THE DAMPING OF SECOND SOUND NEAR THE SUPERFLUID TRANSITION IN HE 4 by BRADLEY J . ROBINSON B . S c , U n i v e r s i t y of Toronto, 1972 B.Ed., U n i v e r s i t y o f Toronto, 1973 M.Sc, U n i v e r s i t y of B r i t i s h Columbia, 1976 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Department of Physics) We accept t h i s t h e s i s as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA May 1981 ^) Bradley J . Robinson In p r e s e n t i n g t h i s t h e s i s in p a r t i a l an advanced degree at further of the requirements the U n i v e r s i t y of B r i t i s h Columbia, I agree the L i b r a r y s h a l l make it I fulfilment freely available for this thesis f o r s c h o l a r l y purposes may be granted by the Head of my Department his representatives. of this thesis for It financial gain s h a l l not be allowed without my Depa rtment U n i v e r s i t y o f B r i t i s h Columbia 2075 Wesbrook P l a c e V a n c o u v e r , Canada V6T 1W5 Date or i s understood that copying or p u b l i c a t i o n written permission. The that reference and study. agree t h a t p e r m i s s i o n for e x t e n s i v e copying o f by for JLUAJ^ to /<?sn i ABSTRACT 4 The attenuation of second sound near the superfluid transition in He has been determined by measuring the decay time for free oscillations of plane wave modes in a resonant cavity. The results for both the c r i t i c a l exponent and amplitude of the second sound damping coefficient are consistent with the early predictions of Hohenberg, Siggia and Halperin based on renormalization group theory. However, the damping observed in this work i s less than the recent predictions of a non-linear renormalization group analysis by Dohm and Folk. The measurements cover the temperature interval 1.8 x 10 ^ < t < 2.1 x 10 , where t law for t < 10 c (T^ - T)/T^. Fitting the results to a single power , the c r i t i c a l exponent governing the temperature depend- ence is found to be 0.31 ± 0.05. If the results are constrained to obey the theoretical asymptotic temperature dependence with an exponent of 2 -1 0.288, then the amplitude obtained for the damping i s 3.7 ± 0.4 cm s . This corresponds to a value for the universal amplitude ratio, R^, of 0.11 ± 0.01. For t > 10 the damping departs from the c r i t i c a l behav- iour, and increases to obtain the values previously observed by Hanson -2 and Pellam for t > 10 . ii TABLE OF CONTENTS Page ABSTRACT i LIST OF FIGURES iv ACKNOWLEDGEMENTS v CHAPTER 1 1 INTRODUCTION A. Introduction to C r i t i c a l Phenomena 1 B. Review of Second Sound Damping 3 C. The Dynamics of Superfluid Helium 6 i ) Hydrodynamics and the Damping C o e f f i c i e n t , i i ) Scaling and D,, i i i ) Renormalization Group Theory and D CHAPTER 2 EXPERIMENT •& 15 2 18 24 A. Techniques 24 B. Apparatus 29 I) Cryogenic Apparatus 29 II) Resonator 31 ( i ) Cavity 31 ( i i ) Generator 33 ( i i i ) Bolometer 35 III) Electronics 36 ( i ) Signal E x c i t a t i o n and Recovery 36 ( i i ) Temperature Regulation . . . . 39 ( i i i ) Level Detection 39 C. Procedures and Tests 39 iii Page CHAPTER 3 A3 INSTRUMENTAL SOURCES OF ADDITIONAL ATTENUATION A. Attenuation Due to Viscous Surface Loss, ot . . . . A3 B. Attenuation Due to Heat Conducting Surface CHAPTER A CHAPTER 5 Losses, a and a ' e s C. Discussion of ot , ot , ot Tr e* s ANALYSIS OF DATA FOR THE DECAY RATE, 1/x A6 A8 51 A. Obtaining 1 / T From Decay Curves 51 B. Discussion of the Data for 1 / T 58 C. Results f o r the Damping C o e f f i c i e n t , D^ 69 CONCLUSIONS AND DISCUSSION 75 A. Conclusions 75 B. Discussion 76 iv LIST OF FIGURES Page Figure 1. Cryogenic Apparatus 30 Figure 2. The Resonator, Generator and Bolometer 32 Figure 3. Block Diagram of Main E l e c t r o n i c s 37 Figure A. A Decay of a Second Sound Resonance Y i e l d i n g a Value f o r the Decay Rate 1 / T 52 Figure 5 . An Extrapolation to Zero Power 54 Figure 6. 1/T - 1/T^ - l / ^ ^ , 60 Figure 7. 1/T Figure 8. 1 / T - 1 / T ^ - / 2HP - 1/T^ 1 l / T T 2 f ° Harmonics 1 and 3 r R f o r Harmonics 1, 2, A P f o r H ™ 3 0 1 1 * 0 5 1 a n d 65 3 Over the E n t i r e Range of AT 67 Figure 9. The Results f o r the Second Sound Damping C o e f f i c i e n t . . 71 Figure 10. Summary of Results f o r the Second Sound Damping C o e f f i c i e n t 72 ACKNOWLEDGEMENTS I am grateful to Dr. M. Crooks for his supervision of this work. I also wish to thank Dr. J. Carolan f o T his supervision of the f i n a l stage of this work while Dr. Crooks was on sabbatical. Dr. W. Hardy proposed the technique of measuring the decay times, am indebted to him for this, and several helpful discussions. CHAPTER 1 INTRODUCTION This chapter begins with a b r i e f chronological review of the theore t i c a l advances i n the f i e l d of c r i t i c a l phenomena the motivation f o r t h i s study. and an i n d i c a t i o n of In section B the problem addressed i n t h i s thesis i s described i n more d e t a i l through a review of other work and a discussion of the implications of recent developments. The t h e o r e t i c a l ideas concerning the dynamics o f superfluid helium i n r e l a t i o n to second sound attenuation are presented i n section C. A. Introduction to C r i t i c a l Phenomena A m a t e r i a l with a temperature approaching the c r i t i c a l value of a phase t r a n s i t i o n displays singular behaviour i n a variety of both s t a t i c and dynamic properties. A few examples'of c r i t i c a l points are: the Curie point i n ferromagnets, the superconducting t r a n s i t i o n occurring i n many metals, the c r i t i c a l point of gas-liquid t r a n s i t i o n s , the lambda l i n e or superfluid t r a n s i t i o n i n helium. The i n i t i a l e f f o r t s to describe,the form of c r i t i c a l point singul a r i t i e s are reffered to as c l a s s i c a l or mean f i e l d theories*. These were developed around 1900 and include Van der Waals equation f o r a f l u i d , the Weiss molecular f i e l d theory of ferromagnetism, and the Ornstein-Zernicke equations f o r c o r r e l a t i o n functions. These theories are quantitatively incorrect i n the c r i t i c a l region, but achieve p a r t i a l success i n that they y i e l d appropriate q u a l i t a t i v e behaviour i n the form of s i n g u l a r i t i e s which can be expressed i n terms of c r i t i c a l exponents. the temperature Thus, a s i n g u l a r i t y i n (T) dependence of a physical quantity, A, with c r i t i c a l 2 exponent, a, i s described by A •= A |(T - T >/ o i c a l temperature Landau and A q c T J c a where T £ i s the amplitude of the s i n g u l a r i t y . i s the c r i t In 1937 proposed a general theory of the continuous or second order phase transition. His work r e s u l t s i n the same exponents as the c l a s s i c a l theories and consequently i s i n c o r r e c t . However, he does advance the con- cept of an order parameter - a c e n t r a l element i n modern theories. The generalization to include dynamic properties occurred i n 1954 with the 2 introduction of an equation of motion f o r the order parameter . of t h i s type are employed i n current time-dependent Equations Ginzburg-Landau models. In the m i d - s i x t i e s there evolved a phenomenological treatment of 3 phase t r a n s i t i o n s known as scaling . or This theory predicts r e l a t i o n s h i p s scaling laws which e x i s t among various c r i t i c a l exponents and therefore escapes the l i m i t s of mean f i e l d theory. By the end of the s i x t i e s the extension of scaling to include dynamic properties had been accomplished . Along with scaling there developed a related concept known as u n i v e r s a l i t y . I n i t i a l l y formulated as the law of corresponding states, the hypothesis of u n i v e r s a l i t y means that r e l a t i v e l y few fundamentally d i f f e r e n t types or classes of c r i t i c a l behaviour are s u f f i c i e n t to accommodate a complete categorization of phase transtions. In p a r t i c u l a r , the entire lambda l i n e i n l i q u i d helium i s i n one u n i v e r s a l i t y class and therefore the e f f e c t s of elevated pressure should be mild i n the sense that, f o r example, exponents remain unchanged. During the l a s t decade the renormalization group methods of quantum f i e l d theory have been applied, with great success, to the problem of both s t a t i c ^ and dynamic^ c r i t i c a l phenomena. Renormalization group theory (RGT) provides a more fundamental derivation of scaling, as well as a means of c a l c u l a t i n g c r i t i c a l exponents and the values of c e r t a i n amplitude ratios. On the basis of RGT there i s a concrete formulation of univer- 3 sality in which the equivalence classes are essentially determined by the dimensionality, d, of the system and the number of components, n, in the order parameter. The transformations involved in this theory are suf- ficiently complex that results can usually be evaluated only approximately with expansions in 4-d or 1/n. Indeed, the mathematical structure of the RG as well as its application to physical systems i s the subject of much current research. The i n i t i a l motivation for this work was based on one of the predictions of dynamic scaling related to the damping of second sound near the lambda transition in liquid helium. Second sound i s a propagating mode of thermal transport which appears as a temperature-entropy wave in 4 the low temperature superfluid phase of He. It was f e l t that precise measurements of the c r i t i c a l damping at various pressures would provide a severe test of both scaling and universality. This prediction, the ex- perimental situation regarding second sound damping, and the implications for this work which resulted from the advent of RGT are the subject of the following section. B. Review of Second Sound Damping The earliest measurements of the attenuation, a^, of second sound relevant to this work are those of Hanson and Pellara (HP) in 1954^. From their data i t i s possible to extract the, now more pertinent, damping coefficient D£ by means of the hydrodynamic expression a (03,T) - Js(w /uJ) D (T) (1) 2 2 2 where w i s the angular frequency of the second sound with velocity \x^. This expression i s derived in section C and, as indicated by the notation, in the regime of hydrodynamics D 2 depends only on temperature. The range _2 of temperatures covered by HP are such that AT • T^ - T ^ 10 K and there- 4 fore do not enter deeply into the c r i t i c a l region. However, t h e i r measure- ments are Important i n the i n t e r p r e t a t i o n of the r e s u l t s of t h i s work and w i l l be discussed i n l a t e r chapters. As previously mentioned, t h e o r e t i c a l work i n the l a t e s i x t i e s resulted i n the dynamic scaling p r e d i c t i o n of the c r i t i c a l temperature dependence of D: 2 D = uK (2) 2 where £ i s the c o r r e l a t i o n length for fluctuations of the order parameter. The arguments used to a r r i v e at t h i s r e s u l t are outlined i n section C. In view of the p r o p o r t i o n a l i t y , the experimental v e r i f i c a t i o n of r e l a t i o n (2) w i l l involve as T * T^. only the exponents which govern the behaviour of each quantity From the two-fluid hydrodynamic expression for u 2 (see E q . (26) in section C), the terms with a s i g n i f i c a n t dependence on AT indicate • u„ 2 c P a (p /c ) s p where p s i s the density of the superfluid component and i s the s p e c i f i c heat at constant pressure. The d e f i n i t i o n of a cor- r e l a t i o n length for a power law decay of the c o r r e l a t i o n function at distances, as i s the case f o r helium with T < T^, and large the formulation of hydrodynamics in terms of c o r r e l a t i o n functions y i e l d s £ oc P ^« There- g -h fore, the s c a l i n g prediction (2) becomes D (c p ) . The exponent for p (AT)^ has been found to be C = 0.666 ± 0.006 from an s a 0 a Andronikashvili-type experiment . measurements of u expression for u > 2 2 Alternatively, p (plus other empirical with the r e s u l t Z, c may information and 0.674 ± 0.001 be derived from the hydrodynamic at saturated vapour 9 pressure . The s p e c i f i c heat, c^, increases slowly with a near l o g a r i - thmic dependence on AT as, AT decreases. Thus measurements of D v e r i f y dynamic scaling to the extent that they confirm D, « (c ) ~ ( A T ) ~ ^ (AT) P h 2 C / 2 _ 1 / 3 2 would (3). 5 P r i o r to the beginning of t h i s study an experiment by Tyson*" pro0 vided the only test of r e l a t i o n (3) f o r the case of macroscopic sound*"*". His r e s u l t s give 0.34 ± 0.06 f o r the exponent of D sequently agree with the value 1/3. This might be expected to the exponent which would be calculated from a power law f i t to the data f o r D (AT) over the temperature 2 experiment. and con- However, the r e s u l t s do not confirm the predicted contribution from c^ i n (3). lower, to perhaps 0.28, 2 second range of his I n i t i a l l y one of the objectives of t h i s work was to obtain more p r e c i s e values of D i n order to e s t a b l i s h or deny the presence of 2 0^ and so provide a stringent test of the d e t a i l s of dynamic s c a l i n g . The second objective was to test the u n i v e r s a l i t y of the scaling r e l a t i o n (2) by performing the measurements of c r i t i c a l damping under pressure. was, and s t i l l i s , no such information on D 2 There available. The renormalization group treatment of a dynamic model^ of helium confirms the scaling r e l a t i o n (2), and provides a means of c a l c u l a t i n g the universal constant defined by the r a t i o R estimated value f o r R , 2 2 = D /2u £. 2 2 The t h e o r e t i c a l l y given i n the next section, has been found to be a factor of about f i v e smaller than that indicated by Tyson's data. This discrepancy had serious implications i n the i n i t i a l stages of t h i s work. If the smaller value of R as predicted by theory was i n f a c t correct, then 2 a considerable improvement i n the experimental error, compared to that obtained by Tyson, would be required i n order to resolve the predicted temperature dependence of D « 2 o During the course of t h i s work, Tanaka and Ikushima 3 t h e i r studies on thermal transport i n He - 12 have interpreted 4 He mixtures as evidence i n 13 support of the value of R 2 obtained from Tyson's data. Recently Ablers has reported r e s u l t s on second sound damping which are i n agreement with 6 the value of R 2 I n i t i a l l y calculated by Hohenberg, Siggia and Halperin 14 (HSH) on the basis of RGT. More recently, a nonlinear renormalization group analysis of the dynamics of the superfluid t r a n s i t i o n by Dohm and Folk ( D F ) ^ has resulted i n predictions concerning second sound damping which are also i n agreement with the work of Ahlers; however, t h e i r theory ef f predicts a temperature dependence for the e f f e c t i v e r a t i o R (AT/T^) ef f 2 which enters the equation, D Also, t h e i r predictions for D tending • R 2 2 2 2u £, governing the behaviour of 2 D 2< cover a larger temperature i n t e r v a l ex- beyond the c r i t i c a l region. In view of the t h e o r e t i c a l and experimental status of D 2 outlined above, the r e s u l t s of t h i s work are s i g n i f i c a n t i n that they confirm r e s u l t s of Ahlers. the In a d d i t i o n , since the measurements of t h i s work are of greater accuracy and cover a larger temperature i n t e r v a l , they provide a more severe test of the t h e o r e t i c a l predictions of HSH dicate the behaviour of D 2 and DF, and i n - in the temperature i n t e r v a l between the pre- v i o u s l y e x i s t i n g experimental data i n the c r i t i c a l and n o n - c r i t i c a l regions. C. The Dynamics of Superfluid Helium i ) Hydrodynamics and the Damping C o e f f i c i e n t D^ The hydrodynamics of the two f l u i d model of superfluid helium i s r e viewed here. This theory Is relevant to an understanding of second sound and the mechanisms responsible f o r i t s damping, as well as providing foundation and and the i n t e r p r e t a t i o n of the dynamic models employed i n scaling RGT. The two f l u i d model obtains some microscopic j u s t i f i c a t i o n i n the q u a s i - p a r t i c l e (elementary excitation) theory of Landau^, and more r e cently i n the theory of Hohenberg and Martin*^ based on the assumption of a Bose condensate. This model describes He II i n terms of interpenetrating normal and superfluid components with d e n s i t i e s and p r e s p e c t i v e l y . g The v e l o c i t y f i e l d s associated with these components are V and v ., subject r g to the i r r o t a t i o n a l condition on v s curl v g » 0, (4) provided v and v are below some c r i t i c a l value. n s In the approximation i n 16 18 which d i s s i p a t i v e e f f e c t s are neglected, a t y p i c a l d e r i v a t i o n * of the hydrodynamic equations begins with the conservation laws and an equation s of motion for v_ s a t i s f y i n g (4). Thus | £ + div J «= 0 (5-A) expresses mass conservation i n terms of l i q u i d density p and mass current ~i = ( j j ) which i s the momentum per unit volume; i ik i r — + -s ot ox, k 3 j 8 1 1 = 0 (summation convention) (5-B) i s the statement of momentum conservation where H., i s the momentum flux ik density tensor; the absence of d i s s i p a t i o n i s written as conservation of entropy ffiS- + div F = 0 (5-C) where a i s the entropy per unit mass and F i s the entropy f l u x ; the i r r o t a t i o n a l character of the superfluid v e l o c i t y f i e l d means that the time development of v s a t i s f i e s an equation of the form - ~ ot + grad (Ssvj + h) = 0 where h i s a scalar function. S These, (5-A,B,C,D), (5-D) are eight equations f o r the eight basic v a r i a b l e s p, O, v , i n terms of the yet to be determined s quantities II , F, h. Conservation of energy, ik | £ + div Q - 0 (6) where U i s the energy per u n i t volume and Q i s the energy f l u x density, i s a ninth and hence redundant equation which must be automatically i s f i e d by (5-A,B,C,D). sat- This constraint, the a p p l i c a t i o n of Galilean r e - l a t i v i t y and thermodynamic arguments are s u f f i c i e n t to determine F, h under the assumption that they do not contain d i s s i p a t i v e contributions i n the form of s p a t i a l gradients of thermodynamic v a r i a b l e s . In the determination of F, h the existence of two independent v e l o c i t y f i e l d s i s s i g n i f i c a n t to the thermodynamics since i t i s not gene r a l l y possible to transform to a frame i n which the f l u i d i s at r e s t . Thus, there appears an a d d i t i o n a l conjugate pair of thermodynamic v a r i a b l e s a r i s i n g from the r e l a t i v e i n t e r n a l v e l o c i t y . Consider a Galilean trans- formation r e l a t i n g two frames denoted by subscripts 1 and 2 and with r e l a t i v e v e l o c i t y v^. s i t y are 2 V = v i + v r The r e l a t i o n s f o r v e l o c i t y , momentum and energy den- » J2 = ^1 + P V r' 2 U = U l + V r * ^1 ^ + P V r * A n e n e r Sy density which s a t i s f i e s t h i s transformation i s U = U + v » ( j - p v ) + O S s hpv s where U i s a Galilean invariant and represents o frame i n which the superfluid i s at r e s t . (7) 2 the energy density i n a As U q i s an invariant i t s de- pendence on the basic variables p, C, j , v i s dU_ «= ydp + Td(pa) + w »d(j - p v ) (8) R since j - pv i s i n v a r i a n t . s In (8) w i s the conjugate to j - pv s i s to p and T i s to pa) and serves to define via w " v ~ V n E ' (as y Then, the l a s t term i n (8) states that the d e r i v a t i v e of energy with respect to momentum i s v e l o c i t y . With expressions (7) and (8) for the energy density, the constraint imposed by energy conservation n"ik' F, h. can be used to determine By d i f f e r e n t i a t i n g U with respect to time and then replacing a l l time d e r i v a t i v e s by s p a t i a l derivatives through the use of equations (5-A,B,C,D), i t i s possible to i d e n t i f y the energy f l u x density Q as well as F, h. The algebra can be found i n considerable d e t a i l i n r e f - erence 18. The r e s u l t s become i n t u i t i v e l y appealing when the basic var-»• i a b l e set i s taken to be p, ' -*• p = p s j - P v + P v with p + n n s s n n e Then the expression f o r n (9). is i k * P ik where p i s the pressure. . •+ -V- a, v , v where v i s r e l a t e d to 1 by ' n* s n 6 + P s si sk V v Thus, the momentum f l u x density + p n ni nk v v ( 1 0 ) appears as a natural generalization of » P^ik + s * 8l n e f l u i d hydrodynamics. The r e s u l t f o r the entropy f l u x vector i s F - pov (11) n which means that a l l entropy i s carried by the normal f l u i d . The scalar function h •= V i s the chemical p o t e n t i a l . (12) The quantities y, p, p ^ which now appear are -»• functions of p , O as w e l l as ( v -*• 2 - v ) . Thus, the hydrodynamic equations, neglecting d i s s i p a t i o n are (5-A,B,C,D) with j , F, h given by (9, 10, 11, 12). Before discussing wave solutions to these equations, they w i l l be augmented to include d i s s i p a t i o n i n a n t i c i p a t i o n of obtaining a hydrodynamic expression f o r second sound attenuation. D i s s i p a t i o n i s a consequence of the i r r e v e r s i b l e processes associated with thermal conduction and the v i s c o s i t y or i n t e r n a l f r i c t i o n which r e s u l t s from i n t e r n a l motion. These i r r e v e r s i b l e processes occur when there are departures from equilibrium and cause the system to move towards an equilibrium 6 t a t e characterized by a maximum i n the entropy. Thus, the approach to equilibrium involves entropy production, Z/T, and the genera l i z a t i o n to equation (5-C) i s 10 div (pov + |) •= | n (13-C). The d i s s i p a t i v e contribution to entropy f l u x , q/T, i s , of course, i d e n t i f i e d i n lowest order with thermal conduction. Admitting a d i s s i p a t i v e contrib- u t i o n , h', to the superfluid flow but s t i l l requiring c u r l v g «= 0, the superfluid acceleration equation (5-D) becomes + grad (y + %v + h') «= 0 (13-D). S dt There s t i l l remain the conservation laws f o r mass, momentum and energy. The equation f o r mass conservation or continuity i s unchanged: | | + div j - 0 V i t h a viscous stress tensor the momentum equation i s 3j dt l (13-A). + 3(11 IK dX, k + T l ) k =0 (13-B). Energy conservation i s now | | + d i v (Q + Q») - 0 (14) where Q' i s the a d d i t i o n a l d i s s i p a t i v e energy flux density. The form of Z , q, h', must now be determined. As i n the non- d i s s i p a t i v e case the energy equation (14) must be s a t i s f i e d automatically. By d i f f e r e n t i a t i n g with respect to time the expression f o r the i n t e r n a l energy (7, 8) and replacing time derivatives by s p a t i a l ones through the -*• -+• use of (13-A,B,C,D), a pure divergence term may be i d e n t i f i e d with Q + Q , while the remainder must vanish. *- - ^ This y i e l d s f o r the entropy production - *u - »'K*s - V (»)• k The entropy production must be p o s i t i v e d e f i n i t e and vanish i n equilibrium. This requirement, and those based on G a l i l e a n covariance, are s u f f i c i e n t 11 to determine expressions for the fluxes q, h' which, to f i r s t order i n the deviations ( s p a t i a l derivatives) from equilibrium, involve thirteen independent k i n e t i c c o e f f i c i e n t s . r small since i n the equilibrium large due -+• At t h i s l e v e l v n of small deviations from a non-rotating equilibrium n - v s s i s not considered •+ •+ v - v can be n s However, i n the l i m i t state of s o l i d body r o t a t i o n to the presence of superfluid v o r t i c e s . t h i s work, v - v i s also small. state, as applies to Then there appear k i n e t i c r r coefficients in the fluxes as follows: q = -KV" T (16) where K i s the c o e f f i c i e n t of thermal conduction, - -^tef £f - KjH) • «(hW\-\> ^H) + \i s > (17 + and h' = ,v".p (v" o s s - v) n - C,v"4 4 (18). n In the viscous stress tensor, T.., there are the usual c o e f f i c i e n t s of f i r s t and second v i s c o s i t y , ri and hydrodynamics. Due there appears i n Z,^, which appear in the "normal" f l u i d to the additional degree of freedom allowed by v , s another second v i s c o s i t y , d i s s i p a t i o n generated by r e l a t i v e motion, r e c t i o n , h', v n ~ which determines the v s * The dissipative cor- to the chemical potential contains two more c o e f f i c i e n t s of second v i s c o s i t y £ 3 , £45 however, by the Onsager r e c i p r o c i t y theorem, £^ = g so that there are f i v e independent k i n e t i c c o e f f i c i e n t s . 2 Z > 0 requires that K, n, & , & be p o s i t i v e and ^ < X, ^>y 2 The 2 3 c h a r a c t e r i s t i c s of sound propagation can now be analyzed on basis of the hydrodynamic equations, (13-A,B,C,D), with the (15, 16, 17, 18). The ized form by means of equations for p, O, v , v Also, the substitutions are written i n l i n e a r - 12 p - p + 6p n o - O 60 + Q p - p + 6 p T = V •+ V • ov s s • n OV n T+6T The disturbances, 6, contain the space and time dependence. The e q u i l i brium state, denoted by a subscript 'o', i s one i n which v - v «* 0. The * n s J l i n e a r i z e d , that i s , to order 6, equations including d i s s i p a t i o n are then 35v . 36v , .r-^+P -5—— no 3x s o dx^^ 3t K 3_ , 3 + l C X j 33(pa) 3t , 36v si , = 0 (19-A), A ^ 3<5u r so< sk - *W 3P l 3x ^2 3 x nk o°o 3x, k 3 i r + v 9 S v P ^nk j 6v < 3 T 3x. ( w i 35v . nk > K k Wave i n the form , . and s i m i l a r l y f o r 6o, 6p, 6T, 6v « 5v x, 6v • 6v x. * * * n n ' s s v equations now become, dropping the subscript The l i n e a r i z e d 'o', -iwp + ikp v + ikp v = 0 n n s s (20-A), K -iwp v nn ... k . i ( u t - kx) ~ -iw(t - x/u) 6p = pe « pe J B ) i (19-C), In t h i s approximation 6u i s j u s t the usual 6y = (l/p)Sp - 0 6 T . solutions are attempted . > 36T 3x, i 3p (6v , - &V , ) so sk nk , i k - iwp v + ikp - - k ( 4l - P ?i + S,)^ ss 3. s i 2 n 2 -iw(po + pa) + ikpav «= -k n 2 £ T 1 - k c; p v l s s 2 (20-B), (20-C), 13 -low + i k ( £ - O-T) e - k (C - P ? )v 2 A s 3 - k £ P v 2 n 3 s 6 Thus, there are now four equations f o r four unknowns p, 5, v , v . (20-D). The small v a r i a t i o n s p, T depend i n t h i s approximation on p, O as „ 8T<\ ~ ST-* ~ A dispersion r e l a t i o n , co(k), can be obtained by eliminating v , v in favour of p, o and r e t a i n i n g only f i r s t order terms i n the k i n e t i c coefficients. Equation (20-B) through the use of (20-A) and (20-C) becomes A second equation i s obtained from (20-D) by using (20-B) as an expression for p and subsequently eliminating v^ and v by means of (20-A) and (20-C). g The r e s u l t i s t 3T u) 2 K. p n >~ r 1 , _ . 2 . 4 n . ? P S n K 3T.~ S s -o|£p 3p (22). Consider f o r the moment the approximation i n which d i s s i p a t i o n i s neglected by ignoring the k i n e t i c c o e f f i c i e n t s i n equations (21) and (22). Then they read <Sl k u^ - & - 0 a &) 5 p r k u 2 - o (a, 14 with u - Op/3p) 2 and u 0 - (P /P )o (3T/9a) 2 2 s n p - (p /p )a T/c . 2 s n v The con- d i t i o n f o r the compatibility of (23) and (24) i s that the determinant of the c o e f f i c i e n t s vanish. c-^r - Hk-u k u 2 Thus k^u 2 o- C|g> tfDp tfgla c^)p a 2 = (c - c )/ c P v p At the saturated vapour pressure and f o r the values of AT i n t h i s exper- iment 19 (c - c )/ c << 1, so the approximation that t h i s term i s zero P v p r e s u l t s i n decoupled modes • f - ' l " *-p ^ < > 25 ~ „ » f_J> 13.]* c k " 2 n CO = u u known as f i r s t and second sound. c = c J p - c (26) v The f i r s t i s the usual adiabatic density- pressure sound wave, while second sound i s an entropy-temperature wave at constant density. Now consider the d i s s i p a t i v e equations (21) and (22) i n the decoupled approximation (c - c )/c - 0. * P v p To f i r s t order i n the k i n e t i c c o e f f i c i e n t s the requirement that the determinant of the c o e f f i c i e n t s of p' and o' van- ish y i e l d s dispersion r e l a t i o n s as follows: 2 co co 2 ico s t 4 ^ . 2 — = u - _ _ ( n + p S 2 and 4 W f 4 ^ , 2 p 2 k y n . - p ^ + n 3 f r r r " ) . , _ . n ic + C + — P 2 ( 2 7 ) s Considering the second sound solution, the attenuation can be determined by writing k - k Q PU2 + i a and expanding to f i r s t order. 2 n This gives 8 15 The damping constant D i s defined by 2 a (a),T) - hCw / u^) D (28) 2 2 Therefore i n l i n e a r i z e d hydrodynamics D '2- PP n M T ) - ? r ( f i i + P\ 2 2 i s independent of frequency and - Pttj. + C ) + C A 2 + ^ f ) (29). Thus, on the basis of hydrodynamics alone, measurements of second 6ound damping may only be interpreted i n terms of a rather lengthy combination of thermodynamic properties. D 2 of However, some information concerning can be gained from independent measurement and t h e o r e t i c a l c a l c u l a t i o n the i n d i v i d u a l quantities contributing to D . 2 haviour of D 2 * n p a r t i c u l a r , the be- as T •+ T^ i s expected to be approximately proportional to -1/3 (AT) . The contributions to t h i s divergence are as follows. 2/3 known from experiment that p s a (AT) . The v i s c o s i t y , T), i s measured to be f i n i t e at T^ so i t s e f f e c t on Dj, i - P n, vanishes. s c o s i t i e s z;^, C 2 > £3 It i s The second v i s - are expected, from f i r s t sound attenuation measure- ments and the Landau theory, to vary roughly as (AT) *" and therefore -1/3 contribute (AT) . The strength of the thermal conductivity term i s conjectured. I f i t s behaviour below T^ (which cannot be measured due to -1/3 s u p e r f l u i d i t y ) i s the same as above, then i t Is about (AT) 3 experiments on . I f , as A He - He mixtures suggest, tc i s f i n i t e at T^, then the growth of the s p e c i f i c heat, c, as T •* T^ would cause tc/pc to diminish. The s c a l i n g treatment of c r i t i c a l dynamics as i t r e l a t e s to D 2 i s now discussed. i i ) Scaling and D,, Scaling^ begins with the recognition of the importance of the variable which has the largest f l u c t u a t i o n s near the t r a n s i t i o n and consequently Is most responsible f o r the c r i t i c a l behaviour. This v a r i a b l e , the order par- 16 amter tj;, has a range, £, of c o r r e l a t i o n s i n the f l u c t u a t i o n s which i s d i v ergent as £ • C ( T/T )~ . A 0 V c The d e s c r i p t i o n of f l u c t u a t i o n s of any (Her- mitian) v a r i a b l e A(r,t) i s done i n terms of the c o r r e l a t i o n function c ( r , t ) defined as c ( r , t ) •= h A < ( (A(r,t) - < A ( r , t ) > ) , (A(0,0) - < A ( 0 , 0 ) » } > where the angular bracket denotes equilibrium average and et i s an anticommutator. (30) the curly brack- In dynamic s c a l i n g the Fourier transform of ~A c ( r , t ) i s written i n the form c£(k,C0) = 2lT ft£(k) *> * C^(k) f^ r ( _ 1 -T—) 0>*(k) where the subscripts £ i n d i c a t e a parametric dependence on the dominant order parameter c o r r e l a t i o n length. This expression contains the equal time c o r r e l a t i o n function (30)) (t • 0 i n 00 CrOO - J ^ c (k,o>) E and a shape function, f , such that 00 / _oo The fJ r ( x ) dx = 1 > ^ 0)^ (k) i s the c h a r a c t e r i s t i c frequency defined jff£ (x)dx-| i ? The by . shape function, f , i s determined by the hydrodynamics of the system being considered. equations and The general r e l a t i o n s h i p between the hydrodynamic c o r r e l a t i o n functions has been established by Kadanoff and 20 Martin 17. The , and the s p e c i f i c case of helium has been dealt with i n reference c o r r e l a t i o n function d e s c r i p t i o n i s i n p r i n c i p l e more general than the hydrodynamic d e s c r i p t i o n , and the two of small k and 0) where hydrodynamics a p p l i e s . are equivalent i n the In p a r t i c u l a r , the limit 17 frequencies and damping of the normal modes i n hydrodynamics appear as the poles of the appropriate c o r r e l a t i o n functions. i n the shape function, f , given previously. This structure i s contained Thus, for example, i f the hydrodynamics y i e l d s the frequency and damping of a normal mode expressed 2 in terms of the dispersion r e l a t i o n co 2 2 2 = u k - icok D as i n equation (27), then the c o r r e l a t i o n function description of t h i s mode i s r e f l e c t e d i n a shape function of the type f, k r 1 (x) = — ^ y —5 * k o 9 (x - l ) + y 2 2 2 The c h a r a c t e r i s t i c frequency co^(k) Implicit i n x i s j u s t the frequency of the normal mode co^(k) • uk » and the width y^ i s Dk 2 y = — k uk *= Dk u Now one assumption, of dynamic scaling i s that the shape function for the order parameter c o r r e l a t i o n function depends on k and £ only through the product k£. Thus, i f the normal mode and shape function discussed above correspond to that of the order parameter, then t h i s assumption means, since y ^ i s l i n e a r i n k, that D « u£ (31). In the case of the X - t r a n s i t i o n there are complications which stem from the f a c t that the order parameter i s the average, over a small region 21 of space-time, of the a n n i h i l a t i o n f i e l d operator . As the f i e l d oper- ators are not Hermitian, the order parameter i s complex, that i s , i t has two components. The order parameter c o r r e l a t i o n function then decays at large r according to a power law, p, so that 18 c^(r) »u \$\ ( | ) 2 P for r + » . and t h i s serves as a d e f l n t i o n of the c o r r e l a t i o n length £. Also, as demonstrated by the hydrodynamics, there are two propagating modes. However, i t i s shown i n reference 17 that the order parameter c o r r e l a t i o n function i s dominated by the second sound mode to the extent that (c - c )/c << 1. p v p 3 In addition, i t i s shown that c^(k,w) , which i s not d i r e c t l y observable, has poles which are i d e n t i c a l to those of the c o r r e l a t i o n function of the heat operator q ( r , t ) * U(r,t) <£-^2>p( ,t) r = U(r,t) - <y + To> p(r,t) (32). Thus, i t i s p o s s i b l e to formulate scaling i n terms of the observable f l u c tuations of the heat operator ((dq^= ^Tp do}) which correspond to second sound. The s c a l i n g r e l a t i o n (31) i s then a prediction of the damping of second sound The s i g n i f i c a n c e of t h i s r e s u l t to t h i s work was discussed i n the previous section, B. i i i ) Renormalization Group Theory and D,, The RG treatment of the dynamics of the lambda t r a n s i t i o n i s a work of such magnitude that even a m i l d l y comprehensive development of the prediction for D 2 i s beyond the scope of t h i s t h e s i s . Thus, following a d i s - cussion of the dynamic model which undergoes the renormalization, only the procedure f o r performing the RG transformation i s indicated. diction for D 2 Then the pre- i s given. The dynamic models treated by RG techniques are semi-microscopic i n 19 that they are defined by equations of notion f o r the v a r i a b l e s which r e main a f t e r averaging over length scales which are larger than atomic dimensions but smaller than the c o r r e l a t i o n length f o r the order parameter when T i s near T . One such v a r i a b l e £ be given i s the order f o r which an equation of notion must parameter, ty. The equations f o r the other var- iables r e f l e c t the various symmetries, or equivalently the conservation laws, o f the system being studied. served f i e l d s . In l i q u i d helium there are three con- As i n the two f l u i d hydrodynamics, they are the energy density, U, the mass density, p, and the momentum density, j . A complete semi-microscopic d e s c r i p t i o n o f the dynamics of helium would then Involve, including the order parameter, four f i e l d s . However, i t i s a n t i c i p a t e d that, as a s t a r t i n g point, a two f i e l d model Is adequate since i t i s poss i b l e to incorporate into such a model the c r i t i c a l hydrodynamic mode associated with a f i e l d , m, which couples most strongly to the order parameter. The f i e l d m i s the l i n e a r combination of U and p which produces second sound f o r T < T^ and i s denoted by q i n equation (32). The two f i e l d 6 22 model of helium i s defined by the following stochastic equations ' : ^ L £ - > - » O at F - F F o 0 o Om • e C33-A). n ' v n - Jd x {h (x,t)m + Re( h^(x,t)\J>* )} (33-C), d o m - Jd x (»sr \ty\ + h\Vi>\ + 2 d o - i* * £ o 2 u O M * + ^ m 2 + y m\ty\ ) (33-D). 