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Electron spin resonance studies of reaction intermediates in metallic halides Catton, Richard Carl 1967

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The U n i v e r s i t y of B r i t i s h  Columbia  FACULTY GF GRADUATE STUDIES  VBDC&MME OF THE FINAL 'ORAL EXAMINATION FOR THE DEGREE OF DOCTOR OF PHILOSOPHY OF RICHARD CARL CATTON B„Sc„, U n i v e r s i t y o f B r i t i s h  Columbia  WEDNESDAY, AUGUST 30, a t 3:30 P.M. IN ROOM 225, CHEMISTRY BUILDING  COMMITTEE IN CHARGE Chairman:  1= McT. Cowan  J„W. B i c h a r d  R„ Stewart  L.Go  R„C„ Thompson  Harrison  W„C. L i n E x t e r n a l Examiner: J„E. Wertz Department of C h e m i s t r y U n i v e r s i t y o f Minnesota M i n n e a p o l i s , Minnesota Research S u p e r v i s o r :  L„G, H a r r i s o n  ELECTRON SPIN RESONANCE STUDIES OF REACTION INTERMEDIATES IN METALLIC HALIDES  ABSTRACT The  object  reaction and  of t h i s work was to produce and i d e n t  intermediates  group I I h a l i d e s  i n systems c o n t a i n i n g treated with f l u o r i n e .  systems which a r e s t u d i e s KC.1/F and S r C . l / F 2  2  2  in detail  group I The  are the NaCl and  systems.  In accordance w i t h p r e d i c t i o n s from k i n e t i c s t u d i e s , ESR has shown t h a t treatment of vacuumsublimed NaCl w i t h F  2  a t room  a defect containing  an  the decay of t h i s d e f e c t  temperature produces  unpaired  e l e c t r o n , and that  i s retarded  by C l » 2  The  ESR spectrum i n d i c a t e s an a n i s o t r o p i c g - f a c t o r and unresolved  hyperfine  i s obtained hyperfine  structure,.  In KC1 a spectrum  which i s c l o s e l y s i m i l a r except t h a t the  structure i s partly resolved.  The s p e c t r a  appear c o n s i s t e n t w i t h i n t e r a c t i o n s of the u n p a i r e d e l e c t r o n w i t h CI atoms, r a t h e r  than w i t h F or such  i m p u r i t i e s as 0 or Br, and can be accounted f o r ade3quately  by a model of the d e f e c t  an H-center or something very spectra  as l i n e a r C l ^ ,  similar.  i.e.  F o r NaCl, the  i n d i c a t e t h a t the vacuum-sublimed m a t e r i a l i s  sometimes p a r t l y o r i e n t e d . I t i s s u g g e s t e d that c e n t e r s  with  d e f i c i e n c y are o b s e r v a b l e a t such h i g h  one-electron temperatures  i n vacuum-sublimed m a t e r i a l because t h i s e s s e n t i a l l y p e r f e c t c r y s t a l s devoid  c o n s i s t s of  of s i t e s  which  c o u l d accept The  a second e l e c t r o n from the c e n t e r s .  r e a c t i o n of S r C l ^ powder or s i n g l e  crystal,  w i t h f l u o r i n e at room temperature produces a d e f e c t , s t a b l e a f t e r removal of f l u o r i n e but  located close  to the r e a c t i o n i n t e r f a c e o n l y , which has t r o n s p i n resonance a b s o r p t i o n . i s consistent with  The  t i o n s of the h y p e r f i n e tensor and  lattice„  The  unpaired suggest tal  principal g-tensor  a f o u r - f o l d a x i s of the  tensor components  towards direc-  are  field  two  SrCl^  i n d i c a t e that  the  e l e c t r o n i s l o c a l i z e d on the C l atom,  and  t h a t the atom i s s u b j e c t to a s t r o n g  cations The  spectrum  from an anion s i t e  a n i o n vacancy.  t w o - f o l d axes and  ESR  elec-  a model of the d e f e c t as a  c h l o r i n e atom d e s p l a c e d a neighbouring  The  an  determined c h i e f l y by  two  crys-  nearest-neighbour  which d e f i n e a t w o - f o l d a x i s of the  unpaired  crystal.  e l e c t r o n i s i n an o r b i t a l mainly  p-character  and  i s probably  the one  of  a l i g n e d along a t w o - f o l d a x i s which perpendicular  to the l i n e of  the  cations„ The  single crystal  orientation-dependent and  spectrum, although  l i n e p o s i t i o n s , has  i n t e n s i t i e s resembling  trum.  T h i s suggests  ment of C l atoms and  those  a range  of a  crystal.  line  powder  shapes spec-  of r e l a t i v e d i s p l a c e -  neighbouring  c a t i o n s along  " r e a c t i o n c o o r d i n a t e " which i s p r o b a b l y a x i s of the  having  a  a  four-fold^  GRADUATE STUDIES F i e l d o f Study: Topics  Chemistry  i n P h y s i c a l Chemistry  Seminar i n Chemistry T o p i c s i n Chemical  Chemical Topics  K.B. Harvey .  Physics  Thermodynamics  i n Inorganic  Chemistry  Chemistry o f the S o l i d  CA.  McDowell W.C. L i n C.E. B r i o n J.R. Sams  S p e c t r o s c o p y and M o l e c u l a r Structures  Topic  J.AoR. Coope W.C. L i n  State  i n Organic Chemistry  WoRo C u l l e n R.C, Thompson N.L. Paddock H.C. C l a r k K„B. Harvey A.V. Bree, L.G. H a r r i s o n F„ McCapra L.D. H a l l DoE. McGreer  PUBLICATIONS  LaG.  H a r r i s o n , R„J„ Adams and R.C. C a t t o n , ESR S t u d i e s of a R e a c t i o n I n t e r m e d i a t e i n Vacuum-sublimed NaCl and K C l , J . Chem, Phys., 45, 4023 (1966).  R„C  C a t t o n and L.G. H a r r i s o n , ESR S t u d i e s o f a Reaction Intermediate i n Strontium C h l o r i d e , J . Chem. Phys. ( i n p r e s s , s h o u l d appear about November, 1967).  ELECTRON SPIN RESONANCE STUDIES OF REACTION INTERMEDIATES IN METALLIC HALIDES  by RICHARD CARL CATTON, J R . B . S c , UNIVERSITY OF BRITISH COLUMBIA, 1964.  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY In the Department of CHEMISTRY  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA AUGUST, 1967  In p r e s e n t i n g t h i s  thesis  f o r an advanced degree at  that  the L i b r a r y s h a l l  Study.  thesis  in p a r t i a l  the U n i v e r s i t y of  make i t  British  freely available  I f u r t h e r agree that p e r m i s s i o n  for  the requirements  Columbia,  I agree  r e f e r e n c e and  f o r e x t e n s i v e copying of  this  f o r s c h o l a r l y purposes may be granted by the Head of my  Department or by h fc: r e p r e s e n t a t i v e s .  or p u b l i c a t i o n of t h i s  thesis  Department of The U n i v e r s i t y of B r i t i s h Vancouver 8, Canada  It  is  for f i n a n c i a l  without my w r i t t e n p e r m i s s i o n .  Date  f u l f i l m e n t of  Columbia  understood that  gain s h a l l  copying  not be allowed  - ii ABSTRACT The object of this work was to produce and identify reaction intermediates in systems containing group I and group II halides treated with fluorine. The systems which are studied in detail are the NaCl and KC1 / F  2  and SrClg / ?  2  systems.  In accordance with predictions from kinetic studies, ESR has shown that treatment of vacuum-sublimed NaCl with  at room temperature  produces a defect containing an unpaired electron, and that the decay of this defect is retarded by CX>. The ESR spectrum indicates an anisotropic g-factor and unresolved hyperfine structure. In KC1, a spectrum is obtained which is closely similar except that the hyperfine structure is partly r e solved. The spectra appear consistent with interactions of the unpaired electron with CI atoms, rather than with F or such impurities as 0 or Br, and can be accounted for adequately by a model of the defect as linear C l ^ , i . e . , an H center or something very similar. For NaCl, the spectra i n d i cate that the vacuum-sublimed material is sometimes partly oriented. It is suggested that centers with one-electron deficiency are observable at such high temperatures in vacuum-sublimed material because this consists of essentially perfect crystals devoid of sites which could accept a second electron from the centers. The reaction of SrCLj powder or single crystal with fluorine at room temperature produces a defect, stable after removal of fluorine but located close to the reaction interface only, which has an electron spin resonance absorption. The ESR spectrum is consistent with a model of the defect as a chlorine atom displaced from an anion site towards a neigh-  - ill bouring anion vacancy. The principal directions of the hyperfine tensor and g-tensor are two two-fold axes and a four-fold axis of the S r C l  2  lat-  tice. The tensor components indicate that the unpaired electron is l o c a l ized on the Cl atom, and suggests that the atom is subject to a strong crystal f i e l d determind chiefly by two nearest-neighbour cations which define a two-fold axis of the crystal. The unpaired electron is in an orbital mainl y of p-character and aligned along a two-fold axis which is probably the one perpendicular to the line of the cations. The single-crystal spectrum, although haying orientation-dependent line positions, has line shapes and intensities resembling those of a powder spectrum. This suggests a range of relative displacement of Cl atoms and neighbouring cations along a "reaction co-ordinate" which is probably a four-fold axis of the crystal.  - iv TABLE OF CONTENTS Page T i t l e Page  i  Abstract  i i  Table of Contents  iv  List of Figures  vi  List of Tables  viii  Acknowledgements  ±x  INTRODUCTION A.  Kinetic and Mechanisms of Reactions of Alkali Halides  1  a) Room temperature NaCl / &  1  2  exchange reactions  b) Room temperature oxidation by Clg and  4  c) Production of colour centers by irradiation at low temperatures, and subsequent interactions  7  d) Summary of electron deficient colour centers i n a l k a l i halides relavent to the present work B.  14  Basic Theory of ESR  16  a) Resonance condition  16  b) Hyperfine interaction  17  c) Anisotropy  19  d) ESR spectrum of an unpaired electron interacting with several nuclei in a solid medium  20  i ) Structure of the ESR spectrum  20  i i ) Relationship of components of g and A tensors to the state  - V  -  of the unpaired electron C.  2 2  Previous Structural Determination of Defects in Alkali Halides by ESR  31  a) F-center  3 2  b) V^-center  33  c) H-center  40  d) Other electron-deficient centers  45  APPARATUS AND PROCEDURE A.  ESR Spectrometers  49  B.  Vacuum Sublimation of Ionic Solids  51  C.  Handling of Halogens  55  a) Chlorine drying system  56  b) Fluorine handling system  58  D.  Reaction Apparatus  60  E.  Reaction Procedure  60  GROUP I HALIDES A.  Summary of Systems Investigated for ESR Signals  6  B.  Reaction of NaCl and KC1 with F  65  2  4  a) Analysis of the spectra  65  b) Computations  71  c) Kinetics  77  GROUP II HALIDES A»  Summary of Systems Investigated for ESR Signals  8 3  - v i -  B.  Reaction of SrClg with F  84  2  a) Results  85  b) Discussion  93  i) Model of the defect  95  i i ) g-shifts  98  i i i ) Interpretation of the hvperfine tensor A iv) Reaction mechanism  100 107  APPENDIX A.  Computations  109  a) Fortran U computer programme  109  b) Computed spectra obtained for defect in KC1  112  LIST OF FIGURES INTRODUCTION Figure  1 - Trapped hole centers  13  Figure  2 - Cone, of V-centers in LiF vs annealing  13  temperature Figure  3 - Isotope effects in KC1 of a V -center K  36  (schematic) Figure  U - Spectrum of a Cl  - C l * ^ molecule ion  with H|| (lCK3),**(Vg-center,  39  in KC1)  Figure  5 - H-center (hyperfine interactions) in KC1  43  Figure  6 - Orbital levels of the  44  both the V and H centers K  molecule ion for  -  Vii  -  Easa APPARATUS AND PROCEDURE Figure  7 - Block diagram of a 100 Kc. (6 inch)  50  ESR spectrometer Figure  8 - Block diagram of an E-3 ESR spectrometer  52  Figure  9 - Evaporated film apparatus  54  Figure 10 - Chlorine purification system  57  Figure 11 - Fluorine handling system  59  Figure 12 - Reaction apparatus 1  61  Figure 13 - Reaction apparatus 2  62  GROUP I HALIDES Figure 14 - ESR spectra obtained in F  treated  66  Figure 15 - Growth and decay of the ESR signal  78  2  NaCl andKCl  after F^ treatment GROUP II HALIDES Figure 16 - Computed spectrum of the defect in S r C l Figure 17 - ESR spectra obtained in F  2  2  88  treated SrClg  91  Figure 18 - Models of the defect in S r C l  2  Figure 19 - Energy levels of 3p^ configuration s p l i t  96 97  by a crystal f i e l d Figure 20 - Energy levels of a positive electron in 3p  99  configuration s p l i t by a crystal f i e l d  APPENDIX Figure 21 - Computed spectra obtained for defect in KC1  113  - viii  LIST OF TABLES INTRODUCTION Table  I - Summary of electron-deficient colour centers in a l k a l i halides relevant to the present work  Table  II - Classification of molecule-ions for simple orientations of the crystal in the d.c. magnetic f i e l d  GROUP I HALIDES Table III  r Summary of systems investigated for ESR signals  Table  IV - Important features of spectra and proposed assignments for H-center model  GROUP II HALIDES Table  V - Summary of systems investigated for ESR signals  Table  VI - Distribution of orientations of defect relative to symmetry axes of crystal  Table VII - Parameters of the defect Table VIII- Character of the state of the unpaired electron Table  IX - Character of the state of the unpaired electron from hyperfine splitting and known constants of 3s and 3p orbitals  Table  X - g and A tensor components for CI atoms in various environments  - ix ACKNOWLEDGEMENTS I would like to express ray sincerest thanks to ray research director, Dr. L. G. Harrison, for his invaluable advice, guidance, and enlightening discussions throughout the course of this work. I would also like to thank Dr. C. A. McDowell for the use of his ESR f a c i l i t i e s ; the departmental shops - especially Mr. S. Rak, the glass blower, and Mr. L. de Fehr, the illustrator; Mr. C. R. Byfleet for his informative discussions; Mr. A. Fowler and Mr. R. Wolfe for their assistance in writing programmes; and f i n a l l y to the National Research Council for my studentships.  INTRODUCTION  A. Kinetics and Mechanisms of Reactions of Alkali Halldea Reaction mechanisms i n solids, like those in the much more widely studied f l u i d phases, are l i k e l y to involve intermediates with unpaired electrons. Such free radicals have not dominated the f i e l d of solid state kinetics as they have that of gas phase kinetics because transport and nucleatlon, which are specific to the solid phase, have required prior attention. Nevertheless, such defects have been discussed in a number of studies ( l - 5 ) , although the analogy to gas-phase mechanisms i s commonly obscured by the conventional use of the terms "trapped electron" and "trapped hole", rather than "free radical", i n the solid phase, a) Room temperature NaCl / C l  2  exchange reaotions  Alkali halides were chosen for mechanistic studies of exchange reaotions (in Dr. J.A. Morrison's laboratory at N.R.C. Ottawa and later in Dr. L.G. Harrison's laboratory here) since more i s known about the structures of their ionic and electronic defects than i s the case for any other class of solids. Thus there i s some hope of obtaining precise information on the role of these defects i n reaction mechanisms. The f i r s t sodium chloride particles used i n the study of the exchange reaction, NaCl^ / Cl^, were made by evaporating sodium chloride i n dry nitrogen at atmospherlo pressure ( 6 ) . The experiments (7) showed that there was a rapid surface exchange at room temperature (like that found i n oxygen-metallic oxide systems ( 8 ) ) , followed by a slower extension of the process into the bulk (which has been followed for a period of days). This indicated (9) that structural irregularities must be present at least in the surface layer i t s e l f and possibly over a larger region. In order to avoid surface disordering by adsorption of atmospheric gases  sodium chloride particles were later made by evaporation in vacuo (10 to 10*"®mm Hg ). It was expected that the avoidance of surface contamination by vacuum evaporation would greatly reduoe the rate of the exchange or even suppress i t altogether* However* the exchange (lO) was found to be much more rapid than i n previous experiments and to involve the bulk of the solid i n a process not kinetically seperable from surface exchange Harrison. Hoodless, and Morrison (10) found that the rate of exchange obeys a power law of the form C  a  ot  n  i n which a i s independent of speci-  f i c surface. They postulated that the unusual rate law probably indicates the reaction i s self-Inhibited by the blocking of vacancies by chlorine atoms or molecules taken up from the gas phase. The defect structures thus formed should be similar to some of the trapped-hole defects which have been studied i n X-irradiated a l k a l i halides* Known as V-centers, a l though other letters are used for some of them. Harrison ( l l ) devised a tentative mechanism of this type. The meohanism was very speculative since i t was based on limited experimental evidence and since i t i s known to be d i f f i c u l t to produce V-centers i n alkali chlorides by additive coloration (l2)« Adams and Harrison (13)» again using vacuum-sublimed sodium chloride, confirmed the t  n  rate law and the laok of dependence of the  rate of exchange on speoifio surface. They attempted to eluoidate the role of defeots i n the reaction meohanism from the pressure and temperature dependence of the exchange rate* and from the exchange kinetics of sodium chloride samples into which electronic defects had been deliberately introduced by X-irradiation and fluorination. The kinetics of exchan ge were completely changed from a power law to a second order rate law of  .  - 3 -  the formt [ l/(  C^-  C ]  -  [ l/Cj  -K t Q  In the light of this more extensive evidence, mechanisms i n volving eleotronic defects were postulated to account for both observed rate laws* In both oases, the proposed kinetic schemes involved a "molecular" trapped hole defect and an "atomic" trapped defect. It was proposed that pretreatment, with fluorine or X-irradiation, produces an "atomic"defeot (a single trapped hole, probably asooiated with vacancies and hence mobile) which decays bimplecularly, rapidly in vacuo and more slowly after introducing chlorine. To explain the probable composition of the defect and i t s decay by the desorption of chlorine in terms of a chemical equation, Harrison's (14) chemical notation for 7-centers was used. In this notation, the customary ion sign i s omitted for an ion on i t s proper site. If these signs are replaced by suffixes (a) and (o) for anion and cation sites respectively, the V-centers acquire superscripts which represent the.number of trapped positive holes. The untrapped hole i s designated x ( a ) ; x(a) i s the halide ion; a +  c  +  i s an anion vacancy,  a cation vacancy, and ac a vacancy pair. The postulated second order decay of the "atomic" defect can  now be written (l3)« 2Cl(a) (ac )" +  2  > Cl (g) + 4ac 2  It was proposed that the decay of the defects i s retarded i n the presence of chlorine by a reversible absorption of chlorine molecules which  _ 4 also serves as a step In the exchange: 2Cl(a) (ac )~ + C l ( g ) . ~ = = : [ C l U ) ^ ^ ) " " +  2  ?  ]  2  Harrison suggested that the possible presence of"atomic" defects merits further study by electron spin resonance ( l 3 ) « Following these proposals, Adams and Catton using vacuum sublimed sodium chloride, treated with fluorine i n situ i n an electron spin resonance spectrometer, detected unpaired spins. A peak at g= 2.017+0.005 was found but was not intense enough for any weaker hyperfine lines to be visible. The peak grew after the fluorine was removed, reached a maximum, and then decayed rapidly. When chlorine was introduced the decay slowed down and a reasonable second-order plot was indicated i n agreement with Harrison's prediction (l3). ( This work, which was eventually published, together with the later studies reported i n this thesis, under the names of Harrison, Adams and Catton (l5)» i s given here as introductory material because i t was principally the work of Adams as a brief postdoctoral project. The present author assisted i n the experiments but was not yet registered as a graduate student.) b) Room temperature oxidation bv chlorine and fluorine If a solid alkali halide i s exposed to a more electronegative halogen, one expects that electrons w i l l be removed from the anion band to leave positive holes. These positive holes may be trapped i n various ways: (i) by electrostatic attraction to point imperfections (e.g. cation vacancies i n the bulk); ( i i ) by covalent bond formation i n a small group of halogen atoms to form trapped hole centers which can usually be detec-  - 5 ted by U.V. spectroscopy (e.g. V-centers or H-centers); ( i i i ) at growing halogen nuclei, leading to completion of the simple halogen displacement reaction. The following discussion w i l l deal mainly with oxidation reactions which produce V-centers as a product. Spectroscopic work, using single crystals of alkali halides, on the systems K l / C l  2  ( l 6 ) , NaCl/Fg (l?), and KBr/Clg (18) i n a l l cases  gave U.V* absorption bands which appeared to correspond to known bands produced i n the same a l k a l i halides by X-irradiation or treatment with the parent halogen. In the case of the NaCl/Fg reaotion (Adams (l7)), polished crystals gave a large V band at 2150 &, which indicates a V^ oenter, supposed by Seitz to contain one trapped hole, (and which therfore should have an unpaired electron) and very l i t t l e chlorine was produced. For a similar crystal with a scratoh across i t the reaction took place to a greater extent with the formation of a lot of chlorine (both in the gas phase and trapped i n the crystal) and a small V band at 2270 &Y indicating a V  2  center which Seitz postulated to oontain two trapped holes. In  the polished crystal, the growth of the V^ center followed a sigmoid rate curve and the accelerating region of the sigmoid curve was usually parabolic (16). The KBr/Clg reaction, studied by Catton (18), was analogous to the latter type of behaviour observed i n the NaCl/F system. The pro2  duction of V-centers i n this system occured only after nucleation had taken place. The V-band observed at 2700 £ was suggested as representing a "molecular center". This band was comparable with the known (19,20,21) V  band observed i n U.V. studies of O r doped with Br. at high tempera-  - 6 tures. The K l / C l  2  system studied by Bird (16) produced l i t t l e  I; 2  the prinoipal products were V-centers which were regarded as various forms of molecular or atomic iodine dissolved in the l a t t i c e . The absorption peaks were at 2800 £, whioh was believed to be an "atomic" center, and at 3600 X, ascribed to a "molecular" center. These two peaks were also observed by Hollwo (19), and Hersh (20) i n KI doped with Ig. The 2800 X band was associated by Seitz with his V^ model and the 3600 2 band with his model of a V  2  center (22). Hersh, however, suggested (20)  that both peaks may arise from 1^ (a linear triatomio molecule-ion) whioh i s in disagreement with Bird who said they arise from different centers ( 2 3 ) . It should be noted that an electrically neutral structure of 1^ would contain two oation vacancies and, except for the covalent bonds of I~,  would correspond to Seitz's model of a "V  2  center" (24).  Baijal ( 2 4 ) , doing electrical conductivity work on pressed pellets of KI during reaction with chlorine, found that after an i n i t i a l increase i n conductance two types of reactions were observed. For i n i t i a l l y low resistance pellets (high i n i t i a l cation vacancy concentration), a mechanism involving the trapping of positive holes at isolated oation vacancies was postulated. After the i n i t i a l rise of conductivity (due to an excess of oation caused by the formation of l ( a )  2  (c)"" ), there i s a decay of the conductivity which is either  f i r s t or second order. The f i r s t order decay was explained by the recombination of vacancies with the grain boundaries or dislocations yielding iodine. The second order decay, which occurs i n unusually low resistance pellets, was explained by the joining together of the two defects of eq-  - 7 ual  concentration. Baijal also found that with high resistance pellets 2l(a)  +  + 2 ( o T — *  Ka)^(c)"  +  c"  K a ) ^ »Ka)^(o)-  (rarely the case) sigmoid rate curves were obtained. These curves were explained in terms of the formation and growth of nuclei of the reaction products. Sigmoid rate curves of thiB type were also found by Morrison (25) for the KBr/Cl  2  system.  o) Production of colour centers by irradiation at low temperatures, and subsequent interactions F-centers and V-centers are produced simultaneously in the same crystal by irradiation (26). This i s the chief way of producing Vcenters i n a l k a l i halides at low temperatures (most commonly liquid N  2  temperature). The type of mechanism involved i n the production of these centers was f i r s t envisaged (27) as follows. I n i t i a l l y i n ionic crystals there are anion and cation  vacan-  cies. When these orystale are irradiated, there i s excitation across the band gap and electrons are moved from the valence band to the conduction band leaving holes i n the valence band. The processes which may then occur are that electrons from the conduction band may be trapped at anion vacancies forming P-centers, while positive holes in the valence band may be trapped at cation vacancies forming V-centers. Such a mechanism i s an oversimplification. It has been recognized that the concentrations of colour centers formed are usually greatly in excess of the i n i t i a l vacancy concentrations. Thus in the f o r mation of P-centers there must be a vacancy-creating step. For the trap-  - 8 -  ped-hole centers, structural studies i n the last twelve years have shown that the trapping process may often involve covalent bonding without r e quiring the assistance of a vacancy. The following account i s not intended to be a comprehensive survey of the extensive literature on formation and destruction of colour centers, but rather to focus attention on a few aspects relevant to the present work* A great deal of work has been done on the bleaching properties of the colour centers (28,29,30,31) but much less on the kinetics of their chemical interactions. It was shown (31,32,33) that at room temperature i n KBr, KC1, and NaCl, of the known V-bands, the  and the  bands only were observed and that they have different bleaching characteristics (32). The work done by Kflnzig (34,35,36), Wilkins (37), Cohen (38) and others on electron deficient or trapped hole defects, produced at low temperatures by irradiation, has mainly dealt with the structural determination of these centers but not with the kinetics or mechanisms of their chemical interactions. Some general trends of reactivity have been published by Kftnzig (39) on the production of F, Vg and H centers at 20.3°K i n KC1 by irradiation. Kanzig found that: i ) The F-center and V-center concentration grow rapidly at f i r s t which was attributed to the trapping of electrons at already existing traps (halide vacancies) while holes are "self-trapped" (by covalent bond formation to form CI,,, the  center). When the electron traps are  f u l l the rate of growth then becomes smaller for i t now depends on the  defects produced by X-irradiation. i i ) The H-centers grow slowly at the beginlng and faster as radiation damage proceeds which shows that they are not dependent on a l ready existing traps. Since the three centers were observed simultaneously at 20.3° K a mechanism for their production was proposed which agreed with their appearance and disappearance at various temperatures. The proposed mechanism (39) i s l i k e that of Varley (40) and was explained as follows. An halide ion i s multiply ionized by the X-irradiation and hence takes on a positive charge. This positive charge moves freely i n the crystal but f i n a l l y becomes neutralized and settles i n an i n t e r s t i t i a l position forming an H-center. The electron remaining i s trapped at the halide vacancy forming an F-center. In terms of a chemical equation using Harrison's (14) notation (see section A(a)) X(a) irradiation  +  + ne" + X(a) e~ + a  (  f  t  f  l  )  r  ^X^a)  +  &  0  +  (H-center)  > a (F-center)  +  The overall reaction i s 2X(a)  > X (a)  (anions on lattice sites) with x" and conduction e +  g  (H-center)  +  a (F-center)  as mobile intermediates. (Note that for an  i n t e r s t i t i a l species like X , just as for a gas-phase species, effecn+  tive charge and ordinary ionic charge are identical; but because they are different i n the crystal, (n+l) eleotrons are produced with X . The n +  - 10 symbol a  necessitates (n+l)e  for charge-conservation. Thus there is no  d i f f i c u l t y i n using i n t e r s t i t i a l s or gas molecules i n Harrison's notation.) Another possible mechanism assumes that i n t e r s t i t i a l ions are a l ready present. These become unstable at higher temperatures (about 42°K) and release electrons forming H-centers. The released electron was explained away by having i t annihilate a Vg center. Annealing experiments support this mechanism only in KC1 and NaCl. This i s expected since the i n t e r s t i t i a l halide ions would be less stable i n bromides and iodides because of their relatively large size. The chemical equations of this mechanism are: X~ ( i n t e r s t i t i a l ) X e"  +  X(a)  +  > X > X (a)  2  —  e"  (H-center)  2  X(a)  +  > 2X(a) (anions on l a t t i c e sites)  (Vg. center) The equation showing the production of the Vg center would be X(a) irradiation X(a)  +  +  l ( > )  +  X(a)  +  &  -  > X(a)  2  Experiments (41*42) have shown that the optical H band and the optical  band are closely related. Thermal bleaching of the H band  i s connected with the growth of the  band. Teegarden and Maurer (4l)  discovered that the optical H band can be generated i n a KC1 crystal containing  v 1  centers by light absorbed i n the  band at 35°K. Kanzig (39)  found that H centers could also be produced by this method. Experiments  - 11 i n KC1 and KBr indicated that the H center on warming i s transformed into a non-paramagnetic center which i s presumably a V through optical-excitation in the  1  center since  band i t can be changed back into  the H center at 20°K. The proposed meohanism envisages the V^ center as a halogen molecule i n a single anion site: X (a) 2  e  +  5* e~ + a (P center)  XgU)"*"  (V center) 1  = > a , (F center)  The H center releases an electron to form a X molecule, the e~ i s trapi 2  ped nearby an F center to form an F ation with V  1  center which i s neutral. On excit-  light the H center was reproduced; but the F center conc-  entration did not increase. More recently (i960) Kanzig (43) found in studying low temperature (77°K) irradiated LiF that four main trapped hole centers were produoed (the V^, H, Vp, and V^ centers; see figure l ) . The  center  i s a self-trapped hole (F~) localized on two fluoride ions; the H center i s chemically equivalent to an i n t e r s t i t i a l F atom; the Vy center i s a hole trapped at a L i vacancy but the hole i s localized on two fluoride ions so that the F~ molecule-ion has a bent bond; and the V^ center i s a hole localized on three fluoride ions whioh form an isosceles triangle (if). Seitz proposed that the optical absorption band V^ in NaCl, KC1 and KBr might be due to a hole trapped at an alkali vacancy (44)* The Vp center i s a center of this type;but i t s optical absorption band has not been identified. Kanzig (39) pictures the V. center in KC1 and KBr  - 12 as a non-paramagnetic center such as a  molecule in a negative-ion  vacancy (unless the absorption band of the center postulated as X^ overlaps the  band and i s obscurred by i t ) . The model of the  center  (45) corresponds i n a general way to Seitz's (26) model of the center that gives rise to the optical  band i n KC1 and KBr but none of the  absorption bands i n X-irradiated LiP have been established to be the analogue of the  band i n KC1 and KBr. Kingsley (46) studying the bleach-  ing and symmetry properties of the the conclusion that the  center i n X-irradiated KBr came to  center i s a single hole trapped at a single  a l k a l i vacancy. He concluded that the  center may be the antimorph of  the F center whose paramagnetio resonance Kanzig found i n LiF (47). Therefore the Seitz's model for the 7^ oenter and Kingsley's model for the V. center are the same but the centers have different optical absorption bands so they cannot be the same oenter. An investigation (47) was made of the generation and decay of the V , H, V K  t  and Vp oenters found i n irradiated LiF at 77°K as a  function of the annealing temperature (see figure 2). The annealing experiments showed: i ) from 95-10O°K the H oenters are the most unstable and i n decaying they emit eleotrons which annihilate Vg centers causing their concentration to drop. F (a)  •> e  2  e  +  F(a)  2  ^  +  F(a)  2  —> 2F(a)  i i ) from 110-130°K, neglecting V_ oenters destroyed by H cen-  - 13 Figure 1 - Trapped hole centers  O O O © O O O © O © O— @0©0®0@0®0 ©*• o © o© o • o • o • o © o ®o © o@®o © o © o © o©o©o© o© o • Ol'OIOiOIOI • o©q© o • [ \ o © o • O 0!v»|_1» o © 0©0©0©0©0©0 mi0D  Vk  Figure 2 - Cono. of V-centers i n LiF vs annealing temperature  I  80  1—~1  120  1  1 160  1  1  200  1  1  240  (temp. °K)  - 14 ters decaying, the decay of associated with the increase of  centers i s second order* This decay i s and Vp centers which suggests that  some of the migrating holes are trapped at vacancy aggregates and at Li vacancies. i i i ) from 175-185°K the V^ and Vp centers decay together for a time which suggests that the V^ centers release holes which recombine with trapped electrons or transform F~ molecule-ions of V-, centers into F^ molecules. d) Summary of electron-deficient colour centers in a l k a l i halides relevant to the present work The objective of the present work i s to use electron spin resonance as a means of obtaining information on colour centers acting as reaction intermediates. Hence the following table i s intended prima r i l y to summarize the situations i n which unpaired electrons have been detected (by BSR)or postulated (on the basis of kinetic evidence). There are some unresolved conflicts, i n which unpaired electrons are postulated but have not been observed by ESR. These are noted in the table. In general, i t should be noted that no structure i s yet definitely established for any of the V centers with the numerical subscripts (V^, V , V^, V^) which produce the strongest U.V. absorptions in X-irradiated alkali halides, and that no ESR signal has been detected for any of these centers, although structures including unpaired electrons have been postulated for some of them on the basis of less direct evidence.  - 15 Table I  Sample  Method of production of defects  Electron-deficient centers detected by U.V. spectroscopy or postulated from indirect evidence ". ;  Centers detected or postulated as having unpaired, eJlepfrronq  NaCl crystal  irradiated  V^, V^. V^, V^, V^, H  V^ H (structures  KC1  crystal  (low temp)  KBr  crystal  LiP  crystal  irradiated  t  determind by ESR )  V, H  V  K  (low temp)  V  V  H  (structures determined by ESR )  NaCl crystal oxidation  2*  according to  3  Seitz's model; but no ESR absorption is known correlated with the  optical absor-  ption KI  crystal  3 oxidation 0 l  V  T  Assignments of  3  structures very uncertain  KI  powder  Cl  2  oxidation  I ( a ) ^ ( e ) - and l ( a ) ^ ( c ) -  None  2  2  postulated to explain observations on conductivity  KBr  crystal  CI,  Assignments of structures very uncertain  NaCl  vacuum  KC1  sublimed particles  C l ( a ) ( a c ) " and C l U ) ^ ^ ) " * +  2  Seitz's  model  2  oxidation  postulated to account for  postulated; detected  kinetic results; the former i s  H-center ( by ESR;  Seitz's  this thesis )  model  16 B. B a s i c Theory o f E l e c t r o n S p i n Resonance a) Resonance c o n d i t i o n An u n p a i r e d e l e c t r o n i n a paramagnetic s p e c i e s can e x i s t two s p i n s t a t e s , m = +1/2,  which form a d o u b l y degenerate s t a t e  they have the same e n e r g y . When the s p e c i e s i s the degeneracy i s  in  since  put i n t o a magnetic  field  l i f t e d ( i . e . the two s t a t e s have d i f f e r e n t e n e r g i e s )  and i n t e r a c t i o n w i t h e l e c t r o m a g n e t i c r a d i a t i o n o f energy h ^ e q u a l to the gap between the two l e v e l s causes b o t h a b s o r p t i o n ( \-l/2^> ->\+l/2^>) and e m i s s i o n ( |+l/2^>->|-l/2^>) t o o c c u r a t the same t i m e . S i n c e an excess o f the m o l e c u l e s a r e i n the lower energy s t a t e ( |-l/2^>) the n e t r e s u l t is  absorption. The p o t e n t i a l energy o f a magnetic d i p o l e ( o f the e l e c t r o n )  w i t h a magnetic moment/i i n a magnetic f i e l d H i s  E =» -^h • H  expressed as:  ( w h e r e ^ a n d fi a r e v e c t o r s )  w i t h the d i r e o t i o n o f the magnetic f i e l d H t a k e n t o be a l o n g the z a x i s the e x p r e s s i o n becomes  E  • Taa '  z z  The s p i n o p e r a t o r i n t h e magnetic f i e l d can t h e r e f o r e be e x p r e s s e d as  but s i n c e the magnetic moment i s one can w r i t e =» ( c o n s t a n t )  S  p r o p o r t i o n a l to the s p i n quantum number  - 17 =tfh .S.  (where & i s the gyromagnetic ratio)  Therefore #  hi  - +g  s  • S  where g i s the spectroscopic splitting factor, B is the Bohr magnetron 0  ( » eh / 2mo ), j£ i s the spin operator and H i s the magnetic f i e l d . The eigenvalues giving the energy levels for this system can be expressed as B=gP H m 0  z  8  where m i s the electron spin quantum number. The transitions.between g  pairs of these energy levels are allowed only i f they obey the selection rule Am = + 1 and the resonance condition which has to be satisfied for s ~" these transitions i s h9 = gB H 0  z  where 0 i s the Larmor frequency. It was assumed above for simplicity  the g-factor i s isotrop-  i c but this i s not generally the case i n solids as we w i l l see later in section B ( C ) . b) Hyperfine interaction There can also be further interactions between the electron spin and internal magnetic f i e l d s , particularly those due to the magnetism of nuclei i n the same molecule. These magnetic interactions between electron spins and nuclear spins are known as hyperfine interactions because they are responsible for the very small splittings known as hyper-  - 18 -  fine structure i n atomic spectroscopy. In ESR spectra, these interactions sometimes produce very large splittings but they are s t i l l known as hyperf ine structure. The spin angular momentum of a nucleus i s described by a quantum number, I, sion > = N  and i t s magnetic moment,/^, is given by the expres-  gjjP I where g N  i s the nuclear g-faotor and ^  N  ±  a  ^  n  u  c  l  e  a  r  magneton (= en /2mc) where m i s the mass of the proton. Now i f a nucleus with a magnetic moment,/^, interacts with an electronic magnetic moment  the interaction energy can be expres-  sed quantitatively as (48) EJJ ---a/^Ajj r '  5  * A£ • I  where a i s a constant depending on the ground state wave function, r i s the radius of the electron orbit and A i s the hyperfine interaction tensor which w i l l be assumed isotropic for simplicity (see section B(c)  )•  In a magnetic f i e l d the spins are uncoupled and hence the nuclear spin angular momentum i s quantized, separately from the spin angular momentum. The spin hamiltonian i s given as #  s  = gp H S 0  + A S • I  z  The energy levels arising from this hamiltonian canbe expressed as E(m mj) • gp H n> g  e  8  +  A m nij s  The selection rules governing the allowed electron magnetic dipole transitions are given by Am= + 1 and Am^= 0 . These transitions take place s  - 19 at the following frequencies h 9 » gfioH + where  A VBj  can have the values I,  1-1, . . . . -I.  The magnetic f i e l d at  resonance i s H =[hv/gaj  -  [A/gftJmj  The hyperfine term causes splitting of each electronic level into 21+1 components which results i n 21+1 hyperfine lines with the spacing between each line (for constant v ) equal to A/gB gauss. 