UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Some considerations on the phonon theory of second sound Adams, David Kendall 1953

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1953_A8 A3 S6.pdf [ 1.81MB ]
Metadata
JSON: 831-1.0085930.json
JSON-LD: 831-1.0085930-ld.json
RDF/XML (Pretty): 831-1.0085930-rdf.xml
RDF/JSON: 831-1.0085930-rdf.json
Turtle: 831-1.0085930-turtle.txt
N-Triples: 831-1.0085930-rdf-ntriples.txt
Original Record: 831-1.0085930-source.json
Full Text
831-1.0085930-fulltext.txt
Citation
831-1.0085930.ris

Full Text

SOME CONSIDERATIONS ON THE PHONON THEORY OF SECOND SOUND BY DAVID KENDALL ADAMS A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS in the Department of Physics We accept this thesis as conforming to the standard required from candidates for the degree of MASTER OF ARTS Members of the Department of Physics THE UNIVERSITY OF BRITISH COLUMBIA August, 1953 Abstract In investigating the propagation of second sound in a gas of non-interacting phonons, this thesis employs a f i e l d theoretical description of the quantized pressure excitations in l i q u i d helium II, as f i r s t presented by R. Kronig and A, TheHung. The relations derived by these writers are generalized with a Galileo transformation. Also, the assumption i s made that the phonons obey a Bose-Einstein distribution and the resulting expressions compared with those obtained intuitively in the earlier investigations of H. Kramers, Several serious disagreements are found between these two approaches to the problem. The most serious of which is the lack of a plane wave solution for the set of equations describing a phonon gas under a small temperature disturbance. This is a result that is unexpected in a f i r s t order second sound calculation and at present appears as a weakness in the Kronig and TheHung formulation. / Acknowledgement I wish to thank Professor F. A. Kaempffer for suggesting this research problem and for his helpful ad-vice and encouragement throughout the performance of this study. I also greatly appreciate the helpful criticism of the manuscript by Professor J. B. Brown. TABLE OF CONTENTS Chapter I Introduction • 1 Chapter II A field theoretical description of a non-interacting phonon gas, • . • 7 Chapter III Second sound in a system of non-interacting phonons. . • • • • • • I d Appendix I . . « . . • • • • • • • • • • • « « • • 23 Appendix II . . . . . . . . . . . . . . . . . . . 24 Appendix III « • • • • • • • • • • • • • • • . . . 29 6 LIST OF ILLUSTRATIONS Figure I Experimental curves showing the dependence of the velocity of second sound upon temperature <>»<>•••» page 4 Figure Ila The distribution of phonons in liquid helium II at a constant temperature « • • • • • • • • facing page 5 Figure lib The distribution of phonons in liquid helium II which results from a harmonic temperature disturbance • • • • • • • • facing page 5 CHAPTER I Introduction The common isotope of helium, | H e » possesses two distinct liquid phases that are characterized by a transition temperature of 2«19°K, when the liquid is under its saturated vapor pressure. Liquid helium below this temperature, called helium II, has properties that are different from those in any other known liquid, but unlike thejmore common phase: changes of liquids into gases and solids, the transition is not accom-panied by a latent heat* On the other hand, there is a sharp discontinuity in the specific heat and vapor pressure curves at this temperature* Also, there is a marked increase in the rate at which the fluid flows through capillary tubes, but this extreme change in viscosity is not observed when the damping of an oscillating disk is measured* Two other anomalous pro-perties of helium II are known as the fountain effect and second sound* The former is observed when a positive temperature gradient exists between two regions in the fluid connected by a capillary tube* This is found to create a positive pressure gradient, which can result in the flow of a large fountain of fluid* Second sound refers to the ability of helium II to propagate heat pulses by means of local temperature fluctuations, much as ordinary sound is transmitted by variations in fluid .density. The only successful phenomenological theory of helium II advanced