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Thermal conductivity of inert gas solids Hurst , Michael James 1969

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THERMAL CONDUCTIVITY OF INERT GAS SOLIDS by MICHAEL JAMES HURST B.Sc,  University o f Calgary, 1967  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE IN THE DEPARTMENT OF PHYSICS  We accept t h i s thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA OCTOBER, 1969  In p r e s e n t i n g an the  advanced  degree  Library shall  I further for  this  agree  scholarly  by  his  of  this  written  thesis  in p a r t i a l  fulfilment  of  at  University  of  Columbia,  the  make  that  it  permission  purposes  may be  representatives. thes.is  for  freely  It  financial  available for  permission.  Department  of  ^XyS  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  by  the  understood  gain  / c S Columbia  for  extensive  granted  is  British  shall  Head o f  be  requirements  reference copying  that  not  the  of  I agree and this  or  allowed  without  that  Study. thesis  my D e p a r t m e n t  copying  for  or  publication my  ABSTRACT The thermal c o n d u c t i v i t y o f p e r f e c t i n f i n i t e c r y s t a l s o f neon, argon, krypton, and xenon has been c a l c u l a t e d numerically.  I t was assumed that the c r y s t a l s possessed face centred  cubic s t r u c t u r e w i t h the atoms i n t e r a c t i n g i n p a i r s through a Lennard-Jones 12:6 p o t e n t i a l energy f u n c t i o n .  The c a l c u l a t i o n s  considered only the e f f e c t s o f three-phonon i n t e r a c t i o n s .  I t ..  was p o s s i b l e t o s i m p l i f y the c a l c u l a t i o n s by introducing " r e duced" p h y s i c a l q u a n t i t i e s .  The thermal c o n d u c t i v i t y o f each  o f the i n e r t gas s o l i d s considered was obtained from the " r e duced thermal c o n d u c t i v i t y " which was c a l c u l a t e d f o r argon. Agreement with experimental data f o r neon, argon, and krypton was obtained f o r temperatures higher than those f o r which the e f f e c t s o f c r y s t a l s i z e and l a t t i c e d e f e c t s determine the t h e r mal c o n d u c t i v i t y .  This agreement suggests that f o r s u f f i c i e n t l y  high temperatures  the thermal c o n d u c t i v i t y i s determined by  the e f f e c t s o f three-phonon Umklapp processes.  i  TABLE GF CONTENTS Page ABSTRACT  i  TABLE OF CONTENTS  i i i  LIST OF TABLES  .  v  LIST OF FIGURES  vi  ACKNOWLEDGEMENTS  v i i  CHAPTER I.  II.  1  INTRODUCTION Review o f previous s t u d i e s  1  Review o f l a t t i c e dynamics  4  Statement o f the problem  10  Organization o f t h e s i s  11  THEORY OF LATTICE THERMAL CONDUCTIVITY ......  14  Formal theory •  14  R e l a x a t i o n times  13  The t r a n s p o r t equation  13  E l i m i n a t i o n o f the time dependence o f the 20  t r a n s p o r t equation Approximation o f the phonon d i s t r i b u t i o n factor  24  •  26  A formula f o r the r e l a x a t i o n times  27  High and low temperatures III.  THERMAL CONDUCTIVITY OF INERT GAS SOLIDS .... The model  30  •  Interatomic interaction  30  •  30  32  The l a t t i c e iii  iv Phonon frequencies  34  •  Cubic term o f the c r y s t a l p o t e n t i a l energy  39  Relaxation times  40  ....  Thermal conductivity IV.  41 44  CALCULATIONS Preliminary remarks  •  ••  L a t t i c e points  . ••  Sampling points i n the r e c i p r o c a l l a t t i c e  44 44 45  Calculation o f the reduced thermal conductivity  ••  46  Reduced frequency spectrum  46  Reduced group v e l o c i t i e s  46  Contributing phonon processes  49  Reduced cubic c o e f f i c i e n t s  50  Reduced thermal conductivity ••••••••••••  50  Temperature v a r i a t i o n o f the reduced thermal conductivity  V.  54  Umklapp region  54  High temperature region  57  Calculation o f the thermal conductivities •  58  Errors  66  CONCLUSION  BIBLIOGRAPHY  ,  69 73  LIST OF TABLES TABLE I II  Page PHYSICAL DATA FOR THE INERT GASES REDUCED THERMAL CONDUCTIVITY FOR INERT 53  GAS SOLIDS III  33  COMPARISON OF PARAMETERS CALCULATED TO FIT THE REDUCED THERMAL CONDUCTIVITY  56  IN THE UMKLAPP, REGION IV  PRODUCTS OF THE REDUCED THERMAL CONDUCTIVITY AND TEMPERATURE IN THE HIGH TEMPERATURE 59  REGION FOR ARGON  v  LIST GF FIGURES FIGURE 1  Page Histogram of the reduced frequency spectrum  2  47  •  Comparison of experimental and calculated values f o r the thermal conductivity of neon  3  Comparison o f experimental and calculated values f o r the thermal conductivity of argon  4  62  63  Comparison o f experimental and calculated values f o r the thermal conductivity of krypton  5  •  ••  64  Comparison of two sets of calculated values for the thermal conductivity of xenon ......  vi  65  ACKNOWLEDGEMENTS I wish to express my gratitude to Dr. R. Howard f o r h i s suggestion o f t h i s problem and h i s advice on the c a l c u l a tions. I wish to acknowledge the assistance o f the s t a f f o f the University o f B r i t i s h Columbia Computing Centre, p a r t i c u l a r l y Mr. A. N. Keenan. I also wish to acknowledge the f i n a n c i a l assistance of the National Research Council o f Canada.  vii  CHAPTER I INTRODUCTION In d i e l e c t r i c solids a l l o f the electrons are bound to atoms of the l a t t i c e and heat i s transferred by the phonons o f the s o l i d .  The thermal conductivity of d i e l e c t r i c solids i s con-  sidered to be due to processes which interfere with the motion of the phonons.  These processes are r e f l e c t i o n at the c r y s t a l  boundaries, scattering i n the region of l a t t i c e imperfections, and interactions among the phonons. As long as there are few enough phonons, as i s the case at very low temperatures  (see Section I I . 3 ) , the phonons  are  able to encounter the bounding surfaces of the c r y s t a l and imperfections i n the l a t t i c e many times before they encounter i n teractions with other phonons.  Then the contributions to the  thermal conductivity are dominated by the effects of the boundaries and imperfections of the c r y s t a l .  For higher temperatures  the phonons become so numerous that the motion o f any phonon i s governed c h i e f l y by i t s interactions with other phonons.  For  these temperatures contributions to the thermal conductivity due to interactions among the phonons w i l l be dominant. 1.  REVIEW OF PREVIOUS STUDIES Experimental observations (de Haas and Biermasz ( 1 9 3 7  and  1938),  Berman, Simon, and Wilks  1  (1951),  White and Woods  2 (1956 and 1958)) i n d i c a t e t h a t the thermal c o n d u c t i v i t y /(<T) o f d i e l e c t r i c s o l i d s has the f o l l o w i n g p r o p e r t i e s . 1.  At very low temperatures  K(T)  - T  ( l e s s than about  \  5 K) a  (1.1)  where T i s t h e a b s o l u t e temperature o f the s o l i d and a. i s a p o s i t i v e number w i t h a value o f about two o r t h r e e , depending on the sample used i n t h e experiment. 2.  At s l i g h t l y h i g h e r temperatures t h e thermal cond u c t i v i t y a t t a i n s i t s maximum value and begins t o decrease r a p i d l y w i t h i n c r e a s i n g  temperatures.  Values o f the thermal c o n d u c t i v i t y remain dependent on t h e sample being examined, but the v a r i a t i o n i n v a l u e s d i m i n i s h e s as the temperature i n creases. 3.  At h i g h temperatures /C<T> « * ~r  ,  (1.2)  and experimental v a l u e s o f the thermal c o n d u c t i v i t y are independent o f t h e sample used.  F o r s o l i d neon,  argon, and krypton White and Woods (1958) found t h a t the thermal c o n d u c t i v i t i r begins t o have the tempera-  o  t u r e dependence o f (1.2) a t about 15 K. T h e o r e t i c a l c o n s i d e r a t i o n s (See, f o r example, Klemens (1958).) i n d i c a t e t h a t a t very low temperatures the e f f e c t o f t h e s c a t t e r i n g o f the phonons a t t h e c r y s t a l boundaries and near  3  various types of l a t t i c e imperfections i s that the temperature v a r i a t i o n (1.1) should be observed i n the thermal conductivity. Since the size and detailed structure of a r e a l c r y s t a l are c h a r a c t e r i s t i c of p a r t i c u l a r samples, i t i s to be expected that the values of the thermal conductivity should depend on the part i c u l a r sample used i n the experiment. At high temperatures the effect of interactions among the phonons determines the temperature dependence o f the thermal conductivity.  These interactions are v i r t u a l l y independent of  p a r t i c u l a r samples o f a d i e l e c t r i c s o l i d and produce the observed temperature v a r i a t i o n (1.2) f o r extremely high temperatures (See Sections 1.2 and I I . 3 » ) • For temperatures s l i g h t l y higher than that o f the maximum experimental value of the thermal conductivity, the temperature dependence i s quite d i f f i c u l t to determine t h e o r e t i c a l l y . P e i e r l s (1929 and 1956) has investigated the effects of the cubic anharmonic term of the c r y s t a l potential energy (See Section 1.2) and has concluded that f o r these temperatures /><SWJ,  (1.3)  where <5> i s the Debye c h a r a c t e r i s t i c temperature and )T i s a numerical constant of order unity.  Other investigators have  suggested that / ( < T ) - <  (®J  ^pl<S>/fr]\  (1.4)  4  but there has been l i t t l e agreement on the values of the parameters n. and (3 .  Berman, Simon, and Wilks (1951) stated  that ^ i s a constant with a value s l i g h t l y greater than two and depending on the s o l i d being considered.  Klemens (1958)  claimed that n=3 with {<S)//9) as an arbitrary parameter from a t h e o r e t i c a l point of view.  Ziraan (I960) expected  /?=3  and(S=2. .  Julian (1965) has developed a formula o f the form (1.4) which he claimed to be v a l i d for the temperature To  and f o r which  range  T and (S>= 1 . 0 3 0 .  (1.5)  Julian also showed that a non-  zero value o f n i s required i n formula (1.4) i n order that ft be constant with temperature.  However, the strong variation o f  the exponential factor i n (1.4) makes the actual value o f 71 not very c r u c i a l to the problem o f f i t t i n g experimental data. 2.  REVIEW OF LATTICE DYNAMICS Before proceeding with t h i s discussion o f thermal con-  d u c t i v i t y and stating the problem o f t h i s t h e s i s , i t i s useful to review l a t t i c e dynamics b r i e f l y to define terms and establish basic equations.  The theory which i s presented may be found i n  Maradudin, Montroll, and Weiss (1963) and L e i b f r i e d and Ludwig (1961).  In a perfect c r y s t a l l a t t i c e with one atom per unit c e l l and which i s "generated" by the three l i n e a r l y  independent  5 vectors «, , a * , and a  s  , the equilibrium position o f the atom  l a b e l l e d rfi i s R where in, ,yv\ >  a  • w i . f i . t w ^ t + 'Wjfij ,  and yw are integers.  The "reciprocal l a t t i c e "  3  x  (1.6)  corresponding to t h i s "direct l a t t i c e " i s generated by the vectors 5, , b , and  which are defined  K  Si> bj *2K$ij  j  by the r e l a t i o n  / je/,i,3.  (1.7)  ;  Points of the r e c i p r o c a l l a t t i c e are l a b e l l e d J\ and are denoted by the " r e c i p r o c a l l a t t i c e  vectors"  K - Kjbj + K ^ + Kjtj K  where h, , K  2  , and j} are integers. 