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### The velocity of second sound near the Lambda point Johnson, David Lawrence 1969

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`cS\oo 1 THE VELOCITY OP SECOND SOUND NEAR THE LAMBDA POINT by DAVID LAWHENCE JOHNSON B.Sc, The University of B r i t i s h Columbia, 1963 M.So., The University of B r i t i s h Columbia, 1967 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of Physics We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1969 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the r e q u i r e m e n t s f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and Study. I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u rposes may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f Physics  The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada Date May 15, 1969 11 DEDICATED to my wife ELIZABETH whose contribution to education was twofold and without whose help this work would have been impossible. i i i ABSTRACT Direct measurements have been made of the v e l o c i t y of second sound i n l i q u i d helium over the temperature range T A-T from 1.3 x 10"2 K to 5 i 10"6 K. Using previously determined relationships f o r the s p e c i f i c heat, superf l u l d density, and thermal conductivity near the lambda point, consistency has been demonstrated between the measurements, v e l o c i t i e s predicted by superfluld hydrodynamics, and c e r t a i n s c a l i n g law predictions. If TABLE OP CONTENTS Chapter I Introduction and Theory Introduction Experiment The Two Pluld Model C r i t i c a l Point Arguments II Apparatus Cryogenic Apparatus Electronics Second Sound Cavity III Experimental Procedure Second Sound Trials Phase Shift and Dispersion T r i a l IV Data Reduction Extraction of Data from the Charts Analysis of Probable Errors V Results and Analysis Results Comparison with the Hydrodynamic Theory Comparison with the Scaling Prediction VI Conclusions and Discussion Conclusions Discussion Glossary of Symbols References and Footnotes Appendix 1 LIST OP TABLES Table I Physical Quantities near T c v i LIST OF FIGUBES Figure Page 1 ( k, I" 1) Plane 15 2 Cryogenic Apparatus 20 3 Block Diagram of Electronios 22 4 Bolometer Electrode Pattern 25 5 T y p i c a l Recorder Trace 29 6 Experimental Results 39 7 Deviation Plot 42 ACKNOWLEDGEMENTS Dr. M.J.Crooks gave many hours of his time for thought-provoking discussions and helpful supervision of my work, for which I thank him heartily. I am indebted also to Mr. G.Brooks and Mr. R.Weisbach for their technical assistance, to Mr. C.R.Brown for discussions about wave propagation, and to Senor J.BeJar for assistance with the computer programming. Dr. G.Ahlers very kindly gave his thermal cond-uctivity data and provided some insights into the problem of c r i t i c a l phenomena. 1 CHAPTER I. INTRODUCTION AND THEORY (1) INTRODUCTION In the two decades preceding this one, a slowly Increasing collection of experimental data began to indicate remarkable similarities in physical behaviour near the c r i t i c a l points of some otherwise very dissimilar systems. It was realized that transitions such ass - the superconducting-normal metal transition, - the liquid-vapour c r i t i c a l transition of a pure f l u i d , - the order-disorder transition of some metallic binary alloys, - the misolble-immisoible c r i t l o a l point of some binary liquid mixtures, - the ferromagnetic Curie point and the antiferromagnetic Neel point, and ' - the liquid helium lambda point, a l l behaved in a qualitatively similar way. This realization, coupled with a rapidly improving experimental technology, led to a recent stimulation of interest In the theoretical and experimental investigation of c r i t i c a l phenomena. Classical c r i t i c a l point theories such as the Landau theory of the second-order phase transition, the van der Waals» equation for a liquid, and the Weiss molecular f i e l d theory for a ferromagnet, were found both theor-et i c a l l y and experimentally* to be unsatisfactory very 2 close to the c r i t i c a l temperature • It has been known for many years that the temperature dependence of a physical quantity near a c r i t i c a l point may often be described by (T-f c ) f i where T is the absolute temperature, T 0 i s the c r i t i c a l temperature, and e i s a constant " c r i t i c a l e x p o n e n t P a r t of the recent theoretical work has been the development of "scaling laws" which attempt to relate to each other the c r i t i c a l exponents or temperature dependences of different physical quantities. One such theory; applied to liquid helium, predicts a relationship between the velocity of second sound below the lambda point and the thermal d i f f u s i v i t y of the liquid above the lambda point. Classical hydrodynamic arguments, based on the two-fluid model for liquid helium, lead to an expression for the velooity of second sound ln terms of the specific heat of the liquid, and the density of i t s superfluld component* Experimental measurements of this specific heat and superfluld density had been obtained over a temperature range much closer to the lambda point than the range covered by the existing direct measurements of the second sound velocity. The above considerations a l l Indicated i t would be interesting to measure the magnitude and temperature dependence of the velocity of second sound as close to the lambda point as possible. 3 (2) EXPERIMENT The experiment we performed b a s i c a l l y consisted of the measurement of second sound resonances, or thermal standing waves, l n a p a r a l l e l plate resonator (called the cavity) Immersed l n and containing helium I I . The use of resonant a m p l i f i c a t i o n to detect and measure second sound i s well known. In general, the conditions f o r resonance l n a cavity depend on the cavity geometry, the frequency of the waves gener-ated i n the cavity, and t h e i r v e l o c i t y i n the medium contained within the c a v i t y . The usual technique f o r second sound measurements has been to f i x the tempera-ture (and therefore the second sound v e l o c i t y ) and vary the frequency to search f o r the d i f f e r e n t resonance modes. In t h i s experiment, a b a s i c a l l y d i f f e r e n t ap-proach was used i n that the frequency was f i x e d and the temperature was not. The temperature i n t h i s experiment was allowed to d r i f t slowly i n time. As a consequence, the second sound v e l o c i t y i n the l i q u i d within the cavity was a slow function of time. Resonances were therefore ob-served with the frequency and resonator geometry fixe d , separated i n time due to the time dependence of the second sound v e l o c i t y l n the l i q u i d helium. The exper-imental d i f f i c u l t i e s associated with the precise control of the l i q u i d temperature were thus eliminated, or more accurately, were reduced to the problems of preelse measurement of temperature. It was f e l t that with t h i s technique observations of the second sound v e l o c i t y could be made much clo s e r to the lambda point than 1-1 had been done i n previous experiments. J (3) THE TWO FLUID MODEL Many of the properties of l i q u i d helium below the lambda point may be described i n terms of a model which assumes two interpenetrating f l u i d s — the normal f l u i d ( mass Mn, density f>n ) and the superfluld ( mass Ms, densltyyOg), which may move Independently of each other, and which together constitute helium I I , ( mass M = ^ n ^ g (a) The Hydrodynamlc Equations. A system of hydro-dynamic equations f o r helium II may be deduced from the above m o d e l , a n d three assumptions} ( i ) that the su p e r f l u l d f r a c t i o n has zero entropy; ( i i ) that below some c r i t i c a l v e l o c i t y the motion of the superfluld i s l r r o t a t i o n a l ; and ( i i i ) that a l l the conservation laws are v a l i d . Assumption ( l i ) i s written where v. i s the super f l u l d v e l o c i t y . We define j , the n c u r l v s «= 0 , 5 momentum/unit votume of the f l u i d . For small superfluld and normal f l u i d v e l o c i t i e s • and v Q , J can be expanded i n powers of the v e l o c i t i e s and to a f i r s t approximation (neglecting terms i n v 2 ) ~f _ Ar + Aj . J As s y n n The conservation of mass equation i s ^/>/^t + d i v ~ J " * 0 . ....