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Thermal wave propagation in bismuth single crystals at 4 K Brown, Christopher Richard 1969

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THERMAL WAVE PROPAGATION IN B I S M U T H S I N G L E C R Y S T A L S A T 4 K  by  CHRISTOPHER RICHARD BROWN B . Sc. , U n i v e r s i t y of E x e t e r , 1963 M . S c . , U n i v e r s i t y of B r i t i s h Columbia,  1965  A T H E S I S S U B M I T T E D IN P A R T I A L F U L F I L M E N T T H E REQUIREMENTS FOR THE DEGREE O F DOCTOR OF  PHILOSOPHY  in the Department of Physics  We accept this thesis as conforming to r e q u i t e d standard.  the  T H E U N I V E R S I T Y O F BRITISH C O L U M B I A  March,  1969  OF  In p r e s e n t i n g an  this  thesis  advanced degree at  the  Library  I further for  shall  the  his  of  this  agree that  written  of  be  for  for extensive  g r a n t e d by  the  It i s understood  for financial  gain  of  The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, Canada  British  available  permission.  Department  Date  University  permission  representatives. thesis  f u l f i l m e n t of  make i t f r e e l y  s c h o l a r l y p u r p o s e s may  by  in p a r t i a l  Columbia  shall  requirements  Columbia,  Head o f my  be  I agree  r e f e r e n c e and copying of  that  not  the  that  Study.  this  thesis  Department  copying or  for  or  publication  allowed without  my  ii  ABSTRACT  Continuous wave thermal propagation experiments were made with two single crystals of bismuth at frequencies up to 7 kHz. The experiments were performed at temperatures close to 4 K (i. e. close to the dielectric-like thermal conductivity peak).  Accurate  phase shift measurements were made in order to permit the detection of small departures from diffusive propagation. Attenuation measurements were also made.  A summary of some microscopic theories of time-dependent thermal propagation in dielectric crystals is given. It is concluded that, for dielectric crystals in both the "hydrodynamic" and "ballistic" phonon gas regimes, the initial deviations from diffusive propagation will be described by a modified heat equation of the Vernotte type:  with appropriate identifications of the relaxation time.  The possi-  bility that the small numbers of charge carriers present in bismuth might lead to different forms of deviation is explored.  Several types of thin-film insulating layers and superconducting alloy thermometers were investigated. Kodak Photo-Resist was found to be the most useful insulating material. This was used in conjunction with constantan heater films and Pb-In alloy thermometer films. The  iii  heat wave detection system employed a radio frequency thermometer bias current, a radio frequency tuned circuit, an envelope detector and phase-sensitive detection of the audio frequency heat wave signals. Heat wave phase lags were measured with a precision of 1°, using the phase-sensitive detector as a null detector.  The measurements were analyzed in terms of a thermal transmission line model based on the modified heat equation given above. The electrical analogue of T" in such a model is L/R. leakage conductance t e r m / 1  A thermal  (electrical analogue G/C) was included in  the model.  The results at low frequencies were in excellent agreement with those expected on the basis of the transmission line model under conditions of diffusive propagation at high attenuations. Values of the apparent diffusivity obtained from these measurements were in reasonable agreement with the results of D. C. experiments made by other workers on comparable specimens.  The quantity ^/ui  was shown to be  small at all frequencies used.  Phase lag measurements at higher frequencies indicated significant departures from diffusive propagation in both crystals. (The crystals had different orientations.) The measurements in this range suggested a harmonic-wave-like mode of propagation. This mode appeared to break down at the highest frequencies examined. Evidence is presented  iv  to show that the observed deviations reflected thermal properties of the bismuth crystals rather than properties of the thin films, or spurious electrical effects.  The apparent wave velocities were lower, and the corresponding relaxation times were longer than those predicted on the basis of the microscopic theories and from the diffusivity values obtained at low frequencies.  In view of these numerical discrepancies, it is suggested  that the wave-like mode could be a mode peculiar to the bismuth system, rather than the "second sound" mode predicted for ideal dielectrics. Some further experiments are suggested.  Ilk.  \ l  I«M  T A B L E O F CONTENTS  .CHAPTER I  Page Introduction  1  A Review of Some T h e o r i e s Some E a r l i e r T h e r m a l Propagation E x p e r i m e n t s Heat Waves in the Boundary Scattering Regime Summary II  Bismuth  25  Some P r o p e r t i e s of B i s m u t h . . Heat Conduction i n B i s m u t h . The Results of E a r l i e r T h e r m a l ConductivityE x p e r i m e n t s in B i s m u t h The E l e c t r o n - P h o n o n Interaction in B i s m u t h . III  E x p e r i m e n t a l Design and A p p a r a t u s . The T r a n s m i s s i o n L i n e M o d e l The C r y s t a l s H e a t e r s , T h e r m o m e t e r s and Insulators Input and Detection Systems The Cryostat Systems  IV  Experimental Procedures,  Results and A n a l y s i s . .  Experimental Procedures. Results Analysis Discussion V  1 10 17 24  25 31 38 42 47 47 55 57 70 78 85 85" 91 109 123  Conclusions  129  Bibliography  132  Appendix 1  135  vi  LIST O F T A B L E S  TABLE  I  Page  Values of P a r a m e t e r s  Controlling E l e c t r o n -  Phonon Scattering  45  II  Results I M C  94  III  Results 1 P D  99  IV  Results 2 P D  105  vii  LIST O F FIGURES  FIGURE  Page  0  Structure of Bismuth  26  1  Heater and Thermometer Films  61  2  Pb-Bi Transitions  65  3  Pb-In Alloy Thermometer (Ingot D)  67  4  System for Heat Wave Generation and Detection  72  5  Cryostat Details  79  6  Sample Holder  7  A Sample of the Raw Data  89  8  IMC Phase Lag Plot  96  9  IMC Attenuation Plot  97  '  82  10  1PD Phase Lag Plot  100  11  1PD Attenuation Plot  101  12  2MD Phase Lag Plot  103  13  2PD Phase Lag Plot  106  14  2PD Attenuation Plot  107  15  1PD Deviations (A)  115  16  1PD Deviations (B)  116  17  1PD Phase Lag vs Frequency  117  18  2PD Deviations (A)  119  19  2PD Deviations (B)  120  20  2PD Phase Lag vs Frequency  121  viii  ACKNOWLEDGEMENTS  I would like to thank my supervisor, Dr. P. W. Matthews, for his support. Mr.  I am indebted to Prof. R. E. Burgess and  D. L. Johnson for many fruitful discussions on various aspects  of the work, and to Mr. A. E. Burgess for the loan of some equipment. Mr.  R. Weissbach and Mr. G. Brooks assisted in the construction of  the apparatus.  Financial support was received from the National  Research Council of Canada and from my wife, Freda, who, with Mrs. S. and Miss P. Killen also assisted in the preparation of the thesis.  CHAPTER 1  INTRODUCTION  SECTION 1  A REVIEW OF SOME THEORIES  The deficiencies of the usual equation for heat transport  . . . ( i . i)  It  have become of increased interest with the advent of experiments in which the applied perturbations involve times and distances comparable with those characterizing the microscopic thermal processes.  Such  situations can be brought about in rarefield " r e a l " gases or in the rarefied phonon gases found at liquid helium temperatures.  The existence  of two essentially different types of phonon-phonon collisions in crystalline solids and the complications introduced by the various defects always present in the crystals studied make the analogy between the two gases incomplete.  A more complete analogy may be drawn  between a rarefied real gas and the phonons in liquid helium..  The inadequacy of equation (1. 1) is inherent in its parabolic form, which gives no upper limit to the speed of thermal waves as their  frequency is increased.  Thus V e r n o t t e ^ , seeking to amend (1. 1)  to deal with thermal pulse propagation in gases, constructed a phenomenological equation with the required hyperbolic form:  -r  3 t  Which gives  -Al-  +  <jt  = k v  2  T  . ... ( i . 2 ,  as the upper limit to the speed of propagation  of thermal waves.  Equation (1. 2) is based on the usual conservation  equation and a modification of the Fourier heat law,  -k  =  Q  +T  - J ^  . . .  (1. 3)  which describes the steady state as being established in a time of order  T" .  The time "77 is of the order of the mean time between  molecular collisions and the limiting speed is of the order of molecular speeds. (2) Cattaneo  arrived at the same result, (1.2), by using kinetic (3)  theory, and Weymann  has given a unified discussion in which he  derives equations of the form (1. 2) to describe heat conduction, diffusion, and viscous flow in fluids.  Although the fundamental objection (4)  to (1. 1) is that it leads to a violation of causality, and although Kelly has derived a modified diffusion equation of the form (1.2) by starting from a relativistically correct Boltzmann equation, as would be expected from the foregoing, the extra term obtained does not arise from what would normally be termed relativistic corrections.  The derivation of the corresponding modified heat equation for the phonon gas problem is more complicated.  Most treatments  deal with the case in which boundary scattering is not the dominant scattering mechanism, although the special case in which it is dominant has also received attention^' ^' ^'^ following e x p e r i m e n t s ^ in this regime.  (9) Chester  has obtained equation (1.2) for the case of a large  dielectric crystal (i.e. one with negligible boundary scattering) by using macroscopic arguments similar to those of Vernotte^ \  He  identified the relevant relaxation time as the mean free time between collisions causing momentum loss.  The propagation of harmonic  phonon density variations, or "second s o u n d " ^ ^ is predicted for frequencies greater than 1/T  . As Chester remarks, the second sound  wave would become undefined at frequencies greater than the rate for "normal" (i.e. quasi-momentum conserving) collisions, since local equilibrium could not then be established within a wave-length. However, this information is not contained in the model described by equation (1. 2). To improve on equation (1.2), one must examine the phonon gas on a microscopic scale. Microscopic theories fall into two general types: those using Green's function formalism and those using linearized phonon Boltzmann equations. The latter rest on the assumption of particle-like phonons and local equilibrium distribution functions,  n (q, r , t ) , s a y , w h o s e v a l i d i t y i s d i f f i c u l t to j u s t i f y w h e n p e r t u r b a t i o n s i n v o l v e f r e q u e n c i e s g r e a t e r than the n o r m a l p r o c e s s c o l l i s i o n r a t e . The  e q u a t i o n w h i c h i s t o b e l i n e a r i z e d i n the d e v i a t i o n s o f the l o c a l  distribution function f r o m i t s e q u i l i b r i u m value i s  •4f  (  —  '  + V.\-7n(q,r,t)  t )  = C n ( q , r , t ) . . . . (1.4)  w h e r e the c o l l i s i o n o p e r a t o r C i s an i n t e g r a l o p e r a t o r i n q-space. The  d i f f e r e n c e s i n the s e v e r a l t r e a t m e n t s a r i s e m a i n l y f r o m the  forms  a s s u m e d f o r C.  C o n s i d e r a b l e u s e i s m a d e of the e x p a n s i o n  Tex/,  7 6  7 =  {  a  _  Ul-H  (£*/k t)tn/%)+  A,x; -  n  4 sih^fo/zkoi:) .  w h e r e f = 1/ ( e x p (  fk  probability,  /I  temperature  deviation.  T  .  • .  (1.5)  ) -1) i s the e q u i l i b r i u m occupation  i s t h e l o c a l d r i f t v e l o c i t y a n d T^ i s t h e l o c a l  Prohofsky andKrumhansl^  ^ u s e t h e e x p a n s i o n ( 1 . 5) i n  w r i t i n g down m o m e n t equations for number, energy and momentum. The  c o l l i s i o n t e r m s i n t h e s e e q u a t i o n s a r e w r i t t e n i n the r e l a x a t i o n  t i m e a p p r o x i m a t i o n , u s i n g r e l a x a t i o n t i m e s *T = f(1/T^.) + (l/'C^)J and  "7ju i n t h e n u m b e r a n d m o m e n t u m e q u a t i o n s r e s p e c t i v e l y .  ( T  N  and  ~X u  are the relaxation times for normal and momentum-  loss collisions.)  On attempting a plane wave solution of these  moment equations,. Prohofsky and Krumhansl obtain dispersion relations (equations 4. 13 and 4. 14 of (11) ) which correspond to an equation of type (1. 2).  The extra damping of high frequency temperature  waves, which depends on the N-process collision rate, has to be taken in-to account by "grafting on" to the solution an additional attenuation of the "second viscosity" type. (12) On the other hand, Guyer and Krumhansl  , starting from  a Boltzmann equation with the Callaway collision term,  ^  n  '  +  X'V  n  _ _ "  (  n  - nx)  _  "^N  (n - n ) e  ^u (1.6)  (n is the distribution with uniform drift, n is the equilibrium distribution) arrive at the equation  . . . (1.7) It is emphasized in (12) that the terms i w 7^,  Tu <2—\  and  occur in the same order of approximation in the  5 solution.  Equation (1.7) predicts lightly damped second sound propa-  gation only in the frequency range T L < ^ < CA/ which could be vanishingly small in practice.  6  A later approach due to Guyer and Krumhansl  (13)  is based  on Krumhansl's method of expressing the deviations from equilibrium in terms of the eigenvectors of the normal process collision operation N. (The collision operator is split explicitly into two parts, C = N + R.) The advantage of choosing eigenvectors of N, as opposed to C, is that two of them are closely related to the leading terms in the expansion (1. 5) and have known eigenvalues, namely zero. It is assumed that the eigenvectors of N form a complete set and that this set may  be approximated by three eigenvectors, including the  two mentioned above, so that the drift operator D* <3  37  + v. ^7  and  the collision operators R and N are reduced to 3 x 3 matrices.  The solution of the Boltzmann equation is thus reduced to the solution of three simultaneous equations for the coefficients in the expansion of the deviations from equilibrium in eigenvectors of N.  On  eliminating the coefficient of the unknown third eigenvector, two equations in the heat flux and local energy density deviations are obtained.  One of these expresses the conservation of energy for the  phonon system, and the other corresponds to equation (1. 3) with a relaxation time expressed as a function of the elements of D, R, and N and hence of both normal and momentum-loss collision rates.  Guyer  and Krumhansl give expressions for this combined relaxation time for the limiting cases in which momentum-loss collisions with very much higher and very much lower frequency than normal processes.  A  7  "switching function" is derived for interpolation between these extremes.  The dispersion relations obtained in the limit of rapid  normal collisions correspond to an equation like (1.7) apart from the factor (4/5).  The result depends on the assumption of an isotropic phonon spectrum.  It is also assumed that anharmonic effects can be handled  adequately by treating the phonon and dilatational fields separately in the harmonic approximation, and subsequently introducing coupling according to the Gruneisen model.  It is shown that the coupling depends  on the small parameter (Cp I C ) -1. v  In their paper on first and second sound in crystals, Gotze (15) and Michel  consider coupled phonon and dilatation fields from  the outset. Their solution proceeds along the same general lines as (13), with the addition of anisotropy and an elastic wave equation.  An  important difference is that a new leading damping term for second sound is found to arise from the anisotropy. A discussion of the intermediate frequency regime between diffusive heat conduction and second sound is given in terms of the temperature response function (which is treated as function of normalized frequency and damping factor). The possibility of observing departures from diffusive behaviour in this regime depends on there being a well defined second sound resonance in the response function.  8 The Green's function methods, which produce expressions for the temperature and displacement response functions, establish that first and second sound occur on an equal basis as natural modes of the phonon system, but the second sound mode has a weighting of roughly ( (Cp /  )- 1) times that of first sound. (Note that this  factor is very much larger in liquid helium than it is in solids. )  In  the method of Kwok and Martin^ * ^, the response of a disper sionless isotropic system to a fluctuating displacement field is calculated for the case in which momentum-loss collisions occur at a negligible (17) rate.  Sham  derives the first and second sound modes via a  calculation of the phonon self energy of a weakly anharmonic crystal in thermal equilibrium, thus avoiding the introduction of non-equilibrium concepts. His result, which is the same as that of Kwok and Martin apart from the inclusion of the momentum-loss relaxation time  "C^  , is given below:  + C %- O X  1  x  t  C^  -ii>/T,  L  ~u  x  - its  /Ti  D Q ( GW) is the Green's function, the poles of which give the natural T  modes of the phonon system.  The definitions of the other quantities  are: 2 r  =  c  2  c* = c (l-3r/2)/3 ,  (1 + 3r/2)  X  c is the speed of a phonon,  = (Cp/C ) - l '  4  ((1/T ) + f t  Tv^q ), 1  x  (  l / t  2  = (1 - 9r/4)  (1/Ta)+  TV  Cf ) l X  9 c_ and  describe the second sound mode and correspond to a  temperature equation of the form  TP  1  R  (1  9 r  $T  - r> n  -  9r. . — . 2 r-yZ ( - ( -T * N 2 1  1  1  }  m  C  . . . (1.8) It is noted that, disregarding a factor 4/5, (1.8) reduces to (1.7) in the limit of negligibly small r, a limit which applies to most solids at liquid helium temperatures.  To summarize, equation (1.8) is taken to be a good description of thermal propagation in an isotropic dielectric crystal for frequencies much lower than the normal process collision rate i.e. in the "hydrodynamic regime".  Additional damping of the second  sound mode is expected in anisotropic crystals. The addition of a few electrons to the system would strictly require a completely new treatment involving three coupled systems, and might be expected to lead to new modes of propagation. A different type of analysis is required to deal with propagation outside the hydrodynamic regime.  