2 o Some o f the features of t h i s model a r e Indicated now. Langevin noise sources. h The 8 , C are n n In the absence of time dependent applied f i e l d s and h,, these noise sources, when chosen appropriately, ensure that ty m ty and m achieve values consistent with the equilibrium p r o b a b i l i t y d i s t r i b - 20 ution P eq (ty,m) functional F q - e d^dm. / Je The f i r s t three terms i n the represent the usual Ginzburg-Landau expansion i n terms of the order parameter. A s i m i l a r expansion i n the f i e l d m i s truncated a f t e r the f i r s t term. In RGT the higher powers of m i n the expansion are I r r e l evant, while i n the expansion i n powers oftythe i n t e r e s t i n g or n o n t r i v i a l behaviour i s a r e s u l t of the U Q | ^ J ^ term. The i n t e r a c t i o n term, Y ^ l ^ i ^ , in F i s included because a v a r i a t i o n i n m, which i s i d e n t i f i e d as second o ' sound, means there i s a change i n the l o c a l value of AT which i n turn r e quires that ty obtain a new l o c a l equilibrium value. The f i r s t term on the r i g h t hand side of (33-A) indicates that ty i s not a conserved f i e l d i n that i t causes ty to relax (ReT > 0) to a value which minimizes F . The f i e l d o o m, however, i s a conserved quantity since the right hand side of (33-B) can be written as the divergence of a current f o r h^ • 0 and £ r as given i n reference 22. The s i g n i f i c a n c e of the coupling constant g stood by considering the e f f e c t of a uniform Q i n (33-A,B) can be under- time-dependent applied f i e l d which i s conjugate to m, h (x,t) = h ( t ) . Writing the complex (two comm m ponent) order parameter i n terms of a phase angle 4> asty= | ^ | e ^ , then (33-A) gives the e f f e c t of h on <t> as m !£=gh (34), dt °o m that i s , h causes a r o t a t i o n of the order parameter. Although the notation m i n (34) i s more suggestive of an equivalent system of spins known as the 0 0 planar ferromagnet O1 Z J » O/ z ,the corresponding r o t a t i o n equation for helium 21 i 6 a "Josephson" equation C 3 5 ) where y i s the chemical p o t e n t i a l per p a r t i c l e of the f l u i d at r e s t . The 21 connection between (35) and the superfluid a c c e l e r a t i o n equation (5-D) i n the two f l u i d model can be made by the i d e n t i f i c a t i o n ti^ct « mv where m 6 i s the mass of a helium atom, and adding the k i n e t i c energy contribution 2 h m v to the chemical p o t e n t i a l i n (35). The r e s u l t of the coupling, g , O 6 on the hydrodynamics i s that there i s a propagating mode f o r T < T 22 which 24 involves coupled v a r i a t i o n s i n m and cp ' . This i s second sound i n helium while the corresponding mode i n the planar magnet i s a spin wave. The renormalization group transformation i s applied to the cor- r e l a t i o n function formulation of the equations of motion (33-A,B,C,D). I t s i s i s defined by = ^b^b ^ * change w n e r e sa s c a x e x -*• x* «= x/b A -»• A' = Ab \K = b i> ip a = b co •*• co co such that b > 1, and a, z are determined within the theory. R^ applied to the diagrammatic expansion The operation of the equations of motion i s an Integration over wave vectors such that b ^A < k < A and frequencies from _» to +°°. The transformation i s i t e r a t e d , say n times. The requirement that the equations r e t a i n the same form leads to recursion r e l a t i o n s for \f renormalized constants 1 o r i g i n a l set {r ,g , X Q q , g ,X n n' n e ... }. ... } which have developed ' from the An analysis of the fixed points of the recursion r e l a t i o n s , that i s , those l i m i t i n g values [T , g , X which remain unchanged by successive i t e r a t i o n , leads to scaling laws as well as values f o r exponents and c e r t a i n universal amplitude ratios. One such u n i v e r s a l amplitude r a t i o i s *2 " where i s the damping constant of second sound with v e l o c i t y u < > 36 and £ 22 i s the transverse c o r r e l a t i o n length of the order parameter. (In super- f l u i d helium there are two c o r r e l a t i o n lengths, one associated with f l u c tuations i n magnitude, the other with f l u c t u a t i o n s i n phase. The latter IA i s c a l l e d the transverse c o r r e l a t i o n length.) used to evaluate an expansion i n three dimensions. i n e - A - d gives R i z a t i o n of RGT One approximation ^ 0.15. 2 D i f f e r e n t methods may technique using Another method i s a general- to three dimensions and r e s u l t s i n R 2 ^ 0.09. In each case the c a l c u l a t i o n s are based on the simpler symmetric model f o r helium a t i o n (33) with Y (equ- • 0) which i s expected, i n three dimensions, to give q the correct asymptotic behaviour as T of c a l c u l a t i o n i s expected An e x p l i c i t be to give R 2 T^. The accuracy for either method to within a factor of two. expression f o r D ( t ) where t • (T^ - T)/T^ may be ob13 tained by using e m p i r i c a l l y determined expressions f o r u and £ . Ahlers' 9 3 0 387 measurements give • 4.63 x 10 t cm/sec at saturated vapour 2 2 pressure. The same data, i n conjunction with the hydrodynamic expression for u_ and measurements of c i P and o provide the best information on p ( t ) . 5 -2 This may be used to evaluate £ • m £ • 3.57 x 10 t 0*675 8 D 2 cffi ^ 2 (k^T)/ fi Pg( ^ * t h the r e s u l t t v Then, the p r e d i c t i o n f o r D - ( 3 or 5 ) x 10" depending on the two estimates f o r R j . 5 " * 0 t 2 8 8 becomes 2 cm s " (37) 1 I t i s noted that both the amplitude and the exponent are subject to v e r i f i c a t i o n by experiment. The recent treatment by Dohm and Folk (DF)*"^ of the dynamics of the superfluid t r a n s i t i o n y i e l d s new predictions concerning D with the stochastic model employed by HSH B, C, D, with Y Q " 0. 2> They begin and described by equations 33 A, However, t h e i r a n a l y s i s of the f i x e d points of the renormalization group transformation leads DF to predict a temperature dependent e f f e c t i v e r a t i o , R f f 2 ( t ) , which determines D 2 v i a D / 2 u £ • R***'. 2 2 Using thermal conductivity data above T^ to evaluate non-universal par- 23 ef f ameters entering t h e i r theory, DF obtain a value f o r -A at t « 10 of -A about 0.1A. t > 10 D 2 The temperature dependence i s weak over the i n t e r v a l 10 , but stronger f o r t > 10 > . As a simple a n a l y t i c expression f o r i s not a v a i l a b l e i n t h e i r report, the graphical presentation of t h e i r predictions i s reproduced s u l t s of t h i s work. their calculations. i n Chapter A along with a discussion of the r e - They {DF) do not indicate the expected accuracy of 24 CHAPTER 2 EXPERIMENT The f i r s t section of t h i s chapter i s a discussion of techniques used in second sound attenuation measurements with p a r t i c u l a r emphasis on the method chosen f o r t h i s work. Section B i s a discussion of the apparatus. This includes a d e s c r i p t i o n of the cryogenic apparatus, the resonator, and the e l e c t r o n i c s . An o u t l i n e of the experimental procedure i s given i n Section C. A. Techniques There are several methods a v a i l a b l e which may be used to measure the attenuation of macroscopic second sound. used by Hanson and Pellam (HP). The most d i r e c t approach i s that They measure the temperature amplitude, T, of a t r a v e l l i n g second sound wave as a function of the distance, x, between the generator and detector, and determine the attenuation, a, by means of ~'CtX T = T e Q . In another method, as employed by Tyson and Ahlers, the atten- uation i s determined from the frequency dependence of the amplitude of 25 standing waves i n a resonant c a v i t y . A t h i r d technique has been developed which involves an analysis of the shape of second sound pulses and has been used to determine the attenuation under pressure but f o r T much less than T^. Other methods are conceivable. For example, one might expect that attenuation measurements could be made using the amplitude decay of a second sound "tone burst" propagating between r e f l e c t i n g plates. In t h i s experiment a resonance method has been developed i n which the attenuation i s derived from the decay time, T, f o r free o s c i l l a t i o n s of the plane wave modes of a c y l i n d r i c a l c a v i t y . This i s e s s e n t i a l l y the Fourier transform of the technique used by Ahlers and Tyson. In t h e i r experiments 25 a Is related to the f u l l frequency width, Au>, at half maximum of the power, 2 that i s , T , by Ato * 2U2CU In the present work M2 a.2 T determines the temporal —t/T response as T • T^e e l l i n g waves, 1 « l^e . The expression which applies to plane trav- 0 0 1 T « 5 L ^ E K X U 2 T appropriate to a standing , becomes plane wave mode and therefore 1 / T E 2U2O1. Of course, Aw «= 1 / T . In these expressions a i s the t o t a l attenuation due to the bulk helium and contributions associated with the boundaries of the resonant c a v i t y . The decay time method used i n t h i s experiment gains one p a r t i c u l a r l y s i g n i f i c a n t advantage while r e t a i n i n g the benefits of a resonance approach. The reasons f o r choosing a resonance technique i n general are based on the desire to approach as c l o s e l y as possible and achieve a r e s o l u t i o n i n AT on the order of 10*"^ K. Thus, i n addition to the requirement of temp- erature s t a b i l i t y , a small system i s preferred i n order to minimize the pressure increase due to gravity which a l t e r s the value of T^ by about 1.3 x 10~^ K per centimeter of helium. I t i s f e l t that thermal i s o l a t i o n and small s i z e are more e a s i l y achieved with a resonant cavity as opposed to the method of HP which requires a v a r i a b l e propagation path of considerable length to avoid m u l t i p l e r e f l e c t i o n s . Also, the e f f e c t s of f i n i t e second sound amplitude ( r e c a l l the approximations of l i n e a r i z e d two f l u i d hydrodynamics and that v severe as T + T, since then v A s n > V g are proportional to 1) become more can become large due to the vanishing of p . s Therefore, i t i s desirable to use small signal l e v e l s to avoid what may be a d i f f i c u l t i n t e r p r e t a t i o n of large amplitude e f f e c t s . I n addition, the frequency dependence of 02 indicates that a signal with limited frequency content requires l e s s i n t e r p r e t a t i o n than, say, a pulse s i g n a l . A res- onance method provides a continuous wave, narrow band signal to which standard but powerful detection techniques may be applied. Moreover, the resonance i t s e l f r e s u l t s i n an a m p l i f i c a t i o n of the AC e x c i t a t i o n . This 26 i s important i f the second sound i s generated, as i t i s i n t h i s experiment, by the AC e l e c t r i c a l heating of a r e s i s t i v e element. Then there i s a DC component present i n the power spectrum of the e x c i t a t i o n which r e s u l t s in a steady counterflow of v^ and v i s superimposed. g upon which the AC second sound flow However, to the extent that the gain*of the cavity i s very large, the DC flow v e l o c i t i e s are n e g l i g i b l e i n comparison to the AC flow v e l o c i t i e s . As t h i s s e l e c t i v e 'gain' i s not present i n the methods using a t r a v e l l i n g wave, or pulsed second sound, a resonance method i s preferred since i t , i s expected that with a DC counterflow there are cor18 rections to the expression (27) for the attenuation The measurement of decay times as opposed to l i n e widths overcomes a problem related to the l i m i t a t i o n s of temperature s t a b i l i t y and frequency range that are encountered i n t h i s experiment. To understand the nature of the problem, consider the harmonic sequence for plane-wave modes i n a cavity consisting of p a r a l l e l plates separated by a length a. The resonant frequencies are given by co p where p = 1, 2, 3 ... . = uk 0 I p = u (pTT/a) 2 0 During the course of a frequency sweep through some resonance of width Aco^ at co^ suppose that the ambient temperature changes by 6T. Then the second sound v e l o c i t y changes by = (3U2/9T)6T and, therefore, the resonant frequency changes by an amount Sco^ • 6u2kp, or 660^ = (6u2/u2)t0p. as T T^. For small AT, 6U2 becomes large since S^/BT diverges The amplitude response to the driven o s c i l l a t i o n s now a d i f f e r e n t value appropriate to the new resonant frequency approaches co^ + Sco^. Thus, the t y p i c a l temperature f l u c t u a t i o n s , 6T, r e s u l t i n d i s t o r t i o n s of the resonance response curve making i t d i f f i c u l t to determine Aco^. This i s s i g n i f i c a n t when the "temperature noise width", 6co , becomes comparable p 27 to the I n t r i n s i c width (AWp) • a l n p r i n c i p l e t h i s problem can be overcome by using higher harmonics since the i n t r i n s i c width i s expected to vary as 2 2 (AOJ ) • 2u„a « (a) /u„)D-, and the frequency squared dependence w i l l u l p ot2 2 2 p 2 2 0 timately dominate the l i n e a r dependence i n 50)^ = (5u /u )top. 2 Unfortunately, 2 modes which do not correspond to plane waves, but' rather to Bessel functions, complicate the cavity response. The excitation of 'Bessel modes', to be discussed below, makes i t d i f f i c u l t i n t h i s experiment to interpret the resonant structure at the frequencies of the higher harmonics. However, the attenuation, a , can be obtained by measuring the decay times of the 2 w e l l i s o l a t e d , low frequency harmonics. In t h i s method the cavity i s driven at or near the resonant frequency u n t i l the e x c i t a t i o n reaches some desired high l e v e l . recorded. The drive i s then turned o f f and the decay of the e x c i t a t i o n The o s c i l l a t i o n frequency s t i l l fluctuates by Sco^ due to the temperature noise, 6T, but now t h i s does not appear as amplitude noise i n the signal since the response i s not driven but allowed to decay f r e e l y . I t i s only necessary that the bandwidth of the detection system be large enough to accommodate the frequency content of the decay, e ^ ,' and the excursions, 6w , which occur during the decay. The d e t a i l s of the signal P recovery system are found i n the discussion of the e l e c t r o n i c s . t T The general resonant frequencies of a c y l i n d r i c a l cavity of radius r 26 and length a are given by, 2 p,m,n The a mn /. v a a 2 r ' , with m, n «= 0, 1, 2 ... , are solutions to (dJ (ircO/da) • 0, m where J (TO) i s a Bessel function of the f i r s t kind. m are obtained f o r a^g •= 0. The plane wave modes The modes with m or n not zero are loosely r e - ferred to as Bessel modes. For m or n near one, the a are on the order mn of u n i t y , while f o r large m and n approximate values are a^ - m/tT and o 28 - n + Jsn + h when n > m. The dimensions of the cavity i n t h i s exper- iment are such that r - 2.A a, and therefore the lowest resonant correspond to Bessel modes. frequencies The density of the Bessel modes Increases with frequency with the r e s u l t that at the frequencies of the higher plane wave harmonics there may be several Bessel modes having nearly the same frequency as any p a r t i c u l a r plane wave mode. With the use of equation 26 (38) and the tabulated the resonant values of a for m = 0 to 8 and n = 1 to 20, mn frequencies can be calculated to determine which of the plane wave modes are well separated from Bessel modes. Because of the high density of the Bessel modes, the r e s u l t s are s e n s i t i v e to the value of the r a t i o r/a, which i s known with an accuracy of about ± 0.5%. In an- t i c i p a t i o n of the observations on harmonics 1, 2, 3 and A, a c a l c u l a t i o n indicates that the f i r s t and t h i r d harmonics are isolated to the extent that to within ± 1% of t h e i r frequencies there are no Bessel modes. In view of the uncertainty In r / a , t h i s means that harmonics one and three are f r a c t i o n a l l y i s o l a t e d from Bessel modes by at least 0.5% of t h e i r r e spective frequencies. This degree of i s o l a t i o n i s s i g n i f i c a n t since i t i s large compared to the maximum f r a c t i o n a l width of a resonance, Aco/co^, of about 0.05%. However, for harmonics two and four, there are two Bessel modes within ± 0.5% of the frequency the case of harmonic four. of harmonic two, and four i n Thus, i f the Bessel modes are excited, they could influence the response at the second and fourth harmonics. The consequences of the p o s i t i o n of the Bessel modes w i l l be discussed i n more d e t a i l i n r e l a t i o n to the experimental observations. Possible mechanisms which may be responsible f o r the e x c i t a t i o n of the Bessel modes are suggested i n the general discussion of the concluding chapter. 