ABOUT THE GENERATION OF TEXTURAL DEFECTS IN SUPERFLUID 3He-A

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ABOUT THE GENERATION OF TEXTURAL

DEFECTS IN SUPERFLUID 3He-A

J. Kurkijärvi

To cite this version:

JOURNAL DE PHYSIQUE Colloque C6, supplément au n° 8, Tome 39, août 1978, page C6-63

ABOUT THE GENERATION OF TEXTURAL DEFECTS IN SUPERFLUID 3He-A J . Kurkijarvi

Department of Technical Physics and the Low Temperature Laboratory, Helsinki University of Technology, SF-021S0 Espoo IS, Finland

Résumé.- Nous discutons le processus menant aux ruptures de texture dans 3He-A dans un gradient de champ magnétique. Les ruptures apparaissent là où l'aimentation locale a été déviée de sa direc-tion par 90° à l'aide d'un champ alternatif transverse.

Abstract.- I discuss the appearence of textural defects in 3He-A subjected to a magnetic field gradient and locally tipped with a transverse rf-field.

INTRODUCTION.- Satellites to the main NMR line in 3He-A have been observed by several groups /1,2,3,4/ In this note the experiment of Kokko et al / 4 / . is discussed in detail. They tip the nuclear spins lo-cally with the aid of a magnetic field gradient. The field gradient makes the spins'precession rates de-pend on their positions. A tipping pulse at a fixed frequency turns the spins where the tipping frequen-cy is in resonance with the local frequenfrequen-cy. As a result of the dipole-dipole interactions, the A pha-se shift of the local magnetization depends on the angle the magnetization makes with the external field. As soon as the magnetization gets tipped the exact resonance point moves a bit toward the higher field. A tipping pulse therefore leads to a final tipping angle which varies spatially along the field gradient and reaches a maximum slightly on the high field side of the unperturbed resonance. Kokko et al. find that certain large tipping angles result in permanent "holes" in the magnetization i.e. mis-sing resonance signal at certain spatial positions. I argue, using Fomin's adiabatic approach /5/ to the Leggett equations 16/ and generalizing to include

spin diffusion, that the permanent holes appear at places where the tipping angle is exactly 90°.That Kokko et al. found this angle to be slightly larger is also a result of spin diffusion. It may be possi-ble to measure the spin diffusion coefficient in the A phase by observing this difference as a function of the temperature and the pressure.

I also considered the effect of spin supercurrents, or the bending of the d-vector 111, in addition to spin diffusion. It turns out that the coefficient multiplying the corresponding terms in the equations

of motion of the magnetization is too samll 15,Si

to play a significant role. The essential correct-ness of the picture presented here has been checked with a computer simulation program.

My approach does not authorize predictions as to what kind of structures emerge in the holes and call forth the satellite line /9/ after the re-moval of the field gradient. One can say that the d-vector and the Jl-vector get seriously twisted at those positions, and work is in progress towards understanding the resulting defect in detail.

THE ADIABATIC SOLUTION OF THE LEGGETT EQUATIONS.-Fomin has shown /5/ that the Leggett equations for the magnetization M can be reduced to the following set expressed in polar coordinates with the exter-nal magnetic field parallel to the z-axis and a the azimuthal and 8 the polar angle :

M = -I Q 2 (1 + cosB)2sin2<|> (1) o A Ma = -| Q 2 JlcosB + (1 + cosB)cos2<tT[ (2) MS = - i Q fsing (1 + cos6)sin2(f) (3) o A M4> = (l - cosB)Ma + M(l - M) (4) 2 . .

Here £2 is the dipole frequency, dotted quantities are time derivatives, and <)> is a conjugate varia-ble to M and plays in the present connection the role of indicator as to whether the local magneti-zation stays in its stationary adiabatic point. The stationary solution given by Fomin is

* ^ ° ;

M ^ I + I B -

3

M ( 1

^

o s g )

3 (5)

where M (1-cosB) is a constant of the motion. M and <j> oscillate around the stable point essential-ly with the longitudinal (Josephson) frequency of the magnetization. This is easy to understand qua-litatively looking at equations (1) and (4).