(2) The f i r s t equation of motion, using the conservation o f momentum law, i s ^ / f r t +^ l k/^x 1_ « 0 , ...(3) l.k - 1,2.3, where the summation i s ca r r i e d out over the twice repeated subscripts, and where Xj_ are the Cartesian coordinates. T / ^ j . i s the momentum f l u x tensor which f o r small v e l o c i t i e s (neglecting viscous e f f e c t s ) i s 7£j_ « p£(l,l-) + y ° n v n i v n k +y^s^slVgi-, where p Is the pressure, and £(i,-r) i s the Kronecker delta function. Again assuming no d l s s i p a t i v e processes we may-write the conservation of entropy equation ^(/>S)/d t + div "F = 0 , where F i s the entropy f l u x vector and S i s the entropy/ unit mass o f f l u i d . As entropy i s ca r r i e d only by the normal f l u i d , F =y° sv n & n d d(/>S)/at + div (/>Svn) =0. ...(*0 The i n t e r n a l energy o f an incremental mass o f l i q u i d helium i s given by 6 dU - TdS - pdV + GdM ...(5) where V i s the specific volume and G is the Gibbs free energy per unit mass. Let the mass of this bit of helium now be increased at constant volume (hence dV = 0 ) by the addition of particles which carry no entropy (hence dS «B 0), that i s by the addition of superfluld. Then Eq.(5) becomes dU *= GdM and we see that the potential energy per unit mass of the superfluld must be G, and that the acceleration of the superfluld must be dv s/dt * ^ v 8 / d t + (v 8*grad)v g = -gradG. ...(6) The total Gibbs energy NG of the f l u i d i s the sum of the Gibbs energy MG0(p,T) of the stationary f l u i d , and the kinetic energy of the relative motion of the normal and superfluld parts. MG - MG0 + (Pn/ZMjj) , ...(7) where P n « ^ n^n'^s^ i s * n e momentum of the normal f l u i d with respect to the superfluld. Recalling that the masses and densities of the f l u i d fractions are related by Mn-Cn/*)!* c (/n//n+/s)M» w e n o w differentiate Eq.(7) with respect to M, obtaining *MG/dM « G Q - (/^n/Z/^M2) - G Substituting P n ^ ^ C v ^ - T g ) ^ we have and substituting this into Eq.(6) gives dv 8/ot + (v s«grad)Vg = -grad[G 0- (/JQ/2y)) ( ) 2 ] . ...(8) The term (Vg'gpadJVg may be rewritten ( v 8 • grad )"ra - £gradv 8 2 - j ^ x ( c u r l V g Q . . . . (9) Substitution of Eq.(9) and Eq.(l) into Eq.(8) gives + g r a d { G 0 + (Vg2/2) - (/n/2/>) ( v n - T s ) 2 } = 0. .(10) Equations (2), (3), (4), and (10) constitute the system of hydrodynamic equations f o r helium I I . (b) L i n e a r i z a t i o n of the Hydrodynamic Equations, In deriving the v e l o c i t i e s of low amplitude sound; we may assume that v^ and "v^ are small and thaty°, S;- p and T exhibit only small fluctuations from t h e i r equilibrium values JQ* S O » p o a n d To» 1 1 1 6 -Dove equations may then be s i m p l i f i e d . Neglecting terms In "v 2 i n Eqs.(3) and (10), and taxingyOS out of the divergence i n Eq.(4) we obtain the following system of l i n e a r i z e d hydrodynamic equations f o r helium I I : (b/>/bt) + d i v f - 0; ...(2) (dj/dt) + grad p * 0; ...(11) (dy>S)/dt) +/>Sdiv-vn =0; ...(12) (dv g /a t ) + grad G Q - 0. ...(13) (c) Calculation of Sound V e l o c i t i e s . We now make use of Eqs.(2) to (13) to calculate the v e l o c i t i e s of wave propagation. D i f f e r e n t i a t i n g Eq.(2) with respect to time and sub s t i t u t i n g E q . ( l l ) we obtain ^^)/91 2 = div grad p. . . . ( H O o 8 The identity a s / a t s (l£>)0(/*S)/Zt) - (s^nayat) on substitution of Eqs.(2) and (12) gives dS/dt « (S y^^c9)div(v 8-"? n). ...(15) The Gibbs free energy Is G Q « U + pV - TS whence dG Q m -SdT + Vdp * -SdT + (l^o)dp. Rearranging this and taking the gradient we obtain grad p =yOSgradT +^agradG0. Substituting grad p from Eq.(ll) and gradG0 from Eq . (13 ) gives y ^ t v ^ - V g J / d t ) +/>SgradT « 0. ...(16) Differentiating Eq . ( 15 ) with respect to time and substituting Eq.(16) we find <^2S/dt2 «  s2^/B//^ d * v T« ...(17) Equations (14) and (17) govern the propagation of waves in helium II. Our previous assumption of small variations in the thermodynamic observables may be written explicitly S°=/o + / V ( t ) » s = s 0 + V * * * P e P 0 + p v ( t ) a n d T as T Q + T v(t) wherey^, S o, p^, and T Q are independent of time. Under these conditions we may further write P v = Ov/^/>)sfy. + (^p/^Sj0Sv, and T v = a/>r + (^ T^ s> s v-Equations (14) and (17) are then rewritten c> 2/ v/£t 2 «= Op/^/))SVZ/>T + Qp/Zs^S^ and ...(18) ^ % / d t 2 = (/>s^°n)S2{OT/^)s'72>0v + O T / a s j ^ S ^ . (19) 2 where V is the Laplacian operator. 9 We now wish to know i f i t Is possible f o r disturbances i n the thermodynamic var i a b l e s , of frequency co to pro-pagate as wares i n the l i q u i d with v e l o c i t y u. To determine t h i s we now seek simultaneous solutions of Eqs. (18) and (19) of the form f = +/)• erp[lco(t-x/u)] and S « S Q + S»exp[i<_>(t-x/u)] . Substitution of these into the above equations leads to (dT/d^gteS/fcTjayO* - {(u/u 2) 2-l } s « =0, ...(20) -[(u/u 1) 2-l ) r >0» + <dp/dS^ (^-/_p)sS» =0, ...(21) where u| m Op/3/>)3, and The condition f o r the simultaneous s o l u b i l i t y of Eqs. (20) and (21), that the determinant of the c o e f f i c i e n t s be zero, gives {(u/u 1) 2-l l|{(u/u 2) 2-l} = QT/d/>)sOs/m/Q^p/dS)/)(^/dp)s - ( C P - C v ) / C p • where C p and C_ are the constant pressure and constant volume s p e c i f i c heats of the l i q u i d . Using the fact that Cp -BS. C^, we now set (Cp-C_.)/Cp « 0. With t h i s approximation we then obtain the two solutions u - u x - [Qp/<^) s l * , . . . ( 2 2 ) u « u 2 - [TS 2/> s// nC p:i*. ... ( 2 3 ) If u « U j , Eq.(21) shows that S»-»0, and that to the f i r s t order In which we are working the entropy fluctuations vanish. This mode Is a t r a v e l l i n g wave of density fluctuations under adiabatic conditions, I.e. ordinary 10 or f i r s t sound. If u • Ugt Eq.(20) shows that the density fluctuations vanish. This i s a t r a v e l l i n g wave of entropy f l u c t u a t i o n s , and therefore temperature fluctuations, at oonstant density. This i s the mode c a l l e d second sound. (4) CRITICAL POINT ARGUMENTS (a) General Discussion. Recent investigations? of c r i t i c a l point phenomena have l e d to the r e a l i z a t i o n that not only do c e r t a i n p h y s i c a l properties exhibit rather simple behaviours i n the v i c i n i t y of a c r i t i c a l point, but that t h e i r behaviours are very s i m i l a r near the c r i t i c a l points of apparently very d i f f e r e n t phase t r a n s i t i o n s . A r i s i n g from such discoveries i s the idea that each phase t r a n s i t i o n i s describable i n terms of an order parameter p. ( P a r t i c u l a r examples are the magnetization of a ferromagnetic material, the condensate wave function of a supe r f l u l d , and the concentration i n a binary l i q u i d system). This parameter i s a measure of the ordering present i n the system, and near the c r i t i c a l point i t may undergo large fluctuations with small change l n the free energy of the system. The spe c i a l behaviours of the other physical quantities i n the c r i t i c a l region are thought to be related to t h i s large s u s c e p t i b i l i t y to fl u c t u a t i o n s . To describe these fluc t u a t i o n s , a c o r r e l a t i o n function of the general form 11 C?(r) -<[?(r\) - <J(?i)>3[;(r2) - <?