10  SECTION II SOME E A R L I E R  THERMAL  PROPAGATION E X P E R I M E N T S  The conditions for the observation of second sound in a given solid are discussed in detail by Guyer and Krumhansl Briefly, if there is a range of temperature for which the normal process collision rate, 1/T^, is more rapid than the appropriately calculated combined collision rate for momentum-loss processes, 1/T , then lightly damped second sound should be observed at frequencies in the range 1 / T <u> <" 1 / "JJT  •  An interesting result is obtained  (19) when a calculation of steady state heat flow is made  for a finite  sample, supposed to be a long cylinder of radius R, in this range of temperature.  It is found that in the special case where the mean free  paths in an infinite specimen would satisfy  \*/<< R,  R> it is  not correct to introduce the boundary scattering by the usual scheme of adding reciprocal relaxation times.  As a result of the screening effect  of the normal processes, the effect of the boundary scattering is reduced for the bulk of the phonons, so that the effective conductivity has a radial dependence. The resulting laminar flow of heat is termed Poiseiulle flow.  The mean thermal conductivity is given by  11  k = k^^G  (/< ), where k ^  is the conductivity without boundary  2 effects, /"> from 0 to 1.  - (5R  /4  JW ^d. ) and the function G takes values  Because the requirements for the observation of  Poiseiulle flow, S s<<&. , ^^/iX/i^ft , are more demanding than those >  f  for the observation of second sound, its observation implies the existence of a second sound regime and, in addition, implies that the range of temperatures for which second sound may be observed will be extended to lower values. Convincing observations of both Poiseuille flow (19) and second s o u n d ^ ^ have been made with single crystals of solid ^ He. The method described in (20) involves the measurement of the time of flight of heat pulses produced by an input approximating a A  - function.  Propagation speeds and pulse shapes corresponding to diffusive, second sound, and ballistic propagation were obtained by varying the mean sample temperature. The results in the diffusive and second sound (13 18) regions were in good agreement with the theory ' .  The  observation of a temperature echo is advanced as further evidence of (21) second sound, but von Gutfeld and Nethercot report observing such echoes in the ballistic regime during an experiment on sapphire. (22) Ackerman et al.  have demonstrated that the temperatures at which 4  the various modes of propagation occur in solid  He scale with the  Debye temperature i . e. the mode depends on the normalized temperature (T/p^  ), the different values of ©  at different pressures.  ft  being obtained by growing crystals  12  An analysis  (23)  of the shapes of the received pulses in the  second sound region has been used to deduce the normal process relaxation time.  In this connection it is surprising that the heater  and detector response times were unimportant sources of pulse broadening, since both heater and detector were made of I. R. C. carbon reisitor board and it would seem from (20) that the Fourier components in the second sound frequency range would have frequencies of the 5 order of 10  Hz.  However, good agreement was obtained between the  observed pulse shapes and those predicted on the basis of (13) by assuming all the broadening to be due to "normal" collisions.  The  value of "^jk/ is the most important result obtained. At lower temperatures, where the theory of (13) begins to lose its validity and the onset of ballistic propagation occurs, the pulse shapes were not so well explained. It has been observed by von Gutfeld (24) and Nethercot  that propagation in the ballistic region can be quite  complicated for anisotropic specimens, in which a multiplicity of thermal pulses may arise.  The velocities of these pulses are not  given directly by the (first) sound velocities. Such a situation might well occur in solid He. Although second sound is not, in theory, restricted to liquid and solid helium, and in spite of the fact that the conditions for its observation seem to be attainable in several dielectrics, eg N a F ^ ^ , no  13  conclusive evidence for its existence in other solids has been reported. Much effort has been put into heat pulse experiments on alkali halides in the hope of observing second sound.  The temperature range of  interest lies close to the temperature of maximum thermal conductivity, which is about 1 /30th to 1 /40th of the Debye temperature, and, for good specimens, depends on sample size. For alkali halides these temperatures lie in the convenient range 4- 10 K (eg. KC1,^=230 K ; NaCl, 6^ = 310 K ).  Heat pulse experiments employing thin film heaters and  superconducting thin bolometers at 3. 8 and 8. 8 K were performed by A  (25) von Gutfeld and Nethercot  on single crystals of NaCl, KC1,. Kl, and  KBr. (The lower temperature was in the ballistic region).  A l l results  were interpreted as being due to ballistic and/or diffusive propagation. The absence of second sound effects was attributed to some or all of the following: poor crystal structure, chemical impurities, failure to investigate.the exact temperature range required and the possibility that the tendency for normal collisions to be collinear makes them ineffective in establishing equilibrium across the sample.  It is also  noted that all four of the substances used contain mixtures of isotopes. The additional (momentum-loss) isotopic scattering could be important in crystals of the quality required for these experiments, as is shown by the work of Thacher  on L i F . F r o m this point of view, NaF and  C s l are more attractive.  However, D. C. thermal conductivity  (27) results  for good single crystals of C s l ("good" as indicated by dis-  location counts,. X-ray and chemical analysis) in the boundary 3 scattering regime were not well described by the usual T  formula  or by a Poiseiulle flow formula, so that C s l must be considered a (8V doubtful candidate. Preliminary heat pulse result for NaF  were  encouraging, but no evidence for second sound in NaF has been published.  It therefore appears that available alkali halide crystals,  by reason of chemical impurities, structural imperfections and in some instances isotope scattering, are not sufficiently good approximations to ideal dielectrics to exhibit second sound. Single crystals of synthetic "sapphire" i.e. A l 0  are in  many ways ideal dielectric crystals, and have therefore been the subject of careful time-dependent thermal conductivity experiments. Chung  performed heat pulse experiments over the range 3.8- 35 K  (The thermal conductivity maximum was at 30 K ) using radio-frequency electromagnetic coupling to the thin films to achieve very rapid response. Second sound was not observed, but it was found necessary to invoke a modified heat equation to describe the results in the boundary scattering regime, in which the received pulses consisted of narrow directly transmitted parts followed by broader parts due to boundary scattered phonons. The delayed onset of the broader parts of the pulse was not given by a diffusion-type equation. Since these effects occur outside the regime covered by theories such as (13), a separate treat-  (7 6) ment is required  '  (7) .  One such treatment,  which uses a very  15  simple model dielectric crystal, is given below.  Heat pulse results  for Al^O^ crystals have also been published by von Gutfeld and Nether(21) cot  , who covered the temperature range from 4 to 54 K.  The  results showed no evidence of second sound and were interpreted in terms of diffusive propagation and ballistic propagation at the "energy" velocity.  The sharp, directly transmitted contributions to the  received pulses were found to persist up to unexpectedly high temperatures. This result and the absence of second sound were attributed to the predominance of small angle scattering in the normal collisions. The reasons for performing a low temperature heat wave experiment on bismuth follow from the preceding discussion.. Firstly, in spite of its being a semi-metal, bismuth displays a phonon dominated thermal conductivity at low temperatures. The thermal conductivity has a maximum which occurs in the convenient temperature range (28 29) near 4 K  '  and which is apparently of the type expected for good  dielectric crystals (i. e. governed by the temperature dependence of boundary and Umklapp scattering and by the size and structure of the crystals.)  Good single crystals of high chemical purity are readily  obtainable, so that by choosing to work with bismuth one expects to gain chemical purity at the expense of having a few electrons present 17 (about 5x10  per cc. at 4. 0 K ).  Lastly, isotope scattering problems  are eliminated, since bismuth has only one stable isotope.  16  The role of the electrons in thermal transport in bismuth (30)  is not entirely clear (See, for example, Bhagat and Manchon  ),  but since they both transport heat and provide extra scattering, they are obviously not directly comparable with chemical impurities in alkali halides.  As mentioned above, the addition of electrons to the  time-dependent thermal conduction problem demands a major revision (15)  of the theory, perhaps after the style of GOtze and Michel  but with  three coupled systems (electrons, phonons and elastic waves). Further,, the extreme anisotropy of the F e r m i surface leads to highly anisotropic electron-phonon scattering rates. In a D. C. thermal conductivity experiment on bismuth it is necessary to apply a magnetic field in order to examine the electronic contribution to heat transport.  In a heat wave experiment,  it would seem possible for the coupling of the electron and phonon systemjto manifest itself (even in the absence of a magnetic field) as a frequency dependent departure from diffusive heat transport.  It was  therefore thought worthwhile to perform a heat wave experiment without a magnetic field at a temperature close to that of the thermal conductivity peak. The objectives were, firstly, to find out whether any departures from diffusive behaviour could be observed at the attainable heat wave frequencies, and secondly, to classify any observed departures as being of types expected for dielectrics on the basis of the theories discussed, or of types peculiar to bismuth.  17  SECTION HI H E A T WAVES IN T H E BOUNDARY S C A T T E R I N G REGIME  The theories of thermal propagation in phonon gases described above are valid in the hydrodynamic regime i . e. when thermalizing collisions occur rapidly within the phonon gas.  Such  theories are not expected to describe the situation where boundary scattering is the only important scattering process. Casimir's treat(31) ment  of D. C. thermal conduction in this situation is well known. (32)  Ziman  , remarking that heat conduction in this limit may be viewed  as a version of the random walk problem, assumed equation (1. 1) with the boundary scattering value for H , K = 2i^R/3 (  (u  the  average phonon velocity, R the radius of the cylindrical specimen) as a description of time dependent heat flow.  Ziman was concerned  with the explanation of the results of heat pulse experiments in liquid (5) helium.  However Brown, Chung and Matthews  found that both the  liquid helium results and the results for heat pulse propagati on in sapphire are much better described by the modified heat equation, (1.2). A (not very rigorous) derivation of an equation of the type (1. 2) for this problem, based on a greatly simplified physical model, is outlined below. (This part of the thesis has been published as (7) ).  18  We consider a pure dielectric crystal, in the form of an "infinite" cylinder, at a temperature sufficiently low that all phonon scattering may be taken to occur at the surface. The phonon spectrum is assumed to be isotropic and dispersionless. The surface is assumed to be "perfectly rough", in the sense that all phonons are diffusely reflected.  Following Casimir, it is assumed that in the  process of diffuse reflection the surface acts as a black body, absorbing all incident phonons and radiating phonons according to a distribution which is symmetric in the wave vectors boundary temperature.  q and which defines the  The object is to express the phonon state  occupation number for phonons of wave vector q at some interior point in terms of the distribution at the intersection of the boundary and the cross-section containing the point. Then, by applying the condition that no scattering occurs inside the crystal an equation in the distribution function (and hence in the temperature) at the boundary may be obtained.  The symmetric distribution from which phonons are radiated at a boundary point r is supposed to have a time dependence such as to maintain zero net radial energy flux at the boundary.  This  distribution is described by a set of occupation numbers Let ^ h ^i.>£> ^  ^ the distribution at some interior e  point r and let n (q, r, t) be identical with the distribution on the o -- boundary at its intersection with the cross-section containing the  19  point r.  It is assumed that each occupation number may be split  up as n (q, r, t) = n compared with n . o  ( q, r, t) + n  ( q, r, t), where n  is small  n, is related to the net phonon flux in the 1  asymmetric distribution  j> n(qr t, )J .  The wave length of heat waves  will be assumed to be longer than the crystal diameter, 2R. Since there is no scattering inside the crystal, all phonons of wave vector q at point r at time t must have left the same boundary point  at the same time t- ( fr - r / / e  the phonons).  u  )  ( U is the speed of q  Hence  If <9 is defined as the angle between (r  -r ) and the z (cylinder)  axis in a cylindrical coordinate sytem J> , z  , ^> then (since n  has  only z dependence)  »(h > ) *«fash r  t  5  " i  -  ^  fatrt-  /v /<>> & # — r  on the assumption of long wave-length heat waves.  In the absence of scattering inside the crystal, the Boltzmann equation for phonons of wave-vector q reduces to  On substituting the above expansions for n (q r t) into this equation and carrying out the differentiation one obtains  h±(*£0  M&=UL  sjjg..^  K.c*> 6/rt-r/M—  Such an equation, with appropriate  a  , and cos  + terms  in cos £  , may be written  20  for each q .  Two  x ( "2-1J* '  averages, x  and x  are now defined,  <p ) is the average of x over all phonons with wave-  vectors of magnitude q arriving at 2^0^^'' . x z*= z, 0<p<R. Thus  average of x over the disc cos 6  ( - t j * ^ ) is the cos'S  and hence  are zero, and the terms in cos & vanish when the averaging  is performed on the above equation.  If N  (qzt) is defined as the o  number of phonons per unit length of crystal with wave-vectors in the range q, q -f dq, then  N  q  (qzt) and the corresponding energy density E  (z, t)  therefore satisfy the wave equation lt  /r - r I  x  //>- ri  V  &  If the boundary temperature has the form T  /  jz  l  + T^ (z, t) where  JT /' « T then T also satisfies this equation, which has the form 1 o 1 of Vernotte's modified heat equation ( 1 . 2 ) .  For an infinite cylinder  of radius R, Cos  Wfrrl = A  L/zJ  W  in  ?«['P-  1  11 3.  L^+P 1  and TrTZTJ  =JLLJ  *tt*  />c*s t e l  since the solid angle e^Jl subtended at a point z -o^fizG area  . <dA = R  ^fftfrdZfe on the curved surface is  j b y the  21  JJL =  A  *«6  The integrals 6 may be evaluated by the method given in Kennard  for similar  integrals arising in the transport theory of rarefied, gases.  The results are  Thus the equation satisfied by the time dependent part of the boundary temperature becomes  . . . (1.9) •which is to be compared with the equation used by J. B. Brown, Chung and Matthews  (5) .  Equation (1.9) is seen to be a modified heat equation of the type (1.2), with a relaxation time speed of propagation v = u^ j/  J~2!.  *7T = 4R/3u  and a maximum o In view of the result  /. £ - Ig, | =<)(4R/3), the above treatment is expected to begin to fail at wave-lengths such that with  X/4  5:  4R/3.  For heat waves  A = 5R, (1.9) predicts a speed of roughly. .88 (u»/J  rather than 1. 67 U  b  as predicted by the Fourier heat law.  frequencies (1.9) reduces to (1. 1),  2' ) At low  22  =  7)  ill?  with diffusivity D given by D = 2u R/3, as found by Casimir  (31)  o  (5) It should be noted that Brown, Chung and Matthews the limiting speed v^ with the phonon speed  identify  obtained by averaging  the speeds of the directly transmitted pulses over the polarizations: u = £" U & It could be argued that the longitudinal ° i * t  a  phonons govern the arrival time of the leading edge of the "diffusive" pulse, and that u  should therefore be closer to the longitudinal phonon  speed. Regardless of the choice of u , it is implied in the paper by Brown et al, that the ratio of the diffusivity to the limiting speed of propagation, (D / v ) = o  ( t>/v ) should be'2/Z/3, whereas from (1.9), a  ./T  (2R/3). /g \  Chung  gives the shapes for the received and computed heat  pulses in sapphire at 5. OK. scattering region.)  (This was well into the boundary  For his crystal 2R/3 =.33 cm and  /2 (2R/3) = .47 cm. ,  v  A poor fit was obtained by computing the received pulse shape using  D= 2u R/3, v -u , u = T U^/Su"  2  A much better fit was obtained by allowing independent adjustments of both D  and v .  It is interesting, but perhaps  fortuitous, that the values of D and v which give the best fit also give o (D/v ) = . 49. The values of D and v needed to fit the pulses at o o  23  higher temperatures ( 9 K and above) correspond to progressively smaller ratios D/v^ , indicating the increased importance of internal scattering in reducing D.  24  SECTION IV SUMMARY  This chapter may be summarized by considering the results expected from a series of continuous-wave performed at increasing heat wave frequencies.  experiments  Irrespective of  the regime in which the experiments are performed i . e. of the relative rapidity of N processes and other processes, including boundary scattering, it is expected that any departures from diffusive behaviour which are observed will be described initially by a modified heat equation of the general type (1.2). The magnitude and physical significance of the relaxation time will depend on the sample and the temperature, and at higher frequencies the heat equation may require extra terms, as in (1.8).  In the case of samples containing more than  one system of thermal carriers, for example bismuth crystals, these conclusions might require modifying to include the possibility of a new mode of propagation produced by the interaction of the different carrier systems.  C H A P T E R II 'BISMUTH .SECTION I SOME P R O P E R T I E S O F BISMUTH  This section contains a brief discussion of those properties other than the thermal conductivity which are relevant to the experiment; thus the many and remarkable effects which require magnetic fields are not included. been given by Jain and Koenig  (34)  More complete discussions have and by Boyle and Smith  (35)  , upon  whose work parts of this section are based. One description of the bismuth structure uses a rhombohedral unit cell, which may be thought of as a cube which has been slightly distorted by extending a body diagonal. shown i n F i g . ( (0) a.) the angle &<  In the smaller unit cell  would be 60° for a cubic structure..  