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Submitted on 1 Jan 1978
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ORBITAL MOTION OF 3He-A IN THE PRESENCE
OF A HEAT CURRENT
J. Hook
To cite this version:
JOURNAL DE PHYSIQUE Colloque C6, supplément au n° 8, Tome 39, août 1978, page C6-17
ORBITAL MOTION OF
3H
e~A IN THE PRESENCE OF A HEAT CURRENT
J.R. HookPhysios Department, Manchester University, Manchester MIS 9PL3 U.K.
Résumé.- Nous avons résolu numériquement l'équation du mouvement du vecteur _£ pour 1' He-A dans le cas où le fluide est contenu entre deux plans parallèles infinis, entre lesquels circule un courant de chaleur. Des solutions périodiques dans le temps ont été obtenues, donnant une explication pos-sible des mouvements orbitaux observés par Paulson, Krusius et Wheatley /!/.
Abstract.- We have solved numerically the equation of motion of the Jt vector for He-A for the case where the fluid is contained between two infinite parallel planes £iid there is a flow of heat bet-ween the planes. Solutions periodic in time are obtained which provide a possible explanation of the persistent orbital motions observed by Paulson, Krusius and Wheatley /!/.
Since the discovery of the superfluid phases of 3He, the interesting phenomena associated with the spin dynamics have been widely investigated. It has only recently been realized that the orbital dy-namics of 3He-A is also a fruitful source of new
phenomena; the decay of a supercurrent through mo-tion of the Ji texture HI and the persistent orbi-tal motions observed in the sound attenuation mea-surements of Paulson, Krusius and Hheatley III
(referred to as PKW) are two examples of this . The orbital motions described in this paper provide a possible explanation of the experiments of PKW .
We have investigated solutions of the equa-tion of moequa-tion of jt when the fluid is contained between two infinite parallel planes (at 2 = o and Z = d) and there is counterflow w = v - v between
— —n — s
the planes such as would be produced by a heat cur-rent. We assume dipole-locking (valid for d » 6ym) and spatially uniform v , in which case the equation of motion of Z_ in the Ginzburg-Landau region beco-mes ( Bit A - ^ -P + (v .VH = - 51 x («, x V2 «.) -p 3t —n — — — — — S]l
- 4_a x (w.v)i
+ 2£ x (J. x curl 2.) («,.curl l_ + _£.W) + 21 x (l_ x W) (jt.curl l_ - K..W) + 2l_ x V(«,.W) (1)where \i is the orbital viscosity and we have put H/2m = 1. This equation differs form that derived by us in a previous publication / 3 / (referred to as
I) by the inclusion of terms, erroneously omitted in I, arising from the _& dependence of the tensors T and Tn introduced in I. The corrected equation
agrees with that given by other authors, e.g. Hu and Saslow /4/.
We assume that Jl and W vary only in the Z direc-tion ; conservadirec-tion of matter then determines the Z variation of Itf for a given Jt and also implies that v » v in the Ginzburg-Landau region. The solutions described below were all obtained for v = o but inclusion of a small v does not cause
—n —n any significant change. It is convenient to describe
the solutions using the polar angles 8(Z,t) and 4>(Z,t) of _£ with respect to Z. We use the dimension-less units t* = p W*t/2u and Z* = WoZ(2m/t4), where W is the value of Iwjat z = o, d. We consider both
o '—' possible types of boundary conditions :
a) 9 = o at Z = o, d (o - o solutions) ; b) 6 = o a.t Z = o and 6 = TT at Z = d (o - tr
solutions).
The solutions described briefly below were obtained by numerical integration of equation (l')„by methods which will be described in a forthcoming
publica-tion.
sa) o - IT domain wall solutions.- We have found pre-cessing domain wall solutions of equation (1) similar to those discussed in I with 0 = 8(Z) i.e. independent of time.
