H E A T PULSES IN A l ^ SINGLE CRYSTALS A T LOW T E M P E R A T U R E S by DAVID YIH CHUNG © . S c . , National Taiwan University, 1958 M . Sc. , University of British Columbia, 1962 A THESIS SUBMITTED IN PARTIAL F U L F I L M E N T O F THE REQUIREMENTS FOR T H E D E G R E E OF DOCTOR OF PHILOSOPHY in the Department ? f PHYSICS We accept this thesis as conforming to the required standard T H E UNIVERSITY OF BRITISH COLUMBIA June, 1966 In presenting this thesis in pa r t i a l fulfilment of the requirements for an advanced degree at the University of Bri t i s h Columbia,, I agree that the Library shall, make i t freely available for reference and study, 1 further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. The University of B r i t i s h Columbia Vancouver 8 , Canada The University of British Columbia FACULTY OF GRADUATE STUDIES PROGRAMME OF THE FINAL ORAL EXAMINATION FOR THE DEGREE OF DOCTOR OF PHILOSOPHY of DAVID Y„ CHUNG B. Sc,, National Taiwan University, 1958 M, Sc., University of British Columbia, 1962 TUESDAY, JULY 26, 1966 AT 3:30 P.M. IN ROOM 301, HENNINGS BUILDING COMMITTEE IN CHARGE External Examiner: J = Wilks Oxford University Research Supervisor: P, W. Matthews Chairman: I. McT, Cowan J. W. Bichard L. C, Brown P. R. Critchlow F. A, Kaempffer P. W. Matthews L. Young HEAT PULSES IN A1 20 3 SINGLE CRYSTALS AT LOW TEMPERATURES ABSTRACT Heat pulse experiments have been made on AI2O3 single crystals in the temperature range 3 , 8 ° K to 35°K with the aim of gaining further insight into the nature of heat transport in solids at low temperatures. Short heat pulses were produced by heating a thin metal film evaporated on to one end of the crystal. The thermal pulse arriving at the other end of the crystal was detected by an indium film thermometer placed in a c o i l connected to a sensitive radio-frequency bridge, so that the variation of resistance was fi n a l l y displayed on an oscilloscope. The pulses received at low temperatures (3.8°K to 8°K) show two quite separate parts, an i n i t i a l sharp rise followed by a slow rise,, starting at a definite delay time corresponding to the phonon velocity in the medium. The results up to 18°K do not show appreciable variation in delay time,, showing that the heat pulse propagation has not entered a second sound region.. As the temperature increases, the amplitude of the i n i t i a l phonon pulse decreases very much compared with the amplitude of the slow rise. Above 18°K, the small sharp rise can no longer be seen clearly so that the delay time is no longer well defined, and at 30°K only the slow rise is observed. It is found that the conventional theory of heat conduction is inadequate to interpret our results at low temperatures, as i t f a i l s to predict the fi n i t e delay of the i n i t i a l rise of the received pulse. A phenomenological approach is taken, using a modified heat equation which has an el e c t r i c a l transmission line analogy. Using Laplace transforms, a solution is obtained and the results calculated with a computer are compared with the experimental curves. It is found that the pulse shape can be interpreted quite sat i s f a c t o r i l y 3 especially at the lowest temperatures. The thermal, diffusivity, D. for different temperatures is foundj and the apparent thermal conductivity, Ks is calculated and compared with Herman's (1955) results. The solution of the modified heat equation is also calculated for liquid He II at 0,25°K and compared with the heat pulses observed by Kramers et al (1954); very good agreement is obtained, GRADUATE STUDIES Field of Study: Low Temperature Physics Elementary Quantum-Mechanics Electromagnetic Theory Quantum Theory of Solids Low Temperature Physics W. Opechowski G. M, Volkoff R. Barrie J . B, Br own Related Studies: Differential Equations Complex Variables Applied Electronics C. A. Swanson H. A. Thurston W. A. G. Voss PUBLICATIONS D. Y. Chung and P. R. Critchlow - Motion of Suspended Particles in Turbulent Superflow-of Liquid Helium II,. Phys. Rev .Letters, ;i4, 892 (1965) J. B. Brown, D. Y. Chung and P. W, Matthews, Heat Pulses at Low Temperatures, Physics Letters, 21, 241 (1966), DAVID YIH CHUNG. HEAT PULSES IN A l 0 SINGLE CRYSTALS AT LOW u TEMPERATURES. "' Super v i s o r : Dr. J . B. Brown. ABSTRACT Heat pulse experiments have been made on Al^O^ s ^ n g l e crystals in the temperature range 3. 8°K to 35°K with the aim-of gaining further in-sight into the nature of heat transport in solids at low temperatures. Short heat pulses were produced by heating a thin metal film evaporated on to one end of the crystal. The thermal pulse arriving at the other end of the crystal was detected by an indium film thermometer placed in a coil connected to a sensitive radio-frequency bridge, so that the variation of resistance was finally displayed on an oscilloscope. The pulses received at low temperatures (3. 8°K to 8°K-) show two quite separate parts, an initial sharp rise followed by a slow rise, starting at a definite delay time corresponding to the phonon velocity in the medium. The results up to 18°K do not show appreciable variation-lri delay time, showing that the heat pulse propagation has not entered a second sound region. As the temperature increases,the amplitude of the initial phonon pulse decreases very much compared with the amplitude of the slow rise. Above 18°K, the small sharp rise can no longer be seen clearly so that the delay time is no longer well defined, and at 30°K only the slow rise is observed. It is found that the conventional theory of heat conduction is inadequate to interpret our results at low temperatures, as it fails to predict the finite delay of the initial rise of the received pulse. A phenomenological approach i i i i s taken, using a modified heat equation which has an e l e c t r i c a l t r a n s m i s s i o n line analogy. Using Laplace transforms, a solution i s obtained and the results calculated with a computer are compared with the experimental curves. It is found that the pulse shape can be interpreted quite s a t i s f a c t o r i l y , e s p e c i a l l y at the lowest temperatures. The thermal diffusivity, D, for different temperatures is found, and the apparent thermal conductivity, K, is calculated and compared with Berman's (1955) results. The solution of the modified heat equation is also calculated for l i q u i d He II at 0. 25°K and compared with the heat pulses observed by K r amers et al_(1954); very good agreement is obtained. (Brown, Chung and Matthews, 1966). A comparison i s made between heat pulse propagation i n l i q u i d He II below 0. 6°K and i n a d i e l e c t r i c s o l i d . It i s noted that heat pulse t r a n s -m i s s i o n i s very s i m i l a r i n these two media, except that i n the temperature region for second sound propagation, the frequency "window" is very much wider i n He II than i n solids (-if such a window exists i n a solid). This i s one reason why l i q u i d He II is a more favorable medium for second sound than is a so l i d . P r e l i m i n a r y results for heat pulses i n a NaF single c r y s t a l at 3. 8°K and 6. 6°K are given: there i s no observable change i n the velocity of heat pulses at these temperatures, and therefore no evidence for the existence of second sound. iv Some general comments about the s i m i l a r i t y between the propagation properties i n molecular gases and i n a"phonon gas" are given. Some further investigations are suggested. i T A B L E O F C O N T E N T S p a g e A B S T R A C T i i LIST O F I L L U S T R A T I O N S A N D T A B L E S v A C K N O W L E D G E M E N T S . vi C H A P T E R I. I N T R O D U C T I O N 1 C H A P T E R II. H I S T O R I C A L R E V I E W A N D C U R R E N T THEORIES* . . . II. 1. Theory of Ward and Wilks (1952) 4 II.2. Theory of Prohofsky and Krumhansl (1964) . . . 5 II. 3. Dispersion Relation for Second Sound in Solids . . (Guyer and Krumhansl , 1964) 11 II. 4. Other Related Theories 13 II. 5. E l e c t r i c a l T r a n s m i s s i o n Line Analogy and Modified Heat Equation 16 C H A P T E R l U . E X P E R I M E N T A L A R R A N G E M E N T S A N D P R O C E D U R E III. 1. Electronics 23 III; 1. L ( Th6 Input System 23 III. 1.2. Detecting System 23 III. 1. 3. Temperature Measurement and Temperature Stabilizer " . . . 32 III. 2. Vacuum-System and Hydrogen-gas Thermometer 36 III. 3. Mounting of the Crysta l 36 III. 4. Method of Measurement 41 C H A P T E R IV. E X P E R I M E N T A L R E S U L T S A N D DISCUSSION IV. 1. Introduction 45 IV. 2. Low Temperature Results and their Interpretation 46 T V . 2. 1. Modified Heat Equation and its Solutions 46 IV. 2. 2. Experimental Results at 3. 8 K 49 o _o IV. 2. 3. Experimental Results between 4. 2 K and 8 K 52 IV. 2.4. A n Example - Propagation of Heat Pulses in He II Below 0. 6 K 57 IV. 3. Higher Temperature Results and their Interpretation 63 IV. 4. General Comments on the Results of the Modified Heat Equation 77 C H A P T E R V . C O N C L U S I O N S 84 B I B L I O G R A P H Y 88 LIST O F I L L U S T R A T I O N S A N D T A B L E S Figure Page II. 1. (a) Phase velocity or attenuation against frequency . . . 8 (b) Attenuation against frequency showing the frequency < window ; 8 II. 2. Current response to a step voltage for a t ransmission line 20 II. 3. Current response to a step voltage for a t ransmission line 21 III. 1. (a.) Evaporation chamber 24 (b) Details of the arrangement for resistance measurement 24 III. 2. Block diagram of the heating and detecting system . . . . 29 III. 3. Block diagram of detecting system with box-car integrator 31 III. 4. Block diagram of feedback system of temperature control-ler for the crystal 34 III. 5. Vacuum system and Hydrogen-gas thermometer . . . . 37 III. 6. Structure of the can and the position of the crystal ". 39 IV. 1. Pictures of received pulses at 3. 8 ° K for 3, 8 c m long -A l ^ O ^ single crystal , showing a r r i v a l of longitudinal and transverse direct phonons 50 IV. 2. Box-car integrator trace at 4. 7 ° K in A l ^ O ^ 53 IV. 3. N o r m a l i z e d output temperature pulse for A1,,0^ single crystal at 5. 0 ° K 54 IV. 4. Normal ized output temperature pulse for liquid He II at 0. 2 5 ° K 60 IV. 5. N o r m a l i z e d output temperature pulse for A l ^ O ^ single • crystal at 9 . 0 ° K 65 IV. 6. • N o r m a l i z e d output temperature pulse for A1_0 single crystal at 16. 5 K 66 • IV. 7. Normal ized output temperature pulse for A l ^ O ^ single crystal at 1 7 ° K 67 v i F i gure Page IV. 8. N o r m a l i z e d output temperature pulse for A l 0 single c r y s t a l at 18°K ; 68 IV. 9 . N o r m a l i z e d output temperature pulse for A l 0 single c r y s t a l at 23°K . . 69 IV. 10. N o r m a l i z e d output temperature pulse for A l 0 single c r y s t a l at 32°K 70 IV. 11 Thermal diffusivity D vs temperature T • 72 IV. 12. Thermal conductivity K vs temperature T . . . . . . 73 IV. 13. Pi c t u r e s of received pulses for NaF single c r y s t a l at 6. 6°K and 3. 8°K 76 Table I. Computed relaxation times as a function of temperature in NaF (taken from Prohofsky and Krumhansl, 1964) . . 