Joumal of Low Temperature Physics, VoL 101, Nos. 1/2, 1995

Spin Dynamics at the Dipole Unlocking Transition in 3He-A

Henry Hall, John Hook, Richard Haley and Fergus Clancy

Department of Physics and Astronomy, University of Manchester, Manchester M13 9PL, England

The equations decribing the spin dynamics of 3 He-A have been generalised to allow for the effect of susceptibility anisotropy on the Leggett-Takagi damp- ing. Analytical and numerical solutions have been obtained for the configu- ration where 1 is locked parallel to a static magnetic field by boundaries. Our investigations show that the dipole unlocking transition, where d flips from parallel to perpendicular to l, should proceed via a long lived spin precession at a frequency (WL + ~LD ), where ~L is the Larmor frequency and ~LD the Larmor frequency at the dipole unlocking field. Progress in searching for this mode experimentally is reported.

PACS numbers: 6Z5ZLm, 76.60.Es

1. I N T R O D U C T I O N

In parallel with our experiments ] on NMR in 3He-A with 1 locked paral- lel to the steady magnetic field H0 by boundaries we have been investigating the spin dynamics for this situation by numerical simulation. In order to ob- t i n physically sensible answers in our simulations it was necessary to gener- Mise the form of the Leggett-Takagi damping to a superfluid with anisotropic magnetic susceptibility. In addition to the small amplitude motions previ- ously predicted, 2 the simulations revealed a steady large-amplitude preces- sional motion associated with the dipole unlocking transition which is not damped by Leggett-Takagi relaxation. The existence of this mode is readily seen analytically, and we have investigated its excitation and decay numer- ically. So far we have not succeeded in detecting the mode experimentally, and possible reasons for this are discussed.

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0022-2291/95/1000-0117507.50/9 �9 1995 Plenum Publishing Corporation

118 Henry Hall et al

2. E F F E C T OF S U S C E P T I B I L I T Y A N I S O T R O P Y O N T H E S P I N D Y N A M I C S

If we insert into the Leggett equations

= 7S x H + RD (1)

a = 3`a • ( n - 3 ` ~ -1 (s + ~RR.I ) (2)

the A-phase dipolar torque RD = (~24/3` 2) x• x l)d. 1 and the anisotropic susceptibility

Xik = X • ((~ik -- c~did~ ) (3)

or equivMently - 1

Xik = X-d l (6 ik + f l d i d k ) , (4)

with fl = a / ( 1 - c~) = ( X • - Xll)/XII, the susceptibility anisotropy does not appear explicitly in the resulting equations. Thus, although

x - I s ~-= X~_ 1 (S -~ fl(S. d)d) (5)

the second term disappears under the d• operation and we have

f~2 = 3`s x tI + ~ x a ( d x 1)d. 1 (6)

a = 3 ` d x ( H - 3 ` x ~ - t S - a ~ r R ( d x l ) d - 1 ) . 3 ` (7)

We shall show shortly that the last term in equation (7), the Leggett- Takagi relaxation, requires modification for the A-phase because of the sus- ceptibility anisotropy. But it is instructive first to express equations (6) and (7) in terms of the deviation of the spin from equilibrium with a steady field H0 (11 l) by writing

s = -~x. rio + s' = x ~ [Ho - ~ ( H o . d)d] + s'. (s) 3 ̀= 7

If we put r = (3 `2 /x . )S ' , v t [ o = ~;L1, equations (6) and (7) become

= ~ , ((1. a )d + (l . d )a ) + . ~ • 1 + ( G - ~ ) ( a • ~)a. ~ (9) k

cl = [~r + r R f ~ ( d X 1)d-1] x d. (10)

The last term in equation (9) shows explicitly the balance between dipolar torque and susceptibility anisotropy torque, and in the absence of Leggett, Takagi damping we obtain correctly the modes of small oscillation about the

Spin D y n a m i c s at D ipo le Unlock ing Transi t ion 119

configurations d II 1 and d A_ 1. 2 However, with the Leggett-Takagi damping as in equation (10) the configuration d • l is always unstable. Even if we conjecture that in the last term of equation (10) ~ l should be replaced by (~t2A -- aW2L), this merely converts a monotonic instability to an oscillatory one above the dipole unlocking field HD, for which O) L : ~A/O~ 1/2.

We have therefore been led to re-examine the derivation of the Leggett- Takagi damping, paying special attention to susceptibility anisotropy. Pro- ceeding as in Vollhardt and WSlfle 3 and using their notation, we write the equation of motion of d as

~l = 7d x (Heg - Hinst), (11)

where Heft = H0 - 7F~S/x~ is the external magnetic field corrected by Fermi liquid interaction effects and Hinst is the 'instantaneous' effective mag- netic field acting on the Cooper pairs and the excitations; here and below the superscript zero on a susceptibility implies the absence of Fermi liquid corrections. In the absence of dissipation, we can use the Leggett equation (2) for d to identify Hinst as

Hinst = Heft - H + 7 x - 1 S = 7 S. (12)

The dissipation arises because the excitation magnetisation sq, unlike the Cooper pair magnetisation, is unable to follow the field instantaneously but instead, through collision processes, is continously relaxing towards the current equilibrium value

Sqq : Xq~ (13)

where X q~ is the (isotropic) excitation susceptibility. Thus we suppose that sq satisfies

l ( s q Seqq), (14) sq = 7 8 q X I-Ief t -- 7"

where 7" is the appropriate relaxation time. The departure of sq from its equilibrium vaiue produces a contribution to (Heft - Itinst) in equation (11)

- . -1

this can be written

"~T (X~ -1 [Sqq - "~Sqq X Hett] , (15)

where the terms in square brackets have to be evaluated in the dissipationless limit. This is readily achieved by using equations (12) and (13) to evaluate