3  >  (1.8)  As a r e s u l t of equations  (1.6)-(l.d), 2.7t^»*«j«r). The potential energy of the c r y s t a l l a t t i c e £  (1.9) is a  function of the position of each atom i n the c r y s t a l and has i t s minimum value I f the  $  0  when every atom i s i n i t s equilibrium  component of. the displacement o f the atom m from i t s  equilibrium position i s u£ be written  position.  »  t  n  e  c r y s t a l potential energy may  as a Taylor series expansion i n the displacements H  «<<*r  <  .  6 where the s u b s c r i p t " 0 " means t h a t the d e r i v a t i v e s are t o be evaluated f o r the e q u i l i b r i u m p o s i t i o n o f every atom.  The  term  In.  l i n e a r i n the displacements u^  vanishes because t h e r e are no c.  net f o r c e s on any o f the atoms when they are a l l i n t h e i r equilibrium positions.  The q u a d r a t i c term i s c a l l e d the  monic term" and a l l subsequent terms" o r " a n h a r m o n i c i t i e s " .  "har-  terms are c a l l e d "anharmonic The u s u a l method o f c o n t i n u i n g the  a n a l y s i s o f the motion o f the l a t t i c e i s to make t h e "harmonic approximation" by t r u n c a t i n g the s e r i e s ( 1 . 1 0 ) a f t e r the h a r monic  term. In the harmonic approximation, the equations o f motion  f o r the atoms o f the l a t t i c e form a system o f second o r d e r l i n e a r d i f f e r e n t i a l equations i n the displacements u£*  The  s o l u t i o n s o f the equations o f motion have s i n u s o i d a l time pendence, the angular f r e q u e n c i e s u>  de-  o f which must s a t i s f y the  "secular equation"  ( f a ^ ( £ ) - oo S^)j-o  (l.n)  z  where the "wave v e c t o r " t  i s a v e c t o r i n the space o f the r e c i p -  r o c a l l a t t i c e and the "dynamical  •LVs(£)s M  may  t  L  matrix",  J  ^  ^  /o  ,  e  (1.12)  be shown to be u n i t a r y and p o s i t i v e d e f i n i t e so t h a t i t s  eigenvalues are p o s i t i v e and may  be w r i t t e n a s t o .  mass o f each atom o f the c r y s t a l  and  X  i s the  7  £  s  £  -  (1.13)  The secular equation (1.11) i s a cubic equation i n the eigenvalues * o * .  For every vector £  (1.11) and each solution corresponds dynamical matrix,e .  there are three solutions of to an eigenvector of the  These three eigenvalues and corresponding  eigenvectors, or "polarization vectors", are labelled by the "polarization index" j :  c o = tf'W a  £  = e  DCVStfj)*  coVty&tj).  (Tj)  The "normal coordinates"  a.{]tj)  (1.14) of the c r y s t a l are  defined so that  £ e « < J O ) * ^ ) <-'"*'*  u * «  ,  (1.15)  where the summation i s over as many wave vectors y£* as there are atoms i n the c r y s t a l so that (1.15) defines the normal coordinates uniquely.  Using (1.15), the expansion of the c r y s t a l poten-  t i a l energy (1.10) may be rewritten i n terms o f the normal coordinates. (See L e i b f r i e d and Ludwig (1961).)  J J J  where  ff' *) *  c (  h*7?i*j).  c  -'*>>  J') (1.17)  i s called the "cubic c o e f f i c i e n t " and may be shown to vanish unless  The quadratic term of (1.16) i s the potential energy of a c o l l e c t i o n of independent harmonic o s c i l l a t o r s moving i n the normal coordinates  &(Xj)  with angular frequencies  <JO(£])  .  In a  t r a n s i t i o n to quantum mechanics the normal coordinates come coordinate operators  Q.Ct]) and  the vibrations i n the  normal coordinates may be treated as p a r t i c l e s called The operators  et(kj)  "phonons".  are bose operators and the phonons are  bosons* In the absence of anharmonic terms i n (1.16) the phonons are non-interacting p a r t i c l e s .  But fcheCanharmonic terms  do not separate into summations over single normal modes or phonon states.  These terms represent the manner i n which the  phonons may interact with each other. The number of normal coordinates which may be defined uniquely by (1.15) i s 3 A/' where N' i s the number of atoms i n the crystal.  In the case of i n f i n i t e c r y s t a l s , there would be an i n -  f i n i t e number of normal coordinates and an i n f i n i t e c r y s t a l potent i a l energy.  A convenient method o f normalizing the c r y s t a l po-  t e n t i a l energy and making the number of normal coordinates f i n i t e without s i g n i f i c a n t l y a l t e r i n g the d i s t r i b u t i o n of the angular frequencies  oo(k^)  ±  s  to impose " c y c l i c boundary conditions," such  9 that  UAJ^  ,  =  (1.19)  where /V^ (<** 1,2,3) are p o s i t i v e i n t e g e r s so l a r g e t h a t n e g l i g i b l e compared w i t h  is  The number o f normal c o o r d i n -  ates under t h e s e boundary c o n d i t i o n s i s 3A/ where A/ = A/,A/ A/ a  (1.20)  3  i s the number o f atoms i n the " c y c l i c volume" and the term " c r y s t a l p o t e n t i a l energy" means the p o t e n t i a l energy o f a c y c l i c volume o f the c r y s t a l . the  I f A/ i s chosen to be equal to  number o f atoms i n a r e a l c r y s t a l , then the p r o p e r t i e s o f  a c y c l i c volume o f an i n f i n i t e c r y s t a l a r e a very good a p p r o x i mation to the p r o p e r t i e s o f a r e a l The normal c o o r d i n a t e s  crystal. belong t o wave v e c t o r s  £  which have u n i f o r m l y d i s t r i b u t e d and d i s c r e t e v a l u e s , such t h a t  k« The " f i r s t  , «*/,3>3.  s  B r i l l o u i n zone"  (1.21)  of a c r y s t a l l a t t i c e i s that unit  c e l l o f the r e c i p r o c a l l a t t i c e which i s centred on the o r i g i n o f the  c o o r d i n a t e system o f the r e c i p r o c a l l a t t i c e .  v e c t o r k.  which ends on a p o i n t o u t s i d e o f the f i r s t  zone o f the l a t t i c e , a v e c t o r ^ the  For any wave Brillouin  which ends on a p o i n t  inside  f i r s t B r i l l o u i n zone may be found such that  t»I  +H . t  (1.22)  10  Because the wave vector appears i n the d e f i n i t i o n o f the dynamical matrix and the normal coordinates i n an exponential of the form ejp  (1.23)  [  which as a r e s u l t o f ( 1 . 9 ) i s equal to e )c p £ i f . * * J ^  (1.24)  the angular frequencies and normal coordinates corresponding to a wave vector It  are the same as those f o r a wave vector 1L i f  the two wave vectors satisfy; ( 1 . 2 2 ) .  Therefore, only wave vec-  tors which end on points inside the f i r s t B r i l l o u i n zone o f the l a t t i c e need be considered. 3.  STATEMENT OF THE PROBLEM The problem to be investigated i n t h i s thesis i s the  calculation o f the thermal conductivity of the inert gas solids by considering the interactions among the phonons o f t h e i r c r y s t a l s due to the cubic anharmonic term i n the expansion of the c r y s t a l potential energy.  The inert gases were chosen f o r  the c a l c u l a t i o n s for two reasons. 1.  Except f o r helium the inert gases c r y s t a l l i z e i n  face centred cubic l a t t i c e s .  These are p a r t i c u l a r l y  simple l a t t i c e s to deal with mathematically; therefore, the inert gases neon, argon, krypton, and xenon were chosen f o r the calculations. 2.  Experimental  and t h e o r e t i c a l studies have indicated  11 that the interaction among the atoms of inert gases and inert gas solids i s pairwise and c e n t r a l . Lennard-Jones 12:6  The  potential was chosen to represent  the interaction among the atoms of an inert gas s o l i d for the calculations of t h i s study.  (Whalley and .  Schneider ( 1 9 5 5 ) , Grindlay and Howard ( 1 9 6 4 ) . ) In the body of t h i s thesis i t i s assumed that the crystals'being considered have perfect i n f i n i t e l a t t i c e s with c y c l i c boundary conditions. The thermal conductivity of such c r y s t a l s i s due only to interactions among the phonons.  Of the  interactions which could occur i n these c r y s t a l s , only those due to the cubic anharmonic term of the c r y s t a l potential energy are considered since higher order anharmonic terms are not thought to provide s i g n i f i c a n t contributions to the thermal conductivity. (Berman, Simon, and Wilks  (1951).)  The results of the calculations were not expected to provide good agreement with experimental data at low temperatures.  But f o r temperatures  higher than that of the maximum  experimental value of the thermal conductivity, the effects of interactions among the phonons should dominate the contributions to the thermal conductivity. Good agreement was expected for such temperatures. 4.  ORGANIZATION OF THESIS In Chapter II a general theory of thermal conductivity  12  of d i e l e c t r i c s o l i d s i s presented.  The theory i s based on a  relaxation method and most of the chapter i s concerned with the derivation o f a formula f o r the effect o f the interactions among the phonons on the "relaxation time" f o r each phonon state.  Only the interactions allowed by the cubic anharmoni-  c i t i e s are considered. In Chapter I I I the general theory o f Chapter II i s applied to a model of the i n e r t gas s o l i d s .  I t i s shown that  the thermal conductivity of the inert gas s o l i d s may be written as K(T) where C and ®  * Ck (WT)  (I.25)  .  are constants c h a r a c t e r i s t i c of the s o l i d under  consideration.  i s c a l l e d the "reduced thermal  conductivity"  and i f i t i s calculated f o r any set o f values of (&/T  ), the  thermal conductivity of any i n e r t gas s o l i d may be obtained by using equation ( 1 . 2 5 ) with appropriate values o f C and © . Chapter IV describes the procedure by which the thermal conductivities were calculated numerically.  The temperature de-  pendence o f the reduced thermal conductivity i s determined.  For  neon, argon, and krypton, the calculated thermal conductivities are compared with the experimental data of White and Woods ( 1 9 5 # ) . For xenon the calculations are compared with calculations made by J u l i a n  (1965). Chapter V contains a summary o f the material i n the  body o f t h i s t h e s i s and comments on the problem o f the thermal c o n d u c t i v i t y o f the i n e r t gas  solids.  calculating  CHAPTER I I THEORY OF LATTICE THERMAL CONDUCTIVITY In t h i s chapter the effect on the thermal conductivity of d i e l e c t r i c solids due to the cubic anharmonic term of the c r y s t a l potential energy i s considered.  The theory i s taken  from the review a r t i c l e on thermal conductivity by Klemens ( 1 9 5 8 ) . 1.  FORMAL THEORY A d i e l e c t r i c s o l i d with a perfect i n f i n i t e l a t t i c e and  c y c l i c boundary conditions may be treated as a c o l l e c t i o n o f phonons belonging to the phonon modes (^j) , where the wave vector k. has values uniformly distributed throughout the f i r s t B r i l l o u i n zone and the p o l a r i z a t i o n index j has values 1 , 2 , and 3.  I f , as i s true i n a l l p r a c t i c a l cases, the temperature  gradient present i n a d i e l e c t r i c s o l i d i s small enough that the r e l a t i v e change i n temperature over the distance o f one phonon wavelength i s small, the phonons may be considered as l o c a l i z e d wave pickets moving with the group v e l o c i t y  where OJ Cti^) i s the angular frequency of the phonon mode  .  