Chapter 1 Intrinsic Critical Velocities in Superfluid Helium

CHAPTER 1

INTRINSIC CRITICAL VELOCITIES IN SUPERFLUID HELIUM BY

J. S. LANGER

PHYSICS DEPARTMENT, CARNEGIE-MELLON UNIVERSITY, 15213 PITTSBURGH, PENNSYLVANIA AND

J. D. REPPY LABORATORY OF ATOMIC AND SOLIDSTATE PHYSICS, CORNELL UNIVERSITY, NEWYORK14850 ITHACA, CONTENTS: 1. Introduction, 1. processes in superiiuids, 2. 4. Specific models, 19.

- 2. General aspects of a theory of thermally activated - 3. Experimental study of superfluid flow states, 10. -

1. Introduction The occurrence of superflow, that is, persistent, non-dissipative current, is undoubtedly the most dramatic feature of the superfluid phase of liquid helium. Yet, a really complete understanding of the stability of superflow has been very elusive. It has turned out to be remarkably difficult, for example, to predict the conditions under which stability breaks down, or even to specify the mechanism which is responsible for the breakdown. Recent attempts to solve this stability problem have led to modification of some of the most basic ideas concerning the nature of superfluidity. Our purpose in this review is the describe and evaluate some of this recent work. Our main thesis is that stable superflow is a non-equilibrium phenomenon, and must therefore be discussed in the language of the statistical mechanics of irreversible processes. This is something of a departure from previous theories of superfluidity which, have for the most part, been equilibrium theories. But neither recent progress in computing many-body wave functions nor the elegant phenomenological pictures of long-range order in 1

References p . 34

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1.

S. LANGER AND 1. D. REPPY

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quantum fluids are directly applicable to the stability problem. The most nearly relevant result of the equilibrium many-body theory is the low-energy excitation spectrum which, via the Landau criterion, can be used to predict critical velocities. Such critical velocities, however, are always much too large. The trouble seems to be that such calculations still deal only with very small deviations from equilibrium; whereas the experimentally observed onset of dissipative processes, especially near the R point, may involve very large fluctuations. This is the point of view which we shall attempt to establish in this article. In section 2 we shall set down the basic assumptions of a non-equilibrium theory of superflow in as general a manner as possible. We shall avoid restricting the discussion to any specific model, and shall attempt only to provide a theoretical framework for the analysis of experimental data. A survey of relevant experimental results will be presented in section 3. As we shall see, the basic theoretical hypotheses seem to be well verified. In section 4, we shall describe several attempts to compare the experimental data to more specific models of the fluctuations which may occur in a superfluid. This comparison brings to light a number of intriguing theoretical problems which seem to be of considerable fundamental importance. 2. General aspects of a theory of thermally activated processes in superfluids

In this section we shall outline what we believe to be the minimum basic assumptions of a non-equilibrium theory of superflow. Throughout this discussion, the system of interest will be helium in a ring-like container which, for simplicity, we shall take to be equivalent to a cylinder (or a box) of length L and cross-sectional area A , with periodic boundary conditions imposed in the direction parallel to L. Macroscopic currents always flow around the ring or parallel to the axis of the cylinder. It is assumed that the container is packed with small-mesh filter paper, or jewelers rouge, or some other porous substance which will ensure that a normal fluid is always stationary with respect to the container. The porous substance may also effect thermodynamic equilibrium between the helium and its environment. 2.1. SUPERFLUID STATES

Our first major assumption is that any thermodynamically stable or metastable state of the superfluid is uniquely characterized by the velocity of superflow, and that this velocity remains invariant when the system undergoes reversible transformations. Here we are invoking a basic feature of both the

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INTRINSIC CRITICAL VELOCITIES

3

phenomenological model of long-range order in superlluids, and the firstprinciples, microscopic theory. For example, in the most naive statement of the microscopic theory, one says that a superfluid state corresponds to a Bose condensation of a finite fraction of the helium atoms into one singleparticle state of wave-vector, say, k. Then the superfluid velocity is

us = hk/m,

(2.1)

where m is the mass of helium atom. Of course, we know that the many-body wave function is very much more complicated than this, and that particleparticle interactions are actually essential for true superfluid stability. But more complete many-body calculations remain consistent with the idea that each state of stable superflow is characterized by a single well-defined wavevector which, in turn, is directly related to the velocity according to eq. (2.1). This assumption has the following operational signifcance. Suppose, as in most of the experiments to be described in section 3, that the helium is set in motion with velocity u by moving the container at this velocity while the temperature is above the lambda-point. Then the system is cooled to some T well below T,; and, finally, the container is brought to rest. At all times it is understood that the helium remains in thermal contact with the stationary laboratory environment. Presumably, the superflow persists after the container is stopped, and has velocity usz u. The momentum density for this state of the system is j = es(%, (2.2) where esis the superfluid density. (One might also have to allow a us dependence of e..) The crucial implication of our assumption is that, if we now vary the temperature T - within certain bounds - the superfluid retains its velocity us. SpecScally, eq. (2.2) remains valid with us remaining constant and only e, varying reversibly with T. In the simplest microscopic picture, helium atoms enter or leave the condensate, but the characteristic wavevector k remains fixed. This situation is illustrated in the us-T diagram shown schematically in fig. 1. Because of the periodic boundary conditions, the allowed superfluid velocities are given by eq. (2.2) with k =2an/L, where n is an integer. Each of these allowed velocities corresponds to a distinct state of the system, and each such state can be depicted by a solid horizontal line on the diagram. As long as the state persists without decay, the system point may be moved reversibly back and forth along the horizontal line by varying the temperature. References p . 34

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1. S. LANCER AND J. D. REPPY

- --

2.2.

1,

2

/

/

Fig. 1 .

[CH.

/

/

Possible persistent current flow velocities, us, are shown as a function of temperature.

METASTABILITY

Our second major assumption is that any state of non-zero supedow is metastable in the sense that there is always some mechanism, intrinsic to the system itself, by which the system can make a transition to a state of lower velocity and, thus, lower free energy. In other words, there is always a finite probability for vertical transitions in fig. 1 . Such transitions may require, however, that the system undergo highly improbable fluctuations, in which case the transition rate may be extremely slow. In fact, it makes sense to draw the horizontal lines in fig. 1 only when the lifetimes of these states are very much longer than the time required or available for experimental observations. Outside of some region of low temperatures and small velocities, the metastable states will decay more rapidly than they can be measured and, in a sense, do not exist at all. The boundary of this region of the v,-T plane is indicated by the dashed line in the figure. This boundary turns out to be quite sharply defined, and may be identified as the critical velocity curve, V$,C(T). This picture of metastability is directly analogous to the more conventional pictures of, say, a supersaturated vapor3, or a simple ferromagnet whose spins are aligned opposite to the direction of the external field. Let us look at the example of the supersaturated vapor in a little more detail. We know that, if one carefully compresses or cools a vapor through its condensation point, the vapor phase may persist metastably until there appears a droplet of liquid sufficiently large to nucleate condensation of the entire sample. The probability that such a droplet will appear spontaneously - as a thermally activated fluctuation in the vapor - is proportional to the Boltzmann factor exp( - Ea/kBT),where the activation energy Ea is the free energy of the

