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MAGNETIC RESONANCES ASSOCIATED WITH
LINEAR TEXTURES IN SUPERFLUID 3He-A
R. Bruinsma, K. Maki
To cite this version:
JOURNAL DE PHYSIQUE
Colloque C6, supplément au n" 8, Tome 39, août 1978, page C6-57
MAGNETIC RESONANCES ASSOCIATED WITH LINEAR TEXTURES IN SUPERFLUID
3He"A
+R. Bruinsma and K. Maki
Department of Physios, University of Southern California, University Park, Los Angeles, California 90007, U.S.A.
Résumé.- Les fréquences de résonance magnétique associées à trois textures linéaires de superfluide
3He-A dans un cylindre sont déterminées théoriquement. Dans la limite R » £,, où R est le rayon du
cylindre et Ç est la longueur de cohérence dipolaire, on trouve que les fréquences de résonance lon-gitudinale dans un champ axial sont les mêmes que celle d'une texture uniforme, tandis que les fré-quences de résonance transversale ont des déplacements supplémentaires de l'ordre de
Q
A
¥-Abstract.- The magnetic resonance frequencies of three typical linear textures in superfluid 3He-A confined in a long hollow cylinder are determined. In the limit R » g , where R is the radius of the cylinder and E, is the dipolar coherence length, the longitudinal resonance frequencies in an axial magnetic field are shown to be the same as that for the uniform texture, while the transverse fre-quencies have additional shifts of the order of
«A
¥•
The condensate of superfluid 3He-A is
charac-terized in terms of two unit vectors, & and d. In a cylindrical container, there are three typical (&or d)textures, which are shown schematically in figure
1. The first texture a) is the one proposed by
t
\\!;/- b/,«9 VIM
y ^ - ^ ^ - ^
Fig. 1 : Three typical textures in superfluid 3He-A
in a cylinder are shown. Arrows indicate the direc-tion of x. and d vectors
Mermin and Ho /l/ (MH) and is most stable in weak magnetic field (H > 20 Oe), while the other two tex-tures b) and c) contain radial and circular disgy-ration /2/ respectively and are stable in an axial field /3/ (H > 20 Oe).
We shall confine ourselves to the case R » £ (y 10 y ) . In this case I and d are parallel except in the vicinity of the wall of the container.
t . . The present work is supported by the National
Science Foundation under grant number DMR 76-21032
Since this small deviation from the dipole-locking has only a minor effect on the nmr frequencies / 4 / , we shall neglect it in the following.
a) MH Texture. The oscillation of the 3 vec-tor is parameterized in the cylindrical coordinates as :
d = sin (X + g) cos fp + cos (X + g) z +
+ sin (x + g) sin f$ (1)
where X(P) describes the equilibrium configuration
cos | = (1 - a (f)2)/[l • 3 a (f)2 + a2 ^J'*
with
a=
l^Jl
=(l^lf
(2)3 t J5 2
and f and g are small fluctuations. The correspon-ding eigenvalue equations are
*f f - ~ 2 52 ( s i n 2Xp ) _ 1^ P ('+cos2X)sin2xf
+ 2p2 f W + f a n d
V "i
5
" [
p_1
4
(p
°
+ c o s 2
x ) g
p
)
+
+ 2 -2[ g# + ( c o s2x - s i n2 X) g ] ]+ g ( 3 )
which a r e obtained from the spin free energy, f and
g have to satisfy the boundary conditions at the walls
The lowest eigenvalue for f is given by
Xf
= 1 , with f = const. On the other hand g has to be solved va- riationally which yieldsAs is shown elsewhere /5/ the nmr frequencies in an axial magnetic field are given in terms of the above eigenvalues as
w, = (1,)
'I2
Sl, and w t =bi
+X
Sl;] li2 g Afor the longitudinal and the transversal resonance respectively, where w = yH is the Larmor frequency,
0
and Sl is the Leggett frequency. A
The above result shows that the longitudinal resonance frequency is the same as that for a uni- form texture. Note that the transverse field couples with the g mode /4/ with the azimuthal quantum num- ber f 1 .
b) Radial Disgyration. Here
2
fluctuation is given by2
= [sin ($+
f).;
+
cos (4 + f) cos g + sin gz with the eigenvalue equationsI -2
X
f =- -
t2
(f,-(pf ld)+
2 f ) + ff 2 dp P P
$4
(8)and the same equation for g, with the boundary con- dition (4). Here we have agin
Xf
= 1 with f = const., while the g mode is solved exactly in terms of the Bessel function of orderA.
with
a
and x is the first zero of
-
(JJZ (x)).o
axC) Circular Disgyration. This configuration has the same energy as that with the radial disgy- ration, although it does no longer have the axial
with
x
= tan-' 2x[R2
-
x' +yj
The eigenvalue equations are given by
where u and v are defined by x/R = sinh u/(cosh u + cos v) Y/R = sin v/(cosh u + cos v)
Since the circumference of the cylinder is now given
71
by v = f ?, the boundary condition reads
As before we have
Xf
= I with f = const., whileX
g is evaluated approximately as
h = I + 1.22 (:l2, with g = sin v
g (1 5)
So far we have neglected the effect of the wall of the container where dipole-locking between "Rnd
2
are broken, In the case of the MH structure thPs effect is studied. We find that both longitudi-- nal and transverse resonance frequencies have nega-5
tive shifts of the order of (-) in this case. From R
this analysis we may conclude that in the dipole locked textures deviation in magnetic resonance fre- quencies from those of a uniform texture is very
5 2
small and of the order of
(E)
,
where R is the line- ar drhension of the container.References
/I/ Mermin, N.D. and Ho, T.L., Phys. Rev. Lett.
35
(1976) 594/2/ De Cennes, P.G., in Proceedings of the Twenty Fourth Nobel Symposium, edited by B. Lundquist and S. Lundquist. Academic Press, New York, 1973 /3/ Buchholtz, L.J., and Fetter, A.L., Phys. Rev. B
I5 (1977) 5227
-
141 Bruinsma, R. and Maki, K., preprint
151 Maki, K., and Kumar, P., Phys. Rev. Lett.
38
(1977) 557/61 Maki, K., J. Low Temp. Phys. (in press)