10 II. Order of magnitude estimates of f at different temperatures 7 5 III. Comparison between propagation properties i n a molecular gas and phonon "gas" 78 v i i A C K N O W L E D G E M E N T I, wish to express my sincere appreciation to my resear c h supervisor, Dr. J. B. Brown, for his continuing interest,, guidance and i n s p i r a t i o n through-out the duration of this project. I would also l i k e to thank Dr. P. W. Matthews for his help and valuable discussion,, especially i n the last stage of this work. I am also indebted to Mr. C. R. Brown and Mr. J. D. Jones for their help in various stages in performing the experiment. In construction of the cryostat and electronics, I would l i k e to thank Mr. G. Brooks, Mr. R. Weissbach and Mr. J. Lees for their assistance and advice. F i n a l l y I would l i k e to acknowledge helpful discussions with Dr. P. R. Critchlow and Dr. M. Crooks. - 1 -C H A P T E R I. INTRODUCTION One of the most interesting properties of superfluid helium is the occurrence of a thermal transfer mode, second sound. In the language of the two-fluid model, second sound is a density wave i n the normal fluid, accompanied by a density wave of opposite phase in the superfluid," so that the total density remains constant. Since one assumes that there is no entropy i n the superfluid, those regions r i c h e r i n normal f l u i d would have higher entropy density and higher temperature. The wave of normal fl u i d density would appear as a temper-, ature wave. According to Landau's theory (1941, 1947) the normal fl u i d is i d e n t i -fied with the elementary excitations, and below 0. 6°K these would consist mainly of phonons. Then one should obtain the equation for the veloc i t y of , ^ 2 C,2 f s C ( 2 P second sound Cx - m — (since / ° ^ 1 at very low tem-2 3 JO 3 p peratures). It was pointed out by Ward and Wilks (1952) that second sound can be thought of as a density fluctuation i n a "phonon gas". The following discussion i s based on the work of Prohofsky and Krumhansl (1964), using a p a r t i c l e gas to i l l u s t r a t e possible properties of a phonon gas. Consider the propagation of a disturbance in a pa r t i c l e gas. The quantities of interest are: the p a r t i c l e density at eq u i l i b r i u m J^ o • the velocity of an element of the gas it* and the relaxation time ^ . In the absence of density grad-ients or applied forces, a pa r t i c l e current f0 U-* w i l l decay with a typ i c a l - 2 -relaxation time < r £ . We can obtain the following hydrodynamic equation for a particle current Here the fundamental distinction between wave motion (ordinary sound) and diffusion becomes clear. For slow variations of fl Ux with time, the viscous term (second term) dominates and any density fluctuation relaxes by diffusion. When the viscous relaxation is slight and the relaxation time is long, spatial variations of the particle density will proceed in time as wave-like motions. Both diffusive and wave descriptions may therefore apply to density fluctuations in a particle gas. The choice is determined by the frequency in question. The "phonon density" fluctuations can be described in the same way. The characteristic time ' t for decay of a heat current is the relaxation time associated with processes for which the phonon momentum is not con-served. The common experience that temperature fluctuations" obey the diffusive description is a consequence of the low frequency at which they are conventionally studied. If the temperature is low enough, the relaxation time t becomes long compared with the reciprocal of the frequencies of interest. Then if the heat flow is described by an equation analogous to momentum y . , ' eq, (1.1), the diffusion term — • : • u is becoming small, and the wave term (the first term on the left hand side) is dominant; and fluctuations of phonon density can propagate as second sound, which is analogous to ordinary sound in a particle gas. (1.1) - 3 -In t h i s t h e s i s , e x p e r i m e n t s w i t h heat p u l s e s i n A l ^ O ^ s i n g l e c r y s t a l s at l o w t e m p e r a t u r e s have been p e r f o r m e d i n the hope of d e t e c t i n g s e c o n d sound p r o p a g a t i o n i n t h i s s o l i d . H o wever, the m a i n c h a r a c t e r i s t i c s of the o b s e r v e d p u l s e i n the t e m p e r a t u r e range 3. 8°K to 35°K a r e t y p i c a l of a d i f f u s i o n wave r a t h e r than s e c o n d sound. We s h a l l p r e s e n t the r e s u l t s and d i s c u s s i o n i n C h a p t e r IV. The r e s u l t s g ive f u r t h e r i n f o r m a t i o n about the heat t r a n s p o r t i n s o l i d at l o w t e m p e r a t u r e s . In p a r t i c u l a r , we s h a l l c o m p a r e our r e s u l t s w i t h heat p r o -p a g a t i o n i n He II and show why i t i s a m u c h m o r e f a v o r a b l e m e d i u m than a s o l i d f o r o b s e r v a t i o n of s e c o n d sound. F u r t h e r , the l o w e r t e m p e r a t u r e (3. 8°K to 18°K) r e s u l t s w i l l be a n a l y z e d i n some d e t a i l i n t e r m s of an equation w h i c h i s analogous to Eq. (1. 1). - 4 -C H A P T E R II. HISTORICAL REVIEW AND CURRENT THEORIES II. 1. Theory of Ward and Wilks (1952) It i s well known that the two-fluid model has been used successfully to explain the properties of l i q u i d He II. Landau (1941, 1947) obtained the expression for the velocity of second sound at absolute zero, — p s — * ~ (where 0^ is the vel o c i t y of ordinary sound), by his theory of excitations in He II. C l However, it is possible to derive the expression d i r e c t l y f r o m the phonon gas model without using the two-fluid model at a l l . This was done by Ward and Wilks (1952) who followed Landau i n representing l i q u i d helium, near the absolute zero as a phonon gas consisting of the elementary excit-ations for which E = pC^, where E and p are the energy and momentum of the excitations. They also assumed that at these temperatures interactions in the helium are v e r y s m a l l so that the co l l i s i o n s between phonons are elastic and conserve momentum,'('normal processes') . Their derivation gives no indication of the region i n which one may expect the expression C, 2 / j to be valid. C l e a r l y it w i l l f a i l i f (a) the c o l l i s i o n s between phonons do not s atisfy the conservation laws of energy and momentum, and (b) i f the excitations of the l i q u i d cannot be adequately represented by phonons. A l l one can say is that i f at a sufficiently low temperature the specific heat of 3 the liquid, helium varies as T then we should expect the velocity of second C l sound to be • The important feature of this theory l i e s in the fact that this model could also be used to represent a c r y s t a l l i n e s o l i d at - 5 -low temperatures. Peshkov (1947) suggested that second sound might be observed in a c r y s t a l in which the scattering of phonons by inhomogeneities and i r r e g u l a r i t i e s is a minimum. In an ideal c r y s t a l the essential c r i t e r i a for second sound are probably that the number of normal c o l l i s i o n s between the phonons should be sufficient to ensure that the phonon gas is not in the 'Knudsen region' ( i . e. , the region i n which the mean free path is l i m i t e d by c r y s t a l boundaries) while at the same time v e r y few momentum-loss col l i s i o n s should occur. Therefore it might be possible to observe second sound by a suitable choice of c r y s t a l and a temperature range i n which the only scattering of phonons for which momentum is not conserved is at the walls of the specimen (Ward and Wilks suggested Al^O^ at helium temperature) Wilks (1953) pointed out that it would be of interest to determine how pulses of heat are propagated in such c r y s t a l s ; unfortunately their low specific heat combined with a high velocity of sound c a l l for the use of very narrow pulses wh|ch can only be observed by thermometers of extremely small thermal capacity. The work described i n this thesis was an experimental follow-up of this suggestion of Wilks'. II. 2. Theory of Prohofsky and Krumhansl (1964) Prohofsky and Krumhansl (1964) applied the Boltzmann transport equation to phonon wave packets. Associated with the Boltzmann equation is a distribution function giving the density of pa r t i c l e s i n specific modes. This distribution can be expressed as an e q u i l i b r i u m distribution plus s m a l l perturbations when the period of the disturbance is long compared to the normal relaxation time " t , and the mean free path for normal c o l l i s i o n s is short compared to the wavelength of the disturbance. The detailed m i c r o -scopic treatment was given in the o r i g i n a l paper, and here we summarise the results. The total relaxation time t r can be expressed as / - / . / — ~ 7 T — ~ ( 2 . 1 ) where '7~ is the relaxation time for a l l interactions which conserve phonon momentum within the phonon distribution (N-processes). ~Cy is the relaxation time for a l l interactions in which phonon momentum is not con-served ( U -processes), and this is taken to include boundary scattering and im p u r i t y scattering as well as Umklapp processes (phonon-phonon co l l i s i o n s for which phonon momentum is not conserved). Note that by 'phonon momentum' we mean the quantity > where ^ is the wave vector associated with the phonon, and unlike p a r t i c l e momentum, it i s not ne c e s s a r i l y conserved in c o l l i s i o n s . Under the assumption Wt<me)[[,4 ^r2]^ j ( 2.3, The expected second sound occurs when the damping due to U -c o l l i s i o n s is small, i . e. UJ 1^ »/, In this'limit Eqs{2. 3) and (2. 4) lead to - 7 Cz=-^- = (* %>aV'-f{/ - -f(Wu)~+ - J ~ <(^3ve (2-5) and 1 ^ / / (2.6) 2. r T A l s o i n the l i m i t -t 4. ' = V= ( 2 . 7 ) .Eq. (2. 7) i s , i n fact, equivalent to the usual expression for low frequency thermal waves, for which the r e a l and imaginary parts of the wave number are equal, and are proportional to £d (e. g. see Landau and L i f s h i t z , 1959). At higher frequencies when oJ >7^J and there are few U- c o l l i s i o n s in a wavelength, the second sound collective mode pr e v a i l s . The velocity becomes constant at = Ct/J~3^<\. (2. 5)J and the attenuation also becomes constant (Eq* 2.6). This is shown in F i g . II. 1 (a) (the variation with frequency of the attenuation and the phase velocity, are identical i n form). However, for the collective second sound mode to exist, there must be a number of normal co l l i s i o n s within a wavelength, and hence when the wave-length becomes s m a l l compared with the mean free path for normal processes ( i . e. ,0J7I„» I ), the attenuation becomes very large. The same situation occurs i n a molecular gas when the ordinary sound wavelength becomes s m a l l compared with the molecular mean free path. Large dispersion occurs for OJlZ^-^ /. Prohofsky and Krumhansl show that the imaginary part of the wave vector in the second sound region (dJZ:u»/) becomes - 8 -(a) Pkasa Velocity or Attenuation (b) Altenuatiori 7 1 -i UJ Figure TL. / (a). 7%ase velocity cu/h* o r zttenuattorLkj zigninst frequency, given by ey.(2.3)3/id(2.4\ TL.Kb). Attenuation At against fre^vency, as given, by ej. (z. 9), shuicny fre^ne/icy toincfotO, TU'<«J< m'J % for tie propagation of secmd £ounoL. The imaginary part of the wave vector is shown as a function of Cu" in F i g . II. I (b) # It can be seen from eqs.(2.4) and (2. 