29 B. Apparatus I) Cryogenic Apparatus The general features of the cryogenic apparatus are i l l u s t r a t e d i n Figure 1 and described here. ator was suspended The experimental c e l l containing the reson- inside an evacuated container. This i n turn was im« mersed i n a bath of l i q u i d helium (T < T^ ) to provide a- stable thermal 27 environment. The temperature of t h i s outer bath could be regulated -4 better than 10 K over a h a l f hour i n t e r v a l . to Helium from t h i s bath 3 f i l l e d the experimental c e l l (about 15 cm ) through a valve and c a p i l l a r y . A porous s t a i n l e s s steel f i l t e r over the valve entrance kept s o l i d a i r p a r t i c l e s out of the c a p i l l a r y and valve seat. An estimate of the helium l e v e l i n the c e l l was made using a depth gauge that consisted of a c y l i n d r i c a l capacitor that formed part of a tunnel diode o s c i l l a t o r . A therm- ometer and standard r e s i s t o r at the bottom of the c e l l formed the cryogenic part of a bridge c i r c u i t that was primarily used as a temperature con- t r o l l i n g device i n conjugation with a feedback r e s i s t o r wound on the outside of the copper top of the c e l l body. The second sound resonator was held i n a brass frame that enabled the resonator body to be held together by spring loading. The second sound detector (bolometer) i n the reson- ator was a superconducting device. I t s t r a n s i t i o n temperature was trimmed to the desired temperature by a magnetic f i e l d produced by means of a s o l enoid wound on the outside of the vacuum container. A few other features might also be considered as follows: ( i ) A second c a p i l l a r y connected the c e l l to room temperature access. This was a v a i l a b l e f o r pumpimg away excess helium i n case of accidental overfilling. sures. Also, t h i s l i n e would be necessary f o r studies at elevated pres- 30 PUMP LINE INDUCTOR AND TUNNEL DIODE FOR OSCILLATOR CIRCUIT \ L O W TEMP. FILTER 1 V A L V E LEADS CAPILLARY H E A T E R S AND T H E R M O M E T E R S CAPILLARY TO ROOM TEMP. ACCESS FILLING CAPILLARY EXPERIMENTAL OUTER VACUUM CELL BATH JACKET FEEDBACK RESISTOR RESONANT CAVITY CYLINDRICAL CAPACITOR FOR OSC. CIRCUIT DEPTH MONITOR INDIUM •0" MAGNET FPOXY RING SOLENOID FEEDTHROUGH LEADS THERMOMETER S T A N D A R D Figure 1 C r y o g e n i c Apparatus RESISTOR 31 ( i l ) There existed several thermometers and heaters at various locations that were used to e s t a b l i s h i n i t i a l working conditions. Those mounted on the c a p i l l a r i e s were p a r t i c u l a r l y important since I t was necessary to destroy temperature i n s t a b i l i t i e s of an o s c i l l a t o r y character, and magnitude _3 of about 10 K, that were generated by helium i n the c a p i l l a r i e s . i n s e r t i o n of piano wire (0.2 mm diameter wire i n 0.3 mm The i . d . capillary) along with the power input from the heaters overcame these i n s t a b i l i t i e s . (iii) The c e l l , suspended by three s t e e l piano wires, was held secure by using the remnants of a poorly designed heat switch as a clamp. (iv) General purpose e l e c t r i c a l leads made of Advance a l l o y were brought down the vacuum pumping l i n e . Signal leads f o r the bolometer, generator, thermometer and l e v e l indicator were' brought down separate s t a i n l e s s s t e e l tubes. The bolometer leads consisted o f a twisted p a i r of #40 copper wire. Leads into the c e l l were brought through holes i n the bottom brass flange and sealed with epoxy. I I ) Resonator (i) Cavity A side view of the resonator i s shown i n Figure 2a. . Two fused quartz o p t i c a l f l a t s separated by a s t a i n l e s s s t e e l annulus (length 3.0 mm, radius 7.4 mm, cavity. and wall thickness 0.38 mm) Thin (^ 6 x 10 mm) inside formed the c y l i n d r i c a l resonant mylar gaskets glued to the annulus e l e c - t r i c a l l y isolated the bolometer and generator thin films on the f l a t s from the annulus. P o l i s h i n g the ends, with gaskets i n place, ensured that the f l a t s (better than one l i g h t wave f l a t ) were p a r a l l e l to within a few l i g h t waves. I t was hoped that t h i s alignment would r e s u l t i n p r e f e r e n t i a l ex- c i t a t i o n of only plane wave modes. also polished. The inside surface of the annulus was A f t e r assembly a small amount of glycerine was applied to 32 a) BOLOMETER RESONATOR STAINLESS ANNULUS I 1 1 1 1 b)GENERATOR STEEL MYLAR G A S K E T , GLYCERINE \ 1 FLAT ^ GENERATOR FLAT RESONANT CAVITY I JHIN FILM CHROMIUM RESISTIVE c)BOLOMETER ELECTRODE FILM THIN F I L M E L E C T R O D E (NOT USED) I Au-Pb BOLOMETER BIAS C U R R E N T THIN FILM Figure 2 The Resonator, Generator and Bolometer FILM PATH ELECTRODE 33 the corners formed by the annulus and f l a t s . I n t e r i o r was This ensured that the c a v i t y sealed o f f from the external helium i n the c e l l and prevented any possible coupling between the i n t e r n a l c a v i t y modes and those modes that existed outside. Such a coupling could r e s u l t i n an energy loss mech- anism that might be misinterpreted as i n t r i n s i c attenuation. was However, i t s t i l l p o s s i b l e f o r the superfluid to penetrate the gylcerine seal and f i l l the resonator i n approximately four hours. The o p t i c a l f l a t s , with 25 mm cially available. 3.2 mm diameter, were t y p i c a l of those commer- They were quite t h i c k , the f l a t with the generator while the one with the bolometer was 6.4 mm thick. being Both materials, fused quartz glass and s t a i n l e s s s t e e l , used i n the construction of the cavity have low thermal c o n d u c t i v i t i e s . This property resulted i n 6trong r e f l e c t i o n of the second sound at the boundaries and w i l l be discussed i n more d e t a i l i n following chapters. The heat generated i n the cavity es- caped mostly through the r e l a t i v e l y thin annulus walls and raised the temperature of the c a v i t y by about 20 KW *" above the ambient temperature of the c e l l . Considerable e f f o r t went into the construction of the thin r e s i s t i v e films which constitute the bolometer and generator. s p e c i f i c properties described below, i t was Besides possessing f e l t that they should be as close as possible to an i d e a l surface so that "perfect" r e f l e c t i o n occurred 28 at the end p l a t e s . Indeed, recent studies (Aquadag) to superleak comparing t h i n r e s i s t i v e f i l m s (nucleopore) transducers have indicated that the r e f l e c t i o n properties of the former are much simpler to Interpret. o ( i i ) Generator A top view of the generator deposition techniques i s shown i n Figure 2b. have been used i n construction. Conventional vapour The p a r a l l e l 6 t r i p 34 electrodes were deposited f i r s t . P a r t i c u l a r l y robust electrodes were made by depositing a thin layer of chromium with a f i l m of gold on top. Simpler, l e s s expensive but l e s s durable electrodes were also made using only aluminum. Electrodes were t y p i c a l l y 300 nm t h i c k with a r e s i s t a n c e l e s s than 1 U per square. Leads of #40 copper wire were usually attached by simply cold welding with a b i t of indium. The a c t i v e r e s i s t i v e element which gen- erated the second sound was a t h i n uniform f i l m of chromium overlapping the electrodes at the edges. square, independent The resistance of t h i s f i l m was about 43ftper of temperature from 300 K to 2 K. I t s thickness has not been determined accurately, but the resistance would indicate that i t was at least 3 nm, while mechanical measurements gave an upper l i m i t of about 100 nm. The f i l m was s u f f i c i e n t l y robust that i t suffered no damage on contact with the annulus. The p o s i t i o n and size of the annulus r e l a t i v e to the generator are indicated by the dashed c i r c l e i n Figure 2b. The reason f o r choosing the geometry i l l u s t r a t e d i n Figure 2b was to p r e f e r e n t i a l l y excite the plane wave modes of the cavity. The significance of the thickness, d, of the generator f i l m can be appreciated by comparing i t to the length 6 = (2 D/co) enuation of temperature which governs the phase and exponential a t t - o s c i l l a t i o n s at angular frequency co i n a material 29 of d i f f u s i v i t y D . The d i f f u s i v i t y ( D= ic/c where K i s the thermal con- d u c t i v i t y and c i s the s p e c i f i c heat per unit volume) i s d i f f i c u l t to e s t imate f o r what Is probably a p o l y c r y s t a l l i n e chromium f i l m ; however, even a cautious estimate indicates that d « experiment. 6 f o r the frequency range of t h i s This means the generator was t h i n i n a thermal 6ense. There- fore i t was capable of fast response, and the r e f l e c t i o n properties were determined by the glass substrate. 35 ( i i i ) Bolometer The temperature s e n s i t i v e mechanism of the bolometer was the super- 30 conducting t r a n s i t i o n of a gold and lead composite f i l m temperature of t r a n s i t i o n was The center adjusted, by means of a magnetic f i e l d , to the operating temperature of the resonator. The temperature associated with second sound resulted i n corresponding resistance of the f i l m . . excursions v a r i a t i o n s in the By b i a s i n g the f i l m with a constant current the resistance v a r i a t i o n s appeared as voltage changes which i n turn were r e covered by the e l e c t r o n i c s . The gold-lead f i l m s were constructed by depositing 8.0 nm of gold 31 followed by 14 nm of lead . The gold was evaporated from a tungsten ament, the lead from a boat or c r u c i b l e l i n e d with AljO^. were 10 * nm per second. fil- Deposition rates The f i l m s were extremely d e l i c a t e and s e n s i t i v e to chemical attack when l e f t exposed to the atmosphere f o r periods of about a day. The resistance of the f i l m s at room temperature per square. The superconducting t r a n s i t i o n was was about 25 Q -2 t y p i c a l l y 5 x 10 K above T^, and could be lowered to T^ by a f i e l d of about 100 gauss. The electrode and bolometer configuration are shown i n Figure 2c. Apart from the d i f f e r e n c e i n pattern, the electrodes are s i m i l a r to those used i n the generator. f i l m was The operating resistance of a square section of too low to provide an adequate signal l e v e l and impedance match to the e l e c t r o n i c s . To increased by a f a c t o r of remedy t h i s , the resistance of the bolometer was 20 by cutting i t with a s t e e l scribe into the pattern shown i n the f i g u r e . A t y p i c a l current path f o r the bias i s shown by the dotted l i n e . This p a r t i c u l a r pattern was constant chosen to maintain an active area as large as possible without allowing the e a s i l y damageable f i l m s to come into contact with the annulus. The sides of the 36 f i l m were also cut to prevent edge e f f e c t s from reducing the sharpness of the t r a n s i t i o n . A u s e f u l figure of merit f o r a bolometer s i t i v i t y defined by (1/R)(dR/dT). of resistance R i s the sen- For the bolometer used i n t h i s work the most rapid v a r i a t i o n of R with T occurred near the center of the transi t i o n where R had f a l l e n to one-half the high temperature value. There the resistance and s e n s i t i v i t y were 140 f2 and 40 K ^ r e s p e c t i v e l y . As a comparison, the s e n s i t i v i t y of conventional carbon f i l m bolometers i s more than a f a c t o r of ten smaller. The bolometer f i l m was a l s o thermally t h i n and capable of f a s t response. However, i t might be expected that the power d i s s i p a t i o n due to the bias current would have some e f f e c t on the resonance decay, and cons i d e r a t i o n was given to t h i s i n the c o l l e c t i o n of data. I l l ) Electronics A block diagram of the major electronic c i r c u i t r y i s shown i n Figure 3. ( i ) Signal E x c i t a t i o n and Recovery 32 The output, at f / 2 , of a frequency synthesizer was supplied to the generator which produced, by Joule heating, second sound at frequency f . 33 The same f / 2 output served as a reference f o r the l o c k - i n analyzer cavity response s i g n a l , as detected by the bolometer, was amplified . 34 The and fed to the l o c k - i n analyzer which responded to the second harmonic of the o r i g i n a l f / 2 reference. The outputs available from the analyzer were the components of the s i g n a l that were in-phase, I , and out-of-phase, Q, with 35 respect to the reference. These components were squared and summed to 2 produce a s i g n a l , I 2 + Q , which was proportional to the squared amplitude of the second sound i n the c a v i t y . In the measurement of decay times, once 37 TEMPERATURE REGULATION SIGNAL RECOVERY SIGNAL EXCITATION SIGNAL AVERAGER I O F F S E T AND FEEDBACK LOW PASS FILTER I DATA _STOR_AG_E I SWEEP CONTROL + Q 2 I 14 DC IFREQ. LOCK-IN ANALYZER SUPPLY jSYNTH, i TRIGGER H I DECADE TRANSFORMER BOLOMETER BIAS ATTENUATOR c i PRE-AMR i I^ 1 BOLOMETER 3 ) GENERATOR X THERMOMETER AND STANDARD AMR PRE-AMR i THRESHOLD OR DC AC LOCK-IN 2 > FEEDBACK HEATER Figure 3 Block Diagram of Main Electronics 38 the I2 + Q2 s i g n a l reached some preset threshold the AC excitation to the generator was interrupted and replaced with a DC drive which produced equivalent power input to the resonator. i l i t y i n the c a v i t y . This enhanced the temperature stab- Simultaneous with t h i s inerruption, a pulse was 36 sent to t r i g g e r the s i g n a l averager n a l was recorded. After 2 and the decay of the I 2 + Q sig- some predetermined time the AC excitation was again applied to the generator. I f the resonant frequency of the cavity had changed s l i g h t l y due to temperature d r i f t s then the frequency of the synthesizer was manually adjusted by some acceptable small amount to come back onto resonance. The process was repeated u n t i l , by averaging, an acceptable s i g n a l to noise r a t i o was achieved. As previously mentioned, i t was necessary that the bandwidth of the detection system be s u f f i c i e n t l y wide to accommodate, the frequency excursions of the signal during decay, as well as the frequency content of the decay. This requirement s t r i c t l y applied i n the i n i t i a l f i l t e r i n g stages where the narrow bandwidth appeared at the l o c k - i n analyzer. However, following the square and sum operation i t was useful to insert a low pass filter. Since at t h i s stage i t was only necessary to pass the decay s i g - n a l , the "bandwidth" of t h i s f i l t e r could sometimes be l e s s than that of the l o c k - i n analyzer. I n i t i a l l y i t was useful to i d e n t i f y the mode structure of the cavity by recording the response as a function of the drive frequency. threshold device and signal averager were not a c t i v e . A Then the microcomputer stepped the synthesizer and stored the cavity response at each frequency increment. computing The r e s u l t i n g data could be r e a d i l y plotted using the U.B.C. facilities. 