A PHENOMENOLOGICAL MODEL OF COLLAPSED SUPERFLOW AND ORBITAL RELAXATION IN SUPERFLUID 3He-A

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A PHENOMENOLOGICAL MODEL OF COLLAPSED

SUPERFLOW AND ORBITAL RELAXATION IN

SUPERFLUID 3He-A

H. Hall

To cite this version:

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JOURNAL DE PHYSIQUE Colloque C6, supplément au n" 8, Tome 39, août 1978, page C6-15

A PHENOMENOLOGICAL MODEL OF COLLAPSED SUPERFLOW AND ORBITAL RELAXATION INSUPERFLUID

3

He~A

H.E. H a l l

Schuster Laboratory, University of Manchester, Manchester MIS 9PL, England

Résumé.- Un modèle est proposé pour décrire les caractéristiques générales du flux superfluide dans

3He-A dans les conditions où il se relaxe, à un taux limité par la viscosité orbitale, tendant à un

état dans lequel le flux superfluide tend vers zéro sauf dans une couche de surface où des tourbil-lons sans noyaux sont présents.

Abstract.- A model is proposed to describe the gross features of superflow in 3He-A under conditions where it is relaxing, at a rate limited by orbital viscosity, towards a state in which the superflow has collapsed towards zero except in a surface layer where coreless vortices are present.

The equation of motion for v in He-A maybe written /1,2/ as

3v . _. ( dl. 3Q. .' 3t 2m "ij 3t j 3t

2m jkJTk 3x. 3t 3x. ' U ;

where y is the chemical potential in the laboratory frame. By integrating (1) between two points A and B in the liquid we find that

fB fBr 1

3 M 3&

• £ v . d r = ^ - I x r r -dS, - (y - u ) . (2) dt ~s ~ 2m 3t B A

JA 'A** >

The integral on the right hand side of (2) has a simple geometrical meaning. If we represent the di-rection of % at a point in the liquid by a point on the surface of a unit sphere, then the texture along the path AB in the liquid is represented by a line on the surface of the sphere, and the integral on the right hand side of equation (2) is the rate at which this line is sweeping out area. Equation (2) shows that such a dynamic texture must be accompa-nied by either changing v or a chemical potential difference, or both. Thus, in the precessing domain wall considered by Hall and Hook /3/ v is constrai-ned to be constant and there must be a chemical po-tential difference ; the steady precession is accomr panied by steady dissipation. On the other hand, in a toroidal geometry there can be ro change of che-mical potential round a closed path, and equation

(2) becomes

a f

K

f f

3 & "

which is the time derivative of a result obtained by Ho M / . The precessing domain wall now results in decay of circulation, and corresponds to the motion of a boojum singularity on the boundary, as discus-sed by Merinin /5/. In the absence of surface singu-larities the surface circulation is constant, be-cause H cannot change on the boundary.

We may now understand the geometry of the collapse of superflow discussed by Bhattacharyya, Ho and Mermin /6/. Consider a texture in which the

I vector spirals as we pass along the path of inte-gration in equation (3), so that % generates a cone of semi-angle 6. If the spiral makes n complete turns along the path of integration, then as 9 in-creases from 0 to I the surface of the unit sphere is covered n times and the circulation changes by (nh/m). It is thus possible to pass from a uniform texture at the boundary to a uniform texture in the liquid with substantially different circulation through a boundary layer of spiralling texture which may be regarded as an array of two quantum coreless vortices. Consider in particular a situation in which v is initially zero everywhere, and the walls are then brought into motion with velocity u. Tex-tures of the above type will then be favoured ener-getically because we want to minimise the free energy

F' = F - P.u , (4)

where P is the momentum of the liquid /10/. This is accomplished by making the relative superflow j = j - ov as small as possible given the

cons-traints. The balance of kinetic energy and bending energy implies that we choose n to make (j - CTV ) zero in the interior with a boundary layer of

core-3

(3)

less vortices of thickness

h

\./2mu. The problem is

clearly hard to solve in exact detail, especially if

u is time dependent so that n must be time dependent.

-

We therefore adopt a similar approach to that used

in

'H~II containing many vortices, and try to deve-

lop equations describing semi-quantitatively the

mean flow /7,8/. The screening of superflow from the

interior /4/ that we have just discussed may be des-

cribed by a London equation

We define an effective superfluid velocity v by

-

-9

-

j

=

Psi

(Y

-

YJ),

and take

h =

&\./2m(Ts

-

vn)J,

-

r

where we expect

a

% 1 ;

equation (5) is thus a non-

linear London equation for the equilibrium state of

collapsed superflow.

Although there is little doubt that thissta-

te of collapsed superflow is the absolute free ener-

gy minimum, it is believed that for Ginsburg-Landau

free energy parameters near T the state of uniform

texture with superflow is locally metastable

161. This local stability may be a little practical con-

sequence, however, because the free energy barrier

is small, and once it has been surmounted we expect

-

v to adjust itself so as to satisfy equation (5) at

"

s

a rate controlled by the orbital relaxation time.

For time dependent collapsed superflow we therefore

generalise equation (5) to

When the length scale of the texture is

h

the orbi-

tal equation of motion /9/ contains only one time

scale

r

% p / p

(Ts

-

vn)'

;

we therefore take

-

sl

T =

Bp/p (vs

-

vnl2 and expect

B

% 1 .

Equation (6)

sl

thus becomes

As it stands, we expect equation (7) to describe the

main features of the relaxation of the average su-

perflow to its equilibrium state in a toroidal geo-

metry. For a straight tube we may guess from equa-

tion (I), by ignoring the distinction between v and

-

s

v

that an appropriate equation

may

be obtained by

-s7

replacing (aEs/at) by (aFs/at)

+ Vu

in equation

( 7 ) .

In principle it should be possible, by a more detai-

led study of orbital dynamics, to obtain theoretical

estimates of a and

B. But until this is done it seems

best to regard them as parameters adjustable within

reason to fit experiment.

References

/I/ Hall, H.E., Physica

(1977) 68

121 Cross, M.C., 3. Low Temp. Phys.

2 (1977) 165

/3/ Hall, H.E. and Hook, J.R.,

J.

Phys. C

M)

(1977)

L 91

/ 4 /

Ho, T.-L., Preprint (1978)

/ 5 /

Mermin, N.D., Physica

(1977)

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