Two critical velocities in superfluid helium?

Volume

27A, number

TWO

PHYSICS

8

CRITICAL

9 September

LETTERS

VELOCITIES

IN SUPERFLUID

1968

HELIUM?

M. LUBAN and S. MILLER Department

of Physics,

Bar-Ran

Received

It is shown that Amit’s two observable critical

University,

Ramat-Gan,

Israel

24 July 1968

proposed thermodynamic potential for superfluid helium implies the existence velocities. Specific experimental tests of this prediction are proposed.

Very recently Amit [I] constructed a thermodynamic potential function based on the Landau theory of second-order phase transitions which yields the known (TX - T)i temperature dependence of both the superfluid number density p s and the critical velocity Trc of He II just below the lambda point [2]. In this note we show that Amit’s thermodynamic potential predicts a second observable critical velocity for He II with a temperature dependence of the same power-law form. We also note that a modified version of an experiment [2] performed recently by Clow and Reppy could decisively test Amit’s theory. In fact the existence of two critical velocities would explain an effect found by Reppy [3] and which had heretofore not been understood. The thermodynamic potential per unit volume proposed by Amit for temperature T < TX is +(T, us, $4 = *I(T) + [A(T) +%&‘2 + B(T)rc/4 + C(T)Ic/6 ,

+ (1)

where + is real and spatially uniform, m is the mass of a helium atom, A(T) = -~(TA- T)$, B(T) = -fl(T~- T)j, C(T) = +yand (Y, p, y > 0. The physically relevant values of +2, those minimizing cf,for given temperature T and superfluid velocity 21 , are easil found to be rc/2=-(B/3C) + + [(B/3C$(A +fmvsg)/3C]i for 0 vc2 a single value of J/2 minimizes a’, and thus ps is unambiguously defined. (See fig. 1.) For values of us between vcl and vc2, where @ has two minima, the identification of ps

of

Fig. 1. Schematic plot of @-*I as a function of 9 for given T and various values of vs. The curves are symmetric about q/=0 and, in discussing the number of minima, we need not be concerned with the region @< 0. The curve for us < vcl has a single minimum. The inner two curves, corresponding to vcl vc2 the system is normal ( p, = 0) and the corresponding curve has a single minimum at @ = 0.

is not immediately obvious. Amit [l] arbitrarily chose ps =0 for us 2 0~1, and he identified vcl with the critical velocity measured recently by Clow and Reppy [2]. We take exception to Amit’s choice of ps = 0 for us >vcl. In our opinion, the value of ps when us lies between vcl and vc2 depends crucially on the previous history of the system. In fact, we show that both vcl and vc2 could be observed in a revised version of the experiment of Clow and Reppy. We suppose that a persistent current of a prescribed angular momentum or equivalent linear 501

Volume

27A, number

8

PHYSICS

velocity v0 is prepared at some standard temperature T, (< TX). With the container held stationary, the system is slowly heated to TH, T, TB, vf will drop discontinuously from ~0. In the latter case the value of vf is expected to be either Vet or VcI(TR) depending on the rate at which the system is heated towards TH *. The experiment should then be repeated for a second value of v0, leading to a new value of TB. By repeating the experiment for different initial veloci-

* The quantity Auf is very large: Auf = vc2(TR) + Thig is obtained usln the va ues - Vcl(TB) = 0.95 Vo. 3f.ergcm/oK, a (OK)-3, /3=0.58x10a!=1.‘7~10-~~er and y = 1.06x10- 81 erg cm6. These values follow from the experimental results for ps, the discontinuity in C, at T). and v, appropriate to a pore size of 0.211. Notepthat &r valu& for &, p and y differ somewhat from those of Amit 111. The discontinuity hvf could be observed only if one measured uf as a function of TH for fixed v0. There is no indication in ref. 2 that the authors performed measurements in this fashion. Furthermore, it is possible to construct a more general form for which will also yield two critical veloMoreover, cities but such that &J is left arbitrary. if p is negative then oni y one critical velocity occurs.

*****

502

LETTERS

9 September

1968

ties v0 and noting the dependence of TB upon vo, one will obtain the temperature dependence of vc2. Preliminary support for Amit’s proposed Q and the notion of two critical velocities in He II is provided by an earlier experiment of Reppy [3]. Here the system was slowly heated towards some maximum temperature T,, (
References 1. D. J. Amit, Phys. Letters 26A (1968) 466. 2. J. R.Clow and J.D.Reppy, Phys.Rev. Letters 19 (1967) 291. 3. J. D. Reppy, Phys. Rev. Letters 14 (1965) 733.