(r 2)>]> . r - r t - t2 i s used, relating the deviations of the order parameter at point r^ from i t s expectation value there to the deviations at point r^. In general, this correlation function i s a monotonically decreasing function of 1% As such, c r i t e r i a may be set up to define a particular value of |"r| called the correlation length, or range £ of the correlation function. This correlation length ^ is generally a function of temperature, and divergent near the c r i t i c a l point. As noted above, many quantities depend on £ as e — where e i s some exponent (called the o r l t i o a l exponent) which i s fixed i f the sign of 6 i s fixed, and € » (T-T 0)/T c. Table I l i s t s some of the physical quantities pertinent to this experiment, their expected behaviours, and the conventional notation for their c r i t i c a l exponents. Recently l t has been suggested®"12 that the c r i t i c a l indices are not independent of each other. Through the use of pl a u s i b i l i t y arguments and assumptions about the functional forms of thermodynamic observables in the c r i t i c a l region, certain relationships called scaling laws have been proposed between various of the c r i t i c a l exponents. Some of these relationships are subject to experimental verification. It should be noted that none of the scaling laws predict the magnitude of any c r i t i c a l exponent. What they do predict i s the relationship 12 TABLE I Physical Quantities near T Physical C r i t i c a l Expected Quantity Exponent e Behaviour Order parameter fi. <*P> - o, o o • P < P > ~ ilel^.^o Correlation function V C - l r ^ l - ^ 2 - 7 d(r) for a d dimensional —> > —-. r = r x - r 2 system Correlation length. V J ~ € ,e >o or range of C. f V' f ~ l € r v ' , e<o Superfluid density y O g - l e f ^ . e<o >*• 13 of one c r i t i c a l exponent to another, usually ln the form of a sum involving two or more c r i t i c a l exponents* Correlation functions may be defined also for operators or physical quantities a other than the order parameter* Halperin and Hohenberg11 point out that scaling arguments may then be subdivided into static or dynamic scaling depending on whether one assumes time independence or dependence of the correlation function, (^(r) or C^(r,t)), and dynamic scaling into restricted or extended dynamic scaling depending on whether or not only the order parameter correlation function i s expeeted to obey the dynamic scaling laws. (b) A Specific Prediction concerning Seeond Sound. In a recent paper 1 1, Halperin and Hohenberg propose a dynamic scaling hypothesis which leads to a specific prediction that may be tested ln this experiment. They note that a dynamic correlation function C*(r,t) for some operator a, may be Fourier transformed to c|(k,6>) and may i n general be written in the form C*(k,u) « Z T r - c J ^ - f ^ / ^ C k ) ) * ^ * ^ ) ] - 1 , where +00 f f (x)dx «= 1 , loo * _ and where the characteristic frequency u> (k) i s determined by the constraint +1 / f(x)dx = I . '-1 A mental picture of the scaling hypothesis may Ik be obtained as follows. In Pig. 1 we represent the (k,jJ~*) plane* (where k i s the wave number and | i s the inverse c o r r e l a t i o n length). Three regions may be defined i n t h i s planet - Region I. k £ « l , T<T C. The macroscopic c r i t i c a l region, or hydrodynamic c r i t i c a l region, below the c r i t i c a l temperature. In t h i s region phenomena occur over distances r large compared with Hydrodynamic arguments are expected to be v a l i d i n t h i s and region I I I . - Region I I . k £ » l . The mlcroscopio c r i t i c a l region i n which phenomena occur over distances r small compared with |. - Region I I I . k | « l , T>T e # The macroscopic c r i t i c a l region above the c r i t i c a l temperature. r The c r i t i c a l point i s the l i n e = 0. The dynamic s c a l i n g hypothesis has as i t s basic assumption that c£ ( r e s t r i c t e d s c a l i n g ) , or Cf (extended scaling) vary smoothly i n the (k,^" 1) plane except at the o r i g i n , and that e i t h e r of these c o r r e l a t i o n functions i s e s s e n t i a l l y s p e c i f i e d by i t s l i m i t i n g behaviour i n regions I, I I , and I I I . That i s to say, I f the asymptotic  forms of the function In regions I (or III) and II are separately extrapolated to the l i n e Lj (or L^) i n F i g . 1, defined by k£«l, T<T (or T>T ) the two r e s u l t i n g expressions J O c must agree to within a f a c t o r of order unity. The asymptotic forms C* 1, C* 1 1- C * 1 1 1 are defined as 1 5 FIGURE 1 . The ( k, p 1 ) Plane 16 the limiting forms of Cj(k) under the conditions: C*1 : J fixed, k-* 0, T<TC ; caIII .^ f l x e d > w 0 f T > T o . The scaling hypothesis then takes the form of the matching conditions C* 1^) =oCC a I I(k), for k = f _ 1 C * m ( k ) -oc'c* 1 1^), for k - p 1 . whence C a I I I ( k ) * AC^^k), f o r k ^ p 1 ...(24) where oc,oc», A are constants of order unity. Halperin and Hohenberg also propose that similar expressions may be assumed for the characteristic frequencies o a I ( k ) , u> a I I(k), c j a I I I ( k ) in the three regions. We now consider a particular c r i t i c a l transition, the lambda transition of l i q u i d helium. The c r i t i c a l temperature T Q for this transition i s the lambda temperature T A = 2.172 K, and C «= (T-TjJ/T*. The specific prediction we w i l l test concerning * l i q u i d helium evolves from using as operator a the heat operator q(r,t) =E(r,t) + (<E^>;/)(r".t) )/</>> where E(r,t) i s the energy density of the l i q u i d . Halperin and Hohenberg11 assume that the heat operator correlation function c|(k,<o) shall he dominated in region I (£<0) by second sound, and in region III (£>0) by thermal diffusion, and that the asymptotic forms of the 17 characteristic frequencies in these two hydrodynamic regions may then be derived and are given by ^T(t) «= u 2k . (6<0)r ...(25) coqHI(k) = D tk 2 , (€>0), ...(26) where D t = fc^0Cp is the thermal d i f f u s i v i t y of the liquid, and A i s the thermal conductivity. The extended dynamic soaling prediction i s then, using Eqs. (24), (25), and (26), the matching condition for k «=j*~1 (see lines k£ « 1 in Pig. 1) that D tJ" 2(€>0) - Au^" 1(fe<0). ...(27) The two temperature dependent correlation lengths J may be written (see Table I) | ( 0 = ^Jt"u and f (-6) _ ^*^f—-e-|"">'* where £ Q are constants. Equation (27) i s then rewritten u 2(-6) * {K[e:)//>^))'l^l(^)"iyxezv\-e\^\ ...(28) Now making use of a static scaling law 1 2 and the static scaling assertion 1 3 thatyOa i s proportional to J* ~ 1 we see that /.~l-€|£~f-l~ l-e|v' . and therefore f = y* = y . Experimental measurements1^"*1^ have shown £ «= 0.666 * .006 = 2/3. Thus we obtain for the exponent of € in Eq.(28) 2V - y» = 2/3 18 and Eq.(28) becomes U 2(-€) = ( ^ ( 6 ) ^ C p ( 6 ) ) . [ A ^ ( ^ ) - l ] - l t 2 / 3 # This r e s u l t was f i r s t derived by Ahlers. 19 CHAPTER II. APPARATUS (1) CRYOGENIC APPARATUS The cryogenic apparatus; shown l n Pig* 2, consisted of a large outer bath for environmental st a b i l i t y , and a small inner bath containing the seoond sound cavity. The 3 l i t e r outer bath could be temperature regulated to a short term accuracy of about 10 K using a Walker diaphragm regulator* 1 8 The 0*33 l i t e r inner bath was separated from the outer bath on the sides and bottom by a vacuum jacket, and on the top by a 3/8" stainless steel plate. This moderate thermal link gave the inner bath a thermal response time with respect to the outer bath of about one hour* The inner bath was f i l l e d from the outer bath through a porous stainless steel f i l t e r (to prevent solid nitrogen particles from entering) and a small stainless steel needle valve* The inner bath could be pumped via a pumping line (containing also the e l e c t r i c a l leads) terminating i n a stainless steel plug bored with a 1 mm diameter hole. Elec t r i c a l leads were brought through the steel plug by sealing varnish insulated copper wires (AWG #37) into short lengths of stainless steel capillary (1/16" outer diameter, 0.007" inner diameter) using A r a l d i t e 1 ^ epoxy. The wires were thus e l e c t r i c a l l y insulated from the steel but the holes i n the capillaries were sealed. These cased 20 VALVE STEM PUMPING LINE FILTER s s \ \ INNER BATH VACUUM SPACE OUTER BATH PIGUBE 2. Cryogenic Apparatus. A = second sound gen-erator, B = resistance thermometer, C = bolometer, D = Perspex spacers. 