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Optical properties of potassium iodide in the far-infrared Kembry, Kenneth Allen 1974

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O P T I C A L P R O P E R T I E S OF P O T A S S I U M I N THE  IODIDE  FAR-INFRARED  by K E N N E T H A L L E N KEMBRY B.Sc,  University of Alberta,  A THESIS SUBMITTED  1971  I N PARTIAL FULFILMENT  OF  THE R E Q U I R E M E N T S FOR THE D E G R E E OF M A S T E R OF S C I E N C E  i n the Department of Physics  We a c c e p t t h i s t h e s i s a s c o n f o r m i n g t o t h e required standard  THE U N I V E R S I T Y OF B R I T I S H S e p t e m b e r , 1974  COLUMBIA  In  presenting  this  an a d v a n c e d  degree  the  shall  I  Library  f u r t h e r agree  for  scholarly  by h i s of  this  written  thesis at  the U n i v e r s i t y  make  that  it  purposes  for  freely  permission may  representatives. thesis  in p a r t i a l  financial  is  of  Columbia,  British  by  for  gain  Columbia  shall  the  that  not  requirements I  agree  r e f e r e n c e and copying  t h e Head o f  understood  Physics  The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, Canada  of  for extensive  permission.  Department  of  available  be g r a n t e d  It  fulfilment  of  or  that  study.  this  thesis  my D e p a r t m e n t  copying  for  or  publication  be a l l o w e d w i t h o u t  my  ii  ABSTRACT Measurements and  calculations of the f a r ^ i n f r a r e d o p t i c a l  39 properties of K  I at 300, 77, and 12*K are presented.  The measurements  are mainly those of absorption i n crystals of various thicknesses.  The  calculation assumed cubic anharmonicity only, with nearest-neighbour coupling, and the input l a t t i c e ^ y n a m l c a l data were obtained from a shell^model program.  These data were generated with a wavevector  density  of 32000 points per zone, which, was s u f f i c i e n t to give Z-3 cm * resolution. The o v e r - a l l agreement between experiment and theory, i n both: the i n t e n s i t y and structure of the spectra, i s good.  The magnitude of certain calc-  ulated features i s , however, too large, indicating a need to consider nextnearest-neighbour interactions.  Evidence was also found f o r three^phonon  damping, both beyond the two-phonon l i m i t at a l l temperatures and at by 300°K.  From these measurements i t was possible to calculate portions  of the three-phonon damping spectra, which were found to be reasonable. The higher-phonon effects at 300°K d i d not seem to be noticeably more pronounced  than those found i n the much harder Li'F, and arguments  presented to understand t h i s .  are  F i n a l l y , the isotope-induced one-~phonon  processes which occur i n natural KI were calculated.  These are showji to be  small away from the resonance frequency v, and not to be the major damping mechanism at v  e  at low, temperatures^ i n contrast to L i f .  iii  TABLE OF CONTENTS.  Page  Abstract Table  o f Contents  i  L i s t o f Tables  .  i  . . . . v'»»'«  INTRODUCTION  1.1  General  1.2  Motivation f o r the Thesis  IT3  Outline o f the Thesis  Introduction  1 ••••  7  9  EXPERIMENTAL TECHNIQUE  II. 1  The Apparatus  II.2  Sample Preparation  II. 3  Analysis o f Experimental  Section I I I  i  • v'i  Acknowledgements  Section I I  i  1 V  List o f Figures  Section I  i  .. 1 0 11 Data  13  THEORY  111.1  General  Introduction  111.2  D a m p i n g a n d W a v e n u m b e r S h i f t o f t h e TO resonance  16  17  III.2.2  Isotope^Induced  Damping  III. 3  Potential Energies  23  111.4  The S h e l l ^ M o d e l  25  111.5  The C a l c u l a t i o n o f t h e Complex Phase S h i f t  32  111.6  The O p t i c a l P r o p e r t i e s  33  Calculation  22  IV  Page Section IV  F U R T H E R M E A S U R E M E N T S AND C A L C U L A T I O N S OF THE FAR-INFRARED ANHARMONIC O P T I C A L P R O P E R T I E S OF K I B E T W E E N 12 AND  300°K  IV. 1  Introduction  37  IV.2  D a m p i n g a n d F r e q u e n c y S h i f t o f t h e TO r e s o n a n c e .  40  IV.3  Optical Properties  51  IV.4  Bibliography  A. T h e D i e l e c t r i c C o n s t a n t  ......  51  B. T h e A b s o r p t i o n  . ....  54  Coefficient  C. E x p e r i m e n t a l  54  D. R e f r a c t i v e I n d e x a n d R e f l e c t i v i t y  58  Discussion A. T h r e e - P h o n o n D a m p i n g  61  B. T w o - P h o n o n D a m p i n g S t r u c t u r e  66 68  V  L I S T OF  TABLES  Table  Page  II. 1  Input Data f o r the Calculations  15  III. l  S h e l l Model Parameters  28  111.2  Input Data f o r t h e S h e l l Model Program  111.3  Output o f t h e S h e l l Model Program  31  IV. 1  Input Data f o r the Calculations  42  IV.2  Various Reported Values of r ( 0 , j ; v )  IV.3  E x p e c t e d Maxima i n Two-Symmetry-Phonon  0  and Feature Assignments  229  47  0  Combinations 48  Vi  L I S T OF F I G U R E S Figure I. 1  Page Schematic Representaion  o f Phonon D i s p e r s i o n i n t h e  ^100) D i r e c t i o n o f a C u b i c  Lattice with Polar  Diatomic  Basis IT2  A P o r t i o n o f the KI L a t t i c e  .  II. 1  T a i l - p i e c e f o r Mounting Samples f o r Low-Temperature  .  2  .  3  Experiments IV.1  Present  12  C a l c u l a t i o n o f t h e Two-Phonon Damping o f t h e  TO R e s o n a n c e o f K seperately  I a t 3 0 0 K, w i t h b o t h p r o c e s s e s  ( s o l i d l i n e s ) and t h e f e a t u r e s  shown  labeled;  together with the c a l c u l a t i o n of Ref. 2 with both  processes  combined (dashed l i n e ) . . * IV.2  43  Calculated frequency-dependent frequency  s h i g t o f t h e TO  39 r e s o n a n c e o f K' I r e s u l t i n g f r o m t w o - p h o n o n d a m p i n g , a t 3 0 0 a n d 12°K IV.3  44  C a l c u l a t e d t w o - p h o n o n d a m p i n g o f t h e TO r e s o n a n c e o f 39 K  o I a t 12 K v i a d i f f e r e n c e a n d s u m m a t i o n  drawn s e p e r a t e l y ; t o g e t h e r w i t h t h e  processes,  isotope-induced  one-phonon damping o f t h e resonance i n n a t u r a l KI IV..4  Frequency d i s p e r s i o n curves d i r e c t i o n s , generated  IV.5  45  o f KI along major symmetry  b y t h e s h e l l m o d e l , f o r KI a t 95°K..50 39 C a l c u l a t e d r e a l d i e l e c t r i c c o n s t a n t s o f K I a t 300 a n d 12°K 52  vii  Figure  Page 39  IV.6  Calculated imaginary d i e l e c t r i c constants of K at  IV.7  I  3 0 0 a n d 12°K  55  P r e s e n t e x p e r i m e n t a l and c a l c u l a t e d v a l u e s o f t h e 39 « a b s o r p t i o n c o e f f i c i e n t o f K I a t 300, 77,and 12*K? together with the room-temperature experimental f r o m R e f . 2. 140  IV.8  The  a n d 180 cm  data  experimental data obtained between have been omitted f o r c l a r i t y  56  P r e s e n t e x p e r i m e n t a l and c a l c u l a t e d v a l u e s o f t h e 39 a b s o r p t i o n c o e f f i c i e n t o f K I a f 3 0 0 a n d 7f/*?K i n t h e range o m i t t e d f o r c l a r i t y f r o m F i g . IV.7  ....57  39 IV.9  Calculated refractive indices of K 12°K,  I a t 300  together with the room-temperature  and  experimental  data from Ref. 2  59 39  IV.10  C a l c u l a t e d 300 K r e f l e c t i v i t y a n d p h a s e a n g l e o f K Together with the present experimental r e f l e c t i v i t y , e x p e r i m e n t a l d a t a f r o m R e f s . 2 a n d 14 a r e p r e s e n t e d  IV.11  I. the 60  P o r t i o n s o f the three-phonon (and h i g h e r ) damping o f t h e TO r e s o n a n c e o f K  I a t 3 0 0 , 7 7 , a n d 12°K.  The  methods o f o b t a i n i n g these curves are e x p l a i n e d i n the t e x t . .  62  ACKNOWLEDGEMENTS  F i r s t I w o u l d l i k e t o e x p r e s s my g r e a t t h a n k s t o my s u p e r v i s o r Dr. J . E . E l d r i d g e f o r h i s p a t i e n c e and a s s i s t a n c e i n t h e work and e s p e c i a l l y h i s e f f o r t s i n assisting^me t o complete t h e t h e s i s  off-campus.  I w o u l d a l s o l i k e t o t h a n k my w i f e N a n c y f o r h e r p a t i e n c e a n d m o r a l support and Mr. J.A.B. B e a i r s t o f o r h i s f r i e n d s h i p and i n v a l u a b l e discussions. T h i s work was s u p p o r t e d b y t h e N a t i o n a l R e s e a r c h C o u n c i l o f Canada Grant number 67-5653. the  I also wish t o acknowledge personal support i n  form o f a one y e a r N a t i o n a l R e s e a r c h C o u n c i l B u r s a r y i n t h e second  year.  1  SECTION I INTRODUCTION I.1  General  Introduction.  Far i n f r a r e d r a d i a t i o n i n c i d e n t on an i o n i c c r y s t a l  such  as K I , i n t e r a c t s w i t h t h e e l e c t r o n i c d i p o l e f o r m e d b y t h e K I resulting i n absorption o f the radiation. may b e u n d e r s t o o d  The n a t u r e  molecule,  of this  absorption  by c o n s i d e r i n g t h e phonon d i s p e r s i o n i n t h e l a t t i c e .  F i g u r e 1.1 s h o w s t h e p h o n o n d i s p e r s i o n i n t h e [ 1 0 0 ] d i r e c t i o n f o r a t y p i c a l cubic l a t t i c e with a polar diatomic basis.  T h e momentum o f t h e p h o n o n  a t t h e B r i l l o u i n Zone b o u n d a r y , X, i s g i v e n b y :  where X i s t h e wavelength o f t h e phonon, T i s t h e wavevector o f t h e p h o n o n , a n d r ; i s t h e l a t t i c e s p a c i n g a s shown i n F i g u r e 1.2. 0  equals  S i n c e rrj  3.526A f o r K I , t h e momentum o f t h e p h o n o n a t t h e z o n e b o u n d a r y h  is approximately"  8 10  -1 cuT , whereas t h e f a r i n f r a r e d r a d i a t i o n h a s 2 - 1  a momentum g i v e n b y e q u a t i o n  ( J - l ) °f a p p r o x i m a t e l y  h x 10  cm  T h u s , c o m p a r e d w i t h t h e momentum o f t h e p h o n o n s , t h e i n c i d e n t r a d i a t i o n has  e f f e c t i v e l y z e r o momentum.  Therefore,  i f we c o n s i d e r t h e i n t e r a c t i o n  o f th.e i n c i d e n t r a d i a t i o n w i t h , a n u n d a m p e d l a t t i c e ( a n d n e g l e c t i n g  other  t h a n t h e f i r s t o r d e r e l e c t r o n i c d i p o l e moment) i t may b e s e e n t h a t i n o r d e r f o r momentum t o b e c o n s e r v e d ,  t h e i n t e r a c t i o n c a n o c c u r o n l y at' t h e zone  c e n t e r n . ' ^ r :twhere. fJT=o. A l s o , s i n c e t h e i n c i d e n t r a d i a t i o n i s t r a n s v e r s e i n n a t u r e , o n l y t h e t r a n s v e r s e o p t i c ( T 0 1 p h o n o n may' b e e x c i t e d .  Therefore,  i n a n undamped l a t t i c e w i t h o n l y a f i r s t o r d e r e l e c t r o n i c d i p o l e moment one w o u l d f i n d r e s o n a n t  a b s o r p t i o n o n l y a t t h e TO (XTP) w a v e n u m b e r ,  F i g u r e 1.1  Schematic R e p r e s e n t a t i o n o f the Phonon D i s p e r s i o n i n t h e ^100)  direction of a Cubic Lattice with a  Polar Diatomic Basis.  3  F i g u r e .1.2  A p o r t i o n o f the KI l a t t i c e  (three faces  shown)  V Q , where t h e wavemumber, v., o f a wave i s g i v e n by: (.1.2) A b s o r p t i o n a t w a v e n u m b e r s o t h e r t h a n t h a t o f t h e r e s o n a n t TO m o d e may  come a b o u t d i r e c t l y t h r o u g h s e c o n d o r h i g h e r o r d e r t e r m s i n t h e  e l e c t r o n i c d i p o l e moment, o r t h r o u g h c o u p l i n g o f phonon modes t h r o u g h t h i r d o r h i g h e r anharmonic terms i n the p o t e n t i a l energy.  In the  case  of i o n i c c r y s t a l s the off-resonance absorption i s thought to occur mainly through the anharmonic mechanism, s i n c e the observed a b s o r p t i o n i s f o u n d t o be much h i g h e r t h a n i n h o m o p o l a r c r y s t a l s s u c h as d i a m o n d and s i l i c o n , w h e r e a b s o r p t i o n i s due t o t h e s e c o n d - o r d e r d i p o l e moment only.  I n t h i s w o r k , t h e r e f o r e , we r e s t r i c t o u r s e l v e s t o t h e c a s e o f a  damped l a t t i c e w i t h f i r s t - o r d e r d i p o l e moment o n l y . I n s u c h a l a t t i c e i t i s p o s s i b l e f o r t h e TO m o d e t o b e  excited  by o f f - r e s o n a n c e r a d i a t i o n and t h e n d e c a y t o two p h o n o n s , due t o t h e coupling of i o n i c motions process.  i n the l a t t i c e .  T h i s i s known as a two-phonon  The i n i t i a l and f i n a l s t a t e s o f t h e s y s t e m m u s t  e n e r g y and momentum;  conserve  t h u s t h e two p h o n o n s i n v o l v e d i n t h e p r o c e s s must  h a v e t o t a l energy- e q u a l t o t h e e n e r g y o f t h e i n c i d e n t r a d i a t i o n , z e r o t o t a l momentum. p h o n o n s may  T h i s may b e a c c o m p l i s h e d i n t w o w a y s . F i r s t ,  be c r e a t e d o f e q u a l , b u t o p p o s i t e , momentum and t o t a l  and two energy  g i v e n by:  T h i s i s known as a two-phonon summation p r o c e s s . *  W a v e n u m b e r , v ( c m ' ) , i s u s e d a s cm e n e r g y u n i t i n s t e a d o f t h e m o r e c o r r e c t u n i t o f f r e q u e n c y , v ( T e r a h e r t z ) , s i n c e i t i s more c o n v e n i e n t when w o r k i n g w i t h t h e f a r - i n f r a r e d s p e c t r o m e t e r .  s  S e c o n d l y , t h e i n t e r a c t i o n may r e s u l t i n t h e a n n i h i l a t i o n o f an a l r e a d y  e x i s t i n g p h o n o n o f m o m e n t u m hl< a n d w a v e n u m b e r  c r e a t i o n o f a n o t h e r p h o n o n o f m o m e n t u m hlc a n d w a v e n u m b e r case i n order t o conserve energy v\-V, two-phonon d i f f e r e n c e  a  y  x  .  In t h i s  n d t h e s e a r e known as  processes.  The i n t e n s i t y o f t h e a b s o r p t i o n wavenumber  }  V, , a n d t h e  V> b y t w o - p h o n o n  of the incident ;  radiation of  p r o c e s s e s depends b a s i c a l l y on t h r e e  quantities;  t h e number o f p h o n o n p a i r s s a t i s f y i n g t h e e n e r g y a n d momentum r e q u i r e m e n t s , the strength  o f the coupling  the p r o b a b i l i t y o f c r e a t i n g  o f e a c h p h o n o n p a i r t o t h e TO m o d e , a n d o r a n n i h i l a t i n g t h e phonons  involved.  I t i s c l e a r t h a t t h e r e i s an u p p e r l i m i t t o t h e wavenumber w h i c h t h e t w o - p h o n o n p r o c e s s may o c c u r . V  v  '• *>-imo.»+ H m o u t  two h i g h e s t  > where V  I M ( L >  energy phonons that  T h i s wavenumber, V -  at  This limit occurs at  and  a r e t h e wavenumbers o f t h e  can combine through a summation process.  , i s r e f e r r e d t o as t h e two-phonon l i m i t .  (  I t i s i n t e r e s t i n g t o examine t h e b e h a v i o r o f t h e s e two t y p e s o f pro;ee"s''s e's5 a t l o w t e m p e r a t u r e s w h e r e t h e t h e r m a l v i b r a t i o n s r  l a t t i c e become s m a l l .  T  of the  - i t h i s c a s e The t o t a l number o f p h o n o n s e x i s t i n g  i n t h e l a t t i c e i s g i v e n by-;  where k a g a i n r e p r e s e n t s t h e w a v e v e c t o r o f t h e phonon and j r e p r e s e n t s one o f t h e polarization branches of the dispersion represents a particular point  curves.  on t h e d i s p e r s i o n  T h u s t h e p a i r C£jj) curves.  The t e r m  4  n  (k*,j) represents  t h e number o f phonons i n t h e s t a t e ( k , j ) ,  is given by the Bose-Einstein  where  which  distribution:  v C k , j ) i s t h e w a v e n u r n b e r o f t h e s t a t e ( k , j ) , 'k- i s t h e B o l t z m a n D  D  c o n s t a n t , and T i s t h e temperature. ^tot ^ w  I f the temperature,T,  i s very  small  a l s o be v e r y s m a l l and w i l l approach zero as T approaches  zero. Since the occurence  o f a difference process  requires the  o f an a l r e a d y e x i s t i n g p h o n o n , t h e number o f such p r o c e s s e s v a n i s h i n g l y s m a l l as T approaches z e r o .  will  Summation p r o c e s s e s ,  presence become  on t h e o t h e r  h a n d , do n o t r e q u i r e a n y e x i s t i n g p h o n o n s a n d w i l l , t h e r e f o r e , n o t v a n i s h f o r s m a l l T. temperatures  T h u s , t h e t w o - p h o n o n i.fand h i g h e r )  i s almost  e x c l u s i v e l y due t o summation  The a b s o r p t i o n p r o c e s s e s  damping a t low  processes.  described thus f a r occur i f the c r y s t a l  i s composed o f two w e l l d e f i n e d atoms.  However, t h e presence  of impurities i n the  1 a t t i c eide's.trbyss t h e I ' a t t i G e ' C p e f ^ o d . i t i ' t y s a h H R r e l a x e s e t h e - c o n d i t i o n o f momentum Gonservationcb^dtheoiatt^  p' - -  T h i s r e s u l t s i n a one-^phonon b a n d a b s o r p t i o n as w e l l a s p o s s i b l e modes and i n - b a n d  resonances.  replaced b y a chemical  I f one o f t h e i o n s i n t h e h o s t  local  lattice is  i m p u r i t y t h e p o t e n t i a l e x p e r i e n c e d ' r b y - t h i s atom i s  s i g n i f i c a n t l y d i f f e r e n t from that o f t h e i o n i t r e p l a c e s .  I t then  r e q u i r e s a complex f o r c e model t o c a l c u l a t e t h e one-phonon  absorption  due t o t h e s e i m p u r i t i e s . I s o t o p i c i m p u r i t i e s , however, a c t as w e a k l y p e r t u r b i n g mass  7  defects which i f present  i n s u f f i c i e n t q u a n t i t i e s p r o d u c e a one-  p h o n o n b a n d a b s o r p t i o n a r o u n d t h e TO r e s o n a n c e ,  the intensity of the  a b s o r p t i o n depending on t h e phonon d e n s i t y - o f - s t a t e s o f t h e isotope ion.  This process must be considered  KI c o n t a i n s 6.88% K  1. 2  41  f o r t h e KI l a t t i c e  since natural  isotope along with the predominant K  39  isotope.  JNotivation f o r the Thesis This work represents part o f a continuing study o f the l a t t i c e  absorption of alkali-halides being c a r r i e d out by Eldridge (see Bibliography).  Previous  studies of the far-infraredoptical properties  ©f C s J ( E l d r i d g e a n d B e a i r s t o  17  ) and L i F (Eldridge  investigation o f the isotope-induced  13  ) , as w e l l as an  a b s o r p t i o n o f L i F (Eldridge''') have  been c a r r i e d out, assuming only cubic anharmonicity. calculated o p t i c a l properties with various experimental g e n e r a l , good.  The agreement o f data was, i n  A l s o i t was f o u n d t h a t c o n s i d e r a t i o n o f a few c r i t e r i a ,  proposed b y E l d r i d g e , f o r t h e strong c o u p l i n g o f phonon p a i r s , allowed one  t o p r e d i c t f a i r l y w e l l w h i c h phonons w i t h h i g h symmetry w a v e v e c t o r  p o i n t s o r branches would combine t o produce peaks i n t h e damping spectrum. The  r e s u l t s o f these  which suggested First, evidence  investigations brought out several i n t e r e s t i n g points  similar investigations i nKI. i n b o t h C s l a n d L i F i t was f o u n d t h a t b y 300°K t h e r e was  o f three-phonon and h i g h e r damping, both a t h i g h wavenumbers  b e y o n d t h e t w o - p h o n o n l i m i t , a n d b e n e a t h t h e m a i n TO r e s o n a n c e . strength o f the three-phonon processes "hardness";  The  a t 300°K d e p e n d s o n t h e c r y s t a l  a hard c r y s t a l b e i n g one w i t h h i g h c h a r a c t e r i s t i c f r e q u e n c i e s , such  L i F w h o s e ; T O wavenumber, at:n3,0 O eK' i s 3 0 5 cm rf  1  J  .  " S o f t " c r y s t a l s , such as  as  Csl  w h o s e 3 0 0 ° K TO w a v e n u m b e r i s 6 2 cm  , w i l l show more e v i d e n c e o f t h r e e -  p h o n o n d a m p i n g a t 300° K t h a n h a r d c r y s t a l s .  I t was t h o u g h t , t h e r e f o r e ,  t h a t K I , b e i n g a s o f t a l k a l i - h a l i d e w i t h a TO w a v e n u m b e r o f 1 0 9 cm * w o u l d a l s o * s h o w c o n s i d e r a b l e s e v i d e n c e o f h i g h e r o r d e r d a m p i n g a t 300°K. A l s o i n b o t h C s l a n d L i F i t was f o u n d t h a t t h e TO  resonance  d a m p i n g was due t o t w o - p h o n o n " d i f f e r e n c e " p r o c e s s e s o n l y ( o r t h r e e phonon d i f f e r e n c e processes i n v o l v i n g the d e s t r u c t i o n o f a t l e a s t phonon).  one  T h u s , a s t h e t e m p e r a t u r e i s l o w e r e d t h e d a m p i n g a t t h e TO  onance becomes v e r y s m a l l .  S i n c e t h e w i d t h o f t h e TO r e s o n a n c e  res-  peak  i s p r o p o r t i o n a l t o , and the h e i g h t o f the peak i n v e r s e l y p r o p o r t i o n a l to,  t h e damping a t ~ Q , t h i s r e s u l t s i n an e x t r e m e l y s h a r p r e s o n a n t  tion.  absorp-  The s h a r p n e s s o f t h e a b s o r p t i o n i n n a t u r a l L i F was l i m i t e d b y  the isotope-induced-one-phonon f r a c t i o n o f t h e damping a t v  0  process, which accounted f o r a large up t o 300°K.  In K I , h o w e v e r , t h e p r e s e n c e o f a l a r g e b a n d gap b e t w e e n t h e a c o u s t i c and o p t i c branches o f t h e phonon d i s p e r s i o n curves  suggested  t h a t r e l a x a t i o n o f t h e TO r e s o n a n c e b y t w o - a c o u s t i c - p h o n o n - s u m m a t i o n p r o c e s s e s s h o u l d be a b l e t o o c c u r a t v . 0  I n t h i s case no r e a l l y  sharp  r e s o n a n c e s h o u l d o c c u r a t l o w t e m p e r a t u r e s i n KT a n d i t w a s o f i n t e r e s t to  s e e i f t h i s was i n d e e d t h e c a s e . T h i r d l y , i n t h e p r e v i o u s - w o r k c o n s i d e r a t i o n was g i v e n t o i n t e r -  a c t i o n s i n v o l v i n g o t h e r - t h a n - n e x t - n e a r e s t - n e i g h b o u r s . However, the m e t h o d u s e d t o i n c l u d e t h e s e i n t e r a c t i o n s was t h o u g h t t o o v e r a c c e n t u a t e t h e c a l c u l a t e d s p e c t r a l f e a t u r e s , a n d i t was f e l t t h a t t h i s w o r k w i t h K I could support this conclusion.  I t s h o u l d be noted, however, t h a t p r e v i o u s c a l c u l a t i o n s o f t h e f a r i n f r a r e d o p t i c a l p r o p e r t i e s o f K I a t 300°K h a v e b e e n  performed, 2  and measurements o f t h e a b s o r p t i o n s p e c t r u m made, b y Berg a n d B e l l . Their calculations  5  however, were performed  involving only nearest neighbours,  assuming i n t e r a c t i o n s  and agreement between t h e i r  experimental  d a t a a n d t h e o r e t i c a l r e s u l t s was o b t a i n e d b y f i t t i n g t h e t h e o r e t i c a l r e s u l t s t o t h e e x p e r i m e n t a l a b s o r p t i o n maximum. showed no e v i d e n c e o f three-phonon performed  no low temperature  and higher processes a t v . 0  They  measurements due t o t h e d i f f i c u l t y o f  doing so w i t h t h e i r asymmetric 1.3  Their results, therefore,  Michelson interferometer technique.  Outline of the Thesis. P r i o r t o t h e w r i t i n g o f t h i s t h e s i s a p a p e r was w r i t t e n f o r  p u b l i c a t i o n c o n t a i n i n g t h e r e s u l t s and c o n c l u s i o n s o f t h i s work.  It  was f e l t t h a t r e w r i t i n g t h e s e s e c t i o n s w o u l d b e o f l i t t l e v a l u e a n d , t h e r e f o r e , t h i s paper has been i n c l u d e d i n t h e 'thesis, f o l l o w i n g a more complete  d e s c r i p t i o n o f t h e experimental techniques and theory  involved. Section I I describes the experimental techniques involved i n the work. S e c t i o n ITT d e s c r i b e s t h e t h e o r y and c a l c u l a t i o n s which  were  performed. S e c t i o n IV. p r e s e n t s t h e p a p e r p u b l i s h e d i n 1 9 7 3 b y E l d r i d g e and Kembry d e s c r i b i n g i n d e t a i l t h e r e s u l t s and c o n c l u s i o n s o f t h e work.  JO  SECTION I I EXPERIMENTAL TECHNIQUE II. 1  The  Apparatus  The  experiments  were performed  Fourier Spectrophotometer detector.  w i t h a Beckman RIIC  w i t h a s t e p d r i v e a n d Go.l>ayr=ti~  The f a r i n f r a r e d r a d i a t i o n was p r o d u c e d  mercury-arc  by a  FS-720 Golay  quartz-envelope  lamp.  The  r a d i a t i o n passing through  o f 10, 20, o r 40 m i c r o n s  depending  t h e sample was s a m p l e d u s i n g  steps  on t h e wavenumber range o f i n t e r e s t .  The maximum w a v e n u m b e r t h a t may b e r e s o l v e d u s i n g a g i v e n s t e p s i z e , s , is given by: n v  max  _ =  10000 -1 —= cm 2s  where s i s g i v e n i n microns.  (IT A ^ --** -y J  J  T h u s , t h e maximum wavenumbers p o s s i b l e  w i t h t h e s t e p s i z e s m e n t i o n e d a b o v e a r e 5 0 0 , 2 5 0 , a n d 1 2 5 cm ^ ' r e s p e c t i v e l y . If, however, radiati'bnrwithewavenumbers ' '  greaterpthannv t h i s present i tw i l l ~??x max • 1  r  be  i n t e r p r e t e d a s r a d i a t i b n - ' w i t h i w a v e n u m b e r s a l l y i n g i n t h e range-.0 ;6o v , causing the jii- max n  v  '  spectrumeat  thesgplower.owavenumbersctbhappfearystrbngefythan  i t - a c t u a l l y i s . These  c o n t r i b u t i o n s d u e to. r a d d i a t ^ i o n r w i s h s w a v e n u m b e r s h g r e a t e r j x t ' h a n s t h e wavenumber a r e known as " f a l s e e n e r g i e s " . with"  RT>I  wavenumbers h i g h e r than ^  maximum r e s o l v a b l e  I n order t o avoid t h i s problem r a d i a t i o n  were e l i m i n a t e d by u s i n g a high-  d e n s i t y b l a c k - p o l y e t h y l e n e l e n s f i l t e r t o g e t h e r w i t h Beckman a b s o r p t i o n filters.  E l e c t r i c a l RC s m o o t h i n g  helped eliminate the higher The  a l s o acted as a o l o w r p a s s - f i l t e r and  energies.  d a t a was p u n c h e d on p a p e r t a p e , t r a n s f e r r e d t o m a g n e t i c  tape  and f a s t - F o u r i e r a n a l y s e d u s i n g an IBM 360/67 computer. For t h e low temperature a r o t a t a b l e copper  experiments  t h e sample was mounted on  c o l d - f i n g e r , w h i c h was i n t h e r m a l  contact with the  II  c o o l a n t r e s e r v o i r i n a metal dewar. samples  The t a i l - p i e c e t o which t h e  were mounted was o f h e x a g o n a l  measurements t o be done on two samples the system.  design (Figure I I . l )which without removing  t h e dewar  The dewar was f i l l e d w i t h l i q u i d n i t r o g e n o r l i q u i d  and i t was p o s s i b l e t o pump o n t h e c o o l a n t t o p r o d u c e  lower  A copper s h i e l d maintained a t l i q u i d n i t r o g e n temperature the c o l d - f i n g e r t o reduce  enabled from helium  temperatures. surrounded  l i q u i d helium b o i l - o f f and prevent  condensation  of r e s i d u a l water vapor on t h e sample. Using a l i q u i d helium reservoir the temperature o f t h e t a i l p i e c e was m e a s u r e d t o b e 11°K u s i n g a c a r b o n - r e s i s t a n c e (a p r e v i o u s measurement b y Kuwahara (1971) a good thermal c o n t a c t between t h e sample  thermometer  g a v e 12°K ± 2 ° K ) .  To ensure  and t h e t a i l - p i e c e , t h e  samples  were p r e s s u r e mounted t o t h e t a i l - p i e c e w i t h s i l v e r  II.2  Sample P r e p a r a t i o n  grease.  A s i n g l e c r y s t a l o f n a t u r a l K I was o b t a i n e d f r o m Harshaw C h e m i c a l Co. samples  This c r y s t a l could be eleavedceasiliy t o produce  Cd=3mm).  To p r o d u c e  the thick  t h e t h i n s a m p l e s , h o w e v e r , i t was  necessary" t o p o l i s h t h i c k e r , c l e a v e d p i e c e s ; s i n c e t h e sample  would  bend i f cleaved t o o thinly-.  cloth  The p o l i s h i n g was done on a f e l t  using a f i n e alumina g r i t , mixed i n ethylene g l y c o l .  Since KI d i s s o l v e s  q u i t e r a p i d l y i n ethylene g l y c o l t h e process w a s . l a r g e l y an e t c h i n g , but t h e u s e o f t h e p o l i s h i n g wheel m i n i m i z e d rounding o f t h e edges and produced  a good, u n i f o r m sample  quickly.  The t h i n samples  w e r e made  w i t h a s l i g h t wedge shape t o e l i m i n a t e i n t e r f e r e n c e f r i n g e s . The p o l i s h e d samples  w e r e t h e n c l e a n e d (in a c e t o n e a n d r i n s e d i n t r i c h l o r o -  tl  t o dewar  shield  sample clanped a t one end o n l y t o re|iuce s t r a i n upon cooling  Beam h— sample  Figure I I . l  hole i n shield t o to allow passage o f beam  T a i l - p i e c e f o rmounting samples f o r low temperature experiments.  13  ethylene and d r i e d .  T r i c h l o r o e t h y l e n e was u s e d f o r t h e f i n a l r i n s e  s i n c e i t was f o u n d t o h a v e n o n o t i c a b l e a f f e c t on t h e sample. A great deal o f care had t o be taken i n t h e c l e a n i n g and r i n s i n g o f the sample as Attempts  e i t a i r i p i e was f o u n d t o b e n d v e r y e a s i l y when w e t .  w e r e made t o p o l i s h v e r y t h i n s a m p l e s u s i n g a p l a t e w i t h a  removable plug, t o minimize  handling o f t h e sample.  