8) that the conditions for second sound propagation are ^ UJ (2-9) i . e. in order for second sound to be possible in solids, there must exist an appreciable difference between the r e c i p r o c a l normal and r e c i p r o c a l umklapp relaxation times. At low temperatures, the normal relaxation time varies as some power of T,' while the umklapp relaxation time varies exponentially with T . It therefore seems reasonable that, below some temperature, a "window" may appear in the frequency spectrum so that the propagation of second sound becomes possible. A rough evaluation of the relaxation times can be obtained f rom thermal conductivity measurements. The relaxation times for N a F shown in Table I are average relaxation times, computed f r o m the data of Walker (1963). At about 10°K, the calculated H 3nd 7zNdiffer by more than a factor 4 3 5 -1 of 10 , indicating that a frequency in the range 10 ~ 10 sec should satisfy the inequalities of eq. (2.9). In Table I , - 2 7 r - £ = - ^ represents the C~u angular frequency above which heat conduction becomes more l i k e wave propagation than diffusion. Above 15°K, no second-sound effects would be seen for frequencies below I Mc/S. Below 9 K, second sound could be present for frequencies above one cycle/sec, were it not for the decreased probability of normal c o l l i s i o n s . However, it should be noted that the above statements apply only i n an infinite, pure c r y s t a l . In either a pulse experi-ment or a standing-wave experiment, i t is^ essential that a major fraction - 10 -T a b l e I. Computed r e l a x a t i o n t i m e s as a f u n c t i o n of t e m p e r a t u r e in N a F . ( T a k e n f r o m P r o h o f s k y and K r u m h a n s l , 1964). T ° K Tr., (sec) lirl = ~L ( s e c " ) (sec) 20 2.2 x l O " 7 4 . 9 x l 0 6 1.6.x 10" 1 15 5.2 x 1 0 " 6 1.9 x 1 0 5 7.0 x 10"' 12 .1. 1 x 10 9. 5 x 10 2. 1 x 10" 11 3 . 8 x l 0 " 4 2 . 7 x l 0 3 4.3 x 10" 10 1.9 x l O " 3 5 . 4 x l 0 2 5.3 x 10" 9 l . l x l O ' 2 8 . 8 x 1 0 1 9 . 0 x 1 0 ' 7 2.2 4.6 x 10'} 3.2 x^lO'* 1 5 7.9 x l O 2 1 . 3 x l 0 " 4 1.7 x 10" 5 of a p u l s e l e n g t h o r a wavelength be c o n t a i n e d w i t h i n the c r y s t a l . W i t h the 5 -1 u s u a l s i z e of p u r e c r y s t a l s (^ ~ / £ ^ ), f r e q u e n c i e s of the o r d e r of 10 s e c 5 a r e n e c e s s a r y b e c a u s e of the l a r g e v e l o c i t y of sound (10 c m / s e c ) . The g e n e r a t i o n and d e t e c t i o n of these h i g h f r e q u e n c i e s make the o b s e r v a t i o n of s e c o n d sound i n s o l i d s m o r e d i f f i c u l t than i s the case i n l i q u i d He II where l o w e r f r e q u e n c i e s can be used. II. 3. D i s p e r s i o n r e l a t i o n f o r s e c o n d sound i n s o l i d s ( G u y e r and K r u m h a n s l ? 1964) G u y e r and K r u m h a n s l d e r i v e d the d i s p e r s i o n r e l a t i o n f o r s e c o n d sound i n a s o l i d f r o m a m i c r o s c o p i c s o l u t i o n of an a p p r o p r i a t e equation of m o t i o n f o r the d i s t r i b u t i o n f u n c t i o n of a phonon gas. In the d e r i v a t i o n the r o l e p l a y e d b y phenaonenological p a r a m e t e r s c h a r a c t e r i z i n g the phonon gas i s e m p h a s i z e d ; the need f o r a "window" i n the r e l a x a t i o n t i m e s p e c t r u m i s e x p l i c i t l y d e m o n s t r a t e d . W i t h the a i d bf the m i c r o s c o p i c s o l u t i o n f o r 'the phonon gas e m p l o y e d t h e y a r r i v e d at a m a c r o s c o p i c equation. But t h e i r e q u ation d i f f e r s f r o m those of C h e s t e r ' s (1963) i n so f a r as a "window" i s c o n c e r n e d . The p h y s i c a l p i c t u r e of the s e c o n d sound p r o c e s s w h i c h l e a d s to c o n s i d e r a t i o n of the m o t i o n of the phonon d i s t r i b u t i o n f u n c t i o n i s t h i s : we a r e i n t e r e s t e d i n the b e h a v i o u r of a phonon gas i n a l o c a l r e g i o n of space when the l o c a l r e g i o n i s s u b j e c t to a t e m p e r a t u r e p e r t u r b a t i o n of the f o r m TMt) - TO = TT^t)^TZe (2 . io) The t e m p e r a t u r e p e r t u r b a t i o n m a y be r e g a r d e d as b e i n g i n d u c e d b y the r e m a i n d e r of the phonon gas o r as the r e s u l t of the a p p l i c a t i o n of an e x t e r n a l - 12 -r e s e r v o i r . In e i t h e r c a s e the l o c a l r e g i o n of the phonon gas i s to be s u b j e c t e d to t h i s p e r t u r b a t i o n f o r a r e a s o n a b l e l e n g t h of t i m e . The phonon gas i s then to be l o o k e d at w i t h a B o l t z m a n n equation h a v i n g the C a l l a w a y (1959) e x p r e s s i o n f o r the c o l l i s i o n t e r m , ije. w i t h the equation ^ j y . 7 / / — 4 ^ ! L - , 2.n, where N , and N q w i l l be functions, of x, q and t. N , N-^ and N q a r e the d i s t r i b u t i o n f u n c t i o n of the phonon s y s t e m , the d i s t r i b u t i o n of a u n i f o r m l y d r i f t i n g phonon gas, and the l o c a l e q u i l i b r i u m d i s t r i b u t i o n f u n c t i o n r e s p e c t i v e -l y . H e r e "2T i s the r e l a x a t i o n t i m e f o r the n o r m a l p r o c e s s e s ; the n o r m a l p r o c e s s e s r e l a x N to N ^ a d i s t r i b u t i o n f u n c t i o n w h i c h c a r r i e s the same m o m e n t u m c u r r e n t as N . The d i s t r i b u t i o n then r e l a x e s to t h e r m a l e q u i l i -b r i u m b y means of the U - p r o c e s s e s , w i t h r e l a x a t i o n t i m e ^tia We now r e f e r to the r e s t r i c t i o n s on Uj ; w h i c h l i m i t the s o l u t i o n of eq. (2. 10) to-the r e g i o n where one expects s e c o n d sound p r o p a g a t i o n . These are: l ) 2 j ' 2 ^ / « / : the t e m p e r a t u r e p e r t u r b a t i o n has a c h a r a c t e r i s t i c p e r i o d l o n g c o m p a r e d to the t i m e r e q u i r e d f o r t e m p e r a t u r e r e l a x a t i o n b y n o r m a l p r o c e s s e s and the phonon gas c a n f o l l o w the p e r t u r b a t i o n . 2) U'7Ztf>> I : a l a r g e n u m ber of t e m p e r a t u r e o s c i l l a t i o n s o c c u r i n the t i m e r e q u i r e d f o r the m o m e n t u m to r e l a x . The d i s p e r s i o n r e l a t i o n d e r i v e d was 'ff.SL (/_ , [ ^ O J ^ , (zkj]*a«v)]«.m T h i s r e l a t i o n i s e s s e n t i a l l y that of P r o h o f s k y (1964) i . e . eq. (2.2); however, the d amping t e r m , the i m a g i n a r y p a r t , c o n t a i n s both the m o m e n t u m r e l a x -- 13 -ati o n t i m e and the t e m p e r a t u r e r e l a x a t i o n t i m e . 1 ^ , . If the damping of the s e c o n d sound wave i s to be s m a l l , the p e r i o d of the t e m p e r a t u r e p e r t u r -b a t i o n , //V, m u s t be i n the "window" between "frnd *~£u ) t.G.. 't^-^j-*^ The f e a s i b i l i t y of any s e c o n d sound e x p e r i m e n t i n s o l i d s r e s t s on the e x i s t e n c e of t h i s "window". A m a c r o s c o p i c equation f o r heat f l u x w a s d e r i v e d f r o m eq. (2. 11) w i t h the a i d of the m i c r o s c o p i c s o l u t i o n eq. (2. 12), ^ + <&t + (,-£uj^* OW)+~)J%\>T\-o (2,3) where K = The equation f o r ""]"" was U+ 4& - [ > * ^r-o P . u , T h i s m a c r o s c o p i c equation f o r the t e m p e r a t u r e con t a i n s the n e c e s s a r y 'window' f o r s e c o n d sound. II. 4. O t h e r R e l ated T h e o r i e s 1. Dingle's T h e o r y (1952) In h i s paper D i n g l e d e r i v e d the v e l o c i t i e s f o r s e c o n d sound i n d i f f e r e n t m e d i a under the a s s u m p t i o n ( w i t h o u t j u s t i f i c a t i o n ) that s e c o n d sound c o u l d propagate i n these m e d i a . The v e l o c i t i e s w e r e d e r i v e d by u s i n g the analogous e x p r e s s i o n f o r He II, n a m e l y - 14 -where ~J~ was the absolute temperature, the density of the medium, S and C respectively the entropy and specific heat per unit volume, and the effective mass density of the excitations which contributed to the entropy. The conditions under which second sound might be possible in these media were given. These media were ideal gases, insulators, conductors, superconductors and ferromagnets. For details we refer the reader to the original paper. Here the results for insulators will be given. In an insulator, the velocity of second sound was given as c?= f& +ii)/3 (~k (216) where and C ^ are the velocities of longitudinal and transverse sound waves in the medium in question. In this equation the transverse and longi-tudinal sound waves were mixed to give one velocity of second sound. II. 4. 2. Ginzburg's Theory (1962) The possibility of observing convection heat transfer and second "sound" in superconductors were discussed in his paper. The change in the single particle excitation spectrum, and in particular, the existence in super-conductors of exciton excitations, can in principle greatly reduce the role played by impurity scattering. As a result, the attenuation of second "sound" may be substantially diminished, while the convection transfer of heat may increase. From his point of view, second "sound" may occur in lead and mercury. He calculated the velocity of second "sound" in superconducting o 4 lead from his simple model, at T = 3. 9 K, - 2.2 x 10 cm/sec. - 15 -It was pointed out by Bardeen and Schrieffer (1961) that it would be difficult to have second "sound" in superconductors since the normal electrons would damp out the coherent wave in a very short distance. Hence the possibility of second sound in superconductors is still uncertain. It is to be noted here that the second "sound" in superconductors discussed above refers to the propagation of a density fluctuation of normal and superconducting electrons. This is slightly different from the second sound discussed in Sections II. 1, II. 2 and II. 3 where we referred to the propagation of a fluctuation of phonon density. II. 4. 3. Sussmann and Thellung's Theory (1963) Sussman and Thellung considered the case of perfect dielectric crystals in the absence of Umklapp processes. The sample was considered to be large compared with the mean free path of normal processes. The mean free time approximation was used for the phonon distribution function and hydro-dynamical equations for the phonon gas were derived. Heat flow was shown to consist of two parts, one was the usual diffusion heat flow and the other was due to the drift motion of the phonon gas. The velocity of second sound they obtained was C*-±Cl Cl-4rKA:-- (2-17) which is exactly the equation given by Dihgle (eq. 2. 1*6) for an insulator. II. 4. 