 13 (ts) rj (its) 8.313 ±0.532 4.021 ±0.239 6.313 ±0.343 4.545 ±0.289 4.480 ±0.391 3.664 ±0.270 3.421 ±0.109 4.124 ±0.194 6.447 ±0.291 2.779 ±0.233 2.337 ±0.321 2.094 ±0.137 2.424 ±0.160 3.035 ±0.060 5.685 ±0.107 1.857 ±0.174 1.638 ±0.250 1.744 ±0.037 2.188 ±0.068 4.065 ±0.365 8.525 3.802 5.855 4.363 4.421 3.647 3.386 4.131 6.188 2.788 2.464 2.010 2.449 3.171 5.609 1.813 1.671 1.724 2.255 3.965 (ohs - caic) (its) —0.213 0.219 0.458 0.182 0.059 0.017 0.035 —0.007 0.259 —0.009 —0.127 0.084 —0.025 —0.136 0.077 0.044 —0.033 0.020 —0.067 0.100 Table C.1l: The observed and calculated lifetimes, r, for 79 2 Br Appendix C. Phase Shift Data 208 2 B ll+ v’ Br 81 J’ 0 I 3 4 5 6 7 8 F’ = 13 Tobs Tcalc (its) (ts) (obs caic) (us) 8.032 3.209 5.139 3.989 4.045 3.407 2.753 3.031 5.587 2.589 2.317 1.892 2.258 2.967 5.543 1.698 1.577 1.619 2.168 4.136 0.287 —0.018 0.056 —0.055 —0.144 —0.146 —0.207 —0.229 —0.104 0.022 —0.057 0.064 0.163 —0.016 0.091 0.033 0.073 —0.001 —0.004 —0.010 1 8.319 ± 0.346 3 3.191 ± 0.056 1_ 5.195±0.551 1+ 3.934 ± 0.443 3_ 3.902 ± 0.187 3 3.262 ± 0.151 4a 2.546 ± 0.135 Sb 2.802 ± 0.145 7 5.482 ± 0.625 5.. 2.611 ± 0.216 5 2.260 ± 0.184 6a 1.956±0.121 7b 2.421 ± 0.082 8 2.952 ± 0.109 9 5.634±0.117 7_ 1.730±0.172 7÷ 1.650 ± 0.154 9b 1.618±0.256 10 2.164±0.023 11 4.125 ± 0.095 - Table C.12: The observed and calculated lifetimes, T, for 81 2 Br Appendix C. Phase Shift Data 209 81 B ll÷ v’ Br 79 J’ o 1 2 3 4 5 6 10 F’ 1 2 3 0 la 2a 2b 3b 4 3c 4b 5 3a 4c 5b 6 3a 5c 7 4a 5a 5d 6c 8 5a 6a 6d 7c 8b 9 lOd lic 13 = 13 Tobs rj (as) (ps) (obs caic) (as) 8.170±0.547 5.173 ± 0.300 3.420 ± 0.412 5.945±1.053 5.277 ± 0.574 5.794 ± 0.671 5.473 ± 0.638 4.225 ± 0.428 3.857 ± 0.314 4.132±0.454 4.177 ± 0.356 4.562 ± 0.275 4.045 ± 0.331 3.463 ± 0,275 3.871 ± 0.288 4.988 ± 0.249 2.658 ± 0.391 3.136 ± 0.058 5.741 ± 0.247 1.851 ± 0.414 2.539 ± 0.257 1.792±0.524 2.566 ± 0.280 5.767 ± 0.239 1.423 ± 0.342 2.081 ± 0.279 1.387 ± 0.321 2.3 15 ± 0.237 2.800 ± 0.129 5.453 ± 0.166 0.898±0.145 1.303±0.285 2.553 ± 0.199 8.244 5.408 3.473 5.787 5.342 4.859 6.227 4.200 3.945 3.905 3.922 4.640 4.354 3.595 3.615 5.359 2.293 3.195 5.875 1.770 2.886 1.603 2.758 5.963 1.421 2.349 1.378 2.341 2.374 5.565 0.765 1.192 2.653 —0.074 —0.235 —0.053 0.158 —0.065 0.935 —0.753 0.025 —0.088 0.228 0.256 —0.078 —0.308 —0.132 0.256 —0.372 0.366 —0.059 —0.135 0.081 —0.347 0.190 —0.192 —0.196 0.002 —0.267 0.009 —0.026 0.426 —0.112 0.133 0.110 —0.100 - Table CJ3: The observed and calculated lifetimes, r, for 79 ’Br 8 Br
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Hyperfine structure and predissociation of the B ³[pi]₀ + state of bromine Booth, James L. 1994
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Title | Hyperfine structure and predissociation of the B ³[pi]₀ + state of bromine |
Creator |
Booth, James L. |
Date Issued | 1994 |
