A technique for photographing vortex positions in ... - Springer Link

Journal of Low TemperaturePhysics, Vol. 39, Nos. 5/6, 1980

A Technique for Photographing Vortex Positions in Rotating Superfluid Helium* Gary A. Williams~ and Richard E. Packard Physics Department, University of California, Berkeley, California (Received November 28, 1979)

A technique is described for photographing the positions of quantized vortex lines in rotating superfluid helium. Electron bubbles are trapped on the lines, and then extracted through the free surface and accelerated into a phosphor screen. Details of the apparatus are presented, along with examples of the data and the data collection techniques. 1. I N T R O D U C T I O N The existence of quantized vortex lines in rotating superfluid helium is perhaps the most remarkable manifestation of the quantum nature of this fuid. The vortex lines were first postulated by Onsager and Feynman I to resolve an apparent discrepancy in the Landau two-fluid model. 2 In the set of equations of motion for the liquid, Landau had included a restriction on the velocity of the superfluid component in requiring that V x vs = 0. For the case of a rotating cylindrical container (a "bucket") filled with He II, the condition V x Vs = 0 yields the solution Vs = 0. The superfluid component is predicted to remain at rest independent of the bucket rotation. However, this was not the apparent behavior found experimentally. Measurements of the angular m o m e n t u m of the liquid 3 and of the shape of the meniscus in a rotating container 4 indicated that the superfluid component did in fact appear to come into solid body rotation, in contradiction to Landau's assumption of potential flow. The explanation of these experiments came from considerations of the quantum nature of a Bose fluid at low temperatures. L o n d o n 5 first pointed out the similarities between superfluid helium and the condensation (in m o m e n t u m space) of an ideal Bose gas, in which the lowest quantum state is *Work supported by the National Science Foundation, Division of Materials Research. tPrcsent address: Physics Department, University of California, Los Angeles, California. :~Alfred P. Sloan Research Fellow. 553 0022-2291/80/0600-0553503.00/09 1980PlenumPublishingCorporation

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Gary A. Williams and Richard E. Packard

occupied by a macroscopic number of particles. This idea of the superfluid as a coherent quantum state was extended by Onsager and Feynman, ~ who postulated that the wave function describing the superfluid in motion could be written in the form to = e i+to0, where tOois the ground state of the fluid at rest, and the phase 4~ is related to the superfluid velocity field by vs = (h/27rm) V~b. Here h is Planck's constant and m is the mass of the helium atom. Requiring the wave function to be single-valued means that the change in phase around a closed path must be an integral multiple of 2~r Aq5 = + V~b 9 dl=27rn,

n =0, 1,2,...

(1)

This restricts the circulation of the liquid to the quantized values

K = + vs 9 dl = n h / m

(2)

w

This macroscopic quantization has been well verified experimentally, in direct measurements of the circulation, 6 and from an analysis of vortex ring motion in the liquid. 7 The Onsager-Feynman model pictures rotating superfluid helium as being threaded by an array of quantized vortex lines. Each vortex line has a velocity field v~ = KO/2"n'r, where r is the radial distance from the center of the vortex, and 0 is a unit vector in the azimuthal direction. This is the same flow pattern as a classical vortex, except that the circulation K is quantized. Since fl 2_~ 2 the energy of the vortex is E = j ~p~v~ a. K it is energetically favorable for each vortex to have a single quantum of circulation, K = h i m . The superfluid density falls to zero at the core of the vortex line (corresponding to a node in the macroscopic wave function) over a coherence length that has been measured 7'8 to be about 1/~. An array of these vortex lines can simulate solid body rotation if the density N of lines per unit area gives a net circulation equal to the circulation per unit area of solid body rotation, NK = 2o9, or N = 2o9/K. For o9 = 1 sec -~ this is a density of 2000 lines per cm 2, a mean spacing between lines of - 0 . 2 mm. Such an array of vortices satisfies Landau's postulate of potential flow because the velocity field of each vortex satisfies V x v~ = 0, and yet the net velocity field found by summing over all of the vortices is that of solid body rotation, in agreement with experimental results. The existence of such a vortex system was first demonstrated by Hall and Vinen 9 in measurements of second-sound attenuation in rotating helium. The presence of the vortices was also detected through the trapping of negative ions on the vortex lines. An ion current in the liquid moving perpendicular to the rotation axis was found to be reduced, 1~the attenuation being proportional to the rotation speed, and the "lost" current was found to be propagating along the vortex lines. Subsequent measurements of the ion trapping cross section, la lifetime, 12 and mobility ~3 served to verify the