I t was  found,  h o w e v e r , t h a t i t was v e r y d i f f i c u l t t o c o m p l e t e l y c l e a n t h e p o l i s h i n g s o l u t i o n f r o m t h e s a m p l e w h i l e t h e p l u g was i n , a n d r e m o v a l  of the  p l u g w h i l e t h e s a m p l e was wet r e s u l t e d i n damage t o t h e s a m p l e . To d e t e r m i n e  t h e e f f e c t s o f p o l i s h i n g on t h e o p t i c a l p r o p e r t i e s  o f t h e sample)?, m e a s u r e m e n t s o f t h e a b s o r p t i o n a n d r e f l e c t a n c e w e r e performed  on cleaved and p o l i s h e d samples.  No s i g n i f i c a n t d i f f e r e n c e  was d e t e c t e d . Using this technique from  samples were produced  ranging i n thickness  . 0 1 c m t o 1 cm.  II.3  Analysis of Experimental The  Data  experimental data, punched on t h e paper tape i n b i n a r y code, were  then w r i t t e n onto magnetic  tape.  The d o u b l e - s i d e d i n t e r f e r o g r a m was  then  a n a l y s e d u s i n g a f a s t ^ F o u r i e r t r a n s f o r m r o u t i n e t o g i v e an i n t e n s i t y spectrum  o f t h e r a d i a t i o n . I n t e n s i t y s p e c t r a ' were o b t a i n e d f o rt h e  r a d i a t i o n w i t h t h e sample i n t h e beam, I ( v ) , and w i t h t h e sample o u t o f t h e beam, T o ( v ) • formula:  These two i n t e n s i t y s p e c t r a a r e r e l a t e d b y A i r y ' s »  i  It  where a i s the a b s o r p t i o n c o e f f i c i e n t , d i s the sample t h i c k n e s s , R i s t h e p o w e r r e f l e c t a n c e , and k and n a r e t h e e x t i n c t i o n c o e f f i c i e n t and i n d e x o f r e f r a c t i o n r e s p e c t i v e l y . m e a s u r e d a n d i t was  was  f o u n d t h a t on t h e l o w w a v e n u m b e r s i d e o f t h e  r e s o n a n c e R c o u l d be a p p r o x i m a t e d  R -  by the c l a s s i c a l  TO  equation:  (II.s)  U-iT -  a n d  w h e r e eg The  The p o w e r r e f l e c t a n c e  an  ^  e  a r e  v a l u e s o f cq,  frequency t h i s was  l°  t n e  00  w  ^  high frequency  s i d e t h e m e a s u r e d r e f l e c t i v i t y was f o u n d t o be f a i r l y c o n s t a n t  necessary  used.  On t h e  r e s o n a n c e t h a t i t was  *  5%.  optimum when a d s 1  Even using the t h i n n e s t  so s t r o n g l y a t t e n u a t e d  high  B e y o n d 1 4 0 cm  at approximately  reliable sample,  i n the region of  the  i m p o s s i b l e t o o b t a i n absorptioriadatarfr,om;>85,ucm  S e v e r a l runs were done at each temperature  samples and the r e s u l t s were a v e r a g e d t o g i v e t h e f i n a l r e s u l t s - and t o g i v e an e s t i m a t e o f t h e e x p e r i m e n t a l  \  constants.  t o use a wide range o f t h i c k n e s s e s t o o b t a i n  h o w e v e r , t h e s i g n a l was  cm ^.  II.1.  o f t h e m e a s u r e m e n t was  r e s u l t s o v e r t h e e n t i r e r a n g e o f a.  t o 145  dielectric  a n d \Tg a r e g i v e n i n T a b l e  Since the accuracy i t was  an  i—  *  using different experimental  error.  TABLE jTT-fe Input Data for the Calculations  T.O. resonance wavenumber  v  Q  (cm  )  300°K  77°K  12°K  101  107.5*  109.5  a  Static d i e l e c t r i c constant  5.09  High frequency d i e l e c t r i c constant  2.65  c  4.78"  4.68  2.67  2.68  r ( 1 0 ° cm)  3.526  3.501  Compressibility  6 (10" /barye)  8.54  8.00  Repulsive overlap potential parameters  C(10"  45.45  60.20  d  d  12  p(10" Third potential derivative  ergs)  10  0.3495  cm)  8  *N!l. o (r  )(1  °  Coulombic term  o"e /r (10  Repulsive term  Ce" ° /p  2  A  12  1 2  o  r  /p  3  e r 8  (10  1 2  F i r s t potential derivative  *N.T. o  S z i g e t i e f f e c t i v e charge  e*/e  a  See Ref. 2  Obtained by interpolation  )/r  ) / r  o  3 )  3  *N.T.< o o d 0 ( r  "  erg cm" )  Second potential derivative  r  C m  2 ( 1  °  erg cm ) -3  1 2  1 2  "See Ref. 7  «gc«- ) 3  e r 8  C m  "  3 )  1-  b  L a t t i c e constant  o  e  b  b  0.3369  c  3.492  d  7.75  d  71.04 0.3302  -3.845  -4.234  -4.449  0.576  0.592  0.599  -4.421  -4.826  -5.048  0.246  0.267  0.278  0.053  0.053  0.055.  0.72  0.72  0.73  See Ref. 8  It  SECTION I I I THEORY III.l  General  Introduction  I n t h i s s e c t i o n we w i l l p r e s e n t  t h e theory used, and t h e  calculations p e r f o r m e d , i n o b t a i n i n g t h e theoretical r e s u l t s . The t h e o r y u s e s t h e r e s u l t s o f c a l c u l a t i o n s b y W a l l i s a n d M a r a d u d i n ' ^ a n d C o w l e y * ^ w h i c h u s e o n l y t h e f i r s t - o r d e r d i p o l e moment a n d i n c l u d e c o n t r i b u t i o n s from c u b i c t e r m s i n t h e p o t e n t i a l e n e r g y . S o l u t i o n o f t h e many-body p r o b l e m f o r t h i s s y s t e m shows t h a t t h e manybody effects may b e e x p r e s s e d e a c h o f t h e n o r m a l modes.  a s a.complex phase s h i f t  The r e a l part o ft h e phase s h i f t gives t h e  c h a n g e i n e n e r g y Cwavenumber), a n d t h e i m a g i n a r y o f t h e mode.  suffered b y  p a r t g i v e s t h e damping  This damping and wavenumber s h i f t a r e d i s c u s s e d i n S e c t i o n  nr.2.  S e c t i o n ITT.2.1 d i s c u s s e s t h e c o n t r i b u t i o n t o t h e damping due to the i s o t o p i c impurity. S e c t i o n ITT.3 d i s c u s s e s t h e c a l c u l a t i o n o f t h e p o t e n t i a l derivatives. S e c t i o n EXT.4 d e s c r i b e s t h e c a l c u l a t i o n o f t h e p h o n o n  eigendata  u s i n g a s h e l l -model c a l c u l a t i o n . S e c t i o n ITT.5 d e s c r i b e s t h e c a l c u l a t i o n o f t h e complex phase s h i f t and Section I I I . 6 describes t h e c a l c u l a t i o n o ft h e o p t i c a l p r o p e r t i e s .  \1  III.2  Damping and Wavenumber Shift of the TO Resonance. An expression for the complex d i e l e c t r i c constant of a cubic  c r y s t a l with f i r s t - o r d e r electronic dipole moment has been obtained by Wallis and Maraduc Maradudin  18  and Cowley  19  and i s given i n convenient notation  by the expression:  where  and M  are the masses of the K* and I  ions, v i s the volume  of the unit c e l l , c i s the v e l o c i t y of l i g h t , and v wavenumber (cm equal to zero.  (0>Jo>^)  are  t n e  i s the observed  of the TO l a t t i c e resonance with wavevector k e f f e c t i v e l y This expression assumes a c r y s t a l with only two atoms per  unit c e l l and,therefore, only one resonance. r  0  The terms A(0,j ;vjf 0  and  wavenumber-dependent wavenumber s h i f t and the damping of the TO  ance respectively. Since we are concerned here only with the TO  resonance  we w i l l use the abbreviated form A(v) and r(v) to r e f e r to these quantities. * F i n a l l y e^ i s the macroscopic e f f e c t i v e charge associated with the TO mode, given by:  with, e  *  3 equal to the S z i g e t t i e f f e c t i v e charge, Tf the damping of the TO resonance i s assinned to occur through  a two-phonon process with a combined wavenumber of v then the damping 19 may be written (Cowley ] ;  18  f 3")  where V  i s the cubic  coupling  c o e f f i c i e n t w h i c h c o u p l e s t h e TO  resonance t o two o t h e r phonons which, i n o r d e r t o s a t i s f y conservation, crystal with  m u s t h a v e e q u a l a n d o p p o s i t e w a v e v e c t o r s Tc, a n d i n a center-of-inversion  p o l a r i z a t i o n branches  symmetry must b e l o n g t o d i f f e r e n t S(v)  and  i c u l a r two-phonon p r o c e s s o c c u r r i n g , servation  momentum  i s the probability of a partand a l s o s e r v e s as an energy con-  t e r m , as f o l l o w s .  The p r o b a b i l i t y o f a p a r t i c u l a r two-phonon p r o c e s s i s d e t e r m i n e d from t h e o c c u p a t i o n numbers, n, o f t h e two s t a t e s -kjj^  .  involved:_ k , j ^ and  a c c o r d i n g t o q u a n t u m t h e o r y , t h e p r o b a b i l i t y , P, o f c r e a t i o n  or a n n i h i l a t i o n o f a phonon i s g i v e n by (Ziman P  . n-*n+l  a  P  i n-»n-l  ot n a n  20  ):  Ob.  n+1  where n i s g i v e n by e q u a t i o n c r e a t i o n o f two phonons, P ^  Thus t h e n e t p r o b a b i l i t y f o r t h e , ( s u m m a t i o n p r o c e s s ! i.s e q u a l t o t h e p r o b -  a b i l i t y o f c r e a t i o n o f two phonons--minus t h e p r o b a b i l i t y f o r a n n i h i l a t i o n of two phonons;  -  i  r  _  *  -i  AT*  \  / t* < \ (jli.5)  and  the net probability f o rcreation  o f one phonon and a n n i h i l a t i o n o f  a n o t h e r , P^. , ( d i f f e r e n c e p r o c e s s ) i s :  *  "ft,!.")  -  v ^ S ^  (S.fc)  The t e m p e r a t u r e dependence o f t h e summation and  difference  p r o c e s s e s a t h i g h a n d l o w t e m p e r a t u r e may b e f o u n d b y c o n s i d e r i n g and (1.5)  i n t h e case o f t h e high and low temperature l i m i t s o f f o r n(k*,j).  P  + +  equation  A t h i g h t e m p e r a t u r e s n ( k * , j ) may b e w r i t t e n a s : 1  1 +- c J r . v f c , y )  ^  ...  -1  which leads t o :  *OsD *  (Hi.*)  T  so t h a t a t h i g h t e m p e r a t u r e s b o t h summation and d i f f e r e n c e p r o c e s s e s a r e p r o p o r t i o n a l t o T.  As T a p p r o a c h e s z e r o , however, t h e p r o b a b i l i t y f o r  a two-phonon d i f f e r e n c e p r o c e s s becomes v a n i s h i n g l y s m a l l , whereas t h e p r o b a b i l i t y f o r a summation p r o c e s s remains f i n i t e .  At very  eratures, therefore, the difference processes w i l l disappear summation processes w i l l  summation  and d i f f e r e n c e  i s incorporated  i n t o t h e S(v)  d e l t a , as shown b e l o w i n t h e f i n a l  processes:  processes:  and t h e  remain.  Energy conservation of a KrDi'raeer  l o w temp-  t e r m b y means  f o r m s o f S(\))  for  I t may b e s h o w n t h a t t h e c o u p l i n g o f t w o p h o n o n modes t o t h e TO m o d e o c c u r s t h r o u g h  t h e t h i r d d e r i v a t i v e o f t h ep o t e n t i a l energy  and t h a t t h e s t r e n g t h o f t h e c o u p l i n g depends l a r g e l y o n t h e r e l a t i v e m o t i o n s o f t h e i o n s i n t h e t w o p h o n o n modes i n v o l v e d , a n d t h e masses of the ions.  The r e l a t i v e m o t i o n s o f t h e i o n s inetheotw.oophojiqn  are given b y t h e eigenvectors  o f t h e equations  ;mqdes  o f motion o f t h e ions  and may b e f o u n d u s i n g a s h e l l m o d e l c a l c u l a t i o n . The g e n e r a l  expression f o rthe cubic coupling c o e f f i c i e n t f o r  a rocksalt structure i sgiven by:  y  *fe t 1 4w, (l,k- l; k') Til < ,p * 1^-^) - A f  where N i s t h e number o f u n i t c e l l s , h•  x(L,K) t h ep o s i t i o n vector o f t h e  K t h t y p e o f i o n i n t h e L t h u n i t c e l l , M' t h e m a s s o f t h e K t h t y p e o f i o n , a n d tt^^a  (L^,*-'*-')  t n e  t  n  i d r  Cartesian d e r i v a t i v e o f t h e p o t e n t i a l energy  b e t w e e n t h e t w o i o n s a t L K a n d E'K' ( t h e f i r s t potential energy Taylor expansion). of tile eigenvector R. /j , t  t  anharmonic term i n t h e  m * C k , j , . ) i s t h e << t h c o m p o n e n t t  t  t  f o r t h e Kth. t y p e i o n when d i s t u r b e d b y t h e mode  T h e d e l t a f u n c t i o n A:Qt^+-  *- k^I  conserves  momentum b y b e i n g s e t  e q u a l t o o n e when t h e w a v e v e c t o r sum equals- z e r o o r a r e c i p r o c a l - l a t t i c e vector, and b y being s e tequal t o zero otherwise. r e c i p r o c a l - l a t t i c e -vectors does n o t a l t e r t h e f i n a l  The i n c l u s i o n o f t h e f o r m o f r(v)  a n d t h e r e f o r e , w i t h o n l y t h e TO r e s o n a n c e ( 1 ^ - 0 ) 1 b e i n g e x c i t e d ,  a n d A(v) then  ai  k = 2  -k^ and V  1  becomes V  J  c u b i c s y m m e t r y , r(v) incident photon.  1  J  (0, j ;"k*, j j ; - k , j ) . Q  2  Since  the c r y s t a l has  w i l l be i n d e p e n d e n t o f t h e p o l a r i z a t i o n o f Assuming i t i s p o l a r i z e d with i t s wavevector  the along  21 t h e x d i r e c t i o n , ot^may b e s e t e q u a l  t o x ( J o h n s o n and B e l l  ).  t h e e f f e c t i v e f o r c e c o n s t a n t s 4> ..jC"-*', «-'vc'}to r a d i a l d e r i v a t i v e s XB  considering nearest-neighbours  In t h i s expression  o n l y , one  a r r i v e s at the  and  expression:  t h e t h i r d d e r i v a t i v e o f t h e p o t e n t i a l e n e r g y has  g i v e n i n t e r m s o f t h e f i r s t , s e c o n d and  d a m p i n g , r(v),  been  t h i r d r a d i a l d e r i v a t i v e s ; ^'(VV) $"(.*•»')  a n d $'{S?) , w h e r e r ^ i s t h e n e a r e s t - n e i g h b o u r The  Converting  separation.  t h e r e f o r e , d e p e n d s on t h e s t r e n g t h o f  the  c o u p l i n g o f t h e t w o p h o n o n m o d e s t o t h e TO r e s o n a n c e , a n d t h e p r o b a b i l i t y o f c r e a t i o n or d e s t r u c t i o n o f the phonons The  involved.  w a v e n u m b e r - d e p e n d e n t w a v e n u m b e r s h i f t , A ( v ) , may  be 21  c a l c u l a t e d i n a m a n n e r s i m i l a r t o t h a t d e s c r i b e d by J o h n s o n and  Bell  2X 2 a n d B e r g a n d B e l l ' " w h e r e A(v) (111.\1) .  equation  However, i n the c a l c u l a t i o n s performed  t o o b t a i n A(v)  found convenient which  i s g i v e n by an e x p r e s s i o n s i m i l a r  to  here,  by means o f t h e K r a m e r s - K r o n i g  i t was theorem  r e l a t e s t h e r e a l and i m a g i n a r y p a r t s o f the c o m p l e x p h a s e  U s i n g t h i s r e l a t i o n s h i p i t may  be w r i t t e n  shift.  (Ziman *): 1  where n i s a s m a l l number. The w a v e n u m b e r d e p e n d e n t s h i f t s c a u s e d thermal  expansion  Isotope The  Induced  values of  VQ.  Damping.  o n e - p h o n o n a b s o r p t i o n due  t o an i s o t o p i c i m p u r i t y i n  an a l k a l i - h a l i d e has b e e n o b t a i n e d b y M a c D o n a l d , K l e i n and H o w e v e r , t h i s e x p r e s s i o n was resonance.  Martin" . 1 1  derived f o r regions i n the wings of  A s i n g u l a r i t y r e s u l t s at V Q .  r  due  to the impurity, which  t h e r e s t s t r a h l e n p p e a k . T h i s may r r  g i v e n by e q u a t i o n  J  i s continuous  t h e n b e a d d e d t o t h e r_  ( I I I . 3 ) , and used  c a l c u l a t e the o p t i c a l p r o p e r t i e s .  