{"Affiliation":[{"label":"Affiliation","value":"Science, Faculty of","attrs":{"lang":"en","ns":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","classmap":"vivo:EducationalProcess","property":"vivo:departmentOrSchool"},"iri":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","explain":"VIVO-ISF Ontology V1.6 Property; The department or school name within institution; Not intended to be an institution name."},{"label":"Affiliation","value":"Physics and Astronomy, Department of","attrs":{"lang":"en","ns":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","classmap":"vivo:EducationalProcess","property":"vivo:departmentOrSchool"},"iri":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","explain":"VIVO-ISF Ontology V1.6 Property; The department or school name within institution; Not intended to be an institution name."}],"AggregatedSourceRepository":[{"label":"AggregatedSourceRepository","value":"DSpace","attrs":{"lang":"en","ns":"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider","classmap":"ore:Aggregation","property":"edm:dataProvider"},"iri":"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider","explain":"A Europeana Data Model Property; The name or identifier of the organization who contributes data indirectly to an aggregation service (e.g. Europeana)"}],"Campus":[{"label":"Campus","value":"UBCV","attrs":{"lang":"en","ns":"https:\/\/open.library.ubc.ca\/terms#degreeCampus","classmap":"oc:ThesisDescription","property":"oc:degreeCampus"},"iri":"https:\/\/open.library.ubc.ca\/terms#degreeCampus","explain":"UBC Open Collections Metadata Components; Local Field; Identifies the name of the campus from which the graduate completed their degree."}],"Creator":[{"label":"Creator","value":"Brown, Christopher Richard","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/creator","classmap":"dpla:SourceResource","property":"dcterms:creator"},"iri":"http:\/\/purl.org\/dc\/terms\/creator","explain":"A Dublin Core Terms Property; An entity primarily responsible for making the resource.; Examples of a Contributor include a person, an organization, or a service."}],"DateAvailable":[{"label":"DateAvailable","value":"2011-06-17T19:52:35Z","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/issued","classmap":"edm:WebResource","property":"dcterms:issued"},"iri":"http:\/\/purl.org\/dc\/terms\/issued","explain":"A Dublin Core Terms Property; Date of formal issuance (e.g., publication) of the resource."}],"DateIssued":[{"label":"DateIssued","value":"1969","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/issued","classmap":"oc:SourceResource","property":"dcterms:issued"},"iri":"http:\/\/purl.org\/dc\/terms\/issued","explain":"A Dublin Core Terms Property; Date of formal issuance (e.g., publication) of the resource."}],"Degree":[{"label":"Degree","value":"Doctor of Philosophy - PhD","attrs":{"lang":"en","ns":"http:\/\/vivoweb.org\/ontology\/core#relatedDegree","classmap":"vivo:ThesisDegree","property":"vivo:relatedDegree"},"iri":"http:\/\/vivoweb.org\/ontology\/core#relatedDegree","explain":"VIVO-ISF Ontology V1.6 Property; The thesis degree; Extended Property specified by UBC, as per https:\/\/wiki.duraspace.org\/display\/VIVO\/Ontology+Editor%27s+Guide"}],"DegreeGrantor":[{"label":"DegreeGrantor","value":"University of British Columbia","attrs":{"lang":"en","ns":"https:\/\/open.library.ubc.ca\/terms#degreeGrantor","classmap":"oc:ThesisDescription","property":"oc:degreeGrantor"},"iri":"https:\/\/open.library.ubc.ca\/terms#degreeGrantor","explain":"UBC Open Collections Metadata Components; Local Field; Indicates the institution where thesis was granted."}],"Description":[{"label":"Description","value":"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.\r\nA 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: [formula omitted]\r\nwith appropriate identifications of the relaxation time. The possibility\r\nthat the small numbers of charge carriers present in bismuth might lead to different forms of deviation is explored.\r\nSeveral 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 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\u00b0, using the phase-sensitive detector as a null detector. \r\nThe measurements were analyzed in terms of a thermal transmission line model based on the modified heat equation given above. The electrical analogue of \u03c4 in such a model is L\/R. A thermal leakage conductance term \u2a4b(electrical analogue G\/C) was included in the model. \r\nThe 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 \u2a4b\/\u03c9 was shown to be small at all frequencies used. \r\nPhase 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 to show that the observed deviations reflected thermal properties of the bismuth crystals rather than properties of the thin films, or spurious electrical effects. \r\nThe 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.","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/description","classmap":"dpla:SourceResource","property":"dcterms:description"},"iri":"http:\/\/purl.org\/dc\/terms\/description","explain":"A Dublin Core Terms Property; An account of the resource.; Description may include but is not limited to: an abstract, a table of contents, a graphical representation, or a free-text account of the resource."