0  Nuclear electric quadrupole interactions can break down the selection ruleten^-+0, leading to the appearance of extra lines in the spectrum, for toj** +1 orto&j**+2. This effect i s a possibility for CI nuclei, which have a quadrupole moment, and extra lines have been observed i n the ESR spectrum of C10 (49)» hut since such lines would not be 2  resolvable i n our spectra, and since they need not be observed for a complete structure determination, I do not discuss quadrupole effects in this thesis. c) Anisotropy In the previous two sections dealing with the zeeman s p l i t ting and hyperfine splitting, the g-factor and A, the hyperfine coupling constant, were assumed to be isotropic;but this i s not generally the case, especially i n solids. A and g are usually to some extent anisotropic and can be represented by symmetric second-rank tensors which can be diagonalized  - 20 by a suitable choice of co-ordinate system x, y, z to give three principal components g^ gy» S t  and A^, A^,, A^, The principal axes of g and  z  ^ usually, though not necessarily, coincide. If H has direction cosines i , m, n to the principal axes x, y, z then for this orientation (50)  and A - (i  A W . x  2  A y  2  n  2  +  2  A ) / z 2  1  2  A common case i s that of axial symmetry i n which i f the axis of symmetry i s the z axis  e  z  A z  ll " ll  m  g  *x " «y  A  A  x  = A  a  «L  » A  y  ±  and i f d i s the angle between the axis of symmetry and the magnetic f i e l d , H, then the hyperfine coupling constant and the g-factor can be expressed as a function of 0 (5l).  e - t gf, + ( «l - gf, ) ein e ] 2  1 / 2  and  A - [ A |+ ( A - A ) sin e ] 2  2  2  2  1 / 2  d) ESR snectum of an unpaired electron interacting with several nuclei i n a solid medium i ) Structure of the ESR spectrum The hyperfine coupling constant and the g-factor are usually to some extent anisotropic as stated i n section B(o). The hamiltonian for the zeeman splitting i s expressed SB (52)  21  ft J  ^z  » e ( S • g • H ) •«• 6 ( g S H + g S H + g S H ) o = — o °x x x y y y z z z  ..(l)  r  where S , S , S are the spin angular momentum operators along the three • 3c y z axes; H , H , H are the components of the external magnetic f i e l d ; and z y z g . g , g are the spectroscopic splitting factors along the three axes, x y z The spin hamiltonian which i s used in f i r s t order pertubation theory to desoribe the hyperfine interaction in a system of more than one nuolei can be written as (53)  where g  Q  i s the spectroscopic splitting factor of the free electron and  g when i t i s used i s that of the speoies for that partioUlar orientation of the magnetic f i e l d , 6 i s a Bohr magnetron, S i s the spin operator of Q  the eleotron and I*  that of the nuoleus i and A ' a are the hyperfine 1  coupling tensors i n gauss* Combining equation 1 and 2 into a more oompaot form the total spin hamiltonian of the system oan be written: ft - fi S . g . e  +  fi  g B g £ . A, . I 1  o  ..(3)  1  o  Now i f the direotion of the magnetio f i e l d H i s taken to be along the z axis then  #  - «e  S 0  A  +  «  p 0  o f V  A ** 1  -(4)  1  The energy for a given spin state B  D 8 i n  m  m i  i •••]> "  < 8 i m  m  m i  i •"l#.IVl."l  — >  - 22 where m i s the spin quantum number of electrons and m  T  magnetic quantum number ( i ^  '~*i^  o  f  n  u  c  l  e  u  s  i s the nuclear  *-> generally writis  ten for a first-order approximation as: B  " t «P H + g B 0  z  o  £ A  o  ^  1  ]m  s  ..(5)  With the selection rule for transition Am = +JL and (Anij)^ 0 the resong  anoe oondition i s given by hv « g^H + g B E A ( m )  ..(6)  i  A  Q  o  I  1  The f i e l d for any particular orientation i s given by  and the hyperfine splitting by H  - H = -(g /g) Z A ( m ) i  Z  q  Q  I  i  ..(8)  i i ) Relationship of components of e and A tensors to the state of the unpaired electron In the centers being studied the state of the unpaired electron can be desoribed as a linear combination of hydrogen-like s and p state functions, therefore the following discussion w i l l only include these state functions. The hyperfine interaction consists of two parts (54) which are: an isotropic part which depends only on the non vanishing component of the wave function at the nucleus; and an anisotropic part whioh depends only on the non spherioally symmetric components distant from the  - 23 nucleus. Therefore i f the electronic wave functions are expanded in terms of atomic functions only s components contribute to the isotropic interaction and non s components to the anisotropic interaction. Many different terminologies have been used in relating hyperfine interaction to fractional s and p character. The letters A, B, a and b are used by different authors for different quantities. In this account, I attempt to avoid this confusion by introducing symbols  and  (P as defined below. The isotropic terms, or the Fermi-contact hyperfine splitting i s proportionalto the electron density at the nucleus. The contact of the nucleus and the electron i s not directional-dependent and hence this interaction i s not a function of the angle between the crystal or molecular axis and the magnetic f i e l d (55)• The Fermi-contact term i s expressed for an s state as (56) ^ where ^  M 8 * / 3 ) 0 y i ) ^ (0) 2  •  ..(9)  i s i n gauss,JLL and I are the magnetic moment and the spin of  the nucleus respectively, and  ^(o)  i s the amplitude of the s function  at the nucleus. The dipolar or anisotropic hyperfine interaction i s the i n teraction due to the magnetic f i e l d at the eleotron caused by the presence of a nucleus near the electron but not right at the electron. This term averages out to zero i n systems whioh have random tumbling of the species present. This term i s very important in solids for i t depends on the angles between the principal axes and the magnetic f i e l d . The dipolar  hamiltonian i s (55)  Wdipolar= where  ftfr.fc^Hi  *  'Cl  ' x)/r ) ..(10) 2  nuolear g-factor  a  e " 8 value of unbound electron 0  |3JJ ° nuolear magneton ^>  :f  9  0  - Bohr magneton  r_ • radius vector from nucleus to the electron ^ S when  9  nuolear spin operator  » electron spin operator  ^dipolar *  s  expanded out the f i r s t term i s #„a[(3cos e - l)/r ] g ^ B B ^ I ^ ) 2  3  ..(ll)  which i s generally used to compute the anisotropic (55) hyperfine tensors i  V  i  V  i  z•  k  For a p^ state using equation 11 A^ - / t j / l / ^ f (3cos e - l ) / r ] r s i n © dr de d^ 2  2  2  ..(12) ... - 4 / 5 ( , A / I ) < r " >  ..(13)  5  I  where ^ r  i s an average value over the radial part of the p function,  also 'Ay-A^--.2/5(/^/l)<Cr" -> 3  ..(14)  Fof convenience I define a quantity P-2/5( /fe/[)<** > 3  ..(15)  - 25 Thus for a single p orbital, designated p^,  A  x  a  2  P  A' = A' = - ( P y z For an eleotron whose state can be described as a linear combination of s and p states, two possible oases arise: (l) axial symmetry, which can be described by a combination of s with a single p orbital (designated p^). The components of the hyp2 erfine tensor A *"e then An ( i . e . A ) and A ( i . e . A » A ). If o. i s x z y x 2 the amount of p character B^ i s the amount of s character i n the ground 2 2 a  x  state atomic ware functions centered on nuclei i , and i f Xj(a^ + 0^) «= 1 then the components of the hyperfine tensors would be (using equations 9, 13, 14, and 15) A„ = f3  +  2a (P  ..(16)  A^-B ^  -  a (P  ..(17)  2  2  2  (2) the three principal components a l l different ( i . e . Ay # A  z  #  )• The description of this situation requires that there should  bo a spin density i n an s state and two separate mutually perpendicular 2 2 2 2 2 2 p states (eg. p and p ) in the amounts 6 , a , and a where B +a +0 =1, X z x z x z This situation w i l l only arise i n a crystal f i e l d which l i f t s the degeneracy of p^ and p » (A oombined state c^p^ + c p z  2  z  represents merely a  p state identical to p or p but with a differently oriented principal x  z  axis. This i s not the situation discussed here.) The components of the hyperfine tensor then become (using equations 9» 13» 14, and 15) A » B ^ 2  s  V  (2« -« )(P 2  2  ..(18)  - 26  4  +  l P  a )(  A » B J  -  (  A — P z  +  (2« -« )(P z z  2  Y  2  ^  2  -  (19)  ..(20)  2  Mow i f the experimental values for the hyperfine tensors are put into equation 16 to 20 they oan be solved for the amount of s and p character i n the bonds formed, provided that the numerical value of the ratio ^  I {P i s known. (3) The above treatment suggests, as i s usually assumed, that t  an anisotropic hyperfine tensor with A^  1 A  ~2A  Y  A  1  ~2A indicates spin A  density i n p^ only. It has been pointed out by Tinkham (57) that this i s 2  2  2  not necessarily the case. Suppose that fraotions a , a , a of the unpaired electron are i n p^, p , and p . Then.y  g  t  A » (2.2- « z y 2  ..(21)  1  «2A = K Z y1 A • (2.*- a 2  Z  z  ..(22) ..(23)  2 2 If ve now represent axial symmetry by writing a • a and introducing y z 2 2 2 ' 2 the symbol £ " ° " ° ' components of A are functions of 6" only. t  x  n  e  z  A'-[2(? ](P  ..(24)  A «[-S ](P  ..(25)  A' » [-6 ] (P  ..(26)  2  2  Y  2  1  Thus when the anisotropic hyperfine tensor satisfies the condition A^ J I 2 2 -2A_ » -2A , this means only that a = a , not that both quantities are y z y z f 2 2 2 2 zero. The quantity obtainable from ^ i s not actually oc^ but £ *>a^ " z ° 3  a  This in fact represents the minimum value of 2  2  •  consistent with experi-  mental results, since oc^ must be positive or zero. Thus the usual analysis given i n the preceding section assuming that a hyperfine tensor axially symmetric about the x axis can be interpreted in terms of spin density i n p  x  only actually gives a minimum value for the p character of  the state of the unpaired electron. Similarly, for the case without axial symmetry, the quantii'  ties which can be determined from the components of 4  a  r  e  t  n  e  differences  in spin density between the least-occupied p state and the other two. Assuming: Sx2 and that Sz2 b*the y i - p i s least occupied, we define the two quantities y a2 - ay2 =» 6r2 x. x and ..(27)  a  2  z  -a - S 2  y  2  z  Then A* - (2 £ Z  A -( y  -  6 )(p  ..(28)  +  6^ )(p  ..(29)  -  S )(p  ..(30)  Z  6\  A - ( 2 £ y  2  2  2  2 2  2  x  The magnetic interactions involving the orbital angular momentum of the unpaired electron are the sole cause for the deviation of the g values from 2.0023 (the free electron value) (55)• The part of the magnetic moment arising from the orbital motion of the electrons i s modi f i e d by the chemical environment i n an atom, molecule or crystal, there-  - 28 fore the g value likewise depends upon the chemical environment (48). In a solid, since the orbital part of the magnetic moment depends on the crystal f i e l d , i t s magnitude i s usually different for different directions of H and shows an angular variation which follows the symmetry of the crystal f i e l d . The total g value (spin +. orbit) may be anisotropic by an amount which depends on the magnitude of the orbital contribution to the moment and on the asymmetry i n the crystal f i e l d . The theory of the g-shifts i s described i n numerous books such as those written by Slichter (58), Fake (59), and Carrington and McLachlan (60). The following aocount follows mainly that of Carrington and McLachlan which gives the most precise definitions of the "ficticious spin" and the "anisotropic g-factor". The g-shift whioh i s observed i n solids i s related to the spin-orbit coupling which mixes the ground and excited states thereby introducing a small contribution of the otherwise quenched angular momentum to the electronic magnetic moment (58). The g-shift can be exemplified quantitatively by considering the case of an atom with one p electron which i s described by the hamiltonian #  T  - ft  x  + V  x  +\L  • 8 + kfi . g . |  ..(31)  where j t y ^ contains the principal energy terms (the kinetic energy and free atom potential), 7^ i s the potential due to the charge around the atom, \ L • 23 i s the spin-orbit interaction and B j[ • g • £ i s the elec0  tronic zeeman interaction. In this expression, g represents the experimental observation that the interaction of H with the electron spin i s often apparently anisotropic. This could not be so i f S, represented the  - 29 true spin S because the interaction of S, with H i s necessarily isotropic with the constant of proportionality g . The " f i c t i t i o u s spin" S actualQ  l y represents the true spin of S modified to allow for small residual effects of the orbital angular momentum, and the significance of S i s found by comparing the " effective spin hamiltonian" as written above with the true hamiltonian written i n terms of J±, S>, and g o In the presence of a magnetic f i e l d H there w i l l be an interaction between H and the orbital angular momentum L. and therefore the "true hamiltonian" for the magnetic interaction i s now written as  #- 4,1 • k + e p* l • s 0  (32)  o  where £ i s the true spin. Let us now consider a potential  of such a nature that i t  l i f t s the orbital degeneracy of the once degenerate p states of the atom. For example, a suitable potential would be provided by the following arrangement of charges around the atom X:  + b  + b  X  .a. X  The resultant energy levels shown below are a l l two fold degeneratebe-  cause of the electron spin*  —  _ _ r f ( r ? A  —  n  (the spaeings are equal i f a b; see referenoe (58); two orbitals remain a  degenerate i f b=0;  the defeot i n S r C l studied i n this thesis corresponds 2  to an intermediate case, b&). The wave function suoh as xf(r)/i con' m (y& " «iP)»  tain an orbital function xf(r) and a spin function jj, Now looking at the effeot of \&  B  • £ on the p states i t i s  found that the matrix elements of the form <Cxf(r)| L  |xf(r)^> . repre-  g  senting the expectation value of L^, vanish. This shows that the angular momentum i s quenohed and has no f i r s t order effoots on the states. However, i t i s found i n terns of pertubation theory that the epin-orbit ooupling (\£  • £) mixes very small proportions of various excited states  into the ground state. The type of modified wave function  calcu-  lated by f i r s t order pertubation i s xf|r) where  ^  > x  < w \\k  • £ | x>  /B -B ] wf(r) w  B  ..(33)  i s spin degenerate. The perturbed states are no longer eigen-  states of the true spin £• In an external magnetic f i e l d BJ[ • g • £ of equation 31 becomes  # '• fpigf++  «yV  +  *M  ..(34)  - 31 For the hamiltonian  of equations 32 and 34 to mean any.  A  Av  /s  thing the corresponding matrix elements of B^lg^S^ + gyS + g S ) and y  those of  BjlCl^  z  z  + ly + L + g ^ + g^S + g S ) formed by these hamiltonz  y  6  2  ians acting on the perturbed states must be identical. By equating these matrix elements the principal components of the g-tensor can be expressed i n a generalized way using only f i r s t order terms i n \ as  y  v -  ^  ^ i < 0 ^ f l s l ^ > < ^ ^ l ^ . -  ,  .  The g-shifts obtained for an unpaired electron i n a p  -  l  ,  5  )  x  orbital are therefore  g y  S - -2 \ / ( B - B ) &  z  ..(36)  x  In these expressions only first-order energy terms have been considered. Others such as Inui, Harasawa, and Obata (62) have taken into account the change of the wave function due to spin-orbit interaction which r e sults i n the addition of (negative) l/energy  2  terms i n the above expres-  sions for the g-shifts. C. Previous Structural Determinations of Defects i n Alkali Halides by ESR The use of electron spin resonance has made i t possible to determine the structures of a variety of defects i n alkali halides (mainl y X-irradiated single crystals at low temperatures). Models for the e l ectron deficient or trapped hole type defects have been proposed by Seitz (26). Varley (63)» Nagameya (64) and others but the structures of these defects f i r s t observed using U.V. and Visible spectroscopy were  - 32 -  uncertain and many of them s t i l l remain so. The structures of the defects now known include; the Vg center (36), or X~ ion(Xis a halogen atom); the H center (35), an X~ "crowdion" on a single site, so that i t could be designated X^ ; the  center (45), an F^  ion distorted from linearity  by a neighbouring group of vacancies (see figure l ) ; a mixed halogen structure PCl" i n KC1 : KP mixed crystal (37); and the superoxide ion 0~ as an impurity in KC1 (65,66). a) The F-center The F-center was the f i r s t colour center to have i t s structure determined unequivocally. The structure, determined by electron spin resonance, was found to agree with the model proposed by deBoer (67) which depicts an F-center as an electron trapped at a halide ion vacancy. The paramagnetic resonance of the F-center (68) i n KC1 was found to consist of a single line of nearly gaussian shape with a width of 56 gauss and a g value of 1.995 . Kahn and K i t t e l (69) said that the g-shift showed that the electron spends a large portion of i t s time near the ions surrounding the trapping s i t e . Kip, K i t t e l , Levy andPortis (70) theoreti c a l l y determined the line shape and width of the resonance line in KC1 using deBoer's model and found that they were i n good agreement with exp39 erimental values. Also using K  41  Cl and K  C l , they showed that the hyper-  fine interaction of the F-center electron with the nuclear spins of i t s six neighbouring cations was mainly responsible for the broadening of the resonance peak. The line width change due to the different isotopes of potassium was not exactly i n the ratio of magnetic moments. This was attributed to a weaker interaction of the F-center with the twelve next  - 33 nearest neighbour halide ions. Linewidths in NaCl and KBr calculated from KC1 results using the ratio of hyperfine coupling constants also gave remarkable agreement with experimental line widths. Feher (7l) using electron spin double resonance technique, which gave better resolution, experimentally determined the hyperfine interaction of the Fcenter electron i n KC1 with i t s neighbours (K ) and i t s next nearest +  neighbours (Cl~). Also the hyperfine interactions in LiF and NaF were observed directly (72,73). With the structural determination of the F-center showing that i t consists of an electron trapped at a halide vacancy, the search for an equally fundamental imperfection consisting of a hole (electron deficiency) trapped at an alkali vacancy, which Seitz (22) called the "antimorph of the F-center" (26), was i n i t i a t e d . Seitz speculated that the antimorph of the F-center might be the center responsible for the 7^ absorption band (74) but this has never been proved. Indeed no ESR signal has yet been found which can be correlated with the 7^, Y^t 7^» or 7^ absorption bands; but the search for such signals has brought about the structural determinations of the following trapped hole center. b) The 7^-center The f i r s t analysis of the electron spin resonance of a 7 center was reported by Kflnzig (75) who obtained an ESR spectrum from single crystals of KC1, NaCl, KBr and LiF irradiated at -180°C. The center observed was thought to be associated with the optical 7^ band since i t bleached at about the same temperature. Later Delbecq, Smaller and Yuster (76) and Lambe and West (77), working independently, showed that the  halogen" center which Kanzig had discovered was a new center ( now usuall y called the Vg center). The ESR spectrum obtained for the Vg center i n alkali chlorides (KCl) i s composed of sets of seven almost equally spaced lines which have an intensity ratio of 1:2:3*4:3:2:1 • These were recognized as a r i s ing from the interaction of an electron or hole equally with two equivalent chlorine nuclei of spin 3 / 2 . In the case of an equal interaction with two nuclei (subscripts 1 and 2 ) , equation 8 of section B(d) becomee  H  z " o " " « / H  (  g )  0  1  ^ l\ m  +  and for two nuclei of spin 3/2 for which each m^ can assume the values - 3 / 2 , -1/2, +l/2, + 3 / 2 , equation 1 indicates seven equally-spaced lines. The relative intensities of these are given by the number of different combinations of (BJ)^ and (m^^ giving rise to the same line and this gives intensities in the ratio 1:2:3:4:3:2:1 • Chlorine, however, has two isotopes: Cl'*"', which i s 75$ abundant and Cl  , 25$ abundant. Therefore three types of molecule-ions are  possible: C1 3 5 -C1 3 5 , C1 3 5 -C1 3 7 , and C1 3 7 -C1 3 7 which are 9 / l 6 , 6 / l 6 , and l / l 6 abundant respectively and allcan give rise to an electron spin resonance spectrum. Cl "* and C l 3  Ay^i  =083)  3 7  have slightly different magnetic moments ([/U^/  and since the hyperfine coupling i s proportional to the nuc-  lear magnetic moments, /Jb , the following relation holds.  [ A / A ] - CA /> ] 3 5  3 7  35  37  35 37 35 37 where A and A are the hyperfine splittings of CI and CI respectively. The molecule-ions C 1 - C 1 35  35  and C 1 - C 1 37  37  both give rise to  seven-line spectra with line intensity i n the ratio I:2:3s4!3s2:l but the intensity of the C 1 - C 1 3 7  3 7  spectrum w i l l be l / l 6 that of the C 1 - C 1 3 5  3 5  37 37 spectrum. The hyperfine splitting for the CI -CI molecule-ion w i l l be 35 35 less than that of the CI -CI molecule-ion as shown by A  3 7  • .83 A  3 5  3 7  3 5  The spectrum of the C l / ' - C l  molecule-ion w i l l be different  from the other two spectra since there are now two non-equivalent nuclei and hence the degeneracies of the lines in the previous seven line spectra are removed. The configuration (3/2,1/2) and (1/2,3/2) no longer appear at the same f i e l d and therfore theline for l i k e nuclei of degeneracy two w i l l s p l i t into two lines of equal intensity and so on; The intensity of each C 1 - C 1 ^ line i s 6/9 that of the outermost line of the 37  CI  3 5  -CI  3  3 5  3 7  3 5  spectrunu The position of the lines in the CI -CI spectrum 35 35 are displaced from the CI' -CI l i n e s t h i s displacement can be expressed as equal to 35 0.17 A mj 3  7  37 35 where m^ i s the nuclear magnetic quantum number of the CI nucleus 35 35 35 k i s the hyperfine splitting of the C l ^ - C l ^ spectrum, and 0.17 i s JJ  the value of ( A - A ) / A 3 5  3 7  3 5  (see figure 3 ) .  Kanzig (36) found that there was more than one set of lines for a particular orientation of the crystal. This was explained by the  - 36 Figure 3 - Isotope effects i n KC1 of a V -oenter (schematic) v  1000  abundance  35 (  35  3  2  1  0  -1  -2  35 37  *7 37  N- 83  1  L  IJlLJJ  -3  - 37 existence of more than one orientation of the centers within the crystal, together with the anisotropy of the hyperfine splitting and the g-factor, which causes the seven line spectrum to have a different spacing and a different center at each orientation* The spectra were consistent with random orientation of centers along directions of the type (lOO), with approximate axial symmetry about the molecular axis, the hyperfine tensor having i t s largest principal component  along the molecular axis,and  smaller components i n the perpendicular directions, with A =A although x y the x and y directions are not crystallographically equivalent. The unpaired electron giving rise to the spin i s a p electron in a c r 3p antibonding orbital (with some s-admixture). Therefore the molecular orbital of the molecule ion i s made up essentially of a combination of 3p functions, one centered on each halogen nuclei which have their rotational symmetry along the line joining the nuclei, which i s a  (lio)  direction. The principal axes of the g-tensor (x, y, z) with respect to the molecule are shown i n the following diagram in which the y axis points out of the pageo x  z  + +  -  (llO)directio:  -f-  -  +  (100)direction  - 38 Now i f the angle 9 between the molecular axis and the d.c. magnetic f i e l d i s changed there w i l l be two effects: (l) there w i l l be a g-shift displacing the oenter of the hyperfine patterns* and(2) there w i l l be a change in the magnitude of the hyperfine splitting. For any orientation of the crystal i n the magnetic f i e l d at least two spectra w i l l be observed simultaneously. When the magnetic f i e l d i s parallel to the (lOO) axis of the crystal the spectrum obtained (see figure 4) i s a combination of the 90° spectrum (1/3 of the molecule-ions) and the 45° spectrum (the remaining 2/3). Table II  below indicates the spectra expected for certain  special directions of the magnetic f i e l d . One can readily see from figure Table Direction of f i e l d (10  0)  ( 1 1 0 )  (111)  II Anffle e  Abundance  45°  4  90°  2  0° 60°  1 4  90°  1  35.26°  3  90°  3  4 that the hyperfine splitting decreases as the angle 9, between the bond axis and the magnetic field.increases.  0=45°  67=90° Figure 4 - Spectrum of a Cl  -Cl  molecule ion with H // [lOO] , * * ( V -center, i n KC1 )  - 40 Numerical data on the g tensor and hyperfine tensors are given later i n this thesis in table IV, page 69. The V  K  center in alkali fluorides was f i r s t discovered by  Kanzig (34) i n LiF and later studied by Bailey (77) in LiP, NaP, KP, RbP, and CsP. Bailey found that the  center was stable in a l l the alkali  fluorides at liquid nitrogen temperature but unstable at room temperature. The results for LiP agreed with Kanzig s results (36). He also found in 1  studying the series of fluorides that the line width of the  spectrum  depends rather heavily on the magnetic moment of the nearest neighbour atoms (the alkalies)<, c) The H-center After Compton and Klick (78) showed that the H center must be different from any of the models previously suggested by Seitz (26) and Yarley (63), Kanzig and Woodruff studied the paramagnetic resonance of the H center in KC1 (35) and then i n KBr and LiP (39) and derived a detailed model of the H center. The H centers were produced by irradiating single crystals at 20°K and were studied at low temperatures. In determining the structure of the H center Kanzig used similar arguments to those he used in determining the structure of the  center (see previous  section). In order to study the H center, the derivative of the dispersion signal was used since i f the spectrometer was tuned to measure pure absorption, the H center resonance, the Vg center resonance and the P center resonance, which appear together i n these spectra, were completely saturated at 20°K by a very low microwave power. The H center spectrum taken at the same orientation as the  - 41 Vg center spectrum i s very similar to the Vg resonance. In the H center spectrum, however, there i s a group of lines (with the main seven lines of the Vg center s p l i t i n a ratio of 1:2:3:4:3:2:1) corresponding to each line i n the Vg spectrum. The primary splitting (between the center lines of adjacent groups) for the H center was about 10$ larger than the splitting i n the Vg center and showed the same orientation dependence. The complete.analogy of the orientation effects for the H center spectra to those of the Vg center spectra indicated that the axis of the major hyperfine interaction was the same for both types of centers namely the (lio)  axes of the crystal. The secondary splitting (between adjacent lines within a  group) i n the H oenter speotrum causes each line of the basic halogen,, speotrum to be s p l i t into a miniature speotrum of the same kind (see f i g 35 ure 5 for the primary and secondary hyperfine splitting of the CI  35 -CI  molecule-ion i n the H center). The oomplex speotrum can be interpreted as arising from ident i c a l strong interactions of the unpaired spin with two equivalent chlorine nuclei ( l and 2) as i n the Vg spectrum and identical weak interactions of the unpaired spin with two other equivalent chlorine nuolei (3 and 4). For this case, equation 8 of section B(d) becomes H  z  - H - - ( g / g ) [ A ( ( » ) + (mj)2) + A ^ U - T J J + U ^ ) ] o  1 2  0  1  1  ..(2)  The f i r s t term i n equation 2 as i n equation 1 of section C(c) gives rise to seven equally spaced lines of intensity 1:2:3:4:3:2:1 • The second term gives rise to seven equally spaced lines for each of the f i r s t part  - 42 which have an intensity ratio of 1:2:3:4:3:2:1 (see figure 5). The unpaired electron was found to spend 4-10$ of i t s time on ions 3 and 4. The four halide ions involved i n the H center were found to be i n a straight line along the (lOO) direction in the crystal (see figure l ) . A similar type of oenter was found i n KBr but i t was hard to determine the secondary splitting because of the existence of two equall y abundant isotopes of bromine (Br  and Br  ) and also additional  splitting due to second order hyperfine and quadrupole effects which arise beoause of the high magnetid.moment of bromine* In LiF a oenter with the same structure as an H center was found and since there i s only one isotope of fluorine  with a nuc-  lear spin of 1/2 the hyperfine spectrum of the H oenter i s rather simple* The strong interaotion of the unpaired electron with two equivalent f l u orine nuclei (like the Vg oenter F~ (34)) gives rise to three lines with an intensity ratio of 1:2:1* The weak interaction with the next two f l u orine nuolei, whioh are also equivalent, splits the f i r s t lines into a miniature (secondary) pattern of three lines with the intensity ratio 1:2:1* This experiment was done using the derivative of absorption so that the F center resonance was completely saturated. The H center and Vg center spectrum, however, appeared together* The g-shift found for the H oenter, previouely discussed, was about half the g-shift found for the Vg center. This difference can be explained using the general discussion i n the last part of section B(d) for the specific case of a C l " molecule ion. Figure 6 below repre-  - 43 Figure 5 - H-center (hyperfine interactions) i n KC1 primary (2 C l , I » 3)  secondary splitting (2 C l , I=» 3) t  >  scale expanded x9  - 44 senta the orbital energy levels (79) of the X~ molecule-ion for both the Vg and the H center (36,38) as a function of internuclear distance Figure 6 H  K  F T  np cr u E  E nprt  g  np«  E  E  u  6 decrease internuclear distance in C l  2  molecule-ion  ^  i n the Cl~ molecule-ion. The g-ehifts are produced by the optically f o r i t bidden transitions E_ and E shown in figure 6 which arise from the ady x mixture of orbitally excited states with the groundstate of the moleculeion (36). The g-shifts using a modified form of equations 36 of section B(d) are (62,80):  «y- o g  e  where a  z  6  2\a /E* 2  ..(3)  o  i s the fraction of p character i n the bond joining the two  ohlorines i n the Cl~ molecule-ion. For axial symmetry, the degeneraoy of the « levels i s not  - 45 l i f t e d by the crystalline f i e l d (the difference between E and E vanishx y es) and therefore E « E which can now be called E . Equations 3 show x  y  g  that the g-shift i s inversely proportional to the forbidden transition energy E and since E for the H oenter i s about twice as large as E for g g g the Vg center, the g-shift for the H center i s about half that of the Vg center. The difference i n g-shifts between the two centers also indicates that the interaudear distance of the two nuclei in the halogen,, molecule -ion i s less i n the H center than i n the Vg center (see figure 6). This led Kanzig to a model for the H center which i s shown in figure 1 along with the model of the Vg center. The H center can be described (39) as a halogens-molecule ion located in a halide vacancy. The resulting crowding accounts not only for the reduced internuclear distance as compared to the Vg center, but also for the additional interactionsof the unpaired electron with two more halide ions. This configuration may be characterized as a "crowdion" (8l) stabilized by a hole. (The term "crbwdion" was coined by Paneth as a descriptive designation for a short linear region of compression.) The paramagnetic center previously discussed was called the E oenter since i t had the production and bleaching characteristics of the optical H band (39)« The antimorph of this center, an i n t e r s t i t i a l L i , has been found i n irradiated LiF (82). d) Other electron-deficient centers Although most attention has been focussed on the Vg and H centers, other electron deficient centers have since been discovered and their structures have been determined by electron spin resonance. Host  - 46 of these centers have been found in irradiated LiP at low temperatures. Two of these are the V  t  ( P ^ ~ ) and the Vp (bent F p  centers. The other  centers found i n LiF (excluding the Vg and H centers) were of such weak intensity that they could not be identified. Also in a mixed crystal of KC1-KF, an FCl"" center has been identified. More research i s being done 1  using fluorides than other halides since the spectra obtained are simple and are interpretable even for rather complex centers. This i s because fluorine has only one isotope, F"^, with a nuclear spin of l/2 and a large nuclear magnetic moment which gives large hyperfine splittings. The V  p  center found by Kanzig (83) in irradiated LiF at 77°K  was found to be the closest approach yet known to the antimorph of the F center although the similarities with Seitz*s " V^" model are limited. The Vp center i s a hole trapped at an L i vacancy but the hole i s l o c a l ized on two fluoride ions so that an  molecule-ion results, with a bent  bond(see figure l ) . The hyperfine spectrum of the Vp center i s closely related to that of the self-trapped hole or Vg center in LiF (84). The spread and anisotropy of the hyperfine spectrum are about the same as that of the Vg center indicating that the electron is localized almost f u l l time on the fluoride ions. The basic spectra, however, consist of four lines of roughly equal intensity which shows that the two fluoride ions are non-equivalent with respect to the magnetic f i e l d . There are special orientations of the molecule where there i s s t i l l the familiar three-line pattern of intensity ls2sl which showB that the two fluoride ions are equivalent i n these instances. The main difference between the V_ and the V 'center i n LiF i s that i n the latter case the two fluorine  - 47 nuclei are equivalent for every direction of the magnetio f i e l d so that the l : 2 s l pattern always results (see reference 3 6 ) . This center has not yet been observed i n KC1, KBr, or NaCl. The  v t  center (P ^) which i s also produced by irradiating LiP  at 20°K was found by Cohen, Kanzig and Woodruff (33,45). This center r e sembles Seitz*a model of the been carried out on the center to the  center but optical studies have not yet  oenter so that a definite assignment of the  band i s not warranted. The model proposed for the  center (supported by ESR studies (33)) consists of an electron deficiency localized on three fluoride ions which form an isosceles triangle (see figure l ) . ESR studies showed (33) that the lines in the  center split  i n a way similar i n detail to the Vj, center except, that since there are three nuclei of spin l/2 in the  center, there are twice as many lines.  The basic spectra mainly consist of eight lines of equal intensity whioh are accounted for by three non-equivalent fluoride ions. There are also cases where the magnetic f i e l d i s in such a position that the two outer fluorine ions become equivalent thus producing basic spectra of six lines with the intensity sequence 1:2:1:1:2:1 where two lines are degenerate. This center has not yet been identified i n other alkali halides. Since mixed halogen molecules such as PCI, BrCl and IC1 are well known and since stable P^, C l ~ and Br,, species have been observed i n crystals, i t seemed reasonable to Wilkins (37) that stable PCI , BrCl and IC1  ions might be produced i n mixed alkali halide crystals. Wilkins  and Gabriel (37) produced an PCI  ion i n a mixed KC1-KP crystal at l i q -  uid nitrogen temperature by irradiation. The PCI  ion produced was  found to be oriented along the ( i l l ) direction unlike the V" ( C l K  2  and  F^) centers which are oriented along the (110) direction. The FC1  ion  can be described as an i n t e r s t i t i a l fluorine atom or an (FC1 ) ion trapped at an anion vacancy The main feature of the spectra i s the double 0  set of four equally spaced lines of equal intensity. These lines are produced by the unpaired electron interacting more or less equally with an F  -  ion of spin l/2 and a Cl~ ion of spin 3 / 2 . The resonance absorption  line i s s p l i t by the F" into two lines which in turn are both s p l i t by Cl  into four lines of equal intensity. The more intense lines in the / 19  spectra were found to be due to the (F  -Cl  surements enabled the lines due to the (F  35\) -Cl  ion and intensity mea)  ion also to be iden-  t i f i e d . The ratio of the two sets of lines w a B in accordance with their relative isotropic abundance. The separation of the two sets of lines was found to be consistent with the relative magnetic moments of the chlorine isotopes. From the viewpoint of studying chemical reactivity, i t i s a l i t t l e unfortunate that the most extensive information on electron deficient defects i s for the fluorides, which are the most resistant to loss of electrons by chemical attack. If fluorine i s being used as an oxidizing agent, however, i t i s important to know the ESR spectroscopic charact e r i s t i c s of fluorine-containing centers, since these are very l i k e l y to appear as intermediates.  APPARATUS AND PROCEDURE  - 49 A. Electron Spin Resonance Spectrometers Two X-band spectrometers were used in the course of the work. One was equipped with a varlan 6 inch magnet while the other instrument, a varlan E3 bench model, had a U inch magnet. The spectrometer with the 6 inch magnet, which is similiar to the varian V-4500 ESR spectrometer, was built in the Chemistry Department with modifications to give more sensitivity and stability. A block diagram of this spectrometer is shown in figure 7. The microwave power is provided by a water cooled klystron oscillator which delivers about 300 milliwatts over i t s tuning range, 8.6 to 10 Gc./sec. The klystron frequency is stabilized by an automatic frequency control (AFC) system which locks the klystron on the resonant frequency of the cavity. The klystron i s coupled to a wave guide which transfers the microwave output via a magic - T bridge and accessory equipment to a rectangular r e flector type cavity that resonates in the TE 102 Mode. The microwave cavity is situated in the 2.5 inch gap between the poles of a 6 Inch water cooled electromagnet that supplies the variable f i e l d . A 100 Kc./seo. modulating magnetic f i e l d i s superimposed on the changing magnetic f i e l d by a pair of auxiliary coils mounted one on each side of the cavity. The resonance absorption signal is detected by a microwave crystal detector using a s i l i c o n diode which is situated In one of the arms of the magic T bridge. The signal is passed through an amplifying system to a phase sensitive detector and thence to a recorder which graphically records the f i r s t derivative absorption signal. The sensitivity of the spectrometer is -13  approximately 3 x 10  AH moles of electron spins where^H is the peak  width in gauss at half height. The magnetic f i e l d was measured using a  - 50 -  Figure 7 - Blook diagram of a 100 Ko. ( 6 inoh ) ESR spectrometer  AFC UNIT  TERMINAL LOAD  1  LEFLECTOR POWER SUPPLY  KLYSTRON  PHASE SHIFTER  ISOLATOR  ATTENUATOR  —  MAGIC TEE  — CRYSTAL — DETECTOR MAGNETOKETER  NMR PROBE CAVITY  100 Ko RECEIVER  c R0  SIGNAL GENERATOR  MODULATION COILS MAGNET COILS 100 Kc ^ 1 0 0 Ko ISCI OSCILLMODULATOI ATOR  FREQUENCY COUNTER PHASE SHIFTER  PHASE DETECTOR  RECORDER  INTEGRATOR  - 51 proton resonance magnetometer. The magnetometer consists of a probe c o i l containing glycerol which is inserted in the magnet gap beside the cavity. The c o i l is connected to a marginal oscillator which is frequency modulated at 20 c./seo. The proton resonance signal is displayed on an oscilloscope and a signal generator is tuned to zero beat. The frequency of the signal generator is measured by an electronic counter. The varian E3 bench model is nearly completely a solid state unit. The block diagram is shown in figure 8. The spectrometer consists of three main parts: (i) a console system which contains the console power supply, the 100 Kc./sec. modulation unit, the magnetic f i e l d regulator, and a recorder; ( i i ) a 9.5 Gc./sec. microwave bridge which consists of a microwave system for excitation and observation of the ESR signal, an AFC which stabilizes the klystron, a klystron d.c. power supply and a preamplifier for signal amplification; ( i i i ) a U inch water cooled magnet system which has an accessible air gap of 1.2 inches in which the cavity s i t s . The spectrometer has a sensitivity of 10^* AH spins. The advantages of this spectrometer are: frequency can be read off the frequency dial on the microwave bridge; the magnetic f i e l d can be read off the console panel; and the recording system (fixed chart and moving pen with adjustable zero for the vertical axis) is more convenient than a continuous strip chart when comparison of successive scans is needed to confirm faint details, to follow kinetic effects, or to observe changes in spectrum with orientation of a crystal. B. Vacuum Sublimation of Ionic Solids Solid samples of various types were used, including analytical reagent grade powders without further treatment, and single crystals from  KLYSTRON MICROWAVE BRIDGE  AFC MODULATION UNIT 100 Kc  X  PREAMP  OSCILLOSCOPE AND CHECKOUT UNIT  1 PROBE  CAVITY 3.4 kG  VJ1 PO  MAGNET  l _ .  