The two atoms per unit cell are seen to l i e on the extended body diagonal. In the convenient system of cartesian coordinates given in F i g . ( (0) a. ) there i s two-fold rotational symmetry about the x axis and three-fold rotational symmetry about the z (trigonal) axis. This partial description of the symmetry w i l l be adequate for the discussions to follow. The complete symmetry is 3m  2  (International).  (SchoefLies) or  - a6  STRUCTURE OF BISMUTH  (35))  (From  tx' r 57.23  a) U n i t  Cell [z  (Trigonal).  y  (Bisectrix)  (Binary)  b) F e r m i S u r f a c e (30))  Send-axes (From  -22  .Holes: lik = l +,lU,V5 x 10 1  E l e c t r o n s : -Til: = j. -,79,7.9 1  FIGURE  -1 cm gm sec  x 10 (0)  -22  cm -gm s e c  27  The unusual transport properties of bismuth are due to (equal) small numbers of free electrons and holes (about 5x10  17  per cc. at 4 K) resulting from the overlap of a full valence and an empty conduction band in what would otherwise be a semi-conductor. The shape of the F e r m i surface (Fig. (0) b.) is reflected in the marked anisotropy of the transport properties. This representation of the F e r m i surface, with three electron ellipsoids and one hole (34) ellipsoid was firmly established by. Jain and Koenig  .  The results  of de Haas-van Alphen experiments imply a slight tilting of the electron ellipsoids out of the x-y plane and, consequently, diagonal terms in the effective mass tensor.  off-  In the parabolic approxi-  mation the ellipsoids are described in terms of the inverse effective mass tensors for electrons and holes as follows, 2m  E/G  =  (XHH ky] 4 oi^ky  2m E/h = fly (k^+kf)  -f- c<« h*  x  4 2<x&* kyki  4- A±kx  There are wide variations in the published values of the components (35) of oC  ft.  and  .  The following values are given by Jain and  Koenig: oCj, = 119, <*yj = 1.31, K  o<,  2  =102,  ZOC^T  =• 8. 6, /i, = 14. 7,  = 1. 09 in units of m o The electrical resistivity of bismuth decreases with decreasing temperature in spite of a decrease in the number of carriers. Single crystals of high purity and good structure may have (36 37) resistivity ratios (^3eoy^^-x ) °^ ^h order of 500 ' , where e  28  Jz^e^lQ  -4  The resistivity of such crystals shows a T  Jl-cm.  2  (37) temperature dependence  at liquid helium temperatures (indi-  cating that electron-phonon scattering is dominant) and is anisotropic. Garcia and Kao  found the resistivity ratio of slab-like mono-  crystalline samples to be a function of thickness for thicknesses in the range . 04 to 3. 1 mm, millimetres.  implying mean free paths of the order of  It was suggested that a plateau in the resistivity ratio  vs.. thickness curve could have resulted from the mean free path for holes being much shorter than that for electrons. To summarize, one expects electron-phonon collisions to provide the dominant scattering for electrons in good single crystals at 4 K, with increasing contributions from dislocations if samples are aged by thermal cycling. Knowledge of the specific heat is required in order to convert the diffusivities measured in the heat wave experiments to thermal conductivities. The electronic contribution to the specific heat is negligible at 4K, so that over the range of temperatures for which heat wave experiments were performed (4. 0-4. 5K) the specific heat can be represented by C = 1944 ( T/t$t>  )  j/mole-K. However,  because the Debye temperature t9> varies rapidly at low temperatures (from a minimum of 9 5 K at 10 K to 120 K at 2 K), it is more ;  convenient to take values of C form the curves given in Ref. 38. An estimate of the quantity ( ( C /C^) - 1) is also required, since this is a measure of the strength of the coupling between elastic  29  waves and the harmonic phonon systems in the Gruneisen approximations..  We use the formula  -  1  CV  Cv  where M is the molecular weight, and  c  e  cK. the volume coefficient  of thermal expansion, u the speed of sound and C^ the molar specific heat. The room temperature longitudinal acoustic velocities  (39)  5  5  . = 1.97x10 c m / s e c , v. - 2.54x10 cm/sec, trig Ir-mary 5 2 a 10 u, . •= 2. 57x10 cm/sec, from which a mean value of u , u = 5. 6x10 are  u  2 . 2 cm  /sec  is obtained. The value of oC at 4 K may be estimated in  the Gruneisen approximation via <K (4) /C (4) = <X (300) / C (300) 3 , - , . - ' K - . . C « ; . . . , , K - W . 1 1 6 Cp(300)*. 12 jK" gm" from (38) we find ( ( Cp/C^-1) 10" at 4 K. T  a  W  n  g  f  (  M  I  t  a  l  P  M  J  The coupling of the phonon system to elastic waves is therefore very weak at the temperatures of interest and will henceforth be neglected. In view of this result and the preceding discussion of the scattering of the electrons, it appears that the heat waves may be considered to propagate in two coupled systems: the electrical carriers and the phonons. The lumping together of electrons and holes in one system is justified by the fact that the electrical carriers make a small (  <S~'/. ) contribution to the D. C. thermal conductivity. The  dynamical properties of the holes and electrons are, of course, quite different, as is shown by the large thermoelectric power^^,  The  thermoelectric power would be zero were the electrons and holes to differ only in the signs of their charges.  30  The properties of the thermal conductivity tensor are discussed in the remainder of the chapter.  31  SECTION II H E A T CONDUCTION IN BISMUTH  In this section, the form of the thermal conductivitytensor for bismuth is obtained via irreversible thermodynamics (42)  by specializing the discussion given by Drabble and Goldsmid to the case of bismuth.  Although the thermal conductivity is not  the quantity which is directly measured in a heat wave experiment, the discussion indicates the various contributions to the total heat fLux in such an experiment. Suppose that, for small deviations from equilibrium, one may write  where the X  L  are generalized forces which produce the flows J £ ,  thereby reducing the entropy, S, below its equilibrium value; The basic theorem of irreversible thermodynamics is that if the rate of entropy production by irreversible processes can be written as  then  where /3 is the magnetic field.  32  It is assumed in the following discussion that local equilibrium exists during irreversible processes and that the temperature T and electrochemical potential ables.  are continuous v a r i -  Then, if just one group of particles is considered in isolation  (for example, the electrons from one ellipsoid in bismuth) and if the flux densities of particles and energy are j (r) and w(r),  one  • (42) may write +  These equations may be written in terms of the electric current density i and the phenomenological coefficients  UJ  • (2-1)  The phenomenological coefficients are: Electrical conductivity  Seebeck coefficient  1(d)—  &IBJ  ^  ~  —  ~h  (  L  —  @J  > Peltier coefficient  7T (0J =  T Q.(?d)  Thermal conductivity  k (&) —  —  ^  (n(B)—<2  )MM.  33  It is noted that the thermal conductivity tensor k is defined for i = o  (Eqn. 2. 1), but this condition is not necessarily satisfied  inside a sample undergoing a heat wave experiment.  If the contributions of several independent groups of carriers (for example the three electron ellipsoids and the hole ellipsoid) are added by writing  I ~J  t  ±i  > \ A / ~ £i Qt.  then the combined coefficients are  and  The extra terms in the thermal conductivity may be important in some problems, for example they describe "bipolar thermodiffusion" in which electron-hole pairs transport energy without disturbing the electrical neutrality. However it is known that at 4 K the total contribution of charge carriers to (D. C.) heat transport in bismuth is small, and we shall therefore lump the charge carriers together, obscuring such effects.  The form of the thermal conductivity tensor for bismuth is now deduced, using a model with two interacting groups of heat carriers, namely the phonons and the combined electrical systems. (Thus phonon-drag effects are included, but in the crudest approximation. ) In this case we still have where  4de  \7. £ - O  but now  \7-^>fO^O  and fop are the heat current densities in the charge  34  carrier and phonon systems. Following Drabble and Goldsmids' method,  we put  \7. e ~  -c/^^Vwhere  tU^e/oft  is the rate  at which energy is transferred from the electrons to the phonons (in unit volume). The equations for the flows are  = IS  L V (*)  71  cit  ( S  \  Onsager relations for  *— Ik  . Application of the  gives  , L»  _  ,  co  j  i  —  , (i)  I  are used.  L?'Vtyf)  ^/  giving 82 coefficients  leaving 46 independent  +  L-iU  <£>  _  I  &  *z  * &  cs)  | W7 _  I  («  =  , ft;  4  W  before crystal symmetry arguments  . The equations describing relaxation toward equilibrium  of the electron and phonon systems may be re-written to introduce the relaxation time  AT* At  _  (~7>  -Te )  Crystal symmetry imposes restrictions on the components of each L  (n)  We need to invoke only the 3-fold rotation symmetry  about the z axis and the 2-fold rotation symmetry about the x axis. For the L / ^ to be invariant under the combined rotation operator for 11  these two rotations they must have the form,  IL  j L  L  -L  \ o but now  oo  \  o  °  L  2 2  / for all  the Onsager relations require  The matrix of coefficients L  fn}  L  (n)  therefore has the form  lijy L i "  tl' ^  p  Ml' il'  il  tl! 21  il',  Clearly k, b , 7T and Q also have this symmetry in the absence of a magnetic field and k has only two independent components which may  be designated  kn  and  k±  , (taking orientations with  respect to the trigonal axis).  If *Tep were not an important relaxation time for the two (3) subsystems, they could be treated independently, in which case L and L / ^ would be zero (no phonon drag). The fluxes and phenomenological coefficients for the electrical system would be given by equations 2. 1 and 2. 2 by making the transformation,. L  (4)  9(5) 1/  .  36  For the phonon system,  The total thermal conductivity tensor would be T  F  "T"-  T  1  1  1  However, T<p could well be an important relaxation time near 4 K and then L / ^ and L ^ ' would be non-zero i . e. phonon drag effects might appear. Writing "Tg — Tp  and comparing the new  (primed)  coefficients in the presence of phonon drag with the previous ones,  Cl = Q -h A  =  rr  ~  L  ~L L  +  The expression for A  L  is slightly more complicated and  it is no longer possible tto write it as the sum of an electronic part and a phonon part. Sondheimer  (43) (44)  , and Hanna and Sondheimer  (45)  have discussed this problem in general. They find phonon drag to have no effect ontthe thermal conductivity.  There are strong phonon drag contributions to the thermoelectric effects in bismuth at just under 4  K^^.  Although the thermal conductivity tensor is not expected to be affected, it is possible, in principle, to observe some effect in a heat wave experiment where electric currents may flow inside the specimens.  Anomalous heat wave propagation might also occur  37  at frequencies near  ( '/'"Z"et )i. , where  ( £?f>  ) i is the  relaxation time characterizing the interaction of the phonons with the  ellipsoid, since at these frequencies the coupling of the  subsystems would begin to break down. These effects are expected to be small in view of the small total contribution of the charge carriers to the thermal conductivity.  38  SECTION III T H E RESULTS O F E A R L I E R T H E R M A L CONDUCTIVITY E X P E R I M E N T S ON BISMUTH  The first measurements of the thermal conductivity of (  bismuth at liquid helium temperatures were made by Shalyt  Z8\  , with  the object of establishing the nature of the thermal conductivity peak implied by measurements at higher temperatures.  Thefe measurements  were made on a cylindrical single crystal of radius 1. 8 mm. , with a resistivity ratio of 50.  The chemical purity was not specified. The  orientation was such as to give a value of k t , as defined above. The k  maximum value of observed was 17. 5 watts/cm K at 4. 0 K and the values A  between 2. 3 and 3. 0 K were consistent with mean free paths being limited by boundary scattering.  Since no measurements were made  between 4. 0 and 14. 0 K, the shape of the high temperature side of the peak was not determined.  Over the temperature range 2. 3 to 4. 0 K  the conductivity was found to change by less than 1% on the application of a 4. 2 kOe tranverse magnetic field, indicating a negligible contribution to the heat currents from the electrical carriers.  It was concluded that  the thermal conductivity peak was defined by boundary scattering and the temperature dependence of phonon-phonon Unklapp collisions, as is the case in good dielectric crystals.  39 Some r e s u l t s obtained by White and Woods c r y s t a l l i n e s a m p l e s support Shalyt's c o n c l u s i o n s .  (29)  from poly-  The p u r e r s a m p l e  e x a m i n e d e x h i b i t e d the exponential t e m p e r a t u r e dependence expected on the high t e m p e r a t u r e side of the t h e r m a l c o n d u c t i v i t y peak i f U p r o c e s s e s a r e dominant.  P e a k c o n d u c t i v i t i e s o c c u r r e d at about 5 K  and w e r e a p p a r e n t l y d e t e r m i n e d by boundary s c a t t e r i n g i n the i n d i v i d u a l crystallites. (40) Steele and B a b i s h k i n of  k  tl  o b s e r v e d a s m a l l o s c i l l a t o r y dependence  on t r a n v e r s e m a g n e t i c f i e l d s for f i e l d strengths up to 13 kOe.  The o b s e r v a t i o n s w e r e made on a 9 9 . 9 9 % p u r e single c r y s t a l of 1.7 diameter at 1. 6 K. higher f i e l d s .  mm  The d e c r e a s e i n the m e a n c o n d u c t i v i t y was 4 % at the  It i s i n t e r e s t i n g to c o m p a r e Shalyt's value f o r k± at 2. 3 K  w i t h that p r e d i c t e d f r o m Steele and Babishkin's value of J<|/ at 1.6 K on the a s s u m p t i o n that boundary s c a t t e r i n g l i m i t e d the mean f r e e paths ( i . e . we s c a l e the value as R T i m p l i e s that  ku  The r e s u l t i s  k  ^- kn w h i c h  i s at l e a s t p a r t i a l l y c o n t r o l l e d by some other m e c h a n i s m .  D e t a i l e d i n v e s t i g a t i o n s of the o s c i l l a t o r y effects have been made r e c e n t l y by Bhagat and M a n c h o n ^ ^ , who u s e d 99. 9 9 9 9 % p u r e single c r y s t a l s w i t h r e s i s t i v i t y r a t i o s i n the range 300-400.  Thermal  c o n d u c t i v i t i e s are quoted for t e m p e r a t u r e s between 1. 3 and 2. 0 K.  The  zero f i e l d m e a s u r e m e n t s again i n d i c a t e d k ± to be about t w i c e as l a r g e as  Ai/ and, i n addition,  k  lf  was o b s e r v e d to have a m o r e r a p i d  3 t e m p e r a t u r e v a r i a t i o n than T .  One cannot a s c r i b e the different  40  magnitudes of  h  u  and  kj.  to differences in crystal quality,  because the "trigonal" specimen j| Un ) had a higher resistivity ratio than the "bisectrix" specimen ( Ux. ), whose thermal conductivity indicated a mean free path limited by boundary scattering.  Small  transverse magnetic fields (up to 500 Oe) were found to produce decreases of 5% in the conductivity in the bisectrix direction (H in the trigonal direction) and 11% in the conductivity in the trigonal direction (H in the binary direction) at a temperature of 1. 3 K.  These results  were thought to reflect the magnitudes of the electronic contributions to the thermal conductivity.  Bhagat and Manchon attribute the anisotropy of the temperature dependence and magnitude of the thermal conductivity at these temperatures to electron-phonon scattering. They argue that the wavevectors of the electrons and typical phonons at these temperatures have comparable magnitudes and this fact, together with the extreme anisotropy of the F e r m i surface, permits the conditions for electron-phonon scattering to be satisfied only in certain directions in k-space.  With  rising temperature, progressively smaller areas of the F e r m i surface will be involved in scattering with the typical phonons. The results were fitted to the formula. _ / _ _ _ £ _ ,  _±_  in which a and b measure the relative strengths of boundary scattering  41 The best values of a and b were roughly equal for  k  lt  This argument and its implications for the conductivity at 4. 0 K  i . e. at its peak value, are examined in some detail in the  following section.  42  SECTION IV T H E  ELECTRON-PHONON  INTERACTION IN BISMUTH  For the purposes of obtaining a rough estimate of the importance of electron-phonon scattering at 4 K (i. e. in the region (47, 48, 49) of the thermal conductivity peak), an analysis due to Ziman will be applied.  Ziman was considering scattering by small numbers  of free electrons arising from impurities in dielectrics or in semiconductors. The electron-phonon scattering problem is greatly simplified when the density of charge carriers is low and the relevant wavelengths are long, as is the case with semiconductors. is then just the shift in the energy  The scattering potential  E(k) of the electron states which is  produced by the nearly uniform strain corresponding to a long wavelength phonon when the electron density adjusts itself to maintain charge neutrality.  The deformation potential,  of proportionality obtained when the energy shift is expressed as a linear function of the strain.  In the case of spherical energy surface s  only dilatations (i. e. longitudinal phonons) are involved in electron-phonon "normal" collisions.  Contributions from tranverse phonons appear  when the method is extended to non-spherical energy surfaces.  43 O n the assumption,? that the electrons are in e q u i l i b r i u m the mean free path  A . ^ of a phonon of wave-vector  deformation potential approximation  where  is the m a s s density,  jo  phonon occupation number.  0  (49)  i s , in the  ,  4  n  q  the speed of sound and  acts like a  ^  the  & -function.  It w i l l be assumed that this expression is v a l i d for bismuth. The conditions of low c a r r i e r density and long wavelengths are  satis-  fied, but the energy surfaces are far f r o m s p h e r i c a l . The deformation potentials for the two types of c a r r i e r are of opposite sign but of roughly equal magnitude, namely.2. % eV  B y use of the properties of the for  & -functions, the expression  m a y be reduced to  in which the m i n i m u m value of k which contributes to the integral is k - l(4/L)-rf /filtl  0  Z i m a n (considering a s p h e r i c a l F e r m i surface)  points out that the strength and temperature dependence of the depends on the relative magnitudes of fakp  .  