}],"DigitalResourceOriginalRecord":[{"label":"DigitalResourceOriginalRecord","value":"https:\/\/circle.library.ubc.ca\/rest\/handle\/2429\/35570?expand=metadata","attrs":{"lang":"en","ns":"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO","classmap":"ore:Aggregation","property":"edm:aggregatedCHO"},"iri":"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO","explain":"A Europeana Data Model Property; The identifier of the source object, e.g. the Mona Lisa itself. This could be a full linked open date URI or an internal identifier"}],"FullText":[{"label":"FullText","value":"T H E R M A L W A V E P R O P A G A T I O N 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 C H R I S T O P H E R R I C H A R D B R O W N B . Sc. , University of Exeter, 1963 M . S c . , University of B r i t i s h Columbia, 1965 A THESIS 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 O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F D O C T O R O F P H I L O S O P H Y in the Department of Physics We accept this thesis as conforming to the requited standard. 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 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the r e q u i r e m e n t s f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I a g r e e t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and Study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department of The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada Date 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 i i i 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\u00b0, 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. A thermal leakage conductance term\/ 1 (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 reason-able 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. I l k . \\ l I\u00abM T A B L E O F C O N T E N T S . C H A P T E R Page I Introduction 1 A Review of Some Theories 1 Some E a r l i e r T h e r m a l Propagation Experiments 10 Heat Waves in the Boundary Scattering Regime 17 Summary 24 II Bismuth 25 Some Properties of B i s m u t h . . 25 Heat Conduction in Bismuth. 31 The Results of E a r l i e r T h e r m a l Conductivity-Experiments in Bismuth 38 The Electron-Phonon Interaction in Bismuth. 42 III Experimental Design and Apparatus. 47 The T r a n s m i s s i o n Line Model 47 The Crystals 55 Heaters, Thermometers and Insulators 57 Input and Detection Systems 70 The Cryostat Systems 78 IV Experimental Procedures, Results and A n a l y s i s . . 85 Experimental Procedures . 85\" Results 91 Analysis 109 Discussion 123 V Conclusions 129 Bibliography 132 Appendix 1 135 vi LIST O F T A B L E S T A B L E Page I Values of Parameters Controlling E l e c t r o n -Phonon Scattering 45 II Results I M C 94 III Results 1PD 99 IV Results 2 P D 105 vii LIST OF 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 ' 82 7 A Sample of the Raw Data 89 8 IMC Phase Lag Plot 96 9 IMC Attenuation Plot 97 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. I am indebted to Prof. R. E. Burgess and Mr. 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 It . . . ( i . i) 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 \"real\" gases or in the rare-fied 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 Vernotte^, seeking to amend (1. 1) to deal with thermal pulse propagation in gases, constructed a phenomenological equation with the required hyperbolic form: -r + -Al- = k v 2 T . ... ( i . 2 , 3 t < j t Which gives 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 experiments^ 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 sound\"^^ 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 ) , say, whose 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 when 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 equation w h i c h i s to be l i n e a r i z e d i n the d e v i a t i o n s of the l o c a l d i s t r i b u t i o n f u n c t i o n f r o m i t s e q u i l i b r i u m v a l u e i s \u2022 4f ( \u2014 ' t ) + V . \\ - 7 n ( q , r , 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 f o r m s a s s u m e d f o r C. C o n s i d e r a b l e use i s made of the e x p a n s i o n Tex\/, 7 7 = {a _ Ul-H A , x ; - (\u00a3*\/knt)tn\/%)+ 4 sih^fo\/zkoi:) \u2022 6 . . . (1.5) where f = 1\/ (exp ( fk T ) - 1 ) i s the e q u i l i b r i u m o c c u p a t i o n p r o b a b i l i t y , \/I i s the l o c a l d r i f t v e l o c i t y and T^ i s the l o c a l t e m p e r a t u r e d e v i a t i o n . P r o h o f s k y and K r u m h a n s l ^ ^ use the e x p a n s i o n ( 1 . 5) i n w r i t i n g down moment equations f o r number, e n e r g y and momentum. The c o l l i s i o n t e r m s i n t h e s e equations 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 the n u mber and m o m e n t u m equations r e s p e c t i v e l y . ( T and ~X are the relaxation times for normal and momentum-N u 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 e) \" \"^N ^ 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 T u < 2 \u2014 \\ and i w 7 ^ , 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 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 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 two of them are closely related to the leading terms two mentioned above, so that the drift operator D* <3 + v. ^ 7 37 the collision operators R and N are reduced to 3 x 3 matrices. and The solution of the Boltzmann equation is thus reduced to the 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 Cv) -1. 