4. Theories of Griffin (1965), Guyer(1965), Kwok and Martin (1966), and Enz (1966) The theories mentioned here are still in the stage of development. - 16 -A l l these papers were in the form of letters, therefore the actual detailed theories using Green's function techniques were not clearly shown. They suggested that a light scattering experiment which looks at the width of the first sound doublet in an isotropic solid would afford an excellent possibility for second sound detection. No quantitative features were given. II. 5. Electrical Transmission Line Analogy and Modified Heat Equation In this section we are going to discuss the macroscopic equation for heat transmission from a different approach which clearly shows the important difference between the ordinary diffusion' equation and the newly obtained equation. Following Osborne (1951) and Ulbrich (1961), one could make an analogy between the problem of heat conduction and an electrical transmission line and therefore obtain a macroscopic equation. The equation obtained is very similar to the one discussed in II. 3 obtained from a microscopic point of view. The Fourier equation for heat conduction •"•'>'• waves proportional to the square root of the frequency, and thus increases without limit for increasing frequency. However, it is evident that there must be an upper limit to this velocity. To remove this contradiction (2.18) (where the thermal diffusivity D = K / C ) predicts a velocity for heat Ulbrich (1961) made an analogy between Vernotte's (1958) modified heat equation and the equation of an electrical transmission line, iand predicted a finite upper limit to the velocity of the Fourier components bf a heat pulse. He made the analogy by considering T, Q (heat flux), 1/K and f'Cy as equiv-alent to V, i , R and C respectively. A new quantity t was introduced in analogy to the electrical quantity L / R . The modified heat equation As was pointed out by Ulbrich (1961) no physical basis was proposed for the quantity , and whether or not "cr actually existed could be only determined by experiments on the mechanism of heat transport in solids. It is part of the purpose of this thesis to show that the quantity T_T does exist and has a specific interpretation in the problem of heat transfer in solids at low temperatures. Recently Chester (1963) applied equation (2. 19) to the problem of propagation of second sound in solids. This application of the transmission line analogy to solids was already familiar to Dr. J. B. Brown, and was one of the chief motivations for the present work. He found that there is a transition frequency f = 1 / 2 T T 1 Z . , above which the second sound should be c observable. From eq. (2.20) one obtains f = t/27T^C = i^/^-TrZ) where 1fo = Q isjthe. second sound velocity, and ' ts 'Zr^ for this case. In fact the analogy is not perfect and to cover the whole frequency region, one must consider what happens when the wavelength becomes less than the mean free path for normal collisions. This was pointed out by Ward and Wilks (1951) and by Chester (1963), but was subsequently included in a mathemat-(2. 19) gives the limiting velocity of propagation for high frequencies (2.20) - 18 -ical argument by Prbhofsky and Krumhansl (1964) (see section II. 2). Osborne (1951) proposed to use equation (2. 19) to describe pulses of second sound in liquid He II below 0. 6°K in order to explain in a general way the observed finite delay time. Dingle (1952) used a similar formulation describing the combined effect of viscosity and conduction in liquid He II. No quantitative discussion was given. Using a Laplace transform, Osborne (1951) obtained the solution of the electrical transmission line equation for the response to a unit h~~-function current input at x = o, and found that for the end of the sample, x = JI .22 ( -/926) shoving tke effect cf increasing v/if. -21-(a) Solid line: Inductive cable* Dotted line -non-inductive cdtte ( Taken froTn Carson, / 92^), 1.0 M>rmli£edO.% 0.6 OA 0.2 0 Inductive cable, Q^o. (Taken fw7n lOeb&r, / ?£"£). Figure. 2L3. Current response ('arbitrary units) to a step voltage -for 3 transmission u*e. - 22 -scale to facilitate comparison. We are going to show s i m i l a r behavior in the propagation of heat pulses in d i e l e c t r i c solids at low temperatures in Chapter IV. To give a complete picture we reproduce another .useful graph in F i g . II. 3(b) (Weber 1956). This is a plot of the current response to a step exponential function predominates and that for CT(-^r-J>2 the B e s s e l function dictates a current r i s e above the i n i t i a l sharp r i s e . It should be noted that the analogy to heat pulse t r a n s m i s s i o n replaces the heat pulse by a current pulse, and the temperature by voltage, as used in eqs. (2.21) and (2.22). The usual t r a n s m i s s i o n line cases shown in F i g s . II. 2 and II. 3 are for current response to a step voltage input, but the f o r m of the voltage response to a current step input is very s i m i l a r . These figures were useful i n understanding i n a qualitative way the observed behavior of heat pulses, in p a r t i c u l a r in understanding the results of K r a m e r s et al (1954) for l i q u i d helium, which have never been adequately explained. The t r a n s m i s s i o n line analogy w i l l be inadequate at very high frequencies when the extra t e r m of eq (2. 14) in Guyer and Krumhansl's analysis becomes important, i . e . for CO » / . voltage for a t r a n s m i s s i o n line (non-leaky), on the amplitude vs.at plot (rather than amplitude v s . y ^ p l o t ) . It i s evi dent that for ger F i g u r e I I I . 3 . B l o c k uiugrai:. of the d e t e c t i n g system w i t h Box-car i n t e g r a t o r . - 32 -signal was then recorded for different frequencies. F o r pulse work, the output from the oscilloscope was fed into a box-car integrator to improve the signal-to-noise ratio. The integrator and moving-gate c i r c u i t r y / were described by Hardy (1964) and Bridges (1964), based on the o r i g i n a l design of Blume (1961), and we refer to these authors for a detailed description. The only m o d i f i -cation made in the present case was to use a faster pulser unit which made it possible to reduce the gate width to 0. 1 juusec The picture of the heat pulses built up by the box-car integrator, was traced out on a chart r ecorder ( F i g . III. 3). III. 1. 3. Temperature measurement and temperature s t a b i l i z e r . Two ways of measuring the temperature of the c r y s t a l w i l l be described here. Two systems for st a b i l i z i n g the temperature w i l l also be described in some detail, one operating on the c r y s t a l itself, and the other controlling the temperature of the can i n which the c r y s t a l is mounted. A H^ gas thermometer used for cali b r a t i o n and cross checking the tem-peratures w i l l be discussed in the next section. III. 1. 3. 1. Thermocouple. A Thermocouple of gold plus 0. 03 atomic % i r o n vs s i l v e r (Johnson Matthey & Co. Ltd. , ) was used. One junction of the thermocouple was immersed in the helium bath alltthe time, the other end being placed in the can :-mar the c r y s t a l where the temperatures were to be measured. With a Hewlett Packard D. C. Micro-volt-ammeter the thermal e.m.f. - 33 -could be read off and the corresponding temperature obtained from the cal i b r a t i o n chart. The sensi t i v i t y of these thermocouple wires was around 12 J J L V ~ / ° K at helium temperature, (for detailed calibration and per-formance see Berman and Huntley (1963)). III. 1. 3.2. Carbon Thermometer and Boyle-Brown temperature s t a b i l i z e r . A 5<6 ohm carbon radio r e s i s t o r was used as a thermometer, mounted on top of the copper can. An A. C. (1 KC/$ ) Wheatstone bridge was used to measure its resistance. The unbalanced signal was fed into a p r e a m p l i f i e r followed by an audio-amplifier. The output was connected with a coaxial cable to the 500 ohm constantan wire heater and was displayed on a s m a l l oscilloscope. With this Boyle-Brown (1954) s t a b i l i z e r one could stabilize the temperature to within a fraction of a degree at tem-peratures between 4. 2°K and about 10°K. F o r higher temperatures the power output from the system was too s m a l l to be effective and moreover, the temperature se n s i t i v i t y of the carbon r e s i s t o r was reduced. Therefore the s t a b i l i z e r m erely served as a thermometer, and the 500 ohm heater was operated by a manually controlled source. III. 1. 3. 3. Temperature controller. To stabilize the temperature of the c r y s t a l i t s e l f inside the can to within one-tenth of a degree, a rather complicated feedback system was constructed and operated quite s a t i s f a c t o r i l y . The block diagram of this temperature controller (as it w i l l be c a l l e d to distinguish it from the tem-perature s t a b i l i z e r described in the last sub-section) is shown in Fi g . III.4. The system worked i n the following way. The Anderson bridge was 50 Kc/s O s c i l l a t o r ~ ~ t ; Anderson b r i d g e Indium f i l m , I.F., a m p l i f i e r Tek. 545A O s c i l l o s c o p e P u l s e r o r Audio f r e q . ^ O s c i l l a t o r J C h a l l e n g e r O s c i l l a t o r ^ G a i n c o n t r a l f — » — y $ v v -O p e r a t i o n a l a m p l i f i e r — 4 5 V o l t s "A Cathode f o l l o w e r F l o a t i n g power supply-To c o n t r o l g r i d o f V 7 and v., -40 V o l t F i g u r e II1.4. B l o c k diagram o f feedback system of the temperature c o n t r o l l e r . balanced i n i t i a l l y at 30 Mc/s, then a slight, change in temperature would unbalance the bridge as indicated by the D. C. meter at the output of IF amplifier. With a bias of about -9 volts to cancel out the-'+9 volts D. C. bias at the output at balance, any change in D. C. le v e l due to unbalancing of the bridge was amplified by a P h i l b r i c k 2-k-w operational amplifier with a cathode follower output. Gain control was obtained by adjusting the amount of feedback of the operational amplifier. The output of the cathode follower was connected in series with a floating DC power supply in such a way that a net DC voltage of about -40 volts could be fed into the control g r i d of the last two tubes of the Challenger transmitter ( F i g . III. 4). Hence this controlled the power input to the heater on the c r y s t a l , and gave a fine control on its temperature. The p r i n c i p l e of operation of this system is quite simple. The t r i c k y thing is to make the proper adjustments, but once it was adjusted properly for one temperature, it would lock i t s e l f to that temperature for quite a long time, (20 min. or longer). The key adjustments were (1) the bridge balance, (2) the I F amplifier gain control, (3) the DC bias, (4) the gain control setting for operational amplifier, and (5) the voltage setting of the DC power supply. (1) and (2) would affect the size of the signals which we were measuring-operating too close to the balance would reduce the signals, operating too far from balance would reduce the sensitivity, and therefore it was very c r i t i c a l to have the proper settings. (3) and (4) could be adjusted before-hand so that the net DC was amplified to proper magnitude for (5). The adjustment of (5) depended on how much - 36 -R F h e a t i n g o n e w i s h e d t o a p p l y to t h e c r y s t a l a t a g i v e n t e m p e r a t u r e . T o o m u c h p o w e r w o u l d i n c r e a s e t h e h e a t i n g , m a k i n g i t d i f f i c u l t t o s t a b i l i z e t h e t e m p e r a t u r e ; t o o l i t t l e p o w e r w o u l d g i v e t o o s m a l l s i g n a l s . T h e a d j u s t -m e n t s w e r e t h e r e f o r e t e m p e r a t u r e - d e p e n d e n t , a n d e x p e r i e n c e a n d p r a c t i s e w e r e n e e d e d t o a c h i e v e t h e b e s t p o s s i b l e p e r f o r m a n c e . III. 2 . V a c u u m S y s t e m a n d H ^ T h e r m o m e t e r . A c o n v e n t i o n a l v a c u u m s y s t e m a n d c r y o s t a t w e r e s e t u p ( s e e F i g . III. 5). T w o d o u b l e w a l l e d g l a s s d e w a r s w e r e u s e d . A r o t a r y p u m p , f o l l o w e d b y a H g d i f f u s i o n p u m p , w e r e u s e d t o l o w e r t h e p r e s s u