scattering  and the F e r m i momentum,  T h r e e cases are considered:  a) when A/r  f  is v e r y much l a r g e r than  ^>/r  8  , so that the lower l i m i t  on the integral m a y be taken to be zero, to obtain  44  J_  _  b) when  (r»*) £J<<\)  ,  A  /n*** there  is s m a l l e r than  cr-  is little scattering except  at temperatures for which the dominant wave vector c) when  £A"p is greater than  temperatures <^  (  0  ^  Uj*&  gives  case a) applies at low  ) , but with r i s i n g temperatures and hence r i s i n g  , the scattering rate r i s e s to a m a x i m u m at the temperature  for which k  o  is zero and then falls off as  T  -2. 5 .  Detailed c a l c u l -  lations are given i n Ref. (47) and n o r m a l i z e d curves i l l u s t r a t i n g case c) are given i n R e f s . (47) and ( 4 9 ) . In o r d e r to extend the p r e c e d i n g analysis to the p r o b l e m at hand, a c o m p a r i s o n is made between the v a r i o u s values  ^ ^ f  of the  s e m i - a x e s of the e l l i p s o i d s i n bismuth ( F i g . 0(b) ) and the c o r r e s p o n d i n g values of  ^ ka  .  Thus only certain areas of the F e r m i surface  are  c o n s i d e r e d . . The effective m a s s e s are obtained by inverting the tensors whose components are given i n Section I. tudinal sound v e l o c i t i e s ' ^ are used.  The r o o m temperature l o n g i -  A value of q given by <£= Z'S^RT/tt  u  is considered, this being the value corresponding to the peak i n the energy density v s . wave-number curve on a simple Debye model, and the t e m p e r a t u r e chosen is 4 K .  The magnitudes of v a r i o u s quantities of  interest (see the e x p r e s s i o n for k ) are given in Table I, in which is the temperature at which k  is zero i . e. the temperature for which o  the scattering due to the given states reaches a m a x i m u m .  TABLE I  V A L U E S O F P A R A M E T E R S CONTROLLING ELECTRON-PHONON S C A T T E R I N G  Crystal Axis  , 22 mux 10  Carrier  hqxlO  22  2hk xlO  22  T K  F  M  _ *  Trigonal  1. 66  11  90  Electrons  . 04  11  15. 8  Holes  . 154  61  28  61  158  Holes  Bisectrix  3. 85  Electrons Binary  Holes  . 154  61  28  Electrons  . 019  61  10. 8  4. 8 .8 1. 8 11 1. 8 .7  F r o m the tabulated values of T  it appears that scattering m ° by holes with wave-vectors in the trigonal direction reaches its maximum r c  in the region of the thermal conductivity peak.  In view of Bhagat and  (30)  Manchon's observations typical dimensions 3mm)  that  k  for their best specimens (of  was determined by boundary scattering between  1. 3 and 2. 0 K, only the effects of the above mentioned holes need be considered at 4 K.  It therefore appears that the magnitude of the thermal  conductivity peak might be determined by hole-phonon scattering if samples are made sufficiently large, pure and strain-free. One would expect this hole-phonon scattering to produce a higher thermal resistance in the trigonal direction than in perpendicular directions thereby enhancing the effects due to the anisotropy of the elastic constants.  It is not clear how Bhagat and Manchon's finding that  is less than  can be explained on the basis of electron-phonon scattering using the values of  given above.  Better agreement would be obtained, without  changing the qualitative conclusions about the thermal conductivity peak, if the  were scaled down slightly, corresponding to a higher value  of q , such as is expected from the true phonon spectra.  Since data on  (51) the phonon spectra for bismuth are available  , detailed computer  calculations of the electron-phonon scattering rates are feasible. Finally, we may safely ignore another form of electronphonon interaction at 4 K, namely phonon assisted electron-hole recom(52) bination  , since the phonons involved have energies corresponding to  temperatures of about 43 K and 130 K. It seems that the most likely deviation from dielectric-like behavior at the thermal conductivity peak is the limitation of the magnitude of the peak by hole-phonon scattering. This might be 3 accompanied by a lower-than T  temperature dependence at temperatures  just below that for peak conductivity.  47  C H A P T E R III  E X P E R I M E N T A L DESIGN AND  APPARATUS  SECTION I T H E TRANSMISSION LINE M O D E L  A comprehensive analysis of heat pulse propagation in a system consisting of a dielectric solid, a thin film heater and a thin (53) film thermometer, has been published recently by Kwok  , who uses  the phonon Boltzmann equation to provide a microscopic description of the processes involved.  The macroscopic model described below  appears to be consistent with that analysis, given that the experiments to be described were not made is the short time regime i.e.. the signal frequencies were lower than the phonon-phonon collision frequencies. The application of transmission line models to second sound experiments in liquid helium is natural in view of the obvious electrical analogues, and the low damping and high reflection coefficients. (54) Osborne  used this method to analyse a liquid helium experiment  covering the temperature range over which second sound propagation breaks down due to increasing phonon mean free path. At the low  48  temperature end of the range, the damping was no longer small and a modified heat equation of the type (1.2) appears to have been required. A transmission line analysis assuming the modified heat equation has been successfully applied to heat pulse propagation in the long mean (5)  free path regime in both liquid helium and sapphire.  In Chapter I it was argued that, if sinusoidal heat inputs are applied to a crystal, the initial deviations from diffusive propagation observed at high frequencies will be described by the modified heat equation in both long and short mean free path situations (The physical significance of T  will be different in the two cases). The modified  heat equations will therefore be assumed in the model used here.  The samples used in the experiments described below were surrounded by helium gas and mounted in such a way as to minimize thermal coupling to other bodies. The model assumes that the time independent component of the temperature distribution is determined by the power input at the heater film and heat leaks over the curved surface of the cylindrical samples. was 7 5 % of the total surface area).  (The area of the curved surface It will be seen that this assumption  entails the addition of a leakage conductance to the electrical analogue. It is now necessary to reconcile the existence of radial heat fluxes with the assumption of plane wave propagation between heater and thermometer.  The inconsistency is not serious for the following reasons.  49  F i r s t l y , the t h e r m a l r e s i s t a n c e to r a d i a l flow is expected to be dominated by the b i s m u t h - h e l i u m gas boundary resistance, s m a l l r a d i a l temperature gradients within the s a m p l e .  leaving  Secondly the  active areas of the heater and thermometer f i l m s extended over only 25% of the end face areas, the heat current Q (x,  and were c e n t r a l l y p l a c e d .  t) and excess temperature  &  A n equation for  (x,  with r e s p e c t to the temperature of the surrounding gas, on the basis of the above assumptions.  t) m e a s u r e d is now d e r i v e d  Since the modified heat equation  is assumed, the appropriate equationffor energy conservation and heat current are,  J>Cv&  + T^i  _  _  =  _  &  . . . 3.1 (a)  -  UA 2L.  . . . 3. I (b)  in which it has been a s s u m e d that the heat leak per unit length is a l i n e a r function of the excess t e m p e r a t u r e : Q specific heat per unit m a s s , area,  \\  J>  r  =  <r  (C  (x, t)  v  is the  the density, A the c r o s s sectional  the t h e r m a l conductivity. )  E l i m i n a t i n g Q f r o m equation 3. 1  we have  ^A.  where  = { £ / J>  A C ^ a n d K = k / J>  -r  ^>  .  6  . . . 3. 2  Q (x, t) satisfies  50  Equation 3. 2 is the general transmission line  the same equation. equation if we regard  &  and  Q  as equivalent to voltage and current.  The solutions are conventionally written in the form.  OiJi^)=z  A  -r =  oL -f- c <jt>  e  -f-  «e  On substituting one of these solutions into 3.2, the following expressions are obtained for the attenuation coefficient wave number  Oi\ and the  '•  The corresponding quantities for the analogous electric line are,  in which  R, C, L and G are the resistance, capacitance, inductance  and conductance per unit length. Thus ( 1 /RC) is the analogue of the diffusivity hi , L/R is the analo gue of T .  The analogy between L/R and "7  and G/C is the analogue of > and in general between  Vernotte's equation ( 1 . 2) and the "equation of telegraphy", has been (55)  discussed by Ulbrich  .  According to the model used here, the heat  wave experiment is analogous to an experiment in which the voltage is measured at the open-circuit termination of a transmission line having a current source.  The expression for the output temperature fluctuation, in terms of the input heat current fluctuations, ^ (f>J d  d;f length L  for a specimen  may be obtained by direct calculation or by analogy. It  is  J> A C /ZUW^J Sink v  1  (*L)CtfyQt CcsUHdL) SiJ(tL)  The objectives of the experiment are to determine  ^  and (given large enough values of L>~~C ) ~C , at a given temperature, from measurements over a range of heat wave frequencies. quantities have to be deduced via values of OC (co)  and  These (^>>)> which  in turn are obtained from measurements of the relative amplitude and phase of quantity,  &  ( L , Co ) with respect to Q (c , L> ).  The third unknown  , must therefore be determined or eliminated.  It is perhaps unnecessary to remark that a "D. C. " heat flow of magnitude at least as great as  ( c ) is also generated at the heater  and the corresponding steady state temperature distribution is superimposed on the A. C. solution given above. employed here, the magnitude of  According to the model  determines the mean temperature  52  rise (of any point on the sample) resulting from a given heater power level.  It is therefore important to know whether, for values of  large enough to permit reasonable power inputs, (^** / co ) may  be treated  as a small quantity at the lowest heat wave frequencies of interest. D. C. experiments suggested that both conditions would be satisfied in the experiments described below (Gas pressures frequencies ^  ^  10 cm. Hg ,  140 Hz) and this is verified in the analysis.  Given a value for y*i , one may,  with the assistance of a  computer, solve (3.4) for best values of OC ( <o ) and $  by an  iterative method. This procedure was used, but, as is shown below, a simpler, more physically revealing graphical analysis may if the small terms in (3. 3) are kept to first order only.  be made  This level of  approximation was also used in the computer programme to obtain . To first order in (v*< / <0 ) and ( CoT ),  and "X from oi. and  j£  the expressions for  oC and  are  from which,  H  =  r~—7, U  1  H ) l  J ^  *  -^  - ^ U  U±-4*J  ... 3.6  A'  The term ( /  ) is expected to be completely negligible at any  frequency for which ( dt  ) is appreciable.  The philosophy of the graphical method of analysis is as follows.  Since the most probable outcome of the experiments is that  diffusive propagation will be observed under conditions of high attenuations the results should be analyzed on the assumption of this behaviour, but presented in such a way as to emphasize deviations from it.  To be  specific, high attenuation in a sample of length L will be defined by the inequality ( °€L ) ^ (  "  2. 5  , for which values one may write  sinh. (oi^-J  ) = cosh ( oCL ) = (exp( Ui. ) I 2 ) with an error of less than  one percent. Thus no resonance affects will be observed under conditions of high attenuation. (See 3.4). and diffusive  pr*j>tic-n  ,  In the approximation of high attenuation  (d.~$~)  A value of the diffusivity,  >  equation (3.4) gives  H , may therefore be obtained from  the slope of a plot of log ( / 6 (D I J?/1 til I ) C  Y*  •  Deviations from a straight line will occur at high frequencies if ( <0 T ) becomes non-negligible. Deviations will occur at low frequencies if the assumptions ( s^y/i*) << I or ($£L) ^  break down.  54  In the same approximation, the expression for the phase 1 ag  £ ~  is reduced to  (j^J  -f~ 0L  Thus a second estimate of l\  ...  may be obtained from a plot of  vs J4I , the zero frequency intercept of which may  check on the model. specified above.  3. 8  serve as a  Deviations should occur under the circumstances  The expected form of these deviations, and the method  whereby, in principle deviations reflecting the thermal properties of the crystal may be distinguished from response time effects in the heater and thermometer films, are discussed in the analysis of experimental results.  55  SECTION II T H E  CRYSTALS  The two single crystals of bismuth used were obtained from Metals Research (Crystals) Ltd. (Melbourne Royston, Herts, England). As received the specimens were spark-cut cylinders, 2 cm. in length and 1 cm. in diameter, with spark planed end faces. According to the manufacturer, the maximum impurity levels were 1 part in 10^ for the crystal with the trigonal, or "c-axis" parallel to the cylinder axis 7 (hence-forth called the trigonal crystal) and 5 parts in 10  for the  crystal with the bisectrix axis parallel to the cylinder axis (henceforth called the bisectrix crystal). o accurate to within 3 .  Orientations were stated to be  The crystal mountings (Section V) were designed to minimize strains, but any deleterious effects resulting from repeated thermal c y c l i n g ' ^ could not be avoided. The trigonal crystal was subjected to considerably more handling and thermal cycling than was the bisectrix crystal. At the completion of the heat wave experiments, the resistivity ratios of the two crystals were found to be: Trigonal crystal (  3 0 0 / ^ 4) = 100, Bisectrix crystal,{f  -6 300 = 111 x 10  JT.-C»O  (Both crystals)  300^4^1= 60,  56  (A four lead D . C . current and potential method was used). finished dimensions of the crystals  Trigonal crystal:  were:  Length Diameter  Bisectrix crystal:  Length Diameter  1.97 +  .005  cm  . 97 +  . 005  cm  1.97 +  . 005  cm  .895 + . 005  cm  The  57  SECTION HI HEATERS, T H E R M O M E T E R S AND INSULATORS  Thin film heaters and resistance thermometers were used in order to obtain the required response times.  (Elsewhere, laser (21)  beams have been used to launch heat pulses in dielectric crystals  .)  The generation of heat waves in bismuth by use of the skin effect was considered, but inconveniently high frequencies would have been necessary in order to achieve the required spatial resolution.  In  terms of the response times attainable, mutual inductance coupling is better than the direct attachment of leads to the thin films.  An excellent  system, which employed mutual inductance coupling, an amplitude modulated radio-frequency input and a 30 MHz twin-T bridge detection system designed to operate with an indium thermometer, was inherited from D. Y. Chung  .  Unfortunately, the high electrical conductivity  of bismuth made that system unsuitable for the present investigation. Methods depending on direct coupling to films deposited on thin insulating substrates were therefore developed. The greatest difficulties were encountered with the insulating layers between the sample crystals and the heaters and thermometers. These layers were required to present high electrical impedances in conjunction with low thermal capacitances and resistances. Evaporated dielectric films were an obvious solution. Fundamental limitations on  58  the thicknesses of such films were imposed by the need to avoid both excessive field strengths (potentials of up to 10 volts were applied across the heaters) and capacitances sufficient to provide significant admittances at the highest frequencies to be used.  Much time was expended on efforts to produce the required insulating layers by the evaporation of SiO on to polished bismuth surfaces.  Various combinations of evaporation rates, pressures and  substrate temperatures were investigated, and some insulating films were obtained.  The films were poorly bonded to the bismuth and  reticulation or "crazing" was common. greater than the 1  Thicknesses  somewhat  estimated for ideal films were required to avoid  dielectric breakdown or other electrical failures.  In view of these  difficulties, other insulators were investigated. (It was later learned that the use of tantalum boats in the SiO work could have led to the production of some free silicon, with attendant ill effects on the insulating properties.)  Anodic oxide films were also tried and abandoned. with apparent resistances of the order 100k  Films  were formed on the end  faces of a cylinder of polycrystalline bismuth of diameter 1 cm, using dilute NaOH as the electrolyte, initial current densities of under 3mA/ and cell potentials up to 40 volts. The difficulties with these films were apparently due to micro-fissures, or a general porosity.  59 Following these failures it was decided to concentrate on the electrical and mechanical properties of the insulators rather than on the theoretically attainable response times. The first successful insulating layers were sheets of mica, about 5./** thick, prepared by first cleaving under water with a razor blade and then cleaving with Scotch tape.  The sheets obtained in this way were rinsed in hydro-  chloric acid, water and ethyl alcohol, and then stuck to the electropolished end faces of the bismuth crystals with very thin smears of Araldite epoxy resin.  Further cleaving by the Scotch tape method was  possible with the sheets in place, leaving freshly cleaved surfaces as substrates for the evaporated thin film heaters and thermometers.  As  described below, some heat wave experiments were performed with crystals having mica insulation, but the layers were undesirably thick and other materials were sought.  Some preliminary work was carried out with evaporated thin films of CaF, but these were abandoned in favour of Kodak Photo-Resist (KPR), a material intended for use in the fabrication of printed circuits by photo-etching. (Thanks are due to D. L. Johnson for pointing out the possible application of this material.) The photo-resist was applied as an aerosol spray to produce films which, when dried, exposed to ultra-violet light and "developed", became chemically very stable. (Instructions for these procedures are supplied by the manufacturer.)  60 The end faces of the bismuth crystals to be sprayed were polished with cerium oxide and de-greased in an ultrasonic cleaner filled with ethyl alcohol. It was found possible to obtain films as thin as desired by rotating the crystals in a miniature lathe during  spraying.  (The curved surfaces were protected by wrapping the crystals in lead foil. ) When the correct combination of rates of rotation and of drying had been discovered, films could be made flat to within a half-wavelength of sodium light, except for circumferential ridges at the outer edges. In view of the sudden disappearance of the circumferential interference fringes inside the rim, it seems likely that the absolute thickness of the central portions of the better films was less than 1500  R. Evaporated constantan heater films were used in all of the  experiments.  They had the simple zig-zag form shown in Fig. 1.  This configuration was used in order to allow thicker, more reliable films to be employed while still retaining reasonable matching to the 600 J\ output impedance of the driving oscillator.  Most of the heaters  used had resistances in the range 600-1000 -A ; the resistances varied , by less than 2% between room temperature and 4 K. The films were prepared from sources consisting of . 004"  "Advance" wire wound  on tungsten filaments, and were deposited on to substrates cooled with liquid nitrogen.  Resistances were estimated during the evaporations  61 HEATER AND THERMOMETER FILMS  Photo-Resist  I n or Au contacts Constantan leads Constantan heater or Pb-In a l l o y thermometer  Approx. Scale  X 6  1 Bismuth  FIGURE 1  62 from the resistances of films simultaneously deposited on glass monitor strips.  The heat wave results to be presented were obtained by use of thermometers consisting of thin films of lead-indium alloys held at the superconducting transition temperature. of thermometer were also investigated. superconductors is as follows.  Two other types  The reason for not using pure  It is clear that the mid-transition  resistance of the thermometer must be large compared with the resistances of any leads or contact films attached to it, otherwise there will be a serious reduction in sensitivity.  If pure superconductors are  used, it is difficult to produce adequately high thermometer resistances in areas as small as the ones available in this work.  Some investigations were made with a synthetic sapphire crystal equipped with a carbon film resistance thermometer of the type  f 5 6) described by Cannon and Chester  . The problem in this case is to  reduce the thermometer resistances to convenient values.  In the  method developed by Cannon and Chester, suspensions of carbon black in xylene are sprayed over gold contact films provided with arrays of fine scratches arranged to produce parallel combinations of very short, wide carbon resistors.  Carbon films prepared in this way seem to be  in relatively poor thermal contact with the substrates, and there exists the possibility the response times ( 1 0 /*» sec) observed by Cannon and  Chester were partly controlled by the coupling of their films to the liquid helium.  In work with bismuth, such problems would be  aggravated by the added thermal resistances of the electrical insulators. Partly for this reason, but also because higher sensitivities were desired, superconducting alloys were preferred over carbon films, at the sacrifice of the wide range of operating temperatures available with the latter.  Besides providing a reasonably wide choice of operating temperatures, the use of superconducting alloys made it easy to construct thermometers having normal state resistances of the order of 50 A  . Simple zig-zag shapes, as shown in Fig. 1, were used.  Attempts were made to construct thermometers having superconducting transitions in the range 4. 0 - 4. 5 K, within which the thermal conductivity peaks of the bismuth crystals were expected to lie.  On the naive supposition that the transition temperatures of lead-bismuth alloys would lie between those of pure lead and amorphous bismuth or highly compressed bismuth, a lead-bismuth film was prepared (the materials being more conveniently to hand than the literature).  (Lead-bismuth films were attractive because the evapor-  ation rates of the constituents are closely similar.)  This and subsequent  films of the same type were evaporated from tantalum boats on to liquidnitrogen-cooled mica substrates,. using source material from an ingot  64 of composition 78% Pb, 22% B i by weight.  The ingot was made in  vacuo from Cominco 49 grade lead and 59 grade bismuth. (57) Previous work  indicated that alloys of this composition  should have transition temperatures close to 8 K.  This was not known  at the time when the first film was examined and found to exhibit a rapid variation of resistance at about 4. 4 K, which was thought highly satisfactory.  Subsequently, investigations of several films, nominally  of the same composition as the first, revealed two regions of rapid variation of resistance: at about 4. 4 K and at about 8. 0 K.  These films,  operating at the lower transitions, were used to detect heat waves in the "trigonal" bismuth crystal. . The results of four-lead current and potential measurements made on one of these films are shown in Fig.  2.  The relative magnitudes of the two jumps in resistance varied  from film to film.  It is suggested that the above results indicate that  the films were inhomogeneous. The high temperature transition could be ascribed to regions consisting of a lead-bismuth alloy; the low temperature "transition" could conceivably be due to regions of amorphous bismuth.  It is noted that a change of phase occurs in the alloy system  at concentrations close to that employed.  Other workers report that  it is necessary to cool substrates below 20 K in order to form films of amorphous bismuth.  65Pb-Bi  TRANSITIONS o  o co  o  \  o  o W  ••  H  « j  H  o  FIGURE  2  o  EH  66  Whatever the explanation of their behaviour, the leadbismuth films made moderately good thermometers. A change to lead-indium films was made to obtain better control over the operating temperatures of the thermometers. The lead-indium alloys were made from Cominco 49 grade lead and 59 grade indium, melted in tantalum crucibles under a helium atmosphere at 200 A» pressure. Three ingots, designated B, C and D and containing 17. 5%, weight respectively, were made up.  15% and 10% of lead by  Thermometer films of approxi-  o mately 500 A thickness were constructed on liquid-nitrogren-cooled mica and photo-resist substrates, using material from ingots C and D evaporated from molybdenum boats.  A resistance vs. temperature  curve for one such thermometer is given in Fig. 3.  These measurements  were made in the course of heat-wave experiments by means of low power A. C. bridges. The resistance values given include lead and contact resistances. As the heat wave detection system required the thermometers to be biased with currents at low radio frequencies, it was necessary to ascertain the frequency dependence of the transitions.  Measurements  made at frequencies up to 1 MHz indicated the maximum usable bias current frequency to be about 500 kHz;  at higher frequencies, progressive  increases in the apparent resistance in the superconducting state seriously reduced the overall sensitivities of the thermometers. The temperature fluctuations observed in the experiments were very small  ^7 Pb-In ALLOY THERMOMETER (Ingot D)  T K  FIGURE 3  68  compared with the widths of the transitions. The operating points of the thermometers were therefore chosen by scanning the transitions to find the temperature for maximum output signal.  For this reason  the mean, or overall, sensitivities of the thermometers were not immediately useful for making calculations.  The D-shaped thin film contacts were made of indium, except for one experiment in which gold was used.  Constantan leads  of . 004" diameter were attached to the contacts by small spots of S C F 12 micropaint (Micro-circuits Co.).  The complete heaters and thermometers  were as illustrated in Fig. 1.  It is useful to have a simple model to describe the thermal response of the heaters and thermometers.  Equally as important  as the response time estimates made on the basis of such a model is the ability to recognize contributions from response time effects in the apparent phase shift and attenuation of the heat waves detected. A thermometer of the type used may be considered to measure the temperature of a representative point within the volume defined by the active area and the thickness of the device, i . e. , an internal spatial averaging is performed. is therefore appropriate.  A simple lumped constant electrical analogue The thermometer is assumed to behave like  a low-pass filter with a time constant determined by the product of the total thermal capacitance and the combination of thermal resistances  69  coupling the representative point to the surrounding media. On this model, and under the further assumption of equal time constants for the heater and the thermometer, the signals detected are expected to have suffered an additional phase shift and a reduction in amplitude by ( /  t-  (t /d)  1  ).  3. tan \ f / f ) Apparent  deviations from diffusive propagation were therefore examined for these frequency dependences.  For the purpose of calculating upper bounds on the response times, one may take the active areas to be the areas of the end faces of the crystals and the only coupling to be that to the crystals. Such calculations are presented in the analysis of results.  Finally, it is perhaps worth remarking that, whereas in this experiment the heaters and thermometers are required to be strongly coupled to their substrates, the reverse situation is required in similar experiments in liquid helium, which therefore employ different types of heaters and thermometers e. g. carbon resistors.  70  SECTION IV INPUT AND  D E T E C T I O N SYSTEMS  The simplest type of detection system applicable in the present experiment is the basic D. C. biased bolometer circuit.  In  this system, the thermometer, . which is characterized by its temperature coefficient of resistance a large load resistance,  is placed in series with r, across which the output voltage V is  developed by the bias current I.  The voltage sensitivity obtained with  this scheme is  (14  RAJ  indicating that r should be made as large as possible.  A major disadvantage of the above type of detection system in the present application is that no distinction is made between genuine signals produced by heat waves and spurious electrical signals at the same frequency.  The distinction is possible in principle, because  electrical pick-up is superimposed on the bias current, whereas heat waves modulate the bias current.  It is therefore possible to remove  the genuine signal to a high frequency, and to filter out the spurious low frequency signals, by using an A. C. bias in conjunction with a tuned detection system. In this case it is still necessary to have a large load, /z/, for good voltage sensitivity, but one may the depth of modulation, given by  also have totconsider , where  is the  amplitude of the assumed sinusoidal variations of the thermometer  71  resistance about its mean value R . o The system which eventually evolved is shown in Fig. 4. In this system, the thermometer is coupled by mutual inductance to a parallel L C circuit driven at its resonant frequency f through the c  load C^.  If R^  is the resistance effectively introduced into the  resonant circuit by its coupling to the thermometer, then the impedance 2 / Q R , which, from the previous  of this circuit at resonance is  c  discussion, is to be small with respect to  / / Orr fc Cd J  Clearly, for effective filtering it is necessary to have f v  much larger than the highest heat wave frequency, 2f^, and to have a high Q.  Given a sufficiently high ratio f /2f it is unnecessary to c o  calibrate the high frequency part of the system at each heat wave frequency. The upper limit on £  £  was determined in this experiment  by the frequency dependence of the superconducting transition, mentioned above.  Using these criteria, a carrier frequency of 500 kHz was  chosen. Apart from filtering, two other considerations govern the choice of Q, which is varied by means of the adjustable mutual inductance coupling to the thermometer. Firstly, the coupling must be sufficiently strong that the effective thermometer resistance R^  dominates any  resistance already in the L C circuit. Secondly, if it is required to examine a range of heat wave frequencies without re-tuning the radio frequency part of the system, the bandwidth about f  must be large  Insulation  Heater Film  Variable Phase  @f\  0  Oscillator  Bismuth R  0@  sample  ?  0  To A.C. Bridge >1>  2nd Harmonic Generator 2f@  2f  6  Ref. S i g . JB6 M  Q  a Kl  Lock-In Amplifier  S i g i @ 2f  CRU, Pre-Amp,  73  compared with the bandwidth of the heat waves.  A Q of approximately  25 was used, giving a 20 kHz bandwidth. (The highest heat wave frequency examined was  7 kHz).  A diode envelope detector was used to recover the heat wave signals. The requirement that this stage should introduce negligible phase shifts at the heat wave frequencies, together with the normal design considerations, led to the choices : R 0^  = 50 ^<«A  p  S 50 pf (See Fig. 4). The audio frequency signals were passed  through a Princeton CR4 pre-amplifier, which incorporated an adjustable band-pass filter, and thence to a Princeton JB6 coherent amplifier.  A limit on the maximum carrier amplitude, and hence on  the bias current, was imposed by the value of the maximum tolerable radio frequency leakage into the pre-amplifer. The effects of excess carrier amplitude are discussed in the following chapter.  The heater film was supplied at frequency f g  and voltage V  from the 0° output of a variable phase oscillator (Feedback Ltd. VPO 230) having three outputs: 0°, 90° and variable phase, all at 600 -A. impedance.  A reference signal of low amplitude (to minimize pickup  in the thermometer circuit) was supplied from the variable phase output via a second harmonic generator to the coherent amplifier. The JB 6  coherent amplifier has two output meters which give the i n -  phase and quadrature components of a coherent incoming signal with  74  respect to the reference signal. These outputs may be considered to represent projections of the signal vector on to a pair of orthogonal axes rotating with the same angular frequency as the signal.  In  order to make use of the superior performance of the variable phase oscillator (errors with respect to the 0° output were less than 1° and .25  o  ° for the variable phase and 90 outputs respectively) the JBo was  used as a null detector in the phase measurements. Phases were measured by rotating the phase of the reference signal to obtain a succession of zero in-phase and quadrature outputs. At each null for one meter, an amplitude measurement was obtained from the deflection of the other. In this manner, several .pairs of amplitude and phase measurements were made at each heat wave frequency, with a consequent reduction of both systematic and random errors.  In the course of each experimental run, a range of heat wave frequencies was examined at constant heater input, carrier frequency and carrier amplitude, with the thermometer held at a given point on its transition. The voltage sensitivity of the detection system in a run was determined from the attenuation plot obtained in the graphical analysis described in Section I.  In Section I an equation, (3. 4),  relating the output temperature amplitude & (L) to the input heat 0  current amplitude, Qi'c), was obtained.  However, the ratio ^J^/<^ ip) 0  is not directly obtainable from the measurements made, since it involves the voltage sensitivity. The attenuation plot actually used was  75  a plot of log  (^Jt)  vs/Twhere Z= J  /  $  V  *  . / l * +<«  l  is the mean magnitude of the signal vector as given by the meter deflections of the coherent amplifier, g is the actual gain of the coherent amplifier and pre-amplifier combination, and V i s the rms heater voltage. The relationship between Z and A/(Lu~Z  -  6c(L)  where N is the calibration factor for the output meters of the coherent amplifier, (in volts per unit deflection), and R  is the heater resistance.  F r o m equation (3.7) it is apparent that, having obtained the attenuation plot, one may deduce a value for the voltage sensitivity from the zero frequency intercept. A typical value for the voltage sensitivity in the later runs was  The complete: system was found to be reasonably linear with respect to small changes in the power input to the heater film but a constant heater power of the order of & /nW was maintained during experimental runs in order to obtain the best accuracy in attenuation measurements. (Temperature control was also improved in this way.) Attenuation measurements therefore involved noting the output meter deflections and input attenuation settings on the JB6 coherent amplifier, and the gain and band-pass filter settings on the CR4 pre-amplifier. After each experimental run, both-\units were calibrated for gain and phase shift at the frequencies which had been employed. This was done by means of a rather tedious procedure involving the use of the variable  76 p h a s e o s c i l l a t o r as a s o u r c e o f r e f e r e n c e s i g n a l (without the s e c o n d harmonic  generator) and d u m m y heat wave signal.  I n the c o u r s e  o f c a l i b r a t i o n , a p e c u l i a r i t y o f t h e JB6 c o h e r e n t a m p l i f i e r , w a s t a k e n i n t o a c c o u n t , n a m e l y t h a t i t w a s n o t p o s s i b l e at a l l f r e q u e n c i e s t o t u n e both r e f e r e n c e and signal channels f o r m a x i m u m gain.  P h a s e shift m e a s u r e m e n t s made during c a l i b r a t i o n w e r e r e p r o d u c i b l e w i t h an e r r o r of + the v a l u e o f a n y angle ^  ^  t  $  1° . A s m e n t i o n e d  was actually deduced f r o m estimates of  a n d c/f 3^  ) $\~Tr  previously,  i n o r d e r to reduce s y s t e m a t i c  e r r o r s due t o b o t h the v a r i a b l e p h a s e o s c i l l a t o r a n d t h e c o h e r e n t a m p l i f i e r . T h e c a l i b r a t i o n o f g a i n s w a s t h o u g h t to b e a c c u r a t e to w i t h i n 2 % . M a n y o f the heat w a v e f r e q u e n c i e s w e r e c h o s e n to be the g e o m e t r i c m e a n s o f t h e 3db f r e q u e n c i e s o f t h e b a n d - p a s s f i l t e r i n t h e p r e - a m p l i f i e r , t h e r e b y m i n i m i z i n g the p h a s e s h i f t c o r r e c t i o n s t o b e m a d e , b u t o t h e r f r e q u e n c i e s h a d t o be u s e d a s w e l l , i n o r d e r to a c h i e v e a f a i r l y u n i f o r m c o v e r a g e o f the  JT  1  scale.  E r r o r s i n the f r e q u e n c i e s , w h i c h w e r e  s e t a n d m o n i t o r e d w i t h a G e n e r a l R a d i o t y p e 1191 c o u n t e r , w e r e n e g l i gible.  