ON THE FREE ENERGY OF VORTICES IN ROTATING He II

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Submitted on 1 Jan 1978

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ON THE FREE ENERGY OF VORTICES IN

ROTATING He II

L. Campbell, R. Ziff

To cite this version:

JOURNAL D E PHYSIQUE

Colloque

C6,

supplement au n°

8,

Tome

39,

aout

1978,

page

C6-222

ON THE FREE ENERGY OF VORTICES IN ROTATING He II

*

L.J. Campbell and R. Ziff

Los Alamos Scientific Laboratory, University of California Los Alamos, New Mexico 87545 USA

Résumé,- Nous avons calculé les configurations stables et meta—stables et les énergies libres d'un arrangement de 2 à 18 tourbillons dans un récipient tournant d'hélium superfluide. Nous ayons aussi obtenu l'énergie d'activation pour passer d'une configuration à l'autre..

Abstract.- We have calculated the stable and metastable configurations and free energies of -vortex patterns for 2 to 18 vortices in a rotating bucket of superfluid helium. We have also found the free energy barrier between various vortex patterns.

1. INTRODUCTION.- In a rotating cylindrical bucket of He II, the superfluid velocity field consists of a two-dimensional array of quantized vortices, each with a circulation K = h/ro. For a given angular velo-city u and bucket radius R the optimum number N and pattern of these vortex lines can be determined by minimizing the free energy function F / ' ] / . For small N the patterns are simple, permitting analytic calculation of the free energy as done by Hess for N = 1 to 8 / ] / . We have extended these results to N = 18 in response to recent experimental progress in seeing such patterns /2/. This was done by a com-puter routine which iteratively moved N initially random vortices inside a circular boundary toward the direction of decreasing free energy until the motion sufficiently converged. Stauffer and Fetter used a similar procedure for certain values of N /'3/, We also extended this procedure to find the barrier in F between certain states.

2. STABLE PATTERNS.- Curves of F, in units of pK2/4ir, as a function of 0), in units of IC/2TTR2, are plotted

in figures 1 and 2 for N = 2 to 18. The ratio of a, the vortex core radius, to R enters F as the additive term -N log a/R. Dropping this term (by setting a/R = 1) we get the "raw data" of F which is plotted in figure 1. For larger N there is more than one local minimum for F and, correspondingly, more than one pattern. The patterns of lower F were always found more frequently by our random procedure than those of higher F. Presumably, these metastable patterns are also experimentally accessible. Their

TH 1 1 1 1

0 10 SO 30 4 0

Angular Velocity

Figure 1

F's are too close, however, to be distinguished in the figures.

Choosing a typical value of a/R = 1 0 we get the curves of F in figure 2. Note that for a given to there are many lines of F near the minimum. "Thus, it should be common for (metastable) patterns of "incorrect" values of N to occur experimentally.

Figure 2

Work performed under auspices of USD0E As to decreases, each curve in figure 1 ends

at a point w (N) where that pattern becomes unstable. This occurs when F, which is normally a minimum in the 2N-dimensional space of the vortex coordinates, becomes, instead, a saddle point. The determination of w (N) is computationally quite delicate but our

errors in w (N) should not exceed "5 ; for N = 2 to 6 our values for wc(N) agree to ?.l with the analytic results of Havelock / 4 / .

3. SADDLE POINTS AND FREE ENERGY BARRIERS.- To pass between various stable and metastable patterns of either the same or different N, a free energy barrier must be surmounted at a saddle point in the free energy function. These free energy barriers have not heretofore been calculated. We were able to calculate this quantity by slowly moving one vortex from its position in a stable (or metastable) pat- tern to a radius characteristic of a second pattern while monitoring the change in free energy. This method of "adiabatic" displacement allows the re- maining N

-

local free energy minimum consistent with the chan- ging position of the first vortex. If a path exists between the two patterns with the given choice of

vortex displacement then a saddle point in the free energy is found. An apparent maximum in the free energy must be traversed a few times in both direc- tions to verify that it represents a saddle point because the majority of all paths lead only to a distortion of the initial pattern. In principle, more than one saddle point may exist between two patterns and the ones we list in Table I may not always be the ones of lowest energy. In table I the

(meta) stable states are listed in order of incre- asing free energy : F l , F2, F$.

Table I

Reduced free energy of vortex patterns and saddle points.

References

/l/ Hess, G.B., Phys.Rev.

161

Bull.Arn.Phys.Soc.

3

/3/ Stauffer, D., and Fetter, A.L., Phys.Rev.

168