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The low temperature thermal conductivity of cesium iodide Johnson, David Lawrence 1967

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i  THE  LOW TEMPERATURE THERMAL CONDUCTIVITY OF CESIUM IODIDE by DAVID LAWRENCE JOHNSON  B.Sc., The U n i v e r s i t y of B r i t i s h Columbia, I963 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE  REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE  i n the Department  >  Physics  We accept t h i s required  THE  t h e s i s as conforming to the  standard  UNIVERSITY OF BRITISH COLUMBIA February, I967  In presenting t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study.  1 f u r t h e r agree that permission.for  extensive  copying of t h i s  t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s representatives.  I t i s understood that copying  or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n permission.  Department of The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8 , Canada  iii  ABSTRACT: The thermal c o n d u c t i v i t y of. three c r y s t a l s of cesium i o d i d e r a n g i n g i n s i z e from three to eight m i l l i m e t e r s diameter was measured i n the temperature range L l 5 ° K to 5.40°K. Thermal c o n d u c t i v i t y measurements were made u s i n g the thermal potentiometer method. Differences  i n the thermal c o n d u c t i v i t y of the  three samples were i n t e r p r e t e d i n terms of phonon s c a t t e r i n g from the boundaries of the c r y s t a l s , and from i n t e r n a l s t r u c t u r e  defects*  ii  DEDICATED to ray f a t h e r FRANCIS HENRY JOHNSON who i s a humanist but w i l l enjoy i t r e g a r d l e s s .  iv  TABLE  OF  CONTENTS'  Chapter  Page  I  Theory Introduction S i z e Dependent Thermal C o n d u c t i v i t y C h o i c e o f C a l as a n E x p e r i m e n t a l S u b s t a n c e Calculations of k f o r C s l Experiment  1 1 7 9 10 16  II  Apparatus Thermal Experimental Electronics  18 18 21 27  III  IV  Chamber  P r o c e d u r e and T e c h n i q u e Introduction Genera-! E x p e r i m e n t a l P r o c e d u r e Sample Temperature Measurement P r o c e d u r e : C a l i b r a t i o n and D a t a T e c h n i q u e Power D e p e n d e n c e o f T h e r m o m e t e r T e m p e r a t u r e Lead Resistances-. Sample Geometry Measurement Data Reduction M a n o m e t r y and Tj vs. T 5  Corrections  8  C a l i b r a t i o n P r o c e d u r e and P r o g r a m Conductivity Point Analysis Conductivity Calculation V  R e s u l t s and I n t e r p r e t a t i o n P r e s e n t a t i o n of Results I n t e r p r e t a t i o n of Results  33 33 3J4. 35 ty.0  IJJI I4J4.  X£> \\$  -14,9  53 57 60 62: 62 67  B i b 1 i o gr. a p h y  77  Appendix  79  V  LIST OP TABLES Table I  Page Average and P a r t i c u l a r Sound Velocities i n Csl  li;  II  Thermometer Power Dependence  %1  III  T j vs. T  51  IV  Thermal C o n d u c t i v i t y of C s l : Sample #1  63  V  Thermal C o n d u c t i v i t y of C s l : Sample #2  6I+  VI  Thermal C o n d u c t i v i t y of C s l : Sample #3:  65  VII  k/T^ v s . Diameter of Samples:  68  As I  Thermal Expansion Properties-; of B u t y l P h t h a l a t e  8l  5 Q  vi  LIST OF FIGURES  Figure  Page  1  Helium Pumping and C o n t r o l System  19  2  Experimental Chamber  22A  3  D e t a i l of Lower End of Sample and Mounting Technique  22B  [j.  D e t a i l of Thermometer Mounting  26  5  AC B r i d g e B l o c k Diagram  28  6  DC Power Siupply f o r Sample Heater  J2.  7  Thermometer Power Dependence  \\Z  8  Tj - T  52  9  Thermal C o n d u c t i v i t y of Cesium  5 6  vs. T  5 Q  66  Iodide Samples 10  k/T^ v s . D and D  11  k/D  2  2:  v s . Temperature  69  71  vii  A cknowle d gement s I would  l i k e to thank Dr. M.J,, Crooks f o r h i s  o r i g i n a l suggestion  of t h i s r e s e a r c h p r o j e c t , and f o r h i s  t h o u g h t f u l s u p e r v i s i o n of my work. I am indebted  a-.lso to Mr. R. Weisbach and Mr. G. Brooks  f o r t h e i r t e c h n i c a l a s s i s t a n c e , and to Dr. P.R..  Critchlow.  and Mr. C.R. Brown f o r d i s c u s s i o n of the t h e o r e t i c a l questions. P a r t i c u l a r thanks are due to my mother, Mrs. F.H. f o r the e x c e l l e n c e f o r her h e l p and  Johnson  of her t y p i n g , and to my wife E l i z a b e t h  patience.  CHAPTER  (1)  INTRODUCTION  The  successful  possible igation  the of  realized  that  would  at  be  at  low  mechanisms  least  of  theoretical  transfer  the  THEORY  liquefaction  combined  heat  I.  partially  of  helium  and  in  experimental  temperatures. heat  made  1908  transfer  different  from  It in  investwas  a  those  quickly  metal  in  a non-  m e t a l .. The in  groundwork  dielectric  mechanical crystal.. (a)  crystals  analysis (Eeierls  Crystal  the  atoms  are  linked  of  the  or  ectric  was  laid  the  interaction  of  Beierls., of  heat  transfer  with his  lattice  quantum  waves  in  :  Vibrations.  In  a. d i e l e c t r i c  the  interatomic  energy the  of  the  crystal  and  these  forces is  p o s s i b l e n o r m a l modes  in  thought of  crystal vibrations the to  crystal. be  vibration  of  lattice. neglect at  mode o f  a  192 9)  coupled by  all  crystal  vibration  by  theory  vibrating,  in  we  current  are  contained  If  the  crystal  thermal  crystal  of  Lattice  The t o t a l  the  for  the  zero  point  temperature frequency  exp  T°K,  energy, the  is  (h^/KT)  - 1  then  average  for  a.  energy  dielin  a  This energy may be considered as b e i n g made up of 1/  (hv^/KT) - l\ quanta  energy hv?..  of v i b r a t i o n a l energy, each w i t h  These qiuanta of v i b r a t i o n a l energy have come  to be c a l l e d  phonons. and t h e i r d i s t r i b u t i o n with r e s p e c t  to the frequency \) i s c a l l e d m a t e r i a l at temperature  T K  the phonon spectrum of the  0  One can e a s i l y s l i p  o  i n t o a b i l l i a r d b a l l concept of  phonons, as one can w i t h photons, but i n both cases care must be e x e r c i s e d (b)  Heat Flow.  i n the a p p l i c a t i o n of t h i s  concept,  Consider groups of phonons w i t h angular  f r e q u e n c i e s nearc0(=2^9} and wave numbers near q>  The group  v e l o c i t i e s ; v(q\) of these packets w i l l be g i v e n by dt^/d-cy* If the phonon spectrum of the m a t e r i a l i s such that t h e r e are  a number N(q\) phonons i n mode qj, the heat c u r r e n t w i l l be "€ = N(^)T>WV(«B) L  In  b U  (1)  thermal e q u i l i b r i u m , the net f l o w of phonons i n any  d i r e c t i o n i s zero, or N (q}) = N (-"q:). Q  However, i f a temperature  0  gradient e x i s t s , there w i l l  be a d e n s i t y g r a d i e n t i n the phonons, r e s u l t i n g i n a net flowj.  This tends to a l t e r the phonon p o p u l a t i o n at a, g i v e n  p o i n t from N (q<) to N(q;)„ 0  When the s i t u a t i o n i s reached whereby t h i s tendency to a l t e r N ( q ) i s balanced by the tendency of s c a t t e r i n g proQ  cesses to r e s t o r e the e q u i l i b r i u m p o p u l a t i o n K ( q ) , m steady Q  3 s t a t e c o n d i t i o n i s a t t a i n e d w i t h a f i n i t e heat flow QL and a f i n i t e temperature  g r a d i e n t tfT..  The magnitude of the heat flow, and hence of the thermal c o n d u c t i v i t y , i s determined by the amount of departure of N(q}' from N (qj). (c)  Boltzmann Equation.,  The  c o n d i t i o n whereby the d r i f t  of phonons due to a temperature  g r a d i e n t i s balanced by th©  processes which s c a t t e r them i s expressed i n the  Boltzmann  equation.. (2) A. complete  s o l u t i o n of t h i s equation, and the r e s u l t i n g  cal-  c u l a t i o n of the phonon spectrum, would r e s u l t i n complete p r e d i c t a b i l i t y of the thermal c o n d u c t i v i t y of a d i e l e c t r i c material. itly,  In g e n e r a l t h i s e q u a t i o n cannot be solved  and v a r i o u s approximations must be used.  The  explicapprox-  imation which has found most a p p l i c a t i o n i s the (d)  A d d i t i v e R e l a x a t i o n Rate Approximation.  In the r e l a x -  a t i o n time method of s o l u t i o n of (2) i t i s assumed that the r e t u r n of a. phonon d i s t r i b u t i o n N(q;) to i t s e q u i l i b r i u m t r i b u t i o n N (q;) i s e x p o n e n t i a l w i t h time. Q  This can be  expressed as;  where 1/T(q) i s termed  the r e l a x a t i o n rate..  In any r e a l c r y s t a l , there may be s e v e r a l processes  dis-  tending to r e s t o r e each w i t h i t s . own  the e q u i l i b r i u m relaxation rate.  mutually n o n - i n t e r f e r i n g , ation rates  to be  1  *  1  £, (q)  Scattering i n t e r f e r i n g are  then we  would expect t h e i r r e l a x -  +  _1  processes which are (Carruthers,  - £j  isotope  - 2Tp  point  - £j>  dislocation  non-  scattering,  scattering, scattering, and  normal or N-process.  (e)  Normal and  the  interatomic  Umklapp Processes. forces  In a l l r e a l c r y s t a l s ,  are anharmonic.  This  i n t e r a c t i o n between phonons i n the body of an "perfect" actions  (ft)  considered to be  (mass-difference)  - 2 ^ Umklapp^ or U-processes, N  * ______  1961);  boundary s c a t t e r i n g ,  defect  ;  Z_(<_)  £_(<_)  -  - 2T  I f these processes; are  additive,  = 1  r(q)  phonon d i s t r i b u t i o n ,  and  infinite  crystal.  or " c o l l i s i o n s "  important.  The  first  interatomic  force  At low  leads to  an  otherwise  temperatures i n t e r -  i n v o l v i n g three phonons are  the most  anharmonic term i n an expansion of  i s the  cubic  term, and  the  t h i s term leads  to three phonon i n t e r a c t i o n s . In phonon c o l l i s i o n s such as while phonon wave number may  be  these, energy i s conserved  e i t h e r conserved or  altered  by a r e c i p r o c a l l a t t i c e v e c t o r b\  Note that the number of  phonons i s not conserved.  then w r i t e the c o n s e r v a t i o n  J  We may  relations,  (5a), q-j. * qij. = q 3  (5b),  + It,  (5c).  "q\ + q.z = q^. I f "& i s a r e c i p r o c a l l a t t i c e v e c t o r (5b) is  called  collision  an Umklapp or U-process, while i f b = 0 (5c)  c o l l i s i o n i s called (f)  the  the  a Normal or N-process.  S o l u t i o n v i a R e l a x a t i o n Time Approximation.,  I f we  con-  s i d e r a m a t e r i a l which e x h i b i t s no s c a t t e r i n g mechanisms', other than those of sec, (e) above, then we may l a x a t i o n time approximation (3) equation to  (2).  use the r e -  to s o l v e the Boltzmann  Doing so leads (Berman, 1965;  Callaway,  a thermal c o n d u c t i v i t y g i v e n by  where C(q;) i s the c o n t r i b u t i o n to the s p e c i f i c heat from phonon packets of wave number qu (g) Debye S o l i d Assumption.  Let us now  suppose that the  phonon group v e l o c i t y i s a constant c independent phonon wave number q„  This i s r e f e r r e d  of the  to as the Debye  approximation 60(q) = cq., A "Debye s o l i d "  i s a. model i n which there i s a constant  1959)  phonon v e l o c i t y c and a c h a r a c t e r i s t i c temperature Using t h i s model, one can d e r i v e Callaway, 1 9 5 9 )  (Makinson,  ..  1938;  a thermal c o n d u c t i v i t y g i v e n by £>/T  k  =  Kft  . < -l). T  3  e  \r  xk e (e -l)  o where K i s Boltzmann's x =~hcD/lH1  t  £*  R  (7),  dx  X  x  2  constant, t i i s Planck's constant,  i s a r e l a x a t i o n time and <^c""^. i s some average  value of the i n v e r s e v e l o c i t i e s of sound and p o l a r i z a t i o n s i n a c r y s t a l .  over a l l d i r e c t i o n s -  This e x p r e s s i o n w i l l  be  r e f e r r e d to as the Debye approximation to Callaway's t h e o r y , (h)  Temperature  V a r i a t i o n - o f Thermal  r e f e r e n c e to ( 6 ) , we may temperature  how  Conductivity.  With  d e f i n e three d i s t i n c t types of  v a r i a t i o n i n the thermal c o n d u c t i v i t y .  ( i ) I/T r e g i o n .  At h i g h temperatures  mechanism i s phonon-phonon c o l l i s i o n . h i g h enough, say g r e a t e r than  the main s c a t t e r i n g  :If the temperature i s  then the dominant phonon  wave numbers are s u f f i c i e n t l y great so that a l a r g e  fraction  of a l l phonons are a v a i l a b l e f o r combinations of the type  (5b).  Umklapp i n t e r a c t i o n s are then the major s c a t t e r i n g p r o c e s s . The r e l a x a t i o n r a t e Z^  1  i s p r o p o r t i o n a l to the phonon d e n s i t y  and hence to absolute temperature, Einstein statistics).  ( s i n c e phonons obey Bose-  The dominant phonon modes make a con-  stant c o n t r i b u t i o n to the s p e c i f i c heat and hence by we o b t a i n k °C I/T.  (6)  7  ( i i ) Exponential region.  As the temperature  the s p e c i f i c heat term i n (6) decreases,  but  decreases,  the number of  phonons a v a i l a b l e to p a r t i c i p a t e i n U-processes d e c r e a s e s exponentially*  The  s c a t t e r i n g time thus i n c r e a s e s so r a p i d l y  w i t h temperature drop; as to be n e a r l y e x p o n e n t i a l , and (6) the thermal c o n d u c t i v i t y i s t h e r e f o r e ko< ( i i i ) Boundary s c a t t e r i n g region..  by  exp(a/T)»  At very low  temper-  atures the s c a t t e r i n g time 2T w i l l i n c r e a s e to the p o i n t where it  i s comparable w i t h the time taken f o r phonons to cross?  the e n t i r e c r y s t a l without  scattering.  When t h i s  occurs,  the m a j o r . s c a t t e r i n g mechanism i s . t h e boundaries of sample.  We  may  the  then w r i t e , i f c i s the phonon v e l o c i t y  D i s the diameter of the c r y s t a l , £ = D/c»  At these  and  temper-  atures  the s p e c i f i c heat i s u s u a l l y w e l l i n t o the T-^ r e g i o n ,  and by  (6) we may  t h e r e f o r e writ© kcKDT  (2) (a)  (8)..  3  SIZE DEPENDENT THERMAL CONDUCTIVITY,  T h e o r e t i c a l Work.  The f i r s t  satisfactory  theoretical  e x p l a n a t i o n of s i z e dependent thermal c o n d u c t i v i t y Casimir's  attempt  (Casimir 1938)  to e x p l a i n the  was  experimental  r e s u l t s of de Haas and Bierraasz (de Haas and Biermasz., 1938).. Casimir a p p l i e s the standard a phonon gas*  He  blackbody r a d i a t i o n theory  to  considers a l o n g c y l i n d r i c a l tube of diameter  D,. w i t h p e r f e c t l y b l a c k w a l l s  ( i . e . a l l phonon r e f l e c t i o n s ^  8 are  d i f f u s e ) at a; temperature  low enough that the o n l y phonon  s c a t t e r i n g mechanism i s the boundaries of the c r y s t a l . .. C a s i m i r o b t a i n s the r e s u l t of  2//ff as noted  ( a f t e r making a c o r r e c t i o n  i n [Berman et a l , 1953J), k = 2 TT K^ !  DT  2  I5?i  (9)  3  3  where the term i n square b r a c k e t s i s an attempt  to f i n d a  value f o r the average v e l o c i t y of sound over a l l p o l a r i z a t i o n s and a l l d i r e c t i o n s i n the c r y s t a l .  I f we r e p l a c e t h i s w i t h  the n o t a t i o n <^c" /, we o b t a i n k = 2.7rK^ < c f / D T 2  15-fi  2  (10),  3  3  i n agreement w i t h ( 8 ) . Casimir a l s o obtained the r e l a t i o n between thermal c o n d u c t i v i t y and s p e c i f i c heat k = 1.155  x 10 A PDT 3  /j  (11),  3  where A, i s the constant i n the s p e c i f i c heat e q u a t i o n C  = AT3 JouLE-/em °K", 3  v  and  V  =<c' > <c- > 2  3  (b)  Previous: Experimental Work.  have t r i e d  —  —  (12),  2 / 2  Numerous  experimenters  to match observed boundary s c a t t e r i n g behaviour  w i t h theory.  Measurements on an a r t i f i c i a l  (Berman et a l , 1955)  sapphire c r y s t a l  showed a thermal c o n d u c t i v i t y which was  9 d i r e c t l y p r o p o r t i o n a l t o c r y s t a l diameter, but agreement w i t h the t h e o r e t i c a l magnitude of the c o n d u c t i v i t y was obt a i n e d o n l y a f t e r making c o r r e c t i o n s f o r the f i n i t e  length  of the sample, and f o r a presumed p a r t i a l l y s p e c u l a r phonon r e f l e c t i o n from the b o u n d a r i e s . The  best  was obtained LiF„  experimental agreement w i t h Casimir's by Thatcher (Thatcher,  19&5) u s i n g  C a l c u l a t i o n of k f o r L i F v i a the Casimir  theory  c r y s t a l s of theory  lead's?  to the r e s u l t k = Thatcher obtained  0,172'  DT  3  (13) .  watt/cm°K  the r e s u l t  k = (0,21 + 0.02)DT ,  (l[j.)  n  n = 3,005 ± 0,015. The  20$ d e v i a t i o n i n magnitude from the c a l c u l a t e d  value  wass presumed t o be due to an i n a b i l i t y to c o r r e c t l y c a l c u l a t e the average sound v e l o c i t i e s (30 The  i n an a n i s o t r o p i c  crystal.  CHOICE OF C s l AS AN EXPERIMENTAL SUBSTANCE. " i d e a l " c r y s t a l f o r i n v e s t i g a t i o n of low temperature  Thermal c o n d u c t i v i t y would be a s i n g l e c r y s t a l of c h e m i c a l l y and  i s o t o p i c a l l y pure m a t e r i a l , homogeneous and i s o t r o p i c i n  i t s thermal p r o p e r t i e s , and e a s i l y handled. Since  a m a t e r i a l was d e s i r e d which, i n some temperature  range, e x h i b i t e d phonon s c a t t e r i n g by boundaries only, the  10  e l i m i n a t i o n of point d e f e c t s c a t t e r i n g was i s the m a t e r i a l was  to be very pure.  necessary, that  A l k a l i h a l i d e s have  a t t r a c t e d much i n t e r e s t as pseudo-Debye s o l i d s .  Both  and i o d i n e have only one n a t u r a l l y o c c u r r i n g s t a b l e  cesium  isotope,  and thus the i s o t o p i c p u r i t y of C s l i s guaranteed. Past experience w i t h m a t e r i a l s e x h i b i t i n g the  boundary  s c a t t e r i n g r e g i o n has shown the thermal c o n d u c t i v i t y peak to l i e at approximately 1/30  of the Debye temperature,, &  -  v  For C s l , OT> i s about 125°K, p l a c i n g the expected peak at around lj...l K and thus making the boundary  scattering region  0  e a s i l y a c c e s s i b l e w i t h a. helium-!; c r y o s t a t * C s l has a l s o been suggested as a p o s s i b l e candidate i n the search f o r second communication).. o n l y 1.05,  sound  in solids  (C.R. Brown, p r i v a t e  The r a t i o of atomic weights of Cs to I i s  c l o s e r to one than any other a l k a l i h a l i d e .  It  has been suggested that t h i s p r o p e r t y would i n c r e a s e the p r o b a b i l i t y of f i n d i n g a P o i s e u i l l e flow "window" et a l ,  19&£>)  (Guyer  i n the temperature wave frequency vs. temper-  ature b e h a v i o u r * W (as)  CALCULATION OF K FOR C s l .  Anisotropy.  The major computational problem  i n a c a l c u l a t i o n of "k f o r C s l i s that of f i n d i n g average values f o r the v e l o c i t i e s of sound Csl c r y s t a l l i z e s  i n the  involved suitable crystal.  i n the Cs;Cl s t r u c t u r e and has t h e r e f o r e a  11  simple cubic s t r u c t u r e with one molecule /unit  or two  atoms  cell. The  anisotropy factor  of the e l a s t i c  of a m a t e r i a l i s given i n terms  constants by (15)  1 Using the e l a s t i c  c  1 1  _c  1 2  constants measured at low  ( V a l l i n et a l , 1961+) we  temperature  obtain for Csl (16).  The Houston approximation  (Houston,  I9I4.8; B e t t s et a l ,  1956) i s a method f o r f i n d i n g the approximate value of an angular i n t e g r a l given the value of the integrand i n t h r e e mutually perpendicular d i r e c t i o n s . It i s found  ( B e t t s et a l , 1956) that f o r a n i s o t r o p y  f a c t o r s i n the range ,5<^<1.5,- Houston's approximation i s not a p p r e c i a b l y d i f f e r e n t from approximations  i n v o l v i n g more  known d i r e c t i o n s , f o r the c a l c u l a t i o n of average  velocities;  of sound. (b)  V e l o c i t i e s of Sound i n the <100>. < l l p ) . <111>  Given the three e l a s t i c constants c ^ , density  c-^, ^ > c  a n c 3  Directions;, t 1 i e  the seven " b a s i c " . v e l o c i t i e s r o f sound are g i v e n by  12 A  <100>  ^ ( l O O ) = 2 ^ < 100> = ( j ^ }  B  <no>  ^,<iio>  ^  =f kh c  ^ i o > ^ i n _ ^ i ^  ^<no> = ^ * c <^ n i >  1 / 2 (  ^,(ni> = %2_< m )  c  = /  n C  ^  +  c  +  i 2 y* }  cn  (  %  <  i  n  >  = /hokk  we (c)  elastic  1 2  V/i  Y  +  ( Using the d e n s i t y and  - c  2 c  12  V  constants measured by  /  +  C  IIVCL  )  Vallin,  o b t a i n f o r C s l the r e s u l t s shown i n Table I . Average V e l o c i t i e s of Sound. (i)  Suppose i t i s d e s i r e d to f i n d  J, the angular i n -  t e g r a l of some integrand I ( ^ < p ) , J  i f )<3/l,  = J  then Houston's method s t a t e s t h a t i f I , 1^, f l ;  and  I  c  are the  known values of the integrand i n the d i r e c t i o n s <^100), and <^111^ r e s p e c t i v e l y , then we  can approximate the  <(llo),  integral  by  J =fl(&,f )dS"L S  M.^  1  0  I  a  +  l 6 l  b  +  9I \ C  (17).  13 ( i i ) We wish to f i n d the three averages over d i r e c t i o n and  polarizations,  (18), 12 where The  I  n  = J - j  n , n = (1, 2, 3 ).  sum over i = (1, 2, 3) r e p r e s e n t s the sum of the three  d i f f e r e n t p o l a r i z a t i o n s , two t r a n s v e r s e and one l o n g i t u d i n a l s The  f a c t o r of 12ff i s a n o r m a l i z a t i o n f a c t o r equal to 3 x hjf»  3 from the three p o l a r i z a t i o n s  and  from the t o t a l  solid  angle. The (d)  r e s u l t s of t h i s c a l c u l a t i o n are shown i n Table I .  Conductivity  Calculations.  Using the r e s u l t s obtained  i n the previous s e c t i o n , we may now c a l c u l a t e the t h e o r e t i c a l thermal c o n d u c t i v i t y  of C s l i n the boundary s c a t t e r i n g  region.  At. temperatures s u f f i c i e n t l y low that phonons are s c a t tered time  only at the boundaries of the c r y s t a l the r e l a x a t i o n i n equation ( 7 ) becomes a c o n s t a n t ^ , and;  $ D / T — •  whence the i n t e g r a l i n ( 7 ) may be s e t equal t o 26.0. thus  obtain k = 26.0.  <( c " ) 1  ^r^-n  3  r  b  T  3  ,  We  Ik TABLE I Average and P a r t i c u l a r S^ound V e l o c i t i e s i n C s l Direction  Polarization  Velocity 1*323 x 10^ cm/sec 2J4.IO  ^lOO")  t 1  <iio)>  t  1.323  t  1.436  1  2.31+5  t 1  l.ftOO 2.322  <111>  <c~ > = .,629 x 1<TJ> 1  = (1.59 x 1 0 ) ~ 5  <c" > = .1+17 x 1 0 ~ * 2  (cm/sec)" _  0  = (1.55 x 1 C K ) " < c " > = .287 x 1 0 ~ ^ = (I.52 x 1 0 )  2  (cm/sec)  3  5  Based on lj..2°K values  c  = I+.J12 gm/em3  l l  = 2-737 x 1 0  c^  = =  -793 .825  J  ( V a l l i n et a l , I96I4.) of  /> C12  (cm/sec)"  « '•  1 1  dynes/cm "  2  15  and  substituting fo )rr the cor constants and the c a l c u l a t e d  value  -1\  of ^ c " " ^ we o b t a i n 1  k = 2.57 * 1 © We now take f  b  5  r  b  T  _._  3  to be the "time of f l i g h t " across a  (19).  diameter  of the c r y s t a l , T  = ^(c' )  = .629 x 10"^D sec  1  b  and by s u b s t i t u t i o n i n t o  —  (20)  (19) we o b t a i n  k = 1.62. DT  3  w;att/cm°K  (21).  Working d i r e c t l y from the Casimir equation ( 1 0 ) , and s u b s t i t u t i n g f o r the constants and <(c  }  we o b t a i n  k = 1.70 DT ' watt/cm °K  (22).  3  W.e may a l s o work from C a s i m i r s s p e c i f i c heat equation 1  (11).  The s p e c i f i c heat of C s l has been measured  et a l , 1962) down to 13.-5l° , at which temperature K  found  to be Cp = 1.927 cal/mole°K..  A simple  (Taylor i t was?  calculation  shows that C . -C p  hence C a.t t h i s  y  = .001 c a l / m o l e K  = 1*926 cal/mole°K at 13»5l°K. temperature  ,  a  y  I f we assume that  16 we f i n d A = 57.17 x 10  -6  watt-sec  and (23).  Casimir's s p e c i f i c heat equation then leads to the r e s u l t k = 1.62+ DT  „. ( 2 5 ) .  Comparing the three t h e o r e t i c a l results,w.e may then say that theory p r e d i c t s f o r C s l a thermal c o n d u c t i v i t y i n the boundary s c a t t e r i n g r e g i o n which i s p r o p o r t i o n a l to the sample diameter D, the cube of the absolute temperature, and is; of magnitude k <_? 1.65 DT (5)  3]  (26)..  watt/cm°K  EXPERIMENT.  The experiment The temperature  used rods of C s l of v a r y i n g diameters D„  d i f f e r e n c e A T between two thermometers a  d i s t a n c e L apart was measured d u r i n g the passage  of a known  17 heat c u r r e n t F through the sample,  The thermal  conductivity  was then taken to be k =  =  P  £T  _P_  AT  All  q u a n t i t i e s on the r i g h t  A (27)  . _J_  77 D  2  of equation ( 2 7 ) may be measured,  whence the thermal c o n d u c t i v i t y of the sample may be  found.  CHAPTER (1)  .II. APPARATUS.  THERMAL  (a) Low Temperature P r o d u c t i o n .  The experiment  was  carried  out i n a l i q u i d helium-i; c r y o s t a t capable of r e a c h i n g and controlling  temperatures  from b ..29°K to 1.12°K.. r  Both the i n n e r  l i q u i d h e l i u m dewar and the outer l i q u i d n i t r o g e n dewar were of Pyrex g l a s s , s i l v e r e d slits  except  for:vertical  down both s i d e s and sealed under vacuum.  Liquid  level  was observed u s i n g a f l u o r e s c e n t tube p a r a l l e l to one set of  slits.. The  top of the helium dewar c o n s i s t e d of 18cm. of s i n g l e  t h i c k n e s s Pyrex p i p e , so that the double w a l l e d s e c t i o n of the helium dewar could be kept e n t i r e l y below the l i q u i d n i t r o g e n level..  