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Submitted on 1 Jan 1978
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HELICAL TEXTURES IN 3He-A
A. Fetter
To cite this version:
JOURNAL DE PHYSIQUE Colloque C6, supplément au n° 8, Tome 39, août 1978, page C6-46
HELICAL TEXTURES IN 3He"A
A.L. F e t t e r
Institute of Theoretioal Physics, Department of Physios, Stanford University, Stanford, Ca 94305, V.S.A
Résumé.- On étudie la stabilité de 3He-A avec une texture et vitesse superfluide uniformes. Au-delà
du seuil trouvé par Bhattacharyya, Ho et Mermin, le milieu subit une transformation vers une confi-guration stationnaire en hélice. Cette structure est stable près du seuil relativement à des per-turbations de la forme exp(ik.r).
Abstract.- We study the hydrodynamic stability of 3He-A with uniform texture and superfluid velocity. Beyond the threshold found by Bhattacharyya, Ho, and Mermin, the system undergoes a transformation to a static helical configuration. This structure is stable near threshold with respect to pertur-bations of the form exp(ik.r).
Bhattacharyya, Ho, and Mermin /I/ have re-cently shown that bulk uniform 3He-A with £ parallel
to v can become unstable with respect to small-am-plitude deformations. This instability reflects the competition between the hydrodynamic torque, which favors the distortion, and the elastic and anisotro-py energy, which oppose it. In the dipole-locked regime near T , the weak-coupling Ginzburg-Landau hydrodynamic parameters just suffice to stabilize the uniform texture /l/. At lower temperatures 111,
however, the decreased anisotropy parameter p may induce a transition to other more complicated confi-gurations. The present paper uses the hydrodynamic free energy of bulk dipole-locked 3He-A and the
re-sulting dynamical equations of motion to examine the character of the instability. Just beyond the thres-hold atp , we show that the & vector can assume a
oc
helical structure with an apex angle proportional to(p - p0) and a critical wavenumber proportional
to the superfluid velocity. This new state is shown to be stable with respect to small deformations of the form exp(ik.r).
We start from the hydrodynamic free-energy
density III
f = 4- P vz - 4- P„(£.v )2 + cv .curl I
-2 s s -2 ov ~s ~s
- c ( L vs) ( L c u r l £ ) + I Ks( d i v S )2 +
+ j Kt( L c u r l £ )2+ ^ K ^ L c u r l I)1 (1)
and the corresponding mass-current density J - 3f/Sv = p v - p 1(1.v ) + c curl 1 -~s ~s s~s o ~s
- c £(Lcurl I). (2) o
As in reference 1, the normal-fluid velocity is assumed to vanish. The order parameter is then
spe-cified by the local orientation of a rigid triad of orthonormal vectors relative to some standard orien-tation. A stable configuration is one in which the free energy is a minimum with respect to small chan-ges S t in the direction of the unit vector % and a small rotation 6 <|> about the local direction SL. The-se virtual displacements induce a corresponding change in the superfluid velocity /3-5/
Sv = V 6 $ - 2 x (VI).Si (3) Instead of working directly with the free energy,
which would involve a second-order expansion, we shall use the equivalent dynamical equations
V.Jg = 0 (4)
yl r= V 1 - - M ^ - ^ + 5 X CJ.V)4 (5) d c L3 T . l 311 ~s ~
I
where y is the orbital viscosity and the partial derivatives of f are taken at fixed v .
~s
A static helical configuration is specified by a constant polar angle 6 , and a linearly increasing azimuthal angle $a(z) = Vz with a wavenumber p. Such a structure has translational invariance in the xy plane and is consistent with a uniform superfluid velocity v = wz. The resulting supercurrent j has a helical character, which satisfies equation (4), and the remaining condition of static equilibrium leads to the equation
sin 9 {p w2 cos 6 + w V p -p cos2 6 +
o o o p1- s o o
+ c (2-3 sin2 9 ) 1 + c p2 sin2 9 cos 6 +
O 0 -1 Or 0 0
+ p2coseo H ^ + 2 0^-iy sin29oJ} = 0 (6)
that determines 9 . One obvious solution is 9 = 0 ,
o o which is just the uniform texture. If p is less
1
that the critical value p = KT1 (c + T P " )2 and p is
oc D o 2Ks r
1 near the critical wavenumber p E-w(c +-p")JC,,-l,
0 2 s however, an expansion for small
Bo
reveals a second solutionThe coefficient is positive for typical hydrodyna- 5
mic parameter (it has the value
-
for the weak-cou- 4pling Ginzburg-Landau parameters), and this relation thereforeshows that small-angle helices can be in static equilibrium within the range of wavenumbers
To study the stability of the helical struc- ture, we consider perturbations of the form
(9) where
Equations (4) and (5) may be expanded to first or- der in these small quantities, and the translational invariance in the xy plane permits a separation of variables in the plane-wave form
exp(iq.zr--Ut),
whereq
= x2 + yy. The z dependence is more com- plicated, for the periodic helical structure in ge- neral mixes wavenumbers that differ by integral mul- tiples of p. This effect is proportional to q, howe- ver, and we may study the stability for small q2 by assuming a z dependence of the form exp(ik z). Since the uniform texture below threshold (Po2
pot)
is definitely stable for perturbations with _q+
0, we anticipate that small q2 will remain the most im- portant range just beyond threshold. Our procedure can be shown to be equivalent to a variational cal- culation with trial function 6@, 68, 6$ proportio- nal to the plane wave exp(ik.r).In this approximation, the equation 1.6j = O
-S
is easily solved for the amplitude 6Q. Substitution into the remaining equations gives a pair of homo- geneous algebraic equations for the amplitudes 68 and 64I. The corresponding decay constant uk satis- fies a quadratic equation, the sum of whose roots is positive for small-angle helices whenever equa- tion (8) holds. In contrast, the product of the roots is positive only within the more restricted range of wavenumbers
which therefore serves as the stability criterion. If p satisfies equation (8) yet violates equation
(ll), the resulting helix is unstable with respect to long-wavelength longitudinal deformations (k go),
7,
but it remains stable with respect to transverse ones
(q
# 0). Note that stable helices always exist near threshold, so that the instability found in reference 1 signals the onset of a small deformation rather than a catastrophic collapse.The present calculation suggests several ex- tensions. First, it is important to include the lea- ding spatial harmonic contributions proportional to expi(kz
'
p)z in the solutions for 6@(z), 68(z), 6$(z). The variational basis shows that this impro- ved set of trial functions will improve the estima- te for the decay constant uk, but we do not antici- pate a qualitative change i< the criterion (1 I). Second, it is important to study the region farfrom threshold, when the helical structure itself may become unstable. If this latter situation occursfor perturbations with finite transverse wavenumbers(q
-
f O), then the new configuration would have non- zero vorticity iq x 2(p sine
168. Given the rela-0
tion between dissipation and vorticity transport in superfluids /1,5,6/ this latter question is of par- ticular interest.
Acknowledgement.- I am grateful to M.C. Cross, H.E. Hall, and N.D. Mermin for stimulating corres- pondence. This research was supported by NSF Grant No. DMR 75-08516.
References
/I/ Bhattacharyya, P., Ho, T.-L., and Mermin, N.D.,
Phys. Rev. Lett.
2
(1977) 1290/ 2 / Cross, M.C., J. Low Temp. Phys.
2
(1975) 525 /3/ Ho, T.-L., in Quantum Fluids (Plenum, New York,1977) 97
/ 4 / Brinkman, W.F. and Cross, M.C., to be published /5/ Mennin, N.D. and Ho, T.-L., Phys. Rev. Lett.
36
(1976) 594