Description | Investigations have been carried out in bromine of the hyperfine structure of the B ³II₀u+ and X ¹Σg⁺ electronic states and of the predissociation of the B ³ll₀u+ state by the ¹ll₁u dissociative level. The technique of laser induced fluorescence of a molecular beam was used. ⁷⁹Br⁸¹Br hyperfine spectra were recorded for various B - X vibrational bands (v’ ←v”) with v’ = 11 through 17 and v” = 0, 1, and 2, and for various rotational transitions (J’ ← J”) with J’ from 0 to 11 and J” from 0 to 10. As well, the ⁷⁹Br₂ and ⁸¹Br₂ hyperfine spectra of the (13’ — 0”) and (17’ — 2”) bands over the same range of rotational states were measured. The spectra are well described using one X state parameter: the electric quadrupole coupling constant eqQx; and two B state parameters: the electric quadrupole coupling constant eqQB, and the nuclear spin-rotation constant Csr. The results show that eqQB(⁷⁹Br) = (177.0 ± 0.6) MHz for v’ = 11 and increases by approximately 0.5 MHz per vibrational quantum up to (180.6 ± 1.4) MHz for v’ 17. Similarly the ground state electric quadrupole coupling constant, eqQx(⁷⁹Br) (808.1 ± 1.4) MHz for v” = 0 and increases by about 1 MHz per vibrational quantum to (811.4 ± 1.4) MHz for v” = 2. The hyperfine data also provided a check on the accuracy of some of the published rovibronic constant¹ for each isotopomer. In order to reproduce the observed relative spacings of the transitions for all three isotopomers, the published term values, T₀₀, have to be modified ; this can be done by decreasing the published values of T₀₀ for ⁸¹Br₂ and ⁷⁹Br⁸¹Br by (177 ± 8) MHz and (326 ± 8) MHz, respectively. The phase shift technique was applied to the study of the predissociation of the v’ = 13 B ³II₀u⁺ electronic state of bromine. The lifetimes of individual hyperfine levels were ¹S. Gerstenkorn and P. Luc, J. Phys. France 50, 1417 (1989). measured for the rotational states J’ = 0 - 7 (except for J’ 2) for each isotopomer of bromine. Revised values are given for the radiative decay rate Γrad, the gyroscopic predissociation parameter Cv, and the magnetic dipole predissociation parameter av. The first observation of electric quadrupole predissociation is reported and is characterized by a new molecular parameter, bv. |
Extent | 4323511 bytes |
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Thesis/Dissertation |
Type |
Text |
File Format | application/pdf |
Language | eng |
Date Available | 2009-04-08 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0085651 |
URI | http://hdl.handle.net/2429/6914 |
Degree |
Doctor of Philosophy - PhD |
Program |
Physics |
Affiliation |
Science, Faculty of Physics and Astronomy, Department of |
Degree Grantor | University of British Columbia |
Graduation Date | 1994-05 |
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UBCV |
Scholarly Level | Graduate |
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