Volume 89A, number 4 PHYSICS LETTERS 10 May 1982

ANGLE DEPENDENT FREQUENCY SHIFT IN ANTIFERROMAGNETIC SOLID 3He "~

F. FISHMAN Physik-Department der Technischen Universitiit Mfinchen, 8046 Garching, Fed. Rep. Germany

Received 11 November 1981

The tipping angle dependent frequency shift in antifertomagnetic solid 3He is calculated. It is shown that for some tip- ping angles this shift could be positive as well as negative and depends on the direction of the external magnetic field.

At T c ~ 1 mK and low magnetic fields (less than 420 mT) solid 3He undergoes a phase transition to an antiferromagnetic state [ 1,2]. In order to explain the cw NMR experiment, Osheroff et al. [ 1 ] have propos- ed the antiferromagnetic structure consisting of [ 1, 0, 0] planes of parallel spins arranged in an u p - u p - d o w n - d o w n sequence and the following equa- tions of the spin dynamics

S = 7SX H - I2027-2x0(d • / ) (d X I ) , (1)

d = "yd x (H - , rzolS). (2)

Here S is the spin density, H is the magnetic field, d is the unit vector along the sublattice magnetization at zero magnetic field, X0 is the transverse susceptibi- lity and 3' is the gyromagnetic ratio. The [1,0,0] di- rection is denoted with a unit vector 1. Linearizing these equations around the equilibrium solution (d . l = d " S = O, ! " H / H = cos O , T S = X o H ) , t h e y

have obtained an excellent agreement with the experi- mental data, with [20/21r ~ 800 kHz at T = 0 and I20/21r ~, 500 kHz at T = T c.

However, in order to verify the nonlinezr nature of these equations, one should consider the situation, where the linearization cannot be used. The "turn- offf ' experiment has been proposed by Hu and Ham [3]. Here the large tipping angle experiment will be considered.

Let us suppose that H = (0, 0, H0) and ! = (0,

• Work supported by a grant from the Alexander-yon- Humboldt Foundation.

sin 0, cos 0) and the initial configuration at t = 0 is S = So(0, sin a, cos ~) and d = (1,0, 0), which corre- sponds to the tipping angle ~. Neglecting first the di. polar torque in eq. (1), we get the solution for S and d:

S = So(sin a sin 6ot, sin a cos cot, cos a ) ,

d = (cos2 ~ot + sin2½ot cos 2 ~ t ,

-- sin 2 ~0~ sin 2 ~ t , -- sin ~ sin ~ t ) , (3)

where ~o = 7 H 0. Now let us take into account the dipolar torque in

eq. (1). Assuming that e = ~22/8¢o 2 ,~ 1, we can con- sider the dipolar torque as a perturbation. As in case of superfluid 3He [4], this will give rise to the change of the precessional frequency and to the appearance of the new harmonics in S. Looking for the solutions of eq. (1) with (S 2) = S02 + O(e 2) where < ) means the average over the new period, we obtain the solu- tion for S

S x = S O (sin a[(1 - a)sin [2t - b sin 2~2t + c_sin 3~2t]

+ h sin 412t},

S y = SO {sin a[(1 + a)cos f~t + c+cos 3~2t]

+ f + h cos 412t},

S z = S0(cos a - g l c o s ~2t - g2cos 2~2t - g3cos 3~2t

- g4cos 4 [ 2 0 , (4)

where [2 = w + ~ co and

0 031-9163/82/0000-0000/$02.75 © 1982 North-Holland 203

Volume 89A, number 4 PHYSICS LETTERS 10 May 1982

5w = coo[(4 cos 20 - sin20)cos t~ + sin20] ,

a = e(2 cos 2 l a cos20 - sin21ot sin20) ,

b = e sm o~ sin 2 0 ,

c+_ = e sin 2 ~ct(cos 20 + cos20 __. ~-sin20),

h = ~e sin 20 s ina~a,

f = 2e sin 20(sin4½a - s in2a),

gl = e sin 20(sin oL + ~sin 2 a ) ,

g2 = e sin2a sin20,

g3 = ~-e sin a sin 20 s in2~a,

g4 = e sin20 sin4½ot. (5)

The expression for ~ o is correct only for the tip- ping angles a not very close to rr0r - t~ ~, 4e sin 20), where the perturbational approach could be used. Under this condition all additional harmonics are of order e compared to the fundamental one and vanish for a -+ O.

The frequency shift 6~o does not depend on the tipping angle or, if 0 = 00 = arctg 2 ~- 63 °, and is a de- creasing (increasing) function of a, if 0 < 00 (0 > 00). I f 0 < 01 = arctgx/2 ~ 55 °, the frequency shift changes sign at a = arccos(tg20/tg20 - 4) and its min- imum value is -[22/260 for 0 = 0 and a --> ft.

Comparison between the present result and the ex- periment could give an additional argument for or against the theory of antiferromagnetic solid 3He.

References

[1] D.D. Osheroff, M.C. Cross and D.S. Fisher, Phys. Rev. Lett. 44 (1980) 792.

[2] E.D. Adams, E.A. Schubert, G.E. Haas and D.M. Bakalyar, Phys. Rev