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The damping of second sound near the superfluid transition in ⁴He 1981
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Title | The damping of second sound near the superfluid transition in ⁴He |
Creator |
Robinson, Bradley J. |
Date Created | 2010-03-29T20:52:09Z |
Date Issued | 2010-03-29T20:52:09Z |
Date | 1981 |
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. |
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Thesis/Dissertation |
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Text |
Language | Eng |
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Retrospective Theses and Dissertations, 1919-2007 |
Series | UBC Retrospective Theses Digitization Project [http://www.library.ubc.ca/archives/retro_theses/] |
Date Available | 2010-03-29T20:52:09Z |
DOI | 10.14288/1.0085592 |
Degree |
Doctor of Philosophy - PhD |
Program |
Physics |
Affiliation |
Science, Faculty of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
URI | http://hdl.handle.net/2429/22963 |
Aggregated Source Repository | DSpace |
Digital Resource Original Record | https://open.library.ubc.ca/collections/831/items/1.0085592/source |
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THE DAMPING OF SECOND SOUND NEAR THE SUPERFLUID TRANSITION IN 4HE by BRADLEY J. ROBINSON B.Sc, University of Toronto, 1972 B.Ed., University of Toronto, 1973 M.Sc, University 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 this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA May 1981 ^) Bradley J. Robinson In present ing th is thes is in p a r t i a l fu l f i lment of the requirements for an advanced degree at the Un ivers i ty of B r i t i s h Columbia, I agree that the L ibrary sha l l make it f ree ly ava i lab le for reference and study. I fur ther agree that permission for extensive copying of th is thesis for scho la r ly purposes may be granted by the Head of my Department or by h is representa t ives . It is understood that copying or pub l ica t ion of th is thes is for f inanc ia l gain sha l l not be allowed without my wri t ten permission. Depa rtment The Univers i ty of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date 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 critical exponent and amplitude of the second sound damping coefficient are con- sistent 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 ^ < t < 2.1 x 10 , where t c (T^ - T)/T^. Fitting the results to a single power law for t < 10 , the critical 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 is 3.7 ± 0.4 cm s . This corresponds to a value for the universal amplitude ratio, R̂ , of 0.11 ± 0.01. For t > 10 the damping departs from the critical behav- iour, and increases to obtain the values previously observed by Hanson -2 and Pellam for t > 10 . i i TABLE OF CONTENTS Page ABSTRACT i LIST OF FIGURES iv ACKNOWLEDGEMENTS v CHAPTER 1 INTRODUCTION 1 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 Coefficient, • & i i ) Scaling and D,, 15 i i i ) Renormalization Group Theory and D 2 18 CHAPTER 2 EXPERIMENT 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 Excitation and Recovery 36 ( i i ) Temperature Regulation . . . . 39 ( i i i ) Level Detection 39 C. Procedures and Tests 39 i i i Page CHAPTER 3 INSTRUMENTAL SOURCES OF ADDITIONAL ATTENUATION A3 A. Attenuation Due to Viscous Surface Loss, ot . . . . A3 B. Attenuation Due to Heat Conducting Surface Losses, a and a A6 ' e s C. Discussion of ot , ot , ot A8 Tr e* s CHAPTER A ANALYSIS OF DATA FOR THE DECAY RATE, 1/x 51 A. Obtaining 1 / T From Decay Curves 51 B. Discussion of the Data for 1 / T 58 C. Results for the Damping Coefficient, D^ 69 CHAPTER 5 CONCLUSIONS AND DISCUSSION 7 5 A. Conclusions 7 5 B. Discussion 7 6 iv LIST OF FIGURES Page Figure 1. Cryogenic Apparatus 30 Figure 2. The Resonator, Generator and Bolometer 32 Figure 3. Block Diagram of Main Electronics 37 Figure A. A Decay of a Second Sound Resonance Yielding a Value for the Decay Rate 1/T 52 Figure 5 . An Extrapolation to Zero Power 54 Figure 6. 1/T - 1/T^ - l / ^ ^ , f ° r Harmonics 1 and 3 60 Figure 7. 1/T - 1/T^ - l / T 2 R P for Harmonics 1, 2, A 65 Figure 8. 1/T - 1/T^ - 1/T2HP f o r H 3 ™ 0 1 1 * 0 5 1 a n d 3 Over the Entire Range of AT 67 Figure 9. The Results for the Second Sound Damping Coefficient . . 71 Figure 10. Summary of Results for the Second Sound Damping Coefficient 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 final 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 brief chronological review of the theor- e t i c a l advances in the f i e l d of c r i t i c a l phenomena and an indication of the motivation for this study. In section B the problem addressed i n this thesis i s described i n more detail through a review of other work and a discussion of the implications of recent developments. The theoretical ideas concerning the dynamics of superfluid helium i n relation to second sound attenuation are presented i n section C. A. Introduction to C r i t i c a l Phenomena A material with a temperature approaching the c r i t i c a l value of a phase transition displays singular behaviour in a variety of both static and dynamic properties. A few examples'of c r i t i c a l points are: the Curie point in ferromagnets, the superconducting transition occurring in many metals, the c r i t i c a l point of gas-liquid transitions, the lambda line or superfluid transition i n helium. The i n i t i a l efforts to describe,the form of c r i t i c a l point singu- l a r i t i e s are reffered to as classi c a l or mean f i e l d theories*. These were developed around 1900 and include Van der Waals equation for a f l u i d , the Weiss molecular f i e l d theory of ferromagnetism, and the Ornstein-Zernicke equations for correlation functions. These theories are quantitatively incorrect in the c r i t i c a l region, but achieve par t i a l success i n that they yield appropriate qualitative behaviour in the form of singularities which can be expressed i n terms of c r i t i c a l exponents. Thus, a singularity in the temperature (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 o|(T - Tc>/ T c J a where T £ is the c r i t - i c a l temperature and A q i s the amplitude of the singularity. In 1937 Landau proposed a general theory of the continuous or second order phase transition. His work results in the same exponents as the classical theories and consequently i s incorrect. However, he does advance the con- cept of an order parameter - a central element in modern theories. The generalization to include dynamic properties occurred in 1954 with the 2 introduction of an equation of motion for the order parameter . Equations of this type are employed in current time-dependent Ginzburg-Landau models. In the mid-sixties there evolved a phenomenological treatment of 3 phase transitions known as scaling . This theory predicts relationships or scaling laws which exist among various c r i t i c a l exponents and therefore escapes the limits of mean f i e l d theory. By the end of the sixties the extension of scaling to include dynamic properties had been accomplished . Along with scaling there developed a related concept known as universality. I n i t i a l l y formulated as the law of corresponding states, the hypothesis of universality means that relatively few fundamentally different types or classes of c r i t i c a l behaviour are sufficient to accommodate a complete categorization of phase transtions. In particular, the entire lambda line in liquid helium i s in one universality class and therefore the effects of elevated pressure should be mild in the sense that, for example, exponents remain unchanged. During the last decade the renormalization group methods of quantum f i e l d theory have been applied, with great success, to the problem of both static^ 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 calculating c r i t i c a l exponents and the values of certain 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 is the subject of much current research. The i n i t i a l motivation for this work was based on one of the pre- dictions of dynamic scaling related to the damping of second sound near the lambda transition in liquid helium. Second sound is a propagating mode of thermal transport which appears as a temperature-entropy wave in 4 the low temperature superfluid phase of He. It was felt that precise measurements of the critical 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 is possible to extract the, now more pertinent, damping co- efficient D£ by means of the hydrodynamic expression a2(03,T) - Js(w2/uJ) D2(T) (1) where w is the angular frequency of the second sound with velocity \x^. This expression is derived in section C and, as indicated by the notation, in the regime of hydrodynamics D2 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, their measure- ments are Important in the interpretation of the results of this work and w i l l be discussed in later chapters. As previously mentioned, theoretical work in the late sixties resulted in the dynamic scaling prediction of the c r i t i c a l temperature dependence of D2: D = u2K (2) where £ is the correlation length for fluctuations of the order parameter. The arguments used to arrive at this result are outlined in section C. In view of the proportionality, the experimental verification of relation (2) w i l l involve only the exponents which govern the behaviour of each quantity as T * T^. From the two-fluid hydrodynamic expression for u 2 (see E q . (26) in section C), the terms with a significant dependence on AT indicate • u„ a (p /c ) where p is the density of the superfluid component and 2 s p s c i s the specific heat at constant pressure. The definition of a cor- P relation length for a power law decay of the correlation function at large distances, as i s the case for helium with T < T^, and the formulation of hydrodynamics in terms of correlation functions yields £ o c P g ^ « There- -h fore, the scaling prediction (2) becomes D 0 a (c p ) . The exponent for p a (AT)^ has been found to be C = 0.666 ± 0.006 from an s Andronikashvili-type experiment . Alternatively, p may be derived from measurements of u 2 (plus other empirical information and the hydrodynamic expression for u2> with the result Z, c 0.674 ± 0.001 at saturated vapour 9 pressure . The specific heat, c^, increases slowly with a near logari- thmic dependence on AT as, AT decreases. Thus measurements of D 2 would verify dynamic scaling to the extent that they confirm D, « (c ) ~ h ( A T ) ~ C / 2 ^ ( A T ) _ 1 / 3 (3). 2 P 5 Prior to the beginning of this study an experiment by Tyson*"0 pro- vided the only test of relation (3) for the case of macroscopic second sound*"*". His results give 0.34 ± 0.06 for the exponent of D 2 and con- sequently agree with the value 1/3. However, the results do not confirm the predicted contribution from c^ in (3). This might be expected to lower, to perhaps 0.28, the exponent which would be calculated from a power law f i t to the data for D2(AT) over the temperature range of his experiment. I n i t i a l l y one of the objectives of this work was to obtain more precise values of D 2 in order to establish or deny the presence of 0^ and so provide a stringent test of the details of dynamic scaling. The second objective was to test the universality of the scaling relation (2) by performing the measurements of c r i t i c a l damping under pressure. There was, and s t i l l i s , no such information on D 2 available. The renormalization group treatment of a dynamic model^ of helium confirms the scaling relation (2), and provides a means of calculating the universal constant defined by the ratio R 2 = D 2/2u 2£. The theoretically estimated value for R2, given in the next section, has been found to be a factor of about five smaller than that indicated by Tyson's data. This discrepancy had serious implications in the i n i t i a l stages of this work. If the smaller value of R 2 as predicted by theory was in fact correct, then a considerable improvement in the experimental error, compared to that obtained by Tyson, would be required in order to resolve the predicted temperature dependence of D 2« o 12 During the course of this work, Tanaka and Ikushima have interpreted 3 4 their studies on thermal transport in He - He mixtures as evidence in 13 support of the value of R 2 obtained from Tyson's data. Recently Ablers has reported results on second sound damping which are in 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 transition by Dohm and Folk (DF)^ has resulted in predictions concerning second sound damping which are also in agreement with the work of Ahlers; however, their theory ef f predicts a temperature dependence for the effective ratio R 2 (AT/T^) ef f which enters the equation, D 2 • R 2 2u 2£, governing the behaviour of D 2 < Also, their predictions for D 2 cover a larger temperature interval ex- tending beyond the c r i t i c a l region. In view of the theoretical and experimental status of D 2 outlined above, the results of this work are significant in that they confirm the results of Ahlers. In addition, since the measurements of this work are of greater accuracy and cover a larger temperature interval, they provide a more severe test of the theoretical predictions of HSH and DF, and i n - dicate the behaviour of D 2 in the temperature interval between the pre- viously existing experimental data in the c r i t i c a l and non-critical regions. C. The Dynamics of Superfluid Helium i) Hydrodynamics and the Damping Coefficient D^ The hydrodynamics of the two f l u i d model of superfluid helium is re- viewed here. This theory Is relevant to an understanding of second sound and the mechanisms responsible for i t s damping, as well as providing the foundation and interpretation of the dynamic models employed in scaling and 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 in the quasi-particle (elementary excitation) theory of Landau^, and more re- cently in the theory of Hohenberg and Martin*^ based on the assumption