CH. 1,

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INTRINSIC CRITICAL VELOCITIES

droplet. This free energy, for a droplet of radius r, can be written in the form

E,(r) = -Ar3+Br2, (2.3) where - A is proportional to the difference in free energy between vapor and liquid phases, and appears here with a minus sign because we assume the liquid phase to be stable. The coefficient B is proportional to the surface energy and is positive. Note that E,(r) rises to a peak at rc =2B/3A and then decreases and becomes negative. Droplets with r less than rc will find it energetically favorable to diminish and disappear. On the other hand, once a droplet is larger than r,, it will tend to grow indefinitely, thus initiating the macroscopic condensation. We therefore identify Ea(rc)as the activation energy, E, , appearing in the Boltzmann factor. Clearly, if the coefficients A and B are such that Ea(rc)is large compared to k,T, the transition rate will be very small, and the metastable phase is likely to persist for a very long time. The process we have described above is known as homogeneous nucleation, as opposed to inhomogeneous nucleation in which the droplet forms on a foreign object in the sample or an irregularity on the walls of the container. For example, the droplets formed along the path of a charged particle moving through a cloud chamber are nucleated inhomogeneously. Similarly, in liquid helium, the vorticity which is generated at the boundaries limits the velocity of superflow in a manner which is analogous to inhomogeneous nucleation of condensation. We shall have more to say in section 4 about the way in which a moving vortex line causes transitions from one state of superflow to another; and this later discussion should help to clarify the analogy between the liquid droplet and vorticity in a superfluid. But the phenomenon which we wish to consider in this paper is the analog of homogeneous nucleation, that is, the thermally activated fluctuationswhich might occur spontaneously in a superfluid and initiate decay of the superflow. The experiments that we shall discuss in section 3 are designed specifically to isolate this intrinsic behavior of superfluid helium. 2.3. ERGODIC HYPOTHESIS The final ingredient of our general theory is the ergodic hypothesis. We have already invoked this hypothesis implicitly in the discussion of the supersaturated vapor; but its special role in the present theory merits extra emphasis. The hypothesis, as applied to a system in equilibrium with a constanttemperature bath, states that any configuration which can be achieved by the References p . 34

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J. S. LANGER AND I. D. REPPY

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system will, in fact, be achieved with a probability which depends only on the energy of the configuration. Specifically, the probability of finding the system in a given configuration with energy E is proportional to the Boltzmann factor, exp(E/k,T). The ergodic hypothesis is extremely valuable in the present kinds of calculations because it enables us to avoid making many detailed dynamical calculations. In discussing the supersaturated vapor, for example, we did not need to inquire how the critically large droplet actually formed, We simply started with the knowledge that the system was in the vapor phase, and then computed that the least energetic fluctuation which could nucleate the condensation, i.e. formation of a droplet, would occur with a probability exp[ - JZa(rc)/kBT]relative to the probability of the uniform vapor configuration. Similarly, in the case of the superfluid, we shall not inquire about matrix elements or perturbation-theoretic transition rates, but shall simply assume that there exists some fluctuation which can initiate a transition from one state of superflow to another. We shall be able to make a great deal of progress knowing only the energy of this fluctuation. 2.4. FLUCTUATION RATE

As mentioned in the Introduction, we shall postpone the discussion of specific models of the current-reducing fluctuations until section 4. For the present, let us simply assume that we know the activation energy, say Ea(us, T), for the required fluctuation. The only condition that we shall impose upon E, is that it be independent of the size and shape of the system. That is, the fluctuation must be localized like the liquid drop, and not indefinitely extended like a vortex line of macroscopic length. This condition is clearly required in order for the picture of homogeneous nucleation to make sense. If E, is macroscopically large, the probability of the fluctuation occurring spontaneously is completely negligible. Given the activation energy, E,, we can write down a formula for R, the rate at which the fluctuations are occurring: R = ALvo exp [ -E,(u, , T ) / k ,TI,

where vo is some fundamental frequency of events per unit volume. The firstprinciples evaluation of the quantity vo is a matter of considerable theoretical interest at present. For present purposes, however, we shall need only a rough order-of-magnitude estimate for y o , which we shall take to be something like the atomic collision frequency times the density. Note that we have factored out explicitly the volume of the sample in writing (2.4).

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7

We now make the natural assumption that each fluctuation carries the system only from one metastable state to the neighboring one of lower velocity; i.e. the system jumps down only one step at a time in the u,T diagram. Then the change in velocity per event is (for us>0):

Av,

and eq. (2.4) becomes

h~ _ -- - dus

dt

m

=

-h/mL;

v exp , [-En(vs, T I / TI, ~ ~

which is a differential equation for v,. Equation (2.6) is central to all of the analysis in this paper. 2.5. DECAY OF SUPERFLOW The general properties of the solution of eq. (2.6) can be predicted without detailed calculations. First, note that the prefactor on the right-hand side must be extremely large on the scale of velocities observablein the laboratory. Thus we shall be interested in values of E much larger than k,T - but not necessarily many orders of magnitude large because of the rapid variation of the exponential function in (2.6). Next, note that Enshould be a decreasing function of both us and T because we expect decreasing stability at higher velocities and higher temperatures. Combining these two observations, we can deduce that du,/dt varies rapidly from extremely small to extremely large values as us or T increase. In particular, the right-hand side of (2.6) should be experimentally observable only within a narrow region in the neighborhood of the critical velocity curve in fig. 1. The critical velocity itself can best be defined, at a fixed temperature, as that velocity below which du,/dt is so small as to be unobservable with available experimental apparatus. Alternatively, we can define to be that velocity above which du,/dt is so large that the metastable state cannot be said to persist for observable times. One of the more interesting implications of our theory is that it is not possible to be more precise than this operational definition of u s , To express the above arguments in mathematical language, we denote the characteristic observable value of (du,/dt I by the symbol 9, . Then vs,,(T) can be obtained from eq. (2.6) by solving:

m.

Returning to (2.6), and assuming, as mentioned in the last paragraph, that References p . 34

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3. S. LANGER AND J. D. REPPY

52

we need to know EB(vs,T ) only for velocities near us,,, we expand about v, to obtain do, - z - % exp [a(vs- vs, J], (2.8) dt where

Suppose that the superfluid is initially set in motion at velocity u,, and then warmed to a temperature T corresponding to the point P in fig. 1, P being outside the critical velocity curve. Solving eq. (2.8), we find that the subsequent decay of v, is:

In [I+a+,t expa(v,-v,,)].

c s ( t ) = u0-a-'

(2.10)

For times long compared to (a@,)-' e x p [ x ( ~ ~ , ~ - vwe ~ ) ]have: , v, z vs, - a- in (a@,t),

(2.11)

which exhibits the logarithmic decay or 'cheep' characteristic of this kind of thermally activated process. Note that the fractional decay per decade is

Ak+z* V,

- vS,=

- 2.303

(2.12)

2.6. DRIVEN MOTION Equations (2.10) and (2.11) describe the free decay of superflow in the absence of any driving force. It is also useful to apply our basic formula, (2.6), to the case of driven fiow. Suppose that each helium atom in the system is subject to a force F, which we take to be parallel to us. For example, if the flow is gravitationally induced, then F=mg. Still assuming that the normal fluid remains at rest in the porous medium, we write the equation of motion for the superfluid in the form Y

(2.13)

fluctuations

where the fluctuation term is given by eq. (2.6). In steady-state, the lefthand side of eq. (2.13) must vanish, so that F = hAv, exp [- E,(u,, T ) / k ,T I .