I f the number of phonons the heat current carried by t h i s c o l l e c t i o n o f phonons i s (2.2) 14  15 where  i s Planck*s constant.  uniformly distributed  Because the wave vectors are  throughout the f i r s t B r i l l o u i n zone,  for every phonon mode  which contributes to the heat cur-  rent there corresponds a mode (-Tt quency which also contributes. distribution"  with equal angular f r e -  Therefore, when the "phonon  A/(Hj) i s i s o t r o p i c i n the wave vector, the sum-  mation over )L i n (2.1) vanishes.  In t h i s case, the phonon d i s -  t r i b u t i o n i s the "equilibrium d i s t r i b u t i o n " V^^Xj) and (2.2) becomes Q  *£  L ^<tj)X*oCXj)^-(Xj)  ,  e  (2.3)  As mentioned i n Section 1.2, i n the quantum mechanical treatment the phonons may be treated as bosons, which have the equilibrium d i s t r i b u t i o n  7£(TCj)* fe*pU<ottj)/*r]-/}"' where K i s Boltzmann s constant. f  If *  (2.4)  i s the deviation o f  A / ( £ j ) from 9J>(Ej) then N ( t j ) * T J ( t j ) + r\CS.j)  and  (2.5)  (2.2) may be rewritten as <JN Z*ltij)-K«»Otj)7p<tj)  (2.6)  For a phonon mode which i s not i n equilibrium, i t i s assumed that the return to equilibrium i s exponential and that the following "relaxation law" i s obeyed.  i n time  16  t-(JTj) i s c a l l e d the "relaxation time" o f the phonon mode and depends on no other phonon mode nor on time. In the presence o f a temperature gradient VT  and no  other external disruptive e f f e c t s , the rate o f change o f the phonon d i s t r i b u t i o n may be written as  For small deviations  n(£j) , a f i r s t approximation to the devia-  tions from equilibrium may be obtained by replacing the second term o f (2.8) by * 7 £ a n d the relaxation law (2.7).  WJtJ)  in  comparing the r e s u l t with  I t i s found that  - [v<*> vT] t r f j )  S^j)/kco  (2.9)  where  i s the contribution to the s p e c i f i c heat o f a c r y s t a l due to a phonon mode  . The heat current (2.6) may be rewritten  Q = - 2Z l*(*$-VTh(tj)S(tj)Y(tj)  (2.11)  and compared with the d e f i n i t i o n o f the thermal conductivity tensor Kjt-m.  17  «  - T  </~  (VT)^  (2.12)  to y i e l d  K*y^*H Vu(Kj)tr^Jtj)f^tj) S<tj).  (2.13)  In t h i s thesis, the crystals which are to be i n v e s t i gated have cubic symmetry.  For such crystals the thermal con-  d u c t i v i t y tensor i s proportional to the identity^ that i s (2.14)  = K  or  Then, the thermal conductivity o f d i e l e c t r i c solids with cubic lattices i s x=  4r7-v <Tj)T(Tj) x  scZi).  (2.i6)  Equation ( 2 . 1 3 ) , and i t s p a r t i c u l a r case ( 2 . 1 6 ) , represents a complete formal solution f o r the thermal conductivity o f dielectric solids.  Since both the group v e l o c i t i e s and s p e c i f i c  heat contributions may be calculated from the phonon angular f r e quency spectrum of a c r y s t a l , a l l that i s required f o r the c a l culation of the thermal conductivity of any p a r t i c u l a r d i e l e c t r i c s o l i d i s i t s phonon angular frequency spectrum and the relaxation time f o r each phonon made.  The angular frequencies o f the pho-  nons may be calculated from the secular equation o f the dynamical matrix ( 1 . 1 1 ) .  The relaxation times are calculated i n the next  id section. 2. RELAXATION TIMES In the was  summary b f l a t t i c e dynamics i n Chapter I, i t  s t a t e d t h a t i n t e r a c t i o n s among the phonons o f a c r y s t a l  occur as a r e s u l t o f a n h a r m o n i c i t i e s i n the expansion o f c r y s t a l p o t e n t i a l energy.  The  cubic  anharmonic term i n v o l v e s  summation over t r i p l e s o f phonon modes and s i b l e i n t e r a c t i o n s among three phonons. c a l l e d "three^phonon p r o c e s s e s " .  the a  r e p r e s e n t s a l l pos-  These i n t e r a c t i o n s  are  Higher order anharmonic terms  r e p r e s e n t i n t e r a c t i o n s among l a r g e r numbers o f phonons.  They do  not  (Berman,  contribute  s i g n i f i c a n t l y to the thermal c o n d u c t i v i t y  Simon, and W i l k s (1951).) and formula f o r the r e l a x a t i o n The (195^) g i v e s  transport  are ignored i n the d e r i v a t i o n o f  a  times.  e q u a t i o n.  In h i s review a r t i c l e Klemens  the f o l l o w i n g t r a n s p o r t  t r i b u t i o n M(lCj) , c o n s i d e r i n g  equation f o r the phonon d i s -  o n l y three-phonon p r o c e s s e s . Co  Co  'to  (2.17)  where  A/*  N CFj),  A/'*  WSj"),  A c o = ± co i co 2 co "  /V"*  N(t"y)  (2.1a)  (2.19)  19 /  C  and  S't")  *  ( jj ' J " / i s  crystal  potential The  energy  the in  cubic  coefficient  normal  "resonance factor"  in  the  expansion of  coordinates.  of  the  transport  equation  "c^T* is  the  time  dependent  for  the  and  (!£"j") .  to  the  equal to  t i m e s "t  like  It  of  three  equation zero.  the  delta  are  the  the  only  this  the  selecting only  will  (X'j')t  contributions  interactions  section it  |6oJ~' ,  probability  m o d e s (Xj) ,  significant  from those in  than  function  quantum m e c h a n i c a l  phonons o f  Later  much g r e a t e r  a Dirac  of  ensures that  transport  almost for  interaction  part  -  (2 20)  ^co  with be  shown  resonance factor  that  behaves  those processes  for  which  ACO = O . In  the  corresponding to erators  for  phonons  in  tion"  of  phonons  quantum m e c h a n i c a l t r e a t m e n t  the  operators  the  are  coordinate  harmonic  , Ck'j'), in  of  the  coordinates  oscillators.  They  change o n l y  and  (It'j'Owith  creation  N  , of  ^  allow the  by t h e  For interactions  e a c h mode b e i n g  probability portional  normal  a n y mode t o  one phonon.  modes  (2.21)  ,  the and  number  "creation"  among t h e resultant ^ " ,  one phonon i n  or  three  £.(ltj) op-  of "destruc-  phonon  number  of  respectively, e a c h mode i s  the pro-  to  MJAljH , <oco'Co"  (2.22)  th  20 and the p r o b a b i l i t y f o r the destruction of one phonon i n each mode i s proportional to < M + / ) ( N .  (2.23)  However, the presence of the cubic c o e f f i c i e n t i n the transport equation requires that the wave vectors o f the three i n t e r a c t i n g phonons be related by the condition (1.18).  The r e s u l t o f t h i s  condition i s that (2.22) r e a l l y represents the a n n i h i l a t i o n o f at least one o f the phonons and (2.2 3) r e a l l y represents the creation of at l e a s t one of the phonons.  With the difference be-  tween (2.2 3) and (2.22) as a f a c t o r , the r i g h t hand side of the transport equation i s the probable change i n the number o f at time t due to three-phonon processes.  phonons i n the mode  The problem with the transport equation as i t stands i n (2.17) i s that i t contains time as an e x p l i c i t variable i n a form which prevents i t s elimination by simple algebraic manipulation.  Unless t h i s e x p l i c i t time dependence i s eliminated from  the transport equation, i t i s impossible to use the relaxation law (2.7) to derive time independent relaxation times. Elimination of the time dependence o f the transport equation.  For convenience the transport equation i s written  y—f *  *  "it ** 7  j'j"  where /ij  J  J  '* " J  J'J" includes a l l o f the time independent factors o f the  21 right hand side of (2.17). summation over Tt"  The condition (1.18) permits the  to be eliminated from (2.24) by making the  substitution  R -t-t.' x  (2.25)  so that the transport equation may  t  2 F J ' rf- ? t £  ^  be rewritten as  </"  •  I f a large enough c y c l i c volume V then the summation over the wave vector  £  { 2  -  2 6 )  has been chosen, may  be  satisfactorily  replaced by the following integrations  H  *'  where 7jf  H \tm  *~*fik? J***'  *  i s the gradient operator  cal l a t t i c e and  S'  i n the space of the recipro-  i s the surface &ta*o .  (2.27), equation (2.26) may  **'*<+<*) With the substitution  be rewritten  For 4a*>ffc o , the resonance factor varies with ACO l/r^w)  approximately as  1  (2.29)  but i n the l i m i t as A w - * o , the resonance factor becomes equal to ± X  Therefore,  /£  •  on the assumption of large enough times,"t »  (2.30) IAtol"'  22  which f o r non zero  AW  i s on the order o f  , the resonance  \oo\~'  f a c t o r i s considerably l a r g e r f o r aw--o than f o r  Aooi&  o  • The  i n t e g r a t i o n over but i n (2.28) may be performed by removing every f a c t o r but the resonance f a c t o r from t h a t i n t e g r a l and placing the r e s t r i c t i o n  AO*=O  on a l l f a c t o r s i n the i n t e g r a t i o n  over the surface s' • The r e s u l t i s  The maximum and minimum values o f Aw - 3 cOvw^x , r e s p e c t i v e l y , where  quency o f t h e phonon modes.  are + 3 ^ ^ and  i s the maximum angular f r e -  These, then, are the l i m i t s o f the  £>to i n t e g r a t i o n , but because o f the largeness o f t  , the l i m i t s  may be extended t o i n f i n i t y without s i g n i f i c a n t l y a l t e r i n g the value o f t h e i n t e g r a l but s i m p l i f y i n g i t s c a l c u l a t i o n .  It i s  e a s i l y shown t h a t  (°° ' ; T ^ ^ ^ A  g  VC*  (2.32)  I f t h i s r e s u l t i s s u b s t i t u t e d i n t o ( 2 . 3 1 ) , t h e e x p l i c i t time dependence cancels and the t r a n s p o r t equation may be w r i t t e n as  In any c r y s t a l f o r which the s u b s t i t u t i o n ( 2 . 2 7 ) i s v a l i d , t h e number o f phonon modes i s so great that i t i s pract i c a l l y impossible t o evaluate every angular frequency.  Usually  some method i s found to approximate t h e a c t u a l d i s t r i b u t i o n o f  23 angular  frequencies,  tributed  sample o f  s u c h as  a l l o f t h e wave v e c t o r s .  f e w enough wave v e c t o r s s e n t s too few a n g u l a r well  enough t h a t  stead, i t  It  %.  But any sample w i t h  t o make t h e c a l c u l a t i o n s  frequencies  feasible  c o u l d be p e r f o r m e d .  to t r a n s f o r m the i n t e g r a t i o n back to  i s w e l l known t h a t is  (See  Lighthill  tion  (2.34) becomes  i f $(%)  is  the Dirac  any f u n c t i o n c o n t i n u o u s  (1962).)  hand s i d e o f  P u t t i n g FCx)  s  at x  s  °  » then  I and X~4<*/l**\ , 1  (2.35) may be i n s e r t e d  i n t o t h e r i g h t hand The  a p p e a r b e c a u s e t h e r e s t r i c t i o n A<o=O h a s  on t h e i n t e g r a l s .  used to r e w r i t e the t r a n s p o r t  The s u b s t i t u t i o n equation  equa-  (2.35)  (2.33) w i t h o u t c h a n g i n g t h e e q u a t i o n .  f u n c t i o n need n o t same e f f e c t  a  delta  C^gr)5(£?H. side of equation  In-  .  f u h c t i o n a n d F~(*)  The l e f t  repre-  f o r t h e s u r f a c e S ' t o be known  an i n t e g r a t i o n o v e r i t  is desirable  summation o v e r  considering only a uniformly dis-  delta the  (2.37) may be  as  The s u b s t i t u t i o n %:&U)/lio\ i s used r a t h e r than so t h a t w i t h PCx) l , (2.34) i s d i m e n s i o n l e s s and i n d e p e n d e n t o f t h e p h o n o n modes ffc'j') and (X"j") . The a b s o l u t e v a l u e o f c o i s u s e d t o t h a t when (2.35) i s s u b s t i t u t e d i n t o (2.33) t h e s i g n s o f t h e terms o f t h e summations a r e not c h a n g e d . s  24  A  t o *  or, replacing  O  by  f^?" 