of a Bose condensate. This model describes He II in terms of interpenetrating normal and superfluid components with densities and p g respectively. The velocity f i e l d s associated with these components are V r and vg., subject to the irrotational 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. In the approximation in n s 16 18 which dissipative effects are neglected, a typical derivation * of the hydrodynamic equations begins with the conservation laws and an equation of motion for v_ satisfying (4). Thus s |£ + div J «= 0 (5-A) expresses mass conservation in terms of liquid density p and mass current ~i = ( j j ) which i s the momentum per unit volume; 3 j i 8 1 1 ik i r — + -s = 0 (summation convention) (5-B) ot ox, k is the statement of momentum conservation where H., is the momentum flux ik density tensor; the absence of dissipation i s written as conservation of entropy f f i S - + div F = 0 (5-C) where a is the entropy per unit mass and F is the entropy flux; the i r - rotational character of the superfluid velocity f i e l d means that the time development of v satisfies an equation of the form - ~ + grad (Ssvj + h) = 0 (5-D) ot S where h i s a scalar function. These, (5-A,B,C,D), are eight equations for the eight basic variables p, O, v , in terms of the yet to be determined s quantities II ik, F, h. Conservation of energy, |£ + div Q - 0 (6) where U i s the energy per unit volume and Q i s the energy flux density, i s a ninth and hence redundant equation which must be automatically sat- is f i e d by (5-A,B,C,D). This constraint, the application of Galilean re- l a t i v i t y and thermodynamic arguments are sufficient to determine F, h under the assumption that they do not contain dissipative contributions in the form of spatial gradients of thermodynamic variables. In the determination of F, h the existence of two independent velocity f i e l d s i s significant to the thermodynamics since i t i s not gen- erally possible to transform to a frame in which the fl u i d i s at rest. Thus, there appears an additional conjugate pair of thermodynamic variables arising from the relative internal velocity. Consider a Galilean trans- formation relating two frames denoted by subscripts 1 and 2 and with r e l - ative velocity v^. The relations for velocity, momentum and energy den- sity are V 2 = v i + v r » J 2 = ^1 + P V r ' U 2 = U l + V r * ̂ 1 + ^ P V r * A n e n e r S y density which satisfies this transformation i s U = U + v » ( j - p v ) + hpv2 (7) O S s s where U is a Galilean invariant and represents the energy density in a o frame in which the superfluid is at rest. As U q is an invariant i t s de- pendence on the basic variables p, C, j , v i s dU_ «= ydp + Td(pa) + w »d(j - pv R) (8) since j - pv is invariant. In (8) w is the conjugate to j - pv (as y s s is to p and T is to pa) and serves to define via w " v n ~ V E ' Then, the last term in (8) states that the derivative of energy with respect to momentum is velocity. With expressions (7) and (8) for the energy density, the constraint imposed by energy conservation can be used to determine n"ik' F, h. By differentiating U with respect to time and then replacing a l l time derivatives by spatial derivatives through the use of equations (5-A,B,C,D), i t i s possible to identify the energy flux density Q as well as F, h. The algebra can be found in considerable detail in ref- erence 18. The results become intuitively appealing when the basic var- -»• -*• -V- . •+ iable set i s taken to be p, a, v , v where v i s related to 1 by ' ' n* s n j - P nv + P ev with p + p = p (9). n n s s n s Then the expression for i s n i k * P 6 i k + P s V s i v s k + p n v n i v n k ( 1 0 ) where p i s the pressure. Thus, appears as a natural generalization of the momentum flux density » P^ik + s * n 8 l e f l u i d hydrodynamics. The result for the entropy flux vector i s F - pov n (11) which means that a l l entropy i s carried by the normal f l u i d . The scalar function h •= V - (12) is the chemical potential. The quantities y, p, p ^ which now appear are -»• -*• 2 functions of p , O as well as (v - v ) . Thus, the hydrodynamic equations, neglecting dissipation 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 dissipation in anticipation of obtaining a hydrodynamic expression for second sound attenuation. Dissipation i s a consequence of the irreversible processes associated with thermal conduction and the viscosity or internal f r i c t i o n which re- sults from internal motion. These irreversible processes occur when there are departures from equilibrium and cause the system to move towards an equilibrium 6tate characterized by a maximum in the entropy. Thus, the approach to equilibrium involves entropy production, Z/T, and the gener- alization to equation (5-C) i s 10 div (pov n + |) •= | (13-C). The dissipative contribution to entropy flux, q/T, i s , of course, identified in lowest order with thermal conduction. Admitting a dissipative contrib- ution, h', to the superfluid flow but s t i l l requiring curl v g «= 0, the superfluid acceleration equation (5-D) becomes + grad (y + %v + h') «= 0 (13-D). d t S There s t i l l remain the conservation laws for mass, momentum and energy. The equation for mass conservation or continuity i s unchanged: | | + div j - 0 (13-A). Vith a viscous stress tensor the momentum equation i s 3j 3(11 + T ) l + I K l k = 0 (13-B). d t dX, k Energy conservation is now | | + div (Q + Q») - 0 (14) where Q' i s the additional dissipative energy flux density. The form of Z , q, h', must now be determined. As in the non- dissipative case the energy equation (14) must be satisfied automatically. By differentiating with respect to time the expression for the internal energy (7, 8) and replacing time derivatives by spatial ones through the -*• -+• use of (13-A,B,C,D), a pure divergence term may be identified with Q + Q , while the remainder must vanish. This yields for the entropy production *- - ̂ - * u - »'K*s - V (»)• k The entropy production must be positive definite and vanish in equilibrium. This requirement, and those based on Galilean covariance, are sufficient 11 to determine expressions for the fluxes q, h' which, to f i r s t order in the deviations (spatial derivatives) from equilibrium, involve thirteen -+• independent kinetic coefficients. At this level v - v i s not considered r n s •+ •+ small since in the equilibrium state of solid body rotation v - v can be n s large due to the presence of superfluid vortices. However, in the limit of small deviations from a non-rotating equilibrium state, as applies to this work, v - v is also small. Then there appear kinetic coefficients n s r r in the fluxes as follows: q = -KV" T (16) where K is the coefficient of thermal conduction, \i - -^tef+ £f - KjH) • s«(hW\-\> + ^H) (17> and h' = ,v".p (v" - v ) - C,v"4 (18). o s s n 4 n In the viscous stress tensor, T.., there are the usual coefficients of f i r s t and second viscosity, ri and Z,^, which appear in the "normal" f l u i d hydrodynamics. Due to the additional degree of freedom allowed by v , s there appears in another second viscosity, which determines the dissipation generated by relative motion, v n ~ v s * The dissipative cor- rection, h', to the chemical potential contains two more coefficients of second viscosity £ 3 , £45 however, by the Onsager reciprocity theorem, £^ = g so that there are five independent kinetic coefficients. Also, 2 Z > 0 requires that K, n, &2, &3 be positive and ^ < X,2^>y The characteristics of sound propagation can now be analyzed on the basis of the hydrodynamic equations, (13-A,B,C,D), with the substitutions (15, 16, 17, 18). The equations for p, O, v , v are written in linear- ized form by means of 12 p - pn + 6p o - O Q + 60 p - p + 6 p T = T + 6 T •+ V • OV V • ov n n s s 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 in which v - v «* 0. The * J n s linearized, that i s , to order 6, equations including dissipation are then 35v . 36v , . r - ^ + P - 5 — — = 0 (19-A), 3t no 3x A Kso dx̂ ^ 3 _ , 3Pso<6vsk - *W + ^nk j ( w . B ) i + 3 X j l C l 3x v 2̂ 3x k > 33(pa) 9 S v n k < 3 36T (19-C), 3t Po°o 3x, T 3x. 3x, k i i 36v ^ r - 3p (6v , - &V , ) 35v . , s i , 3<5u 3 r Kso sk nk , nk > ... i i k k In this approximation 6u i s just the usual 6y = (l/p)Sp - 0 6 T . Wave solutions are attempted in the form . i(ut - kx) ~ -iw(t - x/u) 6p = pe « pe , . and similarly for 6o, 6p, 6T, 6v « 5v x, 6v • 6v x. The linearized J * v* * n n ' s s equations now become, dropping the subscript 'o', -iwp + ikp v + ikp v =0 (20-A), K n n s s -iwp v - iwp v + ikp - -k 2( 4l - P ?i + S,)^ - k2c; p v (20-B), n n s s 3. s i 2 n l s s -iw(po + pa) + ikpav «= -k 2 £ T (20-C), n 1 13 -low e + ik( £ - O-T) - k 2(C A - P s ? 3 ) v n - k 2 £ 3 P s v 6 (20-D). Thus, there are now four equations for four unknowns p, 5, v , v . The small variations p, T depend in this approximation on p, O as „ 8T<\ ~ ST-* ~ A dispersion relation, co(k), can be obtained by eliminating v , v in favour of p, o and retaining only f i r s t order terms in the kinetic co- eff i c i e n t s . Equation (20-B) through the use of (20-A) and (20-C) becomes A second equation is obtained from (20-D) by using (20-B) as an expression for p and subsequently eliminating v^ and v g by means of (20-A) and (20-C). The result is t 3T u)2 pn >~ r 1 , _ . ?2 . 4 n . P n K 3T.~ K. S S s - o | £ p (22). 3p Consider for the moment the approximation in which dissipation i s ne- glected by ignoring the kinetic coefficients in equations (21) and (22). Then they read <Sl - & - 0 a & ) p 5 - o ( a , k u^ r k u 2 14 with u 2 - Op/3p) 0 and u 2 - (P s/P n)o 2(3T/9a) p - (ps/pn)a2T/cv. The con- dition for the compatibility of (23) and (24) is that the determinant of the coefficients vanish. Thus Hk-u c-̂ r - o - C|g>a tfDp tfgla ĉ )p k 2 u 2 k^u 2 = (c - c )/ c P v p ~ = u „ » f_J> 13.]* c = c - c (26) *-p c J p v CO k " u2 At the saturated vapour pressure and for the values of AT in this exper- 19 iment (c - c )/ c << 1, so the approximation that this term is zero P v p results in decoupled modes • f - ' l " ^ <25> n known as f i r s t and second sound. The f i r s t i s the usual adiabatic density- pressure sound wave, while second sound is an entropy-temperature wave at constant density. Now consider the dissipative equations (21) and (22) in the decoupled approximation (c - c )/c - 0. To f i r s t order in the kinetic coefficients * P v p the requirement that the determinant of the coefficients of p' and o' van- ish yields dispersion relations as follows: 2 co 2 4 W f 4 ^ , r " ) co2 2 ico p s t 4 ^ . 2 n f r . r . , _ . P n ic and — = u 2 - _ _ ( y n + p S 3 - p ^ + + C 2 + — - ( 2 7 ) k n s Considering the second sound solution, the attenuation can be deter- mined by writing k - k Q + i a 2 and expanding to f i r s t order. This gives PU2 n 8 15 The damping constant D 2 i s defined by a2(a),T) - hCw 2/ u^) D 2 (28) Therefore in linearized hydrodynamics D 2 is independent of frequency and M T ) - ? r ( f i i + P\ - P t t j . + C A ) + C 2 + ^ f ) ( 2 9 ) . ' 2 - PP n Thus, on the basis of hydrodynamics alone, measurements of second 6ound damping may only be interpreted in terms of a rather lengthy com- bination of thermodynamic properties. However, some information concerning D 2 can be gained from independent measurement and theoretical calculation of the individual quantities contributing to D2. * n particular, the be- haviour of D 2 as T •+ T^ i s expected to be approximately proportional to - 1/3 (AT) . The contributions to this divergence are as follows. It i s 2/3 known from experiment that p a (AT) . The viscosity, T), i s measured s to be f i n i t e at T^ so i t s effect on Dj, i - P sn, vanishes. The second v i s - cosities z;̂ , C 2 > £3 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. If i t s behaviour below T^ (which cannot be measured due to - 1/3 superfluidity) i s the same as above, then i t Is about (AT) . If, as 3 A experiments on He - He mixtures suggest, tc i s f i n i t e at T^, then the growth of the specific heat, c, as T •* T^ would cause tc/pc to diminish. The scaling treatment of c r i t i c a l dynamics as i t relates 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 fluctuations near the transition and consequently Is most responsible for the c r i t i c a l behaviour. This variable, the order par- 16 amter tj;, has a range, £, of correlations in the fluctuations which is div- ergent as £ • C 0 ( A T / T c ) ~ V . The description of fluctuations of any (Her- mitian) variable A(r,t) i s done in terms of the correlation function c (r,t) defined as c A ( r , t ) •= h < ( ( A ( r , t ) - < A ( r , t ) > ) , (A(0,0) -<A(0,0)»}> (30) where the angular bracket denotes equilibrium average and the curly brack- et is an anticommutator. In dynamic scaling the Fourier transform of ~A c (r,t) i s written in the form c£(k,C0) = 2lT ft£(k) _ 1 C ^ ( k ) f^ r ( -T—) *> * 0>*(k) where the subscripts £ indicate a parametric dependence on the dominant order parameter correlation length. This expression contains the equal time correlation function (t • 0 in (30)) 00 CrOO - J ^ c E(k,o>) and a shape function, f, such that 00 / fJ r(x) dx = 1 _oo > ^ The 0)^ (k) is the characteristic frequency defined by j f f £ i ? ( x ) d x - | . The shape function, f, i s determined by the hydrodynamics of the system being considered. The general relationship between the hydrodynamic equations and correlation functions has been established by Kadanoff and 20 Martin , and the specific case of helium has been dealt with in reference 17. The correlation function description is in principle more general than the hydrodynamic description, and the two are equivalent in the limit of small k and 0) where hydrodynamics applies. In particular, the 17 frequencies and damping of the normal modes in hydrodynamics appear as the poles of the appropriate correlation functions. This structure is contained in the shape function, f, given previously. Thus, for example, i f the hydrodynamics yields the frequency and damping of a normal mode expressed 2 2 2 2 in terms of the dispersion relation co = u k - icok D as in equation (27), then the correlation function description of this mode is reflected in a shape function of the type 1 y k f, r(x) = — —5 9 o k ^ * (x 2 - l ) 2 + y 2 The characteristic frequency co^(k) Implicit in x i s just the frequency of the normal mode and the width y^ i s co^(k) • uk » Dk2 Dk y = — *= k u k u Now one assumption, of dynamic scaling is that the shape function for the order parameter correlation 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 this assumption means, since y^ is linear in k, that D « u£ (31). In the case of the X-transition there are complications which stem from the fact that the order parameter i s the average, over a small region 21 of space-time, of the annihilation 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 correlation function then decays at large r according to a power law, p, so that 18 c^(r) »u \$\2 ( | ) P for r + » . and this serves as a deflntion of the correlation length £. Also, as dem- onstrated by the hydrodynamics, there are two propagating modes. However, i t i s shown in reference 17 that the order parameter correlation function i s dominated by the second sound mode to the extent that (c - c )/c << 1. 