39 ( i i ) Temperature Regulation The internal reference of a lock-in amplifier was used to excite a 37 bridge circuit consisting of a seven decade ratio transformer and two cryogenic arms containing a carbon resistance thermometer and a temperature 38 The amplified unbalanced signal from 39 the bridge was fed to the lock-in amplifier , the output of which was com40 insensitive reference resistor. bined with a DC offset and then applied perimental c e l l . to the heater wound on the ex- This negative feedback maintained a null signal and reg- ulated the temperature of the c e l l at a value corresponding to the bridge ratio. With this control scheme the balanced bridge ratio could be held fixed for several hours to within the low frequency (0.2 Hz) temperature noise of ± 2 x 10 ^ K. From the measured sensitivity of the system, d i f - ferent values of AT could be obtained by simply changing the bridge ratio to the appropriate value. ( i i i ) Level Detection This circuit i f not shown in Figure 3. Essentially i t was an o s c i l l 41 ator consisting of an LC circuit driven by a tunnel diode . The level sensing component was the capacitance formed by a tube and the inside of the c e l l (see Figure 1). The accuracy of this device, ± 10% of f u l l , was limited by the mechanical stability of the entire cryostat. Nevertheless, i t was found extremely useful during f i l l i n g , and permitted a daily check on the level in the c e l l . C. Procedure and Tests I n i t i a l studies were performed at large values of AT (AT > 2 x 10 K) to determine which modes of the cavity were excited and, in particular, to search for plane wave modes which were well separated from other Bessel modes. By sweeping the frequency through the plane wave modes i t was AO found that the f i r s t and t h i r d harmonics were "clean", while the other harmonics were accompanied by the nearby resonant structure of Bessel nodes. S i m i l a r l y , the time decay of the f i r s t and t h i r d harmonics was governed by a single time constant, while the other harmonics displayed a more complicated behaviour where beating with the nearby resonances was often evident. were obtained Consequently, the time decays of harmonics one and three f o r smaller values of AT. -5 range A x 10 The data covered the temperature -2 < AT < 5 x 10 K, over which the frequency of the fun- damental harmonic varied from 112 to 1,730 Hz. At each temperature several decay curves were recorded f o r d i f f e r e n t values of both the bolometer bias power and the amplitude of the second sound i n the c a v i t y . The e f f e c t s of bolometer power, which ranged from 3.6 x 10 ^ W to 2.5 x 10 ^ W, were usually weak. Amplitude e f f e c t s , how- ever, could be quite severe i n that there was a c r i t i c a l amplitude above which a resonance would decay pendence. very quickly with a strong amplitude de- Below t h i s c r i t i c a l value the decay rate was much slower, a l - though there s t i l l remained a weak amplitude dependence that became more s i g n i f i c a n t as T •*• T^. To stay below the c r i t i c a l amplitude, which be- came smaller f o r decreasing AT, i t was necessary to use second sound with i n i t i a l (i.e. at the beginning of a decay) temperature amplitudes as low —8 as 3 x 10 K rms. hundreds of decays. The recovery of these signals required averaging over The f i n a l r e s u l t s derived from the data f o r the f i r s t and t h i r d harmonics at f i f t e e n values of AT are determined from the ana l y s i s of 250 decay curves, each representing an average of between 16 and A50 decays. The minimum generator power density used to excite the -9 -2 -8 -2 cavity was as low as 3 x 10 W cm and 1 x 10 W cm f o r the f i r s t and t h i r d harmonics r e s p e c t i v e l y , while f o r large AT —6 —2 ^ 10~ W cm" were used to study amplitude e f f e c t s . l e v e l s as high as The treatment of the Al r e s i d u a l amplitude and power dependence i s discussed i n the analysis of results. The temperature 9 ion d i f f e r e n c e , AT, was determined using the express- A2 ' f o r the second sound v e l o c i t y u 2 •= A6.28(AT/T )* 387 X m s~\ (39) and the observed resonant frequency, f ^ , of the fundamental harmonic gives u 2 which by u 2 - 2af x where a i s the known length of the resonator. This method, as opposed to measurements with the thermometer, was used because i t gave a value of AT appropriate to the i n t e r i o r of the cavity which was at a temperature typ- i c a l l y 2 x 10 "* K greater than the surrounding bath which contained the thermometer. Also, i t eliminated the need f o r the tedious, periodic c a l A3 i b r a t i o n of the thermometer which i s known to d r i f t slowly with time. The v a l i d i t y of the procedure to derive AT from f ^ v i a u was checked once by c a l i b r a t i n g the thermometer at the lambda-point using the anomaly i n A3 -5 the warming curve . A value of AT ^ A x 10 K derived from t h i s c a l 2 AA i b r a t i o n point and the measured ( l/R(dR/dT) = 1.27 K of thermometer s e n s i t i v i t y ) was consistent with that derived on the basis _ 1 second sound v e l o c i t y and the estimate of the i n t e r n a l heating i n the cavity. For AT > 2 x 10 K the expression (39) begins to break down and a simple graphic i n t e r p o l a t i o n of the numerical data given by Ahlers was used to determine u (AT). 9 2 While c o l l e c t i n g data, the d r i f t i n g thermometer c a l i b r a t i o n resulted i n a corresponding change i n the value of AT and resonant frequency f o r a fixed value of the bridge r a t i o . This was compensated f o r by adjusting the bridge setting appropriately to maintain a fixed AT within suitable 42 limits, typically ± 2 x l C f K. 6 The uncertainty in AT i s the maximum of ± 3 x 10~ K or ± 0.5% of AT. 6 The major contributions to this error estimate are the uncertainty in the cavity length, a, which enters the above expression for u » and the stab2 i l i t y in AT during the collection of data. It w i l l be evident in the pre- sentation of results in Chapter 4 that this uncertainty in AT i s insignificant in comparison to the error estimates on the damping. During the collection of data i t was realized that a thermal emf -3 3 x 10 ator. -7 V) resulted in a dissipation of about 10 W in the gener- This power was eliminated by using a simple battery circuit to oppose the current driven by the emf. Studies indicated the the thermoelectric power had no effect on the decay curves for the f i r s t and third harmonics. However, when the thermal emf is added to the AC voltage excitation at frequency f, the resulting power spectrum has a contribution at f, as well as the desired 2f component. Thus, when exciting an even numbered harmonic at 2f there would also be present the harmonic at f . In the case of harmonics two and four, i t was found that the nearby resonant structure disappeared when the thermo-electric power, and consequently the coincidental excitation of harmonics one and two respectively, was eliminated. Therefore, with the thermo-electric power absent, some data was collected on the second and fourth harmonics. However, as w i l l be discussed in the analysis of results, for these resonances there s t i l l appears to be some additional loss that is probably related to the existence of the nearby modes that were evident when the two harmonincs, four and two, or two and one, were simultaneously excited. A3 CHAPTER 3 INSTRUMENTAL SOURCES OF ADDITIONAL ATTENUATION This chapter contains a discussion of sources of attenuation of second sound other than that a r i s i n g from the bulk helium. These additional sources of energy loss occur at the boundaries of the resonator and result from thermal conduction and the v i s c o s i t y , ri, of the normal f l u i d . The t o t a l attenuation, a, i s written as a = a„ + a 2 where T\ + a + a e s i s the bulk contribution given by equation (28) on page 15, i s the contribution from viscous drag at the side walls, a g and a g result from thermal conduction at the r e f l e c t i n g end plates and side walls r e s pectively. The development of the expression f o r in considerable d e t a i l by Heiserman and Rudnick duction losses have been treated by Khalatnikov expressions f o r a , a , a rr e s r A5 A6 has been presented , while the thermal con. The derivations of the are outlined i n sections A and B. of the a p p l i c a t i o n of these r e s u l t s to t h i s experiment Some aspects are discussed i n section C. A. Attenuation Due to Viscous Surface Loss, For a plane wave of second sound propagating In a tube, the normal f l u i d , which moves p a r a l l e l to the wall of the tube, i s entrained i n the vicinity of the wall due to viscous i n t e r a c t i o n . This e f f e c t penetrates \- into the f l u i d a c h a r a c t e r i s t i c distance X a v e l o c i t y dispersion and attenuation. (2n/p co) and r e s u l t s i n n This expression f o r X i s obtained from the related problem of an o s c i l l a t i n g plate i n contact with a viscous 18 fluid. In that case the solution i s a viscous d i f f u s i o n wave with X de- termining the normal f l u i d v e l o c i t y a distance x from the plate by 44 v -x/X = v e n no -i(cot - x/X) e To c a l c u l a t e the viscous surface losses f o r second sound propagating along a tube of c r o s s - s e c t i o n a l area A and perimeter B, the l i n e a r i z e d two f l u i d hydrodynamic equations of Chapter 1 are employed. Again, using 5 to denote small q u a n t i t i e s , the non-dissipative equations f o r mass, entropy, and superfluid a c c e l e r a t i o n are: 35p/3t + V«(p n 6v + p 6v ) = 0 n s s (40), p(3So/3t) + o ( 3 6 p / 3 t ) + pa(V«6v ) = 0 n 36v / 3 t = -Vu s (41), (42). The l i n e a r i z e d momentum equation, which i s 3(p 6v + p 6v ) / 3 t = -Vp i n ^ ' n n s s * the non-dissipative approximation, i s modified to include the e f f e c t s of viscous i n t e r a c t i o n with the walls. Choosing the z-axis as the propa- gation d i r e c t i o n and denoting by r the perpendicular coordinate which i s zero at the w a l l , the momentum conservation law including a viscous stress term i s 3(p 6v _ n nz + p 6v ) s sz _ 3v „ B n2 ^ - p - - r, ^ — J V r = Q (A3) . The assumption implied i n w r i t i n g t h i s equation i s that *"he viscous pene t r a t i o n length i s much smaller than the l a t e r a l dimension of the tube and, therefore, the wave fronts are e s s e n t i a l l y plane wave. The approximation -3 -4 i s v a l i d f o r t h i s experiment since X i s t y p i c a l l y 10 to 10 cm. 16 18 For a second sound wave the f l u i d momentum i s zero * , that i s , p v K n n + pv s s (44) Thus, the hydrodynamic equations become 36p/3t » 0 35o/3t « -o3v /3z nz ( 4 5 ). (*6), 45 85v /3t - +a(3T/3z) - l/p(3p/3z) (47), sz 3p/3z - - (B/A)n ( 3 6 v / 3 r ) \ nz (48). rmQ With equations (44, 46, 47, 48) and the approximation 3T/3z ^ (T/c)3a/3z which neglects (3T/3p) for second sound, i t i s easy o to derive the following wave equation for the entropy: 3 6a , s 2 3t 2 n P P 3 6o i B p 2 3 6o 2 2 r=0 3 t 9 r 2 3z 2 2, h where u„ = (p To /p c) i s the speed of second sound. By analogy to the 2 s n oscillating plate problem, a solution to (49) i s attempted in the form x /, 6o «= 6a (1 - e o t i r / X - r / X ' i ( k ' z - cot) ) e , (50) c n N with k ' = k + i a . Substituting (50) into (49) gives 2 , . s B n « i ( k ' z - cot) 2, ,2. .... - T - (1 - i ) w e = u - k ' 6a (51). A A o I P X co 6o + ico pp n Now, for X much smaller than the lateral dimensions of the tube, the approximation 6o = 6a e * ^ ~ o inary components resulting in i s made in (51) with the real and imag- z OJ 2 ) (B/A)(n/X) = n + co(p / p p s co(p /pP )(B/A)(ri/X) and g 2 u ( k «= 2u a^k 2 n - a), n 2 2 z . Solving these equations yields the dispersion co(k) and, for small dispersion whereto- U j k , the attenuation: 1 rB, s 2u, W p P TI 2 T 1 n p x * Using X = (2n/P oo) , then the viscous surface attenuation for a circular n cylinder of radius r, as in this experiment, i s : a T) - (1/ru.) 2 (ps /p) (na^p)* n 5 (52). 46 B. Attenuation Due to Heat Conducting Surface Losses, The temperature and a g excursions associated with second sound r e s u l t i n thermal conduction at the boundaries which diminishes the magnitude of the temperature excursions and, therefore, contributes a source of attenuation. Neglecting d i s s i p a t i o n i n the helium, consider 8 plane wave of second sound propagating i n the z - d i r e c t i o n and incident on a s o l i d body f i l l i n g the half space z > 0. In the second sound wave the energy flow, 3, i n the z d i r e c t i o n through unit area per unit time i s written as , , ikz * - i k z -icot J - (J^e - J e )e y .... (53) N 2 where 3^ and 3 spectively. a r e t n e 2 amplitudes i n the incident and r e f l e c t e d waves r e - The corresponding temperature o s c i l l a t i o n s , T, are given by I - (l/pcu )(3 e 2 + 3 e- *)e- i k 2 i k 1 (54) i a 5 t 2 with c being the s p e c i f i c heat of helium. The desired quantity to be calculated i s the r e f l e c t i o n c o e f f i c i e n t 3 / 3 ^ . At the boundary there 2 are two thermal impedances to be considered, one being the impedance of the 6 o l i d body, the other i s the Kapitza resistance of the surface i t s e l f . The p r o f i l e of the temperature excursions, T', i n the s o l i d body i s det- ermined by the heat equation c"(3T'/3t) - K(3 T73z ) 2 (55) 2 where ic i s the thermal conductivity, and c i s the heat capacity per unit volume of the s o l i d . T'(z-O) - T'e~ o iCt5t The solution to ( 5 5 ) with the boundary condition , i s the d i f f u s i o n wave * r ( 1 / / 2 - i/iu - re-'^r o The amplitude of the temperature not equal to the amplitude, the Kapitza r e s i s t a n c e , 1/G, r i«t excursions, T^, of the wall at z ( 5 6 ) > 1 0 is + J ) / p c u , i n the helium at z • 0 due to 2 2 of the surface. The requirement of contin- 47 u i t y of energy flow at the boundary provides two equations: 3 and 1 ~ 2 " 3 C 1 1 G ( ( 3 - J )e" + 3 2 ) / p C U 2 " o T - -KRe<3T73z) i ( 0 t 2 I m p l i c i t i n w r i t i n g (58) i s an approximation small contribution to 3 2 with respect to j ' e ~ . 2 i a 3 t } of magnitude - J ( 5 7 ) (58). z = 0 which neglects i n (53) a 2 and phase s h i f t e d by TT/2 Substituting (56) into (58), and eliminating T' from (57) and (58), gives o (G/pcu ) 2 2 r = 1 + G(2/cKco)^ (GTPTUT) 1 + - ( 5 9 ) h - r 1 + G(2/c<a>K Using the i n e q u a l i t y G « p c u , which holds f o r the temperature range of 2 t h i s experiment, the r e f l e c t i o n c o e f f i c i e n t becomes ^•=11 (60). p c u ( l / G + /2/cicw ) 2 Although there i s considerable v a r i a t i o n (an order of magnitude) i n the reported measurements of the Kapitza resistance near T^, f o r the largest frequencies i n t h i s experiment the s o l i d body resistance, i s greater than the Kapitza r e s i s t a n c e . The approximation /2/CKGJ , which neglects 1/G, to be discussed i n section C of t h i s chapter, y i e l d s f o r the r e f l e c t i o n coefficient = 1 - 6 = 1 - (2/pcu ) Vc<oV2 2 (61). This r e s u l t i s now used to c a l c u l a t e the attenuation, ot and ct , due to thermal conduction at the ends and 6ide walls of the resonant c a v i t y . Consider a plane wave propagating arated by a distance 'a'. between r e f l e c t i n g end plates sep- I t i s evident that the e f f e c t i v e attenuation, -a a a^, due to r e f l e c t i o n i s such that e « 1 - B. Substituting from (61), 48 and using $ « 1, gives — a - h J — (62). e apcu 2 ' To obtain the attenuation, a , due to thermal conduction at the side s v 2 walls, consider a c y l i n d r i c a l resonator of cross-sectional area A, perimeter B, and symmetry axis i n the z - d i r e c t i o n . The amplitude of the + 3 , temperature excursions, T" , i s proportional to 23^ r e f l e c t i o n , i s approximately which, for strong 2 The amplitude decrease, d f , Q due to thermal conduction through an area Bdz at the side wall i s proportional to ( 3 - 3 )(Bdz/A). 1 Thus, the e f f e c t i v e attenuation i s , dT 3. - 3 , _ 1 o 1 fB>| 2 a Using equation s : 2 217 y T ^ o (61) gives a s = • 1 (2/rpcu.) (CK03/2)* (63) 5 2 for a cylinder of radius r . C. Discussion of ot , a , a Ty e s The inverse of the decay time i s related to the t o t a l attenuation by 1/T = 2u a. 2 From equation (52) the contribution to 1 / T by viscous surface loss i s (1/T n ) = (2/r)(ps/p)(nw/2pn ) (64). 