1^1  tf/tJ  X"  and writing  w  /ffyj" e x p l i c i t l y J . ( 2 .37)  L  A<ut o  The energy o f a phonon with angular frequency co i s  *Kco  , therefore, the condition c W O means that only those i n s  teractions which conserve energy contribute to the rate o f change o f  N(JLJ)  .  Before deriving relaxation times from  (2.37)  i t i s possible to simplify the equation by approximating the factor containing the phonon d i s t r i b u t i o n s N  ,  hi'  , and /V' • /  Approximation o f the phonon d i s t r i b u t i o n f a c t o r . The factor o f ( 2 . 5 7 ) which contains the phonon d i s t r i b u t i o n s i s (2.3*)  [ <N+rt(lv'4/)(A/" )-NN'N"] +/  which may be expanded i n terms o f the deviations h > n ' , and n. . (•n7i'+nM" + n V ) +  +  (2.39)  The term i n ( 2 * 3 9 ) which i s independent o f the deviations from equilibrium may be shown to be zero because o f the condition Ao?= o and the algebraic form o f the equilibrium d i s t r i b u t i o n s  25 (2,4)•  In Section II.1 i t was assumed that the deviations  from equilibrium were small.  Therefore, the term o f (2.39)  which i s quadratic i n the n's may be neglected compared with the l i n e a r terms. nXTd'tTC'-ri)  To f i r s t order, (2.38) i s  + n Y > ? ^ ' i / ) + n"(77+ £  TJ'H)  .  (2.40)  Klemens (1958) gives the following alternative to equation  ( 2 . 9 ) f o r the deviations 1  =  (2.41)  K T ax  where (2.42) 5-  ?T-*.ir * .  (2.43)  a  I t w i l l be shown l a t e r i n t h i s section that the only important feature o f the vector /\ i s that i t i s p a r a l l e l to the temperature gradient VT •  I f the deviations from equilibrium given by  (2.41) are substituted into ( 2 . 4 0 ) , use may be made o f the fact that f o r A «o= o  to show that (2.40) i s equal to  l^^'+TV,).  (2.45)  26  A formula f o r the relaxation times. f i r s t order approximation  In terms of the  to the phonon d i s t r i b u t i o n factor  (2.38), the transport equation i s  • % U K ^ T  ^  (V\TT'+t)  (2.46)  J'J" A formula for the relaxation times follows d i r e c t l y from (2.46) and the relaxation law (2.7).  T=-TT^J«-^|:.fef±Sp |#^V /). l  r +  ,2.47)  J'J" The f a c t o r of (2.47) which contains the vector /I i s X-Tt  Since  '  (2.48)  "X appears i n both the numerator and denominator of (2.48),  only i t s d i r e c t i o n ( p a r a l l e l to the temperature gradient) affects the evaluation of the relaxation times.  In the calculation of  the thermal conductivity of cubic c r y s t a l s using (2.16), the summation over the wave vector k.  includes wave vectors uniformly  distributed throughout the f i r s t B r i l l o u i n zone, so that any choice for the d i r e c t i o n o f A i s equivalent to any other choice. The presence o f the r e c i p r o c a l l a t t i c e vector K.  in  (2.48) allows two types of i n t e r a c t i o n to be distinguished. In "normal processes" or "N-processes"  t^t"=S.  (2.49)  27 These processes do not contribute to the relaxation times or the thermal conductivity because they cause the factor to vanish.  (2.48)  Therefore, N-processes do not allow the phonon  modes to return to equilibrium.  However, they may be useful  i n rearranging the contents of the phonon modes so that the second type of interaction may occur.  "Umklapp processes" or  "U-processes" are those f o r which  TUTt'+£" K a  (2.50)  Those are the only three-phonon processes which contribute to the thermal conductivity of d i e l e c t r i c s o l i d s . 3.  HIGH AND LOW TEMPERATURES For very high temperatures, - K c o  T »  w  A  (2.51)  ,  the temperature dependence o f the relaxation times and the thermal conductivity assumes a very simple form.  The temperature  dependence of equation (2.47) f o r the relaxation times i s through the factor containing the phonon equilibrium d i s t r i b u t i o n s which for temperatures s a t i s f y i n g (2.51) has the form ^'-f^/'-fJ ^  -J^V. *  r  (2.52)  -  Each term i n (2.47) has t h i s temperature dependence.  The result  i s that I  T  °<  T  .  ,  (2.53)  28  The thermal conductivity depends on the temperature through the relaxation times and the s p e c i f i c heat contributions.  But f o r  high temperatures the s p e c i f i c heat contributions are almost constant, S(tj)  *  (2-54)  K,  so that the thermal conductivities have the same temperature dependence at high temperatures as the relaxation times, K(TW  (2.55)  X ".  Therefore, i n the l i m i t of very high temperatures the effect o f the cubic anharmonicities on the thermal conductivity agrees with that found experimentally. However, temperatures s a t i s f y ing ( 2 . 5 1 )  are much higher than the temperature at which the  b e h a v i o r r ( 2 . 5 5 ) i s found experimentally. Therefore, the pred i c t i o n ( 2 . 5 5 ) i s not relevant to the experimental s i t u a t i o n . In order to examine low temperatures i t i s tempting to write, i n analogy with T «  (2.51)  " K c o ^ A  (2.56)  .  Unfortunately there i s no non-zero minimum value f o r the angular frequencies and since the equations f o r the relaxation times and the thermal conductivity contain summations over a l l phonon modes, the expression ( 2 . 5 6 ) has no meaning.  What may be written  though, i s that f o r T «  t  co(tj)/K  (2.57)  29 the equilibrium number o f phonons i n the mode (Jij) i s  y<tj) ~ which for low temperatures  j i s very small.  For "T^K  (2.58) » there  are no phonons i n any mode, the s p e c i f i c heat contribution o f every mode vanishes, and the relaxation times given by (2.47) have f i n i t e non-zero values.  Therefore, the thermal conductivity  due to three-phonon processes vanishes, as i t must i f there are no phonons to carry heat i n the c r y s t a l .  CHAPTER I I I THERMAL CONDUCTIVITY OF INERT GAS SOLIDS In t h i s chapter, the theory of the previous two chapters i s applied  to a model f o r the inert gas s o l i d s .  shown that the problem of calculating the thermal  It i s  conductivity  of a l l of the inert gas solids f o r which the model i s v a l i d may be reduced to the problem of calculating the thermal cond u c t i v i t y of one of the solids and a set of constants for the other s o l i d s . 1.  THE MODEL The theory of Chapters I and I I i s s u f f i c i e n t for the  c a l c u l a t i o n of the thermal conductivity  due to three phonon pro-  cesses f o r any d i e l e c t r i c s o l i d about which enough i s known that the c r y s t a l potential energy may be written as a Taylor series expansion i n the displacements of the atoms of the c r y s t a l from t h e i r equilibrium  positions.  This may be accomplished by defin-  ing the form of the interaction among the atoms and the  structure  of the c r y s t a l l a t t i c e . Interatomic i n t e r a c t i o n .  Whalley and Schneider (1955)  have shown that experimental data f o r gaseous neon, argon, and krypton may be s a t i s f a c t o r i l y calculated  i f the interaction  among the atoms i s pairwise and of the form  30  31 CDC^r)  i s the Lennard-Jones 12:6 potential energy function f o r  a pair of atoms whose centres are separated by a distance V " . The parameters £  and C  to which they apply.  have values c h a r a c t e r i s t i c o f the gas  In terms o f the formula (3«D» <T i s the  value of Y * f o r which the potential energy i s zero and - £ i s the minimum value o f the potential energy. The a t t r a c t i v e part of the 12:6 potential energy, the inverse s i x t h power term, may be derived by considering the i n teraction between the instantaneous dipole moments o f a pair of i n t e r a c t i n g , e l e c t r i c a l l y neutral, chemically inert atoms.  The  inverse twelfth power term i n the 12:6 potential energy has no physical significance.  I t represents a short range repulsive  force which prevents the gas from collapsing, but the p a r t i c u l a r form of t h i s term was chosen f o r eas© o f mathematical manipulation. Grindlay and Howard (1964) have used the 12:6 energy to calculate the l a t t i c e energy at o°K function of s o l i d argon and krypton.  potential  and the Debye  0"  From t h e i r calculations  they obtained values f o r £ and <T which agreed well with the experimentally determined values f o r gaseous argon and krypton. On the basis of t h i s success, i t i s assumed that the potential energy of a c r y s t a l o f the inert gas solids i s  §» -k z i V ^ - )  (3.2)  32  ^(v -p  where led j  i s the 1 2 : 6 potential energy o f the atoms l a b e l -  t  i and  j  .  The prime on the summation means that  vary over every atom i n the c r y s t a l except t * J and cr  i c a l values of £  •  i  and  The numer-  used i n subsequent calculations are  those o f the gaseous state o f the i n e r t gases.  These values  are l i s t e d i n TABLE I . The l a t t i c e .  Except f o r helium the i n e r t gases crys-  t a l l i z e i n face centred cubic l a t t i c e s .  Since t h i s i s a p a r t i c u -  l a r l y simple type of l a t t i c e with which to work, i n what follows i t i s assumed that the i n e r t gas s o l i d s being considered are neon, argon, krypton, and xenon. For a face centred cubic l a t t i c e with l a t t i c e spacing O- , the vectors which generate the l a t t i c e are  a, = f (<*>', /) «  A  *  f  (3.3)  (>> °> o  The corresponding r e c i p r o c a l l a t t i c e i s body centred cubic and i s generated by the vectors  tx* V-g  (3.4)  0,1,-0.  There are two ways i n which the atoms may be l a b e l l e d . The f i r s t method uses the l a t t i c e vectors ( 3 . 3 ) . m, , m  lf  and  the atom l a b e l l e d m  For any integers  has the equilibrium p o s i t i o n  33  TABLE I PHYSICAL DATA FOR THE INERT GASES  t>  a.  c  £  << ( K) m  6)  no  Neon  20.18  0.492  4.35  12.035  92.32  6 4  Argon  39.94  1.69  5.31  12.831  98.42  so  Krypton  83.8O  2.30  5.68  9.7854  75.06  63  131.30  3.11  6.1  8.450  64.82  76.1  Xenon  Dobbs and Jones  «x = "K ("ir.) l/W £ W „ = White Julian  (1957)  , #<,= /.  7.6706  and Woods ( 1 9 5 8 ) (1965)  cr  e  75  e  250 * 8  270  e  732 *  34  *  +^<2 -  (3.5)  3  In the second method the atom l a b e l l e d Fn has the equilibrium position  - " f C^,^ ^ )= K  where wt, ,  , and  f m  3  (3. ) 6  are any integers such that  hn, + "Ma. «f »ij - ub\hCn*- i^CtZ^*^.  (3.7)  In the following analysis, i t i s convenient to use the second method o f denoting the equilibrium positions o f the atoms. Points o f the r e c i p r o c a l l a t t i c e are given by  = K , f , + K a _b A + K 3 b 3 where h, , 2.  , and h  3  (3.8)  are integers.  PHONON FREQUENCIES The angular frequencies  f o r the phonon modes o f the  inert gas s o l i d s may be obtained by solving the secular equation of the dynamical matrix.  For the model o f the i n e r t gas solids  presented i n Section I I I . l the dynamical matrix i s  ~ ( pFp ~ Tl*! "' 71  *  fi  **** ' *  The summation extends over a l l l a t t i c e s i t e s i n the c r y s t a l .  