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-equili-brium 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: D Q ( GWT) is the Green's function, the poles of which give the natural modes of the phonon system. The definitions of the other quantities are: CtX%1- O x -ii>\/T, + C ^ L ~ u x - its \/Ti 2 = c 2 (1 + 3r\/2) c* = c X(l-3r\/2)\/3 , r = (Cp\/C )-l ' 4 ((1\/T f t) + T v ^ q1 ) , l \/ t 2 = (1 - 9r\/4) x ( (1\/Ta)+ T V ClfX) c is the speed of a phonon, 9 c_ and describe the second sound mode and correspond to a temperature equation of the form 1 9 r $T 9r. . \u2014 . 2 r-yZ m (1 - r > n - (1 - (1 - T * 1 N } C 2 TP R . . . (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. 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. To summarize, equation (1.8) is taken to be a good 1 0 SECTION II SOME EARLIER THERMAL PROPAGATION EXPERIMENTS 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 <\" 1 \/ \"JJT \u2022 An interesting result is obtained ( 1 9 ) 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, \/\"> - ( 5 R \/4 JW ^d. ) and the function G takes values from 0 to 1. Because the requirements for the observation of Poiseiulle flow, >Sfs<<&. , ^^\/iX\/i^ft , are more demanding than those 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 sound^^ 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 \u00a9ft being obtained by growing crystals at different pressures. 12 (23) An analysis 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 NaF^^, 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 temper-atures 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 thinAbolometers at 3. 8 and 8. 8 K were performed by (25) von Gutfeld and Nethercot on single crystals of NaCl, KC1,. Kl, and KBr. (The lower temperature was in the ballistic region). All 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 LiF . From this point of view, NaF and Csl are more attractive. However, D. C. thermal conductivity (27) results for good single crystals of Csl (\"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 Csl 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 approxi-mations 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 ) ( 7 ) ment is required ' . 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 temper-atures. 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 Fermi 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 HEAT WAVES IN THE BOUNDARY SCATTERING 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 q and which defines the boundary temperature. 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 distri-bution 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.>\u00a3> ^ ^ e the distribution at some interior 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 ( q, r, t) + n ( q, r, t), where n is small compared with n . n, is related to the net phonon flux in the o 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\/ \/ u ) (U q is the speed of the phonons). 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) \u00bb(h r>t)5 *\u00abfash \" i - ^ fatrt- \/v r\/<>> & # \u2014 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\u00b1(*\u00a30 M&=UL sjjg..^ K.c*>a6\/rt-r\/M\u2014 + terms in cos \u00a3 Such an equation, with appropriate , and cos , may be written 2 0 for each q . Two averages, x and x are now defined, x ( \"2-1 J* '
- ri \/ jzl If the boundary temperature has the form T + T^ (z, t) where JT \/' \u00ab 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, i n Cos Wfrrl = A L\/zJ ?\u00ab['P-W 1 L^+P1- 11 3. and TrTZTJ =JLLJ *tt* \/>c*s t e l since the solid angle e^Jl subtended at a point z -o^fizG j b y the area . ) 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 THE RESULTS OF EARLIER THERMAL CONDUCTIVITY EXPERIMENTS ON BISMUTH The first measurements of the thermal conductivity of ( Z8\\ bismuth at liquid helium temperatures were made by Shalyt , 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 (29) Some results obtained by White and Woods from poly-c r y s t a l l i n e samples support Shalyt's conclusions. The purer sample examined exhibited the exponential temperature dependence expected on the high temperature side of the thermal conductivity peak i f U processes are dominant. Peak conductivities o c c u r r e d at about 5 K and were apparently determined by boundary scattering i n the individual c r y s t a l l i t e s . (40) Steele and Babishkin observed a sm a l l o s c i l l a t o r y dependence of ktl on tranverse magnetic fields for f i e l d strengths up to 13 kOe. The observations were made on a 99.99% pure single c r y s t a l of 1.7 mm diameter at 1. 6 K. The decrease i n the mean conductivity was 4% at the higher fields. It i s interesting to compare Shalyt's value for k\u00b1 at 2. 3 K with that predicted f rom Steele and Babishkin's value of J<|\/ at 1.6 K on the assumption that boundary scattering l i m i t e d the mean free paths (i . e . we scale the value as RT The result is k ^- kn which implies that ku is at least p a r t i a l l y controlled by some other mechanism. Detailed investigations of the o s c i l l a t o r y effects have been made recently by Bhagat and Manchon^^, who used 99. 9999% pure single c r y s t a l s with r e s i s t i v i t y ratios i n the range 300-400. Thermal conductivities are quoted for temperatures between 1. 3 and 2. 