the  E l d r i d g e a n d Howard''' h a v e  used t h i s expression to c a l c u l a t e the e f f e c t i v e damping, ^ t h e TO r e s o n a n c e  by  have not been c a l c u l a t e d s i n c e t h e i r e f f e c t i s compen-  s a t e d by the use o f t h e e x p e r i m e n t a l  III.2.2  , f o r example,  s o  (^)>  throughout  , 2-phonon  i n the complex phase s h i f t As g i v e n by E l d r i d g e and  °f  (v), to  Howard*:  w h e r e p*(.V) i s t h e p h o n o n d e n s i t y o f s t a t e s f o r t h e K* s u b l a t t i c e g i v e n  w h e r e m ( k , j ) i s d e f i n e d a s b e f o r e , a n d y 6)) i s n o r m a l i z e d  such, t h a t  (i.ifc)  III.3  The P o t e n t i a l E n e r g i e s The  ^  calculation of the cubic coupling  coefficient  ( O j j p ^ j j ^ j - k j j ^ ) , ( a b b r e v i a t e d t o V^jj.) f o r t h e p a i r o f phonon m o d e s Tc, j ^ a n d T j j ^ r e q u i r e s t h e k n o w l e d g e o f t h e p h o n o n e i g e n d a t a o f t h e p o t e n t i a l energy,4(r).  the various d e r i v a t i v e s were determined  These  and  derivatives,  i n t h e m a n n e r e m p l o y e d b y E l d r i d g e a n d Howard''' i n w h i c h  the e f f e c t o f other-than-nearest-neighbours,  was i n c l u d e d .  Starting with the general expression f o r the equilibrium p o t e n t i a l due t o t h e a t t r a c t i v e coulomb i n t e r a c t i o n o f t h e e n t i r e l a t t i c e , and t h e short range repulsive o n l y , we h a v e :  rf  *  interaction of t  ^  nearest-neighbours (-err  -«Vp  w h e r e «C i s t h e M a d e l u n g c o n s t a n t , a n d C a n d p m a y b e o b t a i n e d f r o m t h e temperature-dependent experimental  v a l u e s o f t h e c o m p r e s s i b i l i t y , |3 ,  and t h e l a t t i c e c o n s t a n t , r ^ . ( n e a r e s t n e i g h b o u r  separation).  $  (3)  However, i n t h e e v a l u a t i o n o f V  we a r e i n t e r e s t e d  p o t e n t i a l energy due t o a p a i r o f n e a r e s t - n e i g h b o u r s we h a v e :  . /  only.  i n the  In this  case  at  which leads to the t h i r d  derivative, '17'  n  We t h e n a s s u m e t h a t t h e c o u p l i n g o f a l l n e i g h b o u r s w i l l b e o f t h e s a m e form as V  ,but t h a t t h e c o u p l i n g w i l l d e c r e a s e w i t h more d i s t a n t  n e i g h b o u r s due t o t h e d e c r e a s e i n t h e c o u l o m b i c i n t e r a c t i o n . assumed t h a t t h e r e p u l s i v e term w o u l d decrease s u f f i c i e n t l y  It i s also rapidly  t h a t o t h e r t h a n n e a r e s t - n e i g h b o u r s w o u l d n o t c o u p l e t o t h e TO m o d e t h r o u g h t h i s t e r m and i t c o u l d , t h e r e f o r e , be l e f t u n a l t e r e d . Using these assumptions to  4>'"(rc)  a sum o f t h e c o u l o m b i c c o n t r i b u t i o n s  was p e r f o r m e d o v e r s h e l l s o f n e i g h b o u r s t o p r o d u c e t h e m o d i f i e d  p o t e n t i a l d e r i v a t i v e , <f>'^  , where  w h e r e a' =• 3. 8 6 2 . T h i s p r o c e d u r e was l a t e r f o u n d t o b e i n c o r r e c t f o r t w o reasons.  F i r s t , according t o equation ( I I I . U ) , c o u p l i n g can o n l y  occur between ions o f opposite sign.  Thus, t h e summation o f t h e  coulombic term should have been c a r r i e d out over s h e l l s o f o p p o s i t e 17 sign only.  S e c o n d l y , i t has been found by B e a i r s t o and E l d r i d g e  t h a t t h e s h o r t - r a n g e - r e p u l s i v e t e r m does n o t f a l l o f f as q u i c k l y as expected, and t h a t n e x t - n e a r e s t - n e i g h b o u r s s h o u l d have been c o n s i d e r e d in this  term. The  expressions f o r the p o t e n t i a l d e r i v a t i v e s used i i i t h i s  c a l c u l a t i o n , then, are given by:  is  r  fcf  0  v. If  0.%>  The v a l u e s o f t h e s e d e r i v a t i v e s a r e g i v e n i n T a b l e I I . 1 . U s i n g t h e s e v a l u e s o f t h e p o t e n t i a l d e r i v a t i v e s and t h e eigendata s u p p l i e d by t h e s h e l l model program t h e c a l c u l a t i o n o f becomes r e l a t i v e l y s t r a i g h t f o r w a r d .  III.4  The S h e l l Model C a l c u l a t i o n . The b a s i c t h e o r y o f t h e s h e l l model u s e d i n t h i s work i s 22  d e s c r i b e d by Cochran  et a l  .  T h i s t h e o r y assumes t h a t t h e l a t t i c e  i s made up o f i o n s , e a c h s u r r o u n d e i i b y a s h e l l r e p r e s e n t i n g t h e o u t e r electron cloud surrounding the i o n . This s h e l l i s coupled t o the i o n core by a f o r c e constant determined  by the mechanical  p o l a r i z a b i l i t y , d^., o f  the i o n . I n a l a t t i c e composed o f such u n i t s t h r e e types o f i n t e r a c t i o n s are considered, i n a d d i t i o n t o the long range interaction.  These are between neighbouring  short-range coulombic  i o n s , ions and t h e i r  s u r r o u n d i n g s h e l l s , and between n e i g h b o u r i n g s h e l l s . These short-range i n t e r a c t i o n s are given i n terms o f  parameters  22  A a n d B, w h e r e A a n d B a r e d e f i n e d b y C o c h r a n  ,  where e i s t h e e l e c t r o n i c charge, v i s t h e volume o f t h e u n i t  cell,  and  0  (KK')  i s t h e p o t e n t i a l e n e r g y a t i o n K d u e t o i o n K'.  between n e a r e s t - n e i g h b o u r s then are g i v e n by A(12)  Interaction  and B ( 1 2 ) ,  and  i n t e r a c t i o n s between n e x t - n e a r e s t - n e i g h b o u r s are g i v e n by A ( l l ) , A(22),  B(ll),  B(22). In o r d e r t o reduce the number o f independent  short-range  22 parameters  Cochran  et a l  assumed t h a t the s h o r t - r a n g e f o r c e s extend  only to next-nearest-neighbours, that the forces are a x i a l l y  symmetric,  and t h a t n e x t - n e a r e s t - n e i g h b o u r s c o u p l i n g o c c u r s o n l y t h r o u g h t h e The p a r a m e t e r s  then required are:  shells.  A (12), B (12), A (11), B (11), A (22), o  X  J-  o  o  B (22):, A ( 1 2 ) , B ( 1 2 ) , A ^ J . 2 ) , B ( 1 2 ) , A ( 2 1 ) , B ( 2 1 ) , where the  sub-  s c r i p t S denotes  ion-  s  g  g  T  T  T  s h e l l - s h e l l i n t e r a c t i o n s , the s u b s c r i p t T denotes  s h e l l i n t e r a c t i o n s , and t h e s u b s c r i p t I denotes  ion-ion interactions.  I t can be shown, however, t h a t o n l y t e n o f the above p a r a meters  are independent.  These ten independent  parameters  combined  with  the e l e c t r i c a l p o l a r i z a b i l i t i e s , ou, the mechanical p o l a r i z a b i l i t i e s , a n d t h e e f f e c t i v e c h a r g e Z, y i e l d a f i f t e e n p a r a m e t e r  model.  In o r d e r t o generate the phonon e i g e n d a t a the parameter  s h e l l model program  parameters  fifteen  o f G. D o l l i n g , k i n d l y s u p p l i e d t o u s ,  a n d m o d i f i e d b y R. H o w a r d , was u s e d . as an e l e v e n p a r a m e t e r m o d e l ,  d^,  This program  c o u l d a l s o be  used  s i m p l y by a s s i g n i n g f o u r o f the i n p u t  the value u n i t y . The v a l u e s o f t h e i n p u t p a r a m e t e r s  were taken from  Dolling  et a l ^ . They f o u n d , by f i t t i n g t o d i s p e r s i o n c u r v e s f o r KI m e a s u r e d a t 90 K b y i n e l a s t i c n e u t r o n s c a t t e r i n g t h a t t h e s e c u r v e s c o u l d b e  well  21  d e s c r i b e d by an e l e v e n parameter model. in Table  These parameters a r e given  III.l. The c o m p u t e r p r o g r a m s e t s up t h e a p p r o p r i a t e 6 x 6 d y n a m i c a l  m a t r i x f o r t h e t w o i o n i c s p e c i e s f o r a n y g i v e n w a v e v e c t o r k. solution of the secular equation yields  The  s i x eigenvalues and t h e i r  corresponding e i g e n v e c t o r s , where t h e e i g e n v a l u e s a r e t h e f r e q u e n c i e s o f t h e p h o n o n m o d e s ( i n cm *) a n d t h e e i g e n v e c t o r s d e s c r i b e t h e p o l a r i z a t i o n of t h e i o n i c motions.  The program a l s o g i v e s t h e m u l t i p l i c i t y o f t h e  e i g e n v a l u e s , w h i c h i s t h e n u m b e r o f n o n e q u a l w a v e v e c t o r s k' r e l a t e d b y s y m m e t r y t o t h e w a v e v e c t o r k.  To g e n e r a t e t h e e i g e n d a t a used i n t h e  c a l c u l a t i o n s t h i s p r o c e d u r e was r e p e a t e d f o r a g r i d o f w a v e v e c t o r s distributed zone.  uniformly throughout t h e irreducible  7-5- o f t h e B r i l l o u i n  Then b y p e r f o r m i n g t h e a p p r o p r i a t e symmetry o p e r a t i o n s i t i s  possible t o generate eigendata f o r wavevectors throughout the B r i l l o u i n zone. E i g e n d a t a were g e n e r a t e d b y t h i s method f o r915 v a l u e s o f k in the irreducible  ^g- o f t h e B r i l l o u i n z o n e , w h i c h i s e q u i v a l e n t t o  32000 w a v e v e c t o r p o i n t s i n t h e f u l l  z o n e , o r 10 e v e n l y s p a c e d  from t h e zone c e n t e r t o t h e ^ 1 0 0 ^ boundary.  wavevectors  This operation took  a p p r o x i m a t e l y 30 s e c o n d s o f c o m p u t e r t i m e . I n a n o t h e r mode t h e p r o g r a m may b e u s e d t o g e n e r a t e e i g e n d a t a for wavevectors i n a specific direction  i n t h e B r i l l o u i n zone.  o f t h e i n p u t d a t a f o r o p e r a t i o n i n t h i s mode i s shown i n T a b l e C a r d s 1 t h r o u g h 18 a r e t h e i n p u t p a r a m e t e r s f 6 r a f i f t e e n  A  sample  III.2.  parameter  as  TABLE  III.l  S h e l l Model Parameters Parameter  Value  Units  A (12)  1-3.4 ±  .05  ;e^/2v  Bj(12)  -1.0 ±  .2  e /2v  A (ll)  -0.16 ±  .25  e /2v  B (H)  -0001 ±  .06  e /2v  A (22)  -0.29 ±  .24  e /2v  B (22)  .05 ±  .05  e /2v  Z  .92 +  .04  e  2.28 ±  .145  do"  2 4  4.51 ±  .4  do"  2 4  -0.11 ±  .04  e  0.13 ±  .05  e  r  s  S  s  s  l  a  2  a  d  l  d  2  2  2  2  2  2  a*.  TABLE  III.2  I N P U T D A T A FOR T H E S H E L L MODEL  PROGRAM  Card  Data  1  13.4  A(12)  F10.5  2  -1.0  B(12)  F10.5  3  -0.16  A(ll)  F10.5  4  -0.01  B(ll)  F10.5  5  -0.29  A(22)  F10.5  6  0.05  B(22)  F10.5  7  .92  Z  F10.5  8  0.026  a  l  9  -0.11  d  l  10  ,.0514  a  2  F10.5  11  0.13  d  2  F10.5  Data  i  12 13 14  Description  1.  F10.5 F10.5  these parameters are s e t equal t o 1.0 s i n c e t h e p r o g r a m i s t o b e u s e d a s a n 11 p a r a m e t e r s h e l l model.  f  1.  Format  F10.5 F10.5 F10.5  15  1. 1.  16  39.1  M  F10.5  17  126.91  M  F10.5  18  3.526  r  19  0  mode o f o p e r a t i o n - g r i d  20  9  number o f s t e p s  21  1  J  1 k. . . . initial  F10.5  1  10 ^  F10.5  0  10 f i n a li  10  size  16 16  10 F  716  s h e l l model.  I f t h e p r o g r a m i s t o be used as an e l e v e n parameter  m o d e l , c a r d s 1 2 , 1 3 , 1 4 , a n d 15 a r e s e t e q u a l t o 1.0 a s s h o w n .  shell Card  19 s p e c i f i e s i f t h e p r o g r a m i s t o b e u s e d t o c a l c u l a t e e i g e n d a t a f o r t h e ^g- p a r t o f t h e B r i l l o u i n z o n e , o f i f i t i s t o b e u s e d t o f i n d eigendata f o r a single direction. the number o f wavevectors  In t h e former case the card  specifies  d e s i r e d i n t h e [dhG)] d i r e c t i o n o f t h e f u l l  B r i l l o u i n zone, w h i l e i n t h e l a t t e r case t h e c a r d c a r r i e s t h e v a l u e z e r o , as shown. zero, determine  Cardss20  a n d 2 1 , n o t n e e d e d o f c a r d 19 i s n o t e q u a l t o  the p a r t i c u l a r wavevectors  to be c a l c u l a t e d .  f o r which the eigendata are  The f i r s t t h r e e n u m b e r s o f c a r d 21 a r e t h e c o o r d i n a t e s  of the i n i t i a l wavevector,  k. ., m u l t i p l i e d b y a s c a l i n g f a c t o r , F , ' initial tr j & ( s o t h a t k may b e r e p r e s e n t e d b y i n t e g e r c o o r d i n a t e s ) . T h e n e x t t h r e e  numbers a r e t h e c o o r d i n a t e s o f t h e f i n a l wavevector,  k_. , m u l t i p l i e d ' final T h e l a s t n u m b e r i s t h e s c a l i n g f a c t o r , F. n  r  by t h e same s c a l i n g f a c t o r .  C a r d 2 0 g i v e s t h e n u m b e r o f s t e p s t o b e t a k e n i n g o i n g f r o m k. ' . , t o a s initial 6  ^final'  r  ^  directions  e  "*"  ast  c a i  * d may b e r e p e a t e d t o o b t a i n e i g e n d a t a f o r d i f f e r e n t  i n t h e B r i l l o u i n zone.  The o u t p u t f o r t h e i n p u t d a t a o f —»  Table III.2 would  g i v e e i g e n v e c t o r s and eigenvalues f o r k  k *» (.2, . 2 , .2) e t c . t o k *» ( 1 * 1»1) • i s shown i n T a b l e  =(.1,.1,.1);  An example o f t h e f o r m a t t e d  output  III.3.  It s h o u l d be noted t h a t t h i s program cannot be used t o c a l c u l a t e e i g e n d a t a f o r k = 0.  Also,the electrical p o l a r i z a b i l i t i e s  u s e d i n t h i s p r o g r a m a r e p o l a r i z a b i l i t i e s p e r u n i t v o l u m e , s o i t was n e c e s s a r y t o d i v i d e t h e v a l u e s o f a., g i v e n b y D o l l i n g e t a l b y t h e 3 v o l u m e o f t h e u n i t c e l l i n cm .  TABLE  III.3  T Y P I C A L O U T P U T OF S H E L L MODEL PROGRAM 180*K  9 0 , 9 0 , 90; 53.91296 53.91310 68.81201 96.25179 96.25200 128.15515  MULTIPLICITY 0.00000007 -0.00000014 -0.00000014 00.73000413 0.36573493 0.57735205  4  0, 0 0 0 0 0 0 0 7 -0.00000108 -0.00000014 -0,68173623 0. 4 4 9 3 3 7 4 8 0. 5 7 7 3 5 2 2 9  -0.00000021 -0.00000010 -0.00000018 -0.04826769 -0.81507176 0.57735336  eigenvalues (cm"')  0.38912761 -0.71780807 -0.57735062 00. 0 0 0 0 0 0 4 5 -0.0000004.3 -0.00000009  0.42707890 0.69589734 -0.57735062 -0.00000044 0.00000031 -0.00000020  -0.81620497 0.02191186 --0.57735163 0.0 00.00000021 -0.00000016  m (k*,j) y  Eigenvectors  co  S i n c e t h e phonon wavenumbers g e n e r a l l y decrease temperature  i s i n c r e a s e d , i t was n e c e s s a r y  as t h e  t o make a c o r r e c t i o n t o t h e  w a v e n u m b e r s f o r t h e c a l c u l a t i o n o f t h e 300°K r e s u l t s . d e c r e a s i n g t h e wavenumbers o f a l l modes b y 5%, w h i c h  T h i s was done b y i s approximately  t h e s h i f t s e e n i n t h e w a v e n u m b e r o f t h e TO m o d e a s t h e t e m p e r a t u r e i s i n c r e a s e d f r o m 90* K t o 300°K.  III.5  C a l c u l a t i o n o f t h e Complex Phase  Shift.  A s s h o w n i n e x p r e s s i o n ( I I I . 