35 As i t stands i n (3.9), the dynamical matrix must be evaluated separately for every s o l i d to be considered. However, i t i s possible to write the dynamical matrix as a product of a constant c h a r a c t e r i s t i c o f the s o l i d being examined and a matrix independent of the s p e c i f i c properties of the i n e r t gas solids. Neglecting the zero point energy of the i n e r t gas s o l i d s , i t i s possible to calculate the l a t t i c e spacing at 0- , i n terms of the parameter c • 9  o°tc  9  Grindlay and Howard (1964)  have performed t h i s c a l c u l a t i o n and found that f o r a l l solids f o r which the model i s v a l i d a * 0  A  (3.10)  S"VJ CT.  In what follows thermal expansion i s neglected and the r e l a t i o n  (3*10) i s assumed to be v a l i d at a l l temperatures with a  o  re-  placed by A . I f the form o f J2.  given by (3.6) i s substituted  into (3.9), the dynamical matrix may be rewritten as  (3.11) where (3.12)  36  i s the "reduced dynamical matrix"-*-, = ^  ^  (3.13)  i s the "reduced wave vector", and  The reduced dynamical matrix i s not dependent on the p a r t i c u l a r properties of the s o l i d being considered. same eigenvectors €. as does  t  I t has the  but i t s eigenvalues are ~l>  where  •?-ftfr" i s the "reduced frequency". the eigenvalues o f  As with the eigenvalues of  (3.i ) 5  f£) >  are labelled by the p o l a r i z a t i o n  index j , which i s assigned so that  * fy^sftytfj). The phonon modes with j •= 3  (3.16)  are called "longitudinal" modes and  the modes with j = 1 or 2 are called "transverse" modes.  These  names do not necessarily describe the manner i n which the phonons are propagated.  They have been assigned only f o r the purpose o f  distinguishing the phonon mode with the highest phase v e l o c i t y { u>/\%\  ^v/fijl  ) f o r a given reduced wave vector from the other  two modes (Klemens ( 1 9 5 8 ) . ) .  Throughout t h i s chapter several "reduced" quantities are introduced. In most cases they are denoted by a t i l d e above the symbol f o r that quantity.  37 The form of the reduced dynamical matrix as a summation over the l a t t i c e s i t e s o f a face centred cubic c r y s t a l i n equation  (3.12) i s too complicated  i n a closed form. C-upCy)  to allow i t s evaluation  For any reduced wave vector Zj the elements  must be evaluated numerically and the reduced f r e -  quencies obtained by solving the secular equation o f the reduced dynamical matrix.  However, t h i s need not be done f o r every  reduced wave vector, because the symmetry of the c r y s t a l l a t t i c e causes many reduced wave vectors to be equivalent. Both the direct and reciprocal l a t t i c e s of c r y s t a l s of the inert gas solids have the symmetry o f the cubic point group  .  For any element  of the group Sj,  operator O R . i s such that i f y  the symmetry  i s any reduced wave vector  then so i s  The symmetry operators O  t  represent combinations of rotations,  r e f l e c t i o n s , and inversions and t h e i r application to a reduced wave vector changes the corresponding reduced dynamical matrix by an orthogonal transformation, which does not change the reduced frequencies.  For any  '  satisfying  (3«17)  j ) = £ Y y j ) S^'j)***.. The group  (3.18)  (3.19)  represents forty eight symmetry operations which  38 d i v i d e the f i r s t B r i l l o u i n zone i n t o f o r t y eight d i f f e r e n t equivalent  " i r r e d u c i b l e regions".  but  One such i r r e d u c i b l e r e g i o n  contains a l l reduced wave v e c t o r s f o r which * 9 * * y * * & * '  G  (3.20)  3  C a l c u l a t i o n s o f the reduced f r e q u e n c i e s  and p o l a r i z a t i o n  vectors  need be performed o n l y f o r those phonon modes whose reduced wave vectors  lie  i n the i r r e d u c i b l e r e g i o n defined by ( 3 * 2 0 ) .  reduced f r e q u e n c i e s  The  and p o l a r i z a t i o n v e c t o r s o f every other  phonon mode may be obtained from these c a l c u l a t i o n s by means of  equations  (3.18)  and  (3.19).  For c y c l i c volumes o f mflLcroscopic s i z e the number o f phonon modes i s on the o r d e r o f  to  .  Even the r e d u c t i o n i n the  amount o f c a l c u l a t i o n p r o v i d e d by the symmetry o f the l a t t i c e not enough to make the c a l c u l a t i o n o f *5> mode p r a c t i c a l l y p o s s i b l e .  a c r y s t a l with  may be obtained by performing  f o r a network o f u n i f o r m l y d i s t r i b u t e d ,  reduced wave v e c t o r s . of  f o r every phonon  An approximation to the a c t u a l d i s -  t r i b u t i o n o f reduced f r e q u e n c i e s the c a l c u l a t i o n s  and C  sample  T h i s method o f sampling the phonon modes  macroscopic c y l i c volume i s  s o l u t i o n f o r TP and  is  equivalent  to  the  o f every phonon mode i n a c r y s t a l with a  very much s m a l l e r c y c l i c  volume.  39  3.  CUBIC TERM OF THE CRYSTAL POTENTIAL ENERGY The c o n t r i b u t i o n o f the cubic anharmonicities t o the  c r y s t a l p o t e n t i a l energy i s  where  * *'  tfjyyi  '^iwfi  7>«*  Jo ^ &  G  i s the cubic c o e f f i c i e n t defined by (1.17).  (3.22)  e  > r  The r e s u l t o f  applying the model f o r i n e r t gas s o l i d s t o equation ( 3 * 2 2 ) i s C  •ttr  (ffj.)  ' H r.Jf * / *  S*SVO.  A/ i s the number o f atoms i n the c y c l i c volume.  (3.23)  The f u n c t i o n (3.24)  i s responsible f o r the c o n d i t i o n (1.18) which allows only Nprocesses and U-processes.  The other symbols i n ( 3 * 2 3 ) are  defined as  -2<^(¥~W^)W^e ^.2h<? (R )(it-e)(n-90(R-^ ,u  (3.25)  „  z  3 { 2 6 )  The d e r i v a t i v e s which appear i n ( 3 . 2 6 ) are  "* * r ]  <^YR*WIF)V«>=  <p <W«fe=-)V«>- - V * * ! w  3  ^ "  ^1].  (3.27) (3.28)  40 I f the form o f R. given by (3*6) and the reduced wave vectors are introduced into equations (3.23)-( 3.26 ), the following r e 1  duced quantities may be defined,  f  * •  -  «  ^  f  f  J  -  *  K  (  ?  ^  )  ^  S  ^  )  e  *  f  ^  ^  (3.29)  with  ^ 7tf)»  ±te)Vtf*T>  -vtf  (3.32)  so that at  c 4. RELAXATION TIMES I f the angular frequencies* the wave vectors, and the cubic c o e f f i c i e n t i n equation (2.47) f o r the relaxation times are replaced by the corresponding reduced quantities, i t i s poss i b l e to define a reduced relaxation time  T<Kj) where, from (2.47)  v  'j "  UvoJ  £(Jj)  such that  (3.34)  41  and B  (3.36)  ^  a  The reduced relaxation times have the temperature dependence f o r the factor containing the phonon equilibrium d i s t r i b u t i o n s and  /£ which are o f the form  = T YZ.yl-rA Therefore  ^(Jj)=  •  (3.38)  T) •  (3.39)  The reduced relaxation time at the temperature T, o f a s o l i d with <**©<, i s equal to that o f the s o l i d with ec?«c^ at the temperature T  x  ==  <  T  .  (3.40)  Therefore, the reduced relaxation times o f any i n e r t gas s o l i d may be obtained from those of any other i n e r t gas s o l i d by means o f the equality  ?<ji,3)"*te/,£) with T V given by ( 3 * 4 0 ) . 5. THERMAL CONDUCTIVITY I f the phonon group v e l o c i t y ( 2 , 1 ) i s written i n terms of the reduced frequency and reduced wave vector, the reduced group v e l o c i t y i s defined as  42  • £f?F *  « 1"^ .  (3.42)  With the reduced quantities defined so f a r , equation (2.16) f o r the thermal conductivity K may be written to define the "reduced thermal conductivity^ if J  Both the reduced relaxation times and the s p e c i f i c heat contributions depend on temperature through the variable  (3.44)  (o< / T ) As a result,so does the reduced thermal conductivity. If  i s the maximum value o f the reduced f r e -  y^cpc  quency f o r a l l phonon modes, the c h a r a c t e r i s t i c temperature (£) •Js  i s defined to be  (9= <<2?7**x.  (3.45)  7  which i s the high temperature l i m i t o f the Debye (S?-function. In the following analysis the v a r i a t i o n o f ®  with temperature  i s ignored because ( 9 ( T ) deviates from i t s high temperature l i m i t by l e s s than eight per cent (Grindlay and Howard (1964) .) and because the r e s u l t s o f t h i s study are expected to give good agreement with experimental data only f o r high temperatures. The reduced thermal conductivity o f two d i f f e r e n t i n e r t gas s o l i d s with equation  <S>, and (2>«  are related by the  43  ZC&M  » K(®*/TX)  (3.46)  with (3.47) In t h i s chapter a model f o r the i n e r t gas s o l i d s has been presented and applied to the general theory of l a t t i c e thermal conductivity developed i n Chapter I I .  I t has been shown  that every quantity which must be calculated i n order to evaluate the thermal c o n d u c t i v i t i e s , need not be calculated f o r each i n e r t gas s o l i d .  separately  Once the c a l c u l a t i o n of the reduced  thermal conductivity has been performed f o r any p a r t i c u l a r i n e r t gas s o l i d ,  the thermal conductivity of any i n e r t gas s o l i d  be obtained from equations ( 3 . 4 6 ) and  (3*43).  may  CHAPTER IV CALCULATIONS In t h i s chapter the c a l c u l a t i o n o f the thermal cond u c t i v i t i e s o f the inert gas s o l i d s i s outlined. 1.  PRELIMINARY REMARKS A l l o f the data required f o r the c a l c u l a t i o n o f the  reduced quantities introduced i n Chapter I I I were generated from the model o f the inert gas s o l i d s by an electronic computer.  Not u n t i l the thermal conductivities were calculated  from the reduced thermal conductivity was any experimental data required.  Although the value o f Q  f o r argon given i n TABLE I  was used i n the c a l c u l a t i o n s , any other value could have been chosen. L a t t i c e points.  In order to perform the l a t t i c e sums  from which the reduced dynamical matrix and the reduced cubic c o e f f i c i e n t s are calculated, the vectors TYL which l a b e l the atoms o f the face centred cubic l a t t i c e were generated and grouped into s h e l l s o f atoms with the same value o f l*yv./ . In x  the calculations the contributions o f the atoms were considered one s h e l l at a time f o r successively larger values o f l > u l , unx  t i l every single remaining s h e l l would contribute a r e l a t i v e .4/  amount o f l e s s than  \o  to the l a t t i c e sums.  more than the f o r t y seven s h e l l s with 44  In a l l cases not  i m l - 1 0 0 were required 1  45  for the c a l c u l a t i o n s . Sampling points i n the r e c i p r o c a l l a t t i c e .  In order  to approximate the reduced frequency spectrum, a network o f sample reduced wave vectors was set up by d i v i d i n g each o f the coordinate axes o f the space o f the r e c i p r o c a l l a t t i c e into A/» equal i n t e r v a l s and writing 9*  where n, , K \ , and n a  — 3  ^  V.  ,  <*.l)  are integers such that y corresponds to  an allowed wave vector (See Sections 1.2 and I I I . 2 ) .  The vectors  v\ which correspond to reduced wave vectors l y i n g i n the irreduc- , i b l e region (3*20) are r e s t r i c t e d by the r e l a t i o n s  The second o f the r e s t r i c t i o n s (4.2) requires that N be even i f a  rv i s to represent points on the boundaries o f the f i r s t B r i l l o u i n zone.. For t h i s study the value o f N « was chosen to be s i x , giving f o r t y - s i x points i n the i r r e d u c i b l e region defined by (4*2) and 1046 wave vectors i n the whole f i r s t B r i l l o u i n zone. value o f No would have given a better approximation  A larger  to the r e -  duced frequency spectrum, but with A/„= 6 the t o t a l amount o f data generated  i n l a t e r phases of the calculations was great enough  to cause problems because o f limited storage space i n the computer.  