r e o f t h e v a c u u m s y s t e m . T h e p r e s s u r e i n t h e c r y o s t a t c o u l d b e r e a d o f f . f r o m t h e H g a n d o i l m a n o m e t e r s . A d i s c h a r g e t u b e a n d a c o m m e r c i a l P i r a n i G a u g e w e r e u s e d f o r l e a k t e s t i n g . T h e H - g a s t h e r m o m e t e r w a s a s m a l l c h a m b e r a b o u t 1 c c i n v o l u m e i n s t a l l e d o n t o p o f t h e c o p p e r c a n i n w h i c h t h e A l ^ O ^ c r y s t a l w a s m o u n t e d . T h e c h a m b e r w a s m a d e o f c o p p e r a n d w a s s o l d e r e d w i t h W o o d s m e t a l t o t h e c o p p e r c a n , a n d w a s c o n n e c t e d t o t h e m a n o m e t e r b y a f l e x i b l e , s p i r a l t u b e o f 1 / 2 m m s t a i n l e s s s t e e l . T h e t e m p e r a t u r e o f t h e c h a m b e r c o u l d b e r e a d d i r e c t l y f r o m t h e H ^ v a p o u r p r e s s u r e c u r v e . B y m e a n s o f t h e -t h e r m o m e t e r , t h e c a r b o n r e s i s t o r , w h i c h w a s p l a c e d b e s i d e t h e c h a m b e r , c o u l d b e c a l i b r a t e d f r o m 1 2 ° K t o 2 0 ° K . I n t h e a c t u a l e x p e r i m e n t , t h e t h e r m o c o u p l e a n d c a r b o n r e s i s t o r r e a d i n g s w e r e t a k e n at e a c h t e m p e r -a t u r e a n d t h e H ^ - t h e r m o m e t e r w a s u s e d a s a c r o s s c h e c k . III. 3 . M o u n t i n g o f t h e C r y s t a l . - 3 7 -To c r y o s t a t — — D i s ^ a r g e H 2 - g a s thermoraet e r Manometer -*—>• To a i r / Hg d i f f u s i o n pump 6 To a i r R o t a r y pump To c r y o s t a t P i r a n i gauge L i q u i d N i t r o g e n t r a p to a i r Hg O i l Manometers F i g u r e Til.'*>". Vacuum sys tem and H p - gas thermometer . - 38 -The c r y s t a l was mounted in a copper can of 4 cms in diameter, which was fixed with Woods metal to two coaxial cables s p e c i a l l y made from German-silver tubes. The schematic diagram (F i g . III. 6) shows the structure of the can and the position of the ^crystal and the two R F co i l s . Inside the can, two copper plates each about 1 mm. thick, were fixed by Woods metal to one of the german-silver tubes. A hole 1. 1 cm in diameter was made in each plate to accommodate the c r y s t a l . The c r y s t a l was supported by an insulated copper wire, which was soldered onto the copper plate. The two R F coils were connected to the central wires of the coaxial tubes. The coils served as the heater and the ther-mometer c o i l as described in the f i r s t section of this chapter. When direct heating was used, the heater c o i l was replaced by a thin copper wire soldered to the constantan heater f i l m and the supporting copper wire was used as a ground lead. The cap of the can was fixed r i g i d l y to the german-silver tubes. The lower part of the can could be unscrewed. A brass tube (1/2" o. d. , 3/16" i . d. 7 cm long) connected to a copper rod (diameter 1 cm, about 30 cm long) was screwed onto the bottom of the can. The dimensions of the tubes were designed so that when the bottom of the rod was immersed in the l i q u i d helium bath, and a s m a l l amount of power was applied to the con-stantan heater H (Figure III. 6), one could control the temperature of the can to some extent in the range 4. 2°K to 35°K. The whole assembly could be r a i s e d or lowered according to the l e v e l of the l i q u i d helium in the dewar. -19-F i g u r e I I I . 6 . S t r u c t u r e o f t h e can and the p o s i t i o n o f t h e c r y s t a l . - 40 -Key to Figure III. 6 1. Coaxial cables made of German-silver tubes 2. Copper cap 3. Woods metal joint 4. Copper cylinder 5. Copper plates 6. Constantan heater for the can 7. Brass tube (connected with a copper bar to Helium bath) 8. Indium f i l m thermometer 9. R. F. coils 10. A 1 2 ° 3 C I T s t a l 11. Heater for c r y s t a l (metal film) - 41 -III. 4. Method of measurement. I We would l i k e to describe the methods used i n this project in terms of two main categories: f i r s t l y , those used at 3. 8°K at the trans-ition temperature of the indium f i l m , and secondly, those used between 4. 2°K and 35°K. Both continuous waves (C. W. ) and pulses were applied to the c r y s t a l . III. 4.1. Measurements at 3. 8°K After the helium transfer, the helium dewar was pumped by a Kinney mechanical pump to a pressure of about 20 cm of Hg and then the fine control valve of the pumping line was slowly turned down so that the pressure was kept roughly constant. With some practice the pressure could be maintained within a few mm of the Hg of the pressure correspon-ding to the tra n s i t i o n temperature, T^, of the indium f i l m . The heat pulses or C. W. were applied from the beginning of the pumping so that the heat input was already present when the pumping rate was adjusted. When the temperature was f a i r l y constant near T^, the Anderson bridge was adjusted to balance at 30 Mc/S and the output signal was then displayed on an oscilloscope. F o r pulse inputs, pictures'of the traces obtained were taken with a P o l a r o i d camera. With the output from the oscilloscope being fed into the box-car integrator, it was possible to plot the trace on a recorder for more accurate measurements. To achieve even better temperature stability, the c r y s t a l tem-perature controller (which was discussed in detail in the III. 1. 3. 3. ) - 42 -was used for C.W. input. In this case the output from the oscilloscope could be fed into a P A R L o c k - i n detector. The amplitude of the signals for different frequencies were read f r o m the PAR. The results w i l l be discussed in the next chapter. At the transition temperature T^ of the indium f i l m one obtained the highest thermometer sen s i t i v i t y in the whole temperature range. However, due to the non-linearity and the narrow temperature range of the transition, the temperature controller was difficu l t to operate, land moreover the true shape of the r i s i n g pulse was uncertain. In other words, the pulse shape would depend on the operating point on the transition curve. III. 4.2. Measurements f r o m 4. 2°K to 35°K. The main problem in performing the experiment between 4. 2°K and 35°K was to achieve and maintain the desired temperature for a period of time. The procedure was to f i r s t l i f t the can to a certain height above the helium bath and then apply heat by means of the heater H (F i g . III. 6). At the same time the temperatures inside and outside the can were read f r o m the thermocouple and carbon r e s i s t o r respectively. The power for heating was gradually turned down as the temperature gradually increased. With the proper amount of heating one could keep the c r y s t a l temperature constant to within a degree. The R. F. bridge could then be adjusted to balance and the c r y s t a l temperature stabilized by the R F power.as described in section III. 1. 3. 3. After the temperature was stabi l i z e d at the desired value, the measurement was taken with either a P A R L o c k - i n - 43 -a m p l i f i e r or a box-car integrator, depending on whether C. W. or pulses were applied. In p r i n c i p l e at higher temperatures and with large signal inputs, one could get signals up to quite high frequencies (e. g. 100 Kc/s for C. W.). However, in the actual experiment it is very di f f i c u l t due to some p r a c t i c a l limitatio ns . In the f i r s t place, increasing the signal input would cause the whole c r y s t a l to r i s e excessively in temperature, making the temperature sta b i l i z i n g di f f i c u l t . Secondly, for C. W. , the bridge balance and the I F am p l i f i e r gain control must be kept constant for a l l frequencies at one temperature i n order to compare the amplitude at different frequencies. A big temperature change of the c r y s t a l would drive the bridge off balance so that the temperature s t a b i l i z e r would lose c o n t r o l . S i m i l a r trouble occurred for pulses as well, in the course of the integration of output traces on the recorder. Thirdly, in the present set up, which uses R F coils for input and output signals, the direct pick-up due to a large input would be dif f i c u l t to avoid i f a short c r y s t a l were used. These limitations became more serious at higher temperatures when the signal attenuation was l a r g e r . In fact, we have almost reached the l i m i t with the present arrang-ment using thin f i l m thermometers with the R F bridge, I F amplifier and the box-car integrator. It would appear that more accurate measurements of the s m a l l temperature changes at high frequencies involved calls for a new and - 44 -m o r e s e n s i t i v e type of t h e r m o m e t e r . A t t e n u a t i o n m e a s u r e m e n t s w ere made up to 10 K c / s at a r o u n d 20°K i n a s e a r c h f o r a r e g i o n of constant attenuation, as i n F i g . II. 1(a), w h i c h would be e v i d e n c e f o r the e x i s t e n c e of s e c o n d sound. However, b e c a u s e of the d i f f i c u l t i e s m e n t i o n e d above, at 10 K c / s the s i g n a l was down to the n o i s e l e v e l , and o v e r the range of r e l i a b l e m e a s u r e m e n t , the r e s u l t s m e r e l y s e r v e d to c o n f i r m that the attenuation i s p r o p o r t i o n a l to ( f r e q u e n c y ) . M o r e o v e r i n C h a p t e r I V i t w i l l be shown ( T a b l e II) that i t w o u l d be n e c e s s a r y to use f r e q u e n c i e s above 100 K c / s to o b s e r v e s e c o n d sound, o r i n d e e d that the n e c e s s a r y 'window' m a y not even e x i s t . H e n c e the d i s c u s s i o n i n C h a p t e r I V w i l l be l i m i t e d e n t i r e l y to the p u l s e e x p e r i m e n t s . - 45 -C H A P T E R IV. E X P E R I M E N T A L R E S U L T S A N D DISCUSSIONS IV. 1. Introduction Heat pulse and heat flash experiments (Das and Hossain 1966; Parker et al 1961) have been performed at room temperature and higher temper-atures on many different solids. The results have been interpreted quite satisfactorily by the ordinary diffusion equation. However, this equation fails to explain our results for heat pulses at low temperatures because it gives a velocity that tends to infinity as the frequency increases (see II. 5). The reason that the ordinary diffusion equation describes room temperature pulse shape rather well is that the delay time (the time it would take a phonon signal to reach the detector f r o m the generator) is usually negligibly smal l compared with the thermal transient response time of the materials in question. Furthermore , the high frequency components are highly attenuated so that the actual delay time is not sharply defined as it is at low temperatures. In this chapter, the results for heat pulses and continuous-wave signals in an A l ^ O ^ single crystal will be presented and their interpretation will be discussed. The presentation of results wil l be divided by temper-ature regions into three parts: (1) at 3. 8 ° K , (2) 4. 2°K «£ T < 8 ° K , (3) 8 ° K <- T < 3 5 ° K . The results for (1) and (2) and a discussion of their physical significance will be presented in IV. 2. At the end of IV. 2 an example wil l be discussed which illustrates nicely our special interpretation, - 46 -namely the application to pulses in He II below 0. 