3 ) , t h e c a l c u l a t i o n o f r(v) (3) V  from  -* a n d S (v)£equires,ia g d o u b l e  Brillouin  zone.  summation over a l l values o f k i n t h e  As d e s c r i b e d e a r l i e r ,  s h e l l model r e p r e s e n t a t o t a l o f 32000 S i n c e i twould r e q u i r e an extremely f o r a l l 32000 w a v e v e c t o r s ,  the eigendata generated by the wavevector  p o i n t s i n t h e zone.  lengthy calculation t o evaluate  r(v)  i t w a s f o u n(d3 ) t o b e e a s i e r , a n d f a s t e r , t o V ,. ,, w h i c h c o u l d b e a p p l i e d modified c a l c u l a t e a modified coupling term, to a wavevector i n t h e i r r e d u c i b l e p a r t o f t h e B r i l l o u i n zone t o y i e l d 1  r r  t h e t o t a l d a m p i n g , r(v), f o r a l l s y m m e t r y r e l a t e d w a v e v e c t o r s . m o d i f i e d c o u p l i n g t e r m was a c t u a l l y a n a v e r a g e r o t a t i o n and i n v e r s i o n o f t h e coordinates. V  (3)  i t was n e c e s s a r y  of  T h i s /<•  | obtained  under  Due t o t h e h i g h s y m m e t r y o f  I (31 I* t o c a l c u l a t e o n l y t h r e e v a l u e s o f |V | ; one  (31 with V  as g i v e n by e q u a t i o n I I T . U , and two o t h e r s w i t h x r e p l a c e d  v  by y and z r e s p e c t i v e l y . U s i n g t h e m o d i f i e d c o u p l i n g term t h e t o t a l v a l u e o f rCv). f o r a l l m o d e s r e l a t e d b y s y m m e t r y t o t h e t w o m o d e s k , j ^ and-k,j  2  i n t h e i r r e d u c i b l e p a r t o f t h e z o n e i s s i m p l y r(y) f o r k , j  1  and  If  - k , j ^ t i m e s t h e m u l t i p l i c i t y o f k. I t was a l s o found t h a t a ,  50%  reduction i n calculation  could be acheived b y making the summation i n e x p r e s s i o n I I I . 3 which  i s e q u i v a l e n t t o , 2-» ^ The  •  c a l c u l a t i o n o f r (v)  f o r K I f o r 32000 wavevector  the B r i l l o u i n zone, using t h i s technique of computer  took approximately  points i n  30 s e c o n d s  time. The wavenumber d e p e n d e n t wavenumber s h i f t , A ( v ) , w a s c a l c -  ulated using expression some v  beyond which  t a k e n a § .O-lcmOia  n  ( I I I . 13)  It  ( I I I si's) w h e r e t h e i n t e g r a t i o n w a s t a k e n t o r($) i s e f f e c t i v e l y z e r o ( t w o - p h o n o n l i m i t ) a n d TI w a s  w a s  not necessary  t o know t h e c o n s t a n t  i n equation  s i n c e A ( v ) was a d j u s t e d s o t h a t A (V'Q) was e q u a l t o z e r o .  This  d o n e b y s u b t r a c t i n g - A t ( v < j ) . f f o m e a l i ValuesnofhA iCsD.eulaie.'  was  :  A(V ) 0  and zero, from a l l values o f A(v),  III.6  The O p t i c a l P r o p e r t i e s . The  o p t i c a l p r o p e r t i e s examined i n t h i s work are the  and imaginary p a r t s o f t h e d i e l e c t r i c c o n s t a n t , t h e a b s o r p t i o n t h e r e f l e c t i v i t y a n d t h e p h a s e angl'e,and  the r e f r a c t i v e index.  q u a n t i t i e s may a l l b e d e r i v e d f r o m t h e c o m p l e x d i e l e c t r i c through  the u s eo f the f o l l o w i n g macroscopic A  (yo  ft  /  .  * v\ +• =vk  real coefficient, These  constant  electromagnetic  relations:  II  (jll.a-O  3%  III.6.1  The Real D i e l e c t r i c  Constant.  The r e a l p a r t o f t h e d i e l e c t r i c c o n s t a n t may b'e"?derived d i r e c t l y from equation ( I I I . l ) and i s given by: oT  CM*  TTVC* M M"  III.6.2  The Imaginary The  (si  + +  5  Dielectric  Constant.  i m a g i n a r y p a r t o f t h e d i e l e c t r i c c o n s t a n t may a l s o b e  d i r e c t l y obtained from equation ( I I I . l ) and i s given b y :  (m .2  III.6.3  T h e A b s o r p t i o n C o e f f i c i e n t , a. The a b s o r p t i o n c o e f f i c i e n t , a , i s m o s t e a s i l y c a l c u l a t e d  by f i r s t f i n d i n g t h e e x t i n c t i o n c o e f f i c i e n t , k, a n d t h e n u s i n g t h e relation:  The  (u\ .as)  e x t i n c t i o n c o e f f i c i e n t maisbe found u s i n g equations  (III.25) t o  be:  from  which,  oc(v)  -  2jlTTv[\^(yM  - z-'iy)]^  (Ui.30)  III.6.4  The R e f r a c t i v e I n d e x , n . Using equations A  ( I I I . 2 5 ) we f i n d f o r t h e r e f r a c t i v e i n d e x , n ,  ,  *  So t h a t :  (1.3*  and'.  (lu.33)  n - £ 2.k III.6.5  T h e R e f l e c t i v i t y , R, a n d T h e P h a s e A n g l e , <j) T h e p o w e r r e f l e c t i v i t y , R, a n d t h e p h a s e a n g l e , 0 , a r e f o u n d  from the r e l a t i o n  (1.3^  where r , the r e f l e c t a n c e amplitude, i s g i v e n / f o r normal i n c i d e n c e by,  Writing  A  r  re  * f * <l  -  x<  -  n  as r(cos^ - f i s i n ^ )  n -t- i k - \  i t follows *k  OJft  K  and:  ft -  (n-\V  x  +-WT-1  -  that;  (I.3S)  36 SECTION IV  PHYSICAL REVIEW B  F U R T H E R M E A S U R E M E N T S AND  VOLUME 8, NUMBER 2  C A L C U L A T I O N S OF THE  Department  300°K*  Kembry  of Physics  The U n i v e r s i t y o f B r i t i s h Vancouver,  1973  F A R - I N F R A R E D ANHARMONIC  O P T I C A L P R O P E R T I E S OF K I BETWEEN 12 AND J . E . E l d r i d g e a n d K.A.  15 J U L Y  Columbia  B r i t i s h Columbia,  Canada  M e a s u r e m e n t s and c a l c u l a t i o n s o f t h e f a r - i n f r a r e d o p t i c a l p r o p e r t i e s o f K  3 9  I  a t 3 0 0 , 7 7 , a n d 12°K  are presented.  The m e a s u r e m e n t s a r e m a i n l y t h o s e  o f a b s o r p t i o n i n c r y s t a l s o f v a r i o u s t h i c k n e s s e s . The c a l c u l a t i o n  assumed  c u b i c a n h a r m o n i c i t y o n l y , w i t h n e a r e s t - n e i g h b o r c o u p l i n g , and t h e i n p u t l a t t i c e - d y n a m i c a l d a t a were o b t a i n e d from the s h e l l - m o d e l program.  These d a t a  were g e n e r a t e d w i t h a w a v e - v e c t o r d e n s i t y o f 32,000 p o i n t s p e r zone, w a s e s u f f i c i e n t t o g i v e a 2-Seem  - 1  resolution.  which  The o v e r - a l l agreement  between  e x p e r i m e n t and t h e o r y , i n b o t h t h e i n t e n s i t y and t h e s t r u c t u r e o f t h e s p e c t r a , i s good.  The m a g n i t u d e  o f c e r t a i n c a l c u l a t e d f e a t u r e s i s , however,  i n d i c a t i n g a need t o c o n s i d e r n e x t - n e a r e s t - n e i g h b o r i n t e r a c t i o n s . was  a l s o found f o r three-phonon  a l l t e m p e r a t u r e s and a t V  q  damping,  b y 300°K.  both beyond  more p r o n o u n c e d  The hi'gher-phonon  the two-phonon l i m i t  damping s p e c t r a , which were  at  found  e f f e c t s a t 300°K d i d n o t seem n o t i c e a b l y are  F i n a l l y , the i s o t o p e - i n d u c e d one-phonon  p r o c e s s e s which occur i n n a t u r a l KI were c a l c u l a t e d . s m a l l away f r o m t h e r e s o n a n c e f r e q u e n c y V mechanism at v  Evidence  t h a n t h o s e f o u n d i n t h e much h a r d e r L i F , and a r g u m e n t s  presented to understand this.  large,  F r o m t h e s e m e a s u r e m e n t s i t was p o s s i b l e  to calculate p o r t i o n s of the three-phonon t o be r e a s o n a b l e .  too  q  T h e s e a r e shown t o be  and n o t t o be t h e m a j o r  at low t e m p e r a t u r e s , i n c o n t r a s t t o L i F .  damping  37  IV.1.  INTRODUCTION  C a l c u l a t i o n s have r e c e n t l y been p r e s e n t e d  on t h e f a r - i n f r a r e d  1  p r o p e r t i e s o f L i F and n a t u r a l L i F , assuming  optical  only cubic anharmonicity.  7  The  agreement w i t h v a r i o u s e x p e r i m e n t a l d a t a was, i n g e n e r a l , good, and t h e r e emerged s e v e r a l i n t e r e s t i n g p o i n t s which  i t was f e l t c o u l d be  profitably  i n v e s t i g a t e d i n t h e c a s e o f a d i f f e r e n t compound o f t h e same s t r u c t u r e .  One  p o i n t w a s t h a t b y 3 0 0 °K t h e r e w a s e v i d e n c e o f h i g h e r - p h o n o n r e l a x a t i o n , a r i s i n g from q u a r t i c a n h a r m o n i c i t y , e t c . , b o t h a t h i g h wave numbers, beyond the two-phonon l i m i t , and underneath at  V  q  wave number.  t h e m a i n t r a n s v e r s e - o p t i c (TO)  ( T h e w i d t h o f t h e TO r e s o n a n c e  resonance  peak i n t h e imaginary  d i e l e c t r i c constant i s p r o p o r t i o n a l t o , and t h e h e i g h t o f t h e peak i n v e r s e l y p r o p o r t i o n a l t o , the damping a t  V Q  .)  T h e r e a s o n why t h e s e  processes were n o t i c e a b l e i n t h e r e g i o n o f V  q  higher-order  was due t o two f a c t o r s .  First,  t h e c a l c u l a t i o n s showed t h a t r e l a x a t i o n v i a t h e c r e a t i o n o f two-phonons ('summation' p r o c e s s e s ) d i d not o c c u r a t V wave numbers.  q  but started at slightly  higher  Second, t h e magnitude o f t h e damping by two-phonon ' d i f f e r e n c e '  p r o c e s s e s was v e r y s m a l l a t v .  A b o v e some r a n g e o f t e m p e r a t u r e s , p r o p o r t i o n a l  to t h e wave numbers o f t h e phonons i n v o l v e d (through t h e o c c u p a t i o n numbers see Sec. I V ) , three-phonon  damping w i l l i n c r e a s e more r a p i d l y w i t h  than t h e two-phonon damping. with c r y s t a l hardness.  This range o f temperatures  temperature  obviously increases  I n L i F which i s one o f t h e h a r d e s t a l k a l i h a l i d e s ,  3 0 0 °K w a s s u f f i c i e n t t o s h o w t h e t h r e e - p h o n o n  damping a t V . q  I t was  thought  t h e r e f o r e t h a t i n K I , one o f t h e s o f t e s t r o c k s a l t - s t r u c t u r e a l k a l i h a l i d e s , t h e r e may b e e v e n g r e a t e r e v i d e n c e o f t h r e e - p h o n o n  damping a t V  I t should be noted here, however, t h a t Berg and B e l l the o p t i c a l p r o p e r t i e s o f KI a t room temperature  2  q  b y 3 0 0 °K.  have a l r e a d y measured  by t h e v e r y a c c u r a t e method  38  of  asymmetric  Fourier spectroscopy.  Comparison  with the r e s u l t s of  c a l c u l a t i o n , which a l s o assumed o n l y c u b i c a n h a r m o n i c i t y , d i d not show any t h r e e - p h o n o n  their  markedly  effects i n the v i c i n i t y of v , although there  again the expected high-wave-number t a i l .  T h i s r e s u l t was  t o be  was  partly V  expected s i n c e t h e y s c a l e d t h e i r c a l c u l a t e d v a l u e s by f i t t i n g t o t h e peak h e i g h t .  Q  We w i l l r e t u r n t o t h i s p o i n t a g a i n .  A s e c o n d r e s u l t o f t h e L i F w o r k was m a i n r e s o n a n c e was  t h a t , as a l r e a d y m e n t i o n e d ,  the  found t o r e l a x by two-phonon ' d i f f e r e n c e ' p r o c e s s e s  only (or three-phonon  'difference'processes, involving the d e s t r u c t i o n of  at  so t h a t as t h e t e m p e r a t u r e  l e a s t one p h o n o n ) ,  i n L i F becomes e x t r e m e l y sharp. 7  temperature-independent  i s lowered, the  resonance  I n n a t u r a l L i F , c o n t a i n i n g 7.5%  Li ,  the  6  damping by i s o t o p e - i n d u c e d one-phonon p r o c e s s e s  found to l i m i t the sharpness of the resonance account f o r a l a r g e f r a c t i o n of the V  q  was  a t low t e m p e r a t u r e s , and  d a m p i n g up t o 300  °K.  to  In KI, however,  t h e i o n i c mass r a t i o n o f i o d i n e t o p o t a s s i u m i s h i g h e r t h a n t h a t f o r f l u o r i n e to  l i t h i u m , and t h u s t h e TO-mode wave n u m b e r V  the zone-boundary  a c o u s t i c phonons.  r e l a x a t i o n by two-acoustic-phonon at  (Low-energy  t h e i r long wavelength  i s higher with respect to  C o n s e q u e n t l y i t was  expected that  s u m m a t i o n p r o c e s s e s w o u l d be a b l e t o o c c u r  v , so t h a t no r e a l l y s h a r p r e s o n a n c e  pure KI.  q  should occur at low temperatures  a c o u s t i c p h o n o n s do n o t c o u p l e a t a l l s t r o n g l y due and s i m i l a r n a t u r e , w h i c h i n c o m b i n a t i o n produce  small r e l a t i v e displacements of nearest neighbors:  K* 1  1  w i t h the predominant  t h a t once a g a i n i s o t o p e - i n d u c e d one-phonon damping i s p r e s e n t . k n o w n how  great a p a r t t h i s damping would p l a y at V  nor whether  to  very  C o u p l i n g seems t o  commence a t wave v e c t o r s a p p r o x i m a t e l y h a l f w a y a c r o s s t h e B r i l l o u i n N a t u r a l KI d o e s c o n t a i n , h o w e v e r , 6.88%  in  q  at low  K  zone).  3 9  so  I t was  not  temperatures,  i t would become e v i d e n t i n a b s o r p t i o n measurements below  v  n  39  where t h e ' d i f f e r e n c e ' damping a t low temperatures i s s m a l l . the case i n f o r m a t i o n could be obtained  I f this  were  about t h e phonon eigenvectors  at  c e r t a i n symmetry p o i n t s , a s was done f o r L i F . 3  A f i n a l p o i n t t o emerge from t h e L i F work c o n c e r n e d t h e s i z e and assignment o f t h e two-phonon peaks o r f e a t u r e s . summation o f t h e s l o w l y decreasing  By p e r f o r m i n g  a  lattice  e l e c t r o n i c term i n t h e t h i r d d e r i v a t i v e  o f t h e l a t t i c e p o t e n t i a l , c o n s i d e r a t i o n was g i v e n t o t h e m a g n i t u d e o f t h e cubic coupling by a l l neighbors. coupling by a l l neighbors neighbors.  F o r s i m i l i c i t y , however, t h e form o f t h e  was t a k e n t o b e t h e same a s t h a t f o r n e a r e s t  The n e t r e s u l t was t h o u g h t t o b e a n o v e r a c c e n t u a t i o n  calculated spectral features. With regard  T h i s work w i t h KI supports  this  of the conclusion.  t o t h e two-phonon a s s i g n m e n t s o f a n y f e a t u r e s , i t was s e e n t h a t  consideration o f a few c r i t e r i a f o r strong coupling allowed  one t o p r e d i c t  f a i r l y w e l l which phonons w i t h high-symmetry wave-vector p o i n t s o r branches would combine t o produce peaks i n t h e damping spectrum.  Investigation of  these peaks o f t e n showed, however, t h a t l a r g e c o n t r i b u t i o n s from low-symmetry branches were a l s o present. made o f t h e s e importance,  features.  I n t h e p a s t some c a r e l e s s a s s i g n m e n t s h a v e b e e n  The assignments a r e n o t i n h e r e n t l y o f any great  but nevertheless  a r e i n t e r e s t i n g a n d c a n be made f a i r l y  S r i v a s t a v a and Bist * have r e c e n t l y analyzed 1  and N a l .  Some t i m e a g o R e n k  5  absorption peaks i n KI below V  reported q  two-phonon d i f f e r e n c e processes.  