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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Thermal conductivity of inert gas solids Hurst , Michael James 1969
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Title | Thermal conductivity of inert gas solids |
Creator |
Hurst , Michael James |
Publisher | University of British Columbia |
Date Issued | 1969 |
Description | The thermal conductivity of perfect infinite crystals of neon, argon, krypton, and xenon has been calculated numerically. It was assumed that the crystals possessed face centred cubic structure with the atoms interacting in pairs through a Lennard-Jones 12:6 potential energy function. The calculations considered only the effects of three-phonon interactions. It was possible to simplify the calculations by introducing "reduced" physical quantities. The thermal conductivity of each of the inert gas solids considered was obtained from the "reduced thermal conductivity" which was calculated for argon. Agreement with experimental data for neon, argon, and krypton was obtained for temperatures higher than those for which the effects of crystal size and lattice defects determine the thermal conductivity. This agreement suggests that for sufficiently high temperatures the thermal conductivity is determined by the effects of three-phonon Umklapp processes. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2011-05-27 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0084797 |
URI | http://hdl.handle.net/2429/34949 |
Degree |
Master of Science - MSc |
Program |
Physics |
Affiliation |
Science, Faculty of Physics and Astronomy, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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