thus far assumes i t to be composed of two components; (1) a normal component similar to helium above 2*19°K and (2) a superfluid component whose atoms are considered to have only 2 the zero point energy. Also the latter is assumed to have zero viscosity at small velocities and hence to be the part of the fluid capable of moving through fine capillaries. The normal component is considered to have a normal viscosity and is there-fore responsible for the damping of the motion of an oscil-lating disk. The experiments of Andronikashvilli 1 have demon-strated that the fraction of superfluid present in helium II is zero at the transition temperature and that the fraction approaches unity as the temperature of the normal component 2 decreases to OK, Also, Kapitza has performed experiments that indicate that the superfluid is at the absolute zero of temperature, The phenomenon referred to as the fountain effect is therefore attributed to a flow of the superfluid, a flow created when a source of heat is immersed in the liquid. For the fluid to remain in equilibrium under such a temperature disturbance, the cold superfluid component must move toward the source and the warmer normal component away from i t , unless the flow of the latter is restricted by a capillary tube. Second sound is explained by oscillations of the two components relative to one another, and in such a way that the total fluid density is unchanged. This is easily contrasted with ordinary sound in which both components move in phase, creating local fluctuations in the fluid density. In the case of second sound though, the changes in the relative concentration of the two components result in temperature oscillations at a l l points in the fluid. The temperature at a point will in-crease when the density of the normal component'increases, but 3 half a cycle later, when the superfluid displaces the normal component, the temperature will decrease* In this way, the changes in temperature of a fluctuating heat source are propa-gated through the fluid. At present, the only attempt at a not purely phenomeno-logical theory of the peculiar behavior of liquid helium II is 5,6 the theory of L. Landau, in which the normal component of the fluid is identified with elementary excitations called phonons and rotons. These appear with temperature increases in the underlying liquid (i.e. the superfluid) which is assumed to be at 0°K. It was shown by Landau that the irrotational exci-tations in the fluid (the phonons) are quantized standing waves characterized by wave numbers k such that the intrinsic energy of the phonons is hkc, where c = y | l? | is the velocity of small compressional waves in the fluid. This approach is similar to the Debye theory of specific heats in which the elementary ex-citations in a solid are taken to be longitudinal and trans-verse sound waves. In the Landau theory, the transverse waves are replaced by quantized rotational excitations called rotons, which are attributed to the rotational motions of the fluid. Comparisons with experiment (especially measurements of specific heat) indicate that the roton contribution to the properties of liquid helium II is negligible below 0.5°K. Often i t is convenient to consider the phonons and rotons as a gas moving in an "ether" of underlying superfluid. The viscosity of the normal component then arises from the inelastic interactions of these excitations. However, only the p h o n o n - p h o n o n c o l l i s i o n s a r e o f i m p o r t a n c e b e l o w 0.5°K a n d s i n c e s e c o n d s o u n d e f f e c t s a r e s m a l l a b o v e t h i s t e m p e r a t u r e , t h e p r o p e r t i e s o f t h e r o t o n s w i l l n o t b e c o n s i d e r e d f u r t h e r i n t h i s d i s c u s s i o n . T h i s e f f e c t o f t e m p e r a t u r e u p o n t h e o b s e r v e d v e l o -c i t y o f s e c o n d s o u n d (Cg) i s d e m o n s t r a t e d i n t h e t y p i c a l e x p e r i -m e n t a l c u r v e s r e p r o d u c e d i n f i g u r e I . M / S E C . 