15 n This quantity has a strong dependence on AT, and becomes small f o r decreasing AT. In the a n a l y s i s of r e s u l t s , 1/T^ i s evaluated using the 42 following expressions (t = AT/T,): A (i) p /p « (ii) n/n (iii) p 2.534(t)* 6 7 4 - 1 - 5.19(t) x - p(l - p /p) ,85 ° with n x 10" poise _3 with p -0.146 g cm x - 2.47 5 49 The contribution to 1/x from thermal conduction at the resonator ends is 1/T - E (4/apc) ( C K O J / 2 ) (65) 15 where c and K are those quantities f o r fused quartz glass. At the side walls thermal conduction contributes 1/T «= (4/rpc) (cico)/2) s with c and K f o r s t a i n l e s s steel.. 1/T e and 1 / T (66) ls The major temperature comes from c, the s p e c i f i c heat of helium. s dependence In The temperature dependence through c and K i s weaker i n that they can be considered to 47 ' vary p r i m a r i l y with T, not AT. An estimate using representative values for c and ic indicates that 1 / T 6 i s a factor of f i v e greater than 1 / T . e As the a v a i l a b l e information on c, K does not warrant an accurate e v a l uation of 1 / T and 1 / T , and f o r reasons discussed i n Chapter 4, i n the e s f i n a l analysis of the data only the frequency dependence i s used. Thus, + 1/x the thermal conduction losses are treated c o l l e c t i v e l y as 1 / T - *= 1 / T K e s with 1/T- - gOOu* (67). IN The function g(T) denotes the temperature Recall that an assumption (2/CKGO) » 1/G. dependence through c, c, K . involved i n deriving (65) and (66) was The v a l i d i t y of t h i s approximation i s d i f f i c u l t to ass- ess due to the range of reported values f o r 1/G, reflecting i t s variab48 i l i t y with material and d e t a i l e d surface condition. Also measurements of 1/G using an AC method involving coupled second sound resonators suggest that the value of 1/G measured with DC flows. f o r AC heat flow i s much l e s s than that I f the AC data i s taken as being representative of the present s i t u a t i o n , then, at the highest frequency f o r the t h i r d harmonic of t h i s experiment, the above inequality i s s a t i s f i e d by about 50 a f a c t o r of 100 f o r glass and eight f o r s t a i n l e s s s t e e l . In view of the uncertainties involved, the v a l i d i t y of the approximation i s subject to experimental v e r i f i c a t i o n . The d e r i v a t i o n of (67) involved use of (56), which i s s t r i c t l y correct only f o r a r e f l e c t i n g body of i n f i n i t e extent. The approximation to a f i n i t e wall breaks down at s u f f i c i e n t l y low frequencies when the thermal d i f f u s i o n length, (2</cco) , becomes equal to the wall thickness. As i t i s estimated that t h i s occurs at 40 Hz, a factor of three less than the lowest frequency obtained i n t h i s experiment, to apply. equation (67) i s expected 51 CHAPTER A ANALYSIS OF DATA FOR THE DECAY RATE, 1 / T The method used to determine the decay rate, 1 / T , from the chart r e cordings of the resonance decay curves i s described i n section A. Section B contains a discussion of several aspects of the r e s u l t s f o r 1 / T , i n cluding the frequency and temperature dependences. are The r e s u l t s f o r presented i n section C. A. Obtaining 1 / T From Decay Curves Figure A i l l u s t r a t e s a decay curve representing the average of 150 i n d i v i d u a l decays of the t h i r d harmonic obtained at a temperature -A AT «= 1.70(± 0.03) x 10 K and frequency 588 Hz. In t h i s example the —8 —2 second sound, generated by an input power density of 2 x 10 W cm , had —8 an i n i t i a l amplitude of 5 x 10 1.2 x 10~^ W. K rms. Bias power i n the bolometer was The spike at the beginning of the trace r e s u l t s from the contribution of noise to the t r i g g e r i n g threshold at which the decays are initiated. As can be seen, t h i s noise remains coherent f o r a r e l a t i v e l y short time and i s ignored i n an extrapolation to time zero when drawing the smooth curve through the trace. The amplitude of t h i s smooth curve i s normalized to unity and i s used to determine the inverse decay time, 1/T, from the slope of a plot of the natural logarithm of signal amp- l i t u d e versus time. It was mentioned the This p l o t i s indicated i n the inset of Figure A . i n Chapter 2 that the decay rates are dependent on power input to the c a v i t y . Consequently, at each temperature several decay curves f o r the f i r s t and t h i r d harmonics were obtained f o r d i f f e r e n t values of the e x c i t a t i o n power and bolometer bias power. Generally, for high values of the e x c i t a t i o n power and bolometer power, the logarithmic In (SIGNAL) o r V •\ 1 SECOND | Figure A rjr-0.80 TIME A Decay of a Second Sound Resonance Y i e l d i n g a Value f o r the Decay Rate 1 / T 53 p l o t s displayed a curvature Indicating a larger value f o r the slope, 1 / T , at the beginning of the decay. At s u f f i c i e n t l y low powers the curvature was not noticeable over the useful amplitude range (about 85% of f u l l scale) of any p a r t i c u l a r decay. However, using even lower generator ex- c i t a t i o n powers indicated that there could s t i l l be a detectable amplitude dependence, a smaller value of 1 / T occurring f o r those decay curves with a smaller i n i t i a l second sound amplitude. AT = 1.70 x 10 A plot of the r e s u l t s for 1/T at K for the t h i r d harmonic i s given i n Figure 5 to i l l u s - t r a t e the nature of the extrapolations involved i n determining the zero amplitude l i m i t f o r 1 / T . At the highest bolometer and e x c i t a t i o n powers, the presence of some curvature i s the most s i g n i f i c a n t contribution to the error estimates. For a given bolometer b i a s and lower e x c i t a t i o n power the curvature diminishes or disappears and, as "goodness" of the f i t to a simple exponential decay improves, the error estimates decrease. The errors are then limited primarily by the uncertainty i n establishing the baseline of the decay curve i n the presence of noise racies i n the squaring c i r c u i t r y . and small inaccu- To obtain data at the lowest second sound amplitudes i t was necessary to use the higher bolometer bias v o l tages i n order to obtain *n adequate s i g n a l . An extrapolation to zero amplitude, that i s , zero e x c i t a t i o n power, indicated by the l i n e i n F i g ure 5, i s used to estimate a "best value" for 1 / T . A S the extrapolation i s subjective and without t h e o r e t i c a l guidance, the associated error estimates, indicated by the shaded region i n Figure 5, are treated generously according to the following c r i t e r i a . The upper l i m i t on the error estimate includes at l e a s t one value f o r 1 / T that has been a c t u a l l y measured. Thus, the upper l i m i t i s determined with confidence. l i m i t on the error estimate i s determined The lower i n a much more q u a l i t a t i v e fashion by simply choosing a more severe extrapolation that i s compatible 54 0.90 h 0.85 h i - 0.80 0.75 0.70 1 2 GENERATOR Figure 5 3 4 POWER ( I 0 " W ) 8 An extrapolation of values f o r 1 / T to zero generator power f o r d i f f e r e n t l e v e l s of bolometer power. The extrapolation to zero amplitude, that i s , zero e x c i t a t i o n power i n the generator, i s used to estimate a best value f o r 1/x. The hatched region indicates the error estimate. -4 data i s f o r the t h i r d harmonic at AT = 1.70 x 10 K. The 55 with the measured values f o r 1/x, or by symmetrically placing the lower l i m i t a distance below the best value which i s equal to the difference between the upper l i m i t and best value. In the sense that t h i s extra- polation procedure goes beyond the actual measurements of the experiment, the lower l i m i t s to the error estimate on 1/x are not determined with the confidence of the upper l i m i t s . Generally, the amplitude effects f o r the f i r s t harmonic were not as s i g n i f i c a n t as those f o r the t h i r d harmonic. Although at high amplitudes curvature similar to that for harmonic three was obtained, at lower ampl i t u d e s the extrapolated correction to zero amplitude was usually s l i g h t or i n s i g n i f i c a n t f o r harmonic one. This observation suggests that the amplitude e f f e c t s , at least at low amplitudes, r e s u l t from additional loss mechanisms that are proportional to the bulk helium l o s s rather than 2 surface losses. h Recall the frequency dependence, co versus co , of the sources of attenuation described i n Chapters 1 and 3. Then, comparing harmonic one to harmonic three, the contribution to 1/x from the bulk i s f r a c t i o n a l l y smaller r e l a t i v e to the surface losses by a factor 9//3~. Therefore, bulk related e f f e c t s should be less s i g n i f i c a n t at lower f r e quencies . The e f f e c t s of bolometer power were of two v a r i e t i e s . In one case the dependence of the decay rate on bolometer power was more s i g n i f i c a n t at larger generator powers, or equivalently, larger second sound amplitudes. At the large amplitudes where curvature was present i n the log- arithmic p l o t s of the decay, increasing the bolometer power resulted i n a more severe curvature. However, at lower amplitudes where the curvature was smaller or not noticeable, the e f f e c t s of changing the bolometer power were also smaller. At the smallest amplitudes a dependence on bolometer power was not resolvable within the accuracy of the measurements and ex- 56 trapolation to zero amplitude. usually not necessary. Thus, corrections for bolometer power were There was, however, a second type of dependence on bolometer power which was of an e n t i r e l y d i f f e r e n t character compared to that described above. For the range of bolometer powers used i n t h i s experiment, there was an anomalous bolometer power at which enhanced occurred. losses The value of the anomalous power decreased through the range of a v a i l a b l e power l e v e l s as AT decreased through the temperature i n t e r v a l -4 4 x 10 ent -3 <_AT <_1 x 10 K. The enhanced losses were not strongly depend- on second sound amplitude i n that, at low amplitudes, the decay curves were governed by single time constants with the logarithmic p l o t s showing no noticeable curvature. The magnitude of the enhanced losses at the anomalous power was about 50% of the zero power losses f o r harmonic one, and therefore the e f f e c t was quite dramatic. In addition, the value of the bolomet er power at the anomaly was the same f o r the f i r s t and t h i r d onics and i n t h i s sense was independent of frequency. harm- Also, the absolute magnitude of the enhanced losses appeared to be about the same f o r the f i r s t and t h i r d harmonics, although t h i s was d i f f i c u l t to determine with p r e c i s i o n since the normal amplitude e f f e c t s confounded the observations. Since the enhanced loss was a sharp function at the anomalous bolometer power, i t was possible, by operating either well above or below t h i s power, to obtain meaningful data. I t i s emphasized that the enhanced losses diminshed with increasing bolometer power above the anomalous value, and for powers s u f f i c i e n t l y removed from the anomaly the r e s u l t s became i n - dependent of bolometer power apart from the e f f e c t s of the f i r s t variety -4 described above. However, as stated above, i n the i n t e r v a l 4 x 10 < _3 AT < 1 x 10 K the anomalous power was within the range of available bolometer powers and systematic e f f e c t s , p a r t i c u l a r l y f o r harmonic one, were observable. When operating within t h i s i n t e r v a l at temperatures 57 AT = 1.02 x 10~ 3 K and AT = 5.94 x lO - 4 K with the bolometer power l e s s than the anomalous value, the best values for 1 / T are obtained by extra-4 polation to zero bolometer power. For AT <^4.23 x 10 K, the anomalous power i s near or below the lowest useable bolometer powers. Then, the best values f o r 1/T correspond to the "high power" data; that i s high power r e l a t i v e to the anomaly. Additional support for the general v a l i d -4 i t y of the "high power" r e s u l t s i s gained at AT = 3.05 temperature x 10 K. At t h i s the r e s u l t s of two a d d i t i o n a l decay curves f o r the second and fourth harmonics are consistent with the f i n a l r e s u l t s for D 2 based on the data f o r the f i r s t and t h i r d harmonics. The mechanisms by which the bolometer power contributes to additional attenuation are not c l e a r . The power dependent e f f e c t s of the f i r s t var- i e t y described above can be q u a l i t a t i v e l y explained i n r e l a t i o n to the second sound amplitude e f f e c t s . I f i t i s accepted that the amplitude e f f e c t s simply r e f l e c t the departures from the zero amplitude requirements of l i n earized hydrodynamics, then the superposition of a DC counterflow, produced by the bolometer power, with the AC counterflow i n the second sound would r e s u l t i n more severe v i o l a t i o n of the requirement of small flow v e l o c i t i e s . Of course, the s i t u a t i o n i n the cavity i s further complicated by the asymmetry, AC versus DC flow, as well as the d i r e c t i o n a l i t y of the flows, most of the DC heat leaving through the side walls while the AC flows are p r i - marily p a r a l l e l to the side w a l l s . I t i s pointed out, however, that the DC flow v e l o c i t i e s are at l e a s t a factor of two l e s s than the rms flow v e l o c i t i e s i n the smallest amplitude second sound wave at the smallest value of AT i n t h i s experiment. The mechanism responsible f o r the add- i t i o n a l losses of the second type described above i s a mystery. curious reader wishes to speculate, he may played by v o r t i c e s . I f the consider the possible r o l e 58 B. Discussion of the Data for 1 / T As discussed i n Chapters 1 and 3, the bulk helium damping, viscous surface l o s s e s , and thermally conducting surface losses contribute to 1 / T as 1/T = 1/T, 1/T + In terms of the damping c o e f f i c i e n t D^, 1/T 2 = 2u a 2 f o r harmonic "p" and resonant 1/T_ K + M 2 the expression f o r 1/ 2 T = (co /u )D - (p*/a) D 2 2 2 length "a". 2 The expressions f o r the surface losses due to v i s c o s i t y and thermal conduction, 1 / T and 1/T_ r e s p e c t i v e l y , T) are given i n equations i n obtaining 1 / T 2 (64, 65, 66, 67). s (68) 2 2 * K The essence of the method involved from the data for 1 / T i s to use the d i f f e r e n c e i n f r e - quency dependence, co face c o n t r i b u t i o n s . versus co , between the bulk contribution and the surTo best i l l u s t r a t e t h i s method and enhance the graph- i c a l presentation of the data, i t i s u s e f u l to compute 1/T according to the p r e s c r i p t i o n i n Chapter 3 and reduce 1 / T to 1/T - 1 / T ^ . r e c t i o n " i s s i g n i f i c a n t f o r large AT. This "cor- However, for AT < 1 x 10 K, 1 / T i s numerically small and comparable to or l e s s than the error estimates on 1 / T . In a d d i t i o n , i t i s u s e f u l to decompose the bulk contribution as follows: 1/T = 2 l / T 2 R + P Here, i/ 2jjp denotes that value of 1 / T -4 3.68 x 10 2 cm erature AT «= 3.2 x 10 K. 2 3.54 ) (69). -1 s , observed by Hanson and Pellam at a temp- Thus, l / T 2 H P -2 equation 2 which corresponds to the minimum T value of D^, A ( l / T (68), i s equal to 3.93 x 10 x 10 * s * f o r harmonic three. i s j u s t a constant which, using -1 s f o r harmonic one and The quantity A ( 1 / T ) represents changes 2 from the minimum value observed by H P . Then the data f o r 1 / T i s further reduced to 1 / T - 1/T^ - I/TJHP which i s j u s t 1/T_ + A(1/T ). 