the absorption  some v e r y s h a r p  accurately.  spectra o f KI  temperature-dependent  a t 5 a n d 6 °K, w h i c h h e a s s i g n e d  to specific  The measurements and c a l c u l a t i o n s  presented  h e r e w i l l r e s o l v e some o f t h e c l a i m s made i n t h e s e t w o c o m m u n i c a t i o n s .  40  IV.2.  D A M P I N G AND  F R E Q U E N C Y - S H I F T OF THE TO  RESONANCE  The f i n a l f o r m s u s e d t o c o m p u t e t h e f r e q u e n c y - d e p e n d e n t  damping ( i n v e r s e  l i f e t i m e ) r ( 0 , J ; v ) , a n d f r e q u e n c y s h i f t A ( 0 , j ; v ) o f t h e TO  lattice  q  r e s o n a n t mode, w i t h n e a r - z e r o wave v e c t o r , i n r o c k - s a l t - s t r u c t u r e d may b e f o u n d i n R e f . 1. may  a l s o be f o u n d t h e r e .  crystals  References to the theory underlying these results I t may b e r e p e a t e d h e r e t h a t t h e i n c l u s i o n o f o n l y  c u b i c a n h a r m o n i c i t y means t h a t a b s o r p t i o n o f i n f r a r e d r a d i a t i o n a t any wave n u m b e r v o c c u r s t h r o u g h o f f - r e s o n a n c e e x c i t a t i o n o f t h e TO r e s o n a n c e a t v , w h i c h t h e n r e l a x e s t o two phonons w i t h e q u a l and o p p o s i t e wave v e c t o r , t o g i v e z e r o f i n a l momentum e f f e c t i v e l y e q u a l t o t h a t o f t h e i n f r a r e d and a combined energy  photon,  ('summation' o r ' d i f f e r e n c e ' ) o f wave number v.  T h e f o r m o f i°(0, j ; v ) , w h i c h w i l l b e a b b r e v i a t e d t o r ( v ) , c o n t a i n s t h e w a v e v e c t o r s , e i g e n v e c t o r s , and e i g e n v a l u e s o f t h e p h o n o n s i n v o l v e d , and the c a l c u l a t i o n o f r(v) r e q u i r e s a summation o v e r a u n i f o r m d i s t r i b u t i o n of wavevectors throughout the B r i l l o u i n zone.  These l a t t i c e - d y n a m i c a l  were o b t a i n e d from a s h e l l - m o d e l program, u s i n g t h e i n p u t parameters D o l l i n g et a l . ,  6  data  of  which they found by f i t t i n g t o frequency d i s p e r s i o n curves  m e a s u r e d a t 90 °K b y i n e l a s t i c n e u t r o n s c a t t e r i n g .  M o d e l I I I was  employed , 7  and t h e d a t a were g e n e r a t e d w i t h a w a v e - v e c t o r d e n s i t y o f 32,000 p o i n t s / z o n e ( 9 1 5 p o i n t s i n t h e i r r e d u c i b l e -7-5- e l e m e n t o f t h e z o n e ) . approximation to the temperature-dependent  As a  first-order  f r e q u e n c y - s h i f t s o f the phonons,  a n o v e r - a l l 5% f r e q u e n c y r e d u c t i o n , a p p r o x i m a t e l y t h a t o b s e r v e d i n t h e m o d e , was  a p p l i e d t o p r o d u c e t h e 3 0 0 °K d a t a .  The f o r m o f t h e f i r s t ,  TO second  and t h i r d r a d i a l d e r i v a t i v e s o f t h e n e a r e s t - n e i g h b o r p o t e n t i a l were a g a i n t a k e n a s i n R e f . 1 w h e r e c o n s i d e r a t i o n was  given to the interaction of a l l  neighbors through the s l o w l y decreasing e l e c t r o n i c term.  A summation over  41  50  'shells' of nearest neighbors  yielded a constant  a'  1  which modifies the magnitude of the Coulombic terms.  e q u a l t o 3.862  0.002,  These d e r i v a t i v e s appear  i n t h e c u b i c - c o u p l i n g c o e f f i c i e n t w h i c h i n t u r n i s p a r t o f r ( v ) , and t h e i r r e s p e c t i v e m a g n i t u d e s may  be s e e n i n T a b l e  iy.Togtogethef.twith all"other  input  1  data used i n the c a l c u l a t i o n s . FiguTV.l  shows t h e 300  and summation p r o c e s s e s Berg and  °K c a l c u l a t e d r ( v ) s p e c t r u m , w i t h t h e d i f f e r e n c e  drawn s e p a r a t e l y , together w i t h the c a l c u l a t i o n of  B e l l , i n w h i c h t h e two p r o c e s s e s  c a l c u l a t i o n w i t h 915  have been combined.  In the  i n d e p e n d e n t w a v e - v e c t o r p o i n t s , t h e s p e c t r u m was  by c o n v o l u t i o n w i t h a n i n e - p o i n t l e a s t - s q u a r e s - f i t t i n g f u n c t i o n , r e s o l u t i o n of 2 or 3 cm .  1 0  smoothed  giving a  No h a n d s m o o t h i n g h a s b e e n p e r f o r m e d .  - 1  present  This  r e s o l u t i o n e x p l a i n s i n p a r t t h e d i f f e r e n c e s b e t w e e n t h e two c a l c u l a t i o n s ( e . g . t h e h e i g h t s and w i d t h s o f t h e 142B e r g - B e l l c a l c u l a t i o n was  and  153-cm  1  peaks),  since  the  p e r f o r m e d w i t h o n l y 48 i n d e p e n d e n t w a v e v e c t o r s .  O t h e r d i s s i m i l a r i t i e s o c c u r due  to the d i f f e r e n t values of p o t e n t i a l d e r i v a t i v e s  used i n the c a l c u l a t i o n s . A f a i r l y important  r e s u l t o f t h i s w i l l be  seen  in the subsection of this section. F i g u T V . 2 shows t h e 300 and r (v) as d e s c r i b e d i n Ref.  s h i f t s A'(v)  1 and a d j u s t e d t o e q u a l  c a l c u l a t e d from  zero at the  observed  V .  resonance frequency Figt:TV.3  12 °K f r e q u e n c y  Q  s h o w s t h e 12 °K T ( v )  s p e c t r a f o r t h e sum  and d i f f e r e n c e  ( n o t e t h e s c a l e c h a n g e a t 1 2 0 cm *) t o g e t h e r w i t h t h e d a m p i n g o f t h e mod e d u e  to isotope-induced  d e s c r i b e d i n Ref.  1.  one-phonon processes,  The p e a k n e a r v r  i n r. o  r^  S Q  processes TO  ( v ) , c a l c u l a t e d as  ( v ) i s d u e t o t h e TO  modes  ISO  a c r o s s the zone, and t h e d i s a p p e a r a n c e  a t 95 cm *, f o l l o w s f r o m t h e p r e s e n c e  of  b a n d g a p - w h i c h extendsitd6$nit'Os69_ncmne.  acBelo.wctbi>s"st-he ^ s m a l l a m p l i t u d e s  the  light K  +  i o n s i n the a c o u s t i c modes, t o g e t h e r w i t h  the  of  a  TABLE l E i Input Data f o r the Calculations  T.O. resonance wavenumber  v  (cm  Q  )  300°K  77°K  12°K  101*  107.5*  109.5  Static d i e l e c t r i c constant  5.09  High frequency d i e l e c t r i c constant  2.65°  L a t t i c e constant  r (10" cm) o  3.526  Compressibility  S (10" /barye)  8.54  Repulsive overlap p o t e n t i a l parameters  C(10"  8  12  p(10" Third p o t e n t i a l d e r i v a t i v e  *^  T t  (r )(10  Coulombic term  a"e /r * ( 1 0  Repulsive term  Ce  2  F i r s t p o t e n t i a l derivative  •N.T. o  S z i g e t i e f f e c t i v e charge  e*/e  Obtained by i n t e r p o l a t i o n  c  )/r  ) / r  o  b  b  60.20 0.3369  c  3.492 7.75 71.04  0.3302  erg cm" )  0.576  0.592  0.599  -4.421  -4.826  -5.048  0.246  0.267  0.278  0.053  0.053  0.055  0.72  0.72  0.73  3  -3, ,12 ao"«gcm-) 2 ( 1  °  1 2  See Ref. 7  e r 8  C m _ 3 )  See Ref. 8  d  d  -4.449  J  *N.T. o o  3.501 8.00  2.68  b  -4.234  J  Second potential derivative  See Ref. 2  1 2  2.67  C  -3.845  - r / p ,/p3 („„12 l O " erg era )  ( r  d  4.68  b  3  erg cm" )  1 2  o  (r  d  0.3495  cm)  0  .  45.45  ergs)  10  8  4.78  C  C  Fig.  1  1  n  IV.1  r — j —  PRESENT CALC.  WAVE  NUMBER  i  r  9 10 k' 1  (cm" ) 1  P r e s e n t c a l c u l a t i o n o f t h e t w o - p h o n o n d a m p i n g o f t h e TO r e s o n a n c e o f K I a t 300°K, w i t h b o t h p r o c e s s e s shown s e p a r a t e l y ( s o l i d l i n e s ) and t h e f e a t u r e s l a b e l e d ; together with t h e c a l c u l a t i o n o f Ref. 2 with both process combined (dashed l i n e ) . 3 9  F i g . IV.2  300 °K 12 °K  20  £ o  v  10  0  (12 °K)  o o  <  0  80 WAVE  NUMBER  160  240  (cm" ) 1  C a l c u l a t e d f r e q u e n c y - d e p e n d e n t f r e q u e n c y s h i f t o f t h e TO resonance o f K I r e s u l t i n g from two-phonon damping, a t 3 0 0 a n d 12°K. v i s t h e o b s e r v e d r e s o n a n c e wave n u m b e r o 3 9  Fig.  i  1  '  IV.3  1  WAVE  1  [  NUMBER  1  r  (cm" ) 1  C a l c u l a t e d t w o - p h o n o n d a m p i n g o f t h e TO r e s o n a n c e o f K I a t 12°K v i a d i f f e r e n c e a n d s u m m a t i o n p r o c e s s e s , d r a w n s e p a r a t e l y together w i t h the isotope-induced one-phonon damping o f t h e r e s o n a n c e i n n a t u r a l K I . The l e f t - h a n d s c a l e r e f e r s t o a l l c u r v e s u p t o 1 2 0 cm"" ." 3 9  1  46  l a r g e wave-number s e p a r a t i o n from  v , the small concentration of the isotopes  and t h e s m a l l mass d e f e c t a l l combine t o make r .  (v) v e r y s m a l l .  (Note  ISO  that i n L i F  and N a C I ,  3  1 1  where l a r g e r e f f e c t s were observed,  n o t o n l y were  t h e r e no b a n d s g a p s , b u t t h e mass d e f e c t was l a r g e i n t h e f i r s t c a s e ,  and  the concentration large i n the second).  little  Consequently  i t appears that  i n f o r m a t i o n may b e o b t a i n e d f r o m t h e i s o t o p e - i n d u c e d but t h a t at V  q  approximately  a t low temperatures  a b s o r p t i o n away from v ,  these damping processes  equal s t r e n g t h w i t h t h e two-phonon summation  w i l l be o f processes.  It i s a l s o o f i n t e r e s t t o compare t h e v a r i o u s v a l u e s o f resonance damping r ( 0 , j ; V ) c a l c u l a t e d o r measured by other authors. q  Table IV.2  lists  some o f t h e s e , a n d i t may b e s e e n t h a t w h i l e t h e r e i s g e n e r a l a g r e e m e n t a t room temperature,  t h e one low-temperature  is f a r l a r g e r than the c a l c u l a t e d values.  experimental  value o f Jones et a l .  Since t h e measurements by Jones  et a l . were p e r f o r m e d on t h i n f i l m s , c o u l d t h i s i n d i c a t e s u r f a c e - m o d e r e laxation?  (Their room-temperature value i s also high). Two-Phonon A s s i g n m e n t s  C r i t e r i a f o r t h e s t r o n g c o u p l i n g o f two phonons have been  given.  1 3  On t h e b a s i s o f t h e s e , T a b l e HV.3 l i s t s o f p a i r s o f h i g h - s y m m e t r y p h o n o n s w h i c h may b e e x p e c t e d  t o produce features i n the r ( v ) spectrum i f the  d i s p e r s i o n slopes are matched ( p a r a l l e l f o r d i f f e r e n c e , opposite f o r summation).  The a p p r o x i m a t e r e d u c e d - w a v e - v e c t o r p o s i t i o n h i s g i v e n  the slope m a t c h i n g i s optimum. from t h e shell-model from L through  Figi.rIV.4 shows t h e d i s p e r s i o n c u r v e s r e s u l t i n g  c a l c u l a t i o n , f o r K I a t 9 0 °K.  W t o X were not measured by neutron  numbers i n F i g . IV.-lnandaWableIIV:v3frefer frequency  where  The E ^ c u r v e s and scattering.  The  those feature  tb.etheO3O0°Kpspeefcanim-r w i t h - , i t s 5 %  r e d u c t i o n , w h e r e a s t h e w a v e n u m b e r s i n T a b l e IV3; w e r e t a k e n  from  4-7  TABLEJy.l Various Reported Values of  r ( 0 , j ;v ) equivalent (cm  Jones et a l . (expt.)  o  Dolling et a l . (theo.)  3.025 (4.2°K)  2.06(300°K)  b  0.183 (5°K) ( i s o t o p i c a l l y pure)  -v2.4 (R.T.)  C  Present results (theo.)  -VL.4 (300°K) (M).6 D i f f + M5.8 Sum)  .  0.16 (12°K) (~°.0.8 Sum + M).oS  0.08  a  See Ref. 11  o  )  3.48 (R.T.)  a  Berg and B e l l (expt.)  r(0,j ;v )  b  See Ref. 6  C  See Ref. 2.  :  I SOtOpe)  TABLE 3JtTt3 Expected Maxima l h Two-Symmetry-Phonon Combinations and Feature Assignments Summation  Difference  Wavenumbers a t , and positions of, (h), maximum expected contribution  Corresponding infrared wavenumber (cm~l) and feature assignment, (n! i f appropriate  114 + 66 (0.6)  180 (11)  Symmetries of combining phonons  Wavenumbers a t , and positions of, (h), maximum expected contribution  Corresponding infrared wavenumber (cm~l) and feature assignment, M> i f appropriate  Vi  111-66 (0.6)  45 (small)  Vs  109-28 (0.7)  0 A  109 + 28 (0.7)  137  116-65 (0.7)  51 .(small)  118 + 66 (0.5)  184 (11)  hh  96-54 (0.5)  ij^ao^dA)  ^(LAU^TA)  Y i L  V 1 L  24 (very small)  116-53  63  (0.6) '.  96 + 54 (0.5)  42 (3)  120-96 (0.5)  96-66 (0.5)  30 (2)  66-54 (0.5)  12 (1)  150 (9) >200 (very small)  .  «  118 + 54 (0.5) .  172 (11)  96 + 66 (0.5)  162 (10)  66 + 54 (0.5)  120 (small)  128-69  59  128 + 69  197 (effective 2-phono  128-54  74.  128 +54  182 (11)  96 + 69  165 (10)  96 + 54  150 (9)  104 + 58  162 (10)  96-69 96-54  W  .(6)at 300°K o v e r a l l max a t 12°K  104-58  '27 42 (3) . 46 (3)  162 (10) ( A + B i n F i g . 4) r-»q and W  4  and 5  oo  49  t h e 90 °K d a t a o f F i g . I V . 4 . Some o f t h e s t r o n g l o w - s y m m e t r y c o m b i n a t i o n s a l s o l i s t e d i n T a b l e W>3  b u t i t was  not f e l t worthwhile  t o l i s t them  Of i m p o r t a n c e  also i s the energy of destroyed  as i s o b v i o u s  b y 12 °K, w h e r e t h e m a x i m u m a r o u n d 8 0 cm * c o r r e s p o n d s  A5 A5 c o m b i n a t i o n  mode i s t h e l e a s t e n e r g e t i c o f  t h e two k i n d s o f p r o c e s s . Of i m p o r t a n c e o b t a i n e d between a c o u s t i c modes.  to  the l i m i t s  i n t h i s r e s p e c t i s the low  Not o n l y i s t h e r e v e r y l i t t l e  do n o t become s i g n i f i c a n t u n t i l h a l f - w a y  see Eq.  (6) i n R e f .  <J>''' ( )  coupling  coupling  Q  Q  e  M  I  b o u n d a r y due  occur.  B o t h A1A3  and  t o t h e m o t i o n o f one  a r e l o c a t e d [see Eq.  difference combinations  Q  are forbidden at the  These produce the  along a C a r t e s i a n d i r e c t i o n where the (6) i n R e f .  1].  Consequently  the  i n L i F were w i t h o p t i c modes (E^E^ and  with  a c o u s t i c modes [ j u s t f a r enough away from t h e o r i g i n f o r t h e  same  nearest  lowest  L'sLs), whereas  S i m i l a r l y i n LiF the  summation p a i r s were the t r a n s v e r s e - o p t i c modes n e a r  i n T(v) t o t a k e e f f e c t ] .  zone  i o n o n l y , and t h e l a t t e r i s s m a l l by v i r t u e  KI t h e y a r e t h e v e r y weak E^Ej a c o u s t i c .  term  <j>" ( r ) ;  I i E ^ ( a c o u s t i c ) combine  (r ) but only weakly, s i n c e the former  kind of r e l a t i v e displacement  in  through  L i k e w i s e E3 c o m b i n e s w i t h  o f t h e p o l a r i z a t i o n d i r e c t i o n s i n v o l v e d i n the modes.  neighbors  that  1] w h i c h i s t y p i c a l l y o n l y 1 0 % o f t h e t h i r d d e r i v a t i v e  any o t h e r mode i n t h i s m a n n e r o n l y . through  combine only  s e c o n d r a d i a l d e r i v a t i v e s [<j>' ( r ) a n d  through which major combinations  r  of  to the zone boundary, A1A5  b u t e v e n a f t e r t h i s no g r e a t c o u p l i n g i s o b t a i n e d . i n v o l v i n g t h e f i r s t and  the  acoustic  b e t w e e n s m a l l w a v e - v e c t o r a c o u s t i c p h o n o n s , as a l r e a d y m e n t i o n e d , so  a term  processes,  all.  