A Technique for Photographing Vortex Positions

555

correctness of the hydrodynamic description of the vortices (extensive reviews of this work can be found in Ref. 14). The detection of single vortex lines in rotating helium was accomplished by Packard and Sanders, ~5 who made sensitive measurements of charge trapped on the lines. They found quantum steps in the amount of charge collected as the rotation was slowly increased from rest and successive vortex lines entered the liquid. Once the existence of vortices was well established, a natural question to ask was how the vortex lines are arranged in the rotating liquid. In equilibrium each vortex should be stationary in the rotating reference frame (and hence in solid body rotation as viewed from the lab frame). If a line deviates from solid body motion, it will be acted upon by mutual friction forces. The mutual friction arises from the scattering of normal fluid excitations by the vortex core, and is proportional to the relative velocity between the vortex line and the normal fluid. 16 The force is dissipative, and tends to drive the vortex back to its equilibrium position. ~7 The precise equilibrium arrangement of the vortices in the array is computed by varying the vortex positions until a minimum in the free energy E - ~o 9 L is achieved, is The energy E and angular m o m e n t u m L due to the vortices are given by E = E E(r~), i

E(ri) =

I 1 2 ~ps[vs(r-ri)] d3r

(3)

L = ~ L(ri),i

L(ri) = I psr • v s ( r - r i ) d3r

(4)

where vs ( r - r i ) = K ~ 2 ~ ' ] r - r i ] and ri is the position of the ith vortex. For a two-dimensional infinite array, Tkachenko showed that a triangular arrangement of vortices had the lowest free energy, 19 and he also showed that such an array was stable against small perturbations, 2~ whereas square and other lattices were unstable. For the case of a cylindrical container of finite diameter the presence of the boundary walls affects the flow field of the vortices. Calculations of the vortex positions taking this into account have been carried out by Hess, 21 Stauffer and Fetter, 22 and Campbell and Ziff. 23 In general the arrangements are found to take the form of concentric circles about the center of the cylinder. Campbell and Ziff 23 give detailed predictions for the equilibrium positions of arrays containing between 1 and 30 vortices. They also describe metastable states of the vortex array, which are local minima of the free energy separated from the equilibrium ground state by substantial energy barriers. The problem of experimentally observing the vortex positions in rotating helium has proven to be quite difficult. The analogous experiment to visualize the magnetic fluxoids in type II superconductors yielded striking

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Gary A. Williams and Richard E. Packard

photographs of a triangular lattice array. 24 A number of proposals to determine the vortex arrangement in helium have been advanced, 25 and several attempted experiments have been reported. 26 It is the purpose of this paper to describe the technique which has allowed the observation of the vortices in superfluid helium. The previous reports of this work 27-3~ have been short letters which have not allowed a full description of the experimental method. Section 2 outlines the basic technique that is used, Section 3 details the apparatus, and Section 4 gives examples of the data collection process.

2. BASIC TECHNIQUE The technique used in this experiment utilizes the fact that electrons can be trapped on the core of a vortex line. An excess electron introduced into liquid helium forms a cavity (or "bubble") with a radius of about 17 ,~.31 This arises from the Pauli repulsion between the electron and the filled shell electrons of the He atoms. In the 1/r flow field of a vortex, the electron bubble experiences a Bernoulli force which drives it to the core of the line. 1 2 The pressure in the liquid near the line is given by P = Po-spsvs, where Po is the pressure at r = oo. The force on the electron bubble in the vicinity of a vortex is then

F= Ib.a.PdA= Ib.v.V P d V = - V Ib.v.89

dV

where d A and dV are elements of the bubble area (b.a.) and volume (b.v.). The bubble becomes trapped in the potential well 32

U(r) = fb.v--89

dar'

where r is the distance from the vortex core to the center of the bubble and r' is the distance from the core to the volume element d3r '. The depth of the well at r = 0 (where ps falls to zero over the vortex core radius a0) is 32

U(O)=Rd