Care was taken never t o admit helium gas or l i q u i d  i n t o the h e l i u m dewar unless i t was cooled to n i t r o g e n temperature.  With t h i s p r o c e d u r e , d i f f u s i o n of helium through the  helium dewar w a l l s was kept to a minimum* a dewar l a s t e d f o r about  I t was found  twenty runs, or r o u g h l y two  that  hundred  hours, b e f o r e i t was necessary to break i t open and r e - e v a c uate the interspace.. Both dewars were supported from the end of a i | " pumping l i n e capped  w i t h a 3" G r i n n e l l - S a u n d e r s rubber  v a l v e . (Fig.. 1, #7)  diaphragm  Pumping of the helium bath was done by a  Stokes Microvac Model I4.9-IO Rotary vacuum pump w i t h a c a p a c i t y of  80 cubic f e e t per minute..  IS VACUUM P U M P  "fcACKlNfr - ? R £ S ? U * r TMffRMOeoUpU? C A U C C  MffftcURY  MAMOMerrR HVANOMCTBTR.  *T*Hf RMOCOUPLff  "RCPffRBNCff PRSSSURr « AULA S T VOLUME*  VBNT  HELIUM  ftSTUffN  LINE"  *6  #5 P R E S S O R * "CONTROLLER  1  MAIN PUMPING LINE"  "^Qs^OVCgSTOMP *7  HBMOK- 4- "DCWAR.  HELIUM PUMPING CONTROL SYSTEM  F16URE I  20  (b) Temperature C o n t r o l . times encountered,  Due  necessary.  apparatus of Walker  (1959)  of a 1"  A. modified  was  adopted.  v e r s i o n of  the  It consisted,  pumping l i n e i n p a r a l l e l w i t h  phragm v a l u e .  equilibrium  (up to 1^0 min.), a s t a b l e method of temp-  e r a t u r e c o n t r o l was  ially,  to the long thermal  the 3"  dia-  A short s e c t i o n of the pumping l i n e was (»002 )  placed by a p i e c e of t h i n walled (nominally s o l d  r t  re-  rubber t u b i n g ,  only f o r the p r e v e n t i o n of d i s e a s e ) , which  acted as a v a l v e c o n t r o l l e d by the r e f e r e n c e pressure c l o s e d volume of gas ence volume ( 1 0  v a l v e #3) bath  surrounding  i t . Connection  l i t e r s ) to manometers, ( F i g . 1 ,  in a  of the  v a l v e #1+)  and helium  facilitated  quick and  easy s e l e c t i o n of any  gas  refer-  #2)  valve  ( F i g . 1,  vacuum l i n e ,  essent-  ( F i g . 1,  line,  desired  temperature.. This device maintained  pressures  800mmHg s t a b l e to b e t t e r than 50u four hours.  The  over p e r i o d s of three  major cause of d r i f t was  room temperature r e s u l t i n g i n s l i g h t pressure  from ..500mmHg to  i n the r e f e r e n c e volume.  through the rubber t u b i n g was  or  slow changes of  changes of the  gas  D i f f u s i o n of helium  gas:  negligible*  Temperature s t a b i l i t y to b e t t e r than 5 m i l l i d e g r e e s at 1 . 2 ° ,  and  tained.  At temperatures above the lambda p o i n t , to  .0I|. m i l l i d e g r e e s at i|.2:°over  temperature g r a d i e n t s input of 7.5  i n the bath,  an hour was  the bath was  m.W. of e l e c t r i c a l power to a 300  the bottom of the dewar..  This r e s i s t o r was  ohm  thus  ob-  prevent  s t i r r e d by  the  r e s i s t o r at  a l s o used a t  h i g h e r powers to a i d i n making upward changes of bath temper -  21 attire, and to b o i l o f f any remaining h e l i u m at the end of a run. (c) Temperature Measurement.  A l l bath temperatures  were  measured w i t h 1cm. bore mercury or b u t y l - p h t h a l a t e o i l manometers.  The vacuum s i d e s of the manometers were pumped  with a s m a l l r o t a r y pump and the b a c k i n g pressure read with a Veeco thermocouple gauge. The manometers were read with a cathetometer  calibrated  at 20°C. and capable of r e a d i n g to . 0 0 5 em. The manometers could a l s o be used,  ( v i a valve #2?.), t o  a c c u r a t e l y read or s e t the pressure i n the c o n t r o l l e r erence  volume.  (20 (a)  ref-  EXPERIMENTAL CHAMBER  Vacuum Can.  The experimental chamber and sample mount-  i n g are shown i n F i g u r e 2 . with gold.  The vacuum can was b r a s s , p l a t e d  The two s e c t i o n s were b o l t e d t o g e t h e r , the  vacuum s e a l b e i n g made w i t h a . 0 5 0 "  lead 0 - r i n g wiped w i t h  vacuum grease. A Consolidated Vacuum Corp. VMF 10 o i l d i f f u s i o n pump working through evacuated  the.can through  pumping l i n e .  a, l i q u i d n i t r o g e n cold t r a p  a . 2 5 " thin-walled stainless  steel  The pumping l i n e was jogged, and i t s i n t e r i o r  coated w i t h f l a t b l a c k l a c q u e r t o prevent room  temperature  r a d i a t i o n from r e a c h i n g the thermometers. The d i f f u s i o n pump could evacuate  the can to a p r e s -  sure of 1 0 ~ t o r r as measured by a room temperature 7  P h i l l i p s gauge between the c o l d t r a p and the can.  C.V.C..  22 A Access  U I U  TUBE" P D * U\QUIP  HE»-ltfM.  9CALS.  UBAO 0 - R 1 N & ,  INDIUM 0-*IN«PUOtPHOd-MONUC  Cfl  C*r  U>IHT>OU>.  SAHPLC  Csl  VACUUM CAN.  HEATER.  EXPERIMENTAL CHAMBER FIGURE 2  2ZB  UQdiT> \46UOM ACCffSS TO C C ? i u M tOOiOtf* U » N O O W •  "PRESSURE -6U ^ O L T S  -PUKTE  INDIUM 0-W»NC VACUUM S C A L  DETAIL: SAMPLE MOUNTING.  FIGURE 3  23 (b) Samples. al  The samples  were obtained from the Harshaw Chemic-  Co. and were " o p t i c a l l y pure" C s l s i n g l e c r y s t a l s . .  c r y s t a l s were machined  i n t o rods 3cm. l o n g .  The  The  crystal  s u r f a c e s were not o p t i c a l l y f l a t but had a t r a n s l u c e n t pearance.  ap-  Sample s i z e s were  Sample No»  Length  Diameter D  Thermometer Spacing 1  (c) lay  2.  3 cm  I(...86mm  .9il.l4.cm  1.  3cm  7.95mm  1.208cm  3.  3cm  2.86mm  .817cm  C r y s t a l Mounting.  The major experimental d i f f i c u l t y  i n a c h i e v i n g i n t i m a t e thermal contact between the  sample  and the h e l i u m b a t h , a n e c e s s i t y i n view of the h i g h heat c u r r e n t s b e i n g passed through the I n i t i a l l y we attempted between the sample  to make good mechanical contact  and v a r i o u s types of copper h o l d e r s  mounted i n s i d e the vacuum can. these methods was  sample.  The e f f i c i e n c y of a l l of  reduced g r e a t l y by the l a r g e  thermal expansion of C s l and copper. thermal expansion AL/L h e l i u m temperature with 1.13$  differential  The t o t a l  linear  of C s l from room temperature to  i s 1.16$  f o r L u c i t e , and  (James et a l , 1 9 6 5 ) ,  as compared  ,32% f o r copper..  The b e s t of the mechanical methods c o n s i s t e d copper f o i l  chuck, t i g h t e n e d around  by a tapered Perspex r i n g . r i n g would  the end of the  Our hope was  of a t h i n sample  that the Perspex  keep the copper i n good contact w i t h the C s l as i t  24  cooled.  This device resulted  i n an unacceptable sample to  b a t h thermal r e s i s t a n c e of about 2500 °K/watt. The  s u c c e s s f u l s o l u t i o n , suggested by a K a p i t z a r e s i s t -  ance experiment  (Johnson et a l , 1963), was  to i n s e r t a C s l  window 2.-I|jrnm t h i c k , i n the top of the vacuum can, and the sample to t h i s u s i n g an epoxy r e s i n . the window and the vacuum can was wire 0 - r i n g and  leak t i g h t  s e a l between  made u s i n g a .030" Indium  a, c i r c u l a r phosphor-bronze  ( F i g . 3 ) , and was system r e s u l t e d  The  glue  pressure p l a t e  i n l i q u i d helium-II.  i n an a c c e p t a b l e sample to bath  This  thermal  r e s i s t a n c e of about  50°K/watt.  (d) Sample Heater.  In making the sample h e a t e r , we were  again faced w i t h the d i f f i c u l t y of making good C s l to metal thermal c o n t a c t . contact was  In t h i s  case however, the q u a l i t y of the  of l e s s e r importance,  poor contact r e s u l t i n g  o n l y i n a h i g h e r heater temperature.  If the heater temp-  e r a t u r e were to r i s e very h i g h , l e t us say 20-25°K, some e r r o r due  to r a d i a t i o n pickup by the thermometers might be  encountered.  The heater was  designed with t h i s  possibility  i n mind. The h e a t i n g element  c o n s i s t e d of about 20' of  constantan wire, w i t h a r e s i s t a n c e of about h e l i u m temperature. square its  long a x i s .  The  1250 ohms at  A. C s l b l o c k .5" l o n g and w i t h  c r o s s - s e c t i o n was  .002"  5mm  cut w i t h grooves p e r p e n d i c u l a r to  heater wire was  wound i n t o these  grooves  i n a matrix of l i q u i d  epoxy r e s i n , and  subsequently epoxied to the samples.  the heater b l o c k With t h i s d e s i g n , the  thermometers could not "see" the h e a t e r winding. er b l o c k to sample epoxy r e s i n b u t t j o i n t had s i s t a n c e of around  The . h e a t -  a thermal r e -  50°K/watt.  The heater leads i n s i d e the can were s i x i n c h l e n g t h s of .OOl;" constantan w i r e , with a thermal r e s i s t a n c e of 10^ K/watt, 10,000 times h i g h e r than the b i g g e s t thermal o  r e s i s t a n c e of the samples. uum (e)  Leads were fed through the vac-  can to the helium bath v i a a. p l a t i n u m - g l a s s s e a l , Thermometers.  B r a d l e y 1/10 watt 33 ohms a;t room  The thermometers were commercial  Allen-  carbon composition r e s i s t o r s , n o m i n a l l y temperature.  F i g u r e 1; shows i n d e t a i l the mounting of the sample thermometers. of  ,l|.5min  increased  A phosphor-bronze  diameter copper wire onto the specimen.  Tension  i n the clip by t i g h t e n i n g the #000 x 1/8" brass;  b o l t r u n n i n g through i t . was  s p r i n g c l i p t i g h t e n e d a loop  One  lead of the sample thermometer  s o l d e r e d to the head of t h i s b o l t , thus a c h i e v i n g i n -  timate thermal contact between the thermometer and  the sample.  Thermometer leads i n s i d e the vacuum can were 6" lengths; of  .OOI4"  constantan wire, and were fed through the can v i a a  p l a t i n u m g l a s s s e a l separate from the one er power.  The  c a r r y i n g the h e a t -  thermometers were t h e r e f o r e i n very bad  contact w i t h the bath, as d e s i r e d .  thermal  was  . 015"CH. THICK "P«O«Pt»©8-BR0Na.6  tpamfr c u r .  . 60* C O N STA NT AN TttEKMOWeTCR u6At>S 0  *0OO* ' / 8 BOUT, NOT, TWO co AS M6RSJ  AT*PVY TENSION TO SPRING CUC.  Hlllllllll A - B / l o tOATT 33 XL RES » SToR SOLt>ffRSO TO HEAP OP * 000 X '/g*" J  "BOLT.  . 0 4 1 C M . "DIA. COPP69. UJ IRE SOLt>£R£P TO SPRING C U ? , UJHICV4 TlOWTRNS u>IRC.  DETAIL: THERMOMETER MOUNTING  FIGURE 4  Z7 Two  other thermometers were o e e s s i o n a l l y employed,  one  h e l d by a copper s t r a p to the i n t e r i o r top of the can  and  t h e r e f o r e i n e x c e l l e n t thermal  and  one  epoxied  to the end  contact w i t h the bath,  of the C s l heater b l o c k i n order to  measure the thermal r e s i s t a n c e of the heater b l o c k to sample joint.. The  two  sample thermometers chosen were the two  out of  twenty whose room temperature r e s i s t a n c e s were most n e a r l y equal,  (3) ELECTRONICS; (a) R e s i s t a n c e Measurement.  Resistances  of the thermometers?  were measured u s i n g a: s e n s i t i v e 33H2: A.C Wheatstone b r i d g e (Fig. 5)»  A primary  assumption i n t h i s experiment i s that  the sample thermometers are at the same temperature as sample.  This assumption i s i n v a l i d  th©  i f the current used to  measure t h e i r r e s i s t a n c e causes a p p r e c i a b l e s e l f - h e a t i n g of the r e s i s t o r s .  The  power generated  measurement was  i n the range 5 - 5 0  to cause measurable s e l f (i) Bridge*  The  v i a c o - a x i a l cable and known arm bridge.  x 10 ^ - 1  watts,  during  insufficient  heating.  r e s i s t a n c e thermometers were connected, a s e l e c t o r switch, to form the  un-  of a Leeds & Northrup model i;735 guarded Wheatstone This has  absolute accuracy 5000.  i n the r e s i s t o r s  a f i v e decade v a r i a b l e r e s i s t a n c e , w i t h u s i n g a l l decades, of . 2 5  However, s i n c e a l l measurements were  ohms i n , say, comparative,.  an  28 COHERENT AMPUF\SR, ANO REFERBNCS" Sl&NAl SOURCE*.  OSCILLOSCOPE".  LOW- LEVEL " P R E - A M R  REFERENCE  S^WAL.3** H*.  VARIABLE C^VftCVTANCe.  OuT-OF- PHASE "BALANCS".  UtAEATSTONE VQWG-E.  O"  :  O  Q  Tue-RMOMereR  SELECTOR SUlnTCH.  "THERMOMETERS ANP Lt APS IN CRYOSTAT.  THERMOMETRY ELECTRONICS  FiGJRE 5  29 u s i n g the same b r i d g e under i d e n t i c a l c o n d i t i o n s , measurements to .01 ohm were q u i t e ( i i ) Low-level output  justified.  Preamplifier.  The out-of-balance o r  s i g n a l of the b r i d g e was f e d v i a a s h i e l d e d cable to  a T e k t r o n i x type  122 p r e a m p l i f i e r .  