6°K, with p a r t i c u l a r reference to the observations of K r a m e r s et al (1954). The results for region ^(3) w i l l be given in IV. 3. The derived thermal conductivity, K, at different temperatures w i l l be presented and compared with Berman's results for s m a l l e r c r y s t a l s . Before commencing the detailed description, we would l i k e to indicate an interesting feature of the results and interpretation for the region of long phonon mean free path. Ward and Wilks (1951) pointed to the possible existence of second sound i n a solid, as well as in l i q u i d He II, for the region of short mean free paths for normal c o l l i s i o n s . Moreover, Fairbanks and Wilks (1955) showed that the mechanism for DC thermal conductivity in l i q u i d He II below 0. 6°K is identical to that for a d i e l e c t r i c s o l i d for the region of long mean free paths l i m i t e d by boundary scattering. It seems fr o m our interpretation that heat transport i n these two media for high f r e - , quencies or short pulses are also very s i m i l a r in the boundary scattering l i m i t . IV. 2. Low temperature results and their interpretation. IV. 2. 1. Modified heat equation and its solution. The F o u r i e r equation for heat conduction l)t ^ (4.1) (where the thermal diffusivity, D = K/^Cv) predicts a velocity for heat waves proportional to the square root of the frequency, and thus increases without l i m i t for increasing frequency. However, it i s evident that there must be an upper l i m i t to this velocity, namely the phonon velocity. - 47 -As was discussed in Chapter II, U l b r i c h (1961) made an analogy with an e l e c t r i c a l t r a n s m i s s i o n line and predicted a finite upper l i m i t to the velocity of the F o u r i e r components of a heat pulse. The modified equation Osborne (1951) proposed to use the same equation to describe pulses of second sound in l i q u i d He II below 0. 6°K, in order to explain i n a general way the observed finite delay time. In this thesis we propose to make a specific interpretation for ' t , applicable in the low temperature l i m i t , and t r y to use equation (4. 2) and its solution to explain the detailed shape of the experimental traces. Following Osborne (1951) we use the Laplace t r a n s f o r m to obtain the solution of equation (4. 2) for the response to a unit ^ - f u n c t i o n input at x = o, and find that for the end of the sample x =.£ (4.2) gives the l i m i t i n g velocity of propagation for high frequencies where and b H(y) = Heaviside's step function = o for y <^ o = 1 for y > 0 are Bess el functions of the f i r s t kind, - 48 -The summation represents an expansion i n terms of a series of echoes, which for p r a c t i c a l use may be terminated at j = 1. The f i r s t t e r m within the summation bracket is the o r i g i n a l 5 "-function propagated at velocity without distortion, but attenuated by a factor • The second t e r m with-in the bracket is a generalization of the solution of equation (4.1) with this important distinction - it is not allowed to come into operation until X =• ~Tf~ i.e. u n t i l the a r r i v a l of the c P -function, and is suppressed at e a r l i e r times by the Heaviside function. We notice that the solution E q (4. 3) i s s l i g h t l y different f r om the one in C a r s l a w and Jaeger (1947) for an e l e c t r i c a l t r a n s m i s s i o n line, because we are interested in the voltage response to a S~-function current input rather than the current response to a voltage pulse input. It is to be emphasized that we are t r y i n g to use the modified heat equation and its solution i n a different situation from the usual case where the equation was used for second sound propagation (Osborne 1951; Chester 1963; Prohofsky and Krumhansl 1964). These authors were concerned with the temperature region of short mean free path (for normal collisions) when IT is the velocity of second sound rather than f i r s t sound as we use. Here we are .'dealing with the problem of direct phonon tra n s m i s s i o n and its associated diffusive s c a t t e r i n g due to the boundaries or imperfections of the c r y s t a l s . It is interesting to see that the boundary scattering case may be represented by a differential equation of the same form as that dealing with the effect of second sound propagation. This gives us the chance to use the same mathe-- 49 -mati c a l solution with quite different values and physical meanings of the parameters involved i n dealing with these two different physical situations. IV. 2.2. Experimental results at T = 3. 8°K At the superconducting t r a n s i t i o n temperature of the indium f i l m we have the most sensitive range of the thermometer, therefore th$ signal is l a r g e r at this temperature than at a l l other temperatures. This gives an opportunity to use very narrow input heat pulses (epg. 200 nanosec) which contain a large proportion of very high frequency F o u r i e r components. Some ty p i c a l pictures taken at this temperature are shown in (a) and (d) of F i g . IV. 1. It i s also possible to trace out the details of the pulse on the recorder through the box-car integrator for better delay time measurements. One longitudinal and two barely resolved transverse pulses were observed. 5 5 Their velocities are = 9. 5 x 10 cm/sec and C^ . = 6. 35 x 10 cm/sec. These values of velocities are for a 3. 8 cm long, 1 cm diameter, 60° cut single c r y s t a l of A1_0 and are i n rough agreement with the results obtained by von Gutfeld and Nethercot (1964) on a Z-cut c r y s t a l . Some ringings or oscil l a t i o n s following the transverse peak were observed ( F i g . IV. 1 (a)). This might be some kind 1 of acoustic resonance of the F i t z g e r a l d (1965) type in the c r y s t a l . No further investigations were made on this point. At 3. 8°K in pure A1_0 the phonon mean free path for U-type co l l i s i o n s is mainly l i m i t e d by the boundary of the sample (or imperfections and impurities in the cr y s t a l ) . Some of the phonons generated at one end of the c r y s t a l go d i r e c t l y to the r e c e i v e r without collid i n g with the wall or imperfections i n the c r y s t a l . This sharp r i s e corresponds to the f i r s t part (a) Time s c a l e : 2 ^ i s e c / c m V e r t i c a l s c a l e : 0 . 2 v / c m (b) Time s c a l e : 2 ^ i s e c / c m V e r t i c a l s c a l e : 0 . 1 v / c m ( c ) Time s c a l e : 5^usec /cm V e r t i c a l s c a l e : 0 . 2 v / c m F i g u r e I V . 1. P i c t u r e ^ o i ' r e c e i v e d p u l s e s a t 3 . 8 ° K f o r 3.& cm l o n g A 1 2 0 ^ , showing a r r i v a l o f l o n g i t u d i n a l and t r a n s v e r s e d i r e c t phonons . - 51 -of the solution (eq. 4. 3). Other phonons are diffusively reflected by the wall or collide with imperfections several times before they reach the detector in a kind of "random walk"(Ziman 1954). This forms the "slow general r i s e " of the received pulse and corresponds to the second part of the solution (eq. 4. 3). If the frequency response of the indium thermometer were perfect, one would get two attenuated o r i g i n a l pulses reproduced at the delay time corresponding to. the longitudinal and transverse vel o c i t i e s . Therefore it is possible to estimate the thermal response time of the thin f i l m thermo-meter and the detecting system f r o m the r i s e time of these individual pulses estimated that the thermal response time of the thermometer and the detec-ting system was shorter than 1 J^SSC. S i m i l a r work was reported by von Gutfeld, Nethercot and A r m s t r o n g (1966). The method which they used to measure the thermal response time was by heating the evaporated thin f i l m s s l i g h t l y above the ambient temperature by the use of a short pulse of light f r o m either a suitably attenuated giant pulse ruby l a s e r or from a GaAs diode. The ambient temperature for the In-Sn f i l m s was usually around 3. 8°K. The t y p i c a l thermal response times they obtained for 1000 A thick In-Sn f i l m evaporated on theZ- or x-cut A l O, were around 16-17 nanosec It is rather di f f i c u l t to interpret the general shape of the pulse c o r r e s -ponding to the second part of the solution, for the following reason. One does not know the actual shape of the "general r i s e " because of the non-(assuming that the input pulse is a perfect Hence it was - 52 -l i n e a r i t y of the transition curve of the r superconducting thermometer near its transition temperature. (Presumably one could reconstruct the pulse: if one did a careful measurement of resistance as a function of temperatures near the tr a n s i t i o n temperature. ) Furthermore, the "general r i s e " part of the received pulse shape depended very much on the position of the operating point on the c superconducting tr a n s i t i o n curve of the indium f i l m . Quite different r i s e times for the general r i s e were obtained i n different pictures of the observed pulses f rom different runs. Occasionally one could get very strange pulse shapes l i k e those shown i n (b) and (c) of F i g . (IV. 1). IV. 2. 3. Experimental results between 4. 2°K and 8°K. In this temperature region we are using the thermometer as a normal metal resistance thermometer. A ty p i c a l box-car integrator trace is shown in F i g . IV. 2. Smoothed traces f r o m the recorder were reproduced on graph paper with normalized amplitude (see F i g . IV. 3). The results in this temperature region showed no change in the delay time of the i n i t i a l sharp r i s e (corresponding to phonon velocity). However, the amplitude of this i n i t i a l sharp r i s e as compared with that of the general r i s e was decreasing with increasing temperature. This behavior is best described with the help of F i g . II. 2 and II. 3, which are the rough plot of the current response to a step voltage input for the t r a n s m i s s i o n line analogue. As the temper-ature increased f r o m 3. 8°K to 8°K the value of the attenuation coefficient (T increases gradually. This results in a decrease of the relative amplitude of the i n i t i a l sharp pulse compared with the general slow r i s e of the second part of the received signal. -J>3-0 $ /O fS 20 Z T 30 77/V£ (/Msec) FiQu.relV.2 Box-car integrator tya.cz at 4. T'K in. AL Oj. TkcortcjiYi oi ike. time scale, is the. start of -fhe l/*sec ijifout heat pulse. In the present case, propagation is by (a) d i r e c t t r a n s m i s s i o n of phonons, (b) scattering f r om the walls, both specular and diffuse, and (c) inter n a l scattering, both s m a l l and large angle. Diffuse and large angle scattering might be expected to be associated with slow propagation through a c r y s t a l in the same way that phonon second sound velocity acquires the factor I /sJ~3 . The presence of the direct pulse maybe seen c l e a r l y at the lowest temperatures (see Figs IV. 1 and IV. 2) and these figures are evidence for our suggestion that it is the phonon velocity (suitably averaged over the modes) that determines the beginning of the general r i s e , so that we make the identification U~0 = C*j W r i t i n g the thermal conductivity as , where [_ is a mean free path, would then suggest O that identification of V0 ~(D/t) with the phonon velocity requires that 3 t be the mean time between momentum-loss c o l l i s i o n s . We interpret the received pulses in the following way. The solution of the modified heat eq. (4.2) has two quite separate parts. By comparing the experimental traces obtained at 5. 