3 p v p In addition, i t i s shown that c^(k,w) , which i s not directly observable, has poles which are identical to those of the correlation function of the heat operator q(r,t) * U(r,t) <£ -^2>p( r,t) = U(r,t) - <y + To> p(r,t) (32). Thus, i t is possible to formulate scaling in terms of the observable f l u c - tuations of the heat operator ((dq^= ^Tp do}) which correspond to second sound. The scaling relation (31) is then a prediction of the damping of second sound The significance of this result to this work was discussed in the previous section, B. i i i ) Renormalization Group Theory and D,, The RG treatment of the dynamics of the lambda transition i s a work of such magnitude that even a mildly comprehensive development of the pre- diction for D 2 i s beyond the scope of this thesis. Thus, following a dis- cussion of the dynamic model which undergoes the renormalization, only the procedure for performing the RG transformation i s indicated. Then the pre- diction for D 2 is given. The dynamic models treated by RG techniques are semi-microscopic in 19 that they are defined by equations of notion for the variables which re- main after averaging over length scales which are larger than atomic dim- ensions but smaller than the correlation length for the order parameter when T i s near T £. One such variable for which an equation of notion must be given i s the order parameter, ty. The equations for the other var- iables reflect the various symmetries, or equivalently the conservation laws, of the system being studied. In liq u i d helium there are three con- served f i e l d s . As in 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 description of the dynamics of helium would then Involve, including the order parameter, four f i e l d s . However, i t i s anticipated that, as a starting point, a two f i e l d model Is adequate since i t is pos- sible to incorporate into such a model the c r i t i c a l hydrodynamic mode ass- ociated with a f i e l d , m, which couples most strongly to the order parameter. The f i e l d m is the linear combination of U and p which produces second sound for T < T^ and i s denoted by q in equation (32). The two f i e l d 6 22 model of helium i s defined by the following stochastic equations ' : ^ L £ > - - » O - i * 0 * £ • e n C33-A). at o Om o v ' n F - F o - Jd dx {hm(x,t)m + Re( h^(x,t)\J>* )} (33-C), F o - Jd dx (»sro \ty\2 + h\Vi>\2 + u O M * + ̂ m 2 + yom\ty\2) (33-D). Some of the features of this model are Indicated now. The 8 , C are n n Langevin noise sources. In the absence of time dependent applied fi e l d s h and h,, these noise sources, when chosen appropriately, ensure that ty m ty and m achieve values consistent with the equilibrium probability d i s t r i b - 20 ution P (ty,m) - e / Je d̂ dm. The f i r s t three terms in the eq functional F q represent the usual Ginzburg-Landau expansion in terms of the order parameter. A similar expansion in the f i e l d m i s truncated after the f i r s t term. In RGT the higher powers of m in the expansion are I r r e l - evant, while in the expansion in powers of ty the interesting or nontrivial behaviour i s a result of the U Q | ^ J ^ term. The interaction term, Y^ l^ i^ , in F is included because a variation in m, which is identified as second o ' sound, means there i s a change i n the local value of AT which in turn re- quires that ty obtain a new local equilibrium value. The f i r s t term on the right hand side of (33-A) indicates that ty is not a conserved f i e l d in 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 for h^ • 0 and £ r as given in reference 22. The significance of the coupling constant g Q in (33-A,B) can be under- stood by considering the effect of a uniform time-dependent applied f i e l d which i s conjugate to m, h (x,t) = h ( t ) . Writing the complex (two com- m m ponent) order parameter in terms of a phase angle 4> as ty = |^|e^, then (33-A) gives the effect of h on <t> as m ! £ = g h (34), dt °o m that i s , h causes a rotation of the order parameter. Although the notation m in (34) is more suggestive of an equivalent system of spins known as the 0 0 O1 O / planar ferromagnet Z J » z ,the corresponding rotation equation for helium 21 i 6 a "Josephson" equation C 3 5 ) where y i s the chemical potential per particle of the fl u i d at rest. The 21 connection between (35) and the superfluid acceleration equation (5-D) in the two f l u i d model can be made by the identification ti^ct « mv where m 6 i s the mass of a helium atom, and adding the kinetic energy contribution 2 h m v to the chemical potential in (35). The result of the coupling, g , 6 O on the hydrodynamics i s that there i s a propagating mode for T < T which 22 24 involves coupled variations in m and cp ' . This i s second sound in helium while the corresponding mode in the planar magnet is a spin wave. The renormalization group transformation is applied to the cor- relation function formulation of the equations of motion (33-A,B,C,D). It s i s is defined by = ̂ b^b w n e r e ^ * s a s c a x e change x -*• x* «= x/b A -»• A' = Ab ip \K = bai> co •*• co = b co such that b > 1, and a, z are determined within the theory. The operation R̂ applied to the diagrammatic expansion of the equations of motion is an Integration over wave vectors such that b ^A < k < A and frequencies from _» to +°°. The transformation i s iterated, say n times. The requirement that the equations retain the same form leads to recursion relations for renormalized constants \f , g , X ... } which have developed from the 1 n en' n ' original set {r , g Q, X q ... }. An analysis of the fixed points of the recursion relations, that i s , those limiting values [T , g , X which remain unchanged by successive iteration, leads to scaling laws as well as values for exponents and certain universal amplitude ratios. One such universal amplitude ratio i s *2 " <36> where i s the damping constant of second sound with velocity u 2 > and £ 22 i s the transverse correlation length of the order parameter. (In super- f l u i d helium there are two correlation lengths, one associated with f l u c - tuations in magnitude, the other with fluctuations in phase. The latter IA i s called the transverse correlation length.) Different methods may be used to evaluate in three dimensions. One approximation technique using an expansion in e - A - d gives R 2 ^ 0.15. Another method i s a general- ization of RGT to three dimensions and results in R 2 ^ 0.09. In each case the calculations are based on the simpler symmetric model for helium (equ- ation (33) with Yq • 0) which i s expected, in three dimensions, to give the correct asymptotic behaviour as T T^. The accuracy for either method of calculation i s expected to give R 2 to within a factor of two. An explici t expression for D 2(t) where t • (T^ - T)/T^ may be ob- 13 tained by using empirically determined expressions for u 2 and £ . Ahlers' 9 3 0 387 measurements give • 4.63 x 10 t cm/sec at saturated vapour pressure. The same data, in conjunction with the hydrodynamic expression for u_ and measurements of c and o provide the best information on p ( t ) . i P 5 -2 2 This may be used to evaluate £ • m (k^T)/ fi Pg( t^ v * t h the result £ • 3.57 x 10 8 t 0*675 cffi^ Then, the prediction for D 2 becomes D 2 - ( 3 or 5 ) x 10"5 t " 0 * 2 8 8 cm s" 1 (37) depending on the two estimates for Rj. It is noted that both the amplitude and the exponent are subject to verification by experiment. The recent treatment by Dohm and Folk (DF)*"^ of the dynamics of the superfluid transition yields new predictions concerning D 2 > They begin with the stochastic model employed by HSH and described by equations 33 A, B, C, D, with YQ " 0. However, their analysis of the fixed points of the renormalization group transformation leads DF to predict a temperature dependent effective ratio, R 2 f f ( t ) , which determines D 2 via D 2/2u 2£ • R***'. Using thermal conductivity data above T^ to evaluate non-universal par- 23 ef f -A ameters entering their theory, DF obtain a value for at t « 10 of -A about 0.1A. The temperature dependence i s weak over the interval 10 > t > 10 , but stronger for t > 10 . As a simple analytic expression for D 2 i s not available in their report, the graphical presentation of their predictions i s reproduced in Chapter A along with a discussion of the re- sults of this work. They {DF) do not indicate the expected accuracy of their calculations. 24 CHAPTER 2 EXPERIMENT The f i r s t section of this chapter is a discussion of techniques used in second sound attenuation measurements with particular emphasis on the method chosen for this work. Section B i s a discussion of the apparatus. This includes a description of the cryogenic apparatus, the resonator, and the electronics. An outline of the experimental procedure i s given in Section C. A. Techniques There are several methods available which may be used to measure the attenuation of macroscopic second sound. The most direct approach i s that used by Hanson and Pellam (HP). They measure the temperature amplitude, T, of a travelling 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 Qe . 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 in a resonant cavity. A third 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 for 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 reflecting plates. In this experiment a resonance method has been developed in which the attenuation i s derived from the decay time, T, for free oscillations of the plane wave modes of a cylindrical cavity. This i s essentially the Fourier transform of the technique used by Ahlers and Tyson. In their experiments 2 5 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 T determines the temporal M2 a.2 — t / T response as T • T^e . The expression which applies to plane trav- e l l i n g waves, 1 « l ^ e - 0 0 1 , becomes T « 5 L^ E K X U2 T appropriate to a standing plane wave mode and therefore 1 / T E 2U2O1. Of course, Aw «= 1 / T . In these expressions a i s the total attenuation due to the bulk helium and contrib- utions associated with the boundaries of the resonant cavity. The decay time method used in this experiment gains one particularly significant advantage while retaining the benefits of a resonance approach. The reasons for choosing a resonance technique in general are based on the desire to approach as closely as possible and achieve a resolution i n AT on the order of 10*"̂ K. Thus, in addition to the requirement of temp- erature s t a b i l i t y , a small system i s preferred in order to minimize the pressure increase due to gravity which alters the value of T^ by about 1.3 x 10~^ K per centimeter of helium. It i s f e l t that thermal isolation and small size are more easily achieved with a resonant cavity as opposed to the method of HP which requires a variable propagation path of con- siderable length to avoid multiple reflections. Also, the effects of f i n i t e second sound amplitude (recall the approximations of linearized two f l u i d hydrodynamics and that v n > V g are proportional to 1) become more severe as T + T, since then v can become large due to the vanishing of p . A s s Therefore, i t is desirable to use small signal levels to avoid what may be a d i f f i c u l t interpretation of large amplitude effects. In addition, the frequency dependence of 02 indicates that a signal with limited frequency content requires less interpretation than, say, a pulse signal. 