This i s an AC-coupled  three stage wide-band a m p l i f i e r w i t h a v o l t a g e gain of 1 0 0 0 . It may be powered from ah AG power supply,' or from., dry ^and Lwet batteries.  As the l a t t e r r e s u l t e d i n a s i g n i f i c a n t  decrease  i n n o i s e g e n e r a t i o n , b a t t e r i e s were used. The  frequency  response of the a m p l i f i e r was  decreased  as much as p o s s i b l e to improve s i g n a l to noise r a t i o . low frequency  c u t - o f f of 8 Hz: and a h i g h frequency  A  cut-off  of 50 Hz were used. ( i i i ) Phase-sensitive Detector. preamplifier,  The output  ( i d e a l l y j u s t the a m p l i f i e d output  of t h e of the  b r i d g e ) , was f e d t o a T e l t r o n i c s model OA - 2 coherent lifier.  A coherent  amp-  a m p l i f i e r i s e s s e n t i a l l y an a c t i v e f i l t e r  system which makes use of a p r i o r i knowledge of the frequency and  phase of i t s input s i g n a l i n order t o measure i t s mag-  nitude.  The b a s i s of t h i s knowledge i s always the f a c t  that  the input s i g n a l i s some f u n c t i o n of a r e f e r e n c e s i g n a l generated by the coherent a m p l i f i e r . The plied  reference s i g n a l ,  (33 Hz, 5 v o l t s p-p), was sup-  through a 1000:1 v o l t a g e r e d u c t i o n as the input t o the  Wheatstone b r i d g e . adjusted  A phase c o n t r o l i n the coherent  amplifier  the d e t e c t i o n phase of the input to compensate f o r  phase s h i f t s i n the e x t e r n a l c i r c u i t r y ,  (bridge, c o a x i a l  30  c a b l e s , e t c . ) , so t h a t the s i g n a l detected in-phase, The  was only the  or r e s i s t i v e , s i g n a l as d e s i r e d . output  of the phase s e n s i t i v e d e t e c t o r was a -500uA  to +500uA meter which, when a l l phase adjustments were c o r r e c t l y made, was d i r e c t l y p r o p o r t i o n a l to the r e s i s t i v e unbalance s i g n a l of the Wheatstone b r i d g e , and was t h e r e f o r e the f i n a l readout f o r b a l a n c i n g the b r i d g e .  S e n s i t i v i t y of  the system was a d e f l e c t i o n of 150uA f o r an unbalance of 1 ohm i n 10,000. In o p e r a t i o n , the zero s e t t i n g of the meter was d i s covered  to be dependent on the gain and phase s h i f t c o n t r o l s .  T h i s was found t o be due t o the presence of .2mV of r e f e r e n c e frequency stage  r i p p l e on the -16 v o l t B- supply of the input  transistors.  filtered  The d e f e c t was cured by removing the  v o l t a g e from the input stages and r e p l a c i n g i t with  a -15 v o l t mercury b a t t e r y . (Iv) Out-of-phase Balance. also incorporated  The coherent  amplifier  a 9 0 ° phase s h i f t switch so that the out-  of-phase component of the input s i g n a l could be measured. An AC b r i d g e  i s completely  i n balance  only when both i n -  phase and out-of-phase components are balanced. phase balance  Out-of-  was accomplished by a v a r i a b l e capacitance to  ground i n p a r a l l e l with e i t h e r the unknown thermometer r e s i s t ance, or the bridge decade r e s i s t a n c e .  31 (v)> S i g n a l Leads. c r y o s t a t , had levels.  A l l l e a d s , i n c l u d i n g those  t o be w e l l s h i e l d e d due  to the low  into  signal  With ah input v o l t a g e to the b r i d g e of 5mV  an out-of-balance  p-p,  of 1 ohm. i n .10,000 generates; an e r r o r  s i g n a l of only .5uV„  The n o i s e l e v e l  p r e a m p l i f i e r was; e v e n t u a l l y reduced  the  :  at the input to the  to 2-5uV random or 60^ Hz,  but as the s i g n a l to n o i s e r a t i o of the coherent a m p l i f i e r was  approximately  1.1000, a good usable s i g n a l was  obtained.  Leads were fed through the c r y o s t a t cap via. a; m u l t i p i n s h i e l d e d p l u g , vacuum sealed w i t h a l a y e r of  Apiezon  Q-compound between the p i n s . (b) Heater Power Supply..  The  sample heater was  a H a r r i s o n L a b o r a t o r i e s model 620l|A. r e g u l a t e d v o l t a g e 0-3:6 v o l t DC power supply, ;  med  by a 10-turn H e l i p o t .  The  ( F i g . 6}, remote program-  ( l e s s than  i t s h i g h long term s t a b i l i t y  over e i g h t hours;)).  (c); Power Measurement.  ohm  The  .002$ at  ( b e t t e r than  A. v o l t a g e d i v i d e r was  lower v o l t a g e s to the 1200  constant  d e s i r a b l e features: of this;  power supply were i t s low r i p p l e v o l t s ) and  powered by  used to  10 .15$  supply  sample h e a t e r .  voltage d i v i d e r incorporated  a  c i r c u i t f o r measuring the heater v o l t a g e and heater c u r r e n t via. a Leeds & Northrup No. iometer volt  (Fig.6).  8662' p o r t a b l e p r e c i s i o n potent-  Reference voltage was  i n t e r n a l standard  cell.  s u p p l i e d by a. 1.0191+0  Each v o l t a g e was  1/[|$, the power t h e r e f o r e b e i n g accurate to  measurable to  1/2$.  HI  I K/l  LJO  I O K A .  Rx  TO LAN W ' R .  R  3  _J  Rj  V V„  R.  T O L*Npor*l.  I  R5  1 9 , 3 5-5"  Su  V = ' 8 7 . 66V mV. s  TO  rn  o  DC  POWER  SUPPLY.  V  33  CHAPTER H I . M  PROCEDURE AND TECHNIQUE  INTRODUCTION  Any experiment  must be performed  under p h y s i c a l  d i t i o n s ; r a t h e r s d m i l a r t o those i m p l i c i t  con-  i n the t h e o r e t i c a l  treatment i t p u r p o r t s to be i n v e s t i g a t i n g .  Necessary ex-  perimental, a-nd p r o c e d u r a l c o n d i t i o n s f o r t h i s experiment a r e : - that the sample be i n thermal steady s t a t e when any measurements  are taken; e i t h e r under zero heat flow-  c o n d i t i o n s f o r c a l i b r a t i o n , or under f i n i t e heat flow/ c o n d i t i o n s f o r c o n d u c t i v i t y measurementsj - that the thermometers must be i n good thermal contact w i t h the sample and bad thermal contact w i t h the helium bath; - that t h e r e be no dependence of the thermometer r e s i s t a n c e s on the power used to measure  them;  - that the measured thermal c o n d u c t i v i t y at a g i v e n temperature  must not depend on the power flow through  the sample % - and t h a t no heat generated  i n the sample heater must  reach the helium bath through anything but the sample, whence (a)) there must be "no" gas i n the experimental chamber, and ( b j the sample h e a t e r leads must have a thermal r e s i s t a n c e l a r g e compared w i t h that of the sample.  34  The  d e s i g n , procedure  and technique of the  experiment  were set up w i t h these f a c t o r s i n mind. (2:))  GENERAL EXPERIMENTAL PROCEDURE  A t y p i c a l run began i n the morning,,  Twenty-fours  p r e v i o u s to t h i s , w i t h both de-wars at room temperature, forepump and d i f f u s i o n pump were turned on, the cold filled,  trap  and the experimental chamber and vacuum s i d e s of  the manometers  evacuated.  Some twelve hours  l a t e r , the helium dewar was  pumped  out to 1 cm Hg of a i r , and the n i t r o g e n dewar f i l l e d . b e f o r e any run the experimental chamber was for at  about  twelve hours  at room temperature  Thus  d i f f u s i o n pumped and twelve  hours  n i t r o g e n temperature. To b e g i n the run, the helium dewar was  filled for  the  evacuated,  then  with one atmosphere of helium gas i n p r e p a r a t i o n  the t r a n s f e r of l i q u i d .  The t r a n s f e r siphon was  f l u s h e d w i t h helium gas., the dewar f i l l e d l i t e r s of l i q u i d helium  (liquid  then  w i t h about J.  l e v e l about 50cm above the  chamber t o p i and the r u n begun. F o r some runs the experimental chamber was  open to the  d i f f u s i o n pump f o r the d u r a t i o n of the run, while f o r others i t was  sealed o f f j u s t p r i o r to l i q u i d helium t r a n s f e r to  permit cryopumping of the can.  3£T  (3))  SAMPLE TEMPERATURE MEASUREMENT  PROCEDURE:  CALIBRATION AND DATA. TECHNIQUE (i)  One of the drawbacks of carbon r e s i s t a n c e  therm-  ometers i s that they tend to change t h e i r c a l i b r a t i o n i f c y c l e d up t o room temperature,  then cooled a g a i n .  In our  experience, t h i s change i n c a l i b r a t i o n took the form of an a d d i t i v e constant i n the r e s i s t a n c e , that constant b e i n g nearly  the same f o r b o t h r e s i s t o r s .  We f e e l t h i s f a c t bears  some i n v e s t i g a t i o n . Over the course of s e v e r a l runs, separated by periods? at room temperature,  one would f i n d s h i f t s  incalibration  of the thermometers of the order of *5 m°K. However, s h i f t s ; i n the d i f f e r e n c e of the c a l i b r a t i o n s amounted to o n l y + .5 m°K and the slopes of the c a l i b r a t i o n curves  remained  v i r t u a l l y constant. In s p i t e of these d i s c o v e r i e s , the d e c i s i o n was madethat each r u n should c o n s i s t of a c a l i b r a t i o n r u n and a thermal c o n d u c t i v i t y r u n .  In p r a c t i c e these were interwoven,  a c a l i b r a t i o n p o i n t and a: c o n d u c t i v i t y p o i n t taken at one temperature, t h e temperature point  then s h i f t e d , a c a l i b r a t i o n  and a c o n d u c t i v i t y point taken t h e r e , and so on. ((ii); R e s i s t a n c e Measurement.  The f i r s t  step i n making  &\ r e s i s t a n c e measurement was t o s e t up the phase relations? of t h e coherent a m p l i f i e r so that i t was indeed* measuring a pure r e s i s t a n c e .  The procedure was as follows::  3<b  (Aj  Set the b r i d g e decade so that the b r i d g e i s d e f i n -  i t e l y out of balance; (B) and  With the coherent a m p l i f i e r a t low s e n s i t i v i t y , i n the "in-phase  ,,:  o r (p mode, a d j u s t the phase s h i f t  to o b t a i n maximum d e f l e c t i o n on the output meter. (C)  Obtain a rough r e s i s t i v e b a l a n c e .  (D)  Set the capacitance decade so there i s a l a r g e  capacitive (E) and  unbalance.  With the coherence  a m p l i f i e r a t low. s e n s i t i v i t y ,  i n the "out-of-phase"  or (p * 9 0 ° mode, adjust t h e  c o n t r o l t o o b t a i n maximum d e f l e c t i o n on the meter, thus s e t t i n g A<p to 9 0 ° . (P)  Balance the b r i d g e c a p a c i t i v e l y .  (G)  Return to the "in-phase" mode and a t h i g h sens-  itivity, (H)  o b t a i n an exact r e s i s t i v e b a l a n c e .  Check.  Apply a l a r g e c a p a c i t i v e unbalance.  the coherent a m p l i f i e r should now be measuring a pure r e s i s t i v e  As  only  component, t h i s should have no e f f e c t  on the meter r e a d i n g . S'dnce the phase s h i f t  i n the e x t e r n a l c i r c u i t r y i s  l a r g e l y a; p r o p e r t y of the geometry t h e r e o f , the phase s e t t i n g procedure need o n l y be f o l l o w e d through once. was  checked  In f a c t i t  o c c a s i o n a l l y and found not to have changed.  R e s i s t a n c e measurements were taken by simply observing the output meter of the coherent a m p l i f i e r and a d j u s t i n g the b r i d g e decades t i l l  i t read z e r o .  37 The b r i d g e contained a m u l t i p l i e r  ( r a t i o arm), s w i t c h , 1  whose x . 1 ( f o r resistances] 3'00 - 1000 ohms); and x 1.0 ( f o r r e s i s t a n c e s 1000 - 10,000 ohms) p o s i t i o n s were used. The c a p a c i t i v e balance was found t o d i f f e r on these two ranges', b e i n g about + 500 pf on one range and -5000 pf on t h e o t h e r . The c a p a c i t a n c e decade was adjusted a c c o r d i n g l y . (iii)  Calibration Points.  The f i r s t  step i n o b t a i n i n g  a c a l i b r a t i o n p o i n t was t o choose the temperature at which i t was to be taken.  C a l i b r a t i o n p o i n t s were u s u a l l y planned  so as t o cover the temperature range of i n t e r e s t  i n roughly  equal i n t e r v a l s of I/T. Having chosen the temperature, the corresponding vapour p r e s s u r e was set i n the r e f e r e n c e volume of the p r e s s u r e controller.  I f the r e s u l t was a drop i n temperature, t h e  pumping system q u i c k l y pumped the dewar down and a s t a b l e temperature was o b t a i n e d . rise  i n temperature,  I f , however, the r e s u l t was a  then the pumping system,was shut o f f  by the c o n t r o l l e r diaphragm.  To o b t a i n r a p i d  stabilization,  heat was a p p l i e d t o the bath h e a t e r u n t i l the d e s i r e d  temp-  e r a t u r e was reached, as i n d i c a t e d by the renewal of gas flow/ through the pumping system.  At a l l temperatures above;  the lambda; p o i n t the bath was c o n t i n u a l l y s t i r r e d by a 7»5 mW' heat  input. Once b a t h e q u i l i b r i u m was obtained ) ( u s u a l l y i n 15 sec7  onds t o 2. minutes) we waited f o r thermal steady s t a t e of the; 1  sample.  