300 200 100 70 50 30 20 10 0.2 _L 0.6 T °K P R E S S U R E 0 IN 1 3 ATM. 1.0 F i g u r e I . E x p e r i m e n t a l c u r v e s s h o w i n g t h e d e p e n d e n c e o f t h e v e l o c i t y o f s e c o n d s o u n d u p o n t e m p e r a t u r e ( a f t e r M a y p e r a n d H e r 1 i n 7 ) A t s u f f i c i e n t l y l o w t e m p e r a t u r e s , w h e r e t h e p h o n o n ^ d e n s i t y i s s m a l l , t h e p h o n o n g a s c a n b e t r e a t e d a s i d e a l a n d i n e l a s t i c p h o n o n i n t e r a c t i o n s n e g l e c t e d . I n t h i s a p p r o x i m a t i o n , s e c o n d s o u n d i n h e l i u m I I c a n b e a t t r i b u t e d s o l e l y t o t h e p o s s i b l e c r e a t i o n a n d a n n i h i l a t i o n o f p h o n o n s i n t h e s u p e r f l u i d a n d t o t h e t r a n s p o r t o f p h o n o n s f r o m o t h e r p a r t s o f t h e f l u i d . F i g u r e Jt e m p l o y s t h i s m o d e l t o d e p i c t . ( a ) a p h o n o n g a s i n a r e c t a n g u l a r Figure IIa. The distribution of phonons in liquid helium II at a constant temperature. Each point represents a fixed number of phonons. Figure l i b . The distribution of phonons in liquid helium II as created by a harmonic current in the heating element. This local temperature disturbance produces a standing temperature wave known as second sound, which is depicted by the fluctuations in phonon density. Each point repre-sents a fixed number of phonons. 5 v e s s e l u n d e r a u n i f o r m t e m p e r a t u r e d i s t r i b u t i o n a n d ( b ) a s t a n d i n g w a v e o f s e c o n d s o u n d a r i s i n g f r o m t h e l o c a l t e m p e r -a t u r e c h a n g e s p r o d u c e d b y a h a r m o n i c c u r r e n t i n t h e h e a t i n g e l e m e n t . T h e m a c r o s c o p i c e f f e c t t h a t t h i s l a t t e r p i c t u r e d e s -c r i b e s i s a t e m p e r a t u r e d i s t r i b u t i o n , w h e r e t h e t e m p e r a t u r e a n t i n o d e s a r e c h a r a c t e r i z e d b y i n c r e a s e s i n t h e p h o n o n d e n s i t y . I n a d d i t i o n t o t h e h y p o t h e s i s a b o u t t h e m i c r o s c o p i c n a t u r e o f t h e n o r m a l c o m p o n e n t o f h e l i u m I I , t h e a s s u m p t i o n i s a l s o made t h a t t h e p h o n o n s o b e y a B o s e - E i n s t e i n d i s t r i b u t i o n . T h i s p r e m i s e i s c a r r i e d o v e r f r o m t h e a n a l o g o u s c a s e o f p h o t o n s i n a n e l e c t r o m a g n e t i c f i e l d , f o r p h o t o n s a r e k n o w n t o o b e y B o s e - E i n s t e i n s t a t i s t i c s , a s a r e a l l e l e m e n t a r y p a r t i c l e s , a t o m s , a n d m o l e c u l e s p o s s e s s i n g i n t e g r a l s p i n . I t i s t h e p u r p o s e o f t h i s d i s c u s s i o n t o t h e o r e t i c a l l y i n v e s t i g a t e s e c o n d s o u n d i n a n o n - i n t e r a c t i n g p h o n o n g a s t o s e e i f s u c h a m o d e l w i l l e x p l a i n c 2 i n t h e r e g i o n b e l o w 0 . 5 ° K . A l t h o u g h a p h o n o n g a s m o d e l h a s b e e n u s e d i n p r e v i o u s t r e a t m e n t s o 9 o f s e c o n d s o u n d b y H . K r a m e r s 0 a n d b y J . W a r d a n d J . W i l k s , t h e s e w e r e d o n e r a t h e r i n t u i t i v e l y . T h e r e f o r e we s h a l l h e r e i n v e s t i g a t e a d e s c r i p t i o n w h i c h e m p l o y s a f i e l d t h e o r y d e v e l o p e d 4 b y R . K r o n i g a n d A . T h e l l u n g . T h e s e w r i t e r s h a v e s u c c e e d e d i n f i n d i n g a s a t i s f a c t o r y L a g r a n g i a n d e n s i t y f o r t h e l o n g i t u d i n a l p r e s s u r e e x c i t a t i o n s i n a n o n - v i s c o u s i r r o t a t i o n a l f l u i d , w h i c h e n a b l e s o n e t o f o r m u l a t e t h e p r o b l e m i n t e r m s o f a v a r i a t i o n a l p r i n c i p l e a n d t h u s p e r m i t s t h e t e c h n i q u e s o f f i e l d t h e o r y t o b e u s e d . H o w e v e r , t o e l i m i n a t e t h e d i f f i c u l t z e r o p o i n t t e r m s t h a t 6 - a r i s e i n t h e q u a n t i z e d t h e o r y , a c o m m u t a t o r , w h o s e c l a s s i c a l v a l u e i s z e r o , h a s b e e n a d d e d t o t h e L a g r a n i a n d e n s i t y . T h i s p r o c e d u r e t a k e a a d v a n t a g e o f t h e w e l l k n o w n f a c t t h a t t h e q u a n t u m m e c h a n i c a l f i e l d f u n c t i o n s a r e n o t u n i q u e l y d e f i n e