21 wires were sealed with epoxj into 1/16" holes through the plug. Film flow heat transfer between the two baths occurred only through the 1 mm hole; and was therefore small. The second sound cavity, near the bottom of the Inner bath, consisted of a 2.5 cm square bolometer separated by 2 mm Perspex spacers from a 2.5 cm square second sound generator (heater). The cavity was oriented horizontally (i.e. second sound propagated vertically) to reduce the gravitationally induoed temperature range of the lambda t r a n s i t i o n 2 0 to 2.5 x 10"?K. The cavity was open on two sides. The thermometer, in the center of the cavity/ was a carbon resistor. (2) ELECTRONICS A block diagram of the electronics i s shown in Fig. 3. An osc i l l a t o r supplied a signal of frequency f/2 to the generator, at which the Joule heating pro-duced a second sound plane wave of frequency f (typically 200Hz to 5KHz) i n the l i q u i d . The same f/2 signal, supplied to a frequency doubler, emerged as a signal necessarily coherent with the second sound, and was used as the reference signal for coherent amplifier x A (Fig. 3) . Thermally induced resistance changes i n the bolometer," biased with a constant DC current (rang-ing from 10 to 30/tA)/appeared as voltage changes a-cross the bolometer load resistor which were amplified 2 2 22 SIGNAL OSCILLATOR PHASE AND TEMPERATURE CHART RECORDERS FREQUENCY DOUBLER FREQUENCY COUNTER I COHERENT AMPLIFIER A P R E - A M P I BOLOMETER BIAS. LOAD I COHERENT AMPLIFIER B I P R E - A M P I RESISTANCE BRIDGE HE IE GENERATOR BOLOMETER THERMOMETER FIGURE 3. Block Diagram of El e c t r o n i c s . 23 and fed to coherent amplifier A. The output from co-herent amplifier A was fed to one channel of a two channel chart recorder. 23 This "second sound trace" was proportional to the produot of the second sound amplitude in the cavity and the cosine of the phase of the received second sound with respect to the reference signal. Coherent amplifier 2^ B generated i t s own reference signal which was also fed to a Wheatstone bridge, one arm of which was the resistance thermometer. The bridge unbalance signal was amplified 2-* and fed to coherent amplifier B. the output of which went to the other channel of the chart recorder. This "temperature trace" was proportional to the resistance difference between the thermometer and the preset value of a precision resistance decade. (3) SECOND SOUND CAVITY (a) Bolometer. Some desirable properties of a bolo-meter for second sound detection are: (1) high thermal sensitivity (1/H)(dB/dT); ( i i ) low electrical resistance B to minimize the problems of impedance matching to transmission cables leading out of the cryostat; ( i l l ) small heat capacity to enable i t to respond to very rapid temperature fluctuations; and (iv) large active area. 24 The carbon film bolometers constructed by Cannon and Chester 2^ exhibit properties (1) and ( i l l ) . Sig-nificant improvement over their design was achieved using phot©fabrication techniques rather than thin film evaporation techniques to prepare the bolometer elect-rodes • The electrode pattern Pig. > was photographically reduoed onto Kodak Ortho Type III film. The scale in Pig. 4 represents the f i n a l size of the bolometer. A two inch square of 1/16" thiok fiberglass was bonded with Ara l d l t e 1 ^ epoxy to a 0.001" thick layer of brass. After cleaning with organio solvents i n an ultrasonic cleaner, the brass was spray ooated with a thin (less than 10~* inch) layer of Kodak KPH photoresist. 2? The photographio negative was contact printed onto the ooated brass using ultraviolet light, and the resulting latent image developed. After development/ the photoresist has the property that a l l parts which were exposed to the light become Insoluble l n most acids/ while the unexposed portions dissolve i n the developer. The plate was then etohed to remove a l l the brass which was unprotected by developed photoresist. The remain-ing photoresist was then removed, leaving the pattern of Pig. 4 i n 0.001" brass bonded to the fiberglass substrate. 25 26 The bolometer was completed by spraying the pre-pared electrodes with a suspension of nominally 16 milli-mioron carbon p a r t i c l e s 2 8 i n xylene. The xylene evaporated leaving an extremely thin layer of semicon-ducting carbon covering the electrodes. Reference to Fig. 4 w i l l show that the bolometer therefore consisted of 64 thin film carbon resistors wired in parall e l , each approximately 2*5 om wide and 0.02 cm long. Con-sidering the film to be a homogeneous slab of bulk graphite; and using Ohm's law and the measured room temperature resistances* a thickness of 0.1 milli-microns i s indicated for the film. As the carbon granules had diameters of the order of 16 milll-mlcrons/ one can conolude that the film was microscopically inhomogeneous and roughly the thickness of one carbon granule. This bolometer design achieved the desired properties of low thermal capacity without excessive electrical re-sistance. The bolometer resistance and sensitivity while operating at the lambda point were ~70KXL and ~3.4 K"1 respectively; giving the second sound detection system a maximum sensitivity of 3 x 10*8K rms/chart inch. The maximum temperature wave amplitude observed i n the experiment was 10"*?K rms. Second sound noise (real and 27 apparent) was less than 3 x 10"^ K n s , The detection system bandwidth was typically 0•25 He• (b) Thermometer- An Allen-Bradley 2^ 1/10 watt carbon resistor; nominally 33 ohms at room temperature? was used as the thermometer. Its resistance and sensitivity at the lambda point were ~ 120011 and (1/B)(dB/dT)A ~ 1.2 K*"^  giving the thermometry system a maximum sens-i t i v i t y of ~10"°" K/chart inch. Low frequency thermal noise (real and apparent) was about 10~^ K peak to peak. The thermal response time of the thermometer was meas-ured i n l i q u i d helium and found to be 22 msec at 4.2K and 5 mseo at 2•OK. (o) Second Sound Generator. The second sound gener-ator was a pleoe of commercial carbon resistance board^ 0 nominally 25 ohms/square at room temperature. Its resistance at liquid helium temperatures was about 50 ohms/square. 28 (1) SECOND SOUND TRIALS Data were acquired l n the following manner. (Re-ference w i l l be made to Pig. 5, a typical section of the output from the two channel chart recorder). (1) With the inner-outer bath valve closed and the outer bath stabilized at a temperature a few m l l l l -degrees below T A, the inner bath temperature was held steady at AT ~ 10" 2 to 5 x IO-^K, where AT « T A -T. The frequeney and coherent amplifier pass-band were set to optimize seoond sound detection, and the sens-i t i v i t y (ohms/chart division) of the thermometry system was measured by making discrete changes l n the zero setting of the Wheatstone bridge and recording i t s out-put. ( i i ) With both recording systems and the second sound generator and bolometer operating; pumping of the inner bath was terminated or reduced allowing the tem-perature to climb slowly up to and through the lambda point, (see Pig. 5 lines #2 and #3), the Inner bath dT/dt being typically of the order of 10" 7 K/sec. The second sound velocity and the temperature were thus functions of time. When velocities occurred such that; 29 FIGURE 5. Typical Recorder Trace 30 for the particular frequency i n use; the cavity was resonant, peaks were recorded on the second sound trace, (trace #2 In Pig. 5). ( I l l ) The Inner bath temperature was Immediately brought slightly below the lambda point, the seoond sound generator and bolometer were turned off, and the temperature was again allowed to rise through the lambda temperature (see traces #1 and #4 in Pig. 5) to establish Rpt, the thermometer resistance at the lambda point. Step ( i l l ) was necessary because the accurate identification of the lambda point i s dependent on know-ing the total power dissipation (P) In the cavity. 