F o r the c a l i b r a t i o n p o i n t s , an a b s o l u t e e q u i l i b r i u m  38 was  desired.  Thermometer r e s i s t a n c e s were measured  they were s t a b l e over a p e r i o d thermal steady s t a t e was  until  of about f i v e minutes.  Sample  u s u a l l y obtained i n 2.0 - 30 minutes.  At such time, the c a l i b r a t i o n p o i n t data were r e c o r d e d . They c o n s i s t e d (1)  time;  (2)  By, R^  of:  ohms, the r e s i s t a n c e s of the sample thermo-  meters as measured by the b r i d g e ; (3)  b cms,  tape  the helium b a t h l e v e l ,  ( i f b a t h temperature was  as measured by m e t r i c  above the lambda p o i n t ) ;  (J4.) T-room ° C , the temperature  of the manometers as  measured by thermometer or manotherm ( i f bath temperature was (5) if  P  (3  c  below the lambda p o i n t ) ; m  e w  Hg or cm o i l ,  the mercury manometer  above the lambda p o i n t , or the o i l manometer  i f below, as measured by the (6)  reading reading  cathetometer;  Check b a c k i n g pressure<2i0 microns, as measured by  the thermocouple This completed  gauge.  the c a l i b r a t i o n point and one then moved'  to another c a l i b r a t i o n p o i n t , or to a c o n d u c t i v i t y p o i n t . (iv)  Conductivity Points.  Again the f i r s t  o b t a i n i n g a c o n d u c t i v i t y p o i n t was at which about  i t was  wanted.  , 1 ° K lower was  helium bath brought  t o choose the  Having done t h i s ,  a  step i n temperature  temperature  set on the pressure c o n t r o l l e r and the. to t h i s  temperature.  3?  Next, heat was  generated i n the sample h e a t e r .  This  was  u s u a l l y a predetermined amount of heat i n the range  0.1  - J,0  mw,  and  c o n t r o l to the We For the  set by a d j u s t i n g the power  correct  supply  value.  now, waited 15 - 30 minutes f o r thermal steady s t a t e . c o n d u c t i v i t y p o i n t s we  w i t h absolute but  was  steady s t a t e  were not  q u i t e so concerned  (no change i n e i t h e r r e s i s t o r ) ,  set as our steady s t a t e c r i t e r i o n that there be no  i n the r e s i s t a n c e d i f f e r e n c e of the two When steady s t a t e was data were r e c o r d e d . (1)  R,  R^  (2)  V,  Vj^ mV,  the two  u  v  voltages  obtained,  They c o n s i s t e d  change  thermometers.  the c o n d u c t i v i t y point of:  ohms, as measured by the as measured by the  bridge;  potentiometer,  used i n c a l c u l a t i n g the power i n the  sample  heater. The measurement of one complete and  one  c o n d u c t i v i t y point was  then  moved on to another c o n d u c t i v i t y p o i n t  or t o a c a l i b r a t i o n p o i n t . ((v) Power Dependence of Thermal C o n d u c t i v i t y . the  c o n d u c t i v i t y p o i n t s were spaced roughly  temperature range of i n t e r e s t .  Normally  e q u a l l y over  However, o c c a s i o n a l l y  wished to know whether the apparent c o n d u c t i v i t y was on the heat current  through the c r y s t a l  ( i t should  To determine t h i s , a* s e r i e s of points wer»,taken by  we dependent  not  be).  lower-  i n g the bath temperature s l i g h t l y between each p o i n t , a p p l y i n g enough power to the  the  and  c r y s t a l to b r i n g i t s temperature  up to the temperature at which the f i r s t  p o i n t was  measured.  Ao One  then obtained a s e r i e s of c o n d u c t i v i t y p o i n t s at r o u g h l y  equal temperatures but taken w i t h i n c r e a s i n g heat (1+)  currents,.  POWER DEPENDENCE OF THERMOMETER TEMPERATURE:  A's mentioned  p r e v i o u s l y , i t i s important i n t h i s  experi-  ment that s e l f - h e a t i n g of the r e s i s t a n c e thermometers due to the measuring c u r r e n t be n e g l i g i b l e . • In p r e v i o u s work With carbon r e s i s t o r s the environment of the r e s i s t o r has been shown to be of importance i n d e t e r mining i t s s e l f - h e a t i n g .  Clement  and Quinnel (1952) found-  a dependence of temperature r i s e on power d i s s i p a t i o n W' g i v e n by dW/dT. = £ . 5 x 1 0 ^ T f o r r e s i s t o r s i n vacuum i n av s o l i d  W/deg copper heat s i n k , whereas'  Berman (190%) found the r e l a t i o n s h i p ) <3W/dT = 3.9 x 10-£ T l . 6  W/deg  f o r a r e s i s t o r i n vacuum cemented t o a copper b l o c k i n c o n t a c t w i t h the helium b a t h . Due  to t h i s s t r o n g geometry dependence, we decided t o  check our thermometers t o ensure they were not heating.. A. r u n was done u s i n g two temperatures, 2.10°K.  l\...22 K a  and  F o r each temperature, the r e s i s t a n c e s of the upper  and lower thermometers',  (as w e l l as a "bath thermometer"' b o l t e d  to t h e i n s i d e of the vacuum can) were measured u s i n g power 1  1  d i s s i p a t i o n s r a n g i n g from 5 x lO"""""""" watts t o 5 % 10"^ w a t t s . -  The value of R taken at the lowest power was designated as R . Q  41 The upper and lower r e s i s t o r s behaved manner and the r e s u l t s Table I I  i n an i d e n t i c a l  f o r the upper r e s i s t o r are shown i n  below, and i n F i g u r e 7„ Table I I .  T = 1;.22°,^R Ru^ou  Thermometer Bower Dependence  = 377.75 ohms  0  W/ watts  R  .14.87 x I 0 "  1.000  T = 2.10°, R  1 0  1.000  2.62: X l O "  2.81;  "  1.000  1.000  7.80  "  1.000  1.000  I+8.7  1.000 .  195  1.000  780  .9995  "  .9970  19.5  .9679  78.O  -9U2I|.  .9996  ,,;  n:  I1.87  xl O  -  w  R  1.95  10+.8  = 151+6.7 ohms  v/ ou  1.000  1.000  c  7  1+5.1+  n  .9980  281;.  it  .9966  1.11+ X lO"?  .9892  1+.5I;  the form  and i f we assume t h e l i n e a r  Mi  139.  n  relationship  1/T = A„ +• B l n (R)]  t»  28.1;  -  dW/dT = C T  n  11.1+  If we assume f o r the upper r e s i s t o r a power of  1 *  .9992  .91+56  1+87  1 0  dependence;  43  f o r the temperature dependence of the r e s i s t a n c e , we  then  o b t a i n the r e l a t i o n s h i p dR  = C B T  dV Integrating,  we  n  +  2  obtain  C B T n*2 Prom previous c a l i b r a t i o n s , B i s known, and thus it  and the two known temperatures we  using  can solve f o r C and  n.  These were found to be C =  2  1.9  n = 1.99 g i v i n g the r e l a t i o n s h i p dV//dT f o r our  =  1.9  5% * 5% = 2 10""£  +  a 10"^  T  2  W./deg  thermometers..  Knowing the power d i s s i p a t i o n to be used i n a l l c a l i b r a t i o n and thermal c o n d u c t i v i t y measurements (point N, P i g . 7), and u s i n g  the derived  power dependence r e l a t i o n s h i p we were  thus: able to show that s e l f h e a t i n g from 1 microdegree at 1.2: i|.2 °K and was  therefore  of our thermometers  °K to about 15 microdegrees at completely n e g l i g i b l e .  varied  44 (5) It  LEAD RESISTANCES; will  be n o t e d  ances  of the upper  iated  leads.  (R^  -  ^ l i ^ The  meter  last  a  r  and h e a t e r  circuit  power  mas  At  seen  generated  resistance, variation  (6)  remained  only  The using  fraction  of  heater  about  1 0 $of t h e t o t a l  3$ o f the h e a t e r level  resistance.  o r t e m p e r a t u r e was leads:.  MEASUREMENT  t h e geometry  a r e a A, and t h e d i s t a n c e  were  made, i t  of t h e sample-- i t s ; between  the ther-  L. diameters; D  to center  a  what  i n t h e sample  of the constantan  of t h e sample were measured  distances  cathetometer.  t o !./<%%  or a travelling microscope. between  t h e t h e r m o m e t e r s were m e a s u r e d using  t o determine  i n the heater.  and about  a micrometer c a l i p e r  Center  the thermo-  a t t h e thermo-  was d o n e  to find  constituted  to establish  i n terms of  with  t h e low t e m p e r a t u r e measurements  cross-sectional mometers;  and a l s o  assoc-  o f t h e leads..  circuited  and measured  S A M P L E GEOMETRY  After  analyzed  This  due t o b a t h  i n the resistance  to the r e s i s t plus: t h e i r  was done  short  respectively.  l4..2°K, t h e l e a d s  Negligible  were  sample  leads  not dissipated  thermometer  refer  the resistances  e  r u n on e a c h  and h e a t e r  total  a n d R^  U  The c a l i b r a t i o n s  Q"li ( f o r t h e t h e r m o m e t e r s ) the  R  and l o w e r t h e r m o m e t e r s  where  leads  meters  that  the copper wires  t o 1/2$, f r o m  holding  s i x directions,  4r  CHAPTER IV.  (1)  MANOMETRY AND  (a)  Introduction.  are  based  T^g  helium-li  The  1958  terms cury  vapour  and  local  measured vapour  gravity pressures  pressures.  these We  We  shall  to  tolerate  we  xtfould  define  scale  conductivity  at varying  corrections  must  to reduce  tolerate  2  bore  room  them  o f mer-  cm/sec . mercury  temperatures  b e made  to these  to accurate  i t necessary  accurately  are working.  a t a. c e r t a i n  T^g  t o make  o f up t o 1 %  i n this  error  however,  allowed  i n this  temperature between  that  Each  thermal decided  T^g and T j o f a t Ii|; K, 0  proportional  an e r r o r  difference,  We  temperature, i . e .  between  are inversely  W.e d e c i d e d  corrected  temperature.  a t 1 ° K a n d I4.0 m i l l i d e g r e e s  difference.  a difference  we  an a b s o l u t e  erature  .1%.  as T j , t h e l e s s  i n which  an e r r o r  raillidegrees  of  on f i n i t e  i960)„  i s given i n  o f 980.-665  d i d not consider  Is, q u o t e d  conductivities,  AT.p  et a l ,  of th©  corrections.  temperature  ate  (van D i j k  gravity  are read  i n order  report  o f helium-lj. i n m i c r o n s  manometers many  i n this  of temperatures  pressure  pressures  quoted  t o , the temperatures  scale  s:cale  a t 0 °C a n d a. s t a n d a r d  butyl-phthalate  10  pressure  international  and  all  CORRECTIONS;  on, b u t n o t i d e n t i c a l  vapour  REDUCTION  A l l temperatures  of the vapour  If  DATA  The to a  o f ,1%  i . e . we  would  thermal temp-  could", b e toler-  some < ^ T j a n d t h e c o r r e s p o n d i n g  (b)  Corrections»  Suppose some d i f f e r e n c e i n meniscus  levels.  of a mercury or b u t y l - p h t h a l a t e manometer has been read w i t h a cathetometer, g i v i n g a "raw vapour p r e s s u r e " . Below is a l i s t  of a l l c o r r e c t i o n s that  could be made, and whether  i n s e t t i n g up T j they were, or were not made.  Following  this, s e c t i o n w i l l be a proof t h a t the r e s u l t i n g T  fulfils; J  i t s requirements. Three working c r i t e r i a , were used to determine whether a c o r r e c t i o n would (1)  or would not be retained:.  the o i l and mercury manometers were to agree to  w i t h i n the 5 0 micron u n c e r t a i n t y of the  cathometometer  reading, (2)) was (3)  i f a c o r r e c t i o n f a c t o r was  time independent i t  ignored, and i f a. c o r r e c t i o n f a c t o r was  always l e s s then th®  50 micron cathetometer u n c e r t a i n t y i t was i g n o r e d . The c o r r e c t i o n s a r e : ( i ) Backing pressure. of the manometers was  The p r e s s u r e i n the vacuum s i d e  always checked.  I t was  than 1 micron and l e s s than 10 microns and was  always  greater  therefore  ignored. ( i i ) Meniscus e r r o r s arisie i f the meniscuses arms of the manometer are of d i f f e r e n t " h e i g h t " of a mercury meniscus  shapes.  on the two  S i n c e the  (the v e r t i c a l d i s t a n c e from  the center point of the column to the p o i n t at which i t touches the w a l l s of the tube) i s t y p i c a l l y 1000 -  1500  77 microns f o r a 1 cm bore tube, t h i s e r r o r can be l a r g e .  The  manometers; were t h e r e f o r e tapped w e l l b e f o r e each r e a d i n g . Experience showed that t h i s brought  the meniscus e r r o r to  w i t h i n the 50 micron cathetometer u n c e r t a i n t y , and the c o r r e c t i o n was i g n o r e d . (iii)  Thermal  expansion of cathetometer.  ometer was c a l i b r a t e d coefficient  at 20 °C, and had a thermal expansion  of 1.2 x 10°°^/t  o  Maximum e r r o r from t h i s  would occur at a bath temperature say 21; °C.  room temperature, the e r r o r i s !;0/u  The c a t h e t -  source  of l;.2°K, and at the h i g h e s t  Under these worst c o n d i t i o n s  Cathetometer  expansion was t h e r e f o r e  ignored. (iv)  H y d r o s t a t i c head c o r r e c t i o n . The measured  vapour  pressure was that of the s u r f a c e of the helium b a t h whereasthe thermometers were as much as 50 cm below t h i s point andl hence the l i q u i d  surrounding them was at a h i g h e r p r e s s u r e .  Maximum e r r o r from t h i s source i s about [;000 microns time dependent. (v) G r a v i t y .  