Of f u r t h e r i n t e r e s t a r e t h e p h o n o n p a i r s w h i c h d e t e r m i n e  combinations  all.  phonon i n the d i f f e r e n c e  a t X a r e f o r b i d d e n ) , s i n c e t h e A5  (combinations  are  first  low-energy sine-modulation  These were then n e c e s s a r i l y above v .  50  Fig.  REDUCED  IV.4  WAVE VECTOR COORDINATE,  h - 2r k 0  F r e q u e n c y d i s p e r s i o n c u r v e s o f KI a l o n g m a j o r symmetry d i r e c t i o n s , g e n e r a t e d b y - t h e s h e l l m o d e l , f o r K I a t 95 °K. T h e s h e l l m o d e l was., f i t t e d to neutron measurements, which d i d not include modes o r t h o s e from L t h r o u g h W t o X.  51  Similar combinations  a r e s e e n a s f e a t u r e s m a r k e d 8 i n F i g . I V . 1. I n K I ,  however,  w h e r e t h e TO m o d e i s s o h i g h , t h e f i r s t s u m m a t i o n p a i r s a r e t h e w e a k a c o u s t i c ones w i t h wave v e c t o r s a p p r o x i m a t e l y  halfway  b e s e e n t h a t t h e m a g n i t u d e s o f <f>" ( r ) a n d the summation damping at V  i n K I , and  q  c a l c u l a t e d v a l u e h e r e and t h a t due  across the zone.  <J>'  Thus i t  may  ( r ) determine to a great  extent  Q  the d i f f e r e n c e i n r ( v ) between  t o B e r g and  the  B e l l l i e s i n the d i f f e r e n t  values used f o r these d e r i v a t i v e s . The  high-wave-number l i m i t of the d i f f e r e n c e processes  highest-energy  l o n g i t u d i n a l - o p t i c mode n e a r t h e z o n e o r i g i n , w i t h a  a c o u s t i c mode, t h e i n t e n s i t y d e c r e a s i n g T h e o r e t i c a l l y the highest  so t h a t t h e  The  c a n come f r o m two  low-energy  o p t i c modes,  t o be v e r y weak j u s t as w e r e t h e  ' e f f e c t i v e ' l i m i t a s i n L i F may  IV13  the  r a p i d l y as the o r i g i n i s a p p r o a c h e d .  summation combination  El£4 f o r i n s t a n c e , b u t t h e s e a r e f o u n d combinations,  i s simply  be t a k e n a s  acoustic LiL'j.  OPTICAL PROPERTIES  o p t i c a l p r o p e r t i e s w i l l be c a l c u l a t e d f o r i s o t o p i c a l l y p u r e  K  3 9  I  o n l y , s i n c e i t h a s b e e n s e e n i n Fig.IV/.3hthatht»hes.isotopesdinduded;ione.Tphonon damping has  a s m a l l e f f e c t away from V  performed.  I t c a n e a s i l y be  q  where the present  measurements were  included i n the c a l c u l a t i o n i f i n the  low-temperature measurements are performed i n the r e g i o n of V KI.  The  r e s o n a n c e p e a k a t 12 °K s h o u l d b e a p p r o x i m a t e l y  q  on  future natural  o n l y h a l f as  sharp  i n n a t u r a l KI as i n the i s o t o p i c a l l y p u r e m a t e r i a l . A.  D i e l e c t r i c Constants  F i g u r e Ws5.os.ho^etb«atefJ.e4i»©l:eG.^ri.on<?Qns.taats e± a t i o 3 0 i L " a n d c a l c u l a t e d as i n Ref.  1.  'static' frequency  12°K,  O n c e a g a i n t h e S z i g e t i e f f e c t i v e c h a r g e e*  been c a l c u l a t e d by u s i n g the measured v a l u e s the  !  s h i f t A'  CO)  o f t h e TO  o f e , e^, q  and  v , and  r e s o n a n c e p r o d u c e d by  has including the  F i g . IV.5  WAVE  NUMBER  (cm"" ) 1  u C a l c u l a t e d r e a l d i e l e c t r i c c o n s t a n t s o f K I a t 3 0 0 a n d 12°K. The p r e d i c t e d f r e q u e n c y _ o f t h e l o n g i t u d i n a l - o p t i c mode w i t h n e a r - z e r o w a v e v e c t o r , v , o c c u r s when e' p a s s e ? t h r o u g h z e r o . 3 9  T n  5.3  cubic anharmonicity  [see Eq. (25) i n R e f . 1 ] . T h i s l a t t e r i s t h e o n l y de-  p a r t u r e from t h e u s u a l harmonic  a p p r o x i m a t i o n u s e d t o o b t a i n e*.  A' ( 0 ) e q u a l t o - 2 . 7 , - 0 . 3 , a n d +0.7 c m "  1  With  a t 3 0 0 , 7 7 , a n d 12 ° K , r e s p e c t i v e l y ,  t h e v a l u e s o f e* s o o b t a i n e d d i f f e r l i t t l e f r o m t h e h a r m o n i c  values  a n d may  8  be found i n T a b l e I V . 1 . A n o t h e r d e p a r t u r e from h a r m o n i c i t y o c c u r s i n t h e wave number a t w h i c h e' p a s s e s t h r o u g h z e r o .  T h i s i s t h e n t h e p r e d i c t e d wave number o f t h e l o n -  g i t u d i n a l - o p t i c m o d e w i t h n e a r - z e r o w a v e v e c t o r , V^Q. imation, the Lyddane-Sachs-Teller  ^LO^o  =  o  e  / £  In t h e harmonic  approx-  (LST) r e l a t i o n s h i p i s o b t a i n e d .  -'  ( 1 )  which w i t h t h e experimental v a l u e s i n Table IVpl-epredicFs^v+g t o be5l45.iand 140 c m  - 1  a t 12 a n d 3 0 0 ° K , r e s p e c t i v e l y .  Upon i n c l u d i n g c u b i c a n h a r m o n i c i t y , Eq. (1) i s m o d i f i e d as shown i n Ref. 1 t o be  !L0  =  vS  fo  (1  +  2 ^ 0 I  e  O L Q  +  2__ . [A  v o  00  With A'(v" )  )  ^  _ , A  ( 0 ) ]  ( 2 )  v o  e q u a l t o -12 c m  - 1  ( 3 0 0 °K) a n d -2.4 c m "  1  (12 ° K ) , and t h e  s a m e v a l u e s o f e , e ^ , a n d V , E q . ( 2 ) p r e d i c t s t h e v a l u e s o f V^Q t o b e q  143 c m "  1  a t 1 2 °K a n d 1 2 8 c m "  1  a t 3 0 0 °K, a s shown i n F i g . I J f . 5 .  T h e o n l y v a l u e o f V^Q m e a s u r e d b y i n e l a s t i c n e u t r o n d i f f r a c t i o n i s t h a t a t 90 °K, e q u a l t o 142 c m . - 1  This does not d i s c r i m i n a t e between t h e  a n d a n h a r m o n i c 12 °K v a l u e s o f 1 4 5 a n d 1 4 3 c m , b o t h o f w h i c h a r e a - 1  higher'as  expected.  harmonic little  54  A t 3 0 0 °K, t h e o p t i c a l m e a s u r e m e n t s o f B e r g a n d B e l l deduced.  I t m a y b e s h o w n t h a t a t V^Q,  o f 128 e m ^  T h i s o c c u r s i n Berg and B e l l ' s  -  Furthermore,  the  temperature  as p r e d i c t e d b y Eq. (2) i s s i m i l a r t o t h a t o b s e r v e d  r e s o n a n t mode v , w h e r e a s t h e t e m p e r a t u r e v a l u e s o f V^Q  data  i n f a r b e t t e r agreement w i t h t h e anharmonic v a l u e  t h a n t h e L S T v a l u e o f 1 4 0 cm '''.  v a r i a t i o n o f vLH  a l l o w V^Q t o b e  w h e r e e' = 0, t h e r e f r a c t i v e i n d e x n ,  e q u a l s t h e e x t i n c t i o n c o e f f i c i e n t k. a t a p p r o x i m a t e l y 1 2 9 cm  2  i s very small.  i n the  v a r i a t i o n i n t h e LST p r e d i c t e d  This gives confidence then i n the calculated  frequency s h i f t s of Fig.IV,2 F i g i r I V . 6 s h o w s t h e i m a g i n a r y d i e l e c t r i c c o n s t a n t s , a g a i n a t 12 a n d 3 0 0  °K.  T h e f a c t o r - o f - 2 0 r e d u c t i o n i n r ( v ) i n g o i n g f r o m 3 0 0 t o 12 °K i s s e e n m o s t c l e a r l y i n t h e e" r e s o n a n c e  peak, whereas i t i s not so obvious i n t h e r e l a t e d  c o n s t a n t s n a n d k ( s i n c e e" = 2 n k ) . B.  Absorption  Coefficient  T h e a b s o r p t i o n c o e f f i c i e n t a f r o m 0 t o 3 0 0 cm * w a s c a l c u l a t e d f r o m  e'  a n d £?' a t 3 0 0 , 7 7 , a n d 12 °K a n d m a y b e s e e n i n F i g . I V . 7 . A l s o s h o w n a r e t h e present experimental data obtained at the three temperatures, temperature  d a t a o f R e f . 2, c o n v e r t e d f r o m k t o a .  t h e d a t a b e t w e e n 140 and 180 cm * -  and t h e room-  For t h e sake o f c l a r i t y  h a s b e e n o m i t t e d a n d may b e s e e n i n F i g . I V . 8  w h e r e 3 0 0 a n d 77 °K c u r v e s h a v e b e e n v e r t i c a l l y d i s p l a c e d . C.  Experimental  The measurements were performed w i t h a s t e p d r i v e and Golay d e t e c t o r . o b t a i n e d from Harshaw Chemical  on an RIIC.  FS 720 F o u r i e r  spectrometer,  The s i n g l e c r y s t a l s o f n a t u r a l KI were  Co. and c l e a n e d and p o l i s h e d t o t h e d e s i r e d  t h i c k n e s s d , w h i c h v a r i e d f r o m 0.01 t o 1.0 cm.  A s l i g h t wedge s h a p e was  Calculated imaginary d i e l e c t r i c constants of K  3 9  I a t 300 and  12°K  Slo  Fig.IV.7  WAVE  NUMBER  (cm  -1  )  Present e x p e r i m e n t a l and c a l c u l a t e d v a l u e s o f t h e a b s o r p t i o n c o e f f i c i e n t o f K I a t 3 0 0 , 7 7 , a n d 12°K, t o g e t h e r w i t h t h e r o o m - t e m p e r a t u r e e x p e r i m e n t a l d a t a f r o m R e f . 2. T h e e x p e r i m e n t a l d a t a o b t a i n e d b e t w e e n 1 4 0 a n d 1 8 0 c m have been omitted f o r c l a r i t y . 3 9  - 1  57 Fig.IV.8  O I  I  I  80  I  120 0  WAVE  NUMBER  I  I  160  I 200  (cm"" ) 1  Present experimental and c a l c u l a t e d values o f t h e a b s o r p t i o n c o e f f i c i e n t o f K I a t 300 a n d 77°K i n t h e r a n g e o m i t t e d f o r c l a r i t y f r o m F i g . 7. The r o o m - t e m p e r a t u r e e x p e r i m e n t a l d a t a from R e f . 2 have been i n c l u d e d . The d a t a f o r each temperature have been v e r t i c a l l y d i s p l a c e d , w i t h the o r d i n a t e a x i s and s c a l e f o r t h e 77°K d a t a o n t h e r i g h t . 3 9  58  produced i n order t o eliminate interference f r i n g e s . sonically cleaned  They were then  i n toluene, r i n s e d i n a l c o h o l , and d r i e d .  measurements o f i n t e n s i t y w i t h and without  ultra-  Transmission  t h e sample i n t h e beam, I and I ,  respectively, are r e l a t e d by I (1-R) 1  =  °  1  2  (l+k /n )e" 2  2  a d  2-2ad 1-R e  ( 3 )  R  where R i s t h e power r e f l e c t i v i t y and a t h e a b s o r p t i o n c o e f f i c i e n t . v , i t was c o n v e n i e n t agreed  t o correct f o rR by using t h e calculated values,  well with experimental  Above V  values.  2 was u s e d .  q  t h e measured  n e g l i g i b l e elsewhere.  reflectivity  q  i n E q . (3) was a b o u t 5%  a t 300 °K, a n d t h e e f f e c t was  The a v e r a g e o f s e v e r a l r u n s was t a k e n w i t h t h e most  r e l i a b l e data obtained i n the region o f ad equal t o u n i t y . t h i c k n e s s e s was n e e d e d .  Thus a range o f  A n e r r o r b a r - i s s h o w n o n t h e 3 0 0 °K d a t a  80 cm ''" a n d t h i s w a s t y p i c a l , a l t h o u g h -  around  a l a r g e r u n c e r t a i n t y should be a s s o c i a t e d  with the highest absorption i n Fig.IVj.&hwhichawastaththeil'imit'of  our. s i g n a l t o  T h e l a r g e r b a r s o n t h e 12 °K d a t a w e r e d u e t o s e v e r a l c a u s e s .  f i r s t was t h e l o w a b s o r p t i o n c o m b i n e d w i t h h i g h r e f l e c t i v i t y . temperature  which  2  T h e maximum e f f e c t o f t h e (1+k / n ) t e r m  i n t h e r e g i o n o f highest a b s o r p t i o n above V  noise.  Below  The s e c o n d was  i n s t a b i l i t y , w h i c h h a s a v e r y m a r k e d e f f e c t b e l o w 9 0 cm  the absorption i s due t o d i f f e r e n c e processes. due t o s m a l l amounts o f C I  -  where  F i n a l l y , a p e a k a t 7 7 cm  1  i m p u r i t y ^ i n t h e H a r s h a w K I was s u b t r a c t e d o u t 1  f r o m t h e d a t a , c a u s i n g some u n c e r t a i n t y i n t h e D.  The  remainder.  R e f r a c t i v e Index and R e f l e c t i v i t y  Fig'..':IV.9 s h o w s t h e p r e s e n t  c a l c u l a t e d values o f r e f r a c t i v e index  t o g e t h e r w i t h t h e 3 0 0 ° K m e a s u r e d v a l u e s f r o m R e f . 2.  T h e 3 0 0 °K c a l c u l a t e d  r e f l e c t i v i t y a n d p h a s e a n g l e a r e s h o w n i n F i g . I V p T O o g g g l e t h e r , with;.,the  C a l c u l a t e d r e f r a c t i v e i n d i c e s o f K I a t 3 0 0 a n d 12°K, the room-temperature experimental data from Ref.2. 3 9  together with  loO  Fig.IV.10  1.0 R  h  0.8  o A  PRESENT CALC. ' PRESENT EXPT. BERG + BELL EXPT. HADNI EXPT.  CO c D  0.6 >h-  o  LU  O  • o  PRESENT CALC. BERG + BELL EXPT.  LU -J  0.4  <  LU CC  LU CO < X  0.2 A  CL  40  80 WAVE  120 NUMBER  160  200  (cm' ) 1  C a l c u l a t e d 300°K r e f l e c t i v i t y and p h a s e a n g l e o f K I . T o g e t h e r w i t h the present experimental r e f l e c t i v i t y , the experimental data from R e f s . 2 and 14 a r e p r e s e n t e d . 3 9  m e a s u r e m e n t s f r o m R e f . 2, a n d t h e r e f l e c t i v i t y m e a s u r e d b o t h b y o u r s e l v e s and b y H a d n i . 1 5  Both o f t h e power r e f l e c t i v i t y measurements areee f a i r l y  w e l l but f a l l below t h e s q u a r e - o f - t h e - a m p l i t u d e measurement o f Berg and Bell.  T h i s l a t t e r measurement i s p r o b a b l y t h e more a c c u r a t e .  We w e r e a b l e  t o s e e t h e two s m a l l r e f l e c t i v i t y f e a t u r e s a t 145 a n d 155 cm apparent i n the other data.  n o t so  The s t r u c t u r e o v e r t h e peak depended on s u r f a c e  p r e p a r a t i o n and i s p r o b a b l y not a good i n d i c a t i o n o f t h e r(v) f e a t u r e s i n that r e g i o n (see Fig.IV)1).  I V T'4. A.  DISCUSSION  Three-Phonon Damping  T h e e v i d e n c e f o r t h r e e - p h o n o n d a m p i n g a r o u n d v a t 3 0 0 °K m a y b e Q  seen  i n F i g . l V , 7 ( f r o m 80 t o 100 c m ) , i n F i g . I V 5 ( f r o m 100 t o 130 c m ) , a n d i n - 1  F i g . I V J I O C f r o m 1 0 0 t o 1 2 5 cm '''). -  - 1  The more d i r e c t e v i d e n c e o f h i g h e r - p h o n o n  damping i s t h e p r e s e n t s e t o f a b s o r p t i o n measurements above t h e two-phonon limit  ( f r o m 200 t o 300 cm ^ ) . a t a l l t h r e e t e m p e r a t u r e s , s e e n i n F i g . I V . 7 .  Assuming t h i s i s indeed three-phonon  ( o r h i g h e r ) d a m p i n g o f t h e TO r e s o n a n c e ,  and n o t due t o s e c o n d - o r d e r e l e c t r i c moments, t h e n t h e s e measurements o f a may b e c o n v e r t e d t o d a m p i n g b y m e a n s o f t h e r e l a t i o n - a ( c m ; 4 v e * ( e +2) 2(M ++M~)v ( v )  . a ( c m  2  }  =  +4vtv  2 °- -2-2 - 2 - 2 2 - ' 9 n v c M*M { [ v - v +2v A A ( v ) ] (v)} oo o oo  W  w h i c h may b e o b t a i n e d f r o m E q . ( 2 0 ) i n R e f . 1 . I f T ( v ) i s r e m o v e d f r o m t h e d e n o m i n a t o r , s i n c e i t s e f f e c t w i l l be n e g l i g i b l e away from t h e r e s o n a n c e a t v , and i f t h e c a l c u l a t e d v a l u e o f r e f r a c t i v e i n d e x n i s used, s i n c e t h i s o w i l l be f a i r l y a c c u r a t e b e y o n d 180 cm \ t h e n Eq. (3) r e l a t e s r ( v ) due t o  F i g . .IV. 11  P o r t i o n s o f t h e t h r e e - p h o n o n ( a n d h i g h e r ) d a m p i n g o f t h e TO r e s o n a n c e o f K I a t ' 3 0 0 , 7 7 , a n d . 