d b y t h e c o r -r e s p o n d i n g c l a s s i c a l q u a n t i t i e s . T h e f o r m e r w i l l a l w a y s h a v e a r b i t r a r y " z e r o p o i n t " t e r m s t h a t c a n b e a d j u s t e d f o r c o n v e n -i e n c e , b y a d d i n g a d d i t i o n a l c o m m u t a t o r t e r m s t o t h e c l a s s i c a l L a g r a n g i a n . I t w i l l b e p o i n t e d o u t t h a t d i s a g r e e m e n t s a r i s e b e t w e e n t h i s f i e l d t h e o r e t i c a l d e s c r i p t i o n a n d t h e p h o n o n g a s m o d e l o f K r a m e r s , w h i c h s e e m s t o h a v e n e g l e c t e d t h e momentum a n d e n e r g y t r a n s p o r t e d b y t h e d r i f t i n g m o t i o n s o f t h e p h o n o n s , i . e . m o t i o n s c r e a t e d by i n t e r n a l t e m p e r a t u r e d i f f e r e n c e s . H o w e v e r , t e r m s d e s c r i b i n g t h e s e p r o p e r t i e s d o r e s u l t f r o m t h e t h e o r y a p p e a l e d t o h e r e , b u t i t i s f o u n d t h a t t h e s e a d d i t i o n a l t e r m s i n c r e a s e t h e c o m p l e x i t y o f t h e s e t o f e q u a t i o n s c a l c u l a t e d t o d e s c r i b e t h e r e a c t i o n o f a p h o n o n g a s t o a s m a l l t e m p e r a t u r e d i s t u r b a n c e . T h e r e s u l t b e i n g t h a t w h i l e K r a m e r s w a s a b l e t o o b t a i n a w a v e e q u a t i o n f r o m s u c h a s e t o f e q u a t i o n s , a p l a n e w a v e s o l u t i o n d o e s n o t s a t i s f y t h e s e t o b t a i n e d h e r e . T h i s i s a r a t h e r u n e x p e c t e d r e s u l t b e c a u s e s e c o n d s o u n d i s c o n s i d e r e d t o b e p r o p a g a t e d b y a p l a n e w a v e m o t i o n , i n a f i r s t a p p r o x i -m a t i o n a t l e a s t . T h e r e f o r e t h e m a i n c o n t r i b u t i o n o f t h i s d i s c u s s i o n i s t o i n d i c a t e s e v e r a l p o i n t s w h e r e t h e t h e o r y o f K r o n i g a n d T h e l l u n g i s i n c o n s i s t e n t w i t h t h a t o f K r a m e r s ' a n d a l s o t o s h o w t h a t t h e f o r m e r d o e s n o t p r e d i c t s e c o n d s o u n d t o b e a p l a n e w a v e p h e n o m e n o n . 8^  CHAPTER II A field theoretical description of a non-interacting phonon gas. Following the assumption that phonons are quantized compressional waves in a non-viscous, irrotational fluid, the hydrodynamics of these pressure excitations will now be inves-tigated. We suppose that external forces are absent, so that the fluid can possess an equilibrium state characterized by a density j> and a pressure To • Longitudinal waves in the fluid will then cause pressure and density deviations from these values From classical hydrodynamics, the effect of these local distur-bances is completely described by the Euler equations of motion' which in the absence of external forces become pq»-ad£-<f 4 £(<3>-<*cl + J«"««l T 5 = 0 (1) f •* div(f grad cj>) = 0 (2) Here 4> is the velocity potential defined by ~$ - - ^ d <$> , which expresses the irrotational property of the fluid. When inte-grated along a streamline, (1) yields the Bernouli equation ( { ^ l ^ r a c f ^ + J ? 4 P (3) while (2) is the equation of continuity expressing the conser-vation of mass. In ( 3 ) , the term [ U is the potential energy per unit mass which, by partial integration, is shown below to be composed of two more fundamental quantities so The f i r s t term is the potential energy per unit mass of an 8 incompressible fluid while W is the contribution due to com-pression. It is only the latter term that is of importance here, where the propagation of compressional waves is being considered* Ah alternate description of the fluid can be obtained in terms of one or more space-time functions lp(x,t) , called field variables* Formally the procedure is to find a function of these variables and their f i r s t space and time derivatives, called the Lagrangian density L, such that the integral is stationary with respect to variations of each of the , This variational principle leads to the following Euler-Lagrange conditions on L, which are called the field equations* In the case at hand, the field equations are given by (2) and (3) and the problem is to find the proper expressions for L