and ia,  It i s therefore included. The pressure exerted by a column of mercury  i s d i r e c t l y p r o p o r t i o n a l to the a c c e l e r a t i o n of g r a v i t y . The value taken f o r l o c a l g r a v i t y  (980.937 cm/sec . ) was from 2  a Department of Mines and T e c h n i c a l Surveys: measurement taken w i t h i n 100 yards of the experiment.  The r a t i o of  t h i s value to the T^g standard g r a v i t y i s 980.. 937  =  1.00028  980.665 g i v i n g a maximum e r r o r i n vapour pressure at l;.2°K of about  *8  2 2 0 microns.  However, the e r r o r i s time-independent  and  is  therefore ignored. (vi)  D e n s i t y . (A.) of mercury  temperature.  The T^Q  due  to constant room  s c a l e i s i n terms of mercury  at  0°C.  I f we assume f o r T j a donstant room temperature o f , say, 23"C,  we  i n t r o d u c e an e r r o r i n the d e n s i t y of mercury of  p (23)^0(0)  = .99583  or an e r r o r of •iil7#.  This r e s u l t s i n  a maximum p r e s s u r e e r r o r of about 3500 microns.  As we have  assumed a constant room temperature, however, the e r r o r i s time-independent  and i s t h e r e f o r e i g n o r e d . (B) of mercury  temperature. assumption  Here we  due to v a r i a b l e room  account f o r the f a l s i t y of the p r e v i o u s  of constant room temperature.  I f the room temp-  e r a t u r e were to vary r a t h e r r a p i d l y , there would be an apparent v a r i a t i o n i n b a t h temperature mercury  density.  We  due to the v a r i a t i o n i n  i n t r o d u c e an a d d i t i o n a l c r i t e r i o n at  t h i s p o i n t , that the apparent change i n b a t h s h a l l not exceed  1/2' m i l l i d e g r e e / h o u r .  temperature  The s i t u a t i o n i s  worst at the lowest temperature of 1.2:°K ( s i n c e the q u a n t i t y (dP /dTg^)/P v  v  i s l a r g e s t there) and our c a l c u l a t i o n i s f o r  a helium bath at that At 1.2°K The  temperature.  the vapour pressure of helium-I|. i s 625 microns.  slope of the vapour p r e s s u r e curve dP /dTg^ i s i|.oI4.I v  microns/millidegree. of mercury  The v o l u m e t r i c expansion  at room temperature  coefficient  i s y S - »85 x 10~*3/ C 0  Using  these v a l u e s , i t i s easy to show that an apparent b a t h temp-  49 e r a t u r e v a r i a t i o n of 1/2 m i l l i d e g r e e / h r r e q u i r e s a room temperature v a r i a t i o n of at l e a s t i+^C/hour,  Changes of room  temperature of t h i s r a t e were never observed t h e r e f o r e  this  c o r r e c t i o n i s ignored, (C)  R a t i o of mercury to b u t y l - p h t h a l a t e .  There must be no d i s c o n t i n u i t y when going from the mercury manometer to the o i l manometer. butyl-phthalate  The r a t i o of the d e n s i t y of  to the d e n s i t y of mercury must t h e r e f o r e be  known a c c u r a t e l y , as a f u n c t i o n of room temperature. could f i n d no published  data on t h i s r a t i o , t h e r e f o r e we  performed a s h o r t , accurate andy^oil( )^Hg( ) T  The  T  f  o  r  We  experiment to e s t a b l i s h ^ ^ ^ T )  f  temperatures between 15 °C and 25°C.  r e s u l t s of t h i s experiment are shown i n Appendix A thermometer was then constructed  using  1.  butyl-phthalate  as the thermometric substance, with a long t u b u l a r b u l b , the shape of h a l f a manometer.  This d e v i c e , which we c a l l a  "manotherm" was used t o read room temperature, on the assumpt i o n t h a t i t s thermal response time most c l o s e l y matched that of the manometers. This c o r r e c t i o n i s necessary and i s t h e r e f o r e  included..  (2) T j VS, T ^ Q . The  helium-l^ vapour pressure  s c a l e T^g i s a r e l a t i o n -  s h i p between temperature and pressure  d e f i n i n g a; correspond-  ence f ( p ) ; T  58  = < 58>f  As shown i n the previous  p  s e c t i o n , vapour pressures f o r  SD the ^dewi  T j s c a l e are obtained by measuring the dewar p r e s s u r e u n <  ^  e x i s t i n g conditions,  e r  and making a; head  correction  where n e c e s s a r y ; Pj = d p  bead c o r r e c t i o n , whence T j = f ( P j ) .  + e  w  The T^g s c a l e vapour p r e s s u r e s , P^g are obtained from P' j (  ew  by i n c l u d i n g the e f f e c t s of b a c k i n g p r e s s u r e , l o c a l g r a v i t y and room temperature; P^g = dew P  r +  g r a v i t y c o r r e c t i o n + cathetometer expansion  +• room temperature c o r r e c t i o n + b a c k i n g p r e s s u r e •fr head  correction.  We now. wish to observe, f o r a t y p i c a l s e t of measuring conditions,  the d i f f e r e n c e between T j and T^g* We w i l l assume  a b a c k i n g p r e s s u r e of 5 microns, a v a r y i n g room temperature of 22 ° C + 2°C, a l o c a l g r a v i t y of 980*937 cm/sec , and f o r 2  s i m p l i c i t y , a zero head c o r r e c t i o n . the  Under these  conditions,  c o r r e c t i o n s are +  (a) B a c k i n g p r e s s u r e . . . . . . . . . . . . . . (b) G r a v i t y  . . . . . . . . .  (c) Room temperature. . . (d) Cathetometer. . . . (e) Head c o r r e c t i o n .  + (..277). x 10  - (3»99 £ .-36) x  p  5 microns  dew .  10~ P(-j  microns  3  ew  * (.021; +_ .021;) x 1 0 ~ P^ew 3  . . . . . . . . . . . . . .  0  microns microns microns  g i v i n g a t o t a l c o r r e c t i o n of . . . . . . . . . .  - (3.69 ± »38) x 10~3 P  d e W j  microns.  SI For the c o n d i t i o n s above we see, t h e r e f o r e , «  dew  P58 = Using selected  - ( 3 . 6 9 ± .38) x 1 0 ~  p d  e  w  values of P'^  F  3  d e w  . + 5 microns.  we have prepared Table 7J£*  below, comparing T j = f ( P j ) and T^g = f ( P ^ g ) .  Table HT  V  K  k*5ooo  P  58  m°K  .00011}!  *k *k  %.0000  U.0037  .OOOI4;  3.5000  3*5030  .0003  3.0  .3  3..0000  3.0021;  .0003  z.k  *3  2.5000  2:.5018 + .0012  1.8 + .2  2.0000  2.0012  .00013  1.2  1..5000  1.5005  .00008  .5  +•.  These r e s u l t s are p l o t t e d  *•  .13 .08  i n F i g u r e 8, from which i t  may be seen that f o r T g r e a t e r than 1;5°K, the r e l a t i o n s h i p between T j - T^g and T.^g i s l i n e a r , of the form . Tj - T g 5  =  m T g 5  +  b.  Such b e i n g the case, l e t us assume a AT^g — T ^ g ( l ) - T^g(2 :), :  and a corresponding/^T  T  FIGURE 8  53 W.e t h e r e f o r e  write  Tj(l)  - T  5 6  (l) = m T  5 6  ( l ) + b,  T j ( 2 ) - T (2.) = m T g(2;) * b, 58  and  5  subtracting, Tj(l)  - T  5 8  ( l ) - Tj(2) + T  5 8  (2)  =ffl ( T  5 8  ( l ) - -1^(2)  )  or ATj  -  =  m AT  From the graph we o b t a i n a slope  5 B  .  (using maximum and  minimum values;) of m = ..0012 + ..0001. It may be seen from the graph that the l a r g e s t e r r o r occurs  at the highest  temperature,, say 14.„5>°K..  P u t t i n g i n the appropriate absolute  f i g u r e s , we o b t a i n an  error T —  — T — T  and  a; f a i r l y  <=: *1% <  r e q u i r e d 1$,  58  constant J.  relative  error  2°  =  .12$ £  r e q u i r e d .1$  Hence we see t h a t the T j temperature s c a l e does fulfil ied  indeed  i t s requirements and we conclude that we are j u s t i f -  i n i t s use.. (3)  CALIBRATION PROCEDURE. AND PROGRAM..  Once a. set of c a l i b r a t i o n p o i n t s a given thermometer on a g i v e n run, ical  absolute  (R,T) i s obtained f o r  some method of mathemat-  i n t e r p o l a t i o n must be found to make i t useful..  One  S4  wishes to f i n d t h a t a n a l y t i c e x p r e s s i o n which best describes? the observed facilitate fits  temperature  dependence of the r e s i s t o r .  t h i s , a computer was used  to perform  least  To squares?  of the input data to v a r i o u s a n a l y t i c e x p r e s s i o n s .  (a) Input data.  Consisted o f :  - N, the number of c a l i b r a t i o n p o i n t s , - (R);, the N values of r e s i s t a n c e measured on the b r i d g e , - ( T ) , the N corresponding c a l i b r a t i o n temperatures calculated  as  i n the p r e v i o u s s e c t i o n , and  - 0,^, the r e s i s t a n c e of the thermometer l e a d s , s i n c e R-Q^ i s the r e s i s t a n c e of the thermometer a l o n e .  T h i s input  d a t a i s of course p e r t i n e n t to only one r e s i s t o r . (b) least  Calculation.  U s i n g the above data, a s i n g l e  squares f i t was performed  t o each of the f o l l o w i n g  expressions. - LINEAR 1/T = A, * B, ln(R-Q. ), L  - QUADRATIC 1/T = A  + B,> ln(R-Q. ) +  2  L  C^ln (R-Q, ), 2  L  - CUBIC 1/T = A- + B l n ( R - Q , J * C ln (R-Q. ) + D l n 2  3  3  3  u  3  3  (R-Qj,  - CUBIC + l/LOG R 1/T = A^ + B^ l n ( R - Q ) * C^ I n (R-Q } • * D^ln3(R-Qj 2  u  +E,/ln(R-Q ), u  iteration  L  - CUBIC + 1 / L O G R + 1 / ( L 0 G R)**2  1/T = A ^ * B ^ l n f R - Q j  + C ln (R-Q l 2  r  u  + D^ln^R-Qj  + E /ln(R-Q ) + F j V l n ^ R - Q j * r  u  F o r each e x p r e s s i o n , the i n t e r m e d i a t e estimate  of the  parameters(&,..., F )  estimate of  c o n s i s t e d of the f i n a l  the parameters of the previous  e x p r e s s i o n , the i n t e r m e d i a t e  estimate f o r the l i n e a r case b e i n g A = 0 , B = 0. For each e x p r e s s i o n , when the f i n a l meters was obtained, \ 1/TFIT| —  estimate  of p a r a -  i t was used to c a l c u l a t e the s e t  the invers;e temperatures corresponding  input s e t R and the estimated  parameters.  to the  The i n v e r s e  temperatures 1/T from the input s e t T were a l s o c a l c u l a t e d . F i n a l l y , f o r each s e t of parameters, a s e t of numbers c a l l e d '*TYPE 2 ERRORS" was c a l c u l a t e d . (c) Output. the output  F o r each e x p r e s s i o n  ( l i n e a r , q u a d r a t i c , etc.))  consisted of?  - Name of e x p r e s s i o n - Intermediate  (linear,  etc.),  estimate of parameters  A, B, C, D e t c . , - F i n a l estimate A, B, C,  D ,  of parameters E, e t c . ,  - Set of N values f o r R  T  R-Q.  - " TYPE 2 ERRORS". !  1/T  1/TFIT  L0G(R-Q )  (d) Analysis.. F o r each thermometer on each r u n we were thus presented  with f i v e p o s s i b l e a n a l y t i c expressions d e s c r i b i n g  i t s behaviour.  I t remained o n l y to p i c k the best one.  There were two c r i t e r i a f o r the best  fit;  ( i ) the average value of the set | l / T - . 1/TFIT \ must be as s m a l l as p o s s i b l e , and ( i i ) the "TYPE 2 ERROR" corresponding t o each parameter should be l e s s than 10% of the magnitude of the parameter. In p r a c t i c e the | 1/T - 1/TFIT( average was h i g h f o r the LINEAR FIT (5-10 x: 1 0 ~ ) , 3  relatively  about two orders of  magnitude lower f o r the QUADRATIC FIT (£-10 x 1 0 ^ ) , and -  showed l i t t l e ions.  i f any improvement f o r h i g h e r order  express-  The "'TYPE 2; ERRORS'" rose w i t h i n c r e a s i n g o r d e r of  e x p r e s s i o n s and g e n e r a l l y exceeded 10% of the corresponding parameter at the CUBIC. The  optimum f i t was n e a r l y always the QUADRATIC, and  o c c a s i o n a l l y the CUBIC.  The LINEAR:, CUBIC + 1/LOG R, and  CUBIC * 1/LOG- R * 1/(L0G R )  expressions were never selected..  2  Magnitudes of a t y p i c a l  set of parameters f o r the upper  r e s i s t o r are A. =  -0.2714.76  B =  +0.0331+91  c =•  +0.0093272  D =  0  E =  0  F =  0  51 (1+)  CONDUCTIVITY POINT ANALYSIS-; T.AT PROGRAM.  On a g i v e n run, the raw data f o r c o n d u c t i v i t y points, c o n s i s t e d of the s e t of quadruplets  (R ,Rj, u  V , V^) — the y  r e s i s t a n c e s of the upper and lower thermometers, and the two v o l t a g e s from the power measurement c i r c u i t  (chap., UL - J).  From these one must o b t a i n the heat current o r power through the sample, and the temperature d i f f e r e n c e between the thermometers. (a) Heat c u r r e n t .  ( F i g . 6)  to the sample h e a t e r ,  Define  V  g  as the v o l t a g e  applied  s  i s proportional to V ,  y  g  *V  = ( 3 R  R  I  g  leads, and current measuring r e s i s t o r .  Then the measured p o t e n t i a l V  Define  V  V  v  0  v  as the current through the sample heater,  and. current measuring r e s i s t o r .  leads,  Then the measured p o t e n t i a l  V^ i s p r o p o r t i o n a l to I , g  But  the t o t a l power P'^ generated i n the sample h e a t e r ,  and  current measuring r e s i s t o r i s j u s t P  t  =  v  s  I  a  =(%J_V R R v  leads  v v v  ±o  i  If we assume that a l l the power generated i n the sample heater ated  i s d e l i v e r e d t o the sample, and a l l the power gener-  i n the leads and c u r r e n t measuring r e s i s t o r i s d i s s i p a t e d  S8 elsewhere, total of  then  power  the r a t i o  Pm  t h e sample  i s just  heater  o f power  the ratio  through  t h e sample  of the average  at low temperature  P, t o  resistance  R^m, t o t h e t o t a l  ill circuit of  R g ^ + Rjjrpjj +• R^.  