0 K. The zero f i e l d measurements again indicated k \u00b1 to be about twice as large as Ai\/ and, i n addition, klf was observed to have a more ra p i d 3 temperature variation than T . One cannot ascribe the different 40 magnitudes of hu 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 wave-vectors of the electrons and typical phonons at these temperatures have comparable magnitudes and this fact, together with the extreme aniso-tropy of the Fermi 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 Fermi surface will be involved in scattering with the typical phonons. The results were fitted to the formula. _ \/ _ _ _ \u00a3 _ , _\u00b1_ in which a and b measure the relative strengths of boundary scattering 41 The best values of a and b were roughly equal for klt 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 THE 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 semi-conductors. 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. The scattering potential is then just the shift in the energy E(k) of the electron states which is produced by the nearly uniform strain corresponding to a long wave-length phonon when the electron density adjusts itself to maintain 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. charge neutrality. The deformation potential, 43 On the assumption,? that the electrons are in equilibrium the mean free path A .^ of a phonon of wave-vector q is , in the (49) deformation potential approximation 4 , where jo is the mass density, n the speed of sound and ^ the phonon occupation number. 0 acts like a & -function. It wi l l be assumed that this expression is valid for bismuth. The conditions of low carr ier density and long wavelengths are satis-fied, but the energy surfaces are far f r o m spherical . The deformation potentials for the two types of carr ier are of opposite sign but of roughly equal magnitude, namely.2. % eV B y use of the properties of the & -functions, the expression for may be reduced to in which the m i n i m u m value of k which contributes to the integral is k0 - l(4\/L)-rftl\/fil- Ziman (considering a spherical F e r m i surface) points out that the strength and temperature dependence of the scattering depends on the relative magnitudes of and the F e r m i momentum, fakp . Three cases are considered: a) when A\/rf is very much larger than >^\/r8 , so that the lower l imi t on the integral may be taken to be zero, to obtain 44 J_ _ (r\u00bb*) \u00a3J<<\\) , A cr-b) when is smaller than \/n*** there is little scattering except at temperatures for which the dominant wave vector ^ gives Uj*& c) when \u00a3A\"p is greater than case a) applies at low temperatures ( 0 ) , but with r is ing temperatures and hence r is ing <^ , the scattering rate r ises to a maximum at the temperature -2. 5 for which k is zero and then falls off as T . Detailed c a l c u l -o lations are given in Ref. (47) and normalized curves illustrating case c) are given in Refs. (47) and (49). In order to extend the preceding analysis to the problem at hand, a comparison is made between the various values ^ ^ f of the semi-axes of the ellipsoids in bismuth (F ig . 0(b) ) and the corresponding values of ^ ka . Thus only certain areas of the F e r m i surface are considered. . The effective masses are obtained by inverting the tensors whose components are given in Section I. The room temperature l o n g i -tudinal sound v e l o c i t i e s ' ^ are used. A value of q given by <\u00a3= Z'S^RT\/tt u is considered, this being the value corresponding to the peak in the energy density vs . wave-number curve on a simple Debye model, and the temperature chosen is 4 K . The magnitudes of various quantities of interest (see the expression 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 maximum. TABLE I VALUES OF PARAMETERS CONTROLLING ELECTRON-PHONON SCATTERING , 22 22 22 Crystal Axis Carrier mux 10 hqxlO _ * 2hk FxlO T K M Trigonal Holes 1. 66 11 90 4. 8 Electrons . 04 11 15. 8 . 8 Bisectrix Holes . 154 61 28 1. 8 Electrons 3. 85 61 158 11 Binary Holes . 154 61 28 1. 8 Electrons . 019 61 10. 8 . 7 From the tabulated values of T it appears that scattering m r c \u00b0 by holes with wave-vectors in the trigonal direction reaches its maximum in the region of the thermal conductivity peak. In view of Bhagat and ( 3 0 ) Manchon's observations that k for their best specimens (of typical dimensions 3mm) 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 electron-phonon 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 CHAPTER III EXPERIMENTAL DESIGN AND APPARATUS SECTION I THE TRANSMISSION LINE MODEL 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. (The area of the curved surface was 7 5 % of the total surface area). 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 ther-mometer. The inconsistency is not serious for the following reasons. 49 F i r s t l y , the thermal resistance to radial flow is expected to be dominated by the bismuth-helium gas boundary resistance, leaving small radial temperature gradients within the sample. Secondly the active areas of the heater and thermometer films extended over only 25% of the end face areas, and were centrally placed. A n equation for the heat current Q (x, t) and excess temperature & (x, t) measured with respect to the temperature of the surrounding gas, is now derived on the basis of the above assumptions. Since the modified heat equation is assumed, the appropriate equationffor energy conservation and heat current are, J>Cv& _ _ & - ... 3.1 (a) + T ^ i = _ UA 2L. . . . 3. I (b) in which it has been assumed that the heat leak per unit length (x, t) is a linear function of the excess temperature: Q =