from which these equa-tions can be derived by means of (5)* This problem was solved by Kronig and Thellung who obtained for the Lagrangian density the function L = f f ^ - i ( r ^ 4 > ) x - w ] (6) which depends upon only one field variable, the velocity potential (p. The Lagrangian density can be seen to represent 9 the deviation of the pressure from equilibrium, since the substitution of (3) for (j> yields L = f lu-w) - - p - P o with the aid of ( 4 ) . Using ( 6 ) , we find that the generalized momentum conjugate to <p , defined by ^  , is equal to the density p . Therefore another important field quantity, the Hamiltonian density H, can now be constructed. H - f * - L - iffowdtP + fV (7) It is seen to possess its normal significance as an energy density, since i t is the sum of the kinetic energy density and the compressional energy density. The conditions for the conservation of energy and momentum can be expressed in terms of the following auxiliary field densities; the density of energy flow S, the momentum density G, and the density of momentum transport T (a stress tensor). These are required to satisfy the relations | H + dws =o (8) ox In a well known reference on field quantization by G. Wentzel, a satisfactory definition of these quantities is shown to be the following. 1 0 It is important to note that the above quantities have the mechanical significance intended. Substitution into (9) yields which expresses the condition for the conservation of momentum and by virtue of ( 2 ) , yields the equation of motion ( 1 ) , Simi-larly ( 8 ) becomes 4 A* v ) ( i K + ? W ) + div; «Mp-?.) = 0 or where the f i r s t term is the rate of change of energy per unit volume and the last term represents the rate of change of work done against the excess pressure on the fluid passing through this unit volume. The advantage of this discussion of the problem in terms of field variables is that a description of the quantized longitudinal excitations can easily be obtained by quantizing the resulting field densities. The quantization is performed by treating the volume integrals of these densities as operators whose eigenvalues give characteristic properties of the phonon fie l d . For example, the eigenvalues of I H are the phonon JV energy levels and the eigenvalues of j (j dK express the expec-Jv 11 tation values of the phonon momenta* For the purpose of eliminating the annoying zero point terms that arise in the quantized theory, the Lagrangian density can be written as follows This is a well known technique that is possible because the classical value of the new commutator term is zero, and hence i t makes no contribution to the field equations (5), Although this commutator does not eliminate a l l zero point terms in the field densities, i t has been chosen so that i t cancels a l l non-constant terms, as will be more apparent later (see page 20 ) e The other field densities defined in (7) and (10) now become H = f r - L = i < H x 4 f W + 4F ' (r-rJ-lf-r.)^**} 5 siJL— ,-nrad4i - Z & K j<j>(r-f.)-(?-f.)4>} Qj = - <^rad <|> Ik =• - f <\rad <f> In order to calculate the eigenvalues of the field, i t is necessary to express these densities explicity in terms of the canonical variables j> and <p . Following the suggestion of Landau. Kronig and The Hung have done this by writing the 12 equation of state in the form of the following expansion where ^ |dPJ = c 0 is the velocity of compressional waves of small amplitude in the fluid. Similarly, we find from (4) and from (2) and (3) j = ?.Va4> -»(p-r.)\T4 + j.. ^ ( 1 2 ) 4 = ^ ( f - f J +-iUr«d $)Ni(jJ^(f-j»j' + higher terms in p-pc (13) Now by use of equations (10-13), i t is possible to separate a l l the field densities into terms of increasing powers in f-fc and £. H - i f M " * b ) x * i f (r-r.)1 + lr->Jd"»d $)•- (14) - Z.. 4>(/>-£j - (f -r0J ?^c/</)J 4 - higher terms in Jo 4 2_ . j«j r»d 4> (?