resistance R^  the leads,  the resistance  R  JJTL  ^  t  resist  i e  of the current  8  1  1  0  6  measuring  resistor.  P  RHT or  C HT R  T  P  P  * HTL * i ) B  =  ;  RJJT ( HT  we k n o w  for  t h e pow.er  P  P ,  therefore  T  through  =  A'll rent  + R  H T  resistances  have  shown b e l o w ,  obtain  (R .  +  H T  H  T  3  L  + R.)  R  of the r e s i s t a n c e s , with  a typical  =  19,355 ohms  R  =  103.69 ohms  =  I|.»961|.ohms  y  Rj_  HT  R  HTL  i> expression  1216.3  =  V  <v V v  R. and t h u s voltages  the heat  cur-  V , V^. v  f o r t h e 8mm s a m p l e , a r e  R3  R  v  t o t h e measured  together  + R  the f i n a l  been measured,  i s proportional Values  R  T ,  t h e sample,  R (R  we  P  + HTL  R  but  R  value  of V  y  and V^„  ohms  =  38.28 ohms,  =  36.507 v  v  V. .  and  therefore  Typically  V  v  P  through (b)  =  6 . 9 5 3 mV, V _  =  36.507  =  1 » 3 3 0 mW, a t y p i c a l  (6.953 x 10 3) (5.239 x 10"3) watts _  between  d i f f e r e n c e .AT,  the upper  and l o w e r T  From the c a l i b r a t i o n  1/T 1/T A  the n  L  value  for the  power  the sample.  Temperature  for  = 5 . 2 3 9 mV g i v i n g  j  is  the temperature  difference  resistors, =  T  - T  L  .  n  p r o g r a m we h a v e  obtained  the  parameters  relations  = Ay + B l n ( R - Q u  = A  simple  L  + B ln(R -Q L  L  temperature  L U  )  + C ln (R -Q  )  + C ln  u  L  of  the upper  2  2  u  (R -Q L  L U  L U  )  + D ^ n  )  + D ln (R -Q  designed  and l o w e r  L  to  (Ry-Q^),  3  3  L  calculate  resistors,  and  L  U  ).  the their  differences.  Input -  -  data.  Consisted  N,  t h e number  Q,£,  the lead  Parameters, that Set  (ii)  L U  p r o g r a m was t h e r e f o r e  temperatures  (i)  u  Output T-y,  of r e s i s t a n c e resistance  the set  values,  f o r that  ( A , B,  resistor;  resistor,  C, D, E,  F)  resistor, (R)  of N resistance  data.  temperatures  of the  temperatures  of the  thermometer,  the calculated  lower  values.  Consisted of;  the c a l c u l a t e d  upper T^,  o f , f o r each  thermometer,  for  GO  - ^ T =  T  L  - u»  the two  T  t l i e  "temperature d i f f e r e n c e s between  thermometers.  As w e l l as the (R^-, R )  p a i r s f o r the c o n d u c t i v i t y  L  points  ( i . e . the r e s i s t a n c e p a i r s taken w h i l e power was  flow-  i n g through the sample), the o r i g i n a l c a l i b r a t i o n p o i n t pairs; were fed to the T, /YD program..  In t h e o r y , s i n c e these  p o i n t s were used to produce the c a l i b r a t i o n parameters, and" were taken under zero heat current  c o n d i t i o n s , a l l AT  responding to these r e s i s t a n c e s should be zero. t h e r e f o r e , the A T  cor-  In p r a c t i c e  c a l c u l a t e d from these r e s i s t a n c e s are a  measure of the t o t a l e r r o r i n c a l i b r a t i o n , program i o n s , and r o u n d o f f .  calculat-  These ATs g e n e r a l l y averaged 0.1  or  0.2 millidegrees. ATs  used f o r c o n d u c t i v i t y measurements ranged  from  1 to 15> m i l l i d e g r e e s . ( 5 ) CONDUCTIVITY CALCULATION. We have a l r e a d y  c a l c u l a t e d the power P through the sample  and the temperature d i f f e r e n c e A T between the (chap.  IV - i|.).  The d i s t a n c e  thermometers  "[_, between the thermometers,  the diameter D of the sample have been measured  (chap.  and  I l l - 6).  The thermal c o n d u c t i v i t y k i n terms of the power, temperature  d i f f e r e n c e , thermometer s p a c i n g and c r o s s - s e c t i o n a l  area of the sample i s k  =  I  AT  .  k  A.  9  61 and  A  =  whence  \  ~  p ffV  ,  k"L  p  AT  7TD  2  The temperature T a t which the c o n d u c t i v i t y was: measured was taken t o be =  T  ?U  Hf  T  L  2 the average temperature between the thermometers. Finally,  s u b s t i t u t i n g e x p r e s s i o n s f o r P and AT,  we ob-  t a i n the thermal c o n d u c t i v i t y \  =  ( 3: r  RHT  (R  H T  + R  H T L  +  R R  +  v  at temperature  T  = T  or  K  = fi V  L  R  v)  T t  v  Vi)  L" U T  / 2',  -  T  (v  v  V  ±  L" U T  where S is; a known constant f o r a g i v e n  sample.  kl 77 D  2  (oZ  CHAPTER V.  RESULTS AND  INTERPRETATION  (1) PRESENTATION OF RESULTS (a)  Data.  Data* f o r the thermal  conductivity calculations  on t h e t h r e e C s l samples were gathered f r o m a s e r i e s of about eighteen l i q u i d helium runs.  The measured t h e r m a l  conduct-  i v i t i e s f o r the samples are shown i n Tables 12", X sndYT, and i n Figure  9.  (b) Errors„  The major u n c e r t a i n t i e s and e r r o r b a r s of the  c o n d u c t i v i t y v a l u e s r e s u l t from the u n c e r t a i n t y i n the measured z^T's used to c a l c u l a t e them. m o d u l i of the AT's a g i v e n r u n was  The  average of the  c a l c u l a t e d f o r the c a l i b r a t i o n p o i n t s on  t a k e n to be the u n c e r t a i n t y i n a l l AT's  t h a t r u n (see Chap„ \\ s e c  0  l+-b).  for  The t o t a l u n c e r t a i n t y i n  the t h e r m a l c o n d u c t i v i t y i n t r o d u c e d by the measurements of power and sample geometry i s about  1$.  Note, however, t h a t t h e r e i s a. s y s t e m a t i c e r r o r p r e s e n t . Sample geometry was measured at room temperature. t h e r m a l c o n t r a c t i o n of C s l from room temperature temperature  i s 1.16$  (James et a l , 1961+) and  should t h e r e f o r e be i n c r e a s e d by 1.16$  The  total  to helium  the r a t i o "L./A  i f i t s low.  v a l u e i s t o be used i n c o n d u c t i v i t y c a l c u l a t i o n .  temperature The  con-  d u c t i v i t y v a l u e s c a l c u l a t e d here are based on the room tempe r a t u r e v a l u e of L,/A  and are t h e r e f o r e low by 1.16$.  63  TABLE 3ST Thermal C o n d u c t i v i t y of Sample #' 1. Room Temperature Diameter D = .795 T K  k w.at t cm°K  6  4*33 4*27 3.81 3.46 3.41; 3*11 2»92 2.69 2.38 2.21 2.01 1.90 1.89 1.86  1* 40 + 0.6  k  32  5' •98 h 13 3 59 7 34 3*88 2.88 1.55 1.41 1.39 .965 1.04 .977  Hr  ±  0.6  1»5  * 0.6 ± 0.3 * 2-.0 i 0.5 ± 0.3 ± o.i o.i o.i  .03 .05 .04  T°K  1.79 1.75 1.74 1.68  1.62  1.55 1.5% 1.52 1.51 1.45 1.40 1.37 1.34 1.29  cm  k watt cm°K  .91+3  .843 .928 .717 .642 .593 .552 .595 .542 .571 .476 .471 .459 .411  * +•  *•  +  .04 .05 .03 .02 .01 .05 .02 .02 .01 .05 .02 .04 .02 .04  £=4  TABLE 3E Thermal C o n d u c t i v i t y of Sample #' 2 Room Temperature Diameter D = T K 6  ..lj.86  k watt  cm °K 5,1+1 1+.17 i+a5 1+.16  1+.03 2„90 2.69 2.59 2.58 2.1+1 2.10 1.97 1.96 1.57 1,1+8 1.1+1 1.26  6.01+  .61+  1.79 * . 13 .16 2 . 0 0 1.91+ .13 1.63 * .15 .819 ;oi .01 .707 .656 .008 .71+2 .01 .630 .01 •i+92 .01 .381+: .005 + . 3 9 7 .005 .229 .005 .196 + .003 .177 .003 .131 .003  em  TABLE "2L • Thermal C o n d u c t i v i t y o f Sample # 3""" Room Temperature D i a m e t e r D «= .286 cm T°K-  k watt cm°K  1+.31; 1+.26 1+.06 3.52-  3-.06 3.05  '2.14-9 2.21 2.11; 2.10 2.05  1.92; 1.85 1.75  *  .1+59 •tt- • 02 ,1+88 .05 .21+6 t. .01 .198 HF .003 .163 +• .003 ' .151+•ft .002 .153 •«• .002 .128 * ..002 .002 • .110 .0971+ + .002 ..002 .0938 .002 ^0797 .001 i-0659 .001 .0573 .001 .0567 .001 .0339 '•tt-  1«.73  ;  I.63 1*52  1.1+1+ l.Il2 ;  1..17  Note::  1.13 + .02 • 10 .778 .01 .867 .10 " .825  Sample #3 i s sample #1 ground down from .795 cm  t o .286 cm by a b r a d i n g w i t h a h i g h speed g r i n d i n g tool.-  10  5.0  3.0  K  Z.O  f  UJATf  ©m-K 1.0  •1  .SO  .SO  40  .10  SAMPLE* SAMPLE * SAMPLE *  +  1 £ 3,  .Off  .03  1.0  J 1.25"  THFRMAI  L J.5  2.0 -p.* 2.5-  3.0  CONDUCTIVITY OF Csl.  4.0  £"".0 FIGURE 9  (c)  Graphical  on  a log-log  ence the k  Analysis.  graph  * AT  best  n  for  each  lines  #1,  k = A  l  n  l k =  seemed  Graphical  A^T i  = .19 ± . 0 2 ,  2 ,  n  n  0  = 2.45 + .1 T  n  \  (1),  and  (2),  where  = .022 + . 0 0 2 , n-  (2) (a) for ence  Size each  Dependence. of the three  predicts graph  ____________„__„ ( 3 ) o  The t h e r m a l samples,  i s the sole  a conductivity  phonon k =  conductivity  thereby  thermal  o f k/T- v s . d i a m e t e r ;  and  OF R E S U L T S  o f a siz:e-dependent  scattering  A  = 2.69 ± .1  INTERPRETATION  results;  where  = .072 + . 0 0 8 ,  3  of these  .  2  A  of t h e form  and  = 2.61. * ..1 T  the e x i s t -  , where  n  2  plotted  sample, and  determination  ledto the following  A  t o suggest  o f a-, r e l a t i o n s h i p  A  k  #3,  they  had been  r e l a t i o n s h i p f o reach  sample.  Sample #2.  Sample  line  therefore  straight  Sample  9)  (Pig.  o f one s t r a i g h t existence  When t h e p o i n t s  i sdifferent  suggesting  conductivity,,  the pres-  When  s c a t t e r i n g mechanism,  BDT3  S  where B  i s some  D f o rthe various  boundary theory  constant.  samples  should  therefore yield the  a s t r a i g h t l i n e of s l o p e B p a s s i n g t h r o u g h  origin. As t h e a c c u r a c y of t h e d a t a improves markedly below t h e  lambda p o i n t , i t was d e c i d e d t o average k/T f o r each sample 3  f o r T £ 2.1°K are  The average v a l u e s of k/T f o r each sample 3  ()  shown i n TableHL" below, t o g e t h e r w i t h t h e s t a n d a r d  d e v i a t i o n o f t h e v a l u e of k/T3  0  TABLE HT Sample #'  (k/r3) ± <f,(^±  2.1'K^  D  c m  1  .166 + .016  .795  2  .0578 ± .0062  .1+86  3  .0185 ±  .286  .0013  These r e s u l t s a r e shown i n F i g u r e 1 0 . I t may be seen t h a t t h e p o i n t s do n o t l i e on a s t r a i g h t l i n e t h r o u g h t h e o r i g i n , and i n f a c t t h e t h r e e p o i n t s cannot be made t o l i e on an s t r a i g h t l i n e a t a l l . The points; seemed t o suggest a p a r a b o l a of t h e form (k/T3) = B' D T 2  d e c i d e d t o p l o t k/T^ v s . D . 2  3  and i t was t h e r e f o r e  T h i s p l o t i s a l s o stoown i n  Figure 10. These t h r e e samples appeared t o g i v e a s i z e dependence p r o p o r t i o n a l t o D ^ and a temperature dependence somewhat lessj than T . 3  We t h e r e f o r e used a; g r a p h i c a l method t o o b t a i n  SIZE  DEPENDENCE-A  FIGURE 10  70  an e m p i r i c a l e q u a t i o n f o r t h e b e h a v i o u r of a l l t h r e e  samples:,  / 2 n of t h e f o r m k = B D T .  The r e s u l t o b t a i n e d  was  k = (0.3I + .02) D T watt/cm°K, where 2;  n  n = 2.58 + .1 The  graph o f k/D  _  v s . T i s shown i n F i g u r e  2  „_ (fy).  11..  Now l e t us compare t h e s e r e s u l t s w i t h t h e c o n d u c t i v i t y predicted  t h e o r e t i c a l l y on t h e assumption t h a t s c a t t e r i n g  f r o m the sample w a l l s i s t h e o n l y phonon s c a t t e r i n g mechanism. If k  0  i s t h e t h e o r e t i c a l t h e r m a l c o n d u c t i v i t y , then f r o m  above (Chap. I , s e c .  - d) we have  ( k / T ) e_ 1.65  D watt/cm°K^  (5).  D f o r the three  samples., t h e n com-  3  0  I f we e v a l u a t e  1.65  pare t h i s w i t h t h e e x p e r i m e n t a l v a l u e s of ( k / T ) f r o m Table-XT,, 3  we o b t a i n t h e f o l l o w i n g r a t i o s of measured t o p r e d i c t e d  therm-  a l c o n d u c t i v i t y j. Sample #1  (k/k )  =  12.7^  Sample #2  (k/k )  =  1*2%  Sample #3  (k/k )  =•  3.9$  0  e  0  -_—— -  .  _-___  (6).  T h i s i s a v e r y l a r g e d i s c r e p a n c y between t h e o r y and e x periment.  We t h e r e f o r e  concluded t h a t boundary s c a t t e r i n g  f r o m t h e w a l l s i s n o t t h e major phonon s c a t t e r i n g mechanism. This  conclusion  i s a l s o borne out by t h e temperature depend-  ence of about T " „  Pure boundary s c a t t e r i n g would l e a d t o  72  a dependence on T- . 3  However, i t may be seen f r o m F i g u r e s  9 o r 10 t h a t k i s v e r y s t r o n g l y s i z e dependent, o r a t l e a s t sample dependent.  We thus conclude t h a t k i s a f f e c t e d b y  sample s i z e , o r i n t e r n a l sample s t r u c t u r e , o r b o t h , (b):  Internal Structure  Dependence.  I f the conductivity  i s b e i n g determined by i n t e r n a l sample s t r u c t u r e  t h e r e are  t h r e e major phonon s c a t t e r i n g mechanisms w h i c h should be considered, ( i ) point defect s c a t t e r i n g , ( i i ) l i n e d i s l o c a t ion  s c a t t e r i n g , and ( i i i ) g r a i n boundary s c a t t e r i n g . B o t h Cs and I have o n l y one n a t u r a l l y o c c u r r i n g :  Csl i s therefore  i s o t o p i c a l l y pure.  isotope.  The C s l used was a l s o  of " o p t i c a l p u r i t y " w i t h r e g a r d t o c h e m i c a l i m p u r i t i e s . may t h e r e f o r e mechanism..  