(2.14)

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INTRINSIC CRITICAL VELOCITIES

9

This equation is more naturally rewritten in terms of a resistivity R:

(2.15)

or a differential resistivity:

Two comments must be made concerning the use of these last two formulae. In the first place, the formulae must not be taken literally right at u,=O where fluctuations are just as likely to increase us as to decrease it. Under laboratory conditions, us is almost always large enough that wrongway fluctuations are completely negligible. Secondly, notice that the crosssectional area A appears explicitly in R. This comes about because we have assumed that each fluctuation, once it occurs, changes the velocity of the entire system in a time which is negligibly short. This is clearly not a good assumption for very large systems. But, for the cases of interest to us, A is probably small enough for the assumption to be valid.

2.7. FORMOF THE ACTIVATION ENERGY Before concluding this section, it will be useful to explore the theoretical implications of one special assumption concerning the activation energy E,(u, , T). This assumption turns out to be particularly useful in the analysis of experimental data. To be specific, we assume that E, has the form: &(vs, T ) = e s ( T ) m Y

(2.17)

with E(us) some function to be determined. The right-hand side of (2.17) is the general form suggested by the hydrodynamical arguments to be presented in section 4. Roughly speaking, eq. (2.17) implies that the fluctuation consists solely of a perturbation of the flow pattern of incompressible superfluid, so that we need compute only the additional kinetic energy associated with this irregularity. The interesting aspect of this assumption is that we can deduce E(v,) simply from the experimental critical-velocity curve, us,c(T).We do this by returning to eq. (2.7) but expressing the critical-velocity curve as T,(u,) instead of u,,JT). Thus

(2.18) References p . 34

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[CH.1,

J. S. LANGER AND I. D. REPPY

03

The knowledge of E(v,) acquired here can then be used to predict the parameter describing the logarithmic decay defined in eq. (2.12). Specifically, (2.19) VS=U.,C

For comparison with experiment, the right-hand side of (2.19) must be expressed as a function of temperature. Since the decay rate away from the critical-velocitycurve is not apriori related to the value of the critical velocity, the prediction provided by (2.19) is not trivial. Favorable comparison of (2.19) with experiment enables us to take seriously the evaluation of the activation energy given by eqs. (2.17) and (2.18), and provides an interesting check on the thermal-activation theory as a whole. 3. Experimental study of supedluid flow states

The observation of persistent currents in liquid helium provides a useful means for study of the stability of superfluid flow states. Indeed experiments of this type have provided part of the motivation for the analysis outlined in sections I and 2, and in addition have provided a test of the theory. 3.1. SUPERFLUID PERSISTENTCURRENTS

The existence of persistent currents in superconductors has been known since earliest studies of superconductivity. Onnes showed that an electric current induced in a ring of superconductor would circulate with an essentially infinite lifetime. In direct analogy one would expect persistent currents to exist in liquid helium, There are, however, certain difficulties. First, the flow velocity is limited by a critical velocity which may be small. Second, there is no external magnetic field such as exists in the superconducting case, so that detection of the current must involve mechanical measurement. Over the years experimental technique has improved, and it has been possible to demonstrate the existence and to investigate the detailed behavior of superfluid persistent currents. We will not describe here the early measurements of persistent currents; the reader is referred to the original papers6 for details. At present technique has advanced to the stage where observations of persistent currents can be used to provide basic information on the properties of the superfluid. In these experiments the basic quantity observed is the angular momentum associated with a current of superfluid helium circulating in an annular container'.

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In making these measurements it is desirable to satisfy several requirements: 1. The flow state must be observed over long periods of time; in this way very small changes may be detected. 2. One should be able to determine the superfluid density within the same experiment, since this quantity may be a function of geometry. This is especially true for very small geometries, such as the unsaturated helium film and porous Vycor glass. 3. It is well to have the normal fluid clamped so that the data may be interpreted in terms of the superfluid velocity fleld. This can be accomplished by studying superflow through porous media and in thin films.

C

Fig. 2. Schematic of supeduid gyroscope. Persistent current with angular momentum Lpis formed in container A. Porous material D fills the container. During rotation u), about the vertical axis, the container deflects against a tungsten fiber. C. The deflection is sensed by the detector B. [J. D. Reppy, Phys. Rev. Letters 14,733 (1965)l.

One method which has proved fruitful is the use of a gyroscopic technique. Fig. 2 gives a schematic view of the apparatus. It consists of an annular container mounted on a horizontal torsion fiber. The container is usually filled with a porous material which serves to clamp the normal fluid. The pore size may be varied to study size effects. When a superfluid persistent current is present in the container the angular momentum can be measured by rotating about the vertical axis. The angular momentum vector, L p , is in the horizontal plane. The torque required to make Lpprecess at angular velocity, w , about the vertical axis is provided by a small deflection against References p . 34

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the horizontal torsion fiber. The deflection of the container is monitored by a capacitor in the tank circuit of an oscillator. The resulting frequency changes of the oscillator gives a measure of the angular momentum of the persistent current. In practice the frequency difference, AF, between clockwise and counterclockwise rotation is measured. This allows deflections caused by centripedal forces to subtract out to the first order. Then L, = ksAF/2w

where the torsion constant, k, is typically on the order of 10' dyne cmlrad, and the sensitivity, s, of the capacitance detector is about 2 x lo-' rad/Hz. The angular velocity, w , may be varied from 0.05 to 0.30 rad/sec. For most measurements the apparatus is immersed in the pumped helium bath. The porous material in the gyroscope container fills through a small hole in the container. However, for experiments on the helium film8 the gyroscope is enclosed in a separate chamber in thermal contact with the bath. In this case the required amount of helium gas in introduced to give the desired thickness of film. The persistent currents are created by tipping the container into the horizontal plane and rotating at a speed w about the vertical axis while cooling through the transition temperature. At some temperature T< TAthe rotation is stopped and the container is slowly tipped back to the position shown in fig. 2. Then the angular momentum is measured. This procedure may be repeated for a number of different rotational speeds. Fig. 3 shows the results obtained for a particular pore size, 10pm. For values of w less than w c , the persistent current angular momentum L, is

A

//I

-

o/o

.* In

5

-2 0

2-

/I

' 0

-

al-

J

-

/o

-----0- -

o-o---o

3-

/ /O /

/

/

i 1

!

i 0 O p pores T.2072 K

I

+ w c = l O + t rprn I

I

1

I

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13

INTRINSIC CRITICAL VELOCITIES

proportional to w. For speeds greater than w, the angular momentum saturates and becomes independent of the initial angular velocity. The values of critical rotational speed required to saturate the angular momentum depends strongly on geometry and temperature. The range extends from about 0.1 rad/sec for 100 p pore size to 50 radlsec for 20 A films.