S e v e r a l f a c t o r s c o m b i n e d to l i m i t t h e m a x i m u m h e a t w a v e f r e q u e n c i e s w h i c h c o u l d b e e m p l o y e d to a b o u t  8 k^z  • Firstly,  b e c a u s e the r a d i o f r e q u e n c y l e a k a g e f r o m the envelope d e t e c t o r p r e v e n t e d u s e o f the m a x i m u m g a i n a v a i l a b l e f r o m the p r e - a m p l i f i e r , the a t t a i n a b l e s i g n a l l e v e l s at h i g h e r f r e q u e n c i e s w e r e too l o w f o r a c c u r a t e m e a s u r e ments.  Obviously this difficulty could have been overcome by  77  additional filtering at the emdope detector, but this would have led to appreciable phase shifts at the heat wave frequencies and the necessity of calibrating this stage at each heat wave frequency. Had this been done, a more serious difficulty would have appeared; the narrow bandwidth of the resonant circuit would have required the retuning of the radio frequency system with changes i n heat wave frequency.  This  would have led to a further reduction in the already slow rate of data acquisition as well as difficulties in analysis. The envelope detector stage was the most important source of signal contamination by noise and by pick-up of stray signals, the most serious of which was the  JL4*  0  reference signal. A superior  demodulation device would be recommended for a system not l i m i t e d by the problems outlined above. The levels of spurious signals were determined at each heat wave frequency during the experimental runs by moving the temperature off to either side of the superconducting transition. The coherent amplifier was operated with a bandwidth of 2/3 Ha:  for most heat wave measurements.  78  SECTION V T H E CRYOSTAT SYSTEMS  The cryostat was designed to permit heat wave measurements to be made in an exchange gas at a pressure different from the vapour pressure of the helium bath, and at a temperature higher than the bath temperature.  To this end, the sample was enclosed in a vacuum can  having a brass rod beneath it to provide a variable thermal resistance to the bath.  The vacuum can was suspended from a . 5" stainless steel  pumping line and three . 125" stainless steel vacuum-tight coaxial lines, all of which passed through the end of a piston at the cryostat top, as shown in Fig. 5.  To achieve vertical motion of the vacuum can, the  piston was driven through a sliding 0-ring seal in the surrounding cylinder by means of a threaded rod and a nut attached to a thrust bearing in the supporting bridge. Flexible couplings were provided between the top of the moving piston and the remainder of the vacuum system, which comprised a Veeco ionization gauge, a high vacuum valve, a thermocouple gauge and a mercury diffusion pump.  This vacuum system was used in some preliminary experiments made in order to determine the order of magnitude and pressure dependence of the quantity •*»  (previously discussed in Section I), which  CRYOSTAT DETAIES  Bearing  O-Ring  Seal  Main Pumping L i n e  C r y o s t a t Top  (Approx. s c a l e x l / 5 )  Coaxial lr  •  c  Lines  P r o v i s i o n f o r Indium Seal Copper Walls  m t  Vacuum Can  ^7  Brass TbeEmal R e s i s t a n c e Rod  (Approx. s c a l e x l / 2 ) FIGURE  (5)  80  is related to the thermal leakage conductance between the sample and its surroundings. F r o m these experiments i t was concluded that the proposed heat wave experiments could be performed in the vapour over the helium bath. The vacuum can was therefore not sealed during the heat wave experiments. The helium bath was connected to a conventional cryostat vacuum system, which included mercury and o i l manometers for vapour pressure measurements. In order to permit the bath temperature to be stabilized when desired, the main pumping line, which was connected to a Kinney mechanical pump, was fitted with a Walker pressure  i * ()  regulator.  58  The method of measurement employed i n the heat wave experiments involved repeated slow passages through the superconducting transition of the thermometer film. Hence there was no demand for accurate long t e r m stabilization of the sample temperature. As the heat wave experiments were performed at temperatures close to 4 K, it was possible to maintain good thermal coupling between the can and the helium bath. Adequate temperature control was achieved during measurements by making small adjustments to the rate of pumping. Occasional adjustments of the height of the can were made between measurements to allow for the drop in helium level. Temperature control during the returning  81  period between heat wave measurements was greatly facilitated by making use of the third (90°) output of the variable phase oscillator to maintain a constant power input to the heater film.  The heat wave experiments typically lasted for roughly 36 hours. An automatic nitrogen filling device (which was liberated from the apparatus of its constructor, J. D. Jones) permitted operations to be suspended at helium temperatures for periods of several hours. The sample holder (Fig. 6.) was designed to permit all delicate operations on the samples to be performed under optimum working conditions at a bench, rather than at the cryostat.  It was  also desired to minimize strains on the samples and, in order to create conditions approximating those assumed in the transmission line model, to minimize thermal contacts between the samples and the sample holder.  This sample holder was held inverted in the vacuum can by its three Amphenol "sub minax" bulkhead connectors, which were mated with female connectors soldered into the ends of the coaxial lines. In this way, simple, quickly demountable, combined electrical connections \ and mechanical supports were provided.  Crystals were lightly held in two V-shaped notches by means of retaining bars fastened down with 2/56 screws, as illustrated in Fig. 6.  Five of the six "point" contacts were made to Lucite. One of  SAMPLE HOLDER  83  the brass retaining bars was left without a Lucite pad in order to ground the crystals.  Electrical connections between the thin film contacts and short lengths of miniature coaxial cable protruding from the bulkhead connectors, were made with . 004" constantan leads of (roughly) 1 cm length. (The joints at the thin films have been described previously, (Fig. 1. ) )  A . 1 watt 33 jx Allen-Bradley carbon resistor, connected  to a loop of thin copper wire, was attached to the midpoint of the sample by tightening the loop about it with a 000 gauge nut and bolt. (This scheme is described in detail by its originator, D. L. Johnson  .)  This resistor was connected to the third bulkhead connector by means of about 10cm of . 004" constantan wire, in an attempt to couple the resistor to the sample temperature rather than sample holder temperature.  These operations were performed before mounting the  sample holder in the cryostat.  An A. C. Wheatstone bridge system, which included a preamplifier with a voltage gain of 6, 000 and a Princeton JB4 Lock-In amplifier, was used to measure the resistance of the (nominally) 33 JL -8 resistor.  Power levels at the resistor were of the order of 10  watts.  A resistance of the order of 500 -°-could be measured to within . 1 -JL without difficulty. This system was operated at various low audio frequencies which were chosen to minimize interference with the heat wave measurements.  84  A "Cryocal" pre-calibrated germanium thermometer was used to calibrate the carbon resistor in helium vapor between 4 K and 7 K.  In this range the germanium thermometer was claimed to be  accurate to within 10m K.  Calibration procedures were checked in  liquid helium by comparison with temperatures calculated from vapour pressures, using the T  scale.  The results were consistent to the  5o  accuracy claimed by the manufacturer i . e. to within 5m K.  85  C H A P T E R IV  E X P E R I M E N T A L PROCEDURES, RESULTS AND  SECTION I E X P E R I M E N T A L  ANALYSIS  PROCEDURES  a) > Operations preceeding measurements at liquid helium temperatures After the insulating layers had been constructed on the samples, the evaporated thin films were deposited in the order: contacts, heater, thermometer. In this way, aging of the thermometer films was reduced.  (It was necessary to break the vacuum between depositions.)  The small carbon resistor and the constantan leads were attached to the sample (mounted in the sample holder)', usually within an hour of breaking the vacuum after the final evaporation. The samples were then mounted in the cryostat and maintained under a vacuum or in a helium atmosphere until the "run".  b) Operations in liquid helium At the commencement of a run, the (unsealed) vacuum can, in its lowest position, was immersed in liquid helium.  A reading of  "R sample", the resistance of the carbon resistor strapped to the crystal, was taken simultaneously with a measurement of the helium  86 vapour pressure.  This reading was used to adjust the calibration  of R sample, if necessary.  When possible, the transition of the thermometer film was traced out at this stage, i . e. with the sample submerged in liquid helium.  The resistance measurements were made in a quasi-static  manner by means of the A. C. bridges, as described previously, and produced results such as those illustrated in Fig. 3.  The next operation involved substituting a decade box for the thermometer films, and an oscilloscope (operating in a D. C. mode) for the CR4 pre-amplifier, in the detection circuit (Fig. 4). With these substitutions, the detection circuit was fine-tuned by making small variations in the carrier frequency to optimize its sensitivity (in voltsyohm) to small changes in the value of the decade box resistance about the previously determined mid-transition resistance of the thermometer film.  At the same time the amplitude of the carrier signal  was adjusted to the higher value allowed by the input limitations of the CR4 pre-amplifier and the greatest modulation depth expected. The detection circuit was then returned to its normal configuration.  The mutual inductance coupling, having been investigated and adjusted in an early run, was infrequently adjusted in subsequent runs. When attempts were made to reproduce the results of a preceding run,  87  no r e - t u n i n g o f t h e r a d i o - f r e q u e n c y s y s t e m s w a s  performed.  c) P r o c e d u r e s c a r r i e d out i n h e l i u m v a p o u r p r i o r to e a c h heat wave m e a s u r e m e n t The  f r e q u e n c i e s o f the s e t o f h e a t w a v e s to be e x a m i n e d i n  e a c h r u n w e r e c h o s e n b e f o r e h a n d a c c o r d i n g to c r i t e r i a o u t l i n e d p r e v i o u s l y ( C h a p t e r III, S e c t i o n I V ) .  These frequencies were  e m p l o y e d i n r a n d o m o r d e r i n the h o p e o f a v o i d i n g t h e a p p a r e n t f r e q u e n c y d e p e n d e n c e o f a n y e f f e c t s a c t u a l l y due to s l o w c h a n g e s o f c o n d i t i o n s i n the c o u r s e o f the r u n . T h e  audio f r e q u e n c y tuning p r o c e d u r e s  are  d e s c r i b e d i n A p p e n d i x I.  d) M e a s u r e m e n t s m a d e at e a c h f r e q u e n c y The i n i t i a l o p e r a t i o n c o n s i s t e d of a slow t e m p e r a t u r e  sweep  t h r o u g h the t r a n s i t i o n w i t h the r e f e r e n c e s i g n a l f r o m the " V a r i a b l e o u t p u t s e t at ments.  0°.  The h e a t e r s u p p l y was  Phase"  a l w a y s at 0° d u r i n g m e a s u r e -  On t h i s sweep, the s t e p w i s e input attenuator of the P h a s e  S e n s i t i v e d e t e c t o r (PSD) was  a d j u s t e d a n d t h e v a l u e of "R  c o r r e s p o n d i n g to t h e m a x i m u m v a l u e o f  /l^-t-Q ' 2  was  sample"  found.  (I and  Q  denote t h e d e f l e c t i o n s o f t h e i n - p h a s e a n d q u a d r a t u r e o u t p u t m e t e r s . ) A t the s a m e t i m e i t was  v e r i f i e d t h a t the s i g n a l b e i n g d e t e c t e d v a n i s h e d  on e i t h e r side of the t r a n s i t i o n .  I n a l l m e a s u r e m e n t s , c h a n g e s h a d to  be m a d e v e r y s l o w l y w i t h r e s p e c t to 3 s e c o n d t i m e c o n s t a n t e m p l o y e d i n the  PSD.  88  In subsequent sweeps, the phase of the reference signal (measured as a "lead" at frequency four conditions, I = 0, Q + ve  ;  f was advanced to produce the o Q = 0, I +_ ve ,  at which the  reference leads and the corresponding values of I or Q  were recorded.  These measurements were always made at the temperature for maximum signal, although variations in phase shift observed on traversing the transitions were very small except at the highest frequencies. Observations were recorded on data sheets of the type shown in Fig. 7. , which contains some typical raw data (transcribed for greater legibility).  Amplitudes and phases are given with a precision  corresponding to 1/4 division of the output meter scale of the PSD phase dial of the VPO f  respectively.  and  The reference leads measured at  are seen to differ by multiples of 45, with small errors. The  quantities in parentheses are the amplitudes I and Q, measured on a scale from 0 to 100.  Note that at 3160Hz the "spurious" signal at  temp eratures (R sample = 485 to 520 -JL ) above the transition was just detectable (+2) at a sensitivity five times greater than that used in the subsequent measurements.  e) Other remarks Typically, between 15 and 20 frequencies were examined in a run. The values of R sample for which maximum signals were observed, were found to be controlled to some extent by the temperature of the helium vapour. For this reason, the temperature assigned to a set of heat wave measurements is that of the mid-transition point of the thin  Heater ! volts ' \ ; rras " I  A SAMPLE OF THE RAW DATA  o  o  9  o  t  0  O Cl 1 <H H . fH  0  Z  <+• •J  T3'~--  . -  CO---  <L'' . • •• <H ii  in sL  O II  i a  —  • ft  a  O M ft o  Pre-Amp <Coherent Quad. Pass Amp Jcor.ipt. Band Sens. j  •h  ft  ^ +  <  4  -  <}  Vj  \  «  1  !  *  i  i  1  t  <"<"»  o  i  1 -H  Mi  ^>  O  h  °  ••  v!o  ro  ft  o  «>  ^  rH ft  1  0  i «  o-  W  .  « r CO O  !  ,  X  o  Oo  FIGURE 7  90 f i l m thermometer.  (Differences in mean temperature from end to  end of the samples were calculated to be at most 5 mK i n the worst case.)  91  SECTION II RESULTS  a) Method of Data Reduction The data will be presented in terms of the mean thermal phase lag, A  ,  (ChapterTJI, Section I), and the quantity log  yt>  2 J?  (Chapter III, Section I) in which Z has the nature of a thermal impedance and f is the heat wave frequency. l\  and  Z JP include thermal  effects due to the heater and thermometer films. The results at 3160 given in Fig. 7 will be used to illustrate the calculation of & z  Z  and  Jil as follows.  First, it is necessary to reduce the four measured "reference leads" to the same octant. Examination of the signs of "In-phase compt. " and "Quad compt. " indicates that the heat wave vector lies in the first quadrant and that the reference leads should be reduced to the first octant.  The mean reduced reference lead must now be doubled and  subtracted from an appropriate multiple of 360° to give the heat wave phase lag, uncorrected for electrical phase shifts, AL  - 360 - 2 (41 + 39. 5 + 42 + 42) = 278° 4  At this pass band and frequency, the electronic phase shift is an additional lag of 4°.  The mean thermal phase lag is therefore 274°.  The quantity log  Z  f ( 2 J? * /(1 A <k ^ 1  calculated straightforwardly: the mean value =  ^Y* ^.^^the mean square heater voltage, V  calibrated gain, g, at the settings used is 7. 65.  /ju^i  1  / ^ V ) is 2  is 58,  , is 4.0, and the (A further correction  for the reduction in gain due to excessive r. f. leakage into the preamplifier is also necessary for this particular point.)  b) E r r o r s The r.m. s. deviations, & A  , were obtained from the formula  , given with the tabulated values Jl^, -'>**which 4  J; and M  i  c  are, respectively, the r.m. s. deviations (4 values) of the thermal phase shifts  ^ and the calibrated electrical phase shifts. This procedure  assumes that systematic errors, due to non-linear division of the phase quadrants in the VPO and non-erthogonality of the channels of the PSD have been effectively randomized. Recall that the orthogonality of the 0° and 90° outputs of the VPO, upon which the pha'se measurements rest ultimately, is accurate to within .25°.  E r r o r s in frequency were always negligible. errors to the quantities l°gjQ (Z considered:  In assigning  ,/~f)> the following sources were  the r.m. s. deviations of the amplitudes measured in the  calibration of the gains, g, and in the heat wave measurements, the mean error of the JB6 input attenuator and the errors in measurement of the heater voltage, V, arising from changes in frequency. (Errors  in the absolute value of V were notimportant. )  The m a x i m u m e r r o r s i n the t e m p e r a t u r e s  a r e a s s u m e d to  be e q u a l t o t h e h a l f - w i d t h s o f t h e t r a n s i t i o n s .  c) . C l a s s i f i c a t i o n o f t h e r e s u l t s The  r e s u l t s a r e c l a s s i f i e d i n t e r m s o f the c r y s t a l used, t h e  type o f i n s u l a t i o n u s e d a n d the t h e r m o m e t e r a l l o y used.  The numbers  1 a n d 2 r e f e r t o t h e c r y s t a l s w i t h c y l i n d e r a x e s p a r a l l e l to t h e t r i g o n a l and b i s e c t r i x axes r e s p e c t i v e l y , . M a n d P i n d i c a t e m i c a a n d p h o t o - r e s i s t i n s u l a t i o n , and C and D r e f e r to the l e a d - i n d i u m a l l o y s ( C h a p t e r III, S e c t i o n III).  d) I M C The  results of five e a r l y experimental runs made with the  same heater and t h e r m o m e t e r f i l m s a r e grouped under I M C , using the n o t a t i o n g i v e n above.  M e a s u r e m e n t s made on the same s a m p l e  at h i g h e r f r e q u e n c i e s t h a n t h o s e r e c o r d e d i n T a b l e I I w e r e r e j e c t e d when i t was d i s c o v e r e d that the detection s y s t e m h a d been overloaded.  N o t e t h a t t h e d i s c r e p a n c i e s b e t w e e n p o i n t s m e a s u r e d at t h e s a m e frequency, but o n d i f f e r e n t r u n s , a r e s i g n i f i c a n t . a t t e m p t s w e r e m a d e to o b t a i n c o m p l e t e runs.  Subsequently  sets ofpoints i n single  continuous  94  T A B L E II RESULTS Heat Wave Frequency  IMC  Thermal Phase Lag A  f (Hz)  °  Reduced Amplitude  m  L o  1. 24  g  i  n  z ^  t  140  161. 5  3.0  140  153. 5  3. 5  316  214  3.5  316  214  3. 5  548  266  2. 0  ..57  . 03  548  270  7.0  .66  . 06  548  260  3. 0  . 635  548  260  3. 0  NA  950  346  2.5  -. 