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Submitted on 1 Jan 1978
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THE INTERNAL STRUCTURE OF A DE GENNES
DISGYRATION IN 3He-A
P. Muzikar
To cite this version:
JOURNAL DE PHYSIQUE Colloque C6, supplément au n° 8, Tome 39, août 1978, page C6-53
THE INTERNAL STRUCTURE OF A DE GENNES DISGYRATION IN 3He"A P. Muzikar
Laboratory of Atomio and Solid State Physios, Cornell University, Ithaca, New Jork 14853 U.S.A.
Résumé.- Nous décrivons la structurelle 3He-A dans un cylindre de longueur infinie et de grand rayon à la surface duquel vg est nulle et S. est radial. Entre la surface et la région centrale vs n'est pas toujours nulle. La région centrale est elle-même formée de la phase polaire.
Abstract.- We describe the structure^of 3He-A in an infinitely long cylinder of large radius subject to the boundary condition of radial % and zero v at the surface. Between the surface and the core region a non-vanishing vs develops, and the core itself is composed of polar phase.
The best configuration of superfluid 3He-A in a large cylinder subject to the usual /l/ boun-dary condition is probably given by the texture first proposed by Anderson and Brinkman 111. In this configuration the radial I vector at the wall is accompanied by a single quantum of circulation in v , which is required by topological considera-tions if a singularity inside the cylinder is to be avoided 13/.
If I is radial at the wall with zero v then s a topologically non-trivial singularity is required somewhere in the interior /3/. We wish to point out that for a very large cylinder the most energeti-cally favorable texture in this case may have some unexpected features. The order parameter then has the form
A . = -4=r (-z.+ij.) (1)
at r = R where R is very large. If it remained of this form all the way to the singular core it would constitute a radial, dipole-locked de Gennes dis-gyration / 4 / .
The problem of finding the best configura-tion satisfying (l)atr=R divides into two distinct parts : a) finding the best texture outside the core region (r % 5 ) , where A . remain locally of the A phase form, and b) finding the best structure for the core of the singularity when r < g.
- a) The key observation to make is that the dipole-locked. vortex
A . = — z (r. + i<|>.) (2)
oil rr a 1 Tx v '
has a lower gradient energy than the disgyration (1) /5/. (We use the weak coupling Landau-Ginzburg gradient free energy, along with the usual bulk
and dipole terms / 6 / ) . As a result, the form (1)
will not be maintained as r decreases from R. Ins-tead the disgyration will start changing into the
vortex (2) :
A . = -sr {2 cos8+r sin6}»{r.cos8-z.sin9+i5.} (3) ai Y2 a a I x Ti
where 8=6(r) and 8 decreases from y to zero as r
decreases 111. The function 8(r) can be exactly de-termined in the infinite R limit, being related to the one kink solution of the sine-Gordon equation.
The transition to 8 = 0 is essentially complete
by r £10-If R, although it can occur over a
shor-ter length scale without much of an increase in
energy.
When r < r the A . will remain in the vortex o ai
form (2) until some length r. comparable to the di-pole length L„ /6/. When r < r. the A . will
chan-D 1 ai ge to the dipole-free radial disgyration, because this form has an even smaller bending energy than
the vortex /5/. Thus
A . = -pr z (r.+i$.), r < r,,
Acti = 72 ^(^iSinB+z^osB t i ^ ) , r < *x (4) where 6 is r at r = r. and drops to zero for
smal-ler r. Estimates of the energy for simple trial
g's indicate that r|%2lY), and that this
transi-tion is indeed energetically favorable.
- b) Various proposals have been made for
the form of the A . when r % £. Fishman and
Privo-rotskii / 8 / suggested that the A . change into the
polar phase form to eliminate the singularity whi-le Mermin /9/ conjectured that an axi-planar core
might be favorable. It is also possible that the
magnitude of the order parameter simply goes to
zero /10/.
These cases correspond to References
Aai = AO (E.cos6+i$.sin6), 6 -+ 0 (polar)
a 1 ' 4
=
9
[ ~ a c o s 6 + i a s i n ~ $ i + i [ T i a c o s 6 + i a s i n ~O
.
~os6-$~P~sin63 6 :o+$
(axi-planar) A = ( E i + i i ) b (r)+ 0 (normal core) (5)4 7
Estimating the energies of these texture leads us to conclude, with Fishman and Privorotskii 181, that the polar core is indeed the best. Al- though the planar core, as pointed out by Mermin, has a lower bulk energy, the polar core is far less costly in gradient energy /lI/.
A consequence of this investigation is that a net equilibrium angular momentum is not peculiar to the absolute non-singular equilibrium texture. The configuration with a topologically non-trivial singularity described here will also have an angu- lar momentum of the same order of magnitude as that calculated by Mermin and Ho 1121.
ACKNOWLEDGEMENT.- N.D. Mermin has been a moderate- ly constant source of inspiration and useful cri- ticism.
/I/ Ambegaokar,V., De Gennes,P.G., Rainier, D., Phys. Rev. (1974) 2676
/2/ Anderson,P.W., Brinkman,W.F., in The Helium Liquids, edited by ,T.Armitage and I.Farquhar, Academic, London (1 975) 406-413, see also 1121 /3/ Toulouse,G., Kleman,M., J. de Physique Lett.
37
(1976) 149
/4/ De Gennes,P.G., Phys. Lett. @ (1973) 271
/ 5 / Brinkman,W.F., Cross,M.C., (to be published) ;
these authors point out the energetic favora- bility (in the weak coupling Landau-Ginzburg gradient free energy) of the vortex in the di- pole locked regime, and of the radial disgyra- tion in the dipole free regime. Note that be- cause of stability conditions on the gradient coefficients (KlZ3>O, K2>0) these conclusions remain true even if the weak coupling assump- tion is not made. These authors also speculate that the radial disgyration might appear in the central region of a line singularity.
/ 6 / Buc~oltz,L.J., Fetter,A.L., Phys. Rev.
B15
(1977) 5225
/7/ While the vortex has a lower energy than the radial disgyration, there remains the possibi- lity that there is a topologically equivalent, dipole-locked texture with an even smaller energy. If so, the conclusions of this paper might be modified.
/8/ Fishman,F., Privortskii,I.A., Journal of Low Temperature Phys.
25
(1 976) 225191 See 1121
/lo/ Buchholtz and Fetter /6/ discusses this possi- bility for a disgyration in a small cylinder /11/ Privorotskii,I.A. private communication to N.D.
Mermin
