Precision Sound Velocity Measurements in He 1976

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PHYSICAL REVIEW B VOLUME I4, NUMBER 9 I NOVEMBER I976Precision sound-velocity measurements in He II t

J. Heiserman, J. P. Hulin,* J. Maynard, and I. RudnickDepartment of Physics, Unitersity of Califurnia, Los Angeles, California 90024(Received 10 May 197Q

Acoustic resonators were used to measure simultaneously the velocities of first, second, and fourth sound inHe II to a.typical precision of O.2?o. Data were taken at temperatures from 1.2 K to the tr line in 50-mKsteps, and at pressures from the saturated vapor pressure to the melting curve in l-bar steps. Temperature andpressure were determined to better than O.lVo. Fourth sound was measured in a coarse (- l-f.m) and fine(- 9O-A) packed powder. Smoothed and raw data values are tibulated for more than 1500 data points.

I. INTRODUCTIONOne of the most significant successes of the two-fluid theory of superfluid helium was the accurateprediction of new modes of sound propagation inthe bulk liquid.r-3 In addition to ordinary soundcomposed of compressional waves (first sound),

there appears a thermal or entropy wave (secondsound), and a pure superfluid wave which propa-gates in a superleak (fourth sound). The veloci-ties of propagation of these modes have beenthoroughly studied along t}te saturated vapor pres-sure (SVP) line, and the velocity expressions ofthe two-fluid theory have been verified.4'sPrior to the work we report here, the pressureas well as temperature dependence of the first-and second-sound velocities was measuredr6'7 butthe full potential of such data had not been fullyexploited, The pressure dependence of fourthsound had not until now been measured. There isa very compelling reason for carefully measuringthe sound velocities: the velocities can be mea-sured easily to high precision, and such data,when supplemented with a small amount of data atSVP, will completely determine the thermody-namics of He tr. It can be shown that the soundvelocities determine the most important thermo-dynamic derivatives directly and that the onlyderivative which must be taken numerically canbe found in two independent ways and hence canbe found accurately. The thermodynamic informa-tion contained in the sound velocities will be thesubject of the following paper.22In this paper we describe an experiment whichuses acoustic resonators to measure simultan-eously the velocities of first, second, and fourthsound to -0.2/e precision, Data were taken attemperatures from 1.2 K to the ). transition tem-perature 71 in 50-mK steps, and at pressuresfrom SVP to the melting curve in l-bar steps.The temperature was measured to within 1 mK,and the pressure readings were accurate to betterthan 0.1/6. Fourth sound was measured in a coarse

packed powder anal in a fine packed powder whichgave significant size effects. The experiment wasintended to be the Iast word in sound velocitymeasurements in He II above 1.2 K.II. EXPERMENTA. General method

The sound velocities were found by measuringthe resonant frequencies of plane-wave modes incylindrical cavities. Long, narrow cavities wereused to produce sequences of plane-wave reso-nances without the interference of radial modes.By measuring a resonant frequency / belonging toa harmonic sequence, the sound velocity C can befound fromC =2lf /m, (1)

where I is the length of the cavity and rz is t}teharmonic number. The velocity of fourth soundgiven by Eq. (1) must be multiplied by a scatteringcorrection, n, which accounts for changes in theeff e c tive co mpres s ibility and scattering (aris ingfrom Kelvin drag effects) due to the powder par-ticles. The determination of yt will be discussedin Sec. IIIA.