195  950  342  2. 0  -. 295  1. 39  .05 .075  .08  .05 .04  The slope of the best s t r a i g h t l i n e through the phase l a g -2 points ( F i g . 8) gives a value 466 c m sec  f o r the d i f f u s i v i t y . The zero  frequency i n t e r c e p t i s v e r y c l o s e to 45°, as p r e d i c t e d (Equation 3.8). A s m a l l a d d i t i o n a l phase l a g ( i . e. an apparent departure f r o m d i f f u s i v e behaviour) i s o b s e r v e d at 950 H z , . The attenuation plot, F i g . 9, i s c l e a r l y a l e s s a c c u r a t e s o u r c e f o r a value of the d i f f u s i t i v i t y  (H  to the square of the slope i n both plots). a l l the points g i v e s a value jf^ =  is inversely proportional  The best s t r a i g h t l i n e through  365 c m sec.  ; f r o m this l o w r e s u l t  and the appearance of F i g . 9, it i s deduced that some e x c e s s attenuation o c c u r r e d at 950 H z . The t h e r m o m e t e r f i l m had a b r o a d t r a n s i t i o n ( a p p r o x i m a t e l y . 3° K a c r o s s ) c e n t e r e d at 4.4° K, w h i c h i s the t e m p e r a t u r e a s s i g n e d to the I M C r e s u l t s .  U s i n g the a p p r o p r i a t e value f o r the s p e c i f i c heat  at 4.4° K, n a m e l y .96 m joule / g m ° K ^ ^ , the t h e r m a l c o n d u c t i v i t y was c a l c u l a t e d to be value, of d i f f u s i v i t y ) .  k^kj-C  -4.4 watts cm-10 K-^ (using the higher  It was concluded, i n view of this r a t h e r low  value, that the t h e r m a l c o n d u c t i v i t y peak l a y at l o w e r  temperatures.  (29) (The "D. C. " r e s u l t s of Shalyt  v  w e r e u s e d to e s t i m a t e the p r o b a b l e  p o s i t i o n and magnitude of the peak.) e) T P D The I P D e x p e r i m e n t r e p r e s e n t e d an attempt to i m p r o v e thin  IMC PHASE LAG PLOT o  \  O  o  CO  \  \ \  o  O  o w  o o  o o  I  \ (Do,  w <  ro M x: eg  O O CvJ  o o H  FIGURE 8  (CH  c/7  IMC ATTENUATION PLOT O  O  O CM  O H  o OJ  O rH bO  o  t-3  o  FIGURE 9  r  98 film response times by using photo-resist insulation, and to operate closer to the thermal conductivity peak by using a different thermometer alloy.  A l l points were obtained in one continuous run except for two  (at 1520 Hz and 2756 Hz) which were obtained after a lapse of 12 hours, during which the sample warmed to room temperature.  The results  are presented in Table III and Figures 10 and 11.  The best straight line (in the least squares sense) through the five lowest frequency points on the phase lag plot (Fig. 10) gives: -2 Diffusivity, K  = 1100 cm sec  , zero frequency intercept = 48^°  An anomaly at about 1500 Hz is noted. Significant deviations from the straight line expected from diffusive propagation are observed above 1700 Hz.  The slope of the attenuation plot (ignoring those points near 1500 Hz which represent an excess attenuation corresponding to the phase lag anomaly) leads to the value for the diffusivity 1160 cm  sec  A temperature of 4. 0 K is assumed for this experiment, for which the thermometer transition ranged from 3. 96 K to 4. 06 K. -3 Taking the value . 49x10  /3«\ j/grrf'K  for the specific heat at 4. 0 K  we find the thermal conductivity to be k =HpC  s  5. 5 watts /cm  K,  which indicates that this experiment (1PD) was performed closer to the  99  T A B L E III RESULTS Heat Wave Frequency  1PD  Thermal Phase Lag  Reduced Amplitude  f (Hz)  140  119. 5  2.75  2. 83  . 17  316  154  2. 5  2. 59  . 11  548  187. 5  1.75  2. 38  . 12  950  227  1. 0  2. 03  . 12  1155  254  2. 0  1.85  . 13  1410  280  3. 0  1. 62  . 13  1410  272  2. 0  1. 66  . 10  1520  308  ' 2. 75  1. 21  . 15  1520  292  1. 0  1. 26  . 12  1600  285  2.25  1. 35  . 13  1730  297  1. 0  1. 27  . 11  1840  304  1. 5  1. 37  . 11  1980  317  1. 5  1. 24  . 11  2209  335  1. 75  1. 28  . 10  2500  356  1. 5  1. 16  . 11  2756  368  2. 5  1. 07  . 11  3160  395  5. 0  .92  . 19  3481  407. 5  4. 5  .85  . 13  160  1PD PHASE LAG PLOT  FIGURE 10  JCI 1PD ATTENUATION. PLOT  102  thermal conductivity peak than was IMC.  The fact that the thermal  conductivity appears to have increased more rapidly than  T suggests  that the thermal conductivity peak for this crystal lay close to 4. 0° K. Subsequent experiments were therefore performed with thermometers made from alloy D.  F r o m the intercept of the attenuation plot (Fig. 11), the specific heat, the dimensions of the sample and resistance of the heater film (940 A ), the voltage sensitivity employed in 1PD  was  i V K.  calculated to be  f) 2MD This run was performed in order to make use of films which had been deposited on rather thick (  ) mica insulation layers  at the time when 1PD was being prepared. The results show an unusually large scatter and are therefore not presented in detail. It appears from the results of the following run that the thermal resistances of the insulating layers used in 2MD  were sufficient to produce a serious  reduction in the apparent diffusivity. (Order of magnitude calculations do not rule out this possibility.)  An approximate straight line fit to the phase lags plotted in 2 Fig. 12 indicates and apparent diffusivity of 1600 cm (corresponding  -1 sec  to a thermal conductivity of just under 8. 0 watts cm  *K  ^00  300 PhaseLa gA°  200 t  M O  100 I  i Ed  0 /? H z  2  104  and a zero frequency intercept of 50°. The temperature is taken to be 4. 0 K.  g) 2PD The results presented under 2PD comprise 19 sets of measurements made in one continuous run. Very thin spin-coated layers of "photo-resist" were used for insulation, in the hope of improving on the 2MD results.  The thermometer transition ranged  from 3.92 K to 4. 02 K, giving a temperature of approximately 3. 97 K for the heat wave experiment.  Amplitude measurements made at the higher of the two preamplifier gains employed had to be adjusted to compensate for the effects of setting the carrier voltage a little too high. This is the explanation of the "adjusted values" of Table IV, which are plotted as open circles in Fig. 14. A careful investigation of these effects showed that no phase shift corrections were necessary.  The phase lag plot (Fig. 13) deviates markedly from a straight lineat:frequencies over 3000 H-*.  The best least squares  z  straight line fit to the values at the 12 lowest frequencies gives: H - 2740 cm  sec , Zero frequency intercept 53 . The best least  squares straight line fit to the 10 unadjusted points on the attenuation . -2 -1 plot (Fig. 14) gives for the diffusivity: f\ = 3500 cm sec  A second run, with a reduced r. f. level, was performed on the same sample in order to check the validity of the amplitude adjust-  105  T A B L E IV RESULTS Heat Wave Frequency f (Hz) 140  2PD  Thermal Phase Lag  Reduced Amplitude  4°  2(6)°  98. 5  1. 5  3. 380  . 090  Log  1 ( )  Z J  316  122  2. 25  3.200  . 080  548  139. 5  1.75  3. 075  . 070  950  168  1. 5  2. 855  . 065  1155  181  1.75  2. 785  . 115  1410  195  1.0  2. 675  . 100  1520  202  1. 75  2. 625  . 080  1730  210  1. 5  2. 650  . 080  1840  218  1. 5  2.520*  . 075  1980  221  2. 0  2. 530  . 065  2209  232  1. 5  2.490  . 085  2500  245  3. 0  2. 370  . 100  2756  261  1. 5  2.365*  . 105  3160  274  2. 0  2. 315*  . 085  3481  290. 5  1.25  2. 175*  . 085  4225  321. 5  3. 0  2.050*  .085  5480  359  2. 25  1. 725*  . 085  5480  354  2. 25  1. 760*  . 075  7000  394  3. 5  1. 450*  . 135  * Adjusted Value  Pha se Lag A °  / oy 2PD ATTENUATION PLOT  FIGURE l\y  108  ments and the ratio of the diffusivity estimates were confirmed (the latter to within 2%). The absolute values of the apparent diffusivities observed in the second run were reduced by almost 10%, perhaps indicating a rapid "aging" of the crystal by thermal cycling. 2 The highest diffusivity observed (3500 cm  -1 sec ) -1  corresponds to an apparent thermal conductivity of 16.8 watts cm -3 -1 (on the assumption of a specific heat of . 49 x . 10  j gm  '-' K  K at 3. 97 K ).  Finally, from the intercept of the attenuation plot, the sample dimensions and the heater resistance of 606 -A , the voltage sensitivity of the detection circuit was estimated to be 6 v K ^.  109  SECTION III  ANALYSIS  a) The apparent diffusivities and thermal conductivities The experiments were not intended to provide accurate values of the thermal conductivities, and did not do so for two reasons. Firstly, as in all experiments of this type, the conductivities have to be deduced by assuming a value for the specific heat, which in this 3 case varies as T . Secondly, the presence of the insulating layers leads, in principle, to low estimates of the diffusivity of the crystals. This point may be clarified by considering the effect on phase shift measurements. As the thickness of the insulating layers is increased it must eventually become necessary to consider the heat waves as propagating through a composite sample having three layers. The apparent diffusivities, K, are deduced from the slopes of the phase lag plots. (The slopes are proportional to (L / f f f the crystal).  ) where L is the length of  At any given frequency, the ratio of the contributions  to the phase lag from the crystal and the insulating films, which are assumed to have equal thicknesses L^ and diffusivities K^, will be ( L / 2 L t j ('J~Hx/hi *) The worst errors are expected from a combination of thick insulating films and high crystal diffusivity, as found in the experiment labelled 2 M D . Order of magnitude calculations suggest that the relatively thick  110  thick mica and Araldite insulation used in 2MD  could have been  responsible for the discrepancies between the diffusivities measured in that experiment and these measured in 2PD.  The insulating  layers used in 2PD were at least one order of magnitude thinner than those used in 2MD. about 20%,  Since the slopes of the phase lag plots differ by  it seems safe to assume that the phase shifts in the insulators  were very small when photoeresist was used.  The conductivities estimated from the 2PD comparable with that found by. Shalyt  (28)  results are  (17. 5 watts cm  -1  K  (-1  )  as the peak conductivity of a crystal of the same orientation and similar resistivity ratio, for which the thermal conductivity peak occurred at 4. 0 K.  The peak conductivity of Shalyt's crystal, which had a radius of 1. 4 mm  compared with 4. 5 mm  for "2", was  apparently  (28)(29) determined by boundary scattering for "2" lay below 4. 0 K.  implying that the peak  Assuming the thermal conductivity to vary  exponentially with temperature at temperatures just above the peak, and assuming the exponent to be (&p /XT  ), as for an ideal dielectric  we obtain 3. 7 K as the lowest estimate of the peak temperature for "2". It is therefore concluded that 1PD and 2PD were performed at temperatures slightly higher than that of the thermal conductivity peak.  Ill  The removal of the discrepancy between the two diffusivity estimates arising from 2PD would require a change of over 10% in the slope of one of the plots Fig. 13, Fig. 14.  It does not seem  possible to justify such a change in view of the close fits obtained.  The lower conductivity in the trigonal direction (1PD)  was  expected, but the ratio of conductivities seems rather large. Crystal "1" had a slightly lower nominal chemical purity, but a higher resistivity ratio than crystal "2".  The photo-resist insulation on "1" was thicker  than that on "2",. although still very much thinner than the mica-Araldite layers used in IMC  and  2MD.  b) Method of analysis of the deviations Expressions for the attenuation constant and wave number, including small correction terms to first order, were derived earlier (Chapter III, Section I):-  . . . 3. 5  The slopes of the phase lag and attenuation plots of Section II are (  /J^  ) and -( & L /f?  ) respectively. The plots are  therefore expected to depart from their straight line form at frequencies such that either (  ) or {/**/to ) becomes "large".  As mentioned  in Chapter III, deviations will also occur at low frequencies if ( C>(.L )  112  becomes too small for Equation 3.4 to be approximated by Equation and the reduced form of Equation 3. 8. and thermometer response effects may  3.7  At high frequencies, heater be important.  On the basis  of the simple model of Chapter III, Section III an apparent increase in attenuation and an additional phase lag 2 u*>  are expected.  None of the plots presented in Section II shows a significant deviation at the lowest frequencies. was always small and  olL  It is therefore concluded that  always large, according to the criteria used.  According to Equation 3. 5, departures from diffusive propagation appearing at high frequencies are expected to take the form of an increase in phase lag (corresponding to a reduction in velocity) and a decrease in attenuation (which would be seen as a deflection upwards in the so-called attenuation plots). thin film response effects may  Therefore, in principle,  be distinguished from true departures  from diffusive propagation by their opposite effects on the attenuation plots.  However, since the attenuation measurements were relatively  poor, we also attempt to discriminate between the possible causes of the phase lag deviation by examining their frequency dependence.  The deviation* , & (&) , are calculated as the differences between the measured phase lags and those predicted from the best straight line fits at low frequencies. a plot of  fety/^Jvc/should  On the response time hypothesis,  be a straight line of slope  ('/A  j,  113  from which the mean response time for the heater and thermometer may be deduced.  Combining equation 3. 5 and the reduced form of 3. 8, neglecting  S>/cj and working to first order in  £JT  , we may  express the measured phase lags A as  ... . 4. 1 where  {ff^Jb^^~Jffi  a n <  ^  phase lag predicted on the  i s  assumption of diffusive propagation. Therefore, on the hypothesis of a departure from diffusive propagation, the initial deviations should be given by £ (A) ~ (^i ~~  (J-^J  • ~c  momentum loss relaxation time, slope,  /TT  , of a plot of  8$)/(A -%) h  An estimate of the relevant , may be obtained from the vs frequency.  (In the calcula-  tions, the measured zero frequency intercept was used instead of -7? ) .  c) 1PD Deviations The effect occurring at about 1500 Hac (see Figs. 10, 11) is obviously not of the type expected - it appears to be a resonance rather than a relaxation time effect-and cannot be analysed by the methods given above. The absence of any corresponding effect in the results of 2PD suggests that the "resonance" may have been a property of the particular crystal rather than of bismuth in general, or of the apparatus.  114  The presence of the attenuation peak and the uncertainties in the amplitude measurements do not permit any definite conclusions to be made concerning the curvature of the attenuation plot at high frequencies.  The initial deviations in the phase lag plot (Fig. 10)  appear to be different in nature from those at higher frequencies, where the phase lag curve tends to become parallel to the low frequency diffusive propagation line.  A Wang Model 320 electronic calculator was used to find the low frequency straight lines, the deviations & (is) , and the best straight line fits to the equations  (modified heat equation)  (Thin film response times)  Neither equation gives a good description of the deviations, as can be seen from the deviation plots Fig.  15, 16.  To illustrate the orders  of magnitudes required for the momentum loss relaxation time, *Z* and  {  c  , the results of fits to the points at 1980, 2209, 2500 and 3160 Hz z  are given:-  1) Best straight line through origin:  T-  2) Best straight line  : T -  3) Lines where slopes differ by one standard deviation from 2  : :  3-<?  \iLt  •>*  T ~~7-& j 3>-^ 4c J3 j a  , ^t ~  3d  ) {c ^ 17 IfL/v ^< h ^  The response time hypothesis gives the poorer fit. Other selections of points give the same orders of magnitude.  A'rVz  lis 1PD DEVIATIONS (A) P l o t t o t e s t m o d i f i e d heat e q u a t i o n h y p o t h e s i s I  1 — —  1  1  o " i o o  o o o  rO  O  o  CM O  rO O  h  7  FIGURE  15  UL 1PD DEVIATIONS (B) P l o t to t e s t t h i n f i l m response time h y p o t h e s i s  o CD  U  F-4  CM  O rH  1  i  oo  vO  o  *  tf CO  O  Jo  •  FIGURE 16  CM'  o  in 1PD PHASE LAG v s . FREQUENCY  ^rOO  S  Si  300  Phase Lag A  c  200  100  0 0  1000  Frequency Hz  2000 FIGURE 17  2500  1*18  Finally, a plot of phase lag vs frequency, Fig. 17, shows that over the frequency range in which the initial high frequency deviations appear in Fig.  10, the phase lags vary linearly with the  frequency and are consistent with acoustic-wave-like propagation with l  a velocity,  LJk>/d/$(rae(J  of roughly .95x10  4  cm sec.  -1  The points  marked with open circles (0) are the phase lags predicted by the best straight line fit to the 5 low frequency points of Fig.  d) 2PD  10.  Deviations  The phase lag deviations are quite pronounced in 2PD (Fig. 13). Once again, after the initial deviations, the phase lag curve turns parallel to the original low frequency line.  In view of the uncertainties  in the amplitude measurements, it is not possible to ascribe a definite curvature to the attenuation plot. -(The adjustment applied to the high frequency points in Fig. bring the points at J?-  14 is the mean of the adjustments required to 3 £"-W Hi'  through the low frequency points. Y  /x  on to the best straight line  In both 1PD and 2PD the  deviations in the attenuation plot predicted from the observed phase lag deviations on either hypothesis are no larger than the uncertainties in the points.  The deviations of the 7 points at the higher frequencies were analyzed according to the two hypotheses, obtained again (Fig.  as for 1PD.  Poor fits were  18, 19), but this time the hypothesis of a failure  n<3  2PD DEVIATIONS (A) P l o t t o t e s t m o d i f i e d heat equation  hypothesis  o o o  vO  X  o o o  X  X X  X X  X  CO  o  vO  o  If  o  CM O  FIGURE 18  o o o CM  2PD  DEVIATIONS  (B)  P l o t t o t e s t t h i n f i l m response time hypothesis  FIGURE  19  t1( 2PD PHASE LAG v s . FREQUENCY kOO  300 1  Phase LagA°  200  100  0  2000  ^000 Frequency Hz FIGURE 20  122  in thin film response gave the better fit. The results were: > / c - 3*ArV*  Tf* £1  1) Best straight line through origin  :  2) Best straight line  :  3) Lines where slopes differ from  :  T" -  2 by one standard deviation  :  {  )-7  c  -loUH>  )  -  A phase lag vs frequency plot (Fig.  » ^  j  £\  A-/-/z  20) is given for the  range of frequencies which includes the initial deviations. Open circles (0) mark the phase lags predicted from the best least squares fit to the 12 low frequency points. Acoustic wave-like propagation with a velocity  L  (t**0  of about 1. 7 x 10  cm sec  is observed at the  frequencies of the initial deviations in Fig. 13, at which the phase lags vary linearly with frequency.  