46 2.  CALCULATION OF THE REDUCED THERMAL CONDUCTIVITY Reduced frequency spectrum.  The reduced dynamical  matrix was c a l c u l a t e d from equation (3.12) f o r each o f the r e duced wave v e c t o r s defined by (4*1) and (4*2) w i t h A/ =4». As 0  a r e s u l t o f the t r u n c a t i o n o f the l a t t i c e sum mentioned i n S e c t i o n IV.1, i t was assumed t h a t the r e l a t i v e e r r o r i n each element o f the reduced dynamical m a t r i x was ± io  » The r e -  duced frequencies were c a l c u l a t e d from the secular equation o f the reduced dynamical m a t r i x and the p o l a r i z a t i o n vectors were obtained by s o l v i n g the equation C  ffj)*<?j), i'/,»,3  9  (4-3)  3  f o r the three mutually perpendicular u n i t v e c t o r s \ c ( y j )|.  j"  Figure 1 i s a histogram o f the reduced frequency spectrum based the sample o f 1046 wave vectors obtained from the f o r t y - s i x vectors n. found from (4.2) w i t h /V ~4 • The maximum 0  value o f the reduced frequency f o r t h i s sample o f wave vectors was 2?MCC*-  7.(*70b.  Reduced group v e l o c i t i e s .  (4.4) The reduced group v e l o c i t i e s  were c a l c u l a t e d from the reduced frequencies by a method o f numerical d i f f e r e n t i a t i o n . phonon mode  (T^L/^'j^  I f the reduced group v e l o c i t y o f the  i s t o be c a l c u l a t e d and i f 1?  i s denoted by /^>n.p .«*>A.«v*e  K * (ftt+J,  n * * M ,  n +p) 3  (4.5)  0-30-  0.25-  if) UJ Q  0-20-  U. O  0-15 Z o  g o  o.io  0-05-  REDUCED  Figure 1.  FREQUENCY, v  Histogram o f the reduced frequency spectrum.  48 then  Each component o f ~€r i n ( 4 * 6 ) i s a three point d i f f e r e n t i a t i o n formula derived from a three point Lagrangian formula.  interpolation  The fact that the point at which the derivatives are  to be evaluated i s a point used to obtain the interpolation formulas causes ( 4 . 6 ) to appear to be three two point formulas. (See, f o r example, Handbook o f Mathematical Functions* National Bureau o f Standards  (1965).)  The error caused by the approximation o f the reduced group v e l o c i t y by the numerical formula ( 4 . 6 ) involves derivat i v e s o f the reduced frequencies and i s much too d i f f i c u l t to evaluate i n terms of the standard error formula. t a i l e d consideration o f the c a l c u l a t i o n o f v  However, de-  f o r a few phonon  modes indicated that the error i n each component of the reduced group v e l o c i t y was about one per cent. In cases when the vectors n ' used i n formula ( 4 * 6 ) l a y outside o f the i r r e d u c i b l e region ( 4 . 2 ) ,  the values o f the r e -  duced frequency used i n ( 4 . 6 ) were those corresponding to rt " such that, f o r some symmetry operator ^  o f the group  and  some B * - d e f i n e d by ( 3 . 3 6 ) ,  K"*  O ^ f ^ V Vog*)  (4.7)  49 d i d l a y i n the i r r e d u c i b l e r e g i o n (4.2). C o n t r i b u t i n g phonon processes.  The o n l y three phonon  processes which c o n t r i b u t e t o the reduced thermal c o n d u c t i v i t y are U-processes which conserve energy, that i s A v « t y t y  t v  = o  (4.8)  $*«/'*$"»8 *$.  (4.9)  t  Because o f the f a c t t h a t only a sparse sample o f a l l o f the reduced wave vectors was considered i n the c a l c u l a t i o n o f the reduced frequency spectrum, i t seemed u n l i k e l y that (4.8) could be s a t i s f i e d e x a c t l y f o r any t r i p l e o f reduced frequencies. Instead, t r i p l e s were sought such that |A?| <S  • S^O.  (4.10)  Since t h e phonon modes corresponding t o  have V» c? , when  they take p a r t i n a three phonon process, the i n t e r a c t i o n i s j u s t a two phonon process.  Therefore the reduced frequencies which  were considered f o r f i n d i n g t r i p l e s s a t i s f y i n g (4.10) had a nonzero lower bound, the minimum value o f the reduced  frequency  spectrum i n Figure 1. The value o f S" was chosen to be much smaller than t h i s minimum value which was 1.187. For S < id" no t r i p l e s o f reduced frequencies s a t i s f y 3  i n g (4.10) and (4.9) were found and f o r S / o " two such t r i p l e s 5  representing 192 U-processes were found.  3  For S 3x/cT  t r i p l e s s a t i s f y i n g (4.10) and (4.9) were found.  s  S  nineteen  They represented  50 the 1080 U-processes which were chosen f o r the f o l l o w i n g c a l c u l a t i o n o f the reduced thermal c o n d u c t i v i t y * A c t u a l l y , only three phonon processes s a t i s f y i n g (4.8) c o n t r i b u t e t o the reduced thermal c o n d u c t i v i t y .  However, be-  cause the reduced frequencies are continuous f u n c t i o n s o f the d i s c r e t e reduced wave v e c t o r s , i t should be p o s s i b l e t o f i n d three allowed reduced wave vectors almost the same as ^ »<J'> and y  such that (4*9) remains s a t i s f i e d and f o r which (4.8)  is satisfied.  The 1080 U-processes used f o r the c a l c u l a t i o n s i n  t h i s t h e s i s may be regarded as convenient approximations t o processes which a c t u a l l y contribute t o the reduced thermal conductivity. Reduced cubic c o e f f i c i e n t s .  The reduced cubic co-  e f f i c i e n t s d ( j j ' j " / were c a l c u l a t e d from (3.33) f o r each o f the 1080 U-processes s a t i s f y i n g (4.10) w i t h S - 3 x / o ~  3  . The l a t t i c e  sums were performed by t r e a t i n g the atoms as p a r t s o f s h e l l s (See S e c t i o n IV.1).  The r e l a t i v e e r r o r i n each reduced cubic  c o e f f i c i e n t was assumed to be ± IO  because o f the t r u n c a t i o n  o f the l a t t i c e sum. The values o f \C \ j J ' j " / I  » which appear  i n the formula f o r the reduced r e l a x a t i o n times, ranged from about io  t o /o , but almost a l l were i n the range /o t o  Reduced thermal c o n d u c t i v i t y .  The reduced r e l a x a t i o n  51  times *fc were not c a l c u l a t e d e x p l i c i t l y .  Instead, the reduced  thermal c o n d u c t i v i t y was c a l c u l a t e d i n two steps. In the f i r s t step the temperature dependent f a c t o r s .appearing i n the formulas f o r the reduced r e l a x a t i o n times ( 3 * 3 5 ) and the reduced thermal c o n d u c t i v i t y (3*43) were c a l c u l a t e d .  In  the second step the s p e c i f i c heat c o n t r i b u t i o n s and the f a c t o r s  were c a l c u l a t e d f o r each o f the 1080 U-processes considered as c o n t r i b u t i n g to the reduced thermal c o n d u c t i v i t y .  These temp-  erature dependent f a c t o r s and the c a l c u l a t i o n s o f the f i r s t step were m u l t i p l i e d and added together according to the r e quirements o f equations (3» 35) and ( 3 . 4 3 ) t o give the reduced thermal c o n d u c t i v i t y . The c a l c u l a t i o n s were performed f o r argon f o r which (C) = 9 8 . 4 2 K (See TABLE I and S e c t i o n I I I . 5 ) f o r temperatures 0  from 5*K t o 500°K. The lowest temperatures considered were lower than the temperature a t which the maximum value o f the experimental thermal c o n d u c t i v i t y occurs, and, t h e r e f o r e , i n the region i n which t h i s a n a l y s i s should not be expected to give good agreement w i t h experimental data. Such temperatures were considered, i n order that the temperature dependence o f the c o n t r i b u t i o n s t o the thermal c o n d u c t i v i t y due t o three phonon processes could be determined i n the temperature region  52  f o r which the e f f e c t s o f c r y s t a l size and l a t t i c e defects become i n s i g n i f i c a n t *  The highest temperatures considered were  f a r beyond the temperatures f o r which any o f the inert gases remain s o l i d , but at these temperatures the  T'  1  dependence of  the thermal conductivity should be more apparent than f o r lower temperatures* Three d i r e c t i o n s f o r the vector ^  (See Section I I . 2 )  were used i n the calculations (4.12)  The values of the reduced thermal conductivity f o r these three directions did not d i f f e r from each other by more than one part i n one hundred at low temperatures and one part i n one thousand at high temperatures. TABLE I I gives the average o f these three values o f the reduced thermal conductivity. been normalized so that f o r T=/.o2®  The values have  /.ooo  y  erg./*K.  The  temperatures f o r which the other i n e r t gas s o l i d s have the same reduced thermal as argon have also been tabulated.  These  temperatures were calculated from the equation (4.13)  where (S> i s one o f the c h a r a c t e r i s t i c temperatures calculated from the parameters  and T ^ K ^ X  • (  See  TABLE 10.  TABLE I I REDUCED THERMAL CONDUCTIVITY FOR INERT GAS SOLIDS  T  G>  <"/<)  ( °K>  ( °K)  0.0508  5  4.69  3.81  3.29  380.8  0.0610  6  5.63  4.58  3.95  115.3  0.0711  7  6.57  5.34  4.61  0.0813  8  7.50  6.10  5.27  0.0914  9  8.44  6.86  5.93  18.13  0.102  10  9.38  7.63  6.59  12.88  0.152  15  14.1  11.4  0.203  20  18.8  15.3  13.2  3.492  0.305  30  28.1  22.9  19.8  2.564  0.406  40  37.5  30.5  26.3  2.123  0.508  50  46.9  38.1  32.9  1.807  0.610  60  56.3  45.8  39.5  1.565  0.711  70  65.7  53.4  46.1  1.376  0.813  80  75.0  61.0  52.7  1.224  0.914  90  84.4  68.6  59.3  1.101  1.02  100  93.8  76.3  65.9  1.000  2.03  200  188.  153.  132.  0.5147  3.05  300  281.  229.  198.  0.3450  4.06  400  375.  305.  263.  0.2593  5.0^  500  469.  381.  329. ,  0.2076  a  V a l u e s normalized so that f o r  50.98 1  9.88  *Wo*8  The value a c t u a l l y c a l c u l a t e d was 1.375x10"° 53  29.23  5.072  100°K, K « 1.000.  / «V.J/°K.  54 3.  TEMPERATURE DEPENDENCE OP THE REDUCED THERMAL CONDUCTIVITY In examining the temperature dependence of the r e -  duced thermal conductivity K , two temperature regions are important.  For temperatures just higher than the temperature at  which the maximum value o f the experimental thermal conductivity occurs, i t i s thought that the most important contributions are those due to interactions among the phonons. These interactions have been shown to be Umklapp processes, and t h i s temperature region i s called the "Umklapp region."  The other important  temperature range i s the "high temperature region"; f o r which the experimental thermal conductivity varies as T~', Umklapp region.  tS ^  (ST  In the Umklapp region,  -&  5  (4.14)  (Julian (1965).), i t i s expected that the temperature dependence of both K  and JC should be of the form A  ( W )  n  <3>/j*W  (4.15)  (See Section 1.1). An attempt was made to f i t K with the curve,, (4.K) by a method o f least squares with the "best" values o f (\ , ft , and {3 to be calculated. unstable.  However, the method was numerically  Therefore, the value of Y\ was chosen a r b i t r a r i l y and  the points  u ^ ^ m i j m n  u.i6)  55  were f i t t e d w i t h the l i n e ^  U A ^ f ( ^ )  (4.17)  by the usual method o f l e a s t squares f i t t i n g .  For each value o f  Y\. the parameters A , ft , and the mean square d e v i a t i o n  's't£u±* were c a l c u l a t e d . value f o r which 5  + *J«(*)J-[J«A++(f) l}*  (4.18)  i  The "best" value o f n  was assumed to be that  had i t s minimum value.  