We  r u l e out p o i n t d e f e c t s c a t t e r i n g as a dominant  Point  d e f e c t s c a t t e r i n g would i n t r o d u c e a  temperature dependence of T"^ i n the t h e r m a l  conductivity.  L i n e d i s l o c a t i o n s c a t t e r i n g can be shown (Klemens, 1958) to i n t r o d u c e a; t e m p e r a t u r e dependence of T^ i n the t h e r m a l c o n d u c t i v i t y . The t h e r m a l r e s i s t a n c e 1*1 k ~ T  i s  L  where N  L  h 2  WK  2  v )f x  2  due t o l i n e d i s l o c a t i o n s '  N,b _____________________ ( 7 ) , 2  7.05  i s t h e average number of d i s l o c a t i o n l i n e s p e r  u n i t a r e a , b i s t h e average Burger's v e c t o r of the d i s l o c a t i o n s , &* i s t h e Gruneis-en parameter f o r the m a t e r i a l , and v i s t h e v e l o c i t y of sound i n t h e m a t e r i a l .  73  I t may a l s o be shown (Klemens,  1956) t h a t g r a i n boundary  s c a t t e r i n g would i n t r o d u c e a temperature dependence o f T^' i n the thermal conductivity..  The t h e r m a l r e s i s t a n c e due t o  g r a i n b o u n d a r i e s was shown t o be a p p r o x i m a t e l y 1  k where C  y  =  3  (8),  C vL,  G  v  i s t h e s p e c i f i c h e a t , v i s t h e average v e l o c i t y o f p  sound and L i s e i t h e r t h e s-ample s i z e , o r E> = lo< where 1 i s t h e average d i s t a n c e between g r a i n b o u n d a r i e s and cx i s t h e average a n g l e of t i l t . Let us suppose t h a t the t h e r m a l c o n d u c t i v i t y o f our samples i s governed scattering.  by some c o m b i n a t i o n of l i n e and p l a n e d i s l o c a t i o n We may t h e n s a y t h a t t h e t h e r m a l r e s i s t a n c e  I/k i s r o u g h l y g i v e n by 1  k  =  G. + T 2  H ..... T3  .  (9),  (where G and H are c o n s t a n t s f o r a g i v e n s-ample), i . e . t h a t the t h e r m a l r e s i s t a n c e i s t h e sum of t h e t h e r m a l r e s i s t a n c e due t o l i n e d i s l o c a t i o n s c a t t e r i n g , and t h e t h e r m a l r e s i s t ance due t o g r a i n boundary s c a t t e r i n g . The measured c o n d u c t i v i t i e s In t h i s experiment were shown t o depend on T »5®, whence t h e measured t h e r m a l r e s i s t a n c e 2  74  i s g i v e n by 1 k Suppose we equate these  (10) .  expressions G  H 3  (11.)  T  and s o l v e f o r G and H by s u b s t i t u t i o n of two say 1.5*K A~' „ 1  and Lj..0 °Kj, u s i n g one  temperatures,  of the measured v a l u e s of  D o i n g t h i s g i v e s v a l u e s of G and H and a r a t i o  G?H of about ls3»  of  Subsequent e v a l u a t i o n of (9) and com-  p a r i s o n w i t h (10) shows t h a t i f k were g i v e n by  equation  (9) i n s t e a d of e q u a t i o n ( 1 0 ) , i t would d i f f e r from (10) by no more t h a n $% over the measured temperature  range.  d a t a does not p e r m i t us t o d i s t i n g u i s h between the two s i b i l i t i e s . (9) and a behaviour  (10) and we may  (9) i s a p o s s i b i l i t y  experiment..  As a. f u r t h e r check on the p o s s i b i l i t y of l i n e and d i s l o c a t i o n s c a t t e r i n g ^ one may  plane  use the c a l c u l a t e d magnitudes  of G and H to e s t i m a t e the d e f e c t c o n c e n t r a t i o n s .  A number  of n u m e r i c a l assumptions must be made, but by c h o o s i n g i n d i v i d u a l , case one may  pos-  t h e r e f o r e conclude t h a t  of the form i n d i c a t e d by  i n the l i g h t of the p r e s e n t  The  an  e s t i m a t e the g r a i n boundary s e p a r a t i o n —  75"  - t i l t " . a n g l e p r o d u c t , and the case of sample #3,  the l i n e d i s l o c a t i o n d e n s i t y .  f o r i n s t a n c e , i f one  c r y s t a l l i t e s ; (of the o r d e r one  o b t a i n s f r o m H and  1 5 ° , and  from G and  Q  of 1/10  assumes l a r g e  of the sample  (8); an average t i l t  In  diameter),  a n g l e of about  (7): a d i s l o c a t i o n l i n e d e n s i t y of about  p.  5 x 10 /cm  ( t a k i n g the average B u r g e r ' s v e c t o r t o be a d i s l o c a t i o n d e n s i t y of 5 x 10  few;  l a t t i c e spacings).  The  i s r a t h e r h i g h , but  experiments on the e f f e c t of d i s l o c a t i o n s ' .  i n L i F have shown ( S p r o u l l et a l , 1959) equation  /cm  t h a t Siemens'  does seem t o produce a v a l u e much too h i g h .  Sproull  •3  noted a d i s c r e p a n c y t h i s f a c t o r we  o b t a i n a value  density typical I t should  of about 5 x 10  f o r sample #3  L  <~ 10^/cm , a d i s l o c a t i o n a l k a l i halide dislocation  i s higher than that  ground down f r o m sample #1,  estimated  the g r i n d i n g p r o c e s s  s t r a i n s and  one  being  consequent d i s l o c a t i o n s .  v a l i d i t y of the above i n t e r p r e t a t i o n c o u l d be  by a n n e a l i n g  crystal.  T h i s would be expected s i n c e sample  quite l i k e l y to introduce  ivity.  reduce N b y 2  u  of a s l i g h t l y s t r a i n e d  f o r sample number one.  The  I f we  a l s o be noted here t h a t the  d e n s i t y estimated  #3' was  of N  .  checked  of t h e samples and r e m e a s u r i n g i t s conduct-  I f d i s l o c a t i o n s c a t t e r i n g i s a dominant mechanism,  the c o n d u c t i v i t y of an annealed sample would be expected t o be  considerably higher  annealing.  t h a n t h a t observed p r e v i o u s  to  the  I t should be noted t h a t i f d i s l o c a t i o n s c a t t e r i n g i s responsible  f o r g o v e r n i n g the t h e r m a l c o n d u c t i v i t y  of these  C a l samples, t h e n any f u n c t i o n a l r e l a t i o n s h i p between sample d i a m e t e r and t h e r m a l c o n d u c t i v i t y  is strictly  coincidental.  The apparent p r o p o r t i o n a l i t y of k to t h e square of the sample diameter f o r our specimens i s r a t h e r would be s u r p r i s e d  s t r i k i n g , and the author  to f i n d that i t i s c o i n c i d e n t a l .  C o i n c i d e n c e s , however, do tend to be s u r p r i s i n g . intend  to i n v e s t i g a t e  the t h e r m a l c o n d u c t i v i t y  We  of samples  of o t h e r s i z e s , and t o i n v e s t i g a t e the e f f e c t of a n n e a l i n g the samples t o r e s o l v e  the q u e s t i o n of the e x i s t e n c e of  sizre dependence and/or i n t e r n a l s t r u c t u r e this  material.  dependence i n  77  BIBLIOGRAPHY Berman, R.., Simon, and Ziman:  P r o c . Roy. Soc. A220. 171 ( 1 9 5 3 )  Berman, R.: Rev. S c i . I n s t . 2£, 91+ (1951+) Berman, R.., F o s t e r , and Ziman: P r o c . Roy. S o c . A231. 130 ( 1 9 5 5 ) Berman, R.: C r y o g e n i c s  297.(1965)  B e t t s , D.D., B h a t i a , and Wyman:: P h y s . Rev. lOJi, 37 ( 1 9 5 6 ) C a l l a w a y , J . : Phya. Rev. 1 1 J , 101+6 (1959) C a l l a w a y , J . : Phys. Rev. 122. 787  (I96I)  C a r r u t h e r s , , P.: Rev. Mod. Phys.. 2JLr  9 2 >  ( 96l) 1  C a s i m i r , H.B.G.: P h y a i c a £, 1+95 (1938) Clement, J.R., and Q u i n n e l : Rev. S c i . I n s t . 2J_, 213 (19520 deHaas, W.J., and Biermasz.:; P h y s l e a  619 (1938)  Guyer, R.A., and Krumhansl:. Phys. Rev. 11+J3, No. 2, 778 (1966) Houston, W.V.r Rev.. Mod. Phys:. 20, 161 (191+8) James, B.¥., and Yates:: C r y o g e n i c s  68 (1965)  Johnson, R . C , and L i t t l e : . Phys. Rev. 130. 596 (I963) Klemens, P.G.r i n Handbuch d e r P h y s i k . B e r l i n 1 9 5 6 ) . Volume 11+  (Springer-Verlag,  Klemens, P.G.: i n S o l i d S t a t e P h y s i c s . New Y o r k 1 9 5 8 ) . Volume 7  (Academic P r e s s , I n c . ,  1  Makinson, R.E.B.:: P r o c . Cambridge P h i l . Soc. 3J|, I+7I+; (1938) P e i e r l s , R.E.: Ann. P h y s i k J , 1055 (1929) Sproull,  R.L., Moss, and Weinstocks J . A p p l . Phys ^ 0 , 331+ (1959)  T a y l o r , A.R., Gardner, and S m i t h : Bureau of Mines Rep.. No. 6157° (U.S. Department of t h e I n t e r i o r , I963)  78  T h a t c h e r , B.D.: PhD, T h e s i s , C o r n e l l U n i v e r s i t y ,  (1965)  V a l l i n , J . , Beckman, and Salama: J . A p p l , Phys. 35. 1222. (196]+) van D i j k , H., D u r i e u x , Clement, Logan, and B r i c k w e d d e : J o u r n a l of R e s e a r c h of t h e N a t i o n a l Bureau of Standards - A. P h y s i c s and C h e m i s t r y 6HA. No. 1 (I960) W a l k e r , E.J.? Rev. S c i . . I n s t . ^ 0 , 831; (1959) Ziman, J.M.t Can. J„ Phys. ^ i j . , 1256  (1956)  79  APPENDIX 1 D e n s i t y . V o l u m e t r i c Expansion and' R e l a t i v e D e n s i t y of B u t y l P h t h a l a t e .  (a)  In order to o b t a i n an accurate correspondence between  the o i l ( b u t y l p h t h a l a t e ) manometer and i t was  the mercury manometer',  d e s i r e d to know the r a t i o of the d e n s i t y of b u t y l  p h t h a l a t e to mercury as a f u n c t i o n of temperature i n the room temperature range.  B u t y l p h t h a l a t e (C^gH^O^) *  s  d i b u t y l p h t h a l a t e , n - b u t y l p h t h a l a t e and p h t h a l i c dibutyl  a  l  s  called  o  acid,  ester.  W.'e were unable t o f i n d measurements of the d e n s i t y of t h i s m a t e r i a l as a f u n c t i o n of temperature.  Furthermore,  the d e n s i t y quoted i n the Handbook of Chemistry and P h y s i c s (196i|.-1965) d i s a g r e e d with that on the b o t t l e  label.  We  there-  \  f o r e decided to perform the experiment (b)  ourselves.  A d e n s i t y b o t t l e w i t h attached c a p i l l a r y was  determine the volume of a. sample of o i l .  The b o t t l e  weighed, then a c c u r a t e l y c a l i b r a t e d u s i n g double water.  The o i l used was  used to was  distilled  21.800 + .001 gm of b u t y l p h t h a l a t e  s u p p l i e d by the F i s h e r S c i e n t i f i c  Company, Lot number 753-885.  Immediately p r i o r to p l a c i n g i t i n the d e n s i t y b o t t l e , o i l was  the  t h o r o u g h l y outgassed by pumping on i t with a r o t a r y  80  vacuum pump f o r a p e r i o d of 21+ h o u r s . The a b s o l u t e d e n s i t y of the o i l was  e s t a b l i s h e d by  measurement of mass and volume at two t e m p e r a t u r e s . o v e r the range 17 - 2%°C  direct  The  was measured by o b s e r v i n g the  density  meniscus  l e v e l of the o i l i n the c a p i l l a r y v i a a. c a t h e t o m e t e r , and s u b s e q u e n t l y computing" the volume of the o i l .  The  experiment  was performed w i t h the e n t i r e d e n s i t y b o t t l e immersed i n a l a r g e water bath.  Temperatures  were measured at v a r i o u s p o i n t s  i n the water b a t h and agreed t o w i t h i n I n computing  .1°C.  the volume, d e n s i t y and e x p a n s i o n c o e f f i c -  i e n t of the b u t y l p h t h a l a t e , t h e t h e r m a l e x p a n s i o n of t h e Pyrex d e n s i t y b o t t l e had t o be c o n s i d e r e d .  The v a l u e of the  v o l u m e t r i c e x p a n s i o n c o e f f i c i e n t f o r P y r e x was t a k e n t o be (Handbook of C h e m i s t r y and P h y s i c s , I96I4.) A(l). The d i r e c t measurement; of the d e n s i t y of b u t y l p h t h a l a t e y i e l d e d the r e s u l t s 3  9  and Measurement of t h e meniscus subsequent  l e v e l i n the c a p i l l a r y ,  and  c a l c u l a t i o n s y i e l d e d the v a l u e s of/Soil* / ^ o i l ^ *  TABLE At: I E x p a n s i o n P r o p e r t i e s of B u t y l P h t h a l a t e oil  =  (Ci^H^O^).  Volume (T°C) - Volume ( Q t ) [Volume ( 0 * 0 x T°C J = (.85 ± .01) x 10" /'C a t 22°C. 3  T°C 0 + 5 10 15  gm/cm  3  /oil< > T  16  1.0652 + .0007 11 1.0606 ~* n 1.0562 tt 1.0517 tt 1.0509  17 18 19 20 21 22 23 2k 25  I..050O  1.0U91 1.0482 1.01+73 1.0465 1.0456 + 1.0447 ~ I.0439 1.0430  27 28  1.0413 I.0404 1.0395 I.O387  26 29 30  .1  I.042I  tt ti it ti ti  .0005 ti tt »i ti ti ti it ti  /S?c  (c,/  /  12.762 + 12.818 12.871 12.926 12.936 12.947 12.958 12.969 12.981 12.991 13.002 13.013 13.023 13.034 13.045 13.055 13.067 13.078 13.088  .007 ti  «  tt n n ti it tt ~ tt tilt n ti ti ti ti ti  12.762 + .007 n 12.807 ti 12.848 n 12.891 tt 12.899 it 12.907 it 12.916 ti 12.925 tt 12.934 tt 12.941 12.950 « it 12.959 12.966 n tt 12.975 12.984 ittt 12.992 n 13.000 ti 13.009 tt 13.017  82  mercury  d e n s i t y v a l u e s used i n the o i l t o mercury  r a t i o c a l c u l a t i o n vrere t a k e n from the 1954  density  Smithsonian  P h y s i c a l Tables (The S m i t h s o n i a n I n s t i t u t i o n , 1 9 5 4 ) . jS  oil  w a s  f° ^ u n  t o  measured temperature  ^  e  e s s e n t i a l l y c o n s t a n t over the  range.  

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