-Po) - ( f - f 0 ) ( ? ^ <jj 4 higher terms in y - f e 13 In this way, as will be more evident later, the second order terms describing a non-interacting phonon gas can be separated from the rest, such as is done in a perturbation calculation. At this point, an observation of Kronig and Thellung becomes very important to the phonon gas model described by this field theory. They remark that the Lagrangian density is invariant with respect to the Galileo transformation X'=X-Oei l' = t = ? (f)'(x,*; = ${K,t) + -10c £ which also retains the meaning of <p' as a velocity potential (see Appendix I), since Classically the significance of this transformation is not completely apparent, but without changing the field equations (2) and (3), i t adds terms to the field densities that account for drifting motions of the pressure excitations, that i s , motions superimposed upon their random motions in the fluid. For example, in (7), the compressional energy density is invariant since i t depends only on f , but the kinetic energy density transforms to Similarly, the momentum density in (14) becomes 14 Quantum mechanically this transformation represents the motion of the phonons with respect to the underlying liquid, and where H included both the intrinsic energy of the phonons and their interaction energies, the transformed Hamiltonian H* also in-cludes the energy due to the translation of the phonons through the liquid with velocity v^. However, the physical meaning of the effect of this transformation on the Hamiltonian and the other field densities is much more evident in the eigenvalues of the transformed field operators. We shall therefore proceed with the calculation of these eigenvalues, from which i t will be seen that the above transformation describes a Doppler effect. Using the relation yr~ad'- <^faA <$> + v)o and classifying the transformed terms according to their degree in <£> and f - f 0 » equations (14) become (15) -j- « • • 15 +• * « sw)=[s , l c'H3'0 J] These new field densities satisfy the conservation equations AMI + d»V S ' = o ^ + - / " < L I ^ = O and since | p + "3o«r*e/ and ^, = ^  , they can be written as follows i H U t J + i / 6 . ? ; W H'<x,t; + V ' ^ ' ^ / ^ - o (i6) 16 Since the zero order terms in (15) can be attributed to the motion of the underlying liquid and have no effect on the second sound results obtained in this discussion and because the eigenvalues of the f i r s t order terms are found to vanish, as a f i r s t approximation i t will be sufficient to consider only those terms in (15) of degree two in the variables (j) andf-?c . However, in performing this approximation, equations (16) must be altered to include second degree terms coming from ^- H l" o i and i G > i.e. from <p and p (see 12, 13). It is convenient to write these terms as follows where the auxiliary densities S and T j m are defined by In this way the form of equations (16) is retained and the conservation equations become bt£! l J + v H ' a l + y 4 s , 0 ) + S") . a ? ) We are now ready to quantize the field by requiring that § and I 7 o satisfy the following commutation relations [<|>ix), 4>tx')J= [ f ix) , = o < 1 8 ) When the field densities are integrated over the volume of the fluid, or an appropriate portion V, they are denoted by W-l^l-k* Jiu* Z.-fc»-« <19> In the quantized theory, the quantities in (19) are called the field operators and have eigenvalues that can be evaluated with the aid of conditions (18). This calculation is carried out in Appendix II, where the following eigenvalues are found for the volume integrals of the second degree terms in (15), and where is introduced as the number of phonons with a wave number K. (^'^=bZ (co*-v..*.)(Ma+N.-a. + l) + i Z^K ( 2 0 ) (J' a ) + J") -- iU Z CK(N< + N-**I) +Vc(fc«)] [wn *t) ZQK These expressions indicate that a system of non-interacting phonons have properties that are consistent with the model proposed in chapter I. The energy eigenvalues show that the phonons have associated with their random motions 1 8 an intrinsic energy hc0k, and a kinetic energy h(v0.k) that arises from impressed motions through the fluid with velocity v Q, It can now be seen that the Galileo transformation actually produces a Doppler shift in the energy eigenvalues. Also, the momentum eigenvalues show that the phonons have an intrinsic momentum hk. Although we find no effects of phonon-phonon scattering and other complex phenomena in this approximation, i t is to be expected that these will appear upon consideration of third and fourth order terms in the field densities. It is important to note that while the energy and momentum eigenvalues are identi-cal with