8/13/2019 1 Trapping of Persistent Currents in Superleaks 1975

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PHYSICAL REVIEW B VOLUME 12, NUMBER 5 1 SEPTEMBER 197bTrapping of persistent currents in superleaks*

Joseph Heiserman and Isadore RudnickDepartment of Physics, University of California, Los Angeles, California 90O24(Received 8 January 1975)

In a toroidal persistent-current apparatus which is partially packed with superleak, high-velocity persistentcurrents are found in the superleak and only barely perceptible currents, at most, are found outside thesuperleak.

Consider an annular cavity filled with superfluidhelium. Persistent currents can be generated, 1and if a typical dimension for the height or widthis a fraction of a cm, the maximum current ob-served is about 1 mm,/sec.2 On the other hand,when the cavity is filled with a superleak consistingof a powder with a nominal grain size of 500 A, 3then persistent currents of 102 cm/sec have beenobserved, a Suppose we now eonsider an annuluswith cross section 0,5x0.5 cm2 (the mear radius,R=5.25 cm) with only the bottom half, or so, filledwith superleak of the grain size mentioned (asshown in Fig, 1). What can we expect in the twocoexisting channels ?5 WiU we have the same per-sistent current in the two channels and, if so, willthey be about 10-1 cm/sec or about 102 cm,/sec ?Or will they be small in the top part and Iarge inthe bottom ? The experiment we report here wasperformed to answer this question and we can statethat the last alternative is observed. Thus we havea high-velocity persistent current coexisting inclose proximity with one of very low (or vanishing)velocity. The persistent currents in the powderare observed not to decay, ard we thus assert thatthey are trapped, or caged, in the superleak' Itis important to recognize that the angular momentumis eonstant despite the fact that there is a free ex-change of superfluid mass across the boundary ofthe superleak,It has been shown that two sound modes can prop-agate in the annular channel.6 One is essentiallya modified second-sound mode rvith a velocity C,given by

12 -e2plL-(t-P\al /r\rr=12;O:T.nJV, \r/where C, is the velocity of second sound, P is theporosity of the powder (the fraction of the volumeof the superleak which is free), and p and ps are thedensities of HeII and the superfluid component, re-spectively. Note that C rt> C, .The other mode is an interpolated first-fourthsound mode with a velocity Crn given by

where C, and Cn are the velocities of first andfourth sound, Note that Cr>Cu>C4,Since both modes have components of first andsecond sound in the free region and fourth sound inthe packed region, first-, second-, or fourth-soundtransducers may be used to generate and detectboth modes. We have used both first and second-sound condenser transducers in our experiments, withcomparable results. We have experimentally verifiedthat Eqs. (1) and (2) arecorrect towithin about 2V0.1When there are persistent current velocities u,and zr, in the free and packed parts of the channel,the Doppter-shifted velocities of these modes ares

c,, = (crr)o +(pn/p)ut, (3)c..=(c..\^*p"(L-d)ut+P(2=_ )duo . (4)14-\Lt4to- p L -rt +P(2_P)d

where the zero subscript denotes the value in theabsence of current and the plus and minus signsrefer to propagation with and against the flow, re-spectively. zr, is determined by measuring thesplitting of the azimuthal C, modes and then zt, bythe splitting of the azimuthal Cro modes. sThe situation is particularly simple when ar>> zr,and L - d is comparable with d, as in our experi-ments. Then

Crr= (Qr,)o *(P,/P)ur,Cr+=(Cu)o *A(p"f p)uo,

whereA= P(2 - P)d(L-d)+P(2-P)d

Persistent currents were generated by two dif-ferent methods.Mettpd A. While rotating at constant speed thesystem is cooled through ?, and brought to rest atsome low temperature. Measurements are madein the stationary state.MethodB. While stationary the system is cooledto some temperature below 7^. It is then rotatedand measurements are made while rotating.The outstanding and striking result of rather ex-tensive measurements was that the Crn mode shows

(5)(6)

(7)

(2)

t2 1739^z 12 n2 p(L - D) + p"Pdtu=LrL46Tpg4\+CFgl

8/13/2019 1 Trapping of Persistent Currents in Superleaks 1975

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JOSE PH HEISERMAN AND ISADORE RUDNICK t2t140

LFIG. 1" Annular resonator for persistent-currentmeasurements" The bottom of the resonator is packedto a depth rJ with a powder of nominal grain size 500 A"L is 5 mm as is the width of the annulus. E-5.25 cm.The velocity of the persistent currents in the packed andfree regions, uo and a1, respectively, can be determinedby measuring the splitting of C11 and C14 modes and usingEqs. (3) and (4), or (5) and (6).considerable splitting and the C, mode showed nosplitting at all, although peak broadening was ob-served, This broadening may be due either to aDoppler shift or the attenuation of second sound dueto the presence of vortex lines.10 Using method Aat T =2.10 "K and rotation speeds whieh produ-cesaturated critical velocities in the powder, themaximum broadening was observed with the maxi-mum d=4 mm. If it was caused by a Doppler shift,then 2r= 1 cm/sec, This we shall see is two ordersof magnitude smaller than ur, If part of the broad-

FIG. 2" Velocity of the persistent current in the packedpowder up as a function of time a-fter its formation. TheIine is a least-squares fit to the experimental points.Within experimental accuracy there is no decay. Thepersistent current in the free region above the powder isL cm/sec or less" The saturated critical value of zro un-der the conditions of the experiment (d = Q mm, 7 = 1. 3K) was 85 cm/sec.