1 S)d) , (16) _- (x"~ +

120 Henry Hall et al

where X p~ is the Cooper pair susceptibility for d • H; note that the pair susceptibility vanishes for d II t t . Using this value of Sqq in expression (15) and the dissipationless Leggett equations to evaluate the time derivatives we find that because of the susceptibility anisotropy the square brackets m equation (10) should contain an additional damping term -7Rfl(d x s )d . s. where s = ("y2/XN)S , /~ (-~ xPOxN/ (xq0xO)) is the susceptibility anisotropy

parameter defined above, vn = 7Xq~ / (X~ 2 and XN (= Xl ) is the normal state susceptibility. The additional term arises because the equilibrium ex- citation magnetisation is not in general parallel to the total magnetisation: the usual damping term comes from the first term in equation (16), and our new term from the second term in equation (16). With the additional term we find that small oscillations about d _k 1 are stable for H0 > HD. as they should be, and small oscillations about d I] 1 are stable for H0 < HD.

For numerical computation and comparison with experiment it is more convenient to use the equations of motion for total spin, which with the modified Leggett-Takagi relaxation are

= ~LS • l + a ~ ( d • 1 )d-I (17)

a = [S--WLl+ rRgt~(d X 1 ) d . 1 - TRZ(d x s )d . s] • d. (18)

We shall see Below that despite susceptibility anisotropy appearing only in the damping term, this controls the stable large amp]_itude motions that occur. If we write the energy density E as

it is readily shown by the use of equations (17) and (18) that the rate of change of energy is given by

dEdt - XNTR72 [ P ~ 4 ( d • 2 1 5 2, (20)

O

which is negative definite, as it should be.

3. S T E A D Y S T A T E S O L U T I O N S

We look for solutions of equations (17) and (18) in which s and d have components in the x-y plane preeessing coherently at angular frequency and constant projections on z, the field direction. We find that there are two solutions of this type with

= ~ r + ~ L D , (21)

Spin Dynamics at Dipole Unlocking Transition 121

where ~LD iS the Larmor frequency at the dipole unlocking field, and

s• = +adidzWLD (22)

(23)

note that the subscript • on 8• d• and (in the following section) S• means perpendicular to z, whereas the subscript s on a susceptibility means per- pendicular to d. Equations (22) and (23) mean that s takes the value that would be in equilibrium, for the given orientation of d, at a field of THD. Moreover, in a frame rotating at the frequency given by equation (21) the ap- parent field is ~=HD. In this sense the mode represents a thermal equilibrium state. The Leggett-Takagi damping is identically zero in this mode.

4. N U M E R I C A L I N V E S T I G A T O N S

If numerical integration of equations (17) and (18) is started with an ini- tial orientation of d far from the equilibrium orientation for the applied field, the system very often settles into the steady precession given by equations (21)-(23) rather than the true equilibrium state. This steady precession is approached in an oscillatory way, with oscillations damped by the Leggett- Takagi relaxation. To enable the simulation to escape from this steadily precessing state we have added phenomenological T1 and T2 relaxation pro- cesses to equation (17), to represent the effects of wall relaxation. With this modification we find that the system has three possible quasi-steady states (i.e. states in which Leggett-Takagi relaxation vanishes), depending on the value of Sz, and passes from one to the other as Sz relaxes following a change of applied field H0. These states are:

1. For XiHD < 7Sz or 7Sz < --XiHD, d precesses in the x-y plane in accordance with the disequilibrium between S~ and Ho, with S• = 0.

2. For xIIHD < 7S~ < Xi/-/D or - X i H D < 7Sz < --xIjHD, d precesses at an angle to H0, with S• ~ 0 in accordance with equations (21)-(23). As T1 causes Sz to relax through this range the system is making a first order phase transition between the states with d II 1 and d • 1

3. For -XIIHD < 7Sz < XIIHD, d is along z as Sz relaxes, with S• = 0.

We have also found in simulations that for H0 > liD the system can be excited into the precessing state (item 2 above) by a substantial RF pulse (rather less than ~) at the negatively shifted resonance frequency.

122 Henry Hall et al

5. A T T E M P T S A T E X P E R I M E N T A L D E T E C T I O N

We have tried two methods of exciting the long-lived precession dis- cussed above, but in neither case have we yet detected any signals.

1. We have looked for transient ringing at the frequency given by equation (21) when H0 is switched from a value above HD to a very much smaller value. The simulations suggest that transient ringing should appear after a waiting period while the magnetisation relaxes into the appropriate range. It is possible that the spin motion in different parts of the sample becomes dephased during the initial waiting period, thereby preventing observation of a signal.

2. We have tried pulse excitation at Ho > HD as suggested by the simu- lations. A worry here is that the negatively shifted resonance that we observe is very much weaker than we expect; we are therefore unsure whether we are coupling properly to the mode we wish to excite.

Another point to be considered is the effect of spatial inhomogeneity on our predicted mode: is it stabilised or de-stabilised by spin currents? This we intend to investigate in the future.

A C K N O W L E D G E M E N T S

We acknowledge useful conversations with Yuriy Blmkov and thank the referee for suggesting that we include the expression for energy dissipa- tion, equation (20). This work was supported by the SERC (now EPSRC) through Research Grant GR/H95839 and through the award of a Research Studentship to RH.

R E F E R E N C E S

I. R. Haley, A. Armstrong, F. Clancy, A. Manninen, J. Hook and H. Hall, paper at this symposium.

2. S. Takagi, Y. Phys. C : Solid State Phys. 8, 1507 (1975). 3. D. Vollhardt and P. W51fle, The Superfluid Phases of Helium 3, p 394 (Taylor

and Francis, London, 1990).