B. Experimental cellThe experimental cell is shown in Fig. 1. Itconsists of a brass cylinder drilled to make four10.16*0.01 cm Iong, 0.6-cm diam resonators.The resonators are interconnected with drilledpassageways and are filled and pressurizedthrough a $-in. copper tube. Drive and pickup

sound transducers seal tJte ends of each resonator.Two of the resonators are empty and used forfirst and second sound. The other two are packedwith powders for fourth sound. One is packedwith a 1-pim aluminum oxide powder to an averageporosity (open volume divided by total volume) of5716 which gives pores just small enough to lock thenormal fluid and give negligible size effects. The38624


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pressureline

Resonotors

@

FIG. 1. Experimental cell.other is packed with -90-A carbon black to aporosity of -60/6 and gives large size effects. Athorough discussion of fourth-sound resonatorsand size effects is given in Rel. 5,The sound transducers were capacitive devices;details are shown in Fig. 2. The first- and fourttr-sound transducers used electreted (charge implant-ed) teflon sheet (1.3x10-3 cm thick) for the oscil-lating element. The stationary element was a con-ducting button epoxied into a mounting flange. Theepoxied surfaces were grooved to add strength andto help ensure a leak-tight seal at high pressures.Pressures in excess of 60 bar could be maintained.The exposed surface of the button was roughenedby sandblasting to improve t}re transducer response.The surface of the teflon sheet facing the resonatorhad a conducting layer of evaporated aluminumwhich was grounded through contact with the brass.cylinder. The transducer was driven by applyinga sinusoidally varying voltage of amplitude -80 Vto the stationary plate.The second-sound transducers were identical tothe first and fourth except that the oscillatingmembranes were porous superleaks (1-pm pore

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element

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Oscilloling elemenl(lhickness exoggeroled )O-ring Aluminized surfoceFIG. 2. Details of the sound transducers,

diameter Nuclepores). Operation of these trans-ducers is described in detail elsewhere.eEach transducer was sealed to the brass cylind-er with an indium O ring and ten 6-32 machinescrews.C. Resonator performance

Typical spectra (pickup amplitude versus drivefrequency) of the resonators are shown in Fig. 3.In the first-sound spectrum IFg. 3(a)] the seventh,eighth, and ninth harmonics were particularlystrong and gave velocities which agreed to within0.2/s. The seventh harmonic consistenUy gave theaverage of the three and was used to give t}te tabu-lated values of the first-sound velocity Cr. Thequality factor, given by Q=f /Lf , where A/ is thefull width at half maximum, was at least 500 atall temperatures and pressures and hence theresonant frequency could easily be determined to0.ZVc.The Q of the second-sound resonances [shown infig. 3(b)] was lower than that of the first-soundresonances, but by using the twentieth harmonicthe sound velocity could be determined to within0.04 m sec-r. Altrough some radial modes ap-peared between the tenth and twentieth harmonics,the twentieth harmonic agreed with the harmonicsequence below the tenth to within 0.2/e.The 1-prm fourth-sound spectrum [fie. a(c)]showed structure due to the acoustic interactionof the packed powder and the fluid. However, if aportion of the spectrum is expanded as shown inFig. 3(d), then the powder structure becomes aslowly rolling background and the fourth-soundresonance appears as a high Q peak. The secondharmonic could be consistently lo'cated to withina few Hz for a precision of 0.4 m sec-r.The 90-A fourth-sound spectrum was similar tothe 1-pm fourth-sound spectrum. However, atsome pressures, a few of the resonance peakswould disappear into the background. Hence it

PRECISION SOUND-VELOCITY MEASUREMENTS IN HC II

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Firslsound

Secondsound

Fourthsound

2000 2500FIG. 3. Tlpical resonator spectra, (a) First sound'

(b) second sound, (c) fourth sound, (d) fourth sound,expanded scale. Harmonic numbers are indicated.was necessary to follow several peaks in orderto obtain data over the complete pressure range.There was also a problem in the shifting of theresonances due to the interaction with the powder,so that some of the resonances did not fall in aharmonic sequence. However, there was alwaysa set of resonances which showed the same tem-perature and pressure dependence, and hencecould be normalized to give a sound velocity ac-curate to an estimated (l-Z)Vo.