0°K ( F i g . IV. 3) with the solution (eq. 4. 3) it i s easy to see that the f i r s t part of the solution corresponds to direct phonon t r a n s m i s s i o n (the i n i t i a l sharp rise) and the second part of the solution gives the general r i s e . Since the direct phonon pulse is attenuated, we concentrate on the second part. With the help of an I B M 7040 computer, the second part of the solution could be calculated with certain values of D and \f0 and then used for comparison with the experimental traces. The exact \f required to do the calculations is not known. Since - 56 -there is only one \T involved in the solution (eq. 4. 3) it is not clear what would be the appropriate average value even for the case of an i s o -tropic sol i d , which has two values of the phonon velocity. One of the methods to estimate D was given by C a s i m i r (1938) where we assume an isotrop i c s o l i d and sum over the longitudinal and transverse modes. Using the values Ifg = 9.4 x 10^ cm/sec and 2/£ = 6. 35 x 10^ cm/sec, the corresponding value of D f r o m eq. (4.4) 5 2 is D= 2.25 x 10 cm /sec. To estimate *~CZ we assume an average value for lf0 i n \T0 - (P/t ) /^ i f = 6. 8 x 10^ cm/sec, the same "average" that i s used i n eq. (4.4). The solution computed from eq. (4.3) is shown in F i g . IV. 3 and contains no adjustable parameters. The agreement with experiment is poor, and much better agreement is obtained by fitting para-meters i f and D. A special program for least square fitting of this highly nonlinear expression was made. However, the solution failed to converge by iteration. Therefore the tedious method of fixing one para-meter while the other parameter was changed by s m a l l steps, and then changing the f i r s t parameter in s m a l l steps and so on, was used. The shape of the curve is quite sensitive to the value of D but not too much to VI , and 5 2 r F i g . IV. 3 shows a curve for D = 3. 15 x 10 cm /sec and U0 = 6. 34 x 5 o 10 cm/sec which fits the experimental curve at 5 K rather well. It is possible that since the C a s i m i r formula for D is for an in f i n i t e l y long per-fect c r y s t a l with i d e a l l y rough walls, the r e l a t i v e l y short sample used here could give a higher effective D (see F i g . IV. 11) esp e c i a l l y i f there is much - 57 -specular reflection at the polished boundaries of the c r y s t a l . Enhanced values of D f r o m this cause were also noticed i n the thermal conductivity measurements of A l 0 by Berman, Foster and Ziman (1955). i IV. 2. 4. An example - Propagation of heat pulses in l i q u i d He II below 0. 6°K Heat pulse experiments i n l i q u i d He II below 0. 6°K (e. g. see Atkins 1959) show very interesting behavior of the propagation of thermal waves. Landau (1941) predicted that the second sound velocity at 0°K should approach "^o/ obtained above at 0. 2 5 K, we made use of the formula (see section II. 5. Ot Chester, 1963) f = — — — - = — ^ — , which gives f =11.8 Kc/sec for K ramers' sample. It can be shown that f obtained f r o m this formula i s approximately equal to the frequency at which the velocity reaches 9 0 % of its maximum value as shown i n the following dispersion curve, which is de-r i v e d f r o m the modified heat equation (4. 2) I V e l o c i t y (log scale) f frequency (log scale) c This value cf f is in? reasaiefcbeagreement with the very approximate-value of c 10 Kc/s which we obtained by extrapolating the continuous wave results of Osborne (1956) to the high frequency l i m i t , ; assumed to be for this tem-perature. Quantitative agreement is not expected for the following reasons: (1) the dimension of the containers used for the two experiments (Kramers et al, 1954 and Osborne, 1956) were very different, and, at these temper-atures, D and lT depend very much on the dimensions as well as the surface condition of the container, (2) the highest frequency in Osborne's experiment was 2 Kc/s, and appreciable e r r o r is unavoidable, when one extrapolates to higher frequencies on the log-log plot. • - 62 -F r o m the continuous-wave results (Osborne 1956) one obtains the following: (1) there was appreciable change in upper l imiting velocity 1/max as the temperature was varied f r o m 0. 2 5 ° K to 0. 5 8 ° K . The velocity decreased gradually as the temperature increased. (2) The transition f r e -quency f became much lower as the temperature increased f r o m 0. 2 5 ° K to 0. 5 8 ° K . In other words, the attenuation constant 0" ( 3£>°K). This i s not unreasonable, since the theory of thermal conductivity indicates that at temperatures higher than the temperature of the peak value of K, the Umklapp-processes dominate, whereas at temper-TEMPERATURE (°K) - 7 3 -TEMPERfrTURE (°K) F i g u r e I V . 1 2 . T e m p e r a t u r e dependence o f t h e t h e r m a l c o n d u c t i v i t y o f A ^ O ^ . x E x p e r i m e n t a l p o i n t s f o r 1 cm d i a m e t e r c r y s t a l . o B e r m a n ' s (1955?) r e s u l t f o r u .254 cm d i a m e t e r c r y s t a l . - 74 -atures far below the peak, boundary scattering is important. Hence the K values should not differ too much i n cr y s t a l s of different size above the tem-perature of thermal conductivity peak. F r o m the values of D (see F i g . TV. 11), one could also deduce the mean free time /~C and hence the c r i t i c a l frequency f above which the change in dispersion would occur. A few f for different temperatures were estimated and are shown in Table II. F o r temperatures lower than 18°K, we use the average phonon veloc i t y for U~ , while above 18°K we assume the value (Landau 1941) for order of magnitude calculations. ^3 F r o m these values of f it is not s u r p r i s i n g that our continuous-wave c results did not show any change in the dispersion. As one goes to frequencies higher than a few Kc/s the attenuation becomes so big that even with a phase sensitive detector and using a short c r y s t a l one could not measure the signal with any accuracy with the present detecting system. The results at 3. 8°K were exceptional as here we were using a sensitive superconducting thermometer and . the signal was observed up to .1.50: K c / s . However, f r o m Table II, f estimated at 3. 8°K was about 216 Kc/s, and one c would have to make considerable improvements to the apparatus i n order to measure this high frequency signal. This kind of information about f at different temperatures which shows c the frequency range one should work i n at a given temperature in order to observe second sound in Al^O^, has not previously appeared i n the l i t e r a t u r e . These f values are much higher than one would have thought, as one - 75 -T A B L E II. O R D E R O F M A G N I T U D E E S T I M A T E S O F f ( = l/ZT'C = V^irP ) A T D I F F E R E N T c T E M P E R A T U R E S V used T empe r atur e s T ( ° K ) Diffusivity f D (cm /sec) (C/sec) 5 o 6. 8 x 10 c m / s e c 3. 8 K 3. 4 x 10" 2.16 x 10" 6. 8 x 10 5 c m / s e c 5 ° K 3. 15 x 10' 2. 33 x 10" 5 o 6. 8 x 10 c m / s e c 9 K 1.6 x 10" 4.54 x 10' 6. 8. x 10 5 c m / s e c 1 6 ° K 8 x 10 9.20 x 10" 6. 8 x 10 5 c m / s e c 3 2 ° K J3" 1.6 x 10 1. 53 x 10 WA (a) Time scale: 10 ^ isec/cm V e r t i c a l s cale: 2v/cm (b) Time scale: 5^usec/cm V e r t i c a l s cale: 2v/cm (c) Time scale: 20^usec/cm V e r t i c a l scale: 2v/cm (d) Time scale: 2^usec/cm V e r t i c a l s cale: i?v/cm (e) Time scale: lyusec/cm V e r t i c a l scale: 2v/cm ( f ) Time scale: 10^usec/cm V e r t i c a l s cale: 2v/cm Figure IV.13. Pictures of received pulses f o r NaF c r y s t a l , (a) (b) ( c ) : Single c r y s t a l of 2 cm long at 6.6°K. (d) (e) ( f ) : Single c r y s t a l of 0 . 9 2 cm long at 3.8°K. - 77 -would hope that at the temperatures just below the thermal conductivity peak temperature the Umklapp-processes should be r a r e and the frequency f should be quite low, just as i n the case of l i q u i d He II near 0. 6°K. F r o m the values predicted by Prohofsky and Krumhansl (1964) for f i n NaF at c different temperatures (see section II. 2) one would expect to see second sound at p r a c t i c a l l y any frequency below the temperature of the thermal conductivity peak. Apparently these values are too optimistic. F r o m the p r e l i m i n a r y results obtained with pulses i n NaF at 3.8°K and 6. 6°K (as shown i n F i g . IV. 13) it was found that the f values estimated at these temperatures were around 110 Kc/s rather than few c/s as predicted by the above authors. It would be interesting to see how f changes with temperature when we approach the temperature of the thermal conductivity peak of these c r y s t a l s . IV. 4. General comments on the results and the modified heat equation. It is interesting to consider the analogy between phonons i n a s o l i d and the molecules i n a gas (see Table III). Three different cases w i l l be discussed here. Case 1: Ordinary sound i s the propagation of density variations i n the molecular gas ( l a ) . In a s o l i d or l i q u i d He II this corresponds to second sound which is the propagation of density variations in the phonon gas (lb). Case 2: In the case of a gas with viscous damping a p a r t i c l e density fluctuation w i l l decay with a ty p i c a l relaxation time 7^JT . In other words, at low frequency {0J1l« / ) any density fluctuation would r e l a x with diffusion (2a). In a s o l i d this corresponds to low frequency - 78 -T A B L E III. COMPARISON B E T W E E N P R O P A G A T I O N P R O P E R T I E S IN M O L E C U L A R GAS AND PHONON 'GAS' a. molecular gas b. phonon 'gas' Case 1. l a . F i r s t sound i n a gas l b . Second sound i n He II or s o l i d Case 2. 2a. Diffusive or viscous damping of sound i n a gas 2b. Diffusive damping of phonon gas - ordinary-thermal conductivity Case 3. 3a. Propagation of a disturbance i n Knud-sen gas 3b. Direct phonon propa-gation i n s o l i d or He II (below 0. 2 5°K) - 79 -thermal diffusion in a phonon gas or ordinary thermal conduction. Case 3: F o r a Knudsen gas of molecules (case 3a) where the number of molecules are smal l and the mean free paths are long, the propagation velocity of a disturbance is approaching individual molecular velocity (e.g. like a molecule in molecular beam). In this case we no longer have a "sound wave" as in cases 1 and 2. It is known f r o m kinetic theory that isothermal compressional waves in an ideal gas of molecules propagate with a velocity C which is equal to 1 / t i m e s the r . m . s. average speed C m Q j ° f the molecules ( i . e . C = h/ ... ,. 6, - 80 -If we introduce the relaxation time . 2 ' - * (4 .7, by combining equations (4. 5) and (4. 6) we have one could readily see that at high frequency (i. e ' ^ r ^< OTodt»/) eq. (4. 8) behaves like a wave equation applicable to case (1). At low frequency ( -1 i^^f » ^ ^ o r ^ < / ) it behaves like a.diffusion equation applicable to case (2). It is to be noted that here if is the propagation velocity for density fluctuation, i . e. the velocity of ordinary sound when applied to a molecular gas, but velocity of second sound when applied to a solid or liquid He II. We note that eq. (4. 