3.2. DETERMINATION OF CRITICAL VELOCITIES AND

SUPERFLUID DENSITY

An average critical superfluid velocity can be obtained by multiplying by the mean radius of the container. The experimental critical velocities which we shall discuss have been defined in this way. It should be noted, however, that what we actually have is a range of velocities which depends on the details of the geometry. The critical velocity has been found to decrease as 500i 0

Vycor

0

0

Ternperoture

(K)

Fig. 4. The critical angular momentum for persistent currents formed in Vycor glass and 500 A filter material as a function of temperature.

the temperature is raised toward the transition9. This makes practical a standard procedure which has been used for all geometries. First, the largest possible persistent current is formed by rotating at a speed well above o, and cooling to the lowest temperature attainable, generally near 1 K, before stopping the rotation. The value of Lp is measured at this temperature, then a sequence of angular momentum values are measured while increasing the temperature. As the temperature is raised the angular momentum decreases, both because of the reduction in the superfluid density and because of an irreversible decrease in the superfluid flow velocity. (See section 2.) An angular momentum vs temperature curve is shown in fig. 4 for flow through 500 A pores and a specimen of porous vycor with pores on the order of 40 A. These curves are very similar to the critical transfer rates observed in traditional flow experiments between two reservoirs. References p . 34

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[CH.1,

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83

To deduce a critical velocity from this information we need to know the temperature dependence of the superfluid density for each geometry. In a persistent current experiment this may be obtained directly in the following waylo: Once a temperature near the transition is reached a sequence of measurements of L, is made while cooling. The superfluid flow velocity is iess than critical once a small reduction in temperature has been made. Therefore the velocity remains constant and the angular momentum L, traces out the temperature dependence of the superfluid density. It is one of the striking features of superfluid helium that the angular momentum of a persistent current can be made to increase by lowering the temperature.* This is a direct consequence of the macroscopic quantum state of the super500% 0

0

3

Vycor

* a

p0.50-00

O

0

.

0

e0

00

0

1.1

a 0

i 1.3

1.5

1.7 T(K)

1.9

I

2.1

.

Fig. 5. Superfluid density measured in 500 A filter material and Vycor glass as a function of temperature. Both curves are normalized using estimates of em/@ at 1.1 K for these geometries.

fluid and illustrates the remarkable stability of such a state when the flow velocity is small enough. Fig. 5 illustrates the data obtained for @, in 500 A and 40 A pores l * . The 500 A curve is nearly identical with e, vs. T for bulk helium while the 40 A curve is very different. Once the temperature dependence of the superfluid density is known, it is possible to remove this dependence from the irreversible angular momentum curves of fig. 4 and to obtain the temperature dependence of the critical velocity for flow through various geometries. The results for a variety of geometries are shown in fig. 6. The critical velocity curves for the 500 A and 2000 A pore sizes bear a striking resemblance to the e, vs T curves for these

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INTRINSIC CRITICAL VELOCITIES

same materials near the transition. In fact it was this similarity that originally suggested the thermal activation explanationla. 9oc

0

. 0

0

A

0

A

0

260

SO08

20008 Vycor

..*.

0

A

8

c E

0

-

Y

Y

’30

b

0 0

-

0

0

0

o

o

o

o

*.

o

A

0 0

-

.

A

A A I

I

I

1,

A l

I

A

l

I

,

I

Fig. 6. Superfluid critical velocities obtained for flow through 5WA, 2 0 A filter materials and Vycor glass as a function of temperature. 0 VYCOR

‘01

6500% Millipore 0 0 0

0.75

0

, ** :

* *

*** & I

1

I

I

I

I

I

In fig. 7 we have shown a plot of v ~ vs, e,~ for the various systems. The approach to a nearly linear dependence for small e, is clear for the 500 A References p. 34

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1. S. LANGER

[CH. 1,s 3

A N D J . D. REPPY

data. However, the 40 A data shows a different behavior, the critical velocity becoming quite small while the superfluid density is still finite. A similar behavior has been observed in unsaturated films8. The consistency of the thermal activation explanation can be tested by observing the decay of persistent currents when the flow velocity is near the critical value13. An example of the behavior shown in fig. 8 where the time dependence of the angular momentum is plotted. At the start of the observation the temperature is raised to bring the current to the critical velocity 280

r r ....

Begin to warm

Control temp.

i

220

-

c

I

..

. I

*

* I

*

.J

curve. After the warming, the temperature is controlled at some higher temperature. The angular momentum is observed to continue to decrease as time progresses. Examination of the decay over several decades of time shows a decay rate linear in the log of time. The relationship is shown clearly in fig. 9 where the fractional change in velocity is plotted against the loga-

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INTRINSIC CRITICAL VELOCITIES

0-

-I

0.05

-. .>

-=-

=->=

a

,500 T-1.363 K

d

75 T=I.J85 K

.

\o\

VYCOR

Time (s8c)-

Fig. 9. The fractional change in superfluid velocity during the decay process is shown for persistent currents in a number of geometries as a function of time.

rithm of time. This behavior is very similar to flux creep in superconductors l4 and provides strong support for the thermal activation idea. It should be noted that the 'critical velocity' is really an arbitrary value and is defined as that velocity for which the decay rate has reached some reasonably small value. Thus in cases where the decay rate is large it is not useful to speak of a critical velocity. We shall return to this later when we discuss onset temperatures in unsaturated fims and vycor glass. 3.3. DETERMINATION OF BARRIERFUNCTIONS

AND PREDICTION OF DECAY RATES

As was shown in section 2, it is possible to deduce the form of the barrier function Ea(us,T ) within the framework of the thermal activation model. When the decay rate has reached an arbitrarily small value @, , then: and

@o =

hImAvo exP I:- E a ( V , c2

c

3

T ) / ~TI B

= rkB

(2.7)

(34

where y=ln(hdv,/m@,). If we assume that the barrier is associated with some hydrodynamic flow state, then we expect the temperature dependence of the barrier can be expressed in a factor &'). Thus, Ea(us, T)= es(T)E(vs).

(34 References p . 34

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1. S. LANGER A N D J. D. REPPY

[CH.1,

53

Combining (3.1) and 3.2) we have

A plot of the experimental quantity eT/@,=(e/yk,)E(o,) against v,,= determines, except for a scale factor, the velocity dependence of E(o,). In fig. 10 we show the data for several geometries: 0.2 Bm, 500 A, 40 A. It is convenient to fit these barrier functions with simple analytic expressions 13. 150 o o

125 100

-

2000A 500% Vycor

-

0

A

Y

75-

v)

*

8

' 50-

O 0

(cm /sec 1 Fig. 10. Experimentally determined barrier functions.

The temperature dependence of the decay rate may be predicted by differentiating the barrier functions and substituting in eqs. (2.10)-(2.12). The only free parameter is a constant factor which may be fixed using the decay rate at a single temperature. Then the rest of the temperature dependence provides a powerful test of the whole thermal activation approach. Fig. 11 shows the observed fractional decay rates, us:d(dvs/d log t ) . The solid lines are the rates predicted from the us,=and e, data for the various geometries. The constant y is determined by the best fit. Essentially the same value of 7 is found for all three sets of data, (46 for 500 A and 2000 A and 43 for the vycor).