T h i s f i t t i n g procedure was a p p l i e d to s e v e r a l temperat u r e i n t e r v a l s , none o f which contained temperatures higher than •  For a l l o f the temperature i n t e r v a l s considered the best  i n t e g e r value o f M  was one.  greater than about l e s s than one. ceeded €>/6  I n t e r v a l s c o n t a i n i n g temperatures  had best values o f v\ which were s l i g h t l y  When none o f the temperatures i n an i n t e r v a l ex-  , the best value o f K  The r e s u l t s o f f i t t i n g £  was s l i g h t l y g r e a t e r than one.  f o r the temperature range —j—  O.OS-OS^'Q  are given i n Table I I I .  ^  OJS3.  (4.19)  The best value o f K was found by estim-  a t i n g from a graph the value o f K T h i s value was 1*1754 0.003.  f o r which S  was minimized.  The best i n t e g e r value was 7t» 1.  The eases *n = 3 and 71*$ are o f i n t e r e s t because these were the values p r e d i c t e d by Ziman (I960) and J u l i a n (1965), respectively.  For n. = 3  Ziman expected that  , w h i l e the  TABLE I I I COMPARISON OF PARAMETERS CALCULATED TO FIT THE REDUCED THERMAL CONDUCTIVITY IN THE UMKLAPP REGION 5  A  K  a  0  0.5112  3.020  4.96x10-3  1.00  2.185  3.423  1.08x10"^  1.05  2.349  2.399  5.75x10-5  1.07  2.419  2.390  4.24x10-5  1.09  2.490  2.381  3.02x10-5  1.10  2.526  2.376  2.52x10-5  1.11  2.563  2.371  2.10x10-5  1.12  2.601  2.367  1.74x10-5  1.15  2.717  2.353  1.12x10-5  1.17£  2.8l*>  2.34  <10-5  1.20  2.921  2.331  1.55x10-5  1.30  3.378  2.267  7.89x10-5  2.00  9.338  2.023  2.57x10-3  3.00  c  8.00  d  39.91 5692.  b  1.737  c  1.23x10-  1.017  d  1.71X10"  2  2  A curve o f the form fi(^) ^xp(^r) was f i t to the r e duced thermal c o n d u c t i v i t y f o r the temperature region 0.0508* /® 5 0.152. a  n  T  n  bInterpolated values. By i n t e r p o l a t i o n the w i t h minimum S i s 1.175 ± 0.003. c  d  value o f  Ziman (I960) expected n=3, J u l i a n (1965) expected f o r t h i s temperature range  <s« 1.030.  56  w  *  8,  57 value c a l c u l a t e d from K was 1.737.  Ziman s p r e d i c t i o n was f  based on s t u d i e s which considered temperatures lower than ^ 6 0 . Since the best value o f n. seems to i n c r e a s e as the temperat u r e s f o r which K i s f i t t e d decrease, i t i s not u n l i k e l y t h a t a temperature i n t e r v a l could be found such that the best value o f n. would be t h r e e . For TL"9 J u l i a n predicted that (3= 1.030 which i s remarkably c l o s e t o the value o f 1.017 c a l c u l a t e d from Al . J u l i a n , however, used smaller values o f (S> f o r argon.  This  f a c t would r e s u l t i n lowering the temperatures i n the i n t e r v a l (4.19).  Such i n t e r v a l s have temperatures lower than the lowest  temperatures f o r which /< was c a l c u l a t e d .  Therefore, i t was  not p o s s i b l e to f i n d the best value o f w. f o r t h e i n t e r v a l (4.19) w i t h the lower values o f ®  .  High temperature r e g i o n .  White and Woods (1958) have  found t h a t f o r temperatures higher than about 15°K, t h e thermal c o n d u c t i v i t i e s o f neon, argon, and krypton vary asT"*'.  In  S e c t i o n I I . 3 i t was shown that such behavior would occur f o r temperatures 7- ^> ®  }  but the values o f ® than 15°K.  (4.20)  f o r the i n e r t gas s o l i d s are a l l much greater  (See TABLE I . ) .  The p r e d i c t i o n s o f S e c t i o n I I . 3  have l i t t l e relevance t o the experimental s i t u a t i o n because temperatures s a t i s f y i n g (4.20) would exceed the m e l t i n g p o i n t s o f  5* the  i n e r t gas s o l i d s . In order to determine the temperature region f o r  which the reduced thermal c o n d u c t i v i t y v a r i e s as T~' > values o f T>  R 6 E N  K (•?)  (4.21)  with —  • «  >  O/S-3.  were c a l c u l a t e d and l i s t e d i n Table IV. for  T  >o.zo3®  ii  22)  These values increase  and seem to approach a constant value o f about  104 ergs f o r T > ©  .  Although the values o f (4.21) vary too  much to e s t a b l i s h an exact 7~~'relationship, an e r r o r o f l e s s than t e n per cent would be incurred i f , f o r T  ±0.&o2®  , the  values o f (4.21) were replaced by t h e i r average value 99.2, and for  the values deviate by l e s s than twenty s i x per  cent from t h e i r average value o f 9 2 . 7 .  For  T<ojs-s.®  , K  varies  so s t r o n g l y w i t h temperature that the assumption o f a T~' dependence would be absurd.  These temperatures d e f i n i t e l y do not  belong to the h i g h temperature r e g i o n , but to the Umklapp r e g i o n . 4.  CALCULATION OF THE THERMAL CONDUCTIVITIES As the f i n a l stage o f the c a l c u l a t i o n s , the reduced  thermal c o n d u c t i v i t y was used t o o b t a i n the thermal c o n d u c t i v i t i e s of the i n e r t gas s o l i d s .  Equation (3*43) could not be used f o r  t h i s purpose because the thermal c o n d u c t i v i t y K  defined i n Sec-  TABLE IV PRODUCTS OF THE REDUCED THERMAL CONDUCTIVITY AND TEMPERATURE IN THE HIGH TEMPERATURE' REGION FOR ARGON*  T  <S>  a  ftc\  0.152  15  75.38  0.203  20  69.29  0.305  30  76.80  0.406  40  84.92  0.508  50  90.38  0.610  60  93.94  0.7U  70  96.31  0.813  80  97.96  0.914  90  99.13  1.02  100  100.0  2.03  200  102.9  3.05  300  103.5  4.06  400  103.7  5.08  500  103.8  F o r other i n e r t gas s o l i d s the values o f r  and T&(£)  are p r o p o r t i o n a l to the values given i n the t a b l e f o r argon (T/r^  0 N  = <W)/  98.42).  59  60 t i o n I I . 1 d i f f e r from the experimental thermal c o n d u c t i v i t y ^ejipt by a f a c t o r w i t h the u n i t s of volume. I f  kexp-t * Vk then two cases may be considered. 1.  For some temperature i n the h i g h temperature r e gion the value o f /(eypt (White and Woods (1958).) may be compared w i t h the value o f K  calculated  from ( 3 . 4 3 ) t o f i n d the value o f V . I t was found, that V ~  io N H  (cm).  X  For macroscopic c r y s t a l s ,  (4.24) N^-io*  3  and l / i s much  l a r g e r than the volume o f a sphere j u s t containing the o r b i t o f P l u t o . 2.  I f i t were the case that V were the volume o f the c r y s t a l , then f o r macroscopic c r y s t a l s (\/-^|cm) a 3  comparison o f /de-ycp-h i n d i c a t e s that K /o*'.  and /< c a l c u  i s too l a r g e by a f a c t o r o f order  Such an overestimation may have been due t o  the f a c t that i n summations over redueed wave vect o r s only a small f r a c t i o n (about /o" cluded i n the c a l c u l a t i o n s .  ) were i n -  T h i s would r e s u l t i n  an underestimation i n the a c t u a l value o f the summation and since the r e c i p r o c a l s o f t h e reduced r e l a x a t i o n times are c a l c u l a t e d from such a summa-  61 t i o n , the values o f f  would be overestimated.  In any case, i t i s apparent t h a t no s e n s i b l e agreement w i t h experimental data may be obtained by using (3»43) t o c a l c u l a t e K . I n S e c t i o n IV. 3 i t was found that K has approximately the s o r t o f temperature dependence that has been p r e d i c t e d theo r e t i c a l l y and found experimentally f o r the thermal c o n d u c t i v i t y . Therefore, the thermal c o n d u c t i v i t i e s o f neon, argon, and krypton were c a l c u l a t e d by f i n d i n g a value o f the constant 8  such that  'QK[TJ  (4.25)  agrees e x a c t l y w i t h the experimental data o f White and Woods (1958) f o r one temperature  i n the high temperature r e g i o n . A l -  though no experimental data e x i s t f o r the thermal c o n d u c t i v i t y o f xenon, J u l i a n (1965) has c a l c u l a t e d t h a t f o r high ^xejtoK  *  ^  temperatures  (to* e > j . / ° / < / s e i . / c M . ) .  (4.26)  The thermal c o n d u c t i v i t y o f xenon was c a l c u l a t e d by f o r c i n g (4.25) and (4.26) t o agree e x a c t l y f o r one temperature i n the high temperature r e g i o n .  The r e s u l t s o f the c a l c u l a t i o n s are  compared w i t h the curves t o which /< was f i t i n Figures 2-^5. I n Figures 2-4, the s o l i d curves are the experimental r e s u l t s o f White and Woods (1958), the dashed l i n e s are high temperature  e x t r a p o l a t i o n s o f the experimental data, and the  dots are the values c a l c u l a t e d from K. , For neon (Figure 2) and argon (Figure 3), there i s order o f magnitude agreement be-  62 lOO  NEON  10  o  UJ  2O UJ Q  <S> I  or UJ o  o-i I 1-5  i  i  I I i l l i  J  I  10  I I I I I I lOO  500  T(°K)  Figure 2- Comparison of experimental and calculated values for the thermal conductivity of neon. The curve represents the experimental data of White and Woods (1958). The dots represent the calculations made for this study.  63 IOOO  ARGON  100  —  10  -  o 5£  15  J  I II Mill  IO  J  I I I I III  I  lOO  •  v  *  I  L  500  Figure 3. Comparison o f experimental and calculated values for the thermal conductivity of argon. The curve represents the experimental data of White and Woods (1958). The dots represent the calculations made for this study.  64 IOOO  KRYPTON  500 Tt°K) Figure 4. Comparison o f experimental and calculated values for the thermal conductivity of krypton. The curve represents the experimental data of White and Woods (1958). The dots represent the calculations made for this study.  65 IOOOI  XENON  lOOr  o in  o o e> 10 cc UJ o  I  1.5  I  I  I I 111  J  IO  I II I Mil 100  J  L  500  Tt°K) Figure the the the  5 . Comparison o f two s e t s o f c a l c u l a t e d values f o r thermal c o n d u c t i v i t y o f xenon. The curve represents c a l c u l a t i o n s o f J u l i a n (1965). The dots represent c a l c u l a t i o n s made f o r t h i s study.  66 tween c a l c u l a t e d and experimental values o f K and q u i t e good agreement f o r 7~.> 50°K.  f o r 7 " ^ 10°K  For krypton (Figure 4)»  the c a l c u l a t e d values f i t the experimental curve very w e l l f o r 7"£.15"K.  For low temperatures, the c a l c u l a t e d values o f K  do not agree w i t h the experimental values a t a l l .  This i s the  region where c r y s t a l s i z e and l a t t i c e defects c o n t r o l the value and behavior o f & • In Figure 5 the s o l i d curve represents the values o f K. c a l c u l a t e d by J u l i a n (1965) f o r xenon, the dashed l i n e i s the l i n e (4*26) and the dots are the values o f K < •  For ' — 30 K the values o f K  c a l c u l a t e d from  calculated i n t h i s analysis  l i e q u i t e c l o s e t o the curve c a l c u l a t e d by J u l i a n .  In the Umk-  kapp r e g i o n , there i s order o f magnitude agreement, w i t h the values o f &• c a l c u l a t e d i n t h i s study l y i n g below J u l i a n ' s curve and causing the disagreement between the best value o f ^  found  i n S e c t i o n IV.3 and the value n « 8 p r e d i c t e d by J u l i a n . 5.  E R R O R S  Except f o r the gross over estimation o f K  suggested  i n S e c t i o n IV.4, the e r r o r i n the c a l c u l a t e d values o f ^- arose from three approximations. 1.  Q u a n t i t i e s c a l c u l a t e d from l a t t i c e sums were computed w i t h a r e l a t i v e e r r o r o f * 1°  •  This  approximation introduced a r e l a t i v e e r r o r o f l e s s than about ±/o~* i n the reduced frequencies and  67  equal to ± lo  -  V  i n the reduced cubic c o e f f i c i e n t s .  