1^ The combined power input to the cavity from the second sound generator; the bolometer, and the thermometer -varied between 25 and 300 microwatts. The thermometer power was 0.1 to 1.0 microwatts. The generator power was always twice the bolometer power so that the DC power inputs (as opposed to the power input at the second sound frequency) of the two were equal, and so that the maximum second sound output signal for a given total power input was obtained. With only the thermometer on, the warming curve (the "temperature trace") showed a zero slope region, or B x plateau (traces #1 and #4 in Pig. 5). With the thermometer, generator, and bol-ometer on, the warming curve broke (point A Pig. 5 ) 31 at a temperature AT*(P) slightly below the true lambda temperature and took up a much higher slope dfi/dt. AT'(P) was measured and found to be approximately proportional to the total power input P to the cavity. It was assumed that the plateau temperature was the true, or zero power; lambda temperature. If, however, the power dependent s h i f t i n the apparent lambda temperature was s t i l l i n effect during measurements made with the thermometer only, i t was i=£T«(10" 6 watt) = 3 x 10" 8 K. 16,20 In previous work, a time dependent shift of the lambda point resistance R A has been observed. The ef-fect i s apparent in the non-zero slope of the RA plat-eaux and of the dashed line l n Fig. 5* The thermometer was calibrated by opening f u l l y the valve between the two baths and measuring thermometer resistance and outer bath vapour pressure for a number of temperatures sl i g h t l y below T A. The data were fi t t e d to the expression log R = A + B/T which was taken to be exact over the c r i t i c a l region. Data were extracted from the recorder traces by noting the thermometer resistance R(t) at the time of occurrence of a peak i n the second sound trace corres-ponding to resonance mode n. The experimental second sound velocity was then u 2 e = 2fd/n, ...(30) 32 (where the subscript e denotes the experimental value for the velocity, and d i s the cavity spacing, i.e. the thickness of the Perspex spacers), and the corres-ponding temperature difference AT was a function of R A ( t ) - B ( t ) . (2) PHASE SHIFT AND DISPERSION TRIAL One run was performed at a fixed AT ~ 5 x 10** to determine (a) whether the generator and bolometer were introducing any detectable phase shifts in the second sound signal, (b) whether the cavity had any observable resonance modes other than the axial modes, and (o) whether any dispersion was observable. A search was made from 20 Hz to 5 KHz (covering the range of frequencies used in the experiment) for the second sound resonances. , Resonances corresponding to the fifteen axial modes up to this frequency were observed;* — and no others. The extrapolated zero frequency phase shift was 0° + 15°t and no departure from linearity i n the frequency-phase shift curve was observed. We concluded that (a) at frequencies up to 5 KHz neither the generator nor the bolometer were near their upper frequency res-ponse limits, and (b) no dispersion was detectable. 33 CHAPTER IV. DATA REDUCTION (I) EXTRACTION OP DATA PROM THB CHARTS. Line #2 i n Pig. 5» the seoond sound trace, shows the peaks which ooourred as the cavity resonated, and contains the velocity information. Lines #1, #3, and #4 contain the temperature information. Data were extracted from the chart as follows. (i) The zero-power lambda line R\ (t); the dashed line i n Pig. 5, was drawn between the RA plateaux of two successive traces of the zero power warming curve (#1 and #4 ln Pig. 5). (II) The relative peak numbers n', n*+ 1, were assigned. The integer n i s the mode number of the resonance. It i s the number of half wavelengths of seoond sound i n the cavity. ( i i i ) The distances ^ i , &nt+ i* • (see Pig. 5) representing B A(t) - R(t) at the times of occurrence of the peaks n', n'+ 1, were measured, and converted to AT nt, AT nt+ i * using the previously measured sens-i t i v i t y (ohms/chart division) of the thermometry system and the previously established thermometer calibration R(T). (Iv) The distance of point A (Pig. 5) below the zero-power lambda lin e was measured and oonverted to AT«(P), P being the total power used during the recording 3* of the trace being examined.31 (v) The absolute peak number n of the f i r s t peak was established. To do this, the second sound velocity had to be known roughly. Equation (30) i f l rewritten n •. 2fd/u 2(AT). . . . (3D Taking the measured AT n, for the f i r s t peak on the chart, a rough value of U2(AT a») was inserted Into Eq. (31) giving n m real number. This real number, say 3.04*7* would be within a few percent of some integer. The absolute peak number n was then taken to be that integer. It should be pointed out that the rough value of U2(-T ni) used to establish n must be erroneous by at least Z\$% before an error oould be inourred in es-tablishing n by this method. Not only must n be an integer? but i t s parity i s known from the direction of deviation of the peak (see Pig. 5 ) from the center line of the chart. Thus, i f the correct n were to be i n -correctly identified as rf, the error i n the value of ugC&Tgt) used to establish T/L would have had to be at least * 2 / ( n*2 )• In this experiment the highest value of n for the f i r s t peak on a chart was n = 6 giving 2\$% as the minimum error in U2(ATn,) necessary to produce an incorreot identification of n. As the experimental results w i l l show a maximum deviation of the measured velocity from theory of about 10#, we conclude that no erroneous identifications of n were made. 35 (vi) The absolute mode numbers of a l l peaks on the chart having been established via (11) and (•) the corresponding experimental velocities u 2 e were calculated using Eq.(30). Por each peak recorded on the charts therefore? a data point (u 2 e# AT) has been established and the raw data have been extracted from the charts. In the course of the experiment 63 charts were measured, producing a total of 276 data points. (2) ANALYSIS OP PROBABLE ERRORS (i) Errors ln u 2 e . Error ln the absolute velocity can arise from the measurements of f and d (Eq. (30) ). As noted above/ the error in establishing n i s presumed to be zero. The cavity spacing d was determined by measuring the thickness of the Perspex spaoers several times with a micrometer and looking at the mean and standard error of the measurements. To obtain the value of d at helium temperatures; the total thermal contraction from room temperature to helium temperature of 1.13/S^2 was subtracted giving the low temperature value of d = 1.913 x 10~3M * 0,6%, The frequency f was measured to * 1Hz using a d i g i t a l counter. The uncertainty i n f was therefore \00%/t which was less than 0,5%, The total probable uncertainty i n the absolute value of the measured velocity was therefore taken to 36 be of the order of !