6

I f

41

wanders away from

4

2

0 , M w i l l grow and @ w i l l g e t a n e g a t i v e time d e r i v a t i v e v i a t h e term M (M-1) i n Equation ( 4 ) . I f , however, t h e system d o e s g e t pushed away from t h e s t a b l e p o i n t f o r some r e a s o n , t h e magnitude of

fi

w i l l change and

3

and w i l l g e t o u t of s y n c h r o n i z a t i o n , i . e . g e t t w i s t e d i n t h e s t a - t i o n a r y frame. T h i s i s e a s i e s t t o s e e from t h e Leg- g e t t e q u a t i o n s i n t h e r o t a t i n g frame b e f o r e t h e ave- r a g i n g h a s been c a r r i e d o u t ouver x-and

a,

which r o t a t e a t t h e (high) Larmor frequency.

I

= - E x Z (8)

When M becomes a p p r e c i a b l y d i f f e r e n t from H , t h e p r e c e s s i o n r a t e s o f 11 and d a l s o become d i f f e r e n t . I t has been assumed that: t h e R-vector i s s t a t i o n a r y and p e r p e n d i c u l a r t o t h e magnetic f i e l d fi a t l e a s t a s long a s t h e system s t a y s i n a d i a b a t i c e q u i l i b r i u m .

SPIN DIFFUSION TERMS.- Assuming t h e m a g n e t i z a t i o n t o obey a c l a s s i c a l d i f f u s i o n e q u a t i o n , new terms appear

i n t h e e q u a t i o n of motion f o r

2

2 2 f4 = D ~ M "

-

~ ( 8 ' ) ~

-

M(a') s i n 6 7 (9) & s i n 8 = D[

(Md'

+

2M'a')sinB

+

2 ~ ~ ' a ' c o s ~ l (10) 2 M; = D[ MB" + 2M'B'

-

M(a') sin8cosf37 (1 1 )

and i n d i r e c t l y f o r t h e q u a n t i t y

4

a s seen i n equa- t i o n ( 4 ) . The ;-texture imposed b t t h e f i e l d g r a d i e n t c o n t r i b u t e s a n o t h e r s i m i l a r s e t of t e r n s which we w i l l i g n o r e i n t h e p r e s e n t q u a l i t a t i v e a n a l y s i s , a s

t h e over a l l c o e f f i c i e n t m u l t i p l y i n g them i s s m a l l e r t h a n D by a n o r d e r of magnitude. I n t h e above equa- t i o n s primes i n d i c a t e s p a t i a l d e r i v a t i v e s i n t h e d i - r e c t i o n of t h e magnetic f i e l d g r a d i e n t .

I n t h e experiment of Kokko e t a l . t h e gra- d i e n t o f

a grows s t e a d i l y .

It e n t e r s squared i n two of t h e d i f f u s i o n e q u a t i o n s , e q u a t i o n (9) and equa- t i o n ( L l ) , and t h e s e terms w i l l end up dominating

2

t h e scene. I n Equation (11) t h e term-DM(uV ) sinBcosB w i l l t u r n t h e magnetization upward i f < 90° and down where

B

> 90'. A t e x a c t l y

6

= go0, n o t h i n g

2 2

w i l l happen and t h e term -DM(at) s i n of e q u a t i o n (9) w i l l wrench t h e s o l u t i o n o u t of t h e a d i a b a t i c s t a b l e ' p o i n t . The d i s a s t e r w i l l n o t t a k e p l a c e e l s e - where a s t h e a n g l e a l o o s e s i t s s i g n i f i c a n c e when B

i s e i t h e r around z e r o o r a. I f t h e t i p p i n g p u l s e i s s t r o n g enough, two h o l e s w i l l appear, and t h e s p i n s between t h e h o l e s w i l l p o i n t downward a t l e a s t f o r a s h o r t while. During t h e u n r u l y period a f t e r

t h e t i p p i n g , r e p o r t e d by Kokko e t a l . , one should be a b l e t o observe a n e g a t i v e A phase s h i f t from

t h e downward m a g n e t i z a t i o n between t h e h o l e s . A f t e r t h e t i p p i n g and b e f o r e t h e (aq)'-terms t a k e o v e r , t h e term DMB" i n e q u a t i o n (11) h a s time t o smooth o u t somewhat t h e t i p p i n g p r o f i l e i n space The ( a ' ) '-terms become e f f e c t i v e r a t h e r suddenly roughly 1000 Larmor c y c l e s a f t e r t h e end of t i p p i n g i n t h e Kokko e t a l . experiments. E s t i m a t i n g B" from t h e t i p p i n g p r o f i l e one g e t s t h e c o r r e c t o r d e r of magnitude%lOO f o r t h e d i f f u s i o n c o r r e c t i o n assu- ming D % 0 . 1 cm2/s which i s e x t r a p o l a t e d normal li-

quid v a l u e / 6 , 8 , 1 0 / .

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