MAGNET POWER SUPPLY  MAGNETIC FIELD REGULATOR  KLYSTRON POWER  CONSOLE POWER SUPPLY  Figure 8 - Block diagram of an E-3 ESR spectrometer  RECORDER AND MAGNETIC FIELD SCAN UNIT  - 53 various sources. Each type of sample is described in detail later. Only the preparation of vacuum-sublimed material is given here, because i t requires an extensive description of apparatus and procedure. The vacuum-sublimed particles were made in an apparatus designed by Adams (85) which is shown in figure 9. The apparatus consists of the main vessel B, side arm A, water cooled condenser C and the scraper G. A,B, and C are made of quartz. The scraper G consists of a tungsten rod H sealed into a pyrex B-10 cone at J and K; the scraper blade is made of platinum and is spot welded to H at I. The side arm A is heated by three heaters L, M, N which are made of chromel-A and are encased in Alumina and glass wool. The rheostat was set to allow a current of about U amps to flow into the heater system L, M, and N. This setting was used for KI, KBr, KC1, and NaCl which have melting points around 800°C and sublime rapidly close to their melting points. The sample to be evaporated (KC1, NaCl, KBr, or KI) is placed in side arm A. The apparatus is connected at 0 to the collector (usually an ESR sample tube - figure 12) and to a vacuum system, and is evacuated by a mercury-diffusion pump to 10~^ mm Hg. The sample is outgassed by heating i t to a temperature below the evaporation temperature (current of about 2.5 amps) while the sample is connected to the pumps. Outgassing is discontinued when the pressure measured by a Veeco ionization gauge remains constant after disconnecting the system from the pumps. Cooling water and heaters (current of A amps) are turned on after completion of outgassing. The sample evaporates and condenses out on the water cooled quartz surface of C. When a thin layer of film has been formed the heaters and water are turned off. The film is scraped off and allowed to f a l l down  - 55 -  past D into the collector. The procedure is repeated until there is about ,5 mm of sample in the sample tube. The sample is then sealed off under vacuum at C in the reaction apparatus (figure 12). The B.29 and B.10 Joints at the top and bottom of vessel B are lubricated with high temperature grease, apiezon T, and are cooled by a stream of air while the heaters are operating. C. Handling of Halogens The transfer of chlorine and fluorine from high-pressure cylinders to pyrex storage bulbs, together with drying in the case of chlorine and removal of HF from fluorine, has been confined to a single extensive vacuum system which is shared by everyone in the laboratory. Since halogens are used in this system, mercury-diffusion pumps are not suitable. A three stage Balzer diffusion pump containing silicone o i l is used. The advantage of silicone o i l over mercury is that i f by accident halogens are passed through the pump only volatile products are formed which do not clog the pump. The stopcocks are lubricated with Kel-F grease with which the halogens do not react. Since this grease is more l i k e l y to streak than hydro-carbon types a l l taps are of the right-angled pattern which is less likely to develop leaks from streaking. For pressure measurements of  &  2  sulfuric acid manometers are employed whioh operate non-1inearly with some air in the closed limb (see figure 10). The vapor pressure of the sulfuric acid is about 7 x 10"^ mm Hg. A chlorine and fluorine disposal line is connected to a l l parts of the system. It is separated from the main vacuum l i n e , but served by the same pump. Chlorine is trapped out by liquid nitrogen in removable  traps and disposed of in aqueous sodium hydroxide. Fluorine is directly absorbed into soda lime. The pressure in the vacuum system is determined using a Veeco-R.G. 75? ionization gauge with a 'non-burn-out' iridium filament. Care must be taken when using this gauge, for the filament is damaged by contact with any halogen (even I ). 2  The pressure obtained using liquid  nitrogen traps in front of the gauges is about 1 x 10~° mm Hg. ft) Chlorine drvftng, system. The chlorine obtained from Matheson Company was quoted as 99.5$ pure. Uncondensable gases and water vapor were removed in the purification system shown in figure 10. The procedure is as follows. Chlorine from the stock cylinder is admitted to the 2 l i t e r bulb A with stopcocks 2, 4, and 5 closed. The chlorine is allowed to flow into the system u n t i l a pressure of one atmosphere is reached as shown on the sulfuric acid manometer F. The non-condensable gases are removed by opening tap U and allowing the chlorine to condense into two liquid n i t r o gen cooled traps (not shown) while the non-condensable gases are pumped away. The chlorine is transferred back into bulb A by condensing i t in K. Tap 4 is then closed. Water vapor is removed from the chlorine by the sulfuric acid bulbs D and E. These, as well as the storage bulb B,are evacuated along with connecting systems back to stopcock 5. Then stopcocks 6, 7, 8, 9, and 12 are closed and 10, 11, 13, and 14 are l e f t open. Stopcock 5 is opened slowly, allowing the chlorine to bubble through the drying traps D and E via C to the storage bulb B. When half the chlorine has bubbled through, the remainder is pulled through by putting a liquid nitrogen dewar at  Figure 10 - Chlorine purification system  - 58 finger L. Stopcock 14. is closed and the dewar is removed. The chlorine is taken from storage bulb B, as required for use in different systems, via M or N. The pressure of the gas can be measured by the sulfuric acid manometer G. b) Fluorine handling system Fluorine was handled in pyrex and quartz systems which were not affected by the fluorine so long as i t was dry. Matheson Company fluorine which was quoted as 93# pure was used. The main impurity is hydrogen fluoride which is removed by passing the fluorine through a sodium fluoride trap (see C  f  figure 11).  The fluorine (cylinder pressure 300 p.s.i.) is admitted to the system via a series of expansion volumes between monel needle valves (86) connected to the pyrex system by copper tubing and a kovar seal at D. The -3 system is i n i t i a l l y evacuated to 10  mm Hg by a rotary pump, flushed with  fluorine and then re-evacuated through the disposal line. Fluorine is then allowed to enter the system (see figure 11) (taps 2, 4, and 5 closed and 1, 3, and 6 open) and its pressure is measured by a spiral gauge A. This consists of a mirror on a hollow pyrex spring which is free to rotate when there are pressure differences between outside and inside surfaces of the spiral. The gauge is used as a null meter. A known pressure of carbon d i oxide is admitted (tap 7) and fluorine is admitted through tap 6 to bring the scale indicator back to its i n i t i a l position. The valves at the fluorine cylinder and tap 3 are closed. Tap 5 is opened and the system is evacuated except for the fluorine in the storage bulb B. The B.10 socket at tap 2 is the point of attachment for reaction systems to be charged with fluorine. The pressure of fluorine usually used is approximately 29 cm of  Hg which is obtained by putting liquid nitrogen at finger E. D. Reaction Apparatus The reaction apparatus shown in figure 12 and figure 13 are both constructed with quartz and pyrex in a design which enables the 6 mm quartz tube to be inserted into an ESR spectrometer. Apparatus 1 in figure 12 has a circulation system which enables the chlorine and fluorine to be circulated over the solid by convection. Apparatus 1 has a glass bead  at 8 which breaks up the sample to allow i t  to pass down the 3 mm inner tube of the circulation system; a constriction at C to permit flame sealing under vacuum; two quartz-pyrex graded seals F and G; and a Kel-F greased right-angled tap at D which allows the attaching of the reaction apparatus at E to other systems (usually a portable vacuum system) without breaking vacuum. Apparatus 2 in figure 13 is a self sufficient reaction vessel for i t has its own storage bulb B for fluorine or chlorine. Kel-F grease is used on a l l joints and the quartz-pyrex junction 1B the B.10 joint below stopcock 1. E. Reaction Procedure -5 The reaction apparatus was evacuated to a pressure of 10 mm Hg as measured by a Veeco-R.G. 75P ionization gauge. The samples were transferred into the apparatus by several methods. Single crystals and powders direct from the storage bottle were transferred in a dry box and the sample tube was afterwards evacuated. Vacuum-sublimed particles were transferred under vacuum. The apparatus was mounted securely in the cavity of the ESR spectrometer. The procedure was then usually as follows: a blank run was  - 61 -  - 63 taken o f the sample (after the spectrometer was tuned) to check i f any s i g nals were i n i t i a l l y present. Fluorine was then added. The time was noted. If a signal appeared its growth was followed by scanning through i t at known times. If no peak appeared within 5-30 minutes the fluorine was r e moved and any peak which appeared thereafter was studied both in its growth and its decay. When a peak appeared after the removal of the fluorine, chlorine was sometimes added to determine its effect on the growth and decay of the ESR signal. In order to confirm faint details of the spectra obtained, many successive scans were done. The procedure described above was used for both powders and crystals. The crystals which produced stable paramagnetic species were carefully removed from the reaction apparatus and mounted with p l i bond glue in a crystal holder. The exposure of these crystals to air did not affect the paramagnetic species - even the partly reacted SrCl^ was no longer deliquescent. The holder consisted of three sides p, q, r which are mutually perpendicular to each other (i.e. one-half of a cubical box). A polystyrene rod was in turn attached perpendicularly to each of the Bides and the crystal was rotated about the vertical axis which was perpendicular to the magnetic f i e l d . Spectra were obtained at 5 to 15 degree i n tervals with an accuracy of + 2 ° .  GROUP I - HALIDES  -  64  -  A. Summary of Systems Investigated for ESR Slnnals Xflble Solid  Type of goUti gampjle  NaCl  vacuum-sublimed powder  III  Treatment  ESR signals signal associated with  &  Cl atoms and assigned as an H-center  KC1 reagent grade powder;  no signal  untreated single crystal (Harshaw)  F  2  no signal  &  irradiation (room temp.)  KBr  vacuum-sublimed powder  ci  2  &  signal unidentifiable for lack of observable hyperfine structure  single crystal (Harshaw)  Cl, &  no signal  irradiation (room temp.)  KI  vacuum-sublimed powder  Cl,  no signal  The vacuum sublimed particles of KC1 and NaCl were prepared by the procedure described in section B of the Apparatus and Procedure section of this thesis.  The surface area of these  particles as measured (86) by the B.E.T. method using krypton o 2 adsorption at 77 K was found to be in the range of 20 to 36 m /g. Morrison and Harrison (87) using an electron microscope found that the particles appear as well formed cubes with sides of about 0.1 micron. a) Analysis of the spectra Figure 14 shows a number of spectra obtained in NaCl and KG1 by treatment with fluorine and development in vacuo as described in detail in the kinetics section (c).  Of the NaCl samples (a) and  (b) contained the greatest amounts of coherent material and (d) appeared to be entirely randomly orientated. to give f a i r l y well powdered samples.  The KC1 films broke up readily The fisze spectra in figure 14.  are a l l adjusted to a standard microwave frequency v" = 9.4679 Gc/sec. and the scales at the bottom of the diagram indicate the position at various orientations of the center line and the line at nuclear magnetic number  =  +1, for the H- center model. The most conspicuous part of a l l the spectra is ABCD,  which is qualitatively of the form to be expected (88) for a single l i n e , the position of which is fixed by an anisotropic but axially symmetric g-factor.  The known centres in single crystals with which w i l l  be compared with the present results are a l l close to axial symmetry; the conventional terminology, Q for the angle between the magnetic f i e l d and the molecular axis of the defect with subscripts signify 0 = 0 ° and 0 = 90° respectively w i l l be used.  i - and II to  On this basis,  -  66  Figure 14 - ESR spectra obtained in F  -  treated HaCl and K C 1  - 67 gj_ should be very close to A and g|| f a i r l y close to D; the latter position is somewhat uncertain, especially for NaCl, in which the region around D shows no resolved structure and i t is not clear whether the peak corresponds to D or D' in the KC1 spectrum.  The g-  values calculated from A and D (table IV) are consistent with the interpretation that ABCD is generated by a line without hyperfine interactions.  Such structures as FC1 , F , C l 2  2  (VR) and Cl^ (H) a l l  have such a l i n e , with g-values comparable to those calculated from the present spectra, except in.the case of Og.  (In table IV the features  indicated are peaks and troughs (single letter in f i r s t column) and spacings between them (two letters in the f i r s t column), the latter being expressed in gauss.  Comparison with computed spectra (see section b )  indicates that the values marked with  should be amended to : g  L  = 2.0231,  g„ = 1.9995, and A ^ = 4.) Further elucidation of the structure of the defect depends on the information which the spectra w i l l yield on hyperfine interactions. The existence of these is most clearly seen in the KC1 spectrum, Fig. 14e. I distinguish a "primary splitting", giving rise to the features marked 12 F and G.  This I c a l l A  , anticipating that I attribute the splitting  to interaction with two Cl nuclei designated 1 and 2.  A l l the features  marked by letters with prime or double prime superscripts I consider to arise from "secondary splitting", which I attribute to interaction with two more distant nuclei 3 and 4» hence designate A , o 12 The primary splitting at 9 = 0 , A , calculated as FD, DG 1  or ^FG, is of the same order a's A  ((  2  in the V^ and H-centres in a l k a l i  chlorides, especially that for the H-centre in KC1.  This value is on  gj_ should be very close to A and g|  f  f a i r l y close to D; the latter  position is somewhat uncertain, especially for NaCl, in which the region around D shows no resolved structure and i t is not clear whether the peak corresponds to D or D' in the KC1 spectrum.  The g-  values calculated from A and D (table IV) are consistent with the interpretation that ABCD is generated by a line without hyperfine interactions.  Such structures as FCl", F^, C l " (VR) and Cl^ ( H ) a l l  have such a l i n e , with g-values comparable to those calculated from the present spectra, except in the case of O".  (in table IV the features  indicated are peaks and troughs (single letter in f i r s t column) and spacings between them (two letters in the f i r s t column), the latter being expressed in gauss.  Comparison with computed spectra (see section b )  indicates that the values marked with  should be amended to : g - 2.0231, L  g,l = 1.9995, and A ^ = 4.) Further elucidation of the structure of the defect depends on the information which the spectra w i l l yield on hyperfine interactions. The existence of these is most clearly seen in the KC1 spectrum, Fig. H e , I distinguish a "primary s p l i t t i n g " , giving rise to the features marked 12 F and G.  This I c a l l A  , anticipating that I attribute the splitting  to interaction with two Cl nuclei designated 1 and 2.  All the features  marked by letters with prime or double prime superscripts I consider to arise from "secondary s p l i t t i n g " , which I attribute to interaction with 34  two more distant nuclei 3 and 4# hence designate A . o 12 The primary splitting at 9 = 0 , A , calculated as FD, DG 12  or -£-FG, is of the same order as A|  (  in the V^ and H-centres in a l k a l i  chlorides, especially that for the H-centre in KC1.  This value is on  -  6 8  -  the one hand much too large to represent the interaction of an (X, ion with neighbouring ions of the host l a t t i c e , and on the other hand much too small to represent hyperfine interaction with a fluorine nucleus (table IV).  The evidence is thus strong that the signal here described  arises from a defect structure of chlorine atoms and is therefore relevant to our predictions from kinetics of chlorine exchange, (In a number of runs on both NaCl and KClj a signal which is probably caused by an oxygen-containing impurity was obtained. line, about 5 gauss wide, at g = 2.007.  This is a single narrow  It appeared to be unaffected  by fluorine and had no influence on the signal under discussion other than to abscure part of the spectrum around the point C.) The secondary splitting is indicated as about 20 gauss at 0 = 0° by the distances from the points D, F and G to their "satellites" with prime superscripts.  There is a possibility of confusion between  splittings arising from more distant ions, as in the H-centre, and those arising from isotope effects within the C l " ion, particularly the species (C1 C1 )7 35  37  Since the nuclear spin of C l i s 11% less than that of C l 3 7  3 5  the degeneracies of the nuclear spin levels are l i f t e d in the mixedisotopic ion to give a splitting 17$ of A ^ the 20 gauss here observed.  2  yhich at 0° is very close to  However, the lines for the mixed isotopic  species are displaced from the corresponding (Cl "*)"" line by half-integral, 3  not integral, multiples of the isotopic splitting (36). is here observed.  This is not what  These lines probably represent a secondary splitting  like that of the H-centre rather than an isotope effect both for this reason and because, as discussed in detail below under "Computations", the intensity ratio of peaks F and A is wrong for the V- centre and  Table IV Important Features of Spectra and proposed assignments for H-center model. Feature  Assignment  Known values in single crystals  Data for KC1  Tensor  H-center *34  9  Component  (NaCl in brackets) (KC1.20 K) 0  2.013  1.953  *l!  2.0015* (>1.9962)  2.0023  2.0010  2.003  2.436  A  107.5  109  101  887  -  -  90°  D  0  0  0°  F D  1, o  0  0°  DG  0,-1  0  A'A  0  1, o 90°  AB  0  —  D D* .  