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 results in an amplification of the AC excitation. This 26 is important i f the second sound is generated, as i t is in this experiment, by the AC e l e c t r i c a l heating of a resistive element. Then there is a DC component present in the power spectrum of the excitation which results in a steady counterflow of v^ and v g upon which the AC second sound flow i s superimposed. However, to the extent that the gain*of the cavity i s very large, the DC flow velocities are negligible in comparison to the AC flow velocities. As this selective 'gain' i s not present in the methods using a travelling wave, or pulsed second sound, a resonance method is preferred since i t , i s expected that with a DC counterflow there are cor- 18 rections to the expression (27) for the attenuation The measurement of decay times as opposed to line widths overcomes a problem related to the limitations of temperature s t a b i l i t y and frequency range that are encountered in this experiment. To understand the nature of the problem, consider the harmonic sequence for plane-wave modes in a cavity consisting of parallel plates separated by a length a. The resonant frequencies are given by co = u 0k = u 0(pTT/a) p I p 2 where p = 1, 2, 3 ... . During the course of a frequency sweep through some resonance of width Aco^ at cô suppose that the ambient temperature changes by 6T. Then the second sound velocity changes by = (3U2/9T)6T and, therefore, the resonant frequency changes by an amount Scô • 6u2kp, or 660̂ = (6u2/u2)t0p. For small AT, 6U2 becomes large since S^/BT diverges as T T^. The amplitude response to the driven oscillations now approaches a different value appropriate to the new resonant frequency cô + Scô . Thus, the typical temperature fluctuations, 6T, result in distortions of the resonance response curve making i t d i f f i c u l t to determine Acô . This i s significant when the "temperature noise width", 6cop, becomes comparable 27 to the Intrinsic width (AWp)a • l n principle this problem can be overcome by using higher harmonics since the intrinsic width i s expected to vary as 2 2 (AOJ ) • 2u„a 0 « (a) /u„)D-, and the frequency squared dependence w i l l u l -p ot2 2 2 p 2 2 timately dominate the linear dependence in 50)̂ = (5u2/u2)top. Unfortunately, modes which do not correspond to plane waves, but' rather to Bessel func- tions, 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 in this experiment to interpret the resonant structure at the frequencies of the higher harmonics. However, the attenuation, a 2, can be obtained by measuring the decay times of the well isolated, low frequency harmonics. In this method the cavity i s driven at or near the resonant frequency u n t i l the excitation reaches some desired high le v e l . The drive i s then turned off and the decay of the excitation recorded. The os 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 this does not appear as amplitude noise in the signal since the response is not driven but allowed to decay freely. It i s only necessary that the bandwidth of the detection system be large enough to accommodate the frequency content of the decay, e t^ T,' and the excursions, 6w , which occur during the decay. The details of the signal P recovery system are found in the discussion of the electronics. The general resonant frequencies of a cylindrical cavity of radius r 26 and length a are given by, 2 a 2 p,m,n /. v a r ' The a , with m, n «= 0, 1, 2 ... , are solutions to (dJ (ircO/da) • 0, mn m where J (TO) i s a Bessel function of the f i r s t kind. The plane wave modes m are obtained for a^g •= 0. The modes with m or n not zero are loosely re- ferred to as Bessel modes. For m or n near one, the a are on the order mn of unity, while for large m and n approximate values are a^o - m/tT and 28 - n + Jsn + h when n > m. The dimensions of the cavity in this exper- iment are such that r - 2.A a, and therefore the lowest resonant frequencies correspond to Bessel modes. The density of the Bessel modes Increases with frequency with the result that at the frequencies of the higher plane wave harmonics there may be several Bessel modes having nearly the same frequency as any particular plane wave mode. With the use of equation 26 (38) and the tabulated values of a for m = 0 to 8 and n = 1 to 20, mn the resonant 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 results are sensitive to the value of the ratio r/a, which i s known with an accuracy of about ± 0.5%. In an- ticipation of the observations on harmonics 1, 2, 3 and A, a calculation indicates that the f i r s t and third harmonics are isolated to the extent that to within ± 1% of their frequencies there are no Bessel modes. In view of the uncertainty In r/a, this means that harmonics one and three are fractionally isolated from Bessel modes by at least 0.5% of their re- spective frequencies. This degree of isolation is significant since i t is large compared to the maximum fractional width of a resonance, Aco/cô , of about 0.05%. However, for harmonics two and four, there are two Bessel modes within ± 0.5% of the frequency of harmonic two, and four in the case of harmonic four. Thus, i f the Bessel modes are excited, they could influence the response at the second and fourth harmonics. The consequences of the position of the Bessel modes w i l l be discussed in more detail in relation to the experimental observations. Possible mechanisms which may be responsible for the excitation of the Bessel modes are sug- gested in the general discussion of the concluding chapter. 29 B. Apparatus I) Cryogenic Apparatus The general features of the cryogenic apparatus are illustrated in Figure 1 and described here. The experimental c e l l containing the reson- ator was suspended inside an evacuated container. This in turn was im- « mersed in a bath of liquid helium (T < T^ ) to provide a- stable thermal 27 environment. The temperature of this outer bath could be regulated to -4 better than 10 K over a half hour interval. Helium from this bath 3 f i l l e d the experimental c e l l (about 15 cm ) through a valve and capillary. A porous stainless steel f i l t e r over the valve entrance kept solid a i r particles out of the capillary and valve seat. An estimate of the helium level in 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 os c i l l a t o r . A therm- ometer and standard resistor 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 in conjugation with a feedback resistor wound on the out- side of the copper top of the c e l l body. The second sound resonator was held in a brass frame that enabled the resonator body to be held together by spring loading. The second sound detector (bolometer) in the reson- ator was a superconducting device. Its transition temperature was trimmed to the desired temperature by a magnetic f i e l d produced by means of a sol- enoid wound on the outside of the vacuum container. A few other features might also be considered as follows: (i) A second capillary connected the c e l l to room temperature access. This was available for pumpimg away excess helium in case of accidental over- f i l l i n g . Also, this line would be necessary for studies at elevated pres- sures. 30 PUMP LINE LEADS INDUCTOR AND TUNNEL DIODE FOR OSCILLATOR CIRCUIT \ L O W TEMP. FILTER 1 VALVE CAPILLARY HEATERS AND THERMOMETERS CAPILLARY TO ROOM TEMP. ACCESS FILLING CAPILLARY EXPERIMENTAL C E L L OUTER BATH VACUUM J A C K E T FEEDBACK RESISTOR RESONANT CAVITY CYLINDRICAL CAPACITOR FOR OSC. CIRCUIT DEPTH MONITOR INDIUM •0" RING MAGNET SOLENOID FPOXY FEEDTHROUGH L E A D S T H E R M O M E T E R S T A N D A R D RESISTOR Figure 1 Cryogenic Apparatus 31 ( i l ) There existed several thermometers and heaters at various locations that were used to establish i n i t i a l working conditions. Those mounted on the capillaries were particularly important since It was necessary to de- stroy temperature i n s t a b i l i t i e s of an oscillatory character, and magnitude _3 of about 10 K, that were generated by helium in the ca p i l l a r i e s . The insertion of piano wire (0.2 mm diameter wire in 0.3 mm 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 . ( i i i ) The c e l l , suspended by three steel 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 alloy were brought down the vacuum pumping l i n e . Signal leads for the bolometer, generator, thermometer and level indicator were' brought down separate stainless steel tubes. The bolometer leads consisted of a twisted pair of #40 copper wire. Leads into the c e l l were brought through holes in the bottom brass flange and sealed with epoxy. II) Resonator (i) Cavity A side view of the resonator i s shown in Figure 2a. . Two fused quartz optical f l a t s separated by a stainless steel annulus (length 3.0 mm, inside radius 7.4 mm, and wall thickness 0.38 mm) formed the cylindrical resonant cavity. Thin (̂ 6 x 10 mm) mylar gaskets glued to the annulus elec- t r i c a l l y isolated the bolometer and generator thin films on the fla t s from the annulus. Polishing the ends, with gaskets in place, ensured that the f l a t s (better than one light wave f l a t ) were parallel to within a few light waves. It was hoped that this alignment would result in preferential ex- citation of only plane wave modes. The inside surface of the annulus was also polished. After assembly a small amount of glycerine was applied to 32 a) RESONATOR B O L O M E T E R F L A T b )GENERATOR I c ) B O L O M E T E R I I 1 \ 1 1 1 1 ^ STAINLESS S T E E L A N N U L U S MYLAR G A S K E T , GLYCERINE GENERATOR F L A T R E S O N A N T CAVITY J H I N F I L M E L E C T R O D E CHROMIUM RESISTIVE FILM THIN FILM ELECTRODE (NOT USED) A u - P b BOLOMETER FILM BIAS CURRENT PATH THIN FILM E L E C T R O D E Figure 2 The Resonator, Generator and Bolometer 33 the corners formed by the annulus and f l a t s . This ensured that the cavity Interior was sealed off from the external helium in the c e l l and prevented any possible coupling between the internal cavity modes and those modes that existed outside. Such a coupling could result in an energy loss mech- anism that might be misinterpreted as intrinsic attenuation. However, i t was s t i l l possible for the superfluid to penetrate the gylcerine seal and f i l l the resonator in approximately four hours. The optical f l a t s , with 25 mm diameter, were typical of those commer- c i a l l y available. They were quite thick, the f l a t with the generator being 3.2 mm while the one with the bolometer was 6.4 mm thick. Both materials, fused quartz glass and stainless steel, used in the construction of the cavity have low thermal conductivities. This property resulted in 6trong reflection of the second sound at the boundaries and w i l l be discussed in more detail in following chapters. The heat generated in the cavity es- caped mostly through the relatively thin annulus walls and raised the temperature of the cavity by about 20 KW *" above the ambient temperature of the c e l l . Considerable effort went into the construction of the thin resistive films which constitute the bolometer and generator. Besides possessing specific properties described below, i t was f e l t that they should be as close as possible to an ideal surface so that "perfect" reflection occurred 28 at the end plates. Indeed, recent studies comparing thin resistive films (Aquadag) to superleak (nucleopore) transducers have indicated that the reflection properties of the former are much simpler to Interpret. o ( i i ) Generator A top view of the generator i s shown in Figure 2b. Conventional vapour deposition techniques have been used in construction. The parallel 6trip 34 electrodes were deposited f i r s t . Particularly robust electrodes were made by depositing a thin layer of chromium with a film of gold on top. Simpler, less expensive but less durable electrodes were also made using only alum- inum. Electrodes were typically 300 nm thick with a resistance less than 1 U per square. Leads of #40 copper wire were usually attached by simply cold welding with a bit of indium. The active resistive element which gen- erated the second sound was a thin uniform film of chromium overlapping the electrodes at the edges. The resistance of this film was about 43 ft per square, independent of temperature from 300 K to 2 K. Its 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 limit of about 100 nm. The film was sufficiently robust that i t suffered no damage on contact with the annulus. The position and size of the annulus relative to the generator are indicated by the dashed c i r c l e in Figure 2b. The reason for choosing the geometry illustrated in Figure 2b was to preferentially excite the plane wave modes of the cavity. The significance of the thickness, d, of the generator film can be appreciated by comparing i t to the length 6 = (2 D/co) which governs the phase and exponential att- enuation of temperature oscillations at angular frequency co in 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 is the thermal con- ductivity and c is the specific heat per unit volume) i s d i f f i c u l t to est- imate for what Is probably a polycrystalline chromium film; however, even a cautious estimate indicates that d « 6 for the frequency range of this experiment. This means the generator was thin in a thermal 6ense. There- fore i t was capable of fast response, and the reflection properties were determined by the glass substrate. 