CH. 1,

41

19

INTRINSIC CRITICAL VEU)CITIES 1.0

I

I

A VyCOf 0

5ooae

A

1-

2000A

$Lo40 a\

A

>20

Q

: ,1-

-

0.01

0.001;

"".\\ 1

I

10

100

1000

There are several qualitative features of the observed decay rate curves that deserve comment. At the lowest temperatures the decay rate increases with increasing temperature. In the case of the 2000 A and 500 A pore sizes the fractional rate u,~~(du,/d log t ) approaches a constant value of about 5% per decade of time near TA,The fractional decay rate for the flow through the vycor glass also increases as the temperature is raised. However, the rate is considerably larger and, in fact, becomes on the order of 30%per decade of time as the onset temperature is approached. With a decay rate this large one can hardly speak of a 'persistent current' and the definition of a critical velocity becomes arbitrary. 4. Specific models

Having examined the experimental evidence for thermally activated dissipative processes in superfluid helium, we turn now to the problem of quantitative explanation of the activation energies and fluctuation rates. We shall start by describing the semi-phenomenological generalization of the Ginzburg-Landau-Gross-Pitaevski theory16 which we have found to be useful in this work. Generalizations of this kind have been hypothesized by Rice1$ and others; and one of the present authors has found a systematic derivation of most aspects of the following phenomenological theory1'. References p. 34

20

1. S. LANGER AND 1. D . REPPY

[CH.I,

84

4.1. PHENOMENOLOGICAL PICTURE

In the usual phenomenological picture, it is assumed that the equilibrium states of the superfluid can be described by a complex-valuedorder parameter, $ ( r ) , which is a function of position r. This function plays the role of a wavefunction for the superfluid component of the system. In particular, the super-

and the number-current density is

The function $ ( r ) is assumed to be smooth and very slowly varying on the scale of the average spacing between helium atoms. More important, it is assumed that J,!I does not jump discontinuously as the system evolves in time. In order to discuss the fluctuation problem, we have to go beyond these conventional hypotheses. In the first place, we must assume that all of the relevent states of the superfluid, not just the equilibrium states, can be described by functions $(r). We also assume that there exists some functional, F { $ ) , with the dimensions of an energy. But, instead of supposing that F { $ ) is the actual free energy when minimized with respect to J,!I, we assume that the probability of finding the system in any state $(r) is proportional to the Boltzmann factor, P ( + ) exp ( - F ( $ ) / k , T ) . (4.3) In other words, the statistical fluctuations of the superfluid, caused by interactions with a constant-temperature heat bath, are visualized as a continuous random motion of the system-point + ( r )in its function space, the neighborhood of any point $ being visited with a frequency proportional to P{$}. The minima of F ( + ) , i.e. the peaks of P ( $ } , locate stable and metastable states as long as these peaks are well separated from one another. Thus, the equilibrium states are still determined by the solutions of the GinzburgLandau equation, 6F/6$ = 0. (4.4) But h,t and F ( $ } have been given more general meanings. In the physical situations of interest to us, we have periodic boundary conditions and, at least in some average sense, translational symmetry. Thus we can predict that eq. (4.4) will have solutions of the form: t,hk(r) = fke"".

(4.5)

CH.

1,

8 41

21

INTRINSIC CRITICAL VELOCITIES

For example, near TAwe might guess that F { $ } can be written

in which case the plane-wave states (4.5) satisfy (4.4) if

-)

j; =1 (A- h’k’ B

.

(4.7)

The solutions of (4.4) will actually locate relative minima of F { $ } only if

tj2F is a positive-definite matrix. In the case of the F { $ } given by eq. (4.6), this second criterion requires h2k2 __ < +A. (4.9) 2m Equation (4.9) is identical to the conventional Ginzburg-Landau criterion for the critical current, which is usually derivedls by noting that the current j = kfk” = k (A- 1 h2kZ ;;;) B

(4.10)

has a maximum at h2k2/2m=+A.The interpretation of eq. (4.9) as a stability condition seems much more satisfactory for present purposes. Consider now a superff uid in a state t,6k. The energetics of the situation are illustrated schematically in fig. (4.1). The fact that the matrix (4.8) has only positive eigenvalues means that, in order to move away from $&, the system must move initially to states of higher F and therefore lower a priori probability. Eventually, the system may find its way to a state of even greater stability than $k, say 9; but, as long as $ cannot jump discontinuously, it must pass through states of low statistical weight P{$} to do this. In fig. 12 we have labelled by the symbol $(r) the point of greatest F along the least improbable path between $k and $k.. Topologically, must locate a saddlepoint of the functional F{$};that is, it must locate the highest pass along the lowest route between the two minima. The probability of the system’s reaching this pass in proportional to exp( -Ea/kBT),where

-w&} ;

Ea = -wl

(4.11)

and therefore we identify E, as the activation energy introduced in section 2. References p. 34

22

3. S. LANGER AND J. D. REPPY

[CH. 1,

84

A

-

+kt

Fig. 12.

Note that, if $(r) is a saddle-point of F, it must satisfy the GinzburgLandau equation, (4.4).Of course, it must not satisfy the stability condition that the matrix (4.8) be positive-definite. In fact, this matrix must have one negative eigenvalue at $ corresponding to the direction in $-space in which the system moves across the saddle-point. Here we can return to the analogy of the liquid droplet discussed in section 2. The droplet that was just large enough to nucleate the condensation was unstable against either uniform expansion or contraction away from the critical radius, R,. Similarly, the saddle-point function $(r) must describe a localized fluctuation in lc/k which has one unstable mode of distortion.

4.2. HYDRODYNAMIC ANALOGY

A N D DIMENSIONAL ANALYSIS

In principle, if we know the functional F { $ ) , the above criteria enable us to calculate $ and the corresponding activation energy E, .In practice, however, we have to resort to further phenomenological ideas and classical analogies. Our next step is to examine the Ginzburg-Landau equation from a hydrodynamic point of viewz0. Following the usual procedure, we write (4.12)

where eSand cp are real functions of

P.

If F { $ ) has the form

(4.13)

CH. 1,

8 41

INTRINSIC CRITICAL VELOCITIES

where U is any function of will yield

23

1 ~ 1 2 , then the variation of F with respect to cp V - j , = 0,

with

(4.14) (4.15)

Finally, if e, is constant, then (4.14) is the basic equation of incompressible hydrodynamics with cp(r) playing the role of the velocity potential. That is, h m

us = -

vrp.

(4.16)

Of course, e, is not apt to be everywhere constant for the fluctuations we want to consider. In fact, we shall present an argument shortly which implies that e, must pass through zero during the nucleation process. But it is conceivable that the region in which e, departs from its uniform value is small and contributes only very little to the energy of the fluctuation, E,. If this is so, then E, consists almost entirely of the kinetic energy of hydrodynamic flow. This will be true, for example, of the vortex model discussed below, in which the energy is determined mainly by the extended velocity field surrounding a relatively small vortex core. Given this very simple hydrodynamic picture, we can predict the velocity and temperature dependence of E, by dimensional analysis21. Suppose that the flow pattern is restricted to a three dimensional region of volume R 3 . Then the energy must have the form E,

GC

eSv2R3,

(4.17)

where o is some characteristic velocity. But the only velocity occurring in our theory is us; and the only other quantities which have dimensions are h and m. The only characteristic length is therefore hlmv,. Substituting this for R,we obtain (4.18)

Note that the relation E,Gcv,-', and the resulting prediction o,(T)oce,(T) are in rough agreement with the observations in section 3. Repeating this analysis for the case in which the fluctuation is constrained to occur within a thin film of thickness d, we obtain Ea cc e,d(h/m)2.