12°K. The m e t h o d s o f o b t a i n i n g t h e s e c u r v e s a r e e x p l a i n e d i n the. t e x t . 3 9  63  three-phonon measured  (or higher) processes, to the a b s o r p t i o n c o e f f i c i e n t  beyond  the two-phonon l i m i t .  a  T h e v a l u e s s o o b t a i n e d may b e  seen  i n F i g . IV". 11,- f r , o m ^ l 8 0 ^ c m ~ o \ i o n w a r d s . The 3 0 0 K K v a i u e s i n - F i g g I V l l between o  s  n  80 and 130 cm" ,  which are  1  r e s p o n s i b l e f o r t h e e f f e c t s m e n t i o n e d above, were o b t a i n e d i n an f a s h i o n b y m e r e l y s u b t r a c t i n g o u r v a l u e s o f F- . •2-phonon }  J  Berg and B e l l  6  (see F i g . 1).  (v) from t h o s e o f 1  T h i s was d o n e b e c a u s e t h e l a t t e r a u t h o r s  o b t a i n e d f a i r l y good agreement the v i c i n i t y of v .  v  approximate  between t h e o r y and e x p e r i m e n t f o r n and k i n  T h e i r agreement  f o l l o w e d from t h e i r peak-height  [ w h i c h d e t e r m i n e d t h e i r v a l u e f o r <j>' ' ( r ) ] a n d t h e i r l a r g e r s e c o n d 1  f i r s t d e r i v a t i v e s , which gave g r e a t e r v a l u e s o f r ( v ) around v  Q  fitting and  as e x p l a i n e d  i n S e c . I V 1.2A T h e f i n a l r e s u l t i n g 3 0 0 °K p a r t i a l c u r v e o f F i g . I V i l l a p p e a r s - v e r y : r e a s o n a b l e , p  b e i n g f a i r l y s m o o t h , a n d p e a k i n g a r o u n d 1 6 0 - 1 7 0 cm . 1  T h i s would correspond  to the c r e a t i o n o f t h r e e a c o u s t i c phonons from r e g i o n s near the c r i t i c a l p o i n t s ( e . g . L ^ , l^, 5 5 cm  1  low-energy  o r x p , w h i c h a l l have wave numbers  around  .  I t may  be n o t e d t h a t , c o n t r a r y t o e x p e c t a t i o n , t h e high-wave-number  t a i l and t h e d i s c r e p a n c y around v  Q  i n KI does n o t a p p r e a r t o be  noticeably  more p r o n o u n c e d t h a n t h e s i m i l a r e f f e c t s f o u n d i n t h e much h a r d e r L i F . may b e u n d e r s t o o d b y c o n s i d e r i n g t h e f o l l o w i n g a p p r o x i m a t e T  relations:  , (v)«:S(v) , /y^v,v~v , 2- p h o n o n '2-phonon 1 2 o'  0  This  0  (5)  K  F  T  u  3- p h o n o n  (v)«Sfv) ^  7  u  3-phonon  where y i s the reduced mass, v^,  /y v-,v v,v , 4  0  p  and  1 2 3 o a r e t h e wave numbers o f t h e  phonons i n v o l v e d , and S(v) a r e terms i n v o l v i n g t h e o c c u p a t i o n numbers  (6)  64  n f L j ) o f the phonons. ' JJ  S(v) , was f u l l y d e f i n e d i n R e f . 1, b u t may 2-phonon  r  v  0  given here f o r t h e summation  be  processes,  ^ 2-phonon-summation  1  J  ' J2 J  ^  J  x6[v-v(t,J )-7(-t,J )], 1  (7)  2  where n t j ) =[ ^ '  J  )  /  C  ^  (8)  V - l ] - . 1  The e q u i v a l e n t t h r e e - p h o n o n - s u m m a t i o n express-ion i s g i v e n  by  S(v),, , . . = [ n ( ^ , J , )n(]L,i„) 3-phonon-summation 1' 1 2 2' v  L  v  , J  1  +nCt ,J )nCt ,J )+nCk ,J )nCt ,J ) 2  2  3  3  1  1  3  3  +n(f ,J )+n(k ,J )+nCk ,J )+l] 1  1  x 6 1 v-v(t  2  x  2  3  3  , j p - v (t 2, j ) -v(t5, j |)l] 2  (9)  3  and t  1  +  f  Equations  2  +  t  3  = 0.  (10)  (5) and (6) do n o t c o n t a i n t h e p o t e n t i a l d e r i v a t i v e s , w h i c h  increase r a p i d l y i n magnitude with i n c r e a s i n g order of d e r i v a t i v e , nor any d e t a i l s o f t h e c o u p l i n g c o e f f i c i e n t s . T h e p o i n t i s t h a t we a r e c o n c e r n e d w i t h t h e r a t i o o f r_ , (v)/r , (v) f o r d i f f e r e n t compounds o f t h e 3-phonon^ 2-phonon ' 0  J  v  r  same s t r u c t u r e , a n d t h e v a r i a t i o n i n t h e r a t i o o f t h e f a c t o r s o m i t t e d , one compound t o a n o t h e r ,  i s much s m a l l e r t h a n t h a t o b t a i n e d  from the f a c t o r s  present. The r a t i o  (R f o r b r e v i t y ) i s t h e n g i v e n a p p r o x i m a t e l y ^ _ 3 - p ,h o n o n ( _v ) ^ S ( v3 )- ,p h ,o n o n ... r F  v  7  2-phonon^  S  J  ^2-phonon  v  ^  '  from  by  65  w h e r e v i s some a v e r a g e o r c h a r a c t e r i s t i c w a v e n u m b e r .  R o b v i o u s l y depends  o n t e m p e r a t u r e a n d t o some e x t e n t o n t h e p r o c e s s , b u t i n t h e h i g h - t e m p e r a t u r e — — 2 — limit S(v)_ a p p r o x i m a t e s t o QcT/chv") w h i l e S ( vJ ) „ , approximates 3 -phonon v v B l *• 2-phonon — —2 t o k^T/e'hv. T h u s R b e c o m e s p r o p o r t i o n a l t o T / y v . I n t h e g r o s s a p p r o x i m a t i o n DD _2 of equal c e n t r a l f o r c e constants f o r a l l a l k a l i h a l i d e s , y v i s constant. v  }  r r  Furthermore, t h e r a t i o o f t h e damping due t o any h i g h e r - o r d e r p r o c e s s t o t h a t d u e t o a n y o t h e r - o r d e r p r o c e s s w i l l c o n t a i n some p o w e r o f y v a n d may t h u s b e e x p e c t e d t o b e t h e same o r d e r o f m a g n i t u d e f o r a l l compounds structure at high temperatures.  o f t h e same  A similar situation exists i n the ratio o f  the peak h e i g h t s o f two-phonon damping t o t h e resonance frequency V  q  at high  t e m p e r a t u r e s , w h i c h may b e s e e n t o b e a p p r o x i m a t e l y e q u a l f o r K I a n d L i F . 3—5 T h i s r a t i o , h o w e v e r , i n v o l v e s y v , and depends much more o n t h e s t r u c t u r e 2-phonon. At v e r y low temperatures  i t i s i n t e r e s t i n g to observe that f o r the  summation p r o c e s s e s , S ( v ) becomes u n i t y and t h u s R s h o u l d b e even g r e a t e r for  L i F than f o r KI ( y ^ l p / y ^ ^ and L ^ p / j ( j ~ ^ ) • v  v  The d i f f e r e n c e p r o c e s s e s ,  which a r e t h e ones r e s p o n s i b l e f o r t h e g r e a t e r m a j o r i t y o f t h e damping a t v , a r e n o t s o s t r a i g h t f o r w a r d . I t may b e a r g u e d , h o w e v e r , t h a t R w i l l  again  be a p p r o x i m a t e l y c o n s t a n t f r o m one compound t o a n o t h e r , s i n c e t h e l a r g e r ^ ^3 V  phonon * * n t  ie  s o  ^  t e r  compounds w i l l b e compensated b y t h e l a r g e r y .  R at these low temperatures w i l l o f course be v e r y s m a l l f o r t h e d i f f e r e n c e processes.  A l t h o u g h 300 ° K may b e c o n s i d e r e d h i g h f o r K I b u t o n l y i n t e r -  m e d i a t e f o r L i F , t h e g e n e r a l s i m i l a r i t y i n t h e 300 °K s p e c t r a o f K I and L i F may t h u s b e u n d e r s t o o d .  66  B.  Two-Phonon Damping S t r u c t u r e  I t may b e s e e n f r o m F i g . I V . 7. t h a t _ n o i d i s t i . n e t f e a t u r e s e x i s t i n t h e d i f f e r e n c e spectrum 75 cm  (below'v ) o f KI a t any t e m p e r a t u r e s .  The hump a r o u n d  p r o m i n e n t a t 77 °K, m e r e l y r e f l e c t s t h e o v e r - a l l m a x i m u m i n t h e  d i f f e r e n c e r ( v ) which then decreases s t r o n g l y u n t i l the summation processes 3  start.  The s h a r p f e a t u r e s r e p o r t e d by R e n k  due t o i m p u r i t i e s . p e a k a t 77 cm  1  Almost c e r t a i n l y CI  5  a t 5 a n d 6 °K m u s t t h e r e f o r e b e  w o u l d be r e s p o n s i b l e f o r t h e  ( t r a c e amounts i n our Harshaw sample gave us a s i m i l a r  It i s also known  1 6  t h a t l a r g e r c o n c e n t r a t i o n s o f CI  s t r u c t u r e a r o u n d 60 cm- .  give rise to  peak).  band  The t e m p e r a t u r e v a r i a t i o n o f t h e p e a k s i s a  1  hard t o understand, a l t h o u g h the two-phonon background dependent.  sharp  is certainly  N e v e r t h e l e s s no s h a r p p e a k s c a n o c c u r i n t h e d i f f e r e n c e  o f any compound a t v e r y low t e m p e r a t u r e s , and i n t h e c a s e o f t h e  little  temperature spectrum  rocksalt-  s t r u c t u r e d o n e s , n o t w o - p h o n o n c o m b i n a t i o n s f r o m X, a s c l a i m e d b y R e n k , a r e a l l o w e d i n any c i r c u m s t a n c e s . I n t h e s u m m a t i o n r e g i o n , Figs.I¥-8jiandoIYaiQ s h ^ w d t wo . d i s t i n c t - f e a t u r e s . T h e p o s i t i o n s o f t h e s e a g r e e v e r y w e l l w i t h t h e o r y a t 77 °K  (161 and 150  a n d s e e m t o a g r e e s a t i s f a c t o r i l y a t 3 0 0 °K, w i t h t h e 5 % r e d u c t i o n t h a t applied  ( 1 5 5 a n d 1 4 5 cm :  The m a g n i t u d e  1  a s l i g h t anharmonic  cm ) - 1  was  frequency shift i s also present).  o f t h e s e c a l c u l a t e d f e a t u r e s , however,  together with the  a t wave numbers j u s t above them, a p p e a r s once a g a i n t o be t o o l a r g e .  'hump' The  c o u p l i n g , assumed t o be b e t w e e n n e a r e s t n e i g h b o r s o n l y , has o v e r a c c e n t u a t e d them, and n e x t - n e a r e s t - n e i g h b o r i n t e r a c t i o n s a r e n e c e s s a r y .  The f e a t u r e s  i n q u e s t i o n h a v e b e e n l a b e l e d 9, 1 0 , a n d 11 b o t h i n F i g . I V . l n a n d - i n - T a b l e I V . 3 I t may b e s e e n t h a t many s t r o n g c o m b i n a t i o n s f r o m n e a r e s t - n e i g h b o r c o u p l i n g h a v e b y c h a n c e c o n t r i b u t e d t o t h e s a m e p e a k s . ^.T.. a n d L ' L  are both  67  c o n t r i b u t i n g t o p e a k 9. E ( L A ) ^ ( T O ) , 4  L  3 i > and c o m b i n a t i o n s from t h e L  branches  l a b e l e d A a n d B i n F i g . I V . 4 l a l l t h e w a y f r o m L">to W - h a v e produced p e a k 1 0 . The f o u r s t r o n g c o n t r i b u t o r s t o t h e hump 11 m a y a l s o b e s e e n i n T a b l e  IV.3.  Thus t h e s e c a l c u l a t e d f e a t u r e s seem t o b e a t l e a s t t w i c e a s s t r o n g a s t h e y should be,  (and p o s s i b l y even more when t h e three-phonon  damping i s i n c l u d e d ) .  The r e a s o n w h y t h e same f e a t u r e s c a l c u l a t e d b y B e r g and B e l l a r e n o t a s l a r g e i s due m a i n l y t o t h e l o w e r r e s o l u t i o n o b t a i n e d f r o m t h e i r  wave-vector  grid. In v i e w o f t h e comments above and i n t h e s u b s e c t i o n o f Sec. I I , t h e a s s i g n m e n t s made b y S r i v a s t a v a and B i s t erroneous.  Furthermore,  4  may b e seen t o b e i n c o m p l e t e , a n d  from a n i n s p e c t i o n o fthe data they obtained, the  usefulness o fthe n u j o l mull technique i n i n v e s t i g a t i n g these damping processes appears t obe l i m i t e d . Moto.add.zd  W h i l e w o r k i n g w i t h C s l , B e a i r s t o and E l d r i d g e  Zn  (to b e p u b l i s h e d i n Can. J . Phys.) r e a l i z e d t h a t no c u b i c c o u p l i n g can  occur  t h r o u g h n e x t - n e a r e s t - n e i g h b o r i n t e r a c t i o n s , s i n c e t h e y a r e i o n s o f t h e same type.  C o u p l i n g can occur through f u r t h e r o p p o s i t e - t y p e - n e i g h b o r  due t o t h e l o n g - r a n g e Coulombic the features c a l c u l a t e d here.  interactions  terms and t h i s w i l l r e d u c e t h e s h a r p n e s s o f I t was a l s o found, however, t h a t the next-  n e a r e s t n e i g h b o r s should b e i n c l u d e d i n the r e p u l s i v e component o f the potential  lattice  energy. ACKNOWLEDGMENTS  The a u t h o r s w o u l d  l i k e t o t h a n k D r . G. D o l l i n g f o r t h e m a i n r o u t i n e s  of the s h e l l - m o d e l program, program,  a n d D r . R. H o w a r d f o r h i s a s s i s t a n c e w i t h t h a t  One o f them (K.A.K.) would  l i k e t oacknowledge  C o u n c i l o f Canada f o r a 1 year graduate f e l l o w s h i p .  the National Research  68  BIBLIOGRAPHY *Work s u p p o r t e d b y G r a n t No. A5653 f r o m t h e N a t i o n a l R e s e a r c h C o u n c i l o f Canada. 1  J.E.  2  J . I . Berg and E.E. B e l l , Phys. Rev. B4, 3572  3  J.E.  ^S.P.  E l d r i d g e and Roger Howard, Phys. Rev. B7, 4752  E l d r i d g e , Phys. Rev. B6,3128  (1973).  (1971).  (1972).  S r i v a s t a v a a n d H.D. B i s t , P h y s . S t a t u s S o l i d i B 5 1 , 8 5  5  K.F.  6  G . D o l l i n g , R.A. C o w l e y ,  (1972).  Renk, Phys. L e t t . 2 1 , 132 (1966). C. S c h i t t e n h e l m , a n d I . M . T h o r s o n ,  Phys. Rev.  147, 577 ( 1 9 6 6 ) . Note that f o rthe parameters  7  t o be compatible w i t h t h e program  as described  i n R e f . 1, t h e e l e c t r i c a l p o l a r i z a b i l i t i e s n e e d t o b e d i v i d e d b y t h e v o l u m e o f t h e u n i t c e l l i n cm  24  8  R . P . L o w n d e s a n d D.H. M a r t i n , P r o c . R. S o c . A 3 0 8 , 4 7 3 ( 1 9 6 9 ) .  9  A.M. Karo a n d J.R. H a r d y ,  H.F. M a c d o n a l d ,  11  Phys. Rev. 129, 2024  (1963).  M i l e s V. K l e i n , a n d T.P. M a r t i n , P h y s . R e v . 1 7 7 , 1292 ( 1 9 6 9 ) .  G . O . J o n e s , D.H. M a r t i n , P.A. M a w e r , a n d C . H . P e r r y , P r o c . R. S o c . A 2 6 1 , 10  I 3  t i m e s 10  A . S a v i t s k y a n d M.J.E. G o l a y , A n a l . Chem. 3 6 , 1627 ( 1 9 6 4 ) .  1 0  1 2  3  (1961).  J . E . E l d r i d g e , Phys. Rev. B6, 1510 (1972).  1 L f  I . G . N o l t , R.A. W e s t w i g ,  R.W. A l e x a n d e r , J r . , a n d A . J . S i e v e r s , P h y s . R e v .  157, 730 (1967). I 5  A . H a d n i , J . C l a u d e l , G. M o r l o t , a n d P. S t r i m e r , A p p l . O p t . 7, 1 6 1 ( 1 9 6 8 ) .  1 6  K . D . M o l l e r a n d W.E. R o t h s c h i l d , F a r I n f r a r e d S p e c t r o s c o p y 1971),ip. 540.  ( W i l e y , New  York,  69  17 18 19 20 21  J.A.B. B e a i r s t o  a n d J . E . E l d r i d g e , C a n . J . P h y s . , 51_, 2 5 5 0  R . F . W a l l i s a n d A .'A. M a r a d u d i n , P h y s . R e v . , 1 2 5 , 1 2 7 7 R.A.  C o w l e y , A d v a n . P h y s . 12_, 4 2 1  J.M.  Ziman, Electrons  K.W.  J o h n s o n and E.E. B e l l , Phys.. Rev., 1 8 7 , 1044  (1973)  (1962)  (1963)  and Phonons (Clarendon P r e s s ,  Oxford,  1960)  (1969)  22  W. C o c h r a n , R.A.  C o w l e y , G. D o l l i n g a n d M.M.  Elcombe, P r o c . Roy. Soc. A293,  433 (1966) J.M. Z i m a n , P r i n c i p l e s o f t h e T h e o r y o f S o l i d s Cambridge,  1969)  (Cambridge U n i v e r s i t y  Press,  

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