35 ( i i i ) Bolometer The temperature sensitive mechanism of the bolometer was the super- 30 conducting transition of a gold and lead composite film . The center temperature of transition was adjusted, by means of a magnetic f i e l d , to the operating temperature of the resonator. The temperature excursions associated with second sound resulted in corresponding variations in the resistance of the film. By biasing the film with a constant current the resistance variations appeared as voltage changes which in turn were re- covered by the electronics. The gold-lead films were constructed by depositing 8.0 nm of gold 31 followed by 14 nm of lead . The gold was evaporated from a tungsten f i l - ament, the lead from a boat or crucible lined with AljO^. Deposition rates were 10 * nm per second. The films were extremely delicate and sensitive to chemical attack when l e f t exposed to the atmosphere for periods of about a day. The resistance of the films at room temperature was about 25 Q -2 per square. The superconducting transition was typically 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 in Figure 2c. Apart from the difference in pattern, the electrodes are similar to those used in the generator. The operating resistance of a square section of film was too low to provide an adequate signal level and impedance match to the electronics. To remedy this, the resistance of the bolometer was increased by a factor of 20 by cutting i t with a steel scribe into the pattern shown in the figure. A typical current path for the constant bias i s shown by the dotted l i n e . This particular pattern was chosen to maintain an active area as large as possible without allowing the easily damageable films to come into contact with the annulus. The sides of the 36 film were also cut to prevent edge effects from reducing the sharpness of the transition. A useful figure of merit for a bolometer of resistance R is the sen- s i t i v i t y defined by (1/R)(dR/dT). For the bolometer used in this work the most rapid variation of R with T occurred near the center of the trans- ition where R had fallen to one-half the high temperature value. There the resistance and sensitivity were 140 f2 and 40 K ^ respectively. As a comparison, the sensitivity of conventional carbon film bolometers is more than a factor of ten smaller. The bolometer film was also thermally thin and capable of fast res- ponse. However, i t might be expected that the power dissipation due to the bias current would have some effect on the resonance decay, and con- sideration was given to this in the collection of data. I l l ) Electronics A block diagram of the major electronic circuitry is shown in Figure 3. (i) Signal Excitation 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 for the lock-in analyzer . The 34 cavity response signal, as detected by the bolometer, was amplified and fed to the lock-in analyzer which responded to the second harmonic of the original f / 2 reference. The outputs available from the analyzer were the components of the signal that were in-phase, I, and out-of-phase, Q, with 35 respect to the reference. These components were squared and summed to 2 2 produce a signal, I + Q , which was proportional to the squared amplitude of the second sound in the cavity. In the measurement of decay times, once 37 SIGNAL EXCITATION SIGNAL RECOVERY DATA _STOR_AG_E S W E E P CONTROL D C IFREQ. SUPPLY jSYNTH, i TRIGGER H THRESHOLD OR DC A C i ATTENUATOR I c X S I G N A L AVERAGER I LOW PASS FILTER I 2 + Q 2 14 I LOCK-IN A N A L Y Z E R i P R E - A M R BOLOMETER BIAS TEMPERATURE REGULATION O F F S E T AND F E E D B A C K I LOCK-IN AMR P R E - A M R i DECADE TRANSFORMER I ^ 1 3 BOLOMETER THERMOMETER AND STANDARD ) GENERATOR > FEEDBACK HEATER Figure 3 Block Diagram of Main Electronics 38 2 2 the I + Q signal reached some preset threshold the AC excitation to the generator was interrupted and replaced with a DC drive which produced equiv- alent power input to the resonator. This enhanced the temperature stab- i l i t y in the cavity. Simultaneous with this inerruption, a pulse was 36 2 2 sent to trigger the signal averager and the decay of the I + Q s i g - nal was recorded. After some predetermined time the AC excitation was again applied to the generator. If the resonant frequency of the cavity had changed slightly 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 signal to noise ratio was achieved. As previously mentioned, i t was necessary that the bandwidth of the detection system be sufficiently wide to accommodate, the frequency excur- sions 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 in 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 lock-in analyzer. However, following the square and sum operation i t was useful to insert a low pass f i l t e r . Since at this stage i t was only necessary to pass the decay s i g - nal, the "bandwidth" of this f i l t e r could sometimes be less than that of the lock-in analyzer. I n i t i a l l y i t was useful to identify the mode structure of the cavity by recording the response as a function of the drive frequency. Then the threshold device and signal averager were not active. A microcomputer stepped the synthesizer and stored the cavity response at each frequency increment. The resulting data could be readily plotted using the U.B.C. computing f a c i l i t i e s . 39 (ii) 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 insensitive reference resistor. The amplified unbalanced signal from 39 the bridge was fed to the lock-in amplifier , the output of which was com- 40 bined with a DC offset and then applied to the heater wound on the ex- perimental ce l l . This negative feedback maintained a null signal and reg- ulated the temperature of the cell 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, dif- 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 cell (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 cel l . C. Procedure and Tests Initial 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 third harmonics were "clean", while the other harmonics were accompanied by the nearby resonant structure of Bessel nodes. Similarly, the time decay of the f i r s t and third 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. Consequently, the time decays of harmonics one and three were obtained for smaller values of AT. The data covered the temperature -5 -2 range A x 10 < 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 for different values of both the bolometer bias power and the amplitude of the second sound in the cavity. The effects of bolometer power, which ranged from 3.6 x 10 ^ W to 2.5 x 10 ^ W, were usually weak. Amplitude effects, how- ever, could be quite severe in that there was a c r i t i c a l amplitude above which a resonance would decay very quickly with a strong amplitude de- pendence. Below this 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 significant as T •*• T^. To stay below the c r i t i c a l amplitude, which be- came smaller for 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. The recovery of these signals required averaging over hundreds of decays. The f i n a l results derived from the data for the f i r s t and third harmonics at fifteen values of AT are determined from the an- aly 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 for the f i r s t and third harmonics respectively, while for large AT levels as high as —6 —2 ^ 10~ W cm" were used to study amplitude effects. The treatment of the Al residual amplitude and power dependence i s discussed in the analysis of results. The temperature difference, AT, was determined using the express- 9 A2 ion ' for the second sound velocity u 2 •= A6.28(AT/T X)* 3 8 7 m s ~ \ (39) and the observed resonant frequency, f^, of the fundamental harmonic which gives u 2 by u 2 - 2af x where a is 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 interior 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 for the tedious, periodic c a l - A3 ibration 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^ via u 2 was checked once by calibrating the thermometer at the lambda-point using the anomaly in A3 -5 the warming curve . A value of AT ^ A x 10 K derived from this c a l - AA ibration point and the measured thermometer sensitivity ( l/R(dR/dT) = 1.27 K _ 1 ) was consistent with that derived on the basis of second sound velocity and the estimate of the internal heating in the cavity. For AT > 2 x 10 K the expression (39) begins to break down 9 and a simple graphic interpolation of the numerical data given by Ahlers was used to determine u 2(AT). While collecting data, the drif t i n g thermometer calibration resulted in a corresponding change in the value of AT and resonant frequency for a fixed value of the bridge ratio. This was compensated for by adjusting the bridge setting appropriately to maintain a fixed AT within suitable 42 limits, typically ± 2 x l C f 6 K. The uncertainty in AT is the maximum of ± 3 x 10~6 K or ± 0.5% of AT. The major contributions to this error estimate are the uncertainty in the cavity length, a, which enters the above expression for u2» and the stab- i l i t y in AT during the collection of data. It will be evident in the pre- sentation of results in Chapter 4 that this uncertainty in AT is insig- nificant in comparison to the error estimates on the damping. During the collection of data i t was realized that a thermal emf -3 -7 3 x 10 V) resulted in a dissipation of about 10 W in the gener- ator. This power was eliminated by using a simple battery circuit to oppose the current driven by the emf. Studies indicated the the thermo- electric power had no effect on the decay curves for the first and third harmonics. However, when the thermal emf is added to the AC voltage ex- citation 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 res- onant structure disappeared when the thermo-electric power, and conse- quently 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 will 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 exist- ence 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 sec- ond sound other than that arising from the bulk helium. These additional sources of energy loss occur at the boundaries of the resonator and result from thermal conduction and the viscosity, ri, of the normal f l u i d . The total attenuation, a, is written as a = a„ + a + a + a 2 T\ e s where i s the bulk contribution given by equation (28) on page 15, is the contribution from viscous drag at the side walls, a g and a g result from thermal conduction at the reflecting end plates and side walls res- pectively. The development of the expression for has been presented A5 in considerable detail by Heiserman and Rudnick , while the thermal con- A6 duction losses have been treated by Khalatnikov . The derivations of the expressions for a , a , a are outlined in sections A and B. Some aspects r rr e s of the application of these results to this experiment are discussed in 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 parallel to the wall of the tube, i s entrained in the v i c i n i t y of the wall due to viscous interaction. This effect penetrates \- into the f l u i d a characteristic distance X - (2n/p co) and results in n a velocity dispersion and attenuation. This expression for X i s obtained from the related problem of an oscillating plate in contact with a viscous 18 f l u i d . In that case the solution i s a viscous diffusion wave with X de- termining the normal f l u i d velocity a distance x from the plate by 44 -x/X -i(cot - x/X) v = v e e n no To calculate the viscous surface losses for second sound propagating along a tube of cross-sectional area A and perimeter B, the linearized two f l u i d hydrodynamic equations of Chapter 1 are employed. Again, using 5 to denote small quantities, the non-dissipative equations for mass, entropy, and superfluid acceleration are: 35p/3t + V«(p 6v + p 6v ) = 0 (40), n n s s p(3So/3t) + o(36p/3t) + pa(V«6vn) = 0 (41), 36v / 3 t = -Vu (42). s The linearized momentum equation, which i s 3(p 6v + p 6v ) / 3 t = -Vp in ^ ' n n s s * the non-dissipative approximation, i s modified to include the effects of viscous interaction with the walls. Choosing the z-axis as the propa- gation direction and denoting by r the perpendicular coordinate which i s zero at the wall, the momentum conservation law including a viscous stress term i s 3(p 6v + p 6v ) _ 3v n nz s sz „ B n2 ^ _ - Vp - - r, ̂ — J r = Q (A3) . The assumption implied in writing this equation i s that *"he viscous pen- etration length is much smaller than the lateral dimension of the tube and, therefore, the wave fronts are essentially plane wave. The approximation -3 -4 is valid for this experiment since X i s typically 10 to 10 cm. 16 18 For a second sound wave the fl u i d momentum is zero * , that i s , p v + p v Kn n s s (44) Thus, the hydrodynamic equations become 36p/3t » 0 ( 4 5 ) . 35o/3t « -o3v /3z (*6), nz 45 85vsz/3t - +a(3T/3z) - l/p(3p/3z) (47), 3p/3z - - (B/A)n (36v n z/3r)\ r m Q (48). With equations (44, 46, 47, 48) and the approximation 3T/3z ̂ (T/c)3a/3z which neglects (3T/3p)o for second sound, i t is easy to derive the following wave equation for the entropy: 3 2 6a , ps B 3 26o i 2 3 2 6o 3t 2 P P n 3 t 9 r r=0 2 3z 2 2, h where u„ = (p To /p c) is the speed of second sound. By analogy to the 2 s n oscillating plate problem, a solution to (49) is attempted in the form t x / , i r / X - r / X ' i ( k'z - cot) , c n N 6o «= 6a (1 - e ) e (50) o with k ' = k + ia . Substituting (50) into (49) gives 2X , . Ps B n « i ( k'z - cot) 2, ,2. .... co 6o + ico - T - (1 - i ) w e = u - k ' 6a (51). pp A A o I n Now, for X much smaller than the lateral dimensions of the tube, the app- roximation 6o = 6a e*^ z ~ is made in (51) with the real and imag- o inary components resulting in OJ2 + co(p /pp ) (B/A)(n/X) = u 2 ( k 2 - a 2) , s n z n and co(p g /pP n)(B/A ) ( r i /X) «= 2u2 a^k . Solving these equations yields the dispersion co(k) and, for small disper- sion where to - U j k , the attenuation: 1 rB, Ps T 1 TI 2u, W p p x * 2 n Using X = (2n/Pnoo) , then the viscous surface attenuation for a circular cylinder of radius r, as in this experiment, i s : a - (1/ru.) (p /p) (na^p)*5 (52). T) 2 s n 46 B. Attenuation Due to Heat Conducting Surface Losses, and a g The temperature excursions associated with second sound result in thermal conduction at the boundaries which diminishes the magnitude of the temperature excursions and, therefore, contributes a source of attenuation. Neglecting dissipation in the helium, consider 8 plane wave of second sound propagating in the z-direction and incident on a solid body f i l l i n g the half space z > 0. In the second sound wave the energy flow, 3 , in the z direction through unit area per unit time is written as , , y ikz * - i k z N -icot .... J - (J^e - J 2e )e (53) where 3^ and 3 2 a r e t n e amplitudes in the incident and reflected waves re- spectively. The corresponding temperature oscillations, T, are given by I - ( l / p c u 2 ) ( 3 1 e i k 2 + 3 2 e - i k * ) e - i a 5 t (54) with c being the specific heat of helium. The desired quantity to be calculated is the reflection coefficient 3 2 / 3 ^ . At the boundary there are two thermal impedances to be considered, one being the impedance of the 6 o l i d body, the other is the Kapitza resistance of the surface i t s e l f . The prof i l e of the temperature excursions, T', in the solid body is det- ermined by the heat equation c " ( 3T ' / 3 t ) - K ( 3 2 T 7 3 z 2 ) (55) where ic is the thermal conductivity, and c i s the heat capacity per unit volume of the solid. The solution to ( 55 ) with the boundary condition T'(z-O) - T'e~ i C t 5 t, i s the diffusion wave o * r - re-'^r (1//2 - i/iu r i « t ( 5 6 ) > o The amplitude of the temperature excursions, T^, of the wall at z 1 0 is not equal to the amplitude, + J 2)/pcu 2, in the helium at z • 0 due to the Kapitza resistance, 1/G, of the surface. The requirement of contin- 47 uity of energy flow at the boundary provides two equations: 31 ~ 32 " G ( ( 3 1 + 3 2 ) / p C U 2 " T o } ( 5 7 ) and C 1 - J 2 ) e " i ( 0 t - -KRe < 3 T 7 3 z ) z = 0 (58). Implicit in writing (58) i s an approximation which neglects in (53) a small contribution to 3 2 of magnitude - J 2 and phase shifted by TT/2 with respect to j' 2e~ i a 3 t. Substituting (56) into (58), and eliminating T' from (57) and (58), gives o (G/pcu 2) 2 = 1 + G(2/cKco)^ r (GTPTUT) ( 5 9 )- 1 + - h r 1 + G(2/c<a>K Using the inequality G « pcu 2, which holds for the temperature range of this experiment, the reflection coefficient becomes ^ • = 1 - (60). 