0  0,-1  0°  14.5  D D"  0  -1,-2  0°  18.0 (l/2 ap =18.5)  G G'  0°  ii  A^and cr (not resolved)  0,-1  0°  -1  1, 0  0°  A  0°  A  0,-1  1 2  |  112.5 8.0*  1  -1  (KCl)  2.0438  0  G»'G  (LiF)  2.0223  0  F F«  (KCl)  2.0251* (2.0256)  A  90°  ^-center  A  3 4  II 3 4  2.9  13.3 (15.0)  —  22.5 20.0  !1 5 4  20.0  >  >  7.3  -  -  -  CA  - 70 requires a strongly anisotropic secondary splitting to account for i t . The proposed value Aj| = 20 gauss is to be compared with the known value for the H-centre in KC1 at very low temperatures, 7.3 gauss. The features representing secondary splittings and linewidtha which remain to be discussed in the NaCl and KC1 spectra likewise show marked broadening relative to low-temj?erature values.  The region AB of the spectra,  together with the partly-resolved peak A* in KC1, represents essentially the 90° spectrum, and the overall extent of this region is too large to be accounted for by the combined effect of primary splitting, secondary splitting and linewidth, i f the 20°K H-centre values are used, i . e . A, " 0 (degree of approximation not, however, specified), A ^ = 2.9 gauss, linewidth =0.8 gauss in KC1.  The H-centre spectrum has not been studied  in NaCl single crystals, presumably because the linewidth would be close to the V^centre value of 4.5 gauss and hence the secondary splitting would not be resolved.  In the spectra, i f the region  ocB in NaCl and the  region BD* D in KC1 is to be interpreted as arising from a secondary splitting of about 20 gauss abscured by the linewidth, a linewidth of about 10 gauss in KC1 or 20 gauss in NaCl seems to be indicated. Although one cannot unequivocally assign the defect under study as an H -centre (the close similarity of  and H spectra, which  arise from rather different variants of Cl^ , is enough to invite caution), the above comparisons of parameters at 20°K and room temperature agree very well with that assignment.  The g-values and the primary  splitting are essentially molecular properties of C l " , and hence likely to remain unchanged from 20°K to room temperature; but secondary splitting and linewidth involve coupling with nuclei outside the molecule-ion,  - 71 -  and are hence l i k e l y to be broadened by increasing thermal motion o f the lattice.  The broadening should be l e s s f o r the secondary s p l i t t i n g than  f o r the linewidth, since atoms 3 and 4 are weakly bonded to the C l ~ ion. b)  Computations Most o f the above argument does not depend on the r e s u l t s  of a precise computation o f the powder spectrum from the proposed parameters; but there are several considerations which make i t desirable to attempt such computations.  The degree of accuracy to which the  g-values are given by points A and D should be deterraained. 12  The extent to  3A  which the region A'AB can be analyzed into Aj_ , A [* and linewidth should be established.  The p o s s i b i l i t y o f accounting f o r the part o f the NaCl  spectrum downfield o f A i n terras of p a r t l y - o r i e n t e d material should be studied.  F i n a l l y , the i n t e n s i t y rations o f the various features o f the  spectrum are o f some i n t e r e s t .  For example, the type o f spectrum shown  i n F i g . 14a or b, i n which the negative peaks at B and D are o f comparable s i z e , i s not uncommon; the spectrum of adsorbed 0~ i s another example (89). Yet the shape generated by a l i n e narrow i n comparison to the gap AD should have the peak at D markedly smaller than that at A; a large D peak i s r e a l l y an anomaly. In a s i t u a t i o n with as many variables as the present one, the establishment o f the dependence of the powder spectrum on each requires many computations with t r i a l values.  For t h i s purpose i t was found  necessary t o use programmes severely r e s t r i c t e d i n t h e i r scope ( i . e . written f o r the  and H s i t u a t i o n s only) but using very l i t t l e computer  time, rather than to r e l y on comprehensive but time-consuming  programmes,  one o f which (90) was available and was used i n one or two computations.  -12  -  No programme was devised which was both sufficiently rapid and adequate in integrating over 9 smoothly, but a programme which required about two minutes in an IBM 70A0 computer was found useful especially for the KC1 spectrum. The features essential to the characterization of the defect are given here. Computations have been made for the ( C l ^ )  -  or  (Crp) ~ 3  ion only; and in the expression for the magnetnetic f i e l d at resonance H  r e s  only terms linear in the magnetic quantum numbers m^ (sum of  magnetic quantum numbers for nuclei 1 and 2) and m^ (the same, for nuclei 3 and U) are retained: H  = Chv/gg)-- ( g ^ g j A ^ - ( g / g ) A V j 1  rea  where  g  and  (A )  2  =g 1 2  2  2  (1)  3  o  4  + ( g - g )sin 9 2  2  (2)  2  = ( A ? ) • ( ( A J ) - (A ; ) 2  2  2  3  2  2  )COS 9  ()  2  3  3A and a similiar expression gives the angular dependence of A . In table IV, 12 and throughout the preceding section, I have written A  or A  for  quantities which should s t r i c t l y have the factor (g /g) attached to them; Q  but for the present work i t is permissible to ignore this factor, since the error thus introduced is never greater than about one gauss, and is comparable to other approximations in the treatment such as the neglect 2 of terms in m and the l i f t i n g of degeneracies in m^ by second-order effects which become noticeable (36) i n single-crystal spectra close to 90°. At any fixed orientation 9, the spectrum of a V center con-» 12  sists of seven lines, with the spacing A 1:2:3:4.»3J2:1, corresponding to values of  R  and intensities in the ratio from +3 to -3; i . e . the  - 73 degeneracy associated with m^ is (U - m^). *  n  ^  e  ^-center, each  line is s p l i t into a similar group of seven described by the quantum number m^; and the lines having Gaussian shape (36), with half-width cr at the point of maximum slope, the intensity of micromave absorption .' at magnetic f i e l d H is given by 1(9) =  -3 Z  -3 L  U - m ) U - m^) exp( - ( H - H ^ ^ ^ ) 1 2  o o )V2^)  If in a powdered sample of n particles, a number dn have orientations between 9 and 9+d9, the probability P-(9) is defined by dn= nP(0)d9. In a random powder, P(9) = sin9. The problem of computing a powder spectrum is then the evaluation of the intergral 3t/2  I -  I  l(9)P(9)d9  (5)  0 and the observed derivative-of-absorption spectrum is dl/dH as a function of H. In the programmes which have been used most extensively, 1(9) is calculated according to equations (l) to (4) for a number of values of 9 spaced at equal intervals, usually 1°, and the results are summed in a histogram of I against H. The effects of preferential orientation on the powder spectrum may be computed by including in P(9) a suitable function to describe the preference. Thus, for exarfple, I have considered the possibility that the defects in each crystal are oriented at random in a l l possible directions of the type (110), like the V^. and H centers, but that the coherent mat e r i a l in the solid samples might have a tendency for (110) to be alllgned  - 74 parallel to the f i e l d H. In this circumstance, a function showing a preference for the angles 0 ° , 60°, and 90°, in the ratios of 1:4:1  is  needed. Accordingly I have written P(0)  where  = sine  • f (l0 x 2  4  (Ute" / - x 1  +  2  l / A  )  )  x = cos9 ( l - cos9). The parameter f increases with the  fraction of oriented material in the sample, the function multiplied by f has f a i r l y sharp maxims at 0 ° , 60°, and 90°, the last-mentioned being somewhat underweighted. A similar function could be devised to represent any desired orientation. In considering in qualitative terms the features of a powder spectrum, i t is perhaps best to think in the f i r s t instance of one line (with small linewidth) and the intensity I  ffl  contributed to the spectrum  by this line at any value H of the magnetic f i e l d . I P(9)d0/dH  is proportional to  , i . e . a line gives the largest contribution where i t i3 mov-  1*03  ing most slowly as 9 changes. Thus, for example, every line in the spectrum, with or without hyperfine interaction, produces a noticeable feature in its 0° position, because although P(9) is zero at 0° in a random sample, dH  /d9 is also zero; but at 90°, the center line of the  1*63  spectrum is stationary and gives a prominent peak, while the hyperfine lines are moving rapidly and contribute very l i t t l e . In my example, computations showed that the shape of the region A'AB of the spectrum is 12 very insensitive to Aj_ , which consequently cannot be found with any accuracy from my results and was taken as zero (H-center value at 20°K). 34 The splitting A ,  on the other hand, changes sufficiently slowly with  9 to produce marked effects in this region of the spectrum. Conclusions from computed spectra are as follows:-  .- 75 (a) A  center should give a powder spectrum with two prominent lines  each like AB, of roughly equal intensity. the line at  1 *  s  The reason for this is. that  subject to two motions as 9 changes, the downfield 12  motion of the center of the spectrum opposed by the diminuation of A  ,  and in consequence is almost stationary over a wide range of 0.  Thus i t  becomes as prominent in the powder spectrum as the center-line  0'  (b) The H-center model, with strongly anisotropic secondary splitting, gives a spectrum similar to that observed in KCl (figure H e ) , in which the region A'AB is the most prominent feature of the spectrum, and the features F, F , A', A B, D, D , D " , 1  1  G " , G, and G* are a l l present in  the computed spectrum although the relative intensities of the experimental spectrum are not precisely reproduced, and the integration over 9 is not smooth enough for the identification of the smallest features, G , G, G', M  to be absolutely certain.  The features which are most important in  establishing the values of the components of the g and A tensors, A, D, F, G, F , G', and G " are uniformly about 3.5 gauss downfield of the expected 1  position, so that in the experimental spectrum g located 3.5 gauss upfield of A.  ±)  for example, is to be  This means that the g-values in table I  should a l l be corrected downwards by 0,0020, but the splittings are estimated correctly.  There is one exception:  computation shows that the  gap A'A is not, as anticipated, equal to A^j_ , but has about twice that 3/  value, so that the value given for Ajj in table I must be halved. values for the parameters of the defect in KCl are:g  = 2.0231 - 0.001  g„  = 1.9995 * 0.001  ±  12 A,  uncertain  The f i n a l  - 76 A  ll  = 110  A  l:l  =  20  =  4  t?  U  3 + +  3 1  J_ 2 cr ~ 10 A  (c)  Apart from the prominent feature D, which is fortunately the  important feature in the determination of g|, the shape of the spectrum (  in that region is very sensitive to the precise values of the parameters, and g|| cannot be determined very precisely from the spectra like those obtained for NaCl, in which this region is completely unresolved. (d)  The intensity ratios, especially of A, B, and D, are roughly correct  for completely random orientation in spectra (c) and (d) for NaCl. Even in the KC1 spectrum, D is proportionately somewhat too large, and in the NaCl spectra (a) and (b) i t is very much too large. The dependence of the shape of the spectrum on orientation distribution is not simple. Thus i t might be thought that the most probable angle could be deduced from the positions of C (generated principally by the lines n^2 ^ =  8 1 1 ( 5 E  (figure  lAb from m^= l ) j but the scales at the bottom of figure 14 indicate 2  that these two positions do not give a consistent result for the most probable angle, which is indicated by C as less than 45° and by E as about 75°.However, computations using the function of equation ( 6 ) , with 6 0 ° as the most probable angle, reproduced the section BCD of spectra (a) and (b) very well. The part of the spectrum downfield of A is very sensitive to the values taken for secondary splitting and linewidth, and many reasonable values give a pronounced dip, instead of a rise, in the region of E. To reproduce the observed spectra, i t is necessary that the lines m^ =l, m^= - 1 , -2 should l i e in the region of E and that the 2  linewidth should be sufficient to destroy the resolution of the hyperfine  splitting even at 0 = 0 ° . For a f u l l study of this point i t is necessary to carry out an extensive set of computations with a wide range of values 1  2  3  /  of A* , A]| ,  3  /  kf"*  t  and  cr,  which would be a project in i t s e l f ,  c) Kinetics Contrary to expectation, the ESR signal did not appear during the exposure of the a l k a l i chloride to F , but shortly after 2  the fluorine was pumped out. The growth of the signal sometimes started immediately after removal of fluorine and sometimes after an induction period of a few minutes. Figure 15a shows two separate runs, one (A) in which a small signal appeared during contact with fluorine and started to grow immediately upon evacuation, and one (B) with an induction period. Run A is from the work of Adams referred to in the introduction to this thesis (section A(c) ) and run B is from ray work. The two growth curves are superlmposable from the start of the rise up to the point of i n f l e c t ion. Figure 15b shows plots of log(peak height) against time for one of these curves (B) and another similar reaction (C), suggesting that the peak height varies as exp(+kt), which would indicate that the defect is formed auto-catalytically. Since the growth period is short, quantitative results to establish the rate law unequivocally are d i f f i c u l t to obtain. The reaction does not appear to be markedly affected by ?^ pressure (between 14 and 4.0 cm Hg) or time of contact (from 3 to 30 minutes). The entire experimental procedure could be repeated several times with similar results, except that the rate of growth of the peak and its maximum size were not reproducible. The best spectra for study of both spectroscopic details and kinetics were obtained sometimes on the f i r s t fluorination, sometimes on the second or third. The reaction of the solid with fluorine  - 78 -  was remarkably slow. In the preliminary work of Adams, two samples were analyzed gravimetrically for fluoride after the experiment. One, after a single fluorination, was found unreacted within the limits of error (about 2%) and another, after several fluoridations of several minutes each at 40 cm Hg, was 25% converted to NaF.(As was previously (l6) r e ported, remarkably low reactivity with F was noted for samples of 2  powdered NaCl of a different type). The growth of the signal was usually followed by the predicted decay, which was usually very rapid and, as predicted, was slowed down or even arrested altogether by the presence of C l preted in terms of blocking by C l combine to desorb a C l  2  2  2  gas. This is to be inter-  of sites at which two defects could  molecule. In the previous detailed analysis of  exchange kinetics (17) these sites were considered to be anion vacancies forming part of the structure of the defect with the unpaired spin. This was postulated as analogous to the F^~ ion (V^. center) known (38) in LiF, Since the present study seems to rule out this structure for the defect, indicating instead an H-center structure, which should not be associated with anion vacancies, the previous kinetic schemes may require some ammendment, with less emphasis placed on vacancies as a means of migration at room temperature. It has been suggested speculatively (39) that positively charged halogen atoms may be mobile even at extremely low temperatures in the KC1 l a t t i c e , and that these may be responsible for the formation of H-centers. Whatever amendments to the mechanistic schemes in the bulk of the solid may be necessary, i t is s t i l l reasonable that desorption of CL, from pairs of defects combining at the surface may be prevented by preadsorption of C l  5  in surface vacanciesj and the prediction that the  decay of the defect should exhibit second-order kinetics is s t i l l valid. The experimental results, however, show that, as exemplified by figure 15d, the decay is first-order when i t occurs in the absence of chlorine. Decay in the presence of chlorine was more d i f f i c u l t to study, since i t required the introduction of chlorine at precisely the right stage in the decay for the subsequent decay to proceed at a convenient rate. Figure 15c shows that the decay represented in curve A of figure 15a follows a second-order law. The growth and decay processes, and the blocking of the former in the presence of Fg and the latter in the presence of Clg, are both explicable in terms of mechanisms involving surface defects. For the autocatalytic growth, i t is suggested that fluorine enters into some state of combination with surface C l " , but that the oxidation of C l " with r e lease of the Cl atom (probably positively charged) into the bulk cannot be completed until a surface vacancy is available to receive F". During contact with the gas, these vacancies may be blocked by molecularlyadsorbed F^. On pumping out, they are made available, and the reaction of chlorine-fluorine complexes commences, producing surface F~, positive holes in the bulk, possibly gaseous molecules, and further anion vacancies in the surface, leading to the autocatalytic effect. For the decay processes, the first-order law suggests that the arrival of a single defect at the surface is the rate-determining step. This may indicate rapid surface migration by way of surface vacancies. When many of these are blocked by the presence of chlorine, the decay requires the simultaneous presence of two electronic defects at an immobile surface site, giving rise to the second-order law.  - 81 By c o n t r a s t , vacancies  i n the b u l k o f  of  the e l e c t r o n i c  is  a c t u a l l y n o known ESR  c i a t e d w i t h the  the vacuum-sublimed c r y s t a l s ,  tronic This  the  s i g n a l i n an a l k a l i h a l i d e s i n g l e  s t a b i l i t y ) . There  i n t r i n s i c a l l y unstable  and t h e  i s no o b v i o u s  transfer  (7)  for  its  assignment  c a r r i e d out o x i d a t i v e l y ,  i n i t s e l f m i g h t be s u f f i c i e n t  a  and n o t by  (There  r e a s o n why  o b t a i n any c o r r e s p o n d i n g s i g n a l from s i n g l e ly-powdered NaCl, which suggests that  i n the  defect,  because the  I  or ordinary  am c o n c e r n e d w i t h a  concentration,  a t room temperature  condenser.  A conspicuous similar  into  a stream o f n i t r o g e n were a c t i v e  g a s was p r o b a b l y a t condensation, defect  surface,  very the  quartz  the e a r l i e r k i n e t i c w o r k ^ ' ^  ( a b o u t 1000  particles  o n l y on the  size  of  that  temperature  of  equilibration of  as t h e y grow on t h e w a t e r - c o o l e d  feature  coarse-  special  low vacancy  by e f f i c i e n t  obto  t h e v a c u u m - s u b l i m e d m a t e r i a l . T h i s p r o p e r t y may b e a  crystals  elec-  X-irradiation.  I have n o t been a b l e crystals  X)  7  was  p r e p a r e d by s u b l i m a t i o n  f o r c h l o r i n e exchange a t  not i n the bulk.  a f a i r l y h i g h temperature  per-  d e s t r u c t i o n on  property of  caused  this  to e x p l a i n the p o s s i b i l i t y o f  serving the H-center i n t h i s m a t e r i a l ; but  asso-  defect  o f an e l e c t r o n to another  My m a t e r i a l d o e s n o t c o n t a i n F - c e n t e r s ,  damage was  absence  crystal  at h i g h temperatures  mechanism proposed  w a r m i n g a b o v e 20°K i n v o l v e s an F - c e n t e r .  