%• (11) Errors l a AT. The experimental value of AT for any point i may be written. ATi « (B 1)(A 1) CfOH)] , where i s the thermometry system sensitivity (ohms/ chart division) for the trace containing point 1, A^ i s the measurement of the temperature trace (chart div-isions), and f(A 1) i s the resistance thermometer calib-ration figure i n ( K/ohm ). The reproducibility of the measurements of B^ depended on the (variable) sensitivity of the system. The probable error in B 1 # based on repeated measurements at a fixed temperature, ranged from negligible to about 1%, We therefore take the probable error in B^ to be * \%. The resistance thermometer calibration figure f (A^) was very nearly a constant equal to (dT/dR)^, the inverse slope of the thermometer resistance vs. temperature calibration curve at the lambda point. As such, i t s accuracy was determined by the accuracy of the calibration curve, and i t s probable error Is estimated as ± 1%, A^ was obtained by identifying two points on the chart and measuring the distance between them. The probable error involved for each point was a r b i t r a r i l y 37 taken to be *0.03 ehart divisions (one inch divisions) and the total uncertainty ln A^ was therefore «*0.06 ohart divisions giving a total fractional error in A 4 of * ( chart divisions) %, The values of measured in the experiment ranged from 0*3 to 9*0 chart divisions giving fractional errors of from 20% to 0*7% with an average of about 2%, The average probable error in the absolute values of the experimental data i s therefore taken to be 1% in the velocity u^ e and 3% i n the temperature difference AT. D 38 CHAPTER V. RESULTS AND ANALYSIS. (1) RESULTS The experimental r e s u l t s are shown In Pig. 6. The points (u 2 e» AT), (The subscript e denoting the experimental value of the v e l o c i t y ) , cover the range AT «= 4.75 x 10"^ K to 1.25 x 10" 2 K• The temperature differences AT and the measured v e l o c i t i e s u„ are l i s t e d i n the f i r s t two ze columns of Appendix I i n order of increasing AT. The data overlap the previous measurements of Peshkov 1 ( AT>1 0 * 3K), of Pearce, Lipa, and Buckingham 2 (PLB) ( AT> 2 x 10"** K ), and of Tyson and Douglass 3 ( AT > 9 x 10"^ K ) • In the ranges of overlap, our data are i n excellent agreement with those of Peshkov and of PLB, i n both cases well within the experimental uncertainties. In the range AT «= 10*"**Kto 1 0" 2Kthe Tyson and Douglass measurements give a u 2 which i s 5# to 7% low compared to other experimental data, and to the predictions of hydrodynamics. (2) COMPARISON WITH THE HYDRODYNAMIC THEORY. The expression f o r the v e l o c i t y of second sound, U 2h = ( T S^Wp ^  •'•(32) was derived above using hydrodynamic arguments. We have here used the subscript h to denote t h i s hydrodynamic v e l o c i t y . Two pieces of experimental work enable one to put Eq. (32) i n an a n a l y t i c form useable i n the 14-16 macroscopic c r i t i c a l region below T^. Measurementa x^ of the superf l u l d density y O Q lead to the empirical r e l a t i o n 39 FIGURE 6. Experimental Results. 40 /*//Oa « 0.699AT"^- 1 , ...(33) where j> = 0.666 * X%, based on experimental data with AT 2s 6 x 10"^ K . Buckingham, Palrbank, and K e l l e r s 1 (BPK) measurements 3 3 of the s p e c i f i c heat are described by the empirical r e l a t i o n C p = 4.55 - 3.00 log 1 Q|AT| - 5.202 ( J / g - K ) , ...(34) where £ • 0 f o r AT > 0 , S « l f o r AT < 0, based on experimental data with I ATI > 10"*^ K . The entropy S of the l i q u i d may be written S(T) = S(T X) + / (C p/T)dT. \ Making a change of va r i a b l e to AT, and wr i t i n g Eq; (34) as C p(AT) ss A + Bin AT ( AT>0) we obtain AT* AT., S(AT) « S(T-) + A? d(AT) + B / i n AT d(AT). ...(35) & (T A-AT) AT (T A-AT) where AT^ • T^ - T^. Expanding the denominators of Eq* (35) In powers of AT/T A and neglecting terms of order (AT/T A) obtain ( T A-AT r1 3? (1/T A)( 1 + (AT/T A) ) , ...(36) with a maximum f r a c t i o n a l error of 0,1% i f T^ = 2.10 K• Substituting Eq. (36) Into Eq. (35), performing the Integrations, and c o l l e c t i n g terms we obtain f i n a l l y S(AT) m S(T X) - ...(37) Aln(T A-AT 1) + B ^ A T j / T ^ d n ^ - l ) + i ( A T 1 / T x ) 2 ( l n A T 1 - i ) l + Aln(T A-AT ) - B((AT /T A)(lnAT -1) + £(AT / T A ) 2 ( l n d T - % ) \ . We set ^ = 2.10 K, and obtain SfTj) from the data of 34 H i l l and Lounasmaa^ using t h e i r value 41 S( 2.10 K) • 1.24 (J/g-K). ...(38) Equation (38) and Eq. (37) with AT « 0 give the entropy at the lambda point S(T X) * 1.55 (J/g-K). Substitution of Eqs. (33). (34). (37)# and (38) Into Eq. (32) gives an analytic form for the hydrodynamic velocity of second sound dependent for i t s accuracy on the validity of the hydrodynamic theory and the extrapolation of Eq. (33) from AT • 6 x 10' 5 K to 10~ 6 K. To compare the experimental data with the hydrodynamic theory, a computer was used to evaluate the above mentioned analytic form of u 2 n using the AT from each of the experimental points ( u. . AT), and to calculate the ze ratio Rj^AT) £ u 2 e(AT)/u 2 h(AT) , for each AT. A deviation plot of logfR^) against AT is shown i n Pig. 7. and the corresponding values of u 2 n(AT) are tabulated i n the third column of Appendix I. The mean value of and i t s 95^ confidence limits are *1 " ^ u 2 e / u 2 h\v * 1 , 0 0 ° * # 0 0 l f * •••(39) In obtaining the above result we have extrapolated Eq. (33). As a check on the self-consistency of this extrapolation, a weighted least squares f i t ^ of the data to the form of Eq. (32) was performed to predict the values of Q and £ ln the expression u 2 e - ( TS 2/C p)*»( QAT-^ - 1 )"*. ...(40) 42 <f-o o d in o d o • CP o ° ° o° u 004 o p -.X. ~ „• ^ o -•o « 8 O m o d I ti 901 CM O 10 O — o l O i O FIGURE ?. Deviation p l o t comparing the d i r e c t measurements, and the calculations from thermal conductivity and s c a l i n g to the hydrodynamic v e l o c i t y . Dots = Rj, = u 2 e / u 2 h » open c i r c l e s = Rg = u2s^ U2h* 43 We obtained Q * O.67 * .02 and j? * +.671 * .004 ( 95% confidence limits) in agreement with the more precise values given above i n Eq. (33). (3) COMPARISON WITH THE SCALING PREDICTION. Recent experimental measurements by Ahlers 1?* 3^ of the thermal conductivity of helium I near the lambda point, combined with our seoond sound velocity measurements in helium II, make possible a cheek of the extended dynamic scaling prediction derived above. Equation (29) may be rewritten K(AT-) * u 2(AT)/>C p(AT - ) { A 5 2(^)-l]| AT - r 2 / 3 , ,..(4l) where AT- indicates T A-T i s negative. Substituting for u 2(AT) in Eq. (41) the hydrodynamic velocity u 2 h(AT), Ahlers performed1? a least squares f i t of the resulting equation to his thermal conductivity measurements, obtaining for the constant A j 2 ^ ) " 1 « ( 0.87 * .06 ) x l O - 8 om. ...(42) In performing this f i t Ahlers used for C p(AT-) and Cp(AT), ( contained ln u 2 n ) , his own measurements of Cp»"^ As the AT dpendences of Ahlers specific heat and of the BPK specific heat are slightly different, his data are not directly comparable with the hydrodynamic velocities we have calculated. We have therefore repeated the calculations using the BPK specific heat and Ahlers' original thermal conductivity data (ft(AT-), AT-), 3 8 44 obtaining for the constant AJo (U )" 1 " ( 0 , 8 6 * *° 6 ) x 1 0" 8 e m" •••( l*3) in agreement with Eq. (42). Rewriting Eq. (41) we find u 2 s(AT) = (fc (AT-)^>Cp(AT-) ) • C Af 2<5i>"13 - 1! AT-| 2 /3 . . . . (^) where the subscript s Indicates the velocity i s a scaling prediction. For eaoh of the thermal conductivity points (K(AT-), AT-), Eq. (44) was evaluated (with substitution of Eqs. (34) and (43) ) and the ratio R2(cxT) s u 2 s(AT)/u 2 h(AT) was calculated. The results are shown in Fig. 7. For the 66 points the mean value of H 2 and i t s 95% confidence limits are Rg = < u 2 s / u z h ) - 1.005 * .004 . . . . (45) 45 CHAPTER VI. CONCLUSIONS AND DISCUSSION. (1) CONCLUSIONS The hydrodynamic and the scaling law predictions of the velocity of second sound were tested by comparison with the experimentally determined values. The average value of the ratio of u 2 e to u g h over the temperature range of the experiment i s given by S i ln Eq. (39). The equality of Hi and unity therefore demonstrates the agreement between the measured velocities and those predloted by two f l u i d hydrodynamics. Further calcu-lations showed, and Fig. 7 indicates, that within the measured temperature range there were no significant departures of R^  from unity. In other words, R± was observed to be independent of temperature. The hydro-dynamic theory therefore appears to properly predict both the magnitude and the temperature dependence of the second sound velocity i n the c r i t i c a l region. A comparison of R^  with Rg implies a comparison of U2 e and U2».with ugh and hence of u 2 e with U2s» The equality of Rj and R2 demonstrates the agreement between the measured velocities and those calculated from dynamic