1 pcu 2(l/G + /2/cicw ) Although there i s considerable variation (an order of magnitude) in the reported measurements of the Kapitza resistance near T^, for the largest frequencies in this experiment the solid body resistance, / 2 / C K G J , is greater than the Kapitza resistance. The approximation which neglects 1/G, to be discussed in section C of this chapter, yields for the reflection coefficient = 1 - 6 = 1 - (2/pcu2) Vc<oV2 (61). This result i s now used to calculate the attenuation, ot and ct , due to thermal conduction at the ends and 6ide walls of the resonant cavity. Consider a plane wave propagating between reflecting end plates sep- arated by a distance 'a'. It i s evident that the effective attenuation, -a a a^, due to reflection i s such that e « 1 - B. Substituting from (61), 48 and using $ « 1, gives — h a - — J (62). e apcu 2 v 2 ' To obtain the attenuation, a , due to thermal conduction at the side s walls, consider a cylindrical resonator of cross-sectional area A, per- imeter B, and symmetry axis in the z-direction. The amplitude of the temperature excursions, T" , i s proportional to + 3 2 , which, for strong reflection, i s approximately 2 3 ^ The amplitude decrease, d f Q , due to thermal conduction through an area Bdz at the side wall is proportional to ( 3 1 - 3 2)(Bdz/A). Thus, the effective attenuation is , dT 3 . - 3 , _ 1 o 1 2 fB>| a s : T ^ 217 y • o 1 Using equation (61) gives a = (2/rpcu.) (CK03/2)* 5 (63) s 2 for a cylinder of radius r. C. Discussion of ot , a , a Ty e s The inverse of the decay time is related to the total attenuation by 1/T = 2u 2a. From equation (52) the contribution to 1/T by viscous surface loss i s (1/T ) = (2/r)(p /p)(nw/2pn)15 (64). n s n This quantity has a strong dependence on AT, and becomes small for de- creasing AT. In the analysis of results, 1/T^ i s evaluated using the 42 following expressions (t = AT/T,): A (i) p /p « 2 . 5 3 4 ( t ) * 6 7 4 ( i i ) n/n x - 1 - 5.19(t) , 8 5° with n x - 2.47 x 10"5 poise _3 ( i i i ) p - p ( l - p / p ) with p -0.146 g cm 49 The contribution to 1/x from thermal conduction at the resonator ends i s 1 / T E - (4/apc) (CKOJ/2) 1 5 (65) where c and K are those quantities for fused quartz glass. At the side walls thermal conduction contributes 1/T «= (4/rpc) (cico)/2) l s (66) s with c and K for stainless steel.. The major temperature dependence In 1/T and 1/T comes from c, the specific heat of helium. The temperature e s dependence through c and K i s weaker in that they can be considered to 47 ' vary primarily with T, not AT. An estimate using representative values for c and ic indicates that 1/T i s a factor of five greater than 1/T . 6 e As the available information on c, K does not warrant an accurate eval- uation of 1/T and 1/T , and for reasons discussed in Chapter 4, in the e s f i n a l analysis of the data only the frequency dependence is used. Thus, the thermal conduction losses are treated collectively as 1/T- *= 1/T + 1 / x K e s with 1/T- - gOOu* (67). I N The function g(T) denotes the temperature dependence through c, c, K . Recall that an assumption involved in deriving (65) and (66) was (2/CKGO ) » 1/G. The v a l i d i t y of this approximation is d i f f i c u l t to ass- ess due to the range of reported values for 1/G, reflecting i t s variab- 48 i l i t y with material and detailed surface condition. Also measurements of 1/G using an AC method involving coupled second sound resonators suggest that the value of 1/G for AC heat flow is much less than that measured with DC flows. If the AC data i s taken as being representative of the present situation, then, at the highest frequency for the third harmonic of this experiment, the above inequality i s satisfied by about 50 a factor of 100 for glass and eight for stainless steel. In view of the uncertainties involved, the val i d i t y of the approximation i s subject to experimental ve r i f i c a t i o n . The derivation of (67) involved use of (56), which i s s t r i c t l y correct only for a reflecting 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 sufficiently low frequencies when the thermal diffusion length, (2</cco) , becomes equal to the wall thickness. As i t is estimated that this occurs at 40 Hz, a factor of three less than the lowest frequency obtained in this experiment, equation (67) is expected to apply. 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 re- cordings of the resonance decay curves i s described in section A. Section B contains a discussion of several aspects of the results for 1/T, i n - cluding the frequency and temperature dependences. The results for are presented in section C. A. Obtaining 1/T From Decay Curves Figure A illustrates a decay curve representing the average of 150 individual decays of the third harmonic obtained at a temperature -A AT «= 1.70(± 0.03) x 10 K and frequency 588 Hz. In this 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 K rms. Bias power in the bolometer was 1.2 x 10~^ W. The spike at the beginning of the trace results from the contribution of noise to the triggering threshold at which the decays are init i a t e d . As can be seen, this noise remains coherent for a relatively short time and i s ignored in an extrapolation to time zero when drawing the smooth curve through the trace. The amplitude of this smooth curve is 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- litude versus time. This plot i s indicated in the inset of Figure A . It was mentioned in Chapter 2 that the decay rates are dependent on the power input to the cavity. Consequently, at each temperature several decay curves for the f i r s t and third harmonics were obtained for different values of the excitation power and bolometer bias power. Generally, for high values of the excitation power and bolometer power, the logarithmic • \ rjr-0.80 In (SIGNAL) o r V 1 SECOND | TIME Figure A A Decay of a Second Sound Resonance Yielding a Value for the Decay Rate 1/T 53 plots displayed a curvature Indicating a larger value for the slope, 1/T, at the beginning of the decay. At sufficiently low powers the curvature was not noticeable over the useful amplitude range (about 85% of f u l l scale) of any particular decay. However, using even lower generator ex- citation powers indicated that there could s t i l l be a detectable amplitude dependence, a smaller value of 1/T occurring for those decay curves with a smaller i n i t i a l second sound amplitude. A plot of the results for 1/T at AT = 1.70 x 10 K for the third harmonic is given in Figure 5 to i l l u s - trate the nature of the extrapolations involved in determining the zero amplitude limit for 1/T. At the highest bolometer and excitation powers, the presence of some curvature i s the most significant contribution to the error estimates. For a given bolometer bias and lower excitation 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 in establishing the baseline of the decay curve in the presence of noise and small inaccu- racies in the squaring c i r c u i t r y . To obtain data at the lowest second sound amplitudes i t was necessary to use the higher bolometer bias v o l - tages in order to obtain *n adequate signal. An extrapolation to zero amplitude, that i s , zero excitation power, indicated by the line in Fig- ure 5, i s used to estimate a "best value" for 1/T. A S the extrapolation is subjective and without theoretical guidance, the associated error es- timates, indicated by the shaded region in Figure 5, are treated gener- ously according to the following c r i t e r i a . The upper limit on the error estimate includes at least one value for 1/T that has been actually meas- ured. Thus, the upper limit i s determined with confidence. The lower limit on the error estimate i s determined in a much more qualitative 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 3 4 G E N E R A T O R POWER ( I 0 " 8 W ) Figure 5 An extrapolation of values for 1/T to zero generator power for different levels of bolometer power. The extrapolation to zero amplitude, that i s , zero excitation power in the generator, i s used to estimate a best value for 1/x. The hatched region indicates the error estimate. The -4 data i s for the third harmonic at AT = 1.70 x 10 K. 55 with the measured values for 1/x, or by symmetrically placing the lower limit a distance below the best value which i s equal to the difference between the upper limit and best value. In the sense that this extra- polation procedure goes beyond the actual measurements of the experiment, the lower limits to the error estimate on 1/x are not determined with the confidence of the upper li m i t s . Generally, the amplitude effects for the f i r s t harmonic were not as significant as those for the third harmonic. Although at high amplitudes curvature similar to that for harmonic three was obtained, at lower amp- litudes the extrapolated correction to zero amplitude was usually slight or insignificant for harmonic one. This observation suggests that the amplitude effects, at least at low amplitudes, result from additional loss mechanisms that are proportional to the bulk helium loss rather than 2 h surface losses. Recall the frequency dependence, co versus co , of the sources of attenuation described in Chapters 1 and 3. Then, comparing harmonic one to harmonic three, the contribution to 1/x from the bulk is fractionally smaller relative to the surface losses by a factor 9//3~. Therefore, bulk related effects should be less significant at lower fre- quencies . The effects of bolometer power were of two varieties. In one case the dependence of the decay rate on bolometer power was more significant at larger generator powers, or equivalently, larger second sound amp- litudes. At the large amplitudes where curvature was present in the log- arithmic plots of the decay, increasing the bolometer power resulted in a more severe curvature. However, at lower amplitudes where the curvature was smaller or not noticeable, the effects 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. Thus, corrections for bolometer power were usually not necessary. There was, however, a second type of dependence on bolometer power which was of an entirely different character compared to that described above. For the range of bolometer powers used in this experiment, there was an anomalous bolometer power at which enhanced losses occurred. The value of the anomalous power decreased through the range of available power levels as AT decreased through the temperature interval -4 -3 4 x 10 <_AT <_1 x 10 K. The enhanced losses were not strongly depend- ent on second sound amplitude in that, at low amplitudes, the decay curves were governed by single time constants with the logarithmic plots showing no noticeable curvature. The magnitude of the enhanced losses at the an- omalous power was about 50% of the zero power losses for harmonic one, and therefore the effect was quite dramatic. In addition, the value of the bolomet er power at the anomaly was the same for the f i r s t and third harm- onics and in this sense was independent of frequency. Also, the absolute magnitude of the enhanced losses appeared to be about the same for the f i r s t and third harmonics, although this was d i f f i c u l t to determine with precision since the normal amplitude effects 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 this pow- er, to obtain meaningful data. It is emphasized that the enhanced losses diminshed with increasing bolometer power above the anomalous value, and for powers sufficiently removed from the anomaly the results became i n - dependent of bolometer power apart from the effects of the f i r s t variety -4 described above. However, as stated above, in the interval 4 x 10 < _3 AT < 1 x 10 K the anomalous power was within the range of available bolometer powers and systematic effects, particularly for harmonic one, were observable. When operating within this interval at temperatures 57 AT = 1.02 x 10~3 K and AT = 5.94 x l O - 4 K with the bolometer power less 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 for 1/T correspond to the "high power" data; that is high power relative to the anomaly. Additional support for the general va l i d - -4 ity of the "high power" results i s gained at AT = 3.05 x 10 K. At this temperature the results of two additional decay curves for the second and fourth harmonics are consistent with the f i n a l results for D 2 based on the data for the f i r s t and third harmonics. The mechanisms by which the bolometer power contributes to additional attenuation are not clear. The power dependent effects of the f i r s t var- iety described above can be qualitatively explained in relation to the sec- ond sound amplitude effects. If i t i s accepted that the amplitude effects simply reflect 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 in the second sound would result in more severe violation of the requirement of small flow velocities. Of course, the situation in the cavity is further complicated by the asym- metry, AC versus DC flow, as well as the directionality of the flows, most of the DC heat leaving through the side walls while the AC flows are p r i - marily parallel to the side walls. It i s pointed out, however, that the DC flow velocities are at least a factor of two less than the rms flow velocities in the smallest amplitude second sound wave at the smallest value of AT in this experiment. The mechanism responsible for the add- iti o n a l losses of the second type described above is a mystery. If the curious reader wishes to speculate, he may consider the possible role played by vortices. 