8) for density fluctuations in a molecular gas can be derived from the hydrodynamic equations, namely the equation of continuity and Euler's equation. By eliminating the density between these two equations one obtains eq. (1.1) for the momentum: the equation of identical form for the density (eq. ( 4. 8)) is obtained by eliminating the momentum instead. If one applies eq. (4. 8) to the phonon number density N, it does not lead directly to an equation for the temperature T, although evidently there is a close connection between eq. (4. 8) and the modified heat equation (4.2). In fact one should derive the equation for T from the Baltzmann transport equation as was done by Guyer and Krumhansl (1964). Now we concentrate on case (3), the Knudsen region in which gas - 81 -molecules (phonons) can transport energy f r o m source to detector with velocities up to the molecular velocity (1st sound velocity). It i s r e a d i l y proved that a Knudsen gas with diffuse scattering at the boundary of an i n f i n i t e l y long cylinder, obeys the diffusion equation ~ ^ = P - j - $ r inhere p ^ f t r r (4.9) where v is the average molecular velocity and r the radius of the cylinder. Casimir's (1938) calculation of the heat flow i n the boundary scattering l i m i t using a black-body radiation argument, leads to essentially the same expression, with the p a r t i c l e density replaced by temperature T and with exactly the .same expression for D. Now C a s i m i r assumes that conditions are close enough to the steady state that there exists a w e l l defined.local temperature and i n this he is followed by Ziman (1954), i.e. Casimir's derivation assumes a 're-radiation' of phonons at the boundaries, whereas a s t r i c t analogy with a Knudsen gas requires a random scattering in which the phonon is scattered l i k e a p a r t i c l e . It seems unlikely that an e q u i l i b r i u m temperature distribution exists along the c r y s t a l boundaries under the conditions of the experiment and so the C a s i m i r approach is of doubtful va l i d i t y . One might expect that one i s dealing with 'hot' phonons and that an analagous expression to eq. (4.9) represents the situation more r e a l i s -t i c a l l y . Now i f we postulate that the detector measures a parameter ''T' and that changes in the l o c a l value of N in the c r y s t a l are reflected in variations of 'T' and that to a f i r s t approximation the relationship is li n e a r , then the expression (4.9) i s identical to the C a s i m i r expression and the question of l o c a l e q u i l i b r i u m i s avoided. - 82 -In order to explain the experimental results it i s necessary to modify equation (4.9), since as i n the high temperature case the velocity cannot increase indefinitely as (frequency) . The same objection can be r a i s e d even in the case of a molecular gas, though we can find no discussion of this point. Hence we adopt the same expedient as U l b r i c h (1961) did for the veloc i t y for high frequency (D/'tr) must be just the phonon velocity which leads to the identification of 2>HX as the 'mean free time' between boundary scatterings, as noted e a r l i e r (section IV. 2. 3). In l i q u i d He II, one can demonstrate a l l the three cases (1), (2) and (3) of Table III rather well. However, it i s diffic u l t to show a l l these ch a r a c t e r i s t i c s in a sol i d . In l i q u i d He II, above 0.9°K case (lb) applies since the value for is v e r y long, and one gets second sound propagation at almost a l l frequencies. When the temperature approaches 0. 6°K, case (lb) and(2b) are both applicable depending on the frequency. F r o m Osborne's (1956) results for continuous waves one could see that the c r i t i c a l frequency f became higher and higher when one decreased the temperature from 0. 6°K to 0. 3°K. It i s to be noted that between 0. 6°K and 0. 25°K the velocity i f gradually changes from second sound velocity to phonon velocity. In other words, when eq. (4.8) is applied here both i f and change with temperatures, un t i l If becomes the phonon velocity. Below 0.25°K (see K r a m e r s et al_1954) we have case (3b). Quite good quantitative agreement can be obtained i n case (3b) for He II. However, due to several different factors i n a solid , it i s not easy to see a l l the c h a r a c t e r i s t i c s in Then the l i m i t i n g - 83 -these three cases, and only qualitative results were obtained. Through the comparison between the two systems (liquid He II and solid) one could get quite useful information and insight into heat transport i n these media. For example, the study of the heat propagation i n a s o l i d in case (3b) (sections IV. 2 and TV. 3) gave us the explanation of heat transport in He II below 0. 6°K where the experimental results had not been adequately studied. F u r t h e r -more, as discussed i n section IV. 2.4, one r e a l i z e d that l i q u i d He II i s a much better medium than a s o l i d for second sound propagation. The relaxation time ' " t is very much longer in l i q u i d He II than that i n a solid. This means one should go to a much higher frequency (and therefore suffer much higher attenuation) in order to see s i m i l a r effects in a soli d . In the present work no evidence was found for the existence of second sound i n solids. However, very recently Ackerman et a l (1966) reported that in s o l i d helium heat pulses were observed to propagate with approximately o 4 second sound velocity at 0. 54'K. Of the 13 He cry s t a l s studied, only 4 which appeared to be single c r y s t a l s actually showed this behavior. It was mentioned that the structure of the a r r i v i n g temperature pulse at the lowest temperature (0. 54°K) was not ent i r e l y understood. In the picture of the received heat pulse at 0. 54°K, a sharp r i s e followed by a slow general r i s e was shown. This behavior, i n fact, could be represented by the solution of the modified heat equation (eq. 4. 8). F r o m their p r e l i m i n a r y results and the discussion i n this section, i t can be concluded that a l l three cases in (b) of Table III can i n fact be observed in d i e l e c t r i c solids, i f a single c r y s t a l sample is available with minimum diffusive scattering. - 84 -C H A P T E R V. CONCLUSIONS In this thesis, the results of a study of heat pulses in A l 0 c r y s t a l s at low temperatures have been presented. These were studied with the ultimate aim of gaining further insight into the nature of heat transport in solids at low temperatures. To achieve this aim, a new way of detecting very short heat pulses without making direct e l e c t r i c a l contact with the sample was developed, with the hope that through the conventional theory of heat, conduction and the current theories of second sound in solids, information about heat pulse t r a n s m i s s i o n could be obtained. Instead, it was found that the conventional theory of heat conduction is inadequate at the lowest temperatures. To explain the finite time of t r a n s m i s s i o n of a heat pulse f r o m the heater to the detector, an additional t e r m should be added to the diffusion equation and this i s true for the boundary scattering region as well as in the region of ordinary thermal diffusion. An attempt was made to use this modified heat equation and its solution, and the consequences of this modification were examined in some detail. The following consequences of this phenomenological approach were established in Chapter IV: (1) The solution of the modified heat equation for the response to a a generalization offliesolution of ordinary diffusion equation but begins at unit one i s the undistorted but atten-uated o r i g i n a l if . The other is - 85 -time t The results for heat pulses in A l 0 from 3. 8 K to 8 °K c l e a r l y showed two quite different parts which were associated with the corresponding parts i n the solution mentioned above. The velo c i t y V was an average phonon velo c i t y over different modes. The solution was also calculated for l i q u i d He II at 0. 25°K and com-pared with the observed heat pulses of K r a m e r s et al(l954). V e r y good agree-ment was obtained. These results have been published i n abbreviated f o r m (Brown, Chung and Matthews, 1966) (2) With the help of a computer, the solution was calculated and com-pared with the received pulses. The thermal diffusivities, D,at different temperatures were found, and the apparent thermal conductivities K were also calculated. The values for K were plotted against T and compared with Berman's (195£) results for A l 0 of different size, (Figure IV. 12). C* j (3) The frequencies, f , at which the change occurred f r o m a l i n e a r 1/2. relationship between velo c i t y or attenuation and (frequency) , were estimated for different temperatures, establishing a lower frequency l i m i t for the onset of second sound, (Table II). (4) The comparison between heat pulse t r a n s m i s s i o n in an A l ^ O ^ c r y s t a l and in He II below 0. 6°"K was discussed and a reason why it is more favorable to have collective phonon waves in l i q u i d He II than i n a s o l i d was also suggested. (5) The s i m i l a r i t i e s between the propagation of electromagnetic waves - 86 -in an e l e c t r i c a l t r a n s m i s s i o n line and heat tr a n s m i s s i o n i n a s o l i d at low temperatures were discussed. The relaxation time, ' t , for momentum-loss c o l l i s i o n s of phonons in a s o l i d i s related to the c h a r a c t e r i s t i c time L/R. in a t r a n s m i s s i o n line. The following conclusion can be drawn from the study of heat pulse tr a n s m i s s i o n i n a s o l i d at low temperatures. The finite propagation velocity and relaxation processes of phonons are responsible for the finite delay time and the shape of the received heat pulses. The ordinary diffusion equation cannot explain this finite delay time. The modified heat equation should be used and i n the low temperature l i m i t , the parameter should be the phonon veloc i t y rather than second sound velocity. To our knowledge, this is the f i r s t time that this kind of interpretation has been applied in heat transport problems in s o l i d and l i q u i d He II at low temperatures. In conclusion, some further experiments w i l l be proposed: (1) It would be interesting to measure the velocity and the attenuation with continuous waves for a number of pure single c r y s t a l s of d i e l e c t r i c s o l i d at low temperatures. By going to high enough frequencies (depending on the c r y s t a l and the temperature in question), one should be able to observe the change-over i n the dispersion curve. With a • superconducting thermometer, it is possible to measure continuous waves up to a few Mc/s. In some di e l e c t r i c c r y s t a l s whose Debye temperatures are low and which therefore have lower temperatures for the thermal conductivity peak, one could, with • superconducting thermometers, measure f at different temperatures up to the region where second sound is most probably to be observed. (2) - 87 -The method of using light scattering i n detecting second sound proposed recently (Guyer, 1965) could be used to detect thermal waves i n l i q u i d He II 4 4 3 o (either pure He or He - He mixture) below 0.6 K. The comparison of the results with those obtained i n a s o l i d would provide more information about the heat transport i n these media. 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