<)>•= (|>(Z) + io*t
The precessional frequency oo of these solutions is shown as a function of 1/d in figure 1. It can be seen that w <* i/d* for small d (u) = tfp W /mud
* s II °
in usual units) but u tends to a constant value for large d (u <* p W 2/U in usual units). For
s|[ °
For s m a l l d* t h e domain w a l l i s i n t h e c e n t r e of t h e channel whereas f o r dX % 17 t h e domain w a l l moves away from t h e middle.
b) o
- o s o l u t i o n s . - I n c o n t r a s t t o t h e s i t u a t i o n
f o r t h e i n c o r r e c t e q u a t i o n s d e r i v e d i n I, t h e 0 =o everywhere s o l u t i o n of e q u a t i o n (1) appears t o
*
be s t a b l e a g a i n s t small p e r t u r b a t i o n s f o r a l l d.
It i s l i k e l y t o become u n s t a b l e i n wide channels a t lower temperatures and can a l s o be rendered un- s t a b l e by t h e a p p l i c a t i o n of a s m a l l magnetic f i e l d i n t h e Z d i r e c t i o n . We have i n v e s t i g a t e d t h i s l a t - t e r p o s s i b i l i t y and w i l l d e s c r i b e some of t h e exo- t i c t e x t u r e s which r e s u l t i n a f u t u r e p u b l i c a t i o n .*
For d
%
21 e q u a t i o n (1) has, i n a d d i t i o n t o t h e8 = o s o l u t i o n , a more i n t e r e s t i n g time dependent o
-
o s o l u t i o n which looks approximately l i k e a o-
IT domain w a l l followed by a IT-
o domain w a l l , although i n t h i s c a s e 0i s
i n f a c t s l i g h t l y time dependent. The two domain w a l l s p r e c e s s i n o p p o s i t e s e n s e s and t h e p r e c e s s i o n frequency which i s shown*
i n f i g u r e 1 , i s approximately independent of d for d*2
24 (corresponding t o w O: ps,,
Wo2/p i n u s u a lu n i t s ) and c l o s e t o t h e frequency of p r e c e s s i o n of a s i n g l e o
-
?r domain w a l l i n a wide channel.The o
-
?r domain w a l l and t h e o- o time
dependent s o l u t i o n would cause p e r i o d i c changes i n sound a t t e n u a t i o n s i m i l a r t o t h o s e seen by PKW. To demonstrate t h i s we p l o t on f i g u r e 2 f o r a t y p i - c a l o-
a domain w a l l t h e s p a t i a l average of R: over t h e channel as a f u n c t i o n of time, which should r e f l e c t changes i n t h e a t t e n u a t i o n f o r sound waves t r a v e l l i n g i n t h e x d i r e c t i o n . The o-
o time dependent s o l u t i o n i s however more c o n s t i - t e n t - w i t h t h e experimental r e k u l t s of PKW i n s o f a r a sa ) The r o t a t i n g magnetic f i e l d s needed t o t u r n on t h e motion would correspond t o t h e need t o r o t a t e R through a l a r g e a n g l e t o s e t up a o
-
IT- o tex-
-
t u r e.
b) w a ps Wo2fu + w a ( 1
-
T / T ~ ) - ~ / ~ , c l o s e t o t h e dependence observed-by PKW. The o r d e r of magnitude of t h e frequency i s a l s o c o r r e c t although d e t a i l e d comparison i s i m p o s s i b l e because of t h e complica-Fig. I : P r e c e s s i o n frequency U* f o r (A) o
-
?r domain w a l l s and (B)*o- o time dependent s o l u t i o n s
a s a f u n c t i o n of I / d.
Fig. 2 : The average v l u e of
2
R2
w i t h i n t h e chan- n e l a s a f u n c t i o g of t f o r a ox- s domain w a l l i n a channel w i t h d = 15.References
/ I / Paulson, D.N., Krusius, M., and Wheatley, J . C . , Phys. Rev. L e t t .
37
(1976) 803.121 Mermin, N.D., Quantum F l u i d s and S o l i d s ed. by Trickey S.B., Adams, E.D., and Duffy, J.W.
(Plenum 1977) p. 3 .
131 H a l l , H.E. and Hook, J.R., J. Phys. C
10
(1977) L91.141 Hu, C.R. and Saslow, w.M., Phys. Rev. L e t t s . 38 (1977) 605.
-
t e d geometry of t h e experiments.
