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Submitted on 1 Jan 1978
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TEXTURES IN SLOWLY ROTATING 3He-A
H. Vidberg
To cite this version:
JOURNAL DE PHYSIQUE Colloque C6, supplément au n" 8, Tome 39, août 1978, page C6-61
TEXTURES IN SLOWLY ROTATING
3He~A
H.J. Vidberg
Research Institute for Theoretical Phyeics University of Helsinki
Siltavuorenpenger 20 B - SF-00170 Helsinki 17, Finland
Résumé.- Nous étudions les textures de 3He-A dans un cylindre infiniment long en rotation lente et nous comparons les énergies libres de trois configurations différentes.
Abstract.- I study textures of 3He-A in a slowly rotating infinitely long cylinder and compare
the free energies of three possible configurations.
The line integral of the superfluid
veloci-ty v around a cylinder filled with He-A is
quanti-zed /l/. This can be used to classify X-textures in
a slowly rotating cylinder. The Mermin-Ho texture
111
has the circulation 2ir in units of •=—, I
consi-zm
der also the next simplest non-singular texture
which has the superfluid circulation 67? and a
sin-gular vortex superimposed on the M-H structure of 4ir.
The critical angular velocity for the 6ir-texture to
become more advantageous than the M-H texture is
approximately 7 — r - , where R is the radius of the
mR/
cylinder. For small R, the singular vortex structure
may be more advantageous in temperatures close to T.
One has to minimize the free energy
func-- > -*•
tional F - w.L, where F is the free energy of the
-*• . •*•
fluid at rest, (0 is the angular velocity, and L the
angular momentum. In the London limit
HI.
F = KA
2fd
3r|4v
2- 2(X.v )
2+ 2v .&cX
-
4(X.v ) (X.^xX)
J [ s s s s
+ 5 {|^xX|
2+ (v".X)
2+ v.{(
l.fyt-t & -t
}}]
t
= « £ j
d
3
r m
?
x
[4^ - 2t(t.v
s
) +
%t
-
2
X(X. vxt)]
(1)
Weak coupling coefficients (near T ) have been
assu--a c
med and v is in units of -=— . The magnetic field is
zero and the dimensions of the container have been
taken to be large on the scale of the dipolar length
allowing the 3-vector to be parallel to X almost
eve-rywhere.
I first consider extremely slow rotation
when a modified M-H texture is stable. I neglect any
dependence on z, the direction of the cylinder axis.
Then X = sin 9(r) r+ cos 6(r) z
•*
1 - cos 6 (r) ~ (2)
V = *—'— A
s r *
and I take X (0) to be parallel with the direction
of rotation. The free energy functional is
numeri-cally minimized with respect to the function 9(r)
with the usual boundary condition of
1
perpendi-cular to the cylinder wall /3/. As compared with
the Mermin-Ho «=0 solution 6(r)
- j T[ I
1*!,
the
^-vec-tor here tends to turn away from the direction of
ro-tation by small angles such as TT/8 at the last
an-gular velocity where the state is stable. If the
direction of rotation.is opposite to 1(0) the M-H
type structure is always unstable against the
flip-ping of
1(0)
to the parallel position. The free
energy as a function of R=mR
2(o/H is plotted in the
figure.
F - S T\
\ \
, \
\
50- -\ \ \\ W
\ ' \ ^
— . N , . \ 'i\. ,—.—,—,—_. m ^ L .
^
> \
^ \
-50- \ ^ \Fig. : The free energy in u n i t s of 2TTKA
2as a
func-t i o n of func-the dimensionless angular v e l o c i func-t y mu)R
2/)i
for the Mermin-Ho t e x t u r e ( ) , . the 6ir-texture
(—) , and the singular vortex t e x t u r e with R/£ = 10
s( — ) and.R/£= 10H ( ) .
When to grows the frir-texture w i l l eventually
become advantageous. I choose the simple v a r i a t i o
-nal form
%
= s i n x ( r ) I c o s @;+
s i n $@I+
cosx
( r )-+
v = 2(1-
c o s x ( r ) ) $ / r,
O Z R ' and'jZ
= s i n ~ ( r );
+
cos ~ ( r )-+
v = (3-
c o s ~ ( r ) ) $/r,
R'IrzRwhere c o n t i n u i t y r e q u i r e s x(R') =
n
,
and t h e boun d a r y c o n d i t i o n x ( R T ) = 37~ 12. The f r e e energy i sp l o t t e d i n t h e f i g u r e , and i t i s seen to go below t h a t o f t h e M-H t y p e t e x t u r e a t R
-
7.The f i n a l t e x t u r e I c o n s i d e r i s a s i n g u l a r v o r t e x l i n e superimposed on t h e M-H s t r u c t u r e . I n t h e v a r i a t i o n a l c a l c u l a t i o n t h e gap i s taken t o r e - cover from t h e v o r t e x w i t h t h e coherence l e n g t h 5(T). The f r e e energy a s a f u n c t i o n o f
R
i s a g a i n p l o t t e d i n t h e f i g u r e w i t h two v a l u e s of t h e parameter R/c(T). I t i s lower t h a n those of both t h e M-H and t h e 6 ~ - t e x t u r e f o r s u f f i c i e n t l y small R o r T c l o s e t o Tc. For R=l mm t h e temperature c r i t e r i o n i s T2
.98 Tc.B e t t e r t r i a l f u n c t i o n f o r t h e r e l a t i v e l y complicated 6 ~ - t e x t u r e , however, may lower i t s ener- gy c o n s i d e r a b l y . N e v e r t h e l e s s , t h e energy of t h e s i n g u l a r v o r t e x s t r u c t u r e would s t i l l be of t h e s a m o r d e r of magnitude d e s p i t e t h e l o g a r i t h m i c term l o g (R/<(T)). T h i s i s c o n t r a r y t o t h e h i g h v e l o c i t y r e - s u l t of Volovik and Kopnin 151. One should t h e r e - f o r e b e c a r e f u l a b o u t any statement about t h e r e l a - t i v e energy of t h e s i n g u l a r v o r t e x s t a t e w i t h o u t de- t a i l e d c a l c u l a t i o n s .
References
/ I / T . L . , Ho, PH.D., t h e s i s 1978, C o r n e l l U n i v e r s i t y (unpublished).
/2/ N.D., Mermin and T.L., Ho; Phys., Rev., L e t t . , 36 (1976) 594.
-
/ 3 /
.V.,
Ambegaokar; P. G.,
de Gennes and D. , R a k e r ; Phys.,Rev.,A9
(1974) 2676./ 4 / L . J . , Buchholtz and A.L., F e t t e r ; Phys.,Rev.,
-
B s (1977) 5225.
