[MATH]-TEXTURE EFFECTS ON MAGNETIC RESONANCE IN SUPERFLUID 3He

(1)

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[MATH]-TEXTURE EFFECTS ON MAGNETIC

RESONANCE IN SUPERFLUID 3He

D. Vollhardt, P. Kumar

To cite this version:

D. Vollhardt, P. Kumar.

[MATH]-TEXTURE EFFECTS ON MAGNETIC RESONANCE

(2)

JOURNAL DE PHYSIQUE Colloque C6, supplément au n" 8, Tome 39, août 1978, page C6-43

I-TEXTURE EFFECTS ON MAGNETIC RESONANCE IN SUPERFLUID

3

H<T

D. V o l l h a r d t and P . Kumar

Department of Physios, University of Southern California, Los Angeles,

California 90007, VSA

Résumé.- Nous calculons les effets sur les fréquences satellites dans les expériences de NMR, dus aux changements dans les variations spatiales du vecteur Z .

Abstract.- We calculate the effects on^the satellite frequencies in NMR due to the changes in spatial variation length scale of the £ vector.

INTRODUCTION.- In the A-phase of superfluid 3He, two vectors characterize the order parameter : the % and the d vector that respectively refer to the orbital and spin degrees of freedom. Their orientation has to be either parallel or anti-parallel and can be different only at the cost of nuclear dipolar inter-action energy. The boundary between the parallel and anti-parallel orientations is referred to as a

soli-ton

IM.

The fast time scale dynamics of the d vector, with % fixed, determines the NMR frequency. In the presence of solitons, it determines the satellite frequencies which appear due to the different dipo-lar torque in the middle of the domain wall. The satellite frequency (defined in terms of R = W./fi, for longitudinal resonance and R^fi. =[uiL- ui-)1'2 for the transverse resonance, where U)Q = yU„ is the Larmor frequency and fl, is the Leggett frequency, representing the strength of the dipolar interaction) rather critically depends on the length scale of the variation of the & vector. Denoting that scale of length by A, for A = °° the satellite frequencies are zero whereas if X is totally determined by the dipolar interaction, X is given by £ //5, where E, is the spin coherence length and the- satellite fre-quencies are those for the composite soliton. If the length scale changes, the NMR frequency becomes an efficient measure of it. One such effect is due to currents, currently being studied by Maki and Vollhardt.

When the spin system goes off equilibrium (by tipping the spin or turning off the field) the d vector rotates. If the off-equilibrium is inhomo-geneous, solitons may be created. At the short time

*Work supported in part by The National Science Foundation under grant No DMR 76-21032.

scale, the & vector may not have time to respond, but over a time period /2/ "\< 10 milliseconds

(Cross-Anderson time) the solitonmay lose its kinetic energy and dump the excess dipolar energy

into the spatial gradients of & and/or diffusive orbital waves. The problem of this time evolution is mathematically quite complicated. However by tracking the length scale X by the NMR frequency, it is possible to glean some information from the experiments /3/. In this paper we establish the functional relationship between X and R. and IL,.

2. TEXTURE RESONANCE.- The calculation proceeds along lines similar to ref. IM with one substan-tial difference. We take

1 =

sin

X

x + c o s

X 9

d = (sini|>x + cos ij) y) cos 8 + sin 9 z (1)

with

E 4 f dzfx

2

+ 4(9

2

+ c o s W ) - A _

C

o s

2

6 cosVl

<• - c o ] z z z g 2 I

± (2)

To determine the static texture we use

X = a sin-1 sech y and 9 = 0 (3)

with X fixed. The energy is determined by varia-tionally minimizing E with respect to a and T) . Hence we obtain

^ ( f V ^ f + V ^

1 **) <*>**

Here f is the energy of the pure d-soliton/I/and

a=j I , n = ^ _ ^ i — (5)

(3)

I I I I

and

I I I

I

=

r d z sech z sech

₍₆₎

0

with

.6

-

In Eq. (6) we present an approximation for 1, good

within 1.3% for

(a).

_X

The magnetic resonance fre-

quencies are the eigenvalues of the fluctuation

operator. They too are determined variationally (in

1

I I I I 1

1

0

1 I

,2 ,4 .6 .8 1.0 1.2

analogy to ref. /I/); here the variational parameter

YEA

isV

:

and

where

a

I]

(a,

6)

5

l 3 z sech z sech 6i

-

and

Fik. 1

:

The width of the a-texture as a function

of the width of the

texture. The arrow

represents the composite soliton

results are valid only near Tc. At lower temperatu-

res, the corrections arise due to Fermi liquid ef-

fects, as has been discussed

141. This in our calcu-

lations can be done by replacing

5

in Eq.(4)

by

4 ( 5 +

1)

where

1

F~

K'--+1

'

1 --

]

_(I-

₎

T

Tc

6 [ 4

1+;ll

] + I F a

3 1

The basic change is reflected in the h dependence of

n

and a.

I

the

2 length

n

snaps out and there is a d*scontinuous ~

i

2 ~

:

.

~h~ satellife frequency ratiaR1 and R as

change to n/5

=

.69 from n/5

-

. I 5

(figure

1).

a function of h. The arrows represent the

1

1 composite soliton

The

NMR

satellite frequencies are presented fop the

I2

(a,B)

=

IOm

dz sechU z sech2 Bz

-

'

6 +

i

entire spectrum of values of

h

in figure

2,

Thesf!

10

-]Iq

+

(22-u

+L)

8

i-

(7)

6

p'

1.62 and

q r 1 . 7 3

_RISRT I I I I I I I

.

_.

-

- \

.

-

t

--.

-

.

_--

_-

1

-

- -_

The approximations for II and I2 given in Eqs. (9a,b)

_Rt

-

--

_-

can be shown to be accurate to within about 2.5% for

R i

all a and

B.

I I I I I

An interesting result is the behavior for small

I I

0 2 ,4 .6 . 8 1.0 1.2 1.4

values of h. If

h

is smaller than the composite

X/E.

solitom value, the length

17,

though it follows

h

as

(4)

We thank Professor K. Maki for stimulating

discussions. Special thanks are due to

8.

Bozler

for discussions of his experiment. D. Vollhardt

also wishes to thank the Studienstiftung des

Deutschen Volkes for a scholarship.

References

/ I /

Maki, K., and Kumar, P., Phys.Rev. B E

(1977) 18; / 2 /

Kumar, P., in "Quantum Fluids and Solids", eds.

B. Trickey, et al., Plenum Press, New York

(1977;

/ 3 /

Lee, D.M., Physica B +

C,

90

(1977) 35 ;

Bozler,

H.M., Bartolac, T., Proc. ULT Hakone Symposium

(1977) 136