(4.19) References p . 34

24

J. S. LANGER AND 1. D. REPPY

[CH, 1,

54

which is independent of L?, . An activation energy of this form would predict no persistent current at any velocity above some well-defined temperature. This is also in rough agreement with the experiments. It must be emphasized, however, that all of this analysis is based on the assumption that variations of e, are unimportant. If this is not true, then the characteristic length associated with variations of the superfluid density, i.e. the ‘coherence length’, must be included in the dimensional arguments, and eqs. (4.18) and (4.19) can no longer be valid. The rough success of these equations, however, leads us to believe that the hydrodynamic models may be fairly accurate. 4.3. VORTICITY We next argue that any hydrodynamic model of the dissipative fluctuation almost certainly involves v o r t i ~ i t y ~To ~ ,see ~ ~this, . we invoke an argument originally used by W.A. Little in a similar contextz4.Consider the superfluid wave function $k(r) given by eq. (4.9, and plot three-dimensionally the real and imaginary parts of J/ versus the component of r parallel to k. The result is a helix of n = Lk/2n turns wound on a cylinder of radius I$tal = (e,/m)*. In order for us to decrease the superfluid velocity, we must decrease the number of turns of this helix. But, as long as we do not let J / ( r ) jump discontinuously and do not relax the periodic boundary conditions, the only way in which we can change the number of turns in the helix is by letting $(r) pass through zero somewhere. That is, we must aliow g, to vary and, in fact, to vanish at some points.

Fig. 13. Phase contours of a single vortex line.

The vorticity is implied by this argument because a point of vanishing $ is automatically a point of undefined phase cp. At such a point, the lines of constant phase can look as shown in fig. 13. In this picture we show the cross-

CH.

1, 8 41

INTRINSIC CRITICAL VELOCITIES

25

section of a vortex line. The line itself is normal to the plane of the paper, and the fluid circulates counterclockwise in the direction of increasing cp. The vortex core is indicated by a shaded circle which is meant to imply that e, departs from its uniform value over a small but extended region. I,$ must actually vanish somewherewithin this region. Quantization of the circulation arises from the requirement that $(r) be single-valued; thus cp can change only by multiples of 271 in going around a flow loop. The vortex shown in fig. 13 is singly quantized, meaning that Acp =2n. The most important aspect of vorticity for our purposes is that, if a singly quantized vortex line passes across a moving superfluid, the average velocity of superflow changes by & h/mL. That is, the system jumps from one meta-

f

-77.

i r o +r

t2*

"S

-27

Fig. 14. Phase contours for a single vortex line with superimposed uniform flow normal to the line.

stable state to a neighboring state in the sense described in section 2 and illustrated in fig. 1. To see this, we superimpose the single vortex line of fig. 13 upon a uniform flow moving from left to right. The resulting contours of constant phase are shown in fig. 14. Here it is clear that, if we traverse the system along a flowline (broken arrows) passing above the vortex, the total phase change along the line is 2n less than if the line lies below the vortex. Thus, if the vortex were formed at the top and then moved to the bottom, the total number of turns in $(r) would decrease by one, with a corresponding change in v, of hAcp/m=hlmL as predicted. It is sometimes easier to visualize this process occurring in a type I1 superconductor, in which case a singly quantized vortex line encloses one References p . 34

26

J. S. LANGER A N D J. D. FLEPPY

[CH.

1, $ 4

quantum of magnetic flux. Suppose that a persistent current is flowing around a loop of superconducting wire, and consider the corresponding magnetic flux which is trapped by this loop. Each time a trapped flux line escapes through the loop, the current is diminished by a corresponding amount. The electromagnetic force produced by a moving flux line is directly analogous to the force on a superhid produced by a moving vortex. What we have described so far is the behavior of a macroscopically long vortex line which might somehow be generated within or at the boundaries of the superfluid, but is certainly not a candidate for the necessarily localized nucleating fluctuation $. The most obvious possibility for the actual $ is a vortex ring, that is, a short vortex line closed upon itself. Suppose for simplicity that the ring is formed by some thermal fluctuation so that it lies in the plane normal to the direction of superflow. Suppose, in addition, that

Fig. 15. Cross section of a vortex ring.

the sense of circulation is such that the phase change obtained in moving across the system along a flow line is 2.n less if the path passes through the center of the ring than if the path passes outside the ring. The configuration is sketched in fig. 15. Phase contours for such a ring can be obtained by rotating fig. 14 about a horizontal axis placed at the top of the diagram. Then, if the thermal fluctuations tend to make the ring expand to macroscopic size and eventually be annihilated at the boundaries, the average velocity of the entire system will be diminished by hlmL. The magnetic analogy may be useful here too. The analog of the vortex ring is a small flux loop oriented in such a way that, if it were to expand out to the boundaries of the superconductor, it would cancel a unit of trapped flux. 4.4.

VORTEX RING MODEL: ANALYSIS

The vortex-ring model can be tested quantitatively by using well-known

CH. 1 ,

5 41

INTRINSIC CRITICAL VELOCITIES

27

results of classical hydrodynamics. Consider a singly quantized vortex ring of radius R moving in a stationary superfluid; and assume that the core of the vortex is a hollow region of radius a very much less than R.We shall see that our theory is by no means sensitive enough to test the details of the vortex core. But the requirement a
q = h(8R/a),

(4.20)

where K=h/m is the vorticity26’26i27. The ring will move with a velocity v, parallel to its axis: K v, = -(7-i). (4.21) 4nR If this ring is placed in a moving superfluid in such a way that its velocity v, is pointed upstream, the vortex energy measured in the stationary frame of reference will be E = EO-POV~, (4.22) where usis the velocity of the superfluid background andp, is the momentum, or ‘impulse’, of the vortex ring. There is actually a certain amount of uncertainty about the proper definition ofp,. We shall takep, to be defined by the relation 00 = dEOldP0 (4.23) and the condition that p , should vanish when R=O. Solving (4.23) in the form (4.24) we obtain

+

po E n@,KR2(1 O(q-’)),

(4.25)

which is consistent with the standard hydrodynamic results. Now consider the vortex energy, given by eq. (4.22), as a function of R. Because E, increases linearly in R and -povs decreases quadratically, E ( R ) must have a peak at some critical radius, say R,, determined by (4.26) Comparison of eq. (4.26) with (4.24) indicates that R, is just that value of R which causes the vortex ring to remain stationary in the laboratory frame References p. 34

28

[CH. 1,

J. S. LANGER A N D J. D. REPPY

54

of reference; and this is consistent with our original requirement that the saddle-point fluctuation IJ be a solution of the time-independent GinzburgLandau equation (4.4). In accord with the earlier discussion, we see that vortex rings smaller than R, will lower their energy by contracting, whereas rings larger than R, will find it energetically favorable to expand indefinitely. Thus the vortex-ring model satisfies the criteria for the nucleating fluctuation. The energy E(R,) can now be identhed as the activation energy, E,. Combining eq. (4.20) and (4.26), we obtain

E, = E(R,) g

e K 3 (q-+)(q+). 16nus

(4.27)

Note that this E, has precisely the form predicted by eq. (4.18) except for the factors involving q, which depends on the core radius. The evaluation of I] presents a bit of a problem in the analysis of eq. (4.27). Not only does q violate the dimensional arguments because of the presence of the temperature-dependent core radius; but this temperature dependence also violates the ansatz for E,, given in eq. (2.17), which we have used as a basis for much of our analysis of the experimental data. We shall argue, however, that q itself ought to be roughly independent of temperature along the critical-velocity curve. Because none of the relevant experimental data , ~ , because I] depends only logarithmically is very far away from U ~ = O ~ and on R/a, we should be able to use the same value for I] everywhere. For the moment, let us assume that constancy of q and solve for the critical velocity from the E, given by eq. (4.27). Using eq. (2.7), we obtain, y

),,,(hv, A

= In

.