those chosen intuitively by Kramers, his expressions for the energy and momentum currents do not contain the terms in v Q, In other words, the Kramers' approach would include in the energy current, ( ^ ' ( y ) + J " ) > only the term £ ^c^[iH^*t) and in the momentum current, (TJ^ + TeV) > only the term is. 2L (Na+N-aThus, as mentioned in the introduction, * i^ -the past investigations have neglected the convective trans-port of energy and momentum due to the impressed motions of the phonons through the underlying liquid. We shall now proceed to investigate the effect of these corrections upon the velocity of temperature waves in the phonon gas. CHAPTER III Second sound in a system of non-interacting phonons By ( 2 0 ) , we can now conclude that N£ is the number of phonons with energy c0tA-v/«-U and i f we assume that the density obeys a Bose-Einstein distribution, the temperature can be introduced by substituting M. =(U)d J K = -!• d*H (21) where M ~and x is the Boltzmann constant» It will be con-venient to expand f(k) in increasing powers of v Q, which can be done i>y means of the following Taylor series -f(^) =-To H ^ C ^ O Cl}^ + + • • • (22) where | 0 = . , & - - H etc. If in (20) we now replace the sum over k by a volume integral and substitute (22) as far as second degree terms in v Q, we obtain - I V<M. + 3. & tic*) ] c\\ 20 We shall now consider the case depicted in figure lib, where a slowly changing temperature source has been added to the fluid. Thus v Q and M will now be functions of the time and the coordinates. For simplicity however, we shall take v Q y = v o z =0, which reduces the problem to one dimension. Assuming the temperature to change slowly enough so that ¥i « I and that only f i r s t degree terms in v Q X are necessary, and also noting that OPK = ^ [ J K ^ K , equations (23) become H' - ^ 0jjK {o • ^ K j < ^ K <24> Note that the non-constant zero point terms in S* + S* * have now vanished, i,e, terms containing v Q X , This cancelation is produced by the commutator that was originally added to the classical Lagrangian on page 11, and therefore explains the choice of the coefficient that was introduced with this extra term. Now, with the aid of the following partial inte-gration 9 21 J"iU,ajK K H c 5«Heaed^K ' ( ( { ^ V o j J - f f ^^a<js««e-4<ie(25) 4 these relations can be simplified to H' = UA±< fuM, d 3K S'x t Si' = - ^ ' ^ ^ [ K M , d^K ( 2 6 ) G « = U-*f±?jV-f. 43K TV * -t-T Ax = ^ U * J « % <*J* where the constant zero point terms have been dropped, since they will vanish later upon differentiation. If equations (26) are now subjected to (17), conditions will be obtained on v Q(x,t) and ^(x,t) that will insure the conservation of energy and momentum. We must realize that this procedure is not completely rigorous since the Bose-Einstein distribution is valid only for systems in equilibrium, but we shall assume that the function is true for slowly varying temperatures. Now, as shown in appendix III, equations (17) become + M I / , » ^ - ^ i ^ S . = 0 ( 2 8 ) These equations will be seen to resemble those conservation conditions calculated by Kramers and differ from them only in the coefficients. Even in this approximation however, this difference is sufficient to make a plane wave solution impossible, as can be seen by substituting harmonic 22 expressions f o r ^ and v_„. This indicates an important OX difference between the field theory formulated by Kronig and Thellung and the phonomenological treatments of Landau and Kramers, The work of the later two predicted a constant second sound velocity c 2 = c 0/f3, which can not be verified here. This is indeed a paradoxical result since the seemingly more fundamental theory of Kronig and Thellung does not yield the physically reasonable results that are attainable from more intuitive approaches, A more recent paper by R. Kronig 1 1 suggests that i t is necessary to include the momentum of the underlying liquid in the momentum equation (28), which would necessitate carrying the term G f^°^ through the above cal-culations. Actually, this suggestion did not come from con-siderations of the field theory but from a phonomenological treatment in which an analogous calculation was made. This proposed alteration adds to (28) the term; * % /~ which s