D. CryostatThe cryostat system is shown in Fig. 4. Thetemperature was regulated with a carbon resistorthermometer and heater in a feedback circuit. Thetemperature was found by measuring the vaporpressure of the helium bath and using the ?utscale.lo The vapor pressure was measured witha quartz Bourdon gaugerr calibrated above 3 Torr

HEISERMAN, HULIN, MAYNARD, AND RUDNICK

l0 kHz

t4

Hz

FIG. 4. Cryostat system'with a mercury manometer and cathetometer andbelow 3 Torr with a calibrated capacitance gauge.uThe temperature could be determined to within 1mK over the entire temperature range.The pressure in the resonators was measuredwith a capacitance gaugel3 factory calibrated tobetter than 0.116 accuracy in a 0-10-bar range. Inorder to realize the fuII 0-25-bar range in theexperiment, the reference pressure of the gaugewas changed at 10 and 20 bar. When the gaugeread 10.000 bar, the reference was adiusted untilthe gauge read 0.000 bar. A second precision high-pressure gauge ensured that the resonator pres-sure was constant to less than 0.1fr during thereference change. Hence the accuracy of. O.llowas maintained over the full 0-25-bar range.Both the vapor pressure and resonator pressuregauges were temperature regulated.

E. ElectronicsThe electronics setup shown in Fig. 5 was used

to measure the first-, second-, and one of thefourth-sound velocities simultaneously. The first-and second-sound resonances had sufficiently highQ's and low backgrounds so that their resonancefrequencies could be tracked automatically usingwave analyzersra in a phase-locked loop. Thefourth-sound resonances had high Q's but the back-ground prevented automatic tracking. The fourth-sound resonances were each found in turn using aspectrum analyzerls with a cathode-ray tube dis-


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14 PRECISION SOUND-VELOCITY MEASUREMENTS IN He II 3865

FIG, 5, Electronics.

play. The analyzer could sweep a frequency rangeand locate the resonance in a matter of seconds.Once the fourth-sound resonance was found a com-puter was signaled, and all tlre resonant frequen-cies together with the cell temperature and pres-sure were simultaneously stored and tabulated.Then the spectrum analyzer was switched to t}teother fourth-sound resonator and the process wasrepeated.

F. ProcedureAt the beginning of a run, the temperature wasregulated at a value close to an integral multipleof 50 mK. The resonator fill line was intercon-nected with the bath vapor and tlte cell was giventime to come to thermal equilibrium' When it wasobserved that the sound velocities were stable, theSVP data were taken. The resonator pressurewas then incremented by 1 bar and the data takingprocedure was repeated. After each pressure in-crement, the sound velocities were found to be-come stable alter only a few minutes.

TABLE I. Velocity of fourth sound in He II in a 90-A packed powder (msec-1).7 (K)P (bar) 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.65 1.70 1.80

130 L24r32 t26134 L27135 127136 128138 t29

svP12

45

158 156161 159165 L62168 166r?t 168r73 t?L173 173r78 L75181 L77782 178184 181186 183187 184189 185190 186191 187L92 187193 187t94 t87194 187194 187194 187193 187193 187L92 186191 185

153 151156 154159 t57162 160165 162166 165168 166170 168L?t 170t73 t72t74 t73L75 t74L76 L74t77 L75178 L76L79 176L79 L76r79 t76179 176t79 t76L79 t76t79 t75178 t74178 L74177 L72L76 170

148 1.44151 1-47154 149t57 L52159 153161 155163 157165 159166 160L67 161168 161169 t62169 163170 163170 163170 163170 163170 163L70 t62170 161169 160168 159t67 158165 156764 L54762 752

140 135t42 137145 139t47 141149 143150 144L52 145153 t47L54 L47155 t48155 t49156 L49156 L49r-56 L49156 148156 L47156 r47r54 145153 L44152 L42150 t4L149 139t46 136t44 133742 130740 127

139 r29140 t28141 t27t4t I27L4L L26r4t t24140 122t40 L20139 118138 115136 tLz134 109L32 106130 L02r28 98t26 93r23 87119 82116lt2

118 103119 104t20 105r20 106t20 106t20 106120 105119 104118 t02LL7 101115 99113 97111 94109 92108 89103 86100

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3866On some runs which immediately followed liquid-helium transfers, the carbon resistor thermometerwas found to drift a few millidegrees. Since theactual temperature was measured with the vaporpressure, this caused no problem except for someinconvenience in tabulating the data.