The r e l a t i v e e r r o r introduced i n t o temperature dependent q u a n t i t i e s was estimated by noting that the r e l a t i v e e r r o r i n (4.27)  e.  was about but because o f the range o f values o f T and ~P used i n the c a l c u l a t i o n s , io  ~  *T  ~  'o  (4.29)  ,  so that the r e l a t i v e e r r o r i n the exponentials ( 4 . 2 7 ) i s l e s s than about i ' d *  3  .  The reduced group v e l o c i t i e s were defined i n terms o f a d e r i v a t i v e which was approximated by a numeric a l d i f f e r e n t i a t i o n formula. T h i s approximation introduced a r e l a t i v e e r r o r o f about one per cent i n the reduced group v e l o c i t i e s . The c o n d i t i o n f o r conservation o f energy i n a three phonon process was replaced by the c o n d i t i o n  I AV 1  -  3x/o-\  (4.30)  The only quantity a f f e c t e d by t h i s approximation was the f a c t o r ^Z + ^ f  (4.31)  68 f o r which the r e l a t i v e e r r o r introduced by (4.30) was about 0.4 per cent. The e r r o r introduced i n t o the c a l c u l a t i o n o f K. by the three approximations was l e s s than three per cent.  Therefore, i f  enough reduced wave vectors were included i n the c a l c u l a t i o n s to determine the c o r r e c t temperature dependence, i f not t h e c o r r e c t v a l u e s , o f K. , then the thermal c o n d u c t i v i t i e s c a l c u l a t ed i n S e c t i o n I¥.4 would have had about t h r e e per cent e r r o r ; except f o r the low temperatures where the e r r o r i s very l a r g e because o f the neglect o f the e f f e c t s o f c r y s t a l s i z e and l a t t i c e defects.  The closeness o f t h e f i t s between experimental and c a l -  culated values o f X f o r high enough temperatures  (See Figures  2-5) suggests that enough reduced wave vectors were considered to o b t a i n the c o r r e c t temperature  dependence.  Throughout t h i s chapter emphasis has been placed on the reduced q u a n t i t i e s introduced i n Chapter I I I .  None o f the q u a l i -  t a t i v e statements made concerning any o f the reduced  quantities  would have t o be changed i n order to apply them to the c o r r e s ponding p h y s i c a l q u a n t i t i e s .  However, these statements would  not n e c e s s a r i l y have any a p p l i c a t i o n t o the experimentally measured q u a n t i t i e s .  For example, f o r low temperatures the reduced  thermal c o n d u c t i v i t y f o r three phonon processes has very l i t t l e to do w i t h the experimental thermal c o n d u c t i v i t y which i s dominated by the e f f e c t s o f c r y s t a l s i z e and l a t t i c e d e f e c t s .  CHAPTER V CONCLUSION The previous chapters o f t h i s t h e s i s have presented the theory and o u t l i n e d the c a l c u l a t i o n o f the thermal conductivi t i e s o f s o l i d neon, argon, krypton and xenon. The theory o f the thermal c o n d u c t i v i t y of d i e l e c t r i c s o l i d s which was presented i n Chapter I I was based on the assumpt i o n o f a r e l a x a t i o n law w i t h the r e l a x a t i o n times evaluated i n terms o f the cubic anharmonic term o f the c r y s t a l p o t e n t i a l energy.  Although higher order anharmonicities could have been  t r e a t e d analogously, t h e i r e f f e c t s were ignored as n e g l i g i b l e compared w i t h the e f f e c t s o f the cubic anharmonicities.  The  thermal c o n d u c t i v i t i e s c a l c u l a t e d from t h i s theory could be expected to g i v e good agreement w i t h experimental data only f o r temperatures high enough t h a t the e f f e c t s o f i n t e r a c t i o n s among the phonons are more important than the e f f e c t s o f s c a t t e r i n g o f the phonons at c r y s t a l boundaries and near l a t t i c e imperfections. As a model i t was assumed that c r y s t a l s o f the i n e r t gas s o l i d s possessed p e r f e c t , face centred cubic l a t t i c e s and atoms which i n t e r a c t e d i n p a i r s through the 12:6  potential  energy f u n c t i o n . The a d d i t i o n a l assumption t h a t thermal expans i o n o f the c r y s t a l s could be neglected permitted the i n t r o d u c t i o n o f a set o f "reduced" q u a n t i t i e s . 69  T h i s f a c t meant that the  70 problem o f c a l c u l a t i n g the thermal c o n d u c t i v i t i e s of f o u r i n e r t gas s o l i d s could be reduced to the problem o f c a l c u l a t i n g four sets of constants and the "reduced thermal c o n d u c t i v i t y " K  of  any one o f the i n e r t gas s o l i d s . The t o t a l number of phonon modes i n a macroscopic c r y s t a l i s so great t h a t even w i t h the use o f a computer, i t i s p o s s i b l e to consider only a very small f r a c t i o n o f them i n any calculations. 'o  In t h i s t h e s i s t h i s f r a c t i o n was so small (about  } t h a t the reduced thermal c o n d u c t i v i t y appeared to be too  l a r g e by a f a c t o r o f order /o  ZI  pendence o f £  However, the temperature  de-  was shown to be approximately that.which has been  p r e d i c t e d and measured. as T  •  For  and f o r o.orc>8&£)*  ~r2io.*ca<S>  o./S"Z  , *  , ^ v a r i e s approximately  has the form  A (Q>T**r[ <3>/£T]. The best value found f o r the parameter n 0.003, the best i n t e g e r value being 1.  (5.1) was shown to be 1.175 £ (See S e c t i o n IV.3).  The thermal c o n d u c t i v i t i e s o f s o l i d neon, argon, and krypton were obtained by m u l t i p l y i n g the values c a l c u l a t e d f o r %  by a f a c t o r which produced exact agreement w i t h the e x p e r i -  mental data of White and Woods (1958) f o r one temperature i n the high temperature r e g i o n . calculations.  Figures 2-4 show the r e s u l t s o f these  F a i r agreement between c a l c u l a t e d and experimental  values o f the thermal c o n d u c t i v i t i e s begins near the temperatures  71 at which the maximum value o f the experimental thermal conduct i v i t i e s occur.  For higher temperatures the agreement improves,  becoming very good at very high temperatures. The thermal c o n d u c t i v i t y o f xenon was obtained by a d j u s t i n g the values o f fC so t h a t f o r one temperature i n the high temperature r e g i o n , the adjusted value o f w i t h the c a l c u l a t i o n s f o r xenon by J u l i a n (1965).  agreed e x a c t l y The agreement  between the two sets o f c a l c u l a t e d thermal c o n d u c t i v i t i e s i s q u i t e good f o r a l l temperatures f o r which K was c a l c u l a t e d , but f o r the lowest temperatures shown i n Figure 5$ the values c a l culated i n t h i s study l a y below the curve of values computed by Julian.  I t was i n t h i s region that (5*1) was f i t t e d  f o r which J u l i a n p r e d i c t e d t h a t 71  s  to K.  and  8.  The b a s i c assumption o f t h i s t h e s i s has been t h a t f o r s u f f i c i e n t l y high temperatures, the thermal c o n d u c t i v i t y o f r e a l d i e l e c t r i c c r y s t a l s i s determined c h i e f l y by i n t e r a c t i o n s among the phonons.  The agreement o f the c a l c u l a t i o n s of t h i s study  w i t h experimental data f o r the thermal c o n d u c t i v i t y o f i n e r t gas s o l i d s seems to v e r i f y the theory o f Chapter I I and to j u s t i f y this  assumption. The l i m i t a t i o n s o f t h i s study suggest a number o f  problems f o r f u t u r e i n v e s t i g a t i o n . 1.  I f these c a l c u l a t i o n s were repeated f o r a l a r g e r f r a c t i o n o f the phonon modes, i t would be p o s s i b l e  72 to decide whether o r not enough cases were considered i n t h i s t h e s i s to o b t a i n the c o r r e c t temperature dependence o f the reduced thermal conductivity. I f the reduced thermal c o n d u c t i v i t y were c a l c u l a t e d f o r lower temperatures than have been considered i n these c a l c u l a t i o n s , i t would be p o s s i b l e t o determine whether o r not a low enough temperature i n t e r v a l e x i s t s f o r which the best value o f n i n formula (5»1) would be 3 o r 8. I t would be i n t e r e s t i n g t o determine the r e l a t i v e s i g n i f i c a n c e o f the e f f e c t s o f h i g h e r order anharm o n i c i t i e s on the thermal c o n d u c t i v i t y . Accurate c a l c u l a t i o n o f summations over phonon modes i s p r a c t i c a l l y impossible f o r three dimensional l a t t i c e s .  I t would be valuable t o f i n d a  method f o r considering  a very l a r g e f r a c t i o n o f a l l  o f the phonon modes i n the c a l c u l a t i o n o f such summations.  Perhaps, though, the concept o f i n t e r -  a c t i n g phonons does not provide the best way o f handling the problem o f c a l c u l a t i n g the thermal conductivity  and some e n t i r e l y d i f f e r e n t approach  should be sought.  BIBLIOGRAPHY Berman, R., Simon, F. £•, and W i l k s , J . , Nature 168, 277 (1951). de Haas, W. J . and Blermasz, T., Physica it, 752 (1937). de Haas, W. J . and Biermasz, T., Physica 5_, 47, 320, 619 (1938). Dobbs, E. R. and Jones, G. 0., Rep. progr. Phys. 20, 516 (1957). Dugdale, J . S. and MacDonald, D. K. C . P h y s . Rev. 98, 1751 (1955). G r i n d l a y , J . and Howard, R., i n L a t t i c e Dynamics. edited by R. F. W a l l i s , (Pergaman Press, New York 1964), A17, p. 129. J u l i a n , C a r l L., Phys. Rev. 137. A128 (1965). Klemens, P. G., i n S o l i d State P h y s i c s , edited by F. S e i t z and D. T u r n b u l l (Academic Press I n c . , New York 1958) v o l . 7, p. 1. Lawrence, D. J . , Stewart, A. T., and G u p t i l l , E. W., Phys. 2L, 1069 (1959).  Can. J .  L e i b f r i e d , G. and Ludwig, W., i n S o l i d State P h y s i c s , edited by F. S e i t z and D. T u r n b u l l (Academic Press Inc., New York 1961) v o l . 12, p. 2 7 5 . L i g h t h i l l , M. J . , I n t r o d u c t i o n to Fourier A n a l y s i s and Generali z e d Functions (Cambridge U n i v e r s i t y Press, London 1952). Maradudin, A. A., M o n t r o l l , E. W., and Weiss, G. H. Theory o f L a t t i c e Dynamics i n the Harmonic Approximation i n S o l i d State P h y s i c s , edited by F. S e i t z and D. Turnbull (Academic Press I n c . , New York 1963) Supplement 3. National Bureau o f Standards, Handbook o f Mathematical Functions, Applied Mathematics Series 55, edited by M. Abramowitz and I . A. Stegun (U.S. Government P r i n t i n g O f f i c e , Washington D.C.  1965)  pp.  882^883.  P e i e r l s , R. P., Ann. Physik J , 1055 ( 1 ? 2 9 ) . P e i e r l s , R. P., Quantum Theory o f S o l i d s (Clarendon Press, Oxford 1956) Chapter 2, Section 4.  73  74  P r i g o g i n e , I . , Non-Equilibrium S t a t i s t i c a l Mechanics ( I n t e r science P u b l i s h e r s , New York 1962) Chapters 1 and 2 . Whalley, E. and Schneider, W. 6 . , J . chem. Phys. 2 ^ , 1 6 4 4 (1955). White, 6 . K. and Woods, S. B., Nature 177. 851 ( 1 9 5 6 ) . White, G. K. and Woods, S. B., P h i l . Mag. J , 785 (1958). Ziman, J . M., Can. J . Phys. 2±, 1256 ( 1 9 5 6 ) . Ziman, J . M., E l e c t r o n s and Phonons (Oxford U n i v e r s i t y Press, London I 9 6 0 ) S e c t i o n s 8 . 1 and 8 . 2 .  

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