t i l l ! i K -i / t does not alter the equation enough to afford a simple solution and hence its necessity can not be verified here. Appendix I If the following Galileo transformation is performed then = ^ ^ T U~ Jxi = ^ <*>' Therefore - x (r«d'4'Y = H - ±{r«<J 4f and the Lagrangian density L is invarlent, as quoted in the text, Appendix II Following the quantization procedure for real wave functions outlined in Wentzel10(#5,6), a l l the operators must be written in Hermitean form. This means that those terms containing the factor <jra<A 4 (ff.) must be made symmetrical by substituting A l s o ( f - f 0 j and <fr are taken to be periodic in a cubic volume V, and the following transformation into momentum space is performed which satisfies the commutation conditions (18). Here the "k's are wave numbers characterizing the Fourier components of and ( f - f 0 J . Substitution in (15) yields, as far as second degree terms, H , u > - i h l 4 . ™ M * H Z ™ * ^ * ' * ( 3 1 ) 2 V ft R' - ± L ±&-L I i *(«aP«- - P * * » ) e 25 K K' - &• * £ £ '«(ft P* "ft- t k ) e i * * , , - S R K Now by integrating these expressions over the volume V, we obtain the field operators. The integration is simplified by noting that j ^ e ^ x is zero unless r o and hence 26 Remembering that ft - and = , equations (31) become the field operators: "K SO K •* K 27 D u e t o t h e e q u a l n u m b e r o f p o s i t i v e a n d n e g a t i v e d i r e c t i o n s o f t h e k v e c t o r s , Z.%, =o a n d t h e t e r m J L ^ - K i s z e r o u n l e s s K" K" ft=K" o T h e r e f o r e a l l t h e d o u b l e s u m s a b o v e . ; ; c a n b e r e p l a c e d b y a s i n g l e s u m o v e r k . F o l l o w i n g t h e t e c h n i q u e o f W e n t z e l ( # 6 ) a n d K r o n i g a n d T h e l l u n g , we now i n t r o d u c e t h e a n n i h i l a t i o n a n d c r e a t i o n o p e r a t o r s a £ a n d a £ d e f i n e d b y a n d s a t i s f y i n g t h e c o m m u t a t i o n r e l a t i o n s T h e s e o p e r a t o r s c a n b e r e p r e s e n t e d b y m a t r i c e s , s u c h t h a t a*a& a n d Oct cu a r e d i a g o n a l m a t r i c e s w i t h i n t e g r a l e l e m e n t s g i v e n b y w h e r e N g i s t h e n u m b e r o f p h o n o n s w i t h w a v e n u m b e r k « F r o m t h e i n v e r s e t r a n s f o r m a t i o n i t c a n b e s h o w n t h a t T h e r e f o r e , u s i n g t h e s e r e s u l t s , t h e t e r m s a p p e a r i n g i n ( 1 5 ) c a n b e e v a l u a t e d a s f o l l o w s | & • £ & IN* * N-H | - Z as(N> - i n K K * 28 from which we see that the field operators have the following eigenvalues, as given in (20) ( y . ' n l ) = h.I(cjc-v .-K)[.N«*N.a*l) * JZC . K (J'(\J") = - ^ v . h|c o K ( N ^ N . f t i l ) ^ { ^ K ^ I M J U M ' < T "V D = - * v.. I K , [i N * + ij t h Z K . ( N » + N-* +') -1 L ^ H Appendix III In substituting equations (26) into the conservation equations (17), the following relations will be used. M^ J« **T)jt* -^ t rK^ , K The last expression results from a partial integration similar to that carried out in (25). Now, from (26), we find = - i \*CIM |* J K 4 - M 3 K v„<U±'= - i hcUTKA h L ^ a v 'ox = no fi r s t degree terms in v o x 30 Therefore combining these terms as required by (17), the common factor j^O-fxcpK cancels and with a few other cancellations we obtain equations (27) and (28), References 1. A n d r o n i k a s h v i l l i , E., J . Phys. U.S.S.R*, 10, 201, 1946. 2. K a p i t z a , P., J . Phys. U.S.S.R., 5_, 59, 1941. 3. Kramers, H., P h y s i c a 18, 653, 1952. 4. K r o n i g , R 0 and T h e l l u n g , A., P h y s i c a 18, 749, 1952. 5. Landau, L., J . Phys. U.S.S.R. 5, 71, 1941. I b i d . 11, 91, 19477 6. Landau, L. and Kh a l a t n i k o v I . , J.E.T.P. U.S.S.R. 19, 637, 709, 1949. — 7. Mayper J r . , V. and H e r l i n , M., Phys. Rev. 89, 523, 1953. 8. Sommerfeld, A., Mechanics of Deformable B o d i e s , Academic Press I nc., New York, 1950. 9. Ward, J . and W i l k s , J . , P h i l . Mag. 42, 523, 1951. 10. Wentzel, G., Quantum Theory o f F i e l d s , I n t e r s c i e n c e P u b l i c a t i o n s Inc., New York, 1949. 11. K r o n i g , R., P h y s i c a 19, 535, 1953. 1 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0085930/manifest

Comment

Related Items