2 Although i t 59 44 i s necessary to correct the HP data to the "T^g" temperature scale , t h e i r .values can be considered trustworthy i n the sense that they have used a method of measurement that y i e l d s only 6mall changes i n D directly. As t h e i r r e s u l t s indicate -2 -2 over the temperature range 1 x 1 0 2 K, A ( 1 / T ) i s expected to be small over the i n t e r v a l . < AT < 5 x 10 Thus, t h e i r r e s u l t s 2 are used to " c a l i b r a t e " the present system i n the sense that they are used to check the v a l i d i t y of the predicted co dependence f o r the surface losses, and to check the mode purity of the resonances co on page 27) . p,m,n ( r e c a l l equation (38) for Before presenting the data f o r 1 / T - 1 / T ^ i t i s emphasized that the subtraction of 1/ ^ T convenience T * n u n i t s of s ~ \ 0.607 at AT «= 31.3 x 10" 0.031 H P + 1/T_ A(1/T ), 2 i n the graphical presentation to remove a large, strongly magnitude and temperature dependence of 1/ ^» 1 B 2 from 1 / T i s done only as a temperature dependent contribution at large AT. of l / ^ * l / T at AT «= 1.02 x 10 3 As an i n d i c a t i o n of the some representative values f o r the f i r s t and t h i r d harmonics are: 0.350, K; 0.133, 0.231 K. at AT - 10.3 x 10~ Also, the subtraction of l / T 2 H 3 P K; 0.018, simply r e - duces the data by the appropriate constant value and provides a convenient way of displaying more c l e a r l y any changes, obtained by HP. A(1/T ), 2 from the minimum value In the f i n a l a n a l y s i s , discussed i n section C, the ap- propriate values of 1 / T . _ and 1 / T 2 HP T) are added to 1/T_ + A ( 1 / T . ) to r e K 2 cover 1 / T . The values of 1 / T - 1 / T - 1 / T _ are shown i n Figure 6 for harmonics T"| Zrir -3 -2 one and three at temperatures such that 1.0 x 10 < AT < 4.6 x 10 K. 0 1 The s o l i d l i n e s are meant as v i s u a l a i d s . are discussed Several features of Figure 6 now. Consider f i r s t the data f o r p • 1, 3. The s o l i d smooth l i n e drawn 60 Figure 6 The reduced decay rate 1 / T - 1/ ^ ~ ^/ 2HP ^ T 3 as a function of AT f o r -2 3.93 x 10 -1 s T 1 x 10~ < AT < 5 x 10~ K. 3 2 o r ^ l / T a r m o n 2 H P ^ c s * a n ^ is -1 f o r harmonic 1, and 0.354 s cous surface l o s s , 1 / T , v a r i e s from 0.350 s f o r harmonic 3. - 1 The v i s - at AT - 31.3 mK to 0.018 n s~* at AT = 1.02 mK f o r harmonic 1, and i s just /T times greater f o r harmonic 3. The m u l t i p l i c a t i v e f a c t o r , /3~, indicated i n the figure Ill- ustrates the 0)^ frequency dependence of the surface losses, 1 / T , 1 / T _ . Also indicated i n the lower portion of the figure i s the temperature dependence, r e l a t i v e to 1 mK, of the surface loss 1 / T _ . The crosses K represent the "observed" temperature dependence and are derived from the data f o r the f i r s t harmonic by removing the frequency dependence with the factor [6J^(AT)/to^(l mK)] . The s o l i d c i r c l e s represent the predicted behaviour as determined by the temperature dependence of the s p e c i f i c heat of helium. ! 1 AT (mK) 1 • 1 " f 62 through the data points f o r p • 1 y i e l d s , a f t e r m u l t i p l i c a t i o n by /3~ as Indicated, the s o l i d l i n e through the data points f o r p * 3. Since the major bulk attenuation contribution has been removed using the H P values, i t i s expected that A ( 1 / T ) i s zero or very small i n the temperature 2 around AT ^ 3 x 10 K. region Thus, on the basis of the good agreement between the p • 3 data and that derived form the p = 1 data using the m u l t i p l i c a t i v e factor of /3~, i t i s concluded that f o r the f i r s t and t h i r d harmonics, the contributions to 1 / T from sources other than bulk damping are pro- h p o r t i o n a l to w . -3 The data f o r the t h i r d harmonic, at temperatures such that 1 x 10 < _2 AT < 1 x 10 harmonic. K, f a l l s below the l i n e derived from the data f o r the f i r s t Because the t h i r d harmonic i s nine times more s e n s i t i v e to changes i n D for 2 than the f i r s t , these deviations indicate a negative value A ( 1 / T ) , or equivalently, the bulk damping i s f a l l i n g below the minimum 2 value observed by HP. Of course, the changes i n D 2 are also present to a lesser extent i n the data f o r the f i r s t harmonic and, therefore, the value of l/^ 2 i s determined by the solution, given i n section C, to two sim- ultaneous equations. F i n a l l y , the temperature dependence of the surface losses i s i n d i c - ated by the crosses (x) i n the lower portion of Figure 6. l i n e through the crosses i s a v i s u a l a i d . follows. and frequency de- The l a t t e r i s removed by d i v i d i n g by the factor -3 {o) (AT)/6J (10 1 This information i s obtained as The data f o r p = 1 contains both a temperature pendence. The smooth 1 ^ K) ] where o^CAT) i s the frequency of the f i r s t harmonic at AT. In t h i s way the surface losses are normalized to the measured value at 1 x 10 K, and any v a r i a t i o n s would r e f l e c t the temperature deh pendence. Of course, t h i s requires that the frequency dependence, co , i s 63 correct and that any s i g n i f i c a n t corrections from A ( 1 / T ) i n the data for 2 p « 1 are accounted f o r . The expected temperature dependence, as pre- dicted by equations ( 6 5 ) and indicated by the c i r c l e s . ( 6 6 ) , due to the s p e c i f i c heat of helium i s Variations i n the properties of the r e - f l e c t i n g materials ( C , K ) have been neglected, and would only account for a small f r a c t i o n of the d i f f e r e n c e between the observed and predicted f r e quency dependence. I t may I t i s not clear what the source of the discrepancy i s . indicate that the theory f o r 1 / T _ i s incomplete and that some conic sideration should be given to the d e t a i l s of the interface between the lium and the s o l i d . For example, the presence of a viscous boundary layer might supress the losses due to thermal conduction. theory f o r 1 / T m a he- y lacking. D e A l t e r n a t i v e l y , the I f the "true" viscous contributions to 1 / T were 4 0 % to 5 0 % of the calculated values f o r 1 / T that have been used in obtaining the data i n Figure 6, then the observed differences would r e sult. In any case, by measuring 1 / T f o r both the f i r s t and t h i r d harmonics at each temperature, knowledge of the temperature dependence quired to obtain i s not r e - D« 2 In a fashion s i m i l a r to Figure 6, the r e s u l t s f o r 1 / T - 1 / 2 H P ~ ^ n, T for harmonics one, two and four are shown i n Figure 7. T The dashed l i n e s , derived from the f i r s t harmonic by m u l t i p l i c a t i o n by /2~ and 2, are ex• _2 pected to represent the contribution of 1/x to 1 / T . At AT ^ 3 x 1 0 K i t i s evident, p a r t i c u l a r l y f o r harmonic four, that there i s some addi t i o n a l loss mechanism present which i s not derivable from the f i r s t harmonic on the basis of an co p r o p o r t i o n a l i t y . I t i s noted, however, that the discrepancy of about 0.25 s -1 for harmonic four at AT ^ 3 x 1 0 -2 K is only about 1 0 % of the t o t a l value f o r 1 / T . I t i s apparent that the d i s crepancy diminishes as AT decreases and, tually insignificant. for AT < 3 x 1 0 K, i t is vir- The r e s u l t s from three a d d i t i o n a l decay curves ob- Figure 7 Similar to Figure 6, t h i s figure shows the reduced decay rate 1 / T - 1 / T T\ - 1 / T _ „ _ for harmonics 1 , 2 , A. The f i g u r e i l l u s t r a t e s 2. HP that there i s an additional loss mechanism present f o r harmonics 2 and A. The a d d i t i o n a l loss decreases with decreasing AT. 65 66 tained for p - 2 , A at AT « 3.05 x 1 0 " K and p - 2 at AT - 2 . 5 7 x 1 0 ~ 4 3 K are consistent with the ultimate r e s u l t s f o r J> derived from the f i r s t 2 and t h i r d harmonics. The source of the a d d i t i o n a l loss appearing i n the second and fourth harmonics at large AT i s probably related to the associated resonant struc- ture which appears when two harmonics, say two and four, are excited simultaneously as described i n Chapter 2 , section C. Mechanisms which could re- sult i n t h i s a d d i t i o n a l loss by e x c i t i n g Bessel modes are suggested general discussion of the f i n a l chapter. i n the As a consequence of t h i s discrep- ancy, only data f o r the f i r s t and t h i r d harmonics has been used i n the final a n a l y s i s f o r D^. The values of 1/T - 1 / T -1/T,_ - 1/T_ + A(1/T.) for the f i r s t and t h i r d harmonics over the e n t i r e temperature range covered i n t h i s iment are shown i n Figure 8. increase i n the value of AT < 1 0 The c r i t i c a l A(1/T ) 2 exper- damping i s evidenced by the f o r the t h i r d harmonic as AT 0. For K, the error estimates r e f l e c t the severity of the extapola- tions and the extent to which the data was collected at any p a r t i c u l a r temperature. For large AT the error estimates are due to the f r a c t i o n a l r e s o l u t i o n ( 1 or 2 % ) i n determining 1 / T with large surface contributions present. At the three smallest values of AT i t i s clear that the error estimates are increasing r a p i d l y . This i s a r e s u l t of the severe extra- polations r e s u l t i n g from the amplitude e f f e c t s described previously. As a matter of consistency with the data at larger values of AT, a best value and lower l i m i t have been estimated at these three smallest values of AT; however, i t i s f e l t that the most s i g n i f i c a n t information contained i n these data points i s the upper l i m i t that these place on the ultimate values f o r D^. 67 Figure 8 of AT. 1 / T - 1 / T ^ - l/ 2Hp f ° T harmonics 1 and 3 over the e n t i r e range Mote that the temperature axis i s logarithmic. damping f o r AT < 10 as AT r 0. The critical r e s u l t s i n the increasing separation of the data The inset i l l u s t r a t e s the frequency and temperature dependent contributions to the p • 1 data r e l a t i v e to 1 mK. The dashed l i n e i s derived by considering only the frequency dependence, while the s o l i d l i n e also includes the expected temperature dependence as determined by the s p e c i f i c heat, c, of helium. 68 69 The inset of Figure 8 i l l u s t r a t e s the predicted and observed behaviour of 1/x ic f o r harmonic one and AT < 1.0 x 10 _3 K. The s o l i d c i r c l e s are the values f o r 1/x - 1/x - l/x„„_ = l / x _ + A ( l / x ) . are derived from the s o l i d c i r c l e s by subtracting the contribution A ( l / x ) 0 The open c i r c l e s 2 using the data for the t h i r d harmonic and the expression given i n the following section. Thus, the open c i r c l e s represent l/x_. The s o l i d l i n e represents the predicted behaviour of 1/x , r e l a t i v e to the value at AT = 1.0 x 10 .3 * K, accounting f o r both the frequency dependence, that i s , di , and the temperature dependence as determined by the s p e c i f i c heat, c, of helium. dependence. The dashed l i n e r e s u l t s from considering only the frequency The temperature dependence of 1/x , as r e f l e c t e d by the deK parture from the dashed curve, i s i n q u a l i t a t i v e agreement with the predicted s o l i d l i n e . Although i t appears that there are systematic dep- artures from the expected behaviour, the accuracy of the measurements i s not s u f f i c i e n t to e s t a b l i s h the precise form of these. At AT = 4.0 x 10~ 5 K i t was d i f f i c u l t , because of noise, to obtain data for the f i r s t harmonic. Consequently, i n the analysis f o r D » a value for l / x _ appropriate to the f i r s t harmonic i s determined by a conic 2 tinuation of the open c i r c l e s (or s o l i d l i n e ) i n the inset of Figure 8. In view of the r e l a t i v e uncertainty i n 1/x f o r p = 3 at t h i s temperature, and the small magnitude of the surface l o s s , t h i s extrapolation does not introduce s i g n i f i c a n t error. C. Results for the Damping C o e f f i c i e n t , The inverse decay time f o r the f i r s t harmonic i s described by an expression of the form 1^ 2 2 k 70 S i m i l a r l y , the expression f o r the t h i r d harmonic i s 1 2 U 2 " (V»2>«3 T 6^3 + V i t h to^ • 3co^, these equations may be solved to obtain ,(3) . , .(3) .(1) i - (D /u )0) - ( i - /3"i ) [1 2 2 2 vhich i s the desired expression f o r 1 / T quantities 1 / T ^ and 1 / T ^ \ (3) 2 . (/3/9)] (70) -1 i° terms of the measured Using the dispersion r e l a t i o n u • u k 2 the resonance condition k a • pTT, the expression r e l a t i n g D- and 1 / T 2 P and (3) 0 2 is 2 D 2 1 ( 3 ) - (a /9TT ) i Z 9 (71). Z ^ 2 (To apply d i r e c t l y to the "reduced" data which has been presented i n F i g (3) ures 6 and 8, the appropriate expression f o r 1 / T .(3) I T .(3) . 1 (3) ,(3) « A i « { i .(3) (1) 4 - / 3 ( i + is 2 (1) A i + ) } [ ! - (/3/9)]- .) 1 2 The expressions 2HP 2 ? (71) 2 are used to £ calculate 2 (70), T T T ments of 1 / T ^ and 1 / T ^ \ 3 T T The numerical values for D temperature are found i n Table A i n Appendix A. from the measureas a function of 2 The r e s u l t s f o r iog^o 2 D are presented i n Figure 9, including the t h e o r e t i c a l predictions of HSH. The r e s u l t s are reproduced i n Figure 10 which also includes the experimental r e s u l t s of HP , 7 Tyson*^, Tanaka and Ikushima* , A h l e r s * , and the theor- e t i c a l predictions of HSH 2 14 and DF 15 . The r e s u l t s of t h i s work are i n good agreement with those of HP and Ahlers. evidently extends to 3 In the c r i t i c a l region, which X « t * (0.5 or 1) x 10 of the renormalization group treatment of HSH. _3 , there i s confirmation _3 However, f o r t < 10 , the L0G, (AT) -4 -2 ! 0 —r T • -3.0 (VI Q -3.2 - I — I O O T THIS WORK — HSH \ -3.4 -3.6 -2 -3 LOG, (t) 0 Figure 9 The R e s u l t s f o r the Second Sound Damping Coefficient 72 i cr> CM I 1 r Q fO fO I I ( a) ooi 2 0 , Figure 10 Summary of Results f o r the Second Sound Damping C o e f f i c i e n t 73 r e s u l t s are not i n agreement with the theory of DF, and Indicate values for which are les6 than t h e i r p r e d i c t i o n s . pendence of l>2* Considering the temperature de- the r e s u l t s are not s u f f i c i e n t l y accurate to resolve any p o s s i b l e deviations from a s i n g l e power law that might be interpreted i n ef f terms of a temperature dependent r a t i o , (t) • D^/2\i^K as predicted % by DF. A more quantitative comparison of t h i s experiment and theory i s achieved by describing the r e s u l t s i n terms of the function D„ • D t 2 oex with D Q e x and y ~ ex Y ' representing the experimental values f o r the amplitude and exponent f o r D^. Considering the evident coherence of the data i n Figure 9, i t i s tempting to use a l e a s t squares f i t to the above function. However, i t i s f e l t that such a treatment, p a r t i c u l a r l y with respect to the s t a t i s t i c a l estimate of a standard e r r o r , i s u n r e a l i s t i c , and possibly misleading, i n view of the p o s s i b l e (systematic) errors as represented by the error bars i n Figure 9. A r e a l i s t i c , although subjective, estimate for y i s ex Y ex « 0.31 ± 0.05 The subjective estimate of e r r o r , ± 0.05, i s determined by evaluating, i n Figure 9, the slopes of l i n e s that are h a l f way between the best f i t and the extreme l i m i t s compatible with the error bars. s u l t s of t h i s experiment Thus, while the r e - do not provide a severe test of a detailed pre- d i c t i o n for the c r i t i c a l temperature dependence, they do provide s i g n i f - cant support f o r the appropriate type of c r i t i c a l behaviour f o r D^ i n the -3 region t < 10 . Since the amplitude, D , i s s e n s i t i v e to the value of y , the OCX best independent estimate for D i 6 obtained by constraining y to the * oex t h e o r e t i c a l p r e d i c t i o n of y « 0.288. Then the value of I > o e x t -gain with 74 subjective consideration given to the possible systematic errors, i s D oex = (3.7 ± 0.4) x 10" 5 cm 2 s" 1 Of more general t h e o r e t i c a l s i g n i f i c a n c e i s the universal amplitude R 2 •? D / 2 u £ . 2 2 This may be evaluated using the expressions f o r £ and ratio, u 2 given at the end of Chapter 1 and the above value f o r D obtained for ° oex r y ex = 0.288, with the r e s u l t R„ -= 0.11 ± 2 ex 0.01 This experimental value l i e s between the t h e o r e t i c a l values of HSH, 0.09 and 0.15, which are expected to be accurate to within a factor of two. ef f The value for R 2 -4 at t = 10 as predicted by DF is-about 0.14. However, as they do not indicate the accuracy of t h e i r c a l c u l a t i o n , i t i s d i f f i c u l t to assess the significance of the difference between R x 2 e a n d R 2 ef f * 75 CHAPTER 5 CONCLUSIONS AND DISCUSSION Section A i s a statement of the major conclusions of t h i s work concerning the c r i t i c a l behaviour of the second sound damping c o e f f i c i e n t , D^. In section B there i s a discussion of several general observations related to t h i s experiment. A. Conclusions 4 The damping of second sound i n superfluid He has been measured over -5 the temperature interval 1 . 