At higher frequencies there is a return  to diffusive propagation, as is also the case for  e)  1PD.  IMC  In the case of IMC,  the slight phase lag deviation is  accompanied by an increase in attenuation, suggesting a response time origin for this effect.  123  SECTION IV  DISCUSSION  The phase lags measured in the experiments 1PD and  2PD  show significant deviations from the behavior expected on the basis of diffusive propagation.  It had been expected that the observed deviations  would be due to the response times of the heater and thermometer or to the initial effects of the term  in an equation of the type (1. 2).  Neither hypothesis fits all the observed high frequency deviations, but Figs. 15, 16, 18, 19 suggest that the deviations are not all of the same type.  We now eliminate the heater/thermometer  response time hypothesis  by consideration of the magnitudes involved.  On the basis of the lumped constant model (Chapter III, Section III)^ a worst-case calculation of the response times may be made by assuming the whole assemblies (rather than just the active areas, (Fig.  l))to respond  and using an upper bound on the thermal resistance  coupling the total thermal capacitance to the crystal. The upper bound on the thermal capacitance of the heater and thermometer assemblies was estimated, from their weights, to be  lO ^ jeu/e /<  At  U'O \< ,  Assuming the thermal conductivity of photo-resist to be as low as that of nylon, and allowing a thickness of 10  M  which is a gross over-  estimate in view of the findings of Chapter III, Section III, one obtains a cut-off frequency  -f  c  ~7i"A/V2 . Roughly the same result is obtained  124  by assuming a boundary resistance due to acoustic mismatch as high as that for a liquid helium-copper interface.  Thus the cut-off  frequencies required to explain the deviations on this hypothesis fall below the pessimistic lower limit calculated here.  Furthermore, the  thermal resistances necessary to achieve the lower limit would reduce the apparent thermal conductivities of the crystals to values of the order . 1 watts cm  K \  This explanation of the deviations is there-  fore rejected.  The phase lag vs frequency plots, Figs. 17, 20, suggest that the initial deviations correspond to the onset of a wave-like mode of propagations (with velocity V ), which subsequently breaks down at the highest frequencies examined. The values of H  and V* for each  sample were obtained from different ranges of frequency independently of any assumptions about possible corrections to Equation 1.1.  If  the mean values of hi for each sample?;are taken it may be seen that, to within experimental error, the velocities V  scale as  This  supports the view that the apparent wave-like mode of propagation reflects a thermal property of the crystals.  The calculation of the T t  from the initial deviation plots  is based on the assumption of the modified heat equation (1. 2) and the approximation that terms in ( CSV ) are kept only to first order.  125  Also on the assumption of Equation 1. 2, but in the high frequency limit, the wave velocity V" is given two of the quantities two methods by which  Obviously only  V", H, 7T maybe obtained independently. The may be obtained are not exactly equivalent  and it is not strictly consistent to apply both to the same range of frequencies.  However, if we classify only those points which are  good fits to straight lines in Figs. 17, 20 as giving "initial" deviations, and re-examine the deviation plots, Figs. 15, 18 considering only such points, then the following results are obtained.  For 1PD, Fig. 15, we have only the three points at 1980, 2209 and 2500 H*, the best line through which has a slope corresponding z  to  / l ^*iu< . F o r 2PD, Fig. 18, we have the four points at  2756, 3160, 348.1 and 4225 H^, the best line through which has a slope corresponding to "77 = 10.5 values of  . These values of ~T > the mean  determined at low frequencies and the values of V"  found from Figs. 17, 20 do in fact satisfy the formula  within  experimental error for both samples. On the basis of (4. 1)>, one expects the deviation plots to indicate zero deviation at zero frequency.  (The deviations predicted  at the frequencies used to determine the low frequency lines should be smaller than the errors in measurement.)  In fact the deviations appear  to commence at threshold frequencies of roughly 1520 and 1750 H » for z i P D and 2PD respectively.  126  It is noted that at the highest frequencies at which signals were detected (when propagation was again apparently diffusive); the velocity  did not exceed the wave velocity  Although the observed initial deviations were broadly consistent with the modified heat equation (1. 2), the values of V  and T"  are quite different from those predicted by any of the microscopic theories discussed in Chapter I.  At temperatures close to the thermal conductivity peak, the predicted wave-like mode is the so-called "second-sound", as observed in solid h e l i u m ' ^ .  The optimum temperature for second  sound propagation is just below that of the thermal conductivity peak, and the frequency range within which second sound may be observed (181  decreases rapidly away from the optimum temperature  .  The  predicted and observed velocity of second sound is solids is M where  /Jj  U. is an appropriate average sound velocity, but the velocities,  V , calculated above are an order of magnitude lower than the sound  (39) velocities in bismuth  .  It is also expected that the  "7* in equation  1.2 (or a more sophisticated version of 1. 2), will be that obtained from the thermal conductivity via the kinetic formula: k = ^ C\, u~ ~Z The values of T" obtained above are roughly 50 times larger than the values corresponding to the apparent thermal conductivities. It appears that, if a wave-like mode of propagation has been detected, it cannot involve the entire phonon system as does second  127  sound in liquid and solid helium.  In view of the small amount of  data obtained on the deviations, it is difficult to offer more than speculations as to their physical origins. The following points are noted.  The relationship V"the ratio  J~V{fi*/x~ '  determining  h(  and  implies that V  where 1%  • We  depends on  is the effective relaxation time expect that ~^g(f will be obtained  by adding reciprocal relaxation times, ~£~ef( * dominated by the Umklapp process time  T^  T.  , and will be If all relaxation  T times except^are combined into  ~Vu  , the expression for V  becomes  The long mean free paths implied by the observed values of T  might be found in the long wave-length longitudinal part of the  phonon distribution.  This is also the only part of the distribution  expected to interact appreciable with the charge carriers at 4. 0 K (see Chapter II) . -  The observation of large phonon drag effects in (46)  bismuth between 3. 5 and 4. 0 K  implies long mean free paths for  long wave-length longitudinal phonons. It is usually held that one  may  calculate the effective thermal conductivity of a combined system of electrons and phonons by adding the conductivities calculated for each (44)  (45)  subsystem on the assumptions that the other is in equilibrium i.e. phonon drag effects do not affect the thermal conductivity. As pointed out in Chapter II, the above arguments may  not be valid in a  was  128  time dependent heat propagation experiment. The thermal conductivitytensor  /c is measured under conditions of zero electrical current,  but these conditions do not apply in heat wave experiments, where alternating thermoelectric currents may be generated and phonon drag effects could be reflected in the heat waves.  If the  T  calculated from the deviations is an electro-  phonon scattering time, then the velocity same ratio,  ~Z~u/(~Tt, l-T*zp)  V" will depend upon the  as does the phonon drag contribution to  (44) thermo-electric power.  It would therefore be of interest to  extend these measurements to cover a range of temperatures between 3. 5 and 4. 0 K in order to look for a correlation between the apparent wave-like heat propagation and the thermo-electric effects.  129  CHAPTER V  CONCLUSIONS  In Chapter I it was argued that an equation of the general form  W  +  )&  "  K  . . . 1.2  will describe the initial deviations from diffusive behaviour observed in a heat wave experiment performed in either the boundary scattering (or ballistic) regime or the second sound regime. The discussion concerning the use of (1. 2) in the boundary scattering regime has been  (7) published previously  .  It was shown in Chapter II that bismuth, on  account of its dielectric-like thermal conductivity peak, its isotopic purity and the high degrees of chemical purity attainable, and in spite of the charge carriers, might be a suitable system for the propagation of second sound. The alternative possibility, that the charge carriers might play a more important role in heat wave experiments than in D. C. thermal conductivity experiments, was considered. A method for performing "audio" frequency heat wave experiments on electrically conducting specimens has been developed.  130  Accurate phase lag measurements and somewhat less accurate attenuation measurements have been made on heat waves in two bismuth single crystals at frequencies up to 4. 0 K.  "7ArV?  a  n  (  j  temperatures close to  An electrical transmission line analogue and a modified heat  equation of the Vermotte type were used as the basis for the analysis of the results. The low frequency results were in excellent agreement with predictions based on the transmission time model and the un- ' modified equation for (diffusive) thermal propagation. These low frequency results were used to obtain values for the thermal diffusivities and to show experimentally that the thermal leakage conductance terms in the transmission line model were negligible.  Deviations from diffusive propagation were observed at some of the higher frequencies employed. Evidence has been presented to show that the observed deviations did in fact reflect thermal properties of the crystals.  The initial deviations implied the onset of a wave-like  mode of propagation which broke down at the highest frequencies examined. This is qualitatively the behaviour expected for second sound in a crystal with a small frequency "window"  it was  deduced that the experiments had not been carried out at the optimum temperature for second sound propagation.  • In spite of the qualitative agreement with predictions, the values of the wave velocities, and the relaxation times deduced on the  131  basis of the modified heat equation, were not consistent with the values predicted from the measured diffusivity values on the basis of any of the microscopic theories of Chapter I. It is suggested that the numerical discrepancies rule out second sound propagation as predicted for the ideal dielectrics discussed in Chapter I, as the explanation of the deviations. A possible explanation of the deviations is that they reflect the presence of the electrical charge carriers, specifically via the phonon drag effect.  The quantity of data analyzed was rather small, reflecting the time spent in developing the experiment and the time required to make the measurements. At the stage at which useful data was obtained the two available single crystals had been subjected to considerable thermal cycling and handling in the course of preliminary experiments. The results obtained thus far indicate that any further experiments should be carried out (with new single crystals) in the temperature range 3. 5 - 4. 0 K in order to examine the deviations closer to the thermal conductivity peak and to look for correlations between the deviations and the phonon drag e f f e c t s ^ ^ .  One might also apply magnetic fields  in order to resolve the contributions to the observed deviations from the charge carriers. Finally, it still appears possible that, with a ten fold increase in heat wave frequency one might be able to observe deviations in quantitative agreement with the prediction of the microscopic theories.  132  BIBLIOGRAPHY  1.  Vernotte,  P . Compt. Rend. , 246 (1958), 3154  2.  Cattaneoy* C . Compt. R e n d . , 247 (1958), 431  3.  Weymann, H . D. , A m . J . P h y s . ,  4.  Kelly,  5.  B r o w n , . J . B . , Chung, D . Y . and Matthews, L e t t e r s , 21_, (1966), 241  6.  Matthews,. P . W . ,  7.  Brown,  8.  Chung, D . Y . , P h . D . T h e s i s (1966), U n i v e r s i t y of B r i t i s h Columbia  9.  Chester,  D . C , A m . J. Phys.,  35_(1967), 488  .36 (1968), 585  Can. J . Physics,  C . R. , P h y s i c a ,  P . W . Physics  45 (1967) 323  35 (1967) 114  M . P h y s . R e v . , 131 (1963), 2013  10.  W a r d , J . C . and W i l k s ,  J . , P h i l . M a g . _43 (1952), 48  11.  Prohofsky, 1403  12.  G u y e r , R. A . and K r u m h a n s l , J . A . , P h y s . R e v . , 133 (1964) 1411  13.  Guyer, R. A . and K r u m h a n s l , J . A . , P h y s . R e v . , 148 (1966)  E . W. and K r u m h a n s l , J . A . , P h y s . R e v . , 133 (1964)  766 14.  K r u m h a n s l , J . A . , P r o c . P h y s . S o c . ( L o n d . )_85 (1965), 921  15.  Gotze, W. and M i c h e l , K . H . , P h y s . R e v . 1_56 (1967), 963  1.6.  Kwok, P . C . and M a r t i n , P . C . , P h y s . R e v . 142 (1966), 495  17.  Sham, L . J . , P h y s . R e v . , 1_56 (1967), 494  18.  G u y e r , R . A . and K r u m h a n s l , J . A . , P h y s . R e v . , 148 (1966), 778  19.  M e z o v - D e g l i n , L . P . , Soviet P h y s i c s J . E . T . P . 22_(1966), 47  20.  A ck nv , .C e r t m a n , B789 . , F a i r b a n k , H . A . and Guyer, R. A . , Ph y se .r mRa e L .eC t t.. ,, B16_(1966),  133  21.  von Gutfeld, R. J. , and Nether cot, A. H., Phys. Rev. Lett. 17_(1966), 868  22.  Ackerman, C. C. , and Guyer, R. A. , (Preprint of article cited as (5) in Ref. (23) )  23.  Ackerman, C. C., and Guyer, R. A. , Solid State Commun. (U.S.A.) 5J1967), 671  24.  von Gutfeld, R.J. and Nether cot, A. H. , Phys. Rev. Lett., (1964),  641  25.  von Gutfeld, R.J. and Nether cot, A. H. , Proc. 9th Int. Conf. on Low Temp. Phys., Columbus, (1964) (New York, Plenum Press, 1965) Pt. B, 1189  26.  Thacher, P., Phys. Rev. , 1_5_6 (1967), 975  27.  Johnson, D. L. , M. Sc. Thesis, University of British Columbia, (1967)  28.  Shalyt, S. , J. Phys. (U. S. S. R.) 8_( 1944), 315  29.  White, G. K. and Woods, S. B. , Can. J. Phys. , 3_3 (1955), 58  30.  Bhagat, S. M. and Manchon, D. D. , Phys. Rev., 164 (1967), 966  31.  Casimir, H. B. G., Physica 5_ (1938), 495  32.  Ziman,. J. M. ,. Phil. Mag., 4_5_(1954), 100  33.  Kennard, E. H., Kinetic Theory of Gases (McGraw Hill, New York, 1938) Chap. 8  34.  Jain,. A. L. and Koenig, S. H. , Phys. Rev. 127 (1962), 442  35.  Boyle,. W. S. and Smith, G. E. , Prog, in Semiconductors 7 (1963), 1  36.  Garcia, N. and Kao, Y. J. , Physics Letters 26 (1968), 373  37.  Friedman, A. N. , Phys. Rev. 159 (1967),  38.  W. A. D. D. Tech. Rept. 60-56 Part II (V. J. Johnson, Ed.)  39.  de Bretteville, A. Cohen, E. R. , Ballato, A. D. Greenberg, I.N. and Epstein, S. , Phys. Rev. 148 (1966), 575  134 40.  Steele, . M. C. and Babishkin, J. , . Phys. Rev. 98 (1955), 359  41.  Callen, H.B.,  42.  Drabble, J.R. and Goldsmid, H.J. "Thermal Conduction in Semiconductors" (Pergamon Press, Oxford, 1961)  43.  Sondheimer, E. H. , Proc. Roy. Soc. 234 (1956), 391  44.  Sondheimer, E. H. , Can. J. Phys. 34 (1956), 1246  45.  Hanna, I.I. and Sondheimer, E.H.-,-Proc. Roy. Soc. 239 (1957), 247  46.  Kuznetsov, . M. E. and Shalyt, S. S. , Sov. Phys. J. E. T. P. Lett. 6^ (1967), 217  47.  Ziman, J. M. , Phil. Mag. 1_(1956), 191  48.  Ziman, J. M. , Phil. Mag. 2_(1957), 292  49.  Ziman, J. M. , "Electrons and Phonons" (Oxford University Press, London, i960)  50.  Jain, A. L. and Jaggi, R. , Phys. Rev. 135 (1964), 708  51.  Smith, D. , Los Alamos Scientific Lab. Rept. L A 3773 (1967) (Unpublished)  52.  Lopez, A. A., Phys. Rev. r75_(1968), 823  53.  Kwok, P. C., Phys. Rev. 17_5 (1968), 1208  54.  Osborne, D. V. , . Phil. Mag. 47_(1956), 301  55.  Ulbrich, . C. W. , Phys. Rev. _12_3(196l), 2001  56.  Cannon, . W. C. and Chester, M. , Rev. Sci. Inst. 38_(1967), 318  57.  Adler, J. G. and Ng, S. C. , Can. J. Phys. 43 (1965), 594  58.  Walker, E. J., Rev. Sci. Inst. 30 (1959), 834  Phys. Rev. 7_3 (1948) 1349  135  APPENDIX I Audio frequency tuning and calibration procedures  The audio frequency tuning was performed as follows (all frequencies being set to within 1Hz by means of the General Radio type 1191 counter). A reference signal and a dummy heat wave signal, both at the chosen heat wave frequency 2f^, were supplied directly from the variable phase oscillator (VPO) to the JB6 phase-sensitive detector (PSD), which was then tuned to receive 2f when operating in o its "Select External" mode. (Select External is a mode in which a reference signal of low amplitude and/or imperfect form is brought up to standard by means of a high "Q" tuned amplifier. ) Tuning the PSD comprises setting the electrical zeros of the output meters, tuning reference and signal channels and orthogonalizing and equalizing the gains of the two output channels. It has already been noted that the signal channel could not always be peaked. The manufacturer's scheme for adjusting the orthogonality was not found adequate, as errors of 4 ° or so were common. These errors could be corrected by means of the accurately orthogonal outputs of the VPO. Phases were established with respect to a reference signal at 0° by setting a null on one output meter while using as large a signal  136  as permitted by the other. Denoting the outputs of the in-phase and quadrature meters by I and Q, the two (equivalent) phase conventions used were: 1=0, Q + ve for a +90  o  o signal; Q = 0, I + ve for a 0 signal.  When the PSD had been tuned to 2f , a second harmonic o generator (diode plus load resistor) was inserted into the reference line and the VPO was tuned to f by peaking the reference signal output at the PSD.  The quality of the resulting reference signal and the value  of f obtained were checked by oscilloscope and frequency counter o respectively. The CR4 pre-amplifier was then inserted into the signal line and adjusted to the required pass band. The phase shift and gain calibrations were obtained by observing the effect on a dummy heat wave signal of inserting the CR4 pre-amplifier between the PSD and the VPO.  The phase shifts, which were  measured by the method used in the actual heat wave measurements, typically showed r.m. s. deviations of 3/4° over 4 readings. The accuracy of the gain calibration was determined by the linearity of the input attenuator of the PSD; the worst error in the ratio of two settings was 5% and the mean error was 2%.  

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