(4.28)

Returning to eq. (4.21), we can compute the radius of the critical vortex ring:

(4.29) and, from R,,obtain an equation for

I]:

(4.30) The only temperature-dependent quantities in (4.30) are radius, a. Experimentally, we know that

e m

2 . 4 4 1 - Wi)*,

e, and

the core(4.31)

CH. 1, § 41

INTRINSIC CRITICAL VELOCITIES

29

where e, is the density of helium at the I-point lo.28. We have already argued that the core radius should be proportional to the correlation length which, by comparison with critical phenomena in other systems, we expect to vary as u(T) g uO(l-T/TA)-,

(4.32)

with a, of order 1-2 A and v approximately $ 29. Thus the combination ae, appearing in (4.30) should be roughly independent of temperature. It follows that q is temperature-independent and that E,, given by eq. (4.27), fits the ansatz (2.17). To complete the theoretical analysis of the vortex-ring model, we compute the fractional decay per decade defined by eqs. (2.9) and (2.12). Using eqs. (2.19) and (4.27), we obtain: (4.33)

4.5. VORTEX-RING MODEL:

COMPAFUSON WITH EXPERIMENT

We have already seen that the thermal-activation theory is in qualitatively good agreement with experiment. The observed decay of us is indeed logarithmic; and the temperature dependence of the decay rate can be predicted from the experimentally deduced velocity dependence of the activation energy. None of these features are specificto the vortex-ring model, however. Even the fact that E(u,) is found to behave roughly like u:' can be said to check only the dimensional analysis; but remember that this analysis is based on some very specific hydrodynamic assumptions. The only way really to check the detailed ring model is to use it to evaluate the actual magnitude of E(v,). We shall see that the numerical agreement is extremely crude. The first step in the numerical analysis is the evaluation of the constant y, originally defined in eq. (2.18). The experimental value of y comes directly from the decay rate given by eq. (4.33). As predicted by this equation, the decay rate does become roughly temperature independent near T,,levelling off at a value of about 5% per decade. Thus we deduce Yaxpt

= 46.

(4.34)

The best theoretical estimates of y turn out to be somewhat larger than this. Characteristic frequencies of processes occurring on an atomic scale are of the order 10"-1012 sec-'. The number-density of helium near T, is about 2 x 10" ~ m - so ~ ;vo must be of the order cm-3 sec-l. It References p . 34

30

1. S. LANGER A N D

I. D. REPPY

[CH.1,

54

seems reasonable to assume that the effective cross-sectional area, A, is not much larger than the pore size in the filter material; since vortex rings will almost certainly be trapped or annihilated as soon as they expand to this size. Thus we estimate A to be about lo-* cm2. Finally, we choose @,, to be 1 cm sec-2, and obtain Ylhwr

53*

(4.35)

The discrepancy between (4.34) and (4.35) is equivalent to a factor of about lo3 in the fundamental fluctuation rate; and it is not clear at present whether this is significant. It must be emphasized that the evaluation of y has nothing specifically to do with the vortex-ring model, but is a fundamental theoretical problem well beyond the scope of this review. For present purposes, it seems safest to adopt the experimental value given in eq. (4.34). Given y, we can return to eq. (4.30) to evaluate the constant q. The result is q E 5k0.5.

(4.36)

where the indicated uncertainty is that due to our lack of knowledge about the core radius. Note that this value of q implies (uiathe original definition of q in eq. (4.20)) that the ring to core ratio, RJa, is about 20, which is almost certainly large enough to justify our use of a hydrodynamic model. Finally, we compare our formula for the activation energy E,, eq. (4.27), with the experimental E(o,) curve. In fig. 10 the data was displayed as a plot of (e/ykB)E(os)uersus us. The curves were fit to simple functions (see ref. 13). As an example, the data for flow through the 500 A material could be expressed as

(~/k,y)E(v,z ) Q - [l0,

3

;

Q,,,, z 623 cm sec-' K.

(4.37)

Using eq. (4.27), we can identify the coefficient of us- : (4.38) where we have used q = 5 in the numerical evaluation. Apparently the activation energy obtained in this way is almost an order of magnitude too large. Equivalently, the magnitude of the critical velocity predicted by eq. (4.28) is too large by the same factor. That is, us, .(T)

4 1- T/TA)*

(4.39)

CH. 1,

8 41

INTRINSIC CRITICAL VELOCITIES

31

both experimentally and theoretically; but

ucCxptz whereas u,

2.4

-Qexpt=

Ta

670 cm/sec;

z 4800 cmlsec.

(4.40)

(4.41)

There is a variety of possible explanations for this discrepancy. In the first place, the quoted value for the experimental velocities are always average velocities of flow through a highly irregular porous material. It is conceivable that the fluctuations we are seeing are almost all being nucleated within constricted regions of the system where the superfluid velocity is anomalously high. That is, the experimental value of u, in eq. (4.40)may be too low. A detailed discussion of this possibility has been given by Fishers0. A second important reason why we might expect the calculated critical velocity to be larger than the experimental one is the fact that, throughout much of the temperature range in which the experiments are performed, the radius of the critical vortex ring is not very small compared to the size of the interstices in the porous material. Specifically, R, z 20(1- T/TA)-3A. (4.42) For example, R, is of the order of 500 A at a value of Ta-T of about 2 x loq2K. Thus we expect the pattern of flow around the vortex ring to be restricted, and the activation energy to be correspondingly reduced. This size might also be responsible for the extra, velocity-independent term which appears in the measured E(v,). It is not at all clear how to obtain such a term theoretically; but at least the pore size provides us with a new length to use in the dimensional analysis. Both of these two effects seem quite plausible; and, although neither effect lends itself easily to numerical evaluation, it is conceivable that a combination of the two might bring the theory into quantitative agreement with experiment. It should be mentioned, however, that there might be more fundamental objections to the theory, still within the framework of the thermal activation picture. It might turn out, for example, that the core of the vortex plays a much more important role than we have assigned to it, and that our simple hydrodynamic approximations are inadequate. It might also happen that the entropy associated with the various orientations and distortions of the critical vortex ring significantly lowers the effective activaReferences p . 34

32

[CH. 1,

J. S. LANCER AND J. D. REPPY

84

tion energy 23. There is now theoretical evidence that E, should contain such terms31, but no detailed calculations have been performed as yet.

4.6. FILMFLOW:

THE VORTEX-PAIR MODEL

A thin film, by definition, is one whose thickness is less than the correlation length. Thus it is not possible for a complete vortex ring to form in such a system. By the same reasoning which led us to the vortex ring, however, we can guess that the saddle-point fluctuation, 3, might consist of a pair of vortex lines perpendicular to the film. The situation is illustrated in fig. 16. 7

@rO -7

-

Symmetry Line

w

c

"I

n

4#

u

b

li

Fig. 16. Pair of vortex lines separated by a distance b.

The separation between the two lines is b, and the core radius is again denoted by a. The two lines have opposite vorticity. Thus, if the separation b increases indefinitely, the superfluid velocity decreases by h/mL, just as in the threedimensional case. When moving in a stationary fluid, the energy and momentum of a vortex-pair are 2s, respectively,

1

E , = - de, K 2in (bla), 271

(4.43) (4.44)

where d is the thickness of the film. If the fluid moves with velocity us and the pair is oriented as in fig. 16, the resulting energy is:again E(b) = E o - P o V s ,

which has a maximum at the separation:

b, = K/2nv,.