8 x 1 0 -2 < t < 2.1 x 10 . In the c r i t i c a l r e - gion the r e s u l t s , as i l l u s t r a t e d i n Figures 9 and 10, are i n good agree14 ment with the i n i t i a l renormalization group treatment of D ever, the r e s u l t s hy HSH 2 . How- do not support a recent renormalization group analysis by DF*'"', the observed values for D 2 being less than the predicted values. The experimental r e s u l t f o r the c r i t i c a l exponent, 0.31 ± 0.05, compares favourably with the predicted asymptotic temperature dependence, that i s , 2 ^ 2^' e*P 0.288. I f the r e s u l t s are constrained to obey exactly the t h e o r e t i c a l temperature dependence, then the experi0 288 mental value f o r the amplitude D defined by D_= D t is oex 2 oex D = (3.7 ± 0.4) x 1 0 cm s" . oex D U w i t n a n o n e n t o f r - 5 2 1 The corresponding value for the universal amplitude r a t i o defined by D /2u ? i s 2 2 R 2 ex 0 In the common i n t e r v a l 10~ A = 0.11 ± 0.01 < t < 5 x 10~\ the r e s u l t s of t h i s work 13 are i n good agreement with those of Ahlers that D 2 -3 . For t > 10 , i t i s observed departs from i t s c r i t i c a l behaviour and increases to obtain the 76 values measured by HP'. B. Discussion In this experiment the attenuation of second sound has been determined by measuring the decay time of plane wave modes in a resonant cavity. As opposed to measuring the resonant line-widths in a swept frequency method, the decay time technique possesses the significant advantage of being virtually immune to low frequency fluctuations in the ambient temperature. Consequently, this method may prove useful for future measurements of second sound damping at much smaller values of AT, as well as in studies of c r i t i c a l damping in other systems. Tyson*^ used electrically thin resistive films to generate and detect 13 second sound in his attenuation measurements. Ahlers porous superleak transducers. , however, used The discrepancy in their results suggested that there may have been some qualitative difference associated with the generation and detection devices. In view of the concurrence of the re- sults of this work, which uses resistive devices, with that of Ahlers, i t appears that there i s not some fundamental disparity between the methods used to transduce second sound. There are, however, differences between the experimental methods of Tyson and this work which may account for the difference in results. Three major differences are described here. One involves Tyson's treatment of the reflection coefficient at the end walls of the resonator. In that experiment the reflection coefficient i s assumed to be independent of frequency, and i t s contribution to the resonance widths i s determined from an extrapolation of the total widths to zero frequency. In contrast, on the basis of this work there i s theoretical and experimental evidence for a frequency dependent reflection coefficient, although at sufficiently high frequencies the presence of a Kapitza res- 77 istance could r e s u l t i n the r e f l e c t i o n c o e f f i c i e n t becoming frequency i n dependent. A second d i f f e r e n c e i s that the input power d e n s i t i e s used by Tyson are at l e a s t an order of magnitude greater than the power l e v e l s used i n t h i s experiment. Although Tyson extrapolates to xero power, this procedure might'introduce systematic errors that could account for the difference i n r e s u l t s . The t h i r d difference i s that i n Tyson's experiment the absence of side walls i n the resonator required a correction to the resonance widths involving d i f f r a c t i o n l o s s . In the resonator of t h i s ex- periment there are side walls present which eliminate d i f f r a c t i o n l o s s , but introduce viscous and thermal conduction l o s s e s . Although the calculated d i f f r a c t i o n l o s s i n Tyson's experiment i s small or i n s i g n i f i c a n t , early studies i n t h i s work on a c a v i t y without side walls gave r e s u l t s that were d i f f i c u l t to i n t e r p r e t on the basis of d i f f r a c t i o n from a plane-wave r e s onance. Indeed, the frequencies of the major resonances did not to a harmonic s e r i e s co correspond - _ f o r p = 0, 1, ... 5, but rather to a series p,0,0 with a Bessel mode character corresponding to co _ . or co . . p, U, 1 p,i,u ft At higher frequencies such that p > 10, the resonances did not display a single mode character, but contained Subsequent studies with c v a r i e t y of boundary conditions at the sides i n - dicated that the resonant 6ide boundary. Chapter 2 was several peaks r e s u l t i n g from overlapping modes. structure was s e n s i t i v e to the d e t a i l s of the As a r e s u l t , the simplest, " i d e a l " , side wall described i n f i n a l l y used. Although care was taken to prevent the e x c i t a t i o n of Bessel modes i n the c a v i t y , they were, nevertheless, excited. A general mechanism respon- s i b l e for t h e i r e x c i t a t i o n i s suggested on the basis of the following observation made i n t h i s experiment. I t was found that Bessel modes i n the v i c i n i t y of a plane wave harmonic were excited, while deep i n the region between the plane wave harmonincs there were no Bessel modes v i s i b l e . This 78 indicates that the e x c i t a t i o n proceeds by way of the plane wave resonance. Once the r e l a t i v e l y large energy excursions i n a resonance are established, a small perturbation at the walls of the cavity 16 capable o f d i r e c t i n g a s i g n i f i c a n t portion of the energy into the e x c i t a t i o n of another mode. As an example, the viscous l o s s occurring at the side walls of the c y l i n d r i c a l cavity could r e s u l t i n a temperature and v e l o c i t y p r o f i l e i n the r a d i a l d i r e c t i o n which i s not f l a t , but instead, curved at the edges near the wall. This p r o f i l e could then e s t a b l i s h an energy flow i n the r a d i a l d i r - ection and excite a Bessel mode. This type of mechanism could account f o r the excess l o s s i n harmonic four, where i t i s observed that the excess loss diminishes as the frequency and AT, and therefore the viscous l o s s , decrease. Other f a c t o r s that could excite Bessel modes by developing ang- u l a r and r a d i a l v a r i a t i o n s i n the cavity include thermal conduction losses through the side walls, and the p o s s i b i l i t y of a v a r i a b l e r e f l e c t i o n coe f f i c i e n t , due to power d i s s i p a t i o n , across the face of the bolometer. Progress has been made i n understanding at the walls o f a resonant the loss mechanisms occurring c a v i t y , although there i s some d i f f i c u l t y i n i n t e r p r e t i n g the observed temperature dependence. several resonant Systematic studies on c a v i t i e s of d i f f e r e n t dimensions and materials would l i k e l y solve t h i s problem and, i n addition, provide some information about the contribution of the Kapitza resistance to the r e f l e c t i o n of second sound. While i t i s doubtful that the knowledge gained from such studies Is i n i t s e l f worth the e f f o r t , the information would be u s e f u l i n optimi z i n g the c a v i t y geometry for further improvements i n the measurement of the c r i t i c a l damping of second sound. Thus, for example, by using a longer c a v i t y for a given radius, one should obtain more clean plane wave modes that are useable f o r attenuation studies. However, i f the o v e r a l l r e s o l u t i o n of the experiment i s to improve, the frequency of the highest useable mode must not decrease due to the interference of nearby Bessel modes. To ensure that the frequency range i s maintained i t might be pos- s i b l e to design the cavity to c r i t i c a l tolerances i n the radius-to-length r a t i o , r/a, i n order to avoid Bessel modes. While consideration must be given to the possible means of Improvement a v a i l a b l e through changes i n the resonator geometry and materials, i t i s also important to overcome the e f f e c t s on attenuation due to second sound amplitude and bolometer power. Although small amplitudes and low power have been used i n t h i s experiment, these, nevertheless, have limited the ultimate accuracy at the smaller values of A T . To recover lower level s i g n a l s , enhancement of the s i g n a l to noise r a t i o could be made with improvements to the bolometer. Experience with several bolometer films suggests that the s e n s i t i v i t y can be increased by at least a factor of two. Also, increasing the bolometer resistance with a more i n t r i c a t e pattern design would increase the s i g n a l l e v e l and provide a better impedance match to the noise f i g u r e of the e x i s t i n g e l e c t r o n i c s , although some care must be taken with t h i s procedure as i t ultimately reduces the a c t i v e area of the bolometer. Another Immediate improvement which, un- f o r t u n a t e l y , was not taken advantage o f , involves using d i f f e r e n t e l e c t r o n i c s i n order to obtain a lower noise f i g u r e at the i n i t i a l stages of amplification. For example, with the bolometer resistance and frequencies 49 of t h i s experiment, an appropriate input transformer and preamplifier would reduce the amplifier noise by about a f a c t o r o f two. Although noti c e a b l e , the extent to which t h i s factor would be r e a l i z e d i n the o v e r a l l s i g n a l to noise r a t i o depends on the strength of other noise sources such as pickup i n the leads and " i n t r i n s i c " bolometer noise. I n terms of future work, a more u s e f u l and f l e x i b l e low noise input that a l s o reduces 80 the e f f e c t of lead pickup would be a low temperature preamplifier. Then, achieving a f i n a l accuracy i n on the l e v e l of a few percent, i t would be possible to provide d e t a i l e d information of the temperature dependence of D^. I t would also be worthwhile to perform the measure- ments at elevated pressures as a test of u n i v e r s a l i t y . 81 REFERENCES 1. H.E. Stanley, Introduction to Phase Transitions and C r i t i c a l Phenomena (Oxford University Press, New York, 1971). 2. L.D. Landau and I.M. Khalatnikov, Dokl. Akad. Nauk SSSR 96, A69 (195A); reprinted i n Collected Papers of L.D. Landau, edited by D. t e r Raar (Gordon and Breach Science Publishers L t d . and Pergamon Press Ltd., New York, 1965). 3. L.P. Kadanoff, W. Gotze, D. Hamblen, R. Hecht, E.A.S. Lewis, V.V. Paliciauskas, M. Rayl, J . Swifts, D. Aspnes, and J . Kane, Rev. Mod. Phys. 39, 395 (1967). A. B.I. Halperin and P.C. Hohenberg, Phys. Rev. 177, 952 (1969); R.A. F e r r e l l , N. Menyhard, H. Schmidt, F. Schwabl, and P. Szepfalusy, Ann. Phys. 47, 565 (1968). 5. K.G. Wilson and J . Kogut, Phys. Rep. 12C, 76 (197A); K.G. Wilson, Rev. Mod. Phys. A7_, 773 (1975); M.E. Fisher, Rev. Mod. Phys. A6, 597 (197A). 6. P.C. Hohenberg and B.I. Halperin, Rev. Mod. Phys. A9_, A35 (1977). 7. W.B. 8. J.A. Tyson, Phys. Rev. 166, 166 (1968). 9. D.S. Greywall and G. Ahlers, Phys. Rev. A 7, 21A5 (1973). 10. J.A. Tyson, Phys. Rev. L e t t . 21_, 1235 (1968). 11. Light scattering measurements probe "microscopic" second sound. See reference 6 and references therein. 12. M. Tanaka and A. Ikushlma, J . Low Temp. Phys. 35, 9 (1979). 13. G. Ahlers, Phys. Rev. L e t t . A3_, 1A17 (1979). IA. P.C. Hohenberg, E.D. S i g g i a , B.I. Halperin, Phys. Rev. B IA, 2865 (1976); E.D. Siggia, Phys. Rev. B 13, 3218 (1976). 15. V. Dohm and R. Folk, Phys. Rev. L e t t . 46, 3A9 (1981). 16. I.M. Khalatnikov, An Introduction to the Theory of Superfluidity (W.A. Benjamin Inc., New York, 1965). The i n i t i a l paper by Landau on the q u a s i - p a r t i c l e theory i s reprinted at the back of t h i s book 17. P.C. Hohenberg and P.C. Martin, Ann. Phys. 3A_, 291 (1965). 18. S. Putterman, Superfluid Hydrodynamics (North Holland Publishing Company, Amsterdam, 197A). Hanson and J.R. Pellam, Phys. Rev. 95_, 321 (195A). 82 19. At AT - 2 x 10'* k, (c - c )/c » 3.6 x l ( f ; 20. a discussion of the s i g n i f i c a n c e of (c - c )/c . P v p L.P. Kadanoff and P.C. Martin, Ann. Phys. 24, 419 (1963). 21. P.W. Anderson, Rev. Mod. Phys. 38, 298 (1966). 22. B.I. Halperin, P.C. Hohenberg and E.D. Siggia, Phys. Rev. B 13, 1299 (1976). 23. T. Matsubara and H. Matsuda, Prog. Theor. Phys. ^6, 569 (1956). 24. B.I. Halperin and P.C. Hohenberg, Phys. Rev. 188, 898 (1969). 25. T. Worthington, J . Yan and J.U. Trefory, J . Low Temp. Phys. 24. 365 (1976). 26. P.M. Morse, Vibration and Sound (McGraw-Hill Book Company Inc., New York, 1948). Also, P.M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw H i l l Book Company Inc., New York, 1953). A large table for the a i s i n M. Abromonitz and I.A. mn Stegun (eds.), Handbook of Mathematical Functions, page 411, (Dover Publications Inc., New York, 1965). 27. E.J. Walker, Rev. S c i . Inst. 30, 834 (1959). 28. D. d'Humieres, A. Launay, and A. Libchaber, J . Low Temp. Phys. 38, 207 (1980). 29. H.S. Carlsaw and J.C. Jaeger, Conduction of Heat i n S o l i d s , (Oxford University Press, 1959). 30. H.L. Caswell, Phys. L e t t . 10, 44 (1964). 31. Measured with a D i g i t a l Thickness Monitor DTM - 200 manufactured by Sloan Technology Corporation, Santa Barbara, C a l i f o r n i a . 32. John Fluke Mfg. Co., Inc., Seattle, Washington; model 6010 A. 33. Princeton Applied Research Corporation (P.A.R.C.), Princeton, New Jersey; model 5204. 34. P.A.R.C. model 114 with 118 option. 35. Teledyne P h i l b r i c k m u l t i p l i e r / d i v i d e r model 4452. 36. Nicolet Instrument Corporation, Madison, Wisconsin; model 1170. 37. E l e c t r o S c i e n t i f i c Industries, Portland, Oregon; model DT 72 A. 38. P.A.R.C. model 114 with 185 option. 39. P.A.R.C. model 112. Z see ref erence 42 f o r 83 40. The feedback c i r c u i t r y was of personal design and made with standard s o l i d state operational a m p l i f i e r s . 41. General purpose tunnel diode 1N3714. A d e s c r i p t i o n of t h i s type of c i r c u i t can be found i n C. Boghosian, H. Meyer, and J . E . Rives, Phys. Rev. 146, 110 (1966). 42. G. Ahlers i n The Physics of Liquid and Solid Helium, edited by J.B. Ketterson and K.H. Benneman (John Wiley and Sons, New York, 1976), Vol. I . 43. B. Robinson, MSc 44. The thermometer was c a l i b r a t e d by vapour pressure thermometry against "1958 He* Scale of Temperatures", United States Department of Commerce, National Bureau of Standards, Monograph 10 (1960). 45. J . Heiserman and I. Rudnick, J . Low Temp. Phys. 22, 481 46. I.M. Khalatnikov, Usp. F i z . Nauk. £0, 69 (1956). (English transl a t i o n a v a i l a b l e i n Univ. of C a l i f o r n i a Radiation Lab. T r a n s l . 675; a l s o , Hydrodynamics of Helium II with U.B.C. l i b r a r y l i s t i n g QD 181 H4 K5 1956). 47. G.K. White, Experimental Techniques i n Low-Temperature Physics (Oxford U n i v e r s i t y Press, London, 1968); WADD Technical Report 60-56 Part II (1960), A Compendium of Properties of Materials at Temperatures (Phase I ) , by V.J. Johnson of the National Bureau of Standards, Cryogenic Engineering Laboratory. Thesis, U.B.C. (1976); see also reference 8. Osborne, P h i l . Mag. 3_, 1463 (1976). 48. N.J. Brow and D.V. (1958). 49. P.A.R.C. model AMI with, for example, P.A.R.C. model 114 with option. Low APPENDIX A TABLE A R e s u l t s f o r Ts^ AT(K) t = AT/T^ D 2 -4 2 -1 (10 cm s ) 4.62 X ID" 2 2.13 X i o " 2 3.86 + 0.7 4.12 X IQ" 2 1.90 X i o " 2 4.39 + 0.6 3.13 X ID" 2 1.44 X i o " 2 3.83 + 0.5 1.97 X IQ" 2 9.07 X i o " 3 3.35 + 0.5 4.74 X i o " 3 3.31 + 0.4 2.38 X i o " 3 3.09 + 0.6 1.18 X i o " 3 2.72 + 0.6 4.70 X i o " 4 3.26 + 0.6 1.03 X i c " 2 5.17 X IQ" 2.57 X i o - 3 3 1.02 X ID" 3 5.94 X i o " 4 2.74 X i o " 4 3.88 + 1.0 4.23 X i o " 4 1.95 X i o " 4 4.24 + 0.5 3.05 X i o " 4 1.40 X i o " 4 4.70 + 0.5 1.70 X i o " 4 7.83 X i o " 5 5.69 + 0.6 9.10 X i o " 5 5.95 X i o " 5 4.00 X i o " 5 4.19 X IO" 5 6.86 + 1.0 2.74 X IO" 5 7.92 + 1.2 5 9.14 + 1.8 1.84 X i o "
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The damping of second sound near the superfluid transition in ⁴He Robinson, Bradley J. 1981
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Title | The damping of second sound near the superfluid transition in ⁴He |
Creator |
Robinson, Bradley J. |
Date | 1981 |
Date Issued | 2010-03-29 |
Description | The attenuation of second sound near the superfluid transition in ⁴He has been determined by measuring the decay time for free oscillations of plane wave modes in a resonant cavity. The results for both the critical exponent and amplitude of the second sound damping coefficient are consistent with the early predictions of Hohenberg, Siggia and Halperin based on renormalization group theory. However, the damping observed in this work is less than the recent predictions of a non-linear renormalization group analysis by Dohm and Folk. The measurements cover the temperature interval 1.8 x 10⁻⁵ ≲ t ≲ 2.1 x 10⁻², where t = (T[sub λ] - T)/T[sub λ]. Fitting the results to a single power law for t < 10⁻³, the critical exponent governing the temperature dependence is found to be 0.31 ± 0.05. If the results are constrained to obey the theoretical asymptotic temperature dependence with an exponent of 0.288, then the amplitude obtained for the damping is 3.7 ± 0.4 cm² s⁻¹. This corresponds to a value for the universal amplitude ratio, R₂, of 0.11 ± 0.01. For t ≳ 10⁻³ the damping departs from the critical behaviour, and increases to obtain the values previously observed by Hanson and Pellam for t ≳ 10⁻². |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | Eng |
Collection |
Retrospective Theses and Dissertations, 1919-2007 |
Series | UBC Retrospective Theses Digitization Project |
Date Available | 2010-03-29 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0085592 |
Degree |
Doctor of Philosophy - PhD |
Program |
Physics |
Affiliation |
Science, Faculty of Physics and Astronomy, Department of |
Degree Grantor | University of British Columbia |
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UBCV |
Scholarly Level | Graduate |
URI | http://hdl.handle.net/2429/22963 |
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