scaling arguements. R2 was also shown to be Independent of temperature. As u 2 s contains a 46 constant term (evaluated i n Eq* (43) ) determined by f i t t i n g to the thermal conductivity data, i t i s this temperature independence of Bg which constitutes the affirmative test of the sealing law prediction. On the basis of these results (Eq. (39) and (45) ) we conclude that in the c r i t i c a l region for AT£ 5 x 10"^K the hydrodynamic prediction Eq. (23) and the scaling law prediction Eq. (29) of the velocity of low amplitude, low frequency second sound have been verified. The verification of the hydrodynamic prediction, and the prediction of the values of Q and j> in Eq. (40) both imply that the extrapolation pf the superfluid density relation Eq. (33) to 5 x 10~^K was valid. The experiment also proved that when properly ap-plied, the method of d r i f t i n g temperatures i s well suited for making measurements of physical phenomena near the c r i t i c a l point. (2) DISCUSSION In the derivation of Eq. (23) i t was assumed ex-p l i c i t l y that the velocities v l and are small, and s n implicitly that inhomogeneitles (in the absence of seoond sound) in the thermodynamic properties of the f l u i d occur only over distances small compared with the second sound wave length (i.e. that k£«l). In other words. 47 the derivation was for low amplitude (v g and w are proportional to the seoond sound amplitude), low fre-quency second sound. In deriving the scaling law prediction, the c r i t i c a l frequency coq(k) was related (Eq. (25) ) to second sound by using the assumption that second sound i s the dominant mode of heat transport in Region I (Pig. 1), defined by kf « 1 . Both the above conditions (low velocities, and k | « l ) were met in the experiment, the highest observed seoond sound amplitude being <~ 10"?K RMS, and the largest value of kj" being ~0 .02. A departure from either the low amplitude, or the low frequency condition would presumably be accompanied by a departure of the observed velocity from the hydro-dynamic value. Velocities of large amplitude second sound (analogous to shock waves i n ordinary sound) different from the hydrodynamic velocity have been observed. 3 9 , "*° In the c r i t i c a l region very near T^ the correlation function range j* i s diverging rapidly with decreasing AT. It would therefore seem interesting, and not im-practical, to make use of higher frequency second sound to enter Region II (Pig. 1) near the c r i t i c a l point 48 i n which kl* i s of the order of, or greater than unity, and i n which two f l u i d hydrodynamics could no longer he presumed v a l i d . 49 GLOSSARY OP SYMBOLS A Constant of order unity. a An operator or phys i c a l quantity other than p. B^ Thermometry system s e n s i t i v i t y (ohms per chart d i v i s i o n . C Constant pressure s p e c i f i c heat. Constant volume s p e c i f i c heat. c* Correlation function f o r a. c* Fourier transform of c£. c*1 " Asymptotic forms of cf i n regions and III ( F i g . 1). c* Correlation function f o r p. d Resonant cavi t y spacing. D t Thermal d i f f u s i v i t y . E Energy density of the l i q u i d . e A c r i t i c a l exponent. t Entropy f l u x vector. t Second sound frequency. G Gibbs free energy per unit mass. T Momentum per unit volume. k Wave number. M Mass of l i q u i d helium sample. M n Mass of normal f l u i d component. M s Mass of superfluid component. 50 n Resonance mode number. (Equal to the number of h a l f wavelengths of second sound i n the c a v i t y ) . n' Relative mode number. Only differences between the n 1 are correct. P Total power d i s s i p a t i o n i n the cavity. P n Normal f l u i d - superfluid r e l a t i v e momentum. p Pressure. * p Order parameter. Q Coe f f i c i e n t of AT i n e x p r e s s i o n . q Heat operator. R(t) Thermometer resistance at time t . R^ E l e o t r i c a l resistance of device at T^. S Entropy per unit mass. T Absolute temperature. T c A c r i t i c a l temperature. T^ The lambda temperature 2.172 K. t Time. U Internal energy per unit mass. u^ V e l o c i t y of f i r s t sound i n l i q u i d helium. u 2 V e l o c i t y of second sound i n l i q u i d helium. u2e u2 d e t e r r a * n e d experimentally. u2h u2 c a l c t * l a t e d from hydrodynamics. u 2 a u 2 calculated from thermal conductivity and s c a l i n g theory. V S p e c i f i c volume ( equal to ). v n Normal f l u i d v e l o c i t y . v s Superfluid v e l o c i t y . 51 S t a t i s t i c a l weight of a data point. Cartesian coordinates ( k =1,2,3 )• Constants of the order of unity. C r i t i c a l exponent of p. A measured distance on a recorder chart. A i s subsequently converted to a temp-erature difference. T A - T . T\ - (the temperature of the sharp dis-continuity i n the slope of the warming curve) The Kroneoker delta function. ( T - T c )/Tc . C r i t i c a l exponent ofytfg. C r i t i c a l exponent of C\$. Thermal conductivity. C r i t i c a l exponents of Correlation length, or range of C?. Momentum flux tensor. Density of liq u i d helium sample. Density of normal f l u i d component. Density of superfluld component. 2=0 i f T<T V 2*1 i f T>TX. Angular frequency. Characteristic frequency for c|. Div grad . 52 REFERENCES AND FOOTNOTES 1. V.P.Peshkov, Soviet Phys. - JETP, U , 580 (I960). 2. C.J.Pearce, J.A.Llpa, and M.J.Buckingham* Phys. Rev. Letters, 20, 1471 (1968). 3. J.A.Tyson and D.H.Douglass, Phys. Rev. Letters, 21, 1308 (1968). 4. D.L.Johnson and M.J.Crooks, Phys. Letters, 27A, 688 (1968). 5. A good review of the two f l u i d model is contained ln: J.Wllks, The Properties of Liquid and Solid  Helium, (Clarendon Press, Oxford, 1967)* Chapter 3. 6. L.D.Landau, J. Phys., Moscow 71 (1941). Reprinted ln: I.M.Khalatnikov, Introduction to the Theory of Superfluidity (V.A.Benjamin, Inc. New York, 1965). 7. A good review paper i s : L.P.Kadanoff, W.Gotze, D.Hamblen, R.Hecht, E.A.S.Lewis, V.V.Palclauskas, M.Rayl, and J.Swift, Rev. Mod. Phys., 22, 395 (1967). 8. M.E.Fisher, J. Math. Phys., 944 (1964). 9. B.Wldom, J. Chem. Phys., 43, 3892 (1965). and 42, 3898 (1965). 10. L.P.Kadanoff, Physics. 2, 263 (1966). 11. B.I.Halperin and P.C.Hohenberg, Phys. Rev., 177. 952 (1969). 12. R.A.Ferrell, N.Menyhard, H.Schmidt, F.Schwabl and P.Szepfalusy, (a) Phys. Rev. Letters, 18, 891 (1967). (b) Phys. Letters, 24A, 493 (1967). and (c) Ann. Phys. (N.I.) , 47., 3&7 (1968). 13. See Reference 12(c), Equation (5.4). 14. J.R.Clow and J.D.Reppy, Phys.Rev. Letters, 16, 887 (1966). 15. J.A.Tyson and D.H.Douglass, Phys. Rev. Letters, 12, 472 (1966); 12. 622(E) (1966). 16. J.A.Tyson, Phys. Rev., 166. 166 (1968). 53 17. G.Ahlers, Phys. Rev. Letters, 21, 1159 (1968). 18. E.J.Walker, Rev. S c i . Instr., 20, 834 (1959). 19. CIBA (A.R.L.) Ltd., Duxford, Cambridge, England. 20. G.Ahlers, Phys. Rev., 121, 275 (1968). 21. Princeton Applied Research Corp. Model 122. Typical s e n s i t i v i t y 0.1 to 1.0 mV rms f u l l scale. Typical in t e g r a t i o n time 1 to 3 seconds. 22. Princeton Applied Research Corp. Model 112 preamp-l i f i e r . Fixed gain of 40db. 23. Hewlett-Packard / Moseley D i v i s i o n Model 7100B. Typi c a l gain 1 to 5 V f u l l scale. 24. T e l t r o n l c s Inc. Model CA-2. Typical s e n s i t i v i t y 5ju.V rms f u l l s c a le. Typical integration time 1 second. 25. Tektronix Inc. Model BM 122. Fixed gains of 40db or 60db. 26. W.C.Cannon and M.Chester, Bev. S c i . Instr., 38, 318 (1967). 27. An Introduction to Photofabrlcatlon using Kodak Photosensitive Resists, Kodak Publication No. P-79. Eastman Kodak Co., Rochester, N.Y. 28. No. 999 Black. Columbian Carbon Co. of New York, N.Y. 29. Allen-Bradley Co., Milwaukee, Wisconsin. 30. IRC Inc., Boone Di v i s i o n , Boone, North Carolina. 