(4.45)

CH. 1, $41

INTRINSIC CRITICAL VELOCITIES

33

Pairs separated by less than b, will tend to coalesce, whereas those separated by more than b, will tend to move even further apart. The corresponding activation energy is E, = E(bc) = d--~ s K 2 [In (L) -11. 2n 2nvsa

(4.46)

Using (4.46) in eq. (2.7), we can solve for the critical velocity

(4.47) where y is the same constant that was defined in eq. (2.18). The dominant temperature dependence of the right-hand side of (4.47) comes from the factor es-' appearing in the argument of the exponential function. As we increase the temperature, eq. (4.47) predicts that u,,,(T) will drop abruptly in the neighborhood of the onset temperature, To,where the argument of the exponential is of order unity. Above To,us,, may become unobservably small. It is instructive to examine the fractional decay rate using the vortex pair barrier function (4.46). This expression for the fractional decay rate is only valid for slow rates of decay of the superflow. However, it may be used to obtain a rough expression for Toin terms of the film thickness and average superfluid density at onset. As the temperature is raised, the factor T/e, in the decay rate grows rapidly and the superflow becomes increasingly dissipative. If the approximate expression for the fractional decay rate is set equal to some arbitrary value, say 50% per decade of time, we obtain the relation,

This expression is in order of magnitude agreement with the data from persistent current experiments in unsaturated filmss and Vycor and the estimates of (e,/e)o obtained by Rudnick and coworkers32 from third sound measurements. Acknowledgments The authors are grateful to J. R. Clow, G. Kukich and R. P. Henkel for use of unpublished data. One of the authors (J. R.) would like to thank the Aspen Center for Physics for hospitality during the period when part of this work was done. Support of the experimental work by the National Science Foundation and the Advanced Research Projects Agency through the Material Science Center at Cornell University is gratefully acknowledged. References p . 34

34

J. S. LANGER A N D J. D. REPPY

[CH. I

REFERENCES (a) N. Bogoliubov, J. Phys. U.S.S.R. 11, 23 (1947). (b) N. M. Hugenholtz and D. Pines, Phys. Rev. 116,489 (1959). (c) W.L. McMillan, Phys. Rev. 138, A 442 (1965). (d) D.Schiff and L. Verlet, Phys. Rev. 160, 208 (1967). te) L. Reatto and C. V. Chester, Phys. Rev. 155, 88 (1967). 2 (a) L. D. Landau, J. Phys. U.S.S.R. 5, 71 (1941). (b) 0. Penrose, Phil. Mag, 42, 1373 (1951). (c) 0. Penrose and L. Onsager, Phys. Rev. 104, 576 (1956). (d) P. W.Anderson, Rev. Mod. Phys. 38,298 (1966). J. Frenkel, Kinetic Theory of Liquids (Dover Publications, New York, 1955)ch. 7. * The question of zero-current resistivity and wrong-way fluctuations is treated more carefully for the case of fluctuations in a superconductor by J. S. Langer and V. Ambegaokar, Phys. Rev. 164,498 (1967). H. Kamerlingh Onnes, Verslag Kon. Akad. Wet. 1413 (1914); Comm., Leiden No. 140b (1914). ( a ) H. E. Hall, Phil. Trans. A m , 359 (1957). (b) W. F. Vinen, Proc. Roy. SOC.(London) A260, 218 (1961). (c) Philip J. Bendt, Phys. Rev. 127, 4441 (1962). (d) J. D.Reppy and D. Depatie, Phys. Rev. Letters 12, 187 (1964). ’ {a) J. Clow, J. C. Weaver, D. Depatie and J. D. Reppy, Proc. of the 9th International Conference on Low Temperature Physics, Columbus, Ohio, 1964, p. 328. (b) J. D.Reppy, Phys. Rev. Letters 14,733 (1965). (c) J. B. Mehl and W. Zimmermann, Phys. Rev. 167, 214 (1968). * R.P. Henkel, G. Kukich and J. D. Reppy, Proc. of the 1 lth Internationat Conference on Low Temperature Physics, St. Andrews, 1968, p. 178. * J. R. Clow and 3. D. Reppy, Phys. Rev. Letters 19, 291 (1967). lo J. R. Clow and J. D. Reppy, Phys. Rev. Letters 16, 887 (1966). 11 (a) G.Kukich, R. P. Henkel, and J. D. Reppy, Proc of the 1 lth International Conference on Low Temperature Physics, St. Andrews, 1968, p. 140. (b) 4OA data obtained by G. Kukich and R. P. Henkel at LASSP, Cornell University; not yet published. l8 J. S. Langer and M. E. Fisher, Phys. Rev. Letters 19, 560 (1967). G. Kukich, R. P. Henkel and J. D. Reppy, Phys. Rev. Letters 21, 197 (1968). See also ref. I l a . P. W. Anderson and Y. B. Kim, Revs. Mod. Phys. 36, 39 (1964). (a) V. L. Ginzburg and L. D. Landau, Zh. Eksperim. i Teor. Fiz. 20, 1064 (1950). (b) E. P. Gross, Nuovo Cimento 20, 454 (1961). (c) V. L. Ginzburg and L. P. Pitaevskii, Zh. Eksperim. i Teor. Fiz. 34, 1240 (1958). [Engl. transl. Sov. Phys. JETP 7,858 (1958)l. L. P. Pitaevsku, Zh. Eksperim. i Teor. Fiz. 40, 646 (1961) [Engl.transl. Sov. Phys. JETP 13,451 (1961)]. l8 T. M. Rice, Phys. Rev. 140, A1889 (1965). J. S . Langer, Phys. Rev. 167, 183 (1968). J. S. Langer and V. Ambegaokar, op. cit. l9 J. Bardeen, Rev. Mod. Phys. 34, 667 (1962).

CH.

11

INTRINSIC CRITICAL VELOCITIES

35

See I. M. Khalatnikov, Introduction to the Theory of Superfluidity (Benjamin, New York, Amsterdam, 1965). 81 This approach was suggested to us by G. V. Chester. W. F. Vinen, Liquid Helium (Academic Press, New York, 1963). S. V. Iordanskii, Zh. Eksperim. i Teor. Fiz. 48, 708 (1965) [Engl. trans]. Sov. Phys. JETP 21,467 (1965)l. 24 W. A. Little, Phys. Rev. 156, 396 (1967). zr Sir Horace Lamb, Hydrodynamics (Dover Publications, N.Y., 1945, orig. pub. 1879) ch. VII. m G. W. Rayfield and F. Reif, Phys. Rev. 136,A1194 (1964). e7 J. S. Langer and M. E. Fisher, Phys. Rev. Letters 19, 560 (1967). ** (a) Tyson and Douglas, Phys. Rev. Letters 17,472,622 (1966). (b) Tyson, Phys. Rev. 166, 166 (1968). z9 (a) B. D. Josephson, Phys. Letters 21, 608 (1966). (b) M.E. Fisher and R. J. Burford, Phys. Rev. 156, 583 (1967). 8o M. E. Fisher, Proc. of Conference of Fluctuations in Superconductors, Asilomar, Calif., March 1968. J. S. Langer, Phys. Rev. Letters 21, 973 (1968). a2 (a) I. Rudnick, R. S. Kagiwada, J. C. Frazer and E. Guyon, Phys. Rev. Letters 20,430 (1968). (b) R. S. Kagiwada, J. C. Frazer, I. Rudnick and D. Bergmann, Phys. Rev. Letters 22, 338 (1968). 2o