58 B. Discussion of the Data for 1/T As discussed in Chapters 1 and 3, the bulk helium damping, viscous surface losses, and thermally conducting surface losses contribute to 1/T as 1/T = 1/T, + 1/T + 1/T_ 2 M K In terms of the damping coefficient D^, the expression for 1/T2 * s 1 / T 2 = 2u 2a 2 = (co 2/u 2)D 2 - (p*/a) 2D 2 (68) for harmonic "p" and resonant length "a". The expressions for the surface losses due to viscosity and thermal conduction, 1/T and 1/T_ respectively, T) K are given in equations (64, 65, 66, 67). The essence of the method involved in obtaining 1 / T 2 from the data for 1/T is to use the difference in fre- quency dependence, co versus co , between the bulk contribution and the sur- face contributions. To best i l l u s t r a t e this method and enhance the graph- i c a l presentation of the data, i t is useful to compute 1/T according to the prescription in Chapter 3 and reduce 1/T to 1/T - 1/T^. This "cor- rection" i s significant for large AT. However, for AT < 1 x 10 K, 1/T i s numerically small and comparable to or less than the error estimates on 1/T. In addition, i t i s useful to decompose the bulk contribution as follows: 1 / T 2 = l / T 2 R P + A ( l / T 2 ) (69). Here, i/ T2jjp denotes that value of 1 / T 2 which corresponds to the minimum -4 2 -1 value of D^, 3.68 x 10 cm s , observed by Hanson and Pellam at a temp- erature AT «= 3.2 x 10 K. Thus, l / T 2 H P is just a constant which, using -2 -1 equation (68), i s equal to 3.93 x 10 s for harmonic one and 3.54 x 10 * s * for harmonic three. The quantity A ( 1 / T 2 ) represents changes from the minimum value observed by H P . Then the data for 1/T is further reduced to 1/T - 1/T^ - I / T J H P which i s just 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 , their .values can be considered trustworthy i n the sense that they have used a method of measurement that yields directly. As their results indicate - 2 -2 only 6mall changes in D 2 over the temperature range 1 x 1 0 < AT < 5 x 10 K, A ( 1 / T 2 ) is expected to be small over the interval. Thus, their results are used to "calibrate" the present system in the sense that they are used to check the v a l i d i t y of the predicted co dependence for the surface losses, and to check the mode purity of the resonances (recall equation (38) for co on page 27) . p,m,n Before presenting the data for 1/T - 1/T^ - l / T 2 H P B 1/T_ + A ( 1 / T 2 ) , i t i s emphasized that the subtraction of 1/T^ from 1/T is done only as a convenience in the graphical presentation to remove a large, strongly temperature dependent contribution at large AT. As an indication of the magnitude and temperature dependence of 1/T^» some representative values of l / 1 ^ * * n units of s ~ \ for the f i r s t and third harmonics are: 0.350, 0.607 at AT «= 31.3 x 10"3 K; 0.133, 0.231 at AT - 10.3 x 10~3 K; 0.018, 0.031 at AT «= 1.02 x 10 K. Also, the subtraction of l / T 2 H P simply re- duces the data by the appropriate constant value and provides a convenient way of displaying more clearly any changes, A ( 1 / T 2 ) , from the minimum value obtained by HP. In the f i n a l analysis, discussed in section C, the ap- propriate values of 1 / T . _ and 1/T are added to 1/T_ + A(1/T.) to re- 2 HP T) K 2 cover 1/T. The values of 1/T - 1/T - 1 / T 0 1 _ are shown in 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. The solid lines are meant as visual aids. Several features of Figure 6 are discussed now. Consider f i r s t the data for p • 1, 3. The solid smooth line drawn 60 Figure 6 The reduced decay rate 1/T - 1/T^ ~ ̂ /T2HP ^ o r ^ a r m o n ^ c s * a n ^ 3 as a function of AT for 1 x 10~3 < AT < 5 x 10~2 K. l / T 2 H P is -2 -1 -1 3.93 x 10 s for harmonic 1, and 0.354 s for harmonic 3. The v i s - cous surface loss, 1/T , varies from 0.350 s - 1 at AT - 31.3 mK to 0.018 n s~* at AT = 1.02 mK for harmonic 1, and is just /T times greater for harmonic 3. The multiplicative factor, /3~, indicated in the figure I l l - ustrates the 0)^ frequency dependence of the surface losses, 1/T , 1/T_. Also indicated in the lower portion of the figure is the temperature dependence, relative to 1 mK, of the surface loss 1/T_. The crosses K represent the "observed" temperature dependence and are derived from the data for the f i r s t harmonic by removing the frequency dependence with the factor [6J^(AT)/to^(l mK)] . The solid circles represent the predicted behaviour as determined by the temperature dependence of the specific heat of helium. ! 1 1 • 1 " f AT (mK) 62 through the data points for p • 1 yields, after multiplication by /3~ as Indicated, the solid line through the data points for p * 3. Since the major bulk attenuation contribution has been removed using the HP values, i t i s expected that A ( 1 / T 2 ) i s zero or very small in the temperature region around AT ̂ 3 x 10 K. Thus, on the basis of the good agreement between the p • 3 data and that derived form the p = 1 data using the multiplic- ative factor of /3~, i t i s concluded that for the f i r s t and third harmonics, the contributions to 1/T from sources other than bulk damping are pro- h portional to w . -3 The data for the third harmonic, at temperatures such that 1 x 10 < _2 AT < 1 x 10 K, f a l l s below the line derived from the data for the f i r s t harmonic. Because the third harmonic i s nine times more sensitive to changes in D 2 than the f i r s t , these deviations indicate a negative value for A(1/T 2), or equivalently, the bulk damping is f a l l i n g below the minimum value observed by HP. Of course, the changes in D 2 are also present to a lesser extent in the data for the f i r s t harmonic and, therefore, the value of l / ^ 2 is determined by the solution, given in section C, to two sim- ultaneous equations. Finally, the temperature dependence of the surface losses i s indic- ated by the crosses (x) in the lower portion of Figure 6. The smooth li n e through the crosses i s a visual aid. This information i s obtained as follows. The data for p = 1 contains both a temperature and frequency de- pendence. The latter i s removed by dividing by the factor -3 ^ {o)1(AT)/6J1(10 K) ] where o^CAT) i s the frequency of the f i r s t harmonic at AT. In this way the surface losses are normalized to the measured value at 1 x 10 K, and any variations would reflect the temperature de- h pendence. Of course, this requires that the frequency dependence, co , i s 63 correct and that any significant corrections from A ( 1 / T 2 ) in the data for p « 1 are accounted for. The expected temperature dependence, as pre- dicted by equations ( 6 5 ) and ( 6 6 ) , due to the specific heat of helium i s indicated by the c i r c l e s . Variations in the properties of the re- flecting materials ( C , K ) have been neglected, and would only account for a small fraction of the difference between the observed and predicted fre- quency dependence. It is not clear what the source of the discrepancy i s . It may indicate that the theory for 1 / T _ i s incomplete and that some con-ic sideration should be given to the details of the interface between the he- lium and the solid. For example, the presence of a viscous boundary layer might supress the losses due to thermal conduction. Alternatively, the theory for 1 / T m a y D e lacking. If the "true" viscous contributions to 1 / T were 40% to 50% of the calculated values for 1 / T that have been used in obtaining the data in Figure 6, then the observed differences would re- sult. In any case, by measuring 1 / T for both the f i r s t and third harmonics at each temperature, knowledge of the temperature dependence is not re- quired to obtain D 2« In a fashion similar to Figure 6, the results for 1 / T - 1/ T2HP ~ ^ Tn, for harmonics one, two and four are shown in Figure 7. The dashed lines, derived from the f i r s t harmonic by multiplication 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, particularly for harmonic four, that there is some add- i t i o n a l loss mechanism present which i s not derivable from the f i r s t har- monic on the basis of an co proportionality. It i s noted, however, that -1 -2 the discrepancy of about 0.25 s for harmonic four at AT ^ 3 x 1 0 K is only about 10% of the total value for 1 / T . It i s apparent that the dis- crepancy diminishes as AT decreases and, for AT < 3 x 10 K, i t is v i r - tually insignificant. The results from three additional decay curves ob- Figure 7 Similar to Figure 6, this figure shows the reduced decay rate 1 / T - 1 / T - 1 /T_„_ for harmonics 1, 2 , A. The figure illustrates T\ 2. HP that there is an additional loss mechanism present for harmonics 2 and A. The additional loss decreases with decreasing AT. 65 66 tained for p - 2 , A at AT « 3.05 x 10" 4 K and p - 2 at AT - 2.57 x 10~ 3 K are consistent with the ultimate results for J>2 derived from the f i r s t and third harmonics. The source of the additional loss appearing in 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 simul- taneously as described in Chapter 2 , section C. Mechanisms which could re- sult in this additional loss by exciting Bessel modes are suggested in the general discussion of the f i n a l chapter. As a consequence of this discrep- ancy, only data for the f i r s t and third harmonics has been used in the f i n a l analysis for D^. The values of 1/T - 1 / T - 1 / T , _ - 1/T_ + A ( 1 / T . ) for the f i r s t and third harmonics over the entire temperature range covered in this exper- iment are shown in Figure 8. The c r i t i c a l damping i s evidenced by the increase in the value of A ( 1 / T 2 ) for the third harmonic as AT 0. For AT < 10 K, the error estimates reflect the severity of the extapola- tions and the extent to which the data was collected at any particular temperature. For large AT the error estimates are due to the fractional resolution (1 or 2 % ) in 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 rapidly. This i s a result of the severe extra- polations resulting from the amplitude effects described previously. As a matter of consistency with the data at larger values of AT, a best value and lower limit have been estimated at these three smallest values of AT; however, i t i s f e l t that the most significant information con- tained in these data points i s the upper limit that these place on the ultimate values for D^. 67 Figure 8 1/T - 1/T^ - l/T2Hp f° r harmonics 1 and 3 over the entire range of AT. Mote that the temperature axis i s logarithmic. The c r i t i c a l damping for AT < 10 results in the increasing separation of the data as AT 0. The inset illustrates the frequency and temperature dependent contributions to the p • 1 data relative to 1 mK. The dashed line i s derived by considering only the frequency dependence, while the solid line also includes the expected temperature dependence as determined by the specific heat, c, of helium. 68 69 The inset of Figure 8 il l u s t r a t e s the predicted and observed be- _3 haviour of 1/x for harmonic one and AT < 1.0 x 10 K. The solid circles ic are the values for 1/x - 1/x - l/x„„_ = l/x_ + A ( l / x 0 ) . The open circles are derived from the solid circles by subtracting the contribution A(l/x 2) using the data for the third harmonic and the expression given in the following section. Thus, the open circles represent l/x_. The solid line represents the predicted behaviour of 1/x , relative to the value at . 3 * AT = 1.0 x 10 K, accounting for both the frequency dependence, that i s , di , and the temperature dependence as determined by the specific heat, c, of helium. The dashed line results from considering only the frequency dependence. The temperature dependence of 1/x , as reflected by the de- K parture from the dashed curve, i s in qualitative agreement with the pre- dicted solid 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 sufficient to establish 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, in the analysis for D 2» a value for l/x_ appropriate to the f i r s t harmonic is determined by a con- ic tinuation of the open circles (or solid line) in the inset of Figure 8. In view of the relative uncertainty in 1/x for p = 3 at this temperature, and the small magnitude of the surface loss, this extrapolation does not introduce significant error. C. Results for the Damping Coefficient, The inverse decay time for the f i r s t harmonic is described by an ex- pression of the form 1 ^ 2 2 k 70 Similarly, the expression for the third harmonic is 1 2 2 U T " (V»2>«3 + 6̂ 3 (70) Vith to^ • 3cô , these equations may be solved to obtain ,(3) . , .(3) .(1) . i - (D2/u2)0)2 - ( i - / 3 " i ) [1 - (/3/9)] - 1 (3) vhich i s the desired expression for 1 / T 2 i° terms of the measured quantities 1 / T ^ and 1 / T ^ \ Using the dispersion relation u • u 2k and (3) the resonance condition k a • pTT, the expression relating D- and 1 / T 0 P 2 2 is 2 2 1 ( 3 ) D 9 - (a Z/9TT Z) i (71). ^ 2 (To apply directly to the "reduced" data which has been presented in Fig- (3) ures 6 and 8, the appropriate expression for 1 / T 2 i s .(3) .(3) (3) ,(3) .(3) (1) (1) I . 