if  a n i o n s o f t h e p u r e c r y s t a l and r e p r e s e n t i n g  s h o u l d be  fect lattice;  suggest an almost complete  d e f e c t u n d e r s t u d y as a n H - c e n t e r . be a c c e p t e d .  with high-temperature center  my r e s u l t s  Since the  i n the r e g i o n  t h e s e p a r t i c l e s m i g h t r e a d i l y be f o r m e d w i t h  room  nitrogen of  higher  concentrations. The m e c h a n i s t i c  schemes p r o p o s e d p r e v i o u s l y a l s o  indicated  of  - 82 the removal of? vacancies from the bulk by adsorption of Clj at the surface.  For Clg, this would be reversible on pumping out; but for  ?2 gas* the sequence of steps suggested above locks the vacancies in the surface as they become occupied by F~ on pumping out.  It is possible  that this process w i l l effectively dispose of a low vacancy concentration in the bulk, but w i l l not remove the vacancies completely i f the i n i t i a l frozen equilibrium state is much above room temperature. These considerations indicate that the relationship between vacancies, electronic defects and chemical reactivity may not be a simple one.  I suggest that electronic defects with unpaired spins are readily  formed in vacuum-sublimed material because i t consists of essentially perfect crystals devoid of trapping sites; and that the formation of electronic defects leads to a high reactivity for chlorine exchange.  Yet  i t is unusual to conclude that the more perfect the crystal, the higher i t s reactivity; and for the chemical reaction with F^, this material was noticeably unreactive.  GROUP II - HALIDES  - 83 A. Summary  of Systems Investigated for ESR Signals Table V  Solid SrCl  2  Type of solid sample  Treatment  ESR signals  reagent grade powder;  signal associated with  single crystal  Cl atoms and assigned as an i n t e r s t i t i a l Cl atom.  single crystal  irradiation  F-center-like spectrum  (low temp.)  observed, but signal not yet identified.  irradiation  no signal,  (room temp.) HgClg  reagent grade powder reagent grade powder  no signal. thermal  signal, basically of six  decomp.  lines,believed to arise from an impurity such as Manganese.  BaCl_  reagent grade powder  thermal  six line spectrum a t t r i -  decomp.  buted to Manganese impurity.  - 84 B. Reaction of S r C l  2  with F^.  Both powders and single crystals of S r C l were used i n the 2  reactions with fluorine* The powder samples were made by heating the hydrated S r C l  2  (Mallinckrodt analytical reagent grade S r C l » 6R" 0 ) at 2  2  about 120°C i n a quartz tube while pumping the water off with a rotary vacuum pump. The samples were heated for about 16 to 20 hours and then sealed off i n storage bottles u n t i l used* The single crystals were made by two methods: (1) by heating the sample under vacuum at a temperature (l20°C) which i s sufficient to remove the waters of hydration. The sample was then melted and cooled very slowly over a period of about two days. The crystal formed by cooling the melt slowly was cleaved along rather i l l - d e f i n e d cleavage planes. (2) by the Bridgman-Stockbarger technique (9l). The crystals grown by this method were generously donated by Dr. J . A. Morrison of the National Research Council. The starting material was SrCl *6H 0 2  2  which was melted with FbCl i n a graphite crucible. The PbCl was used 2  2  to react with the water and prevent the formation of Sr oxide and hydroxide. The excess PbCl and the Pb oxide and hydroxide formed are 2  more volatile the S r C l  2  and therefore evaporated off. The melt in the  graphite crucible was thenlowered through a sharp temperature gradient. A crystal nucleus formed f i r s t at the tip (bottom of the crucible) and continued to grow as the crucible was lowered u n t i l the entire melt solidified. The main precaution i n handling the SrCl  samples was to  - 85 keep them out of contact with moist a i r for SrCl  i s deliquescent.  2  Therefore, the transferring of the samples was carried out in a dry box or in vacuo, ft) He?ul^3 Figure 17 shows the ESR spectra obtained i n S r C l  2  by treat-  ment with F * The spectra are adjusted to a common microwave frequency 2  V=» 9.2442 Gc./sec. The room temperature reaction of S r C l with fluorine pro2  duces a paramagnetic species in powder samples as well as single crystals. The signal due to this center was found to grow in the presence of fluorine; and when the fluorine was removed the growth of the s i g nal stopped. In reactions with powdered samples as wellas single crystals there was an i n i t i a l rapid growth of the signal (in some cases after an induction period varying from 5 to 15 minutes) followed by a slow growth which continued for as long as four hours (reaction was not followed for a longer time). The fluorine pressure usually used was 29 cm of Hg since it. was found that when higher pressures were used the reaction proceeded to such an extent that the single crystal spectra looked exactly like the powder spectra (Fig. 17a ). The reaction of F  with the single crystals of S r C l  2  in-  i t i a l l y starts with the appearance of nuclei (presumably of SrF  2  ) on  2  the surface of the crystal which spread out quickly covering the whole surface of the crystal. After the reaction had been stopped by removal of F , there was usually a sharply defined interface between the r e 2  acted region (which i s pale yellow, probably on account of the presence of Cl  or C1F ) and the remaining unreacted crystal. The center  - 86  -  giving rise to the ESR spectrum was found by sectioning the solid to be present i n the reaction interface and not i n the deeper layers of the unreacted crystal. A weak signal was also found in the outer r e acted layers, but i t did not appear to have strongly-developed orientation-dependent structure. The species formed was found to be very stable even when the crystal was exposed to a i r for periods up to even a month. Evidently the reacted layer of SrFg protects the inner material from moist a i r . The spectra i n figure 17 can be interpreted i n terms of a model i n which the resolved hyperfine interaction of the unpaired e l ectron i s with a Cl atom with a nuclear spin of 3/2 (a more detailed explanation of the model i s given i n the discussion section (b)). The spectra are based on sets of four almost equally spaced lines of equal intensity; but such a set of lines i s not seen undisturbed i n any spectrum which seems to indicate that the centers exist i n a range of situations i n the crystal lattice (as discussed below, the variable displacement seems to be translational rather than orientational ). The nearest approach to a simple four line spectrum i s seen i n the lines EFGH i n figure 17b, i n which H i s parallel to a four-fold axis of the  N  crystal. A rough computation showed ( see figure 16 ) that the main features of the powder spectrum of figure 17a can be accounted for by a hyperfine tensor and a g-tensor which are both axially symmetric about the x-axis. This sort of model reproduced the seemingly extra peak E as well as the peaks ABC and D. The splitting of the four lines ABCD i n the powder were found to be about 80 gauss.  The single crystal spectra showed, however, that the defect was not axially symmetric about any axis of rotation of the s i n gle crystal. This indicates that the hyperfine tensor has different components (A^, A , A^) in three mutually perpendicular principal direcy  tions, whichare identified by the changes i n the spectrum on rotation of the crystal as a four-fold axis (z) and two two-fold axes ( x and y ) of the crystal. The directions and values of A and A were found by r o t x z " ating the crystal to locate special orientations in which maximum and minimum splittings were observed ( refer to Fig. 17b and c ). When H was parallel to a four-fold axis (Fig. 17b ) A was found ( spectrum z EF6H ) to be 16 gauss and with H parallel to a two-fold axis A was x found ( Fig. 17c, ABCD ) to be 92 gauss ( which i s the largest component of the hyperfine tensor )• This spectrum also should show A directly but due to the complexity of the spectrum the value of A had y to be confirmed by a more detailed analysis of the spectra of Figs. 17b and 17c . If one assumes that the defects i n a single crystal are randomly orientated in the sense that the x-axia can be any two-fold axis, any one orientation of the crystal presents the defect to the magnetic f i e l d i n two or three distinct orientations, so that the overall spectrum i s composed of two or three groups of four lines. Table VI shows for the special orientations of Fig. 17b and 17c , the hyperfine s p l i t ting which should be observed and the relative abundance of the defects oriented to give each splitting. The table also gives the direction cosines (i,m,n) of the magnetic f i e l d with respect to the x, y, z axes of the defect.  Figure 16 - Computed spectrum of the defect i n SrCl^ (10 gauss linewidth)  I 3150  I 3200  1 I 3250 3300 H (gauss)  J 3350  1—  3400  - 9 8  -  Table VI Direction cosines  Direction of magnetic f i e l d  (l,.m, n )  H parallel to  ( 0 , 0 . 1 )  4 (figure 17b )  ilfcl/t,  H parallel to  ( 1 , 0 , 0 )  °2 (figure 17c )  ( 0 , 1 , 0 )  Hyperfine  Peaks Abundance  splitting v  X  z  EFGH  1  ABCD  2  A x  ABCD  1  A y  EFGH  1  IJKL  4  A z  C  o)  n  (1/2,1/2,1/21  (l/2A +l/2A ) 2  2  l/2  (IAA^IAAJ+I^A ) / 2  1  2  It i s now possible to evaluate the component A from three y separate four-line spectra. The agreement between these three spectra of »  the A value makes i t possible to determine that the spectrum i n figure y 17c which displays A directly i s EFGH . The value of A 40gauss was y, • y 3  given by a l l three sets of lines. Since the g-factor tensor ( g ) and the chlorine hyperfine splitting tensor ( ^ )  a  r  e  assumed to have the same principal axes, the  principal components of the g-tensor can be obtained from finding the mid-points of the four-line spectra corresponding to A^, A , and A^ . y  The g values and A values obtained for the defect are tabulated below in table VII with the limits of their accuracy. Gardner (92) studying Cl atoms produced by irradiating Cl,, adsorbed on s i l i c a gel obtained values for the components of the gtensor, g  = 2.003 + 0.001 and g ~ g ^2.012 are similar to my values  - 90 Table VII Parameters of the defect e  x  = 2.0028 + 0.0010  g = 2.0194 ± 0.0020 y g • 2.0162 + 0.0010 z  A =92+1 x  gauss  A  = 40 + 2  gauss  A  =16+1  gauss  for a Cl atom i n a SrCIg l a t t i c e . My single crystal results permit a more precise analysis of the defeot and i t s effect on the chlorine 3s and 3p orbitals than was possible in Gardner's work on randomly-oriented Cl atoms . ( A closer analysis of the g values and A values w i l l be given i n the next section (b) under Discussion; and comparative values of parameters for Cl atoms in three different situations are given in table X ). As previously mentioned the lines obtained even in the single crystal have powder like shape. This i s illustrated i n figure 17b and 17c by the spectra having apparently the positive half of the normal derivative peak at low f i e l d and the negative half of the peak at high f i e l d . A complete analysis of the linewidth i s not possible both because of this "powder" character and because of incomplete resolution of some of the lines i n the spectra (Pig. 17c ). The linewidth, however, i s clearly anisotropic and there i s a strong indication that the minimum linewidth  — I  1  3200  1 —  :  — I  3300  1 —  3400  3200  3300  3400  3200  3300  H( gauss) a) POWDER  b) CRYSTAL H  Figure 17 - ESR spectra obtained i n F  II C . A X I S 4  treated SrCl  c)CRYSTAL H  II  c  2  AXIS  3400  - 92 i s seen along the z axis (figure 17b, spectrum EFGH) while the maximum linewidth i s seen along the four-fold axes 1 to z which w i l l be called xy. The linewidth 2cr = 7 + 2 gauss i s estimated as, for example, the interval GG' and the linewidth 2or as AA' or DD' ( a l l on figure 17b ). xy Since the lines are not normal derivative peaks the values obtained for the hyperfine components w i l l depend on where measurements are taken on the "powder-like" peaks* In measuring the hyperfine component A^ using the (powder type) peaks ABCD i n figure 17c one gets a maximum value of 92 gauss when measuring from the tops of the low f i e l d peaks to the bottom of the high f i e l d peaks. In figure 17b the peaks appear to be an intermediate between "powder peaks" and "single crystal peaks" i n character. The value of A , however, shows that the peaks are y close to "powder" peaks since a reasonable value of A was obtained by y taking the hyperfine lines as the tops of A and B and the bottoms of C and D but not by taking those lines to be at the points where the spectrum returns to the baseline (or half-way between, e.g. A and A' ). The component of the g-tensor g does not change, within experimental error, using y any of the above methods for determining A . y The relative intensities of the lines also indicate powder character. The intensities of the lines i n a single crystal should be equal but this i s not the case. The inner lines i n each of the four-line groups are generally more intense than the outer ones. The intensity ratio of the different groups of lines was found to be roughly consistent with the expected abundanceasshown i n table VI. For example, i n figure 17b peak D should be twice asintense as peak H, the ratio calculated from (peak height times linewidth) i s 2.3 to 1; and i n figure 17c the prominence of the peaks IJKL i s roughly consistent with the abundance of four times  - 93 greater than that of ABCD or EFGH. b) Discussion It i s necessary to know the crystal structure of the l a t t i c e one i s dealing with when attempting to construct a model of the defect under study. Strontium chloride has a fluorite l a t t i c e , with a lattice constant of 6.977 X at 26°C. Each anion i s surrounded tetrahedrally by four cations at a distance of 3*021 X and octehedrally by six anions at a distance only 16$ greater, 3.488 %. The anisotropy as shown by the spectra i n figure 17 show clearly that the chlorine atom responsible for the ESR signal oannot be i n the highly symmetrical environment of a normal anion site i . e . a tetrahedral hole of the cation sub-lattice. Neither can i t be i n an octahedral hole, which would otherwise be an obvious place for an i n t e r s t i t i a l atom. Both of these positions would give i s o tropic interaction with the neighbouring anions and cations. Any d i s placement whatever of the Cl atom from an anion site must bring i t within a distance less than 3.488 % of at least one Cl  ion. It i s thus surpris-  ing that the ESR spectra show no evidence of a strong covalent interaction with one or more neighbouring anions leading to the formation of such structures as 01^ which are well-known i n alkali chlorides (36,39) i . e . there i s no indication that the spectra might be basically 7-line like the Vg-center. Estimates of the probable bond length in a free Cl^ ion range (93,94,95) from 2.16 to 2.70 2 and the molecule-ion forms readily in KC1 i n which the separation of anion sites i s 4.450 X. One i s therefore led to propose the presence of an anion vacancy adjacent to the neutral Cl atom. This would permit the atom to  - 94 move from an anion site i n a direction which increases i t s distance from a l l other anions i . e . towards the vacancy along a four-fold axis. This direction i s identified as the z axis, along which both the hyperfine i n teraction and the linewidth have their minimum values. The linewidth i n colour centers has been attributed mainly to unresolved hyperfine interaction between the center and i t s nearest neighbours (36,39,70). In the SrClg defect the nearest neighbours to the Cl atom are two Sr atoms which have no nuclear spin (there i s a 7# abundance of Sr  with 1= 5/2 but i t i s of suoh low concentration that i t s effect may  not be seen). The next nearest neighbours, the C l " ions, must therefore by their interaction with the Cl atom cause the observable linewidths. It i s known that the closer the ions are to the defect the stronger is the interaction and therefore the larger the linewidth; on this basis i t i s possible to correlate the linewidth with the distance between the defect and the surrounding neighbouring anions. The closest anions to the defect l i e , relative to the anion site from which the Cl atom i s displaced, along the four-fold axes perpendicular to z which may be designated (x + y) since they bisect the angles between the two-fold axes x and y. The maximum linewidth , as seen i n figure 17b designated ABCD, was observed with the magnetic f i e l d oriented along the direction (x + y). The minimum l i n e width observed with H parallel to the z axis i s of course consistent with the absence of an anion neighbour i n that direction. On this basis the linewidth of ABCD in figure 17c which i s due to the anions should be equal to some value between 2cr and 2<j- since the anions are closer than for the xy z interaction causing 2(j but farther away from the Cl atom than the i n t e r z acting Cl ions causing the linewidth 2o~»  - 95 i ) Model of the defect We must now consider two possibilities, both illustrated in figure 18. ( In the figure the solid ciroles represent species lying in that plane. For each dotted c i r c l e there are two ions, in planes lying + a /2/2~ above and below the plane of the diagram. X marks vacant anion Q  sites.) The Cl atom may l i e exactly half-way between two anion sites and hence at the mid-point of the line between two neighbouring cations ( f i g ure 18b ), or i t may l i e in an intermediate position as shown in figure 18a. In either case, the 3p orbital of the Cl atom oriented along the cation-cation line should be strongly stabilized electrostatically and hence doubly occupied in the ground state. Thus the two-fold axes x and y are not equivalent i n respect of the environment of the displaced atom. Since the high value of  and the close correspondence of g^ to the free-elec-  tron value (g 2.0023) indicate that the h a l f - f i l l e d state i s p , the line S3  x  of the two nearest-neighbour cations i s identified as the y axis. The relative stabilization of p-states in the other two principal directions i s most easily visualized f i r s t in the situation of f i g ure 18b. P  i s oriented towards two tetrahedral holes of the cation sub-  z  l a t t i c e , marked X i n the diagram, which are the normal anion sites of the SrCl  2  l a t t i c e . P^ i s oriented towards the normally unoccupied octahedral  holes, and should be somewhat less stabilized than p in conformity with z the experimental indication that p^ i s the h a l f - f i l l e d state. The crystal f i e l d splitting of the p-orbitals i s presented below. Figure 19 gives an 5 , energy level diagram for the 3p  case vit also includes the 3s orbital  the position of which i s uncertain).  - 96 Figure 18 - Models of the defect i n S r C l  - 97 Figure 19  5  *x  3  *z  ##  y 3s 3 p  It i s useful to think of the defect as a molecule composed of the Cl atom and two neighbouring cations, with i t s symmetry modified by the remainder of the environment, particularly the cations which determine the positions of the octahedral and tetrahedral holes referred to above. (Evidence on the involvement of orbitals on the nearest cations i s assessed in detail below.) On this basis, the arrangement of figure 18a may be thought of as a  molecule, and i t i s useful to compare my r e -  sults with, for example, the study by Cole (96) of C10 molecule produced 2  as a defect in KCIO^. The coordinate system I use corresponds to that used by Morton (97) and later by Carrington andMcLachlan (60) in discussing ESR spectra of 0^ molecules; Morton changed the assignment of coordinates used by Cole for ClO^ on the basis that the unpaired electron was probably i n an orbital of b^. aymmetry rather than a^ as assumed by Cole. My assignment of the unpaired electron to the p^ orbital, perpendicular to the plane of figure 18a, likewise represents b^ symmetry. If the geometry of the defect i s as shown in figure 18b, i t s symmetry i s nominally higher; i f i e d to D  21l  i f one thinks of three atoms only, mod-  by the rest of the environment. This symmetry has three two-  fold axes, while  n  a  s  only one; but the properties of a  molecule  -  98  -  may be very different in these three directions, while for C^ the orbital v  electronic properties often approximate to axial symmetry about the x axis, 2 as implied by the suggestion of sp  hybridization in the yz plane in f i g -  ure 18a e My results thus seem most consistent with the model of f i g 18b, with the degeneracy of the 3p orbitals completely l i f t e d (energy i n creasing in the order p , p , p as shown i n figure 19 )» the 3s and 3p y z x y doubly occupied, and the spin density mostly i n p^ but partly in p » This z  model i s developed quantitatively below i n terms of the components of the g and A tensors* The observed spectroscopic splitting factors were found to deviate from the free electron value. Table VII gives the measured components of the g-tensor referred to the x, y, z axes of the defect , see figure 18a and 18b. The g^ component i s very close to the free electron value g = 2.0023* The components g and g show a rather large positive o y z deviation from g , indicating that the resonance arises from an electron deficiency i n the halide ion noble gas configuration. The calculation by first-order perturbation of the g-shifts brought about by spin-orbit coupling between p-orbitals whose degeneracy has been l i f t e d by a crystal f i e l d i s well known (58,60,92) and i s discussed briefly in section B(d)(ii) of the introduction of this thesis. Using the model of the defect previously discussed, the p orbitals s p l i t by the 5 crystal f i e l d for the 3p configuration of the Cl atom are shown in f i g 5 6 ure 19* The 3p configuration can also be expressed as a hole i n the 3p complete shell and henoe i t can be treated as a positively charged electron  - 99 in a 3p* configuration. Using this terminology the energy level diagram of figure 19 would have to be inverted, as shown in figure 20 below, figure 20  3 p  T  y  >  A  A zz 3  The relative energies indicated for the p^, p  y  "x  and p^ orbitals correspond  to the experimentally determined g-shifts, as discussed below. The transitions shown i n figure 20 are responsible for the g-shifts since the ground state p^ i s mized with the upper states p ator and the L  g  z  and p  y  by the L  y  oper-  operator respectively. If the energy differences between  the states connected by the spin-orbit interaction are A and A as xy xz shown i n figure 20 and the spin-orbit ooupling constant i s X , the r e sulting shifts from the free electron g value g  Q  are (from equations  36 of the introduction)}  2 where  %  -  i s the amount of p^-character of the unpaired electron.The  values of the g-shifts were found experimentally to be 0.0, 0.0171 and 0.0139 for Ag , Ag  and Ag  respectively. The g-shifts are inversely  proportional to the energy of the transition from the ground state to the higher energy states. The values of the g-shifts thus indicate that the p orbitals are i n the order shown i n figure 20. Using the experi-  - 100 -  mental values for the g-shifts, with \  - -586 cm  the doublet separation of the (ns) (np)  (which is 2/3 of  state and has a negative sign  for the positive hole situation), and UBing a  » 1,0, the maximum upper  limit of the transition energies i s calculated. The calculations show that A = 10.5 eV, and that p l i e s 2 eV below p , i . e . A _ = 8.5 eV, xy z y xz The following section on the interpretation of the hyperfine tensor shows that the state of the unpaired electron (or hole) has nearly completely p-character of which at least 80$ i s due to the p^ orbital. Therefore 2 the value of  has a minimum value of 0.8 which gives a minimum value  for A ^ and A^^ of 8.4 eV and 6.8 eV respectively. The true energy values 2 of the transitions are somewhere between these values for o =» 1. These x figures may be compared with a crystal f i e l d splitting of about 15 eV found i n Gardner's work,(92). Although the splitting i s somewhat smaller (~10 eV) i n the S r C l  2  defect i t i s s t i l l larger than the familiar range  of d-orbital splittings. i i i ) Interpretation of the hyperfine tensor The terminology used below to relate the hyperfine i n t e r action to the fractional s and p character has been explained in section B(d)(ii) of the introduction of this thesis. A , A , and A are used for x y z the principal components of the hyperfine tensor; and A ^  s q  and A^, etc.  for the same data analyzed into isotropic and anisotropic contributions. The symbols ^  and (p were introduced in equations 9 and 15 of the introd-  uction of this thesis to relate these to properties of atomic orbitals and are defined as  2 = ( /D(W )^(o) A  <P =  3  {A /i)(2/5)4;p> 1  3s  n  M  - 101 in  which  for  a C l  5  (.(i./l)  nucleus  5  =  2.76  x  10~ gau33 ?4  cm . 5  The s character of the state of the unpaired electron i s probably chiefly 3s, and for this Whiffen's value (98) of  ^' (o) for Cl ^ /2  3  5  24 -3 which i s 10.64 a.u. or 71.8 x 10 cm  i s used. This i s slightly higher  than the value of 69 x 10 cm~ used by Kanzig and Woodruff (39), and 24  3  gives a maximum valve for  • To determine the greatest possible effect  of any 4s admixture, I have also made some calculations using a minimum value of  calculated from Kanzig and Woodruff's value ^ ( o ) ^  - 5.72 x  10 cm~ . For <^r^ ^>I followed Kanzig et a l . (36,39,) and Gardner (92) 24  3  3  i n using Sternheimer's value (99) of 55.6 x 10 cm"" . Whiffen (98) and 24  3  Morton (97) (who I w i l l c a l l W and M ) give a somewhat smaller value of Oil  6.71 a.u. which i s 45.3 x 10 cm . Adoption of this value would not s i g nifigantly alter any conclusions which were drawn from the calculations. However both values were used for comparison i n table VIII(a) and (b). With the above selection of numerical values one has: = 1662 gauss ( 3s, Whiffen )  s&  "^min ~  1  3  2  S  a u s a  (  4 8  »  S )  K a n z i  (P =61.5 gauss ( Sternheimer ) max ^min  =  5  0  ,  0g  a  U  8  S  ^  W  ^  M  ^  Using the results of the discussion on the occupancy of the p orbitals i n section B(d)(ii) of the, introduction, I assume that the state of the unpaired electron can be represented by contributions from an s-orbital, p , p > and p^, the fraction of the spin density i n the %  y  2 2 2 2 four states being 8 i n s, a i n p , a i n p , and a i n p . However, x -x y y' z *z r  - 102 VIII  Table  Character of the state of the unpaired electron. &) Using s, 3p » and 3p  orbitals  3s o r b ^ a j A z  positive 12.0  9.7  8.0  69.4  83.9  84.0  60.1  61.8  76.4  77.8  29.4  29.5  15.3  15.3  25.5  26.2  13.9  14.2  .037  .030  .037  .030  »464  .378  .464  .378  63.4  52.4  52.4  73.2  71.2  57.6  56.5  1.1  6"  (*)  69.3  (*)  (P/$ assumed  r  14.4  1.3  ^  >  .7  (*)  2  ne«a.1fdve. .8  2  e  4s o r b i t a l  (P from e z p t ' l 6 3 . 5 results  bj Using s, 3p • and 3p orbitals z y 3s orbital A z  positive  4s orbital  negative  positive  negative  5.1  4.2  2.5  2.1  40.8  35.5  24.8  21.1  72.1  72.8  64.2  64.5  45.0  49.0  49.5  52.0  22.8  23.0  33.3  33.4  14.2  15.5  25.7  26.9  .037 assumed  .030  .037  .030  .464  .378  .464  .378  (P from e z p t ' l 35.1 results  34.8  56.1  55.8  56.2  51.7  72.7  69.3  e  (»  2  s\ s 2  y  w  00.  (P/J£)  - 103 as pointed out i n the introduction one can only determine experimentally the difference of the occupation of two of the p-orbitals from that of the third orthogonal one. Therefore aesuming that p  i s least occupied  and using equations 27, 28, 29 and 30 of section B(d) of the introduction the expressions for the components of the hyperfine tensor are: A - B ^ .+ (2<S - S\ ) X X z 2  A  (P  2  \(2&\-6b  Z  Z  (P X  from whioh the quantities relating to orbital occupancy are calculated as  P ^ - (1/3) ( 2  A + A. + A x  y  )  a  S JP = (1/3)( A - A ) 6 (P .-- (l/3)( A - A 2  x  y  Z  Z  z  y  )•  The above expressions may be converted to the corresponding expressions for p  ( i f p^ i s the least occupied state ) by interchanging subscripts  z  y and z on both the A's and the 6~'s. Since each of the above expressions reppresents a positive quantity using the experimentally determined values of A , A , and A X y z from table VII of the results section, one can readily see that A^ must be positive, A must be negative and A ity, A  y  i s indeterminate. I f u l l general-  i s also indeterminate because i f the expressions are set up with  p as the least occupied state instead of p then A must be positive. z y y Table VIII (a) and (b) give calculated values of P , <$ , and x  - 104 ' r for the 3s and 4s cases. The evaluation of these parameters z or y r  requires a value for  (P/^§ • The value of..(P/^S  t a l i for a 3a Orbital i t i s and for a 4s orbital  (P^J 2^  (p^J Jf,^  varies with the s o r b i -  - 0.037 and { ^ / . ^  » 0.464 and (P^J 3^  = 0-030  - 0.378.  Table VIII(a) shows the calculated results assuming that p i s the least occupied p orbital and using A positive. A negative, and x  y  A both as positive and negative ( which I w i l l c a l l case 1 ). z  Table VIIl(b) has p as the least occupied orbital and A^ z  positive, A positive, and A positive and negative ( case 2 ). Both y z tables show that s character i s much increased i f i t i s assumed to be 4s, especially i n oase 2* The two possible values for the s character as seen from table VIIl(b) are then 24.8$ and 40,8$. I consider these to be the least l i k e l y of a l l possible values and therefore indicate with some reservations that the 3s orbitals are mainly:involved. Table IX summarizes the results obtained by using the 3s and 3p orbitals and indicates that 2 the unpaired electron i s of almost pure p character. The value of 6 and 6" or <5 are the minimum value of the amount of p-character since (using z  equations 27 of2 section B(d)(ii) of the introduction ) for case 1: S - « - a x . x y and 2  2  S - a - « z z y 2  2  2  u  2 2 2 Since a i s either positive or zero 6 and 8 are a minimum value of the y x z p-character. It i s d i f f i c u l t to choose between the values obtained using 3s orbitals with A either positive or negative just from the numerioal z  data. I chose, as the most l i k e l y set, the values given i n the right hand column of table IX, which support the model of figure 18b and the indica-  - 105 tiona from the g-shifts that p l i e s lowest of the p-orbitals i n a posiy tion closest to the 3s orbital ( as shown i n figure 19). It i s reasonable that the p orbital i s least occupied by the unpaired electron and i t i s y possible that the 3s. and the 3p form a pair of sp hybrid orbitals. y It should be noted, however, that the calculation using 4s 2 2 give a very much lower value of 6 /P » the two possible values being Z  1.78 and 1.43. Thus the possibility that the orbitals in the yz plane are 2 sp  hybrids, i n conformity with figure 18a, cannot be entirely ruled out,  although a large contribution from the 4s orbital seems unlikely. It i s not possible to determine unequivocally from the a v a i l able data whether there i s substantial derealization on the two nearest cations* The values of (P from the experimental results ( bottom of table IX ) are for most calculations within the range of theoretical estimates ( 61.5 to 50.00 gauss ) suggesting localization of the electron; but table X shows that our values for the components of the A tensor resemble those found i n ClOg more closely than those found by Gardner for a Cl atom adsorbed on SiOg. Those latter values seem a l i t t l e high, both r e l ative to theoretioal values and relative to the,double values of the hyperfine constants found i n the work of Kflnzig et a l on C l " ions (39),which as Gardner points out, should be comparable. Evidently the range of uncertainty i n these comparisons i s such that no definite conclusion can be drawn on this point. In general, however, the similarity between the v a l ues of g and A components for the defect studied here and for ClOg i s striking. In table X the subscripts a, b, c and d are defined as follows:  - 106 Table IX Character of the state of the unpaired electron from hyperfine splitting and known constants of 3s and 3p orbitals Range  in  Values considered  the  calculations  most l i k e l y (p  z  ,  with A negative) 2 2  0.8$ to  s-character, 10 B p-character, 10 (d~ + 6~y 2  10%  J  2  o  r  5.1$  0.8$  94.9$ to 99.2$  99.2$  69.3$ to 83.9$  83.9$  15.3$ to 33.3$  15.3$  2  10 £ 2  2  or 10 6 2  («S or^)/p 2  2  4.5 35.1  (P-( A /l)(2/5)<r;5> /  to 22.6  19.1  to 63.5 G  52.4 G  2  I  Table X g and A tensor components for Cl atoms i n various environments Component  Cl defect i n SrCl. (this work)  CIO. i n KC10 97N ( C o l.96 e * . Horton ) af  g  x  2.0028  2.0036  g  v  2.0194  2.0183  2.0162  2.0088  12  15  y  g A A  A  z i  a  o  2  i " x " iso A  y  a  (0)  - e ^  A  A  y- iso A  ( G )  (  0  8  )  A« = A - A. (G) z z iso  *  0  '  a  5  2  -28  "  a  -27  5  l  b  b  A  2.003° 2.012 ± 0.004 11  58 ° &  E  Cl adsorbed on _. 92, SiO., (Gardner*^)  184 -  9  C  ,  2  °  d  ,  -92 ° '  d  d  - 107 Negative sign assumed for A and A^. y  ^ Assignment of co-ordinate system i s that of Morton, not that f i r s t suggested by Cole* c I have changed the designation of the axis of symmetry from z i n Gardner's paper to x here, to f a c i l i t a t e comparison with the other data* Calculated from Gardner's  B  = 2.58 x 10  cm  = 276 gauss,  assuming axial symmetry. iv) Reaction mechanism It should be recalled that the defects giving rise to the ESR signal are known to l i e close to the reaction interface; though e v i dently not near enough to allow any interaction with the large magnetic moments of the fluorine nuclei. One may envisage the reaction as i n v o l ving the successive removal of cations ( to the growing SrPg crystals ) and discharged anions ( to the gas phase as C l  2  or C1F ). The removal of  each anion provides the necessary situation for displacement of an anion i n the immediately subjacent layer. While the position indicated i n figure 18b has been i d e n t i fied as the most probable one to which my quantitative data refers, i t should be recalled those data refer to the maximum value of the hyperfine interaction, and that the "powder" character of the lines i n the single crystal spectra suggested the existence of a continuous range of values below this maximum. The reaction mentioned above must occur by the f o l lowing reaction mechanism to assure the production of the defect; the Oxidized anions are removed to the gas phase somewhat ahead of the migration of the cations to the growing SrF ' crystals. Thus anion vacancies  - 108 are created near to the reaction interface which enables the defect to be formed. The displacement of the underlying anions towards the vacancy i s aided by the discharge of th© anion which provides the unpaired electron. In turn, the neutralization of the anions loosens the attachment of surrounding cations so that they can start to move towards their ultimate positions in the SrF,, l a t t i c e . The Cl atom approaches the reaction interface ( i . e . the anion vacancy ) along the z axis, which may thus be thought of as a " r e a c t i on coordinate" and may pass through a range of positions of the type illustrated i n figure 18a, exhibiting i t s maximum hyperfine splitting when i t passes between two cations in the special position of figure 18b. It i s possible that in this condition the Cl atom i s at the top of an energy barrier for migration; and in that case the defect I  have studied  i n this work may be called a transition complex. The spectra are thus consistent with the reaction soheme: l/2F (g) 2  +  Cl"(subsurface)  Cl(subsurface)  •  Sr (surface) 2+  Cl ( i n t e r s t i t i a l ) - —  —F~(SrF ) 2  +  Cl(subsurface)  •> C l ( i n t e r s t i t i a l ) (defect under study) >> S r ( S r F j  (disorientation of 2-fold axes) > l/2Cl (g) (determines 4-fold axis for next layer of defects) 2 +  2  - 109 A, Computations a) Fortran A computer programme C  C 1 2 3 A 5 6  THE DEFECT IN ?  TREATED KCl AND NaCl  2  DIMENSION ANS(5000),1(100) DIMENSION HS(20), DER(lOOO),ZN(l000) LOGICAL ITR,JTR EQUIVALENCE(Nil,NI) CALL PLOTS FORMATS FORMAT (8F10.0) F0}MAT(1X,3HI=I3,6HIHI=H,6HH(I)=F10.3,12HARRAY(IHI)=F6.2) FORMAT(IX, 13F10.2) FORMAT(1X,2I10) FORMAT(I6,F10.3 ) FORMAT(1615)  C  INPUT  STATEMENTS  IMAX: THE OVERALL HISTOGRAM SUBDIVISION NI  : NUMBER OF ANGLE CALCULATIONS  NP  : DIVISIONS OF £ NORMAL CURVE  A  :  g  o  B  :(Ai )  C n  :(A* ) - (A* ) 2  E  t gi-  F  :(A^)  G  : (A#)  2  2  2  2  2  2  2  2  g*  2  2  - (A? ) 4  2  CAI(K): h\)/fl  SIG : <T ON NORMAL CURVE DXX : DIVISION OF NORMAL CURVE AS A FRACTION OF <T 2095 WRITE(6,2096) 2096 FORMAT(1H1) READ(5,6)IMAX,NI,NP IF(IMAX.EQ.O) GO TO 2005 WRITE(6,7)IMAX,NI,NP 7 FORMAT(35H NUMBER OF HISTOGRAM SUBDIVISIONS =,I5,5X,31H NUMBER OF 1ANGLE CALCULATIONS =,I5,5X,/A1H NUMBER OF SUBDIVISIONS OF NORMAL 1CURVE = ,15)  APPENDIX  - 110 -  8  9 C 11  READ(5,l)A,B,C,D,E,F,G,CAY WRITE(6,8)A,B,C,D,E,F,G,CAY FORMAT(3H A=,F12.5,5X,3H B=,F12.5,5X,3H C=,F12.5,5X,3H D=,F12.5,/ 15X,3H S=,F12.5,5X,3H F=,F12.5,5X,3H G=,F12.5,5X,3H K=,F12.5) READ(5,1)SIG, DXX WRITE(6,9)SIG,DXX FORMAT(8H SIGMA=,F12.5,5X,5H DX=,F12.5) PI1^3.U159265/(2.0*FL0AT(NI)) I N I T I A L I Z E A R R A Y DO 11 I=1,§000 ANS(I)=0. THE =0. DX=0.5*DXX**2  Y(l)=l. DO 210 1=1,NP 210 Y(I+1)=EXP(-DX»FL0AT(I*I)) WRITE(6,3)(Y(I),I=1,NP) WRITE(6,3) HCC=CAY/SQRT(D) Z2=3.0»(A»SQRT((B+C)/D)+SQRT(F+G)+1.5*SIG) Z=2.0*Z2 FACT=FL0AT(IMAX)/Z HM=HCC-Z2 HMAX=HCC+Z2 DS=DXX*SIG*FACT WRITE(6,12) HM,HCC,HMAX 12 FORMAT(13H H MINIMUM*,F12.5,5X,12H H CENTER=,F12.5,5X,12H H 1MAXIMUM=,F12.5/) WRITE(6,13) 1 3 = FORMAT(68H CENTER(H) DPRIM(H) DSECND(H) CENTR(M) DPRIM(MOD) lDSN(MOD) DX(MOD)) DO 1000 IT=1,NII THE= THE + PI1 SN=SIN(THE) SINTHE=SN**2 COSTHE= 1. - SINTHE. UU=SQRT(F+G*COSTHE) TX=1./SQRT(D+E*SINTHE) TT=A*SQRT(B+C*COSTHE)*TX HH=CAY*TX HV=HH - HM T=TT*FACT U=UU*FACT HC=HV*FACT WRITE(6,3) HH,TT,UU,HC,T,U,DS ITR=.THUE. DO 20 1=0,3 AA=I HT1=U.0 - AA)*(SN) DHP=AA*T HP1=HC+DHP HP2=HC-DHP  - Ill  -  JTR=.TRUE. DO 30 J=0,3 JMAX=4 JD=1 BB=J  HT2=HT1*U.-BB)  37  AOO  44  403 50 49 40 30 20 1000  203  206 130  DHS=BB*U HS(1)=HP1+DHS HS(2)=HP2+DHS HS(3)=HP1-DHS IF (ITR) GO TO 33 IF (JTR) GO TO 3 9 HS(A)=HP2-DHS TDS=0. DO AO K=0,NP. DO 50 JJ=1,JMAX, JD L=(HS(JJ)+TDS) HH=HT2*Y(K+1) IF(L.LE.IMAX) GO TO 400 WRITE(6,A) L L=IMAX CONTINUE ANS(L)=ANS(L)+HH IF(K) 44,50,44 CONTINUE L=(HS(JJ)-TDS) IF(L.GT.O) GO TO 403 WRITE(6,4)L L=l CONTINUE ANS(L)=ANS(L)+HH CONTINUE TDS=TDS+DS CONTINUE JTR-.FALSE. CONTINUE ITR=.FALSE. CONTINUE CONTINUE FACT1=1./FACT DO 203 I=1,IMAX ZN(I)=(FLOAT(I)+0.5)"FACT1+HM CONTINUE IML = IMAX - 2 DO 206 I = 2,IM1 DER(I) F= 6.0*ANS(l+l) - ANS(l+2) - 2.0*ANS(l-l) - 3.0*ANS(l) CONTINUE WRITE(6,130) (I,ZN(I),ANS(I),DER(I),I=1,IMAX) FORMAT(3(5X,I5,3F10.2)) CALL AIiPL0T(ZN,ANS,IMAX,7H I AXIS, 7 , 8.0,7H X AXIS, 7 , 10.0) CALL PLOT(12.,0.,-3) CALL ALPL0T(ZN,DER,IMAX,7H I AXIS, 7 , 8.0,7H X AXIS, 7 , 10.0)  112 -  39 38  2005  CALL PL0T(l2.,0.,-3) GO TO 2095 JMAX=2 GO TO 37 JMAX=1  IF(JTR) GO TO 37 JD=2 JMAX=3 GO TO 37 CALL PLOTND END  b) Computed spectrum obtained for defect In KCl The spectrum obtained, using the Fortran L computer programme found in (a) of this Appendix* for the defect in KCl is shown in figure 21.  113 Figure 21 - Computed spectrum obtained for defect in KCl  A  A  r  BIBLIOGRAPHY  - 114 BIBLIOGRAPHY (1)  S. K. Deb and A. D. Yoffe, Trans. Faraday Soc.  (2)  A. K» Galwey and P. W. M. Jacobs, Proc. Roy. Soo. (London) A25A.  455 (i960); F. 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