III. ANALYSIS AND RESULTSA. Analysis;determination of z

The first- and second-sound velocities (C, andC, respectively) were determined from the reso-nant frequencies using Eq. (1) with I =10.16 cm,m(Crl=1, and m(Cr') =20. The 1-pm fourth-soundveloeity uncorrected for the powder scattering(C;n ) was found using rz =2. The scatteringcorrection z was found by first calculating a low-temperature (1 .187 K) SVP value of the fourth-sound velocity from the thermo-hydrodynamicequationsci=c?lt - + .? (*)' - + ?f#l o,

andp,/p = (1 +coc2z/Ts2)-r . (3)

At SVP and 1.18? K, Fo*O, and the last term inEq. (2) can be ignored. The only data needed hereother than our own C, and C, are S and C, whichwere interpolated from Ref. 16. Note that sincep,/p is small (=0.02), an error of. 5% in Co/52will produce an error of only 0.0570 in Cn. Thescattering correction is taken to be

n =Cn/Cl -oevaluated at 1.187 K and SVP. The calculated

FIG. 6. Velocity of first sound in He II. Solid lines,this experiment; solid circles, Ref. 6; open circles,Ref. 20; crosses at SVP, Ref. 21,

HEISERMAN, HULIN, MAYNARD, AND RUDNICK t4value z =1,228 is in reasonable agreement with theapproximate expressions n=(2- P)t/z, wltere p isthe porosity of the packed powder.Although it can be assumed that the scatteringcorrection for the 1 - pm powder is constant, nosuch assumption ean be made for the 90-A powder.The reason is that as the pressure approaches themelting pressure, solid layers build up on thepowder particles. For the 1-pm powder this hasnegligible effect on the particle and pore size,but in the 90-A powder the fractional change inparticle and pore size is significant and resultsin a change in the scattering correction. Sincethere is no clear theory for calculating the correc-tion, we have tabulated the uncorrected 90-Afourth-sound velocity da-ta.

B. ResultsThe sound-velocity data have been used to deter-mine continuous thermodynamic functions forHeII. As a result, continuous functions fitting theCr, C* and Cn (1 pm) data have been obtained.Since these functions fit the data to within the ex-perimental precision, we have used t}tem to obtainsmoothed values for these sound velocities. In atablerT we present the smoothed values along withthe raw data (in parentheses). The general pre-cisions for the C' C* and Cn (1 pm) values are0.2 msec-l, 0.04 msec-1, and 0.4 msec-r, re-spectively. For the bulk of the data these corre-spond to a precision of.0.210.The accuracy of the data is limited by the errorin the length of the resonators (0.170) and, for the1-pm fourth sound, inthe scattering correction(0.05%). We assume that these are constant overthe temperature and pressure range of the experi-

Temperoture (K)FIG. 7. Velocity of second sound in He II. Solid lines,this experiment; open circles, Ref. 7. Data points ofRef. 7 at elevated presslrres have been shifted down by

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ment, and hence the accuracy is within the 0.2V0precision.Very near the I line the C, (1 prm) resonanceamplitude decreased and the peak could have beenshifted by the background. The accuracy of thetemperature measurement also becomes signifi-cant near ?I. Hence the raw data values of C, andCnf.or T/Ty>0.98 have precisions of -170. How-ever, the functions which generated the smoothedvalues in the table had the correct asymptotic be-havior (as determined from Ah1ers18 and Greywalland Ahlersle) built in. For those few points verynear Q, the smoothed values can be consideredas more accurate than the raw data values,The Cn (SO A) Oata are presented in Table I.The accuracy is estimated at (1-2)7o. The signifi-cance of these data in regard to size effects willbe the subject of a future paper.