1 « A i « { i + 4 - / 3 ( i + A i )}[! - (/3/9)]-1 .) T2 T2HP T2 ? T2 T £ T2 The expressions (70), (71) are used to calculate from the measure- ments of 1 / T ^ and 1 / T ^ 3 \ The numerical values for D 2 as a function of temperature are found in Table A in Appendix A. The results for iog^o D2 are presented in Figure 9, including the theoretical predictions of HSH. The results are reproduced in Figure 10 which also includes the experimental results of HP7, Tyson*^, Tanaka and Ikushima*2, Ahlers* 3, and the theor- 14 15 e t i c a l predictions of HSH and DF . The results of this work are in good agreement with those of HP and Ahlers. In the c r i t i c a l region, which X _3 evidently extends to « t * (0.5 or 1) x 10 , there i s confirmation _3 of the renormalization group treatment of HSH. However, for t < 10 , the -3.0 - I (VI Q O O -3.2 -3.4 -3.6 -4 —r L0G,0(AT) I T -2 ! • THIS WORK — — HSH T \ -3 -2 LOG,0(t) Figure 9 The Results for the Second Sound Damping C o e f f i c i e n t 72 i 1 r cr> Q CM fO fO I I I ( 2a) 0 ,ooi Figure 10 Summary of Results for the Second Sound Damping Coefficient 73 results are not in agreement with the theory of DF, and Indicate values for which are les6 than their predictions. Considering the temperature de- pendence of l>2* the results are not sufficiently accurate to resolve any possible deviations from a single power law that might be interpreted in ef f terms of a temperature dependent ratio, (t) • D^/2\i^K% as predicted by DF. A more quantitative comparison of this experiment and theory i s ~ Yex achieved by describing the results in terms of the function D„ • D t 2 oex ' with D Q e x and y representing the experimental values for the amplitude and exponent for D^. Considering the evident coherence of the data in Figure 9, i t is tempting to use a least squares f i t to the above function. However, i t is f e l t that such a treatment, particularly with respect to the s t a t i s t i c a l estimate of a standard error, i s unrealistic, and possibly misleading, in view of the possible (systematic) errors as represented by the error bars in Figure 9. A r e a l i s t i c , although subjective, estimate for y i s ex Y « 0.31 ± 0.05 ex The subjective estimate of error, ± 0.05, is determined by evaluating, in Figure 9, the slopes of lines that are half way between the best f i t and the extreme limits compatible with the error bars. Thus, while the re- sults of this experiment do not provide a severe test of a detailed pre- diction for the c r i t i c a l temperature dependence, they do provide signif- cant support for the appropriate type of c r i t i c a l behaviour for D^ in the -3 region t < 10 . Since the amplitude, D , is sensitive to the value of y , the OCX best independent estimate for D i6 obtained by constraining y to the * oex theoretical prediction of y « 0.288. Then the value of I> o e x t -gain with 74 subjective consideration given to the possible systematic errors, is D = (3.7 ± 0.4) x 10"5 cm2 s" 1 oex Of more general theoretical significance i s the universal amplitude ratio, R 2 •? D 2/2u 2£. This may be evaluated using the expressions for £ and u 2 given at the end of Chapter 1 and the above value for D obtained for ° r oex y = 0.288, with the result ex R„ -= 0.11 ± 0.01 2 ex This experimental value l i e s between the theoretical values of HSH, 0.09 and 0.15, which are expected to be accurate to within a factor of two. ef f -4 The value for R2 at t = 10 as predicted by DF is-about 0.14. However, as they do not indicate the accuracy of their calculation, i t is d i f f i c u l t ef f to assess the significance of the difference between R 2 ex a n d R2 * 75 CHAPTER 5 CONCLUSIONS AND DISCUSSION Section A i s a statement of the major conclusions of this work con- cerning the c r i t i c a l behaviour of the second sound damping coefficient, D^. In section B there is a discussion of several general observations related to this experiment. A. Conclusions 4 The damping of second sound in superfluid He has been measured over -5 -2 the temperature interval 1.8x10 < t < 2.1 x 10 . In the c r i t i c a l re- gion the results, as illustrated in Figures 9 and 10, are in good agree- 14 ment with the i n i t i a l renormalization group treatment of D 2 hy HSH . How- ever, the results do not support a recent renormalization group analysis by DF*'"', the observed values for D 2 being less than the predicted values. The experimental result for 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 , D2 ^ U2^' w i t n a n e * P o n e n t o f 0.288. If the results are constrained to obey exactly the theoretical temperature dependence, then the experi- 0 288 mental value for the amplitude D defined by D_= D t is r oex 2 oex D = (3.7 ± 0.4) x 10 - 5 cm2 s" 1. oex The corresponding value for the universal amplitude ratio defined by D 2/2u 2? i s R 0 = 0.11 ± 0.01 2 ex In the common interval 10~A < t < 5 x 10~\ the results of this work 13 -3 are in good agreement with those of Ahlers . For t > 10 , i t is observed that D 2 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 critical damping in other systems. Tyson*^ used electrically thin resistive films to generate and detect 13 second sound in his attenuation measurements. Ahlers , however, used porous superleak transducers. 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, it appears that there is 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 is assumed to be independent of frequency, and its contribution to the resonance widths is determined from an extrapolation of the total widths to zero frequency. In contrast, on the basis of this work there is theoretical and experimental evidence for a frequency dependent reflection coefficient, although at sufficiently high frequencies the presence of a Kapitza res- 77 istance could result in the reflection coefficient becoming frequency i n - dependent. A second difference is that the input power densities used by Tyson are at least an order of magnitude greater than the power levels used in this experiment. Although Tyson extrapolates to xero power, this procedure might'introduce systematic errors that could account for the difference in results. The third difference is that in Tyson's experiment the absence of side walls in the resonator required a correction to the resonance widths involving diffraction loss. In the resonator of this ex- periment there are side walls present which eliminate diffraction loss, but introduce viscous and thermal conduction losses. Although the calculated diffraction loss in Tyson's experiment i s small or insignificant, early studies in this work on a cavity without side walls gave results that were d i f f i c u l t to interpret on the basis of diffraction from a plane-wave res- onance. Indeed, the frequencies of the major resonances did not correspond to a harmonic series co - _ for p = 0, 1, ... 5, but rather to a series p,0,0 with a Bessel mode character corresponding to co _ . or co . ft. At higher p, U, 1 p,i,u frequencies such that p > 10, the resonances did not display a single mode character, but contained several peaks resulting from overlapping modes. Subsequent studies with c variety of boundary conditions at the sides i n - dicated that the resonant structure was sensitive to the details of the 6ide boundary. As a result, the simplest, "ideal", side wall described in Chapter 2 was f i n a l l y used. Although care was taken to prevent the excitation of Bessel modes in the cavity, they were, nevertheless, excited. A general mechanism respon- sible for their excitation is suggested on the basis of the following ob- servation made in this experiment. It was found that Bessel modes in the v i c i n i t y of a plane wave harmonic were excited, while deep in the region between the plane wave harmonincs there were no Bessel modes v i s i b l e . This 7 8 indicates that the excitation proceeds by way of the plane wave resonance. Once the relatively large energy excursions in a resonance are established, a small perturbation at the walls of the cavity 16 capable of directing a significant portion of the energy into the excitation of another mode. As an example, the viscous loss occurring at the side walls of the cylindrical cavity could result in a temperature and velocity p r o f i l e in the radial direction which i s not f l a t , but instead, curved at the edges near the wall. This prof i l e could then establish an energy flow in the radial d i r - ection and excite a Bessel mode. This type of mechanism could account for the excess loss in harmonic four, where i t i s observed that the excess loss diminishes as the frequency and AT, and therefore the viscous loss, de- crease. Other factors that could excite Bessel modes by developing ang- ular and radial variations in the cavity include thermal conduction losses through the side walls, and the possibility of a variable reflection co- ef f i c i e n t , due to power dissipation, across the face of the bolometer. Progress has been made in understanding the loss mechanisms occurring at the walls of a resonant cavity, although there i s some d i f f i c u l t y in interpreting the observed temperature dependence. Systematic studies on several resonant cavities of different dimensions and materials would li k e l y solve this problem and, in addition, provide some information about the contribution of the Kapitza resistance to the reflection of second sound. While i t i s doubtful that the knowledge gained from such studies Is in i t s e l f worth the effort, the information would be useful in optim- izing the cavity geometry for further improvements in the measurement of the c r i t i c a l damping of second sound. Thus, for example, by using a longer cavity for a given radius, one should obtain more clean plane wave modes that are useable for attenuation studies. However, i f the overall resolution 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- sible to design the cavity to c r i t i c a l tolerances in the radius-to-length ratio, r/a, in order to avoid Bessel modes. While consideration must be given to the possible means of Improvement available through changes in the resonator geometry and materials, i t is also important to overcome the effects on attenuation due to second sound amplitude and bolometer power. Although small amplitudes and low power have been used in this experiment, these, nevertheless, have limited the ultimate accuracy at the smaller values of A T . To recover lower level signals, enhancement of the signal to noise ratio could be made with im- provements to the bolometer. Experience with several bolometer films suggests that the sensitivity can be increased by at least a factor of two. Also, increasing the bolometer resistance with a more intricate pattern design would increase the signal level and provide a better im- pedance match to the noise figure of the existing electronics, although some care must be taken with this procedure as i t ultimately reduces the active area of the bolometer. Another Immediate improvement which, un- fortunately, was not taken advantage of, involves using different elec- tronics in order to obtain a lower noise figure at the i n i t i a l stages of amplification. For example, with the bolometer resistance and frequencies 49 of this experiment, an appropriate input transformer and preamplifier would reduce the amplifier noise by about a factor of two. Although not- iceable, the extent to which this factor would be realized in the overall signal to noise ratio depends on the strength of other noise sources such as pickup in the leads and " i n t r i n s i c " bolometer noise. I n terms of future work, a more useful and flexible low noise input that also reduces 80 the effect of lead pickup would be a low temperature preamplifier. Then, achieving a f i n a l accuracy in on the level of a few percent, i t would be possible to provide detailed information of the temperature dependence of D^. It would also be worthwhile to perform the measure- ments at elevated pressures as a test of universality. 81 REFERENCES 1. H.E. Stanley, Introduction to Phase Transitions and C r i t i c a l Phen- omena (Oxford University Press, New York, 1971). 2. L.D. Landau and I.M. Khalatnikov, Dokl. Akad. Nauk SSSR 96, A69 (195A); reprinted in Collected Papers of L.D. Landau, edited by D. ter Raar (Gordon and Breach Science Publishers Ltd. 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. Hanson and J.R. Pellam, Phys. Rev. 95_, 321 (195A). 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. Lett. 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. Lett. A3_, 1A17 (1979). IA. P.C. Hohenberg, E.D. Siggia, 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. Lett. 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 quasi-particle theory i s reprinted at the back of this 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). 82 19. At AT - 2 x 10'* k, (c - c )/c » 3.6 x l ( f Z ; see ref erence 42 for a discussion of the significance of (c - c )/c . P v p 20. 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 is in 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. Sci. 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 in Solids, (Oxford University Press, 1959). 30. H.L. Caswell, Phys. Lett. 10, 44 (1964). 31. Measured with a Digital Thickness Monitor DTM - 200 manufactured by Sloan Technology Corporation, Santa Barbara, California. 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 Philbrick multiplier/divider model 4452. 36. Nicolet Instrument Corporation, Madison, Wisconsin; model 1170. 37. Electro Scientific Industries, Portland, Oregon; model DT 72 A. 38. P.A.R.C. model 114 with 185 option. 39. P.A.R.C. model 112. 83 40. The feedback c i r c u i t r y was of personal design and made with standard solid state operational amplifiers. 41. General purpose tunnel diode 1N3714. A description of this type of c i r c u i t can be found in C. Boghosian, H. Meyer, and J . E . Rives, Phys. Rev. 146, 110 (1966). 42. G. Ahlers in 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 Thesis, U.B.C. (1976); see also reference 8. 44. The thermometer was calibrated 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 (1976). 46. I.M. Khalatnikov, Usp. F i z . Nauk. £0, 69 (1956). (English trans- lation available in Univ. of California Radiation Lab. Transl. 675; also, Hydrodynamics of Helium II with U.B.C. library l i s t i n g QD 181 H4 K5 1956). 47. G.K. 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APPENDIX A TABLE A Results f o r Ts^ -4 2 -1 AT(K) t = AT/T^ D 2 (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 1.03 X i c " 2 4.74 X i o " 3 3.31 + 0.4 5.17 X IQ" 3 2.38 X i o " 3 3.09 + 0.6 2.57 X i o - 3 1.18 X i o " 3 2.72 + 0.6 1.02 X ID" 3 4.70 X i o " 4 3.26 + 0.6 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 4.19 X IO" 5 6.86 + 1.0 5.95 X i o " 5 2.74 X IO" 5 7.92 + 1.2 4.00 X i o " 5 1.84 X i o " 5 9.14 + 1.8
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