TABLE I. Measured saturated critical velocities forvarious values of rl and their associated porosities. It isknown that z2uas decreases as P increases (Ref. 9), andit is possible that this accounts almost entirely for thedifference in the values.- ,,'L o.r o"z o": o.+;;-6 *-L. 1

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t2 TRAPPING OF PERSISTENT CURRENTS IN SUPERLEAKS

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current in the superleak with a vanishing one in thefree region, there is a predilection to ascribe thisto pinned vortex lines. From our discussion it isapparent that the really significant pinning occursat the boundary rather than in the body of the super-Ieak. After all, in the two left-hand figures inFig, 3 the vertical vortex lines are in oppositeparts of the apparatus, yet in both cases the per-sistent current resides entirely in the superleak.The equilibrium thermodynamic state is one inwhich solid body rotation is approximated by theoccumence of the necessary density of vortex lines.Although this is always achieved in the free regionand the required nucleation centers are present atthe top surface of the superleak, the vortices donot extend down into the powder in the required den-sity. This is possible because the pinnning is ade-quate for the presence of the horizontal sheath.Our results elearly imply that persistent currentscan exist in a superleak which is not in a container-a bare superleak-since the existence of the currenthinges on the occurrence of a vortex-line sheathat the boundary of the superleak. The bottom right-hand schematic in Fig. 3 shows the vortex-Iine con-figuration in this case. Now vortices terminate onthemselves. If large velocity-persistent currentsoccur in a bare superleak, this opens the possibil-ity of increasing gyroscopic effects by the elimina-tion of unnecessary mass, It, moreover, raisesinteresting questions about the nature of the pres-sure fields which are required to cage the currentswithin the vortex sheaths. We shall be looking intothese matters.Joseph Rudnick derived the results for the Dopplershift, Eqs. (3) and (a). We had numerous valuableconversations with Seth Putterman,

IIETHOO A

[}fl:E-m-H rETHooA [9s*ffi*EIffiEEFIG. 3. Schematic of vortex-line configurations. Thedirection of the persistent current is indicated" The di-rection of rotation of the resonator (Fig. 1) is the sameas that of the current in method A and opposite in methodB. c.lcl is the maximum velocity of rotation of the reso-nator for occurrence of the Landau state in the superleak.favorable in the top left-hand figure. There is asheath of vortex lines at the top surface and almostcertainly in the superleak. The thiekness of thissheath is measured in particle-grain diameters(which are large compared to the coherence lengthof helium) and is therefore small eompared to d.The persistent current is confined to the superleakand the vortex line sheath can be considered neces-sary or responsible for this,In the bottom left-hand figure the absence of per-sistent current in the top free part requires a vor-tex density that gives the solid body rotation of fre-quency oJ. Since the persistent current in the super-leak is in the Landau state, there are no vorticesin the superleak, Again there is a horizontal vor-tex-Iine sheath in the superleak at its top surface.When o > 0.16, there are vortices in the superleakbut with a density lower than that in the top regionand the top surface.In describing the coexistence of a large persistent*Supported by ONR Contract No. N00014-69-A-0200-4014 and National Science Foundation Contract No"DMR7500761 -000.lThe veloclty of the persistent currenb is the differencein the steady-state velocities of the normal and super-fluid components.2J. C. W"uve., Phys. Rev. A 6, 378 $972]r. In carefulexperiments in our orvn laboratory using the Dopplershift of second sound rve have never found evidence fora current exceeding this"3Linde B 0.05-pm alumina polishing powder, Union Car-bide Corp.aJ. S. Langer and J. D. Reppy, in P'rogress in Loro Tem-

PeT ature Physics, edited by C. J. Gorter (North-Hol-Iand, Amsterdam, 1970), Vol. VI, Chap. I.sln previous persistent-current experiments, unavoidably,there were small spaces unfilled with superleak, par-ticularly when Vycor was used U. o. neppy (privatecommunication),l. Thus there is circumstantial evi-dence for persistent currents in cavities incon-rpletelyfilled with superleak.

6J. Rudnick, I. Ruclnick, and R" Rosenbaum, J. LolvTemp. Phys. tA, qt Q9741,TUnpublished"sJoseph Rudnick (private communication).eH. Ko;ima, W. Veith, S. J. Putterman, E. cuyon, and. I. Rudnick, Phys. Rev. Lett. 27,714 (197L).10H. e. HaU and W. Ir. Vinen, Proc. R. Soc. A238,204(1e56).11H" Kolima, W. Veith, ll. cuyon, and I. Rudnick, Pro-ceerl.ings of the Thi.?'teenth Intevnational Conference ofLou) Tenperature Phgsics, Boulder, 1972, edited byW. J. O'sultivan, K" D. Timmerhaus, and E. F. Ham-mel (Plenum, Ne*, York, 1974), Vol. 7, p. 279.l2whether af is very small or identically zero is of noimportance in the present context. If it is the former,from past experience we are sure it can be reduced tozero without significant change in 2r, by counter rotat-ing the resonator \,vith some small velocity and then re-turning it to rest again. In fact, one should be able toget z1 and op rvith opposite signs by this procedure.