fWork supported in part by the NSF Grant No. DMR?500761-000 and the Office of Naval Research, Con-tract No. N00014-75-C-0246.*Present address: Ecole Normale Srpdrieure, Paris V,France.tL. Tisza, J. Phys. Radium 1, 165, 350 (1940).2L. D. Landau, J. Phys. USSR 5, 71 (1941).3J. R. Peuam, Phys. Rev, 73, 608 (1948).aV. P. Peshkov, J. Phys. USSn fO, 389 (1946).5K. Shapiro and I. Rudnick, Phys. Rev. A 137, 1383(1965); M. Kriss and L Rudnick, J. Low Temp. Phys.3, 3s9 (1970).6.i U. Vtgrros and H, A. Fairbank, Phys. Rev. L47, 185(1e66).?R, D. Maurer and M. A. Herlin, Phys. Rev. 81,, 444(1e51).sNuclepore Membranes, Nuclepore Corp., Pleasanton,Ca.94566.9R. Williams, S. E. A. Beaver, J. C. Fraser, R. S.Kagiwada, and L Rudnick, Phys. Letl.29, 279 (1969);R. A. Sherlock and D. O. Edwards, Rev. Sci. Instrum.41, 160s (1970).10I: c. Brickwedde, H. Van Dijk, M. Durieux, J. R.Clement, and J. K. Logan, Natl. Bur. Std. Mon. 10,1 (1960).llPrecision Pressure Test Set Model 145, Texas In-struments, brc., Houston, Texas.l2Baratron Pressure Meter Type 170 with special 0-3-Torr Head, MKS hstruments, Inc., Burlington, Mass.01803.13same as Ref. 12 but with special 0-10-bar head.

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C. Comparison with other experimentsFigures 6 and ? present smooth curves repre-senting a part of our data and points taken fromother experiments. It can be seen in Fig. 6 thatthe C, data of Atkins and Stasior2o deviate by sev-eral percent at high pressures whereas the dataof Vignos and Fairbank6 are in good agreement.The data of Chase2l at SVP agree with our data to-0.lVo.We found that the Crdata of Maurer and Herlinzagree with our data at SYP but are systematically

27o higher at elevated pressures. However, ourC, data agree with that of Greywall and Ahlers to-0.2V0 at SVP and elevated pressures. In Fig. ?,the open circles for elevated pressures representthe data of Mauer and Herlin after being shifteddown by 270.

l4Wave analyzer Tlzpe 19004, General Radio Co., WestConcord, Mass.ltspectrum Ana-lyzer, Model 3580A, Hewlett- Packard,Loveland, Colorado.16R, G. Hussey, B. J. Good, and J. M. Re5.nolds, Phys.Fluids 10, 89 (1967).1?See AIP document No. PAPS PLRBA-L4-3862-9 fornine pages comprising this table. Order by PAPS numberand journal reference from American trrstitute ofPhysics, Physics Auxiliary Publication Service, 335East 45th Street, New York, N. Y. 10017, The priceis $1.50 for each microfiche (98 pages) or g5 forphotocopies of up to 30 pages with $0.15 for each addi-tional page over 30 pages. Airmail additioaal. Makechecks payable to the American Institute of Physics.This material also appears iL Current Physics Mic-rofilm, the monthly microfilm edition of the completeset of journals published by AIP, on the framesimmediately following this journal article. Thesmoothed sound-velocity data may also be found inthe Physical Review article following this one.18G, Ahlers, Phys. Rev. A 8, 530 (1973).1eD. S. Greyrvall and G. AhGrs, Phys. Rev. A7, 2145(1973).20K. R. Atkins and R. A, Stasior, Can. J. Phys.31, 1156(1 953).21C. E. Chase, Phys. Fluids 1, 193 (1958), as tabulatedin Experimental Superfluidity by R. J, Donnelly (Uni-versity of Chicago Press, Chicago, 1967).22J. Ma5rnard, following paper, Phys. Rev. B 14, 3868(1e76).

PRECISION SOUND-VELOCITY MEASUREN,IENTS IN HE II

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Precision Sound Velocity Measurements in He 1976