31* Some data were acquired before the neccesslty to measure zero-power warming curves was r e a l i z e d . On these early charts, the point A was assumed to be the lambda point. As P was known, these data were subsequently corrected by the addition to t h e i r AT values of AT'(P) obtained from extrapolation and i n t e r p o l a t i o n of the subsequent measurements of AT»(P) as a fuc t i o n of P. 32. Thermal Expansion of Technical Solids at Low Temp- eratures, National Bureau of Standards Monograph 29, [United States Department of Commerce, Government P r i n t i n g O f f i c e , Washington, D.C, 1961). Note that Perspex and Plexiglass are both manufacturers trade names f o r polymethylmethacrylate. 54 33* C.P.Kellers, Thesis, Duke University ( i960) unpub-l i s h e d 1 M.J.Buckingham and W.M.Pairbank, In Progress  In Low Temperature Physics. ed. C.J.Gorter (North-Holland Publishing Co.,Amsterdam, I 9 6 D , V o l . I l l , Ch .3 . 34. R.W.Hill and O.V.Lounasmaa, P h i l . Mag., 2, Ser. 8, 1 4 5 (1957) . 35. The "oorrect" weight W N given to a point should be proportional to the inverse of the variance of that point. The best a v a i l a b l e estimate of the variance i s the square of the probable error. The points were therefore assigned weights WQ proportional to ( 2 + 6/A n )" » the number i n brackets being the estimated probable f r a c t i o n a l e r r o r {%) i n the measurement of AT n. 36. G.Ahlers, Phys. Letters, 28A, 507 (1969) . 37 . G.Ahlers, B u l l . Am. Phys. S o c , 1^, 506 (1968) , and to be published. 38. G.Ahlers, p r i v a t e communication. 39 . A.J.Dessler and W.M.Fairbank, Phys. Rev., 1 0 4 , 6 (1956) . 4 0 . D.V.Osbourne, Proc. Phys. S o c , A 6 4 , 1 1 4 ( 1 9 5 1 ) . 55 APPENDIX I EXPERIMENTAL AND THEORETICAL SECOND SOUND VELOCITIES TEMPERATURE . EXPERIMENTAL , , HYDRODYNAMIC DIFFERENCE VELOCITY VELOCITY (K) (M/SEC) (M/SEC) 4.75E-06 0.335 0.321 5.27E-06 0.353 0.334 5.33E-06 0.380 0.335 5.66E-06 0.389 0.342 5.66E-06 0.353 0.342 5.75E-06 0.389 0.344 5.98E-06 0.342 0.349 6.10E-06 0.383 0.352 6.18E-06 0.389 0.354 6.59E-06 0.383 0.362 6.63E-06 0.392 0.363 6.90E-06 0.383 0.368 7.31E-06 0.392 0.376 7.61E-06 0.389 0.382 7.75E-06 0.383 0.384 7.90E-06 0.444 0.387 7.94E-06 0.383 0.388 8.08E-06 0.389 0.390 8.36E-06 0.425 0.395 8.44E-06 0.415 0.396 8.44E-06 0.418 0.396 8.74E-06 0.418 0.401 8.74E-06 0.418 0.401 8.87E-06 0.383 0.404 8.93E-06 0.410 0.405 9.03E-06 0.399 0.406 9.42E-06 0.399 0.413 9.43E-06 0.410 0.413 9.68E-06 0.383 0.417 9.88E-06 0.446 0.420 1.03E-05 0.441 0.426 1.05E-05 0.444 0.429 1.07E-05 0.453 0.432 1.09E-05 0.446 0.435 1.10E-05 0.425 0.437 1 .12E-05 0.465 0.440 1.13E-05 0.446 0.441 1.I5E-05 0.441 0.444 1.17E-05 0.425 0.447 1.23E-05 0.465 0.455 56 MPERATURE . EXPERIMENTAL , , HYDRODYNAMIC FFERENCE VELOCITY VELOCITY (K) (M/SEC) (M/SEC) 1.24E-05 0.478 0.456 1.26E-05 0.465 0.459 1.29E-05 0.511 0.463 1 .31E-05 0.478 0.466 1.42E-05 0.478 0.480 1.45E-05 0.478 0.483 1.46E-05 0.478 0.485 1.46E-05 0.478 0.485 1.47E-05 0.479 0.486 1.48E-05 0.504 0.487 1.49E-05 0.480 0.488 1.50E-05 0.486 0.489 1.51E-05 0.480 0.491 1.52E-05 0.478 0.492 1.52E-05 0.479 0.492 1.52E-05 0.486 0.492 1.52E-05 0.519 0.492 1.54E-05 0.480 0.494 1.57E-05 0.478 0.498 1.58E-05 0.523 0.499 1.61E-05 0.517 0.502 1.61E-05 0.478 0.502 1.62E-05 0.478 0.503 1.63E-05 0.498 0.505 1.66E-05 0.486 0.508 1.68E-05 0.504 0.510 1.73E-05 0.523 0.516 1.76E-05 0.523 0.519 1.78E-05 0.540 0.521 1.78E-05 0.519 0.521 1.80E-05 0.536 0.523 1.80E-05 0.536 0.523 1.81E-05 0.511 0.524 1.83E-05 0.536 0.527 1 .89E-05 0.511 0.533 1.93E-05 0.510 0.537 1.95E-05 0.547 0.539 2.00E-05 0.547 0.544 2.02E-05 0.574 0.546 2.04E-05 0.553 0.548 2.05E-05 0.540 0.549 2.09E-05 0.547 0.553 2.12E-05 0.547 0.556 2.14E-05 0.540 0.558 2.26E-05 0.540 0.569 57 TEMPERATURE DIFFERENCE (K) . EXPERIMENTAL , VELOCITY (M/SEC) , HYDRODYNAMIC VELOCITY (M/SEC) 2.29E-05 0.574 0.572 2.30E-05 0.574 0.573 2.34E-05 0.597 0.577 2.5IE-05 0.574 0.592 2.53E-05 0.597 0.593 2.55E-05 0.574 0.595 2.55E-05 0.589 0.595 2.57E-05 0.574 0.597 2.58E-05 0.574 0.598 2.63E-05 0.618 0.602 2.63E-05 0.589 0.602 2.68E-05 0.623 0.606 2.72E-05 0.597 0.609 2.75E-05 0.574 0.612 2.76E-05 0.622 0.613 2.79E-05 0.618 0.615 2.86E-05 0.598 0.621 2.87E-05 0.622 0.622 2.90E-05 0.618 0.624 2.97E-05 0.638 0.630 2.98E-05 0.638 0.630 2.99E-05 0.639 0.631 3.01E-05 0.638 0.633 3.01E-05 0.598 0.633 3.02E-05 0.599 0.633 3.03E-05 0.638 0.634 3.03E-05 0.598 0.634 3.05E-05 0.599 0.636 3.09E-05 0.638 0.639 3.09E-05 0.639 0.639 3.10E-05 0.648 0.640 3.11E-05 0.638 0.640 3.14E-05 0.618 0.643 3.15E-05 0.639 0.643 3.16E-05 0.638 0.644 3.18E-05 0.638 0.646 3.20E-05 0.648 0.647 3.30E-05 0.639 0.655 3.33E-05 0.648 0.657 3.42E-05 0.670 0.663 3.44E-05 0.639 0.665 3.54E-05 0.656 0.672 3.55E-05 0.646 0.673 3.57E-05 0.702 0.674 3.59E-05 0.670 0.675 58 MPERATURE . EXPERIMENTAL , • HYDRODYNAMIC FFERENCE VELOCITY VELOCITY (K) (M/SEC) (M/SEC) 3.81E-05 0.670 0.690 3.84E-05 0.638 0.692 3.86E-05 0.718 0.694 3.86E-05 0.697 0.694 3.89E-05 0.718 0.696 3.93E-05 0.697 0.698 3.97E-05 0.721 0.701 4.07E-05 0.717 0.707 4.13E-05 0.697 0.711 4.16E-05 0.721 0.713 4.17E-05 0.718 0.714 4.20E-05 0.712 0.716 4.24E-05 0.706 0.718 4.31E-05 0.706 0.723 4.33E-05 0.656 0.724 4.36E-05 0.765 0.726 4.36E-05 0.765 0.726 4.36E-05 0.718 0.726 4.37E-05 0.721 0.726 4.48E-05 0.717 0.733 4.63E-05 0.767 0.742 4.67E-05 0.721 0.744 4.68E-05 0.765 0.745 4.78E-05 0.765 0.751 4.93E-05 0.738 0.760 4.96E-05 0-.765 0.761 4.96E-05 0.765 0.761 4.96E-05 0.765 0.761 5.00E-05 0.765 0.764 5.03E-05 0.765 0.765 5.21E-05 0.765 0.775 5.23E-05 0.767 0.776 5.30E-05 0.778 0.780 5.52E-05 0.765 0.792 5.57E-05 0.765 0.795 5.67E-05 0.778 0.800 6.08E-05 0.820 0.821 6.13E-05 0.765 0.824 6.13E-05 0.797 0.824 6.24E-05 0.797 0.829 6.27E-05 0.820 0.830 6.43E-05 0.797 0.838 6.45E-05 0.830 0.839 6.61E-05 0.865 0.847 6.67E-05 0.836 0.850 59 TEMPERATURE . EXPERIMENTAL , , HYDRODYNAMIC DIFFERENCE VELOCITY VELOCITY (K) (M/SEC) (M/SEC) 6.78E-05 0.836 0.855 6.87E-05 0.850 0.859 6.89E-05 0.852 0.860 6.95E-05 0.842 0.863 6.97E-05 0.862 0.864 6.97E-05 0.836 0.864 7.06E-05 0.865 0.868 7.10E-05 0.865 0.870 7.25E-05 0.850 0.877 7.34E-05 0.862 0.881 7.35E-05 0.852 0.881 7.38E-05 0.861 0.882 7.54E-05 0.865 0.889 7.62E-05 0.893 0.893 7.63E-05 0.852 0.893 8.04E-05 0.893 0.911 8.53E-05 0.957 0.931 8.54E-05 0.957 0.932 8.59E-05 0.957 0.934 9.09E-05 0.918 0.954 9.10E-05 0.957 0.954 9.10E-05 0.957 0.954 9.15E-05 0.957 0.956 9.16E-05 0.957 0.956 9.17E-05 0.918 0.957 9.21E-05 0.957 0.958 9.39E-05 0.959 0.965 9.39E-05 0.957 0.965 9.44E-05 0.959 0.967 9.56E-05 0.957 0.972 9.75E-05 0.957 0.979 9.84E-05 0.957 0.982 9.88E-05 0.957 0.984 1.02E-04 0.985 0.995 1.03E-04 0.957 0.999 1.07E-04 0.985 1 .013 1.22E-04 1.030 1 .064 1.27E-04 1.081 1 .080 1.28E-04 1.081 1.084 1.28E-04 1.052 1.084 1.31E-04 1.081 1.093 1.33E-04 1.090 1 .099 1.33E-04 1.090 1 .099 1.36E-04 1.081 1 .108 1.41E-04 1.150 1.124 60 TEMPERATURE • EXPERIMENTAL . . HYDRODYNAMIC DIFFERENCE VELOCITY VELOCITY (K) (M/SEC) (M/SEC) 1.45E-04 1.150 1.135 1.51E-04 1.149 1.153 1.58E-04 1.1 46 1.173 1.59E-04 1.149 1.175 1.79E-04 1.200 1.229 1.83E-04 1.200 1.239 1.99E-04 1.290 1.279 1 .99E-04 1.280 1.279 1.99E-04 1.280 1.279 2.06E-04 1.279 1.295 2.11E-04 1.279 1.307 2.22E-04 1.379 1.332 2.62E-04 1.379 1.418 2.82E-04 1.480 1.458 3.06E-04 1.530 1.503 3.11E-04 1.530 1 .512 3.98E-04 1 . 720 1 .660 4.35E-04 1.724 1.717 4.43E-04 1.724 1.729 5.27E-04 1.910 1.846 5.57E-04 1.910 I .885 5.88E-04 1.913 1.924 6.01E-04 1.913 1.941 6.05E-04 1.913 1.945 6.11E-04 1.960 1.953 6.62E-04 2.060 2.014 6.64E-04 2.070 2.017 7.25E-04 2.150 2.085 8.70E-04 2.240 2.235 9.15E-04 2.300 2.278 9.22E-04 2.300 2.285 9.74E-04 2.350 2.333 1.03E-03 2.540 2.382 1.08E-03 2.460 2.426 1.20E-03 2.620 2.525 1.23E-03 2.580 2.549 1.39E-03 2.740 2.671 1.67E-03 2.870 2.866 1.75E-03 2.960 2.918 2.01E-03 2.960 3.077 2.03E-03 3.140 3.089 2.17E-03 3.290 3.174 2.65E-03 3.440 3.426 2.65E-03 3.440 3.426 2.95E-03 3.550 3.570 61 TEMPERATURE DIFFERENCE (K) EXPERIMENTAL VELOCITY (M/SEC) HYDRODYNAMIC VELOCITY (M/SEC) 3.18E-03 3.74E-03 3.93E-03 4.62E-03 5.32E-03 5.53E-03 7.35E-03 7.90E-03 9.99E-03 1.15E-02 1.25E-02 3.550 3.920 4.110 4.310 4.440 4.440 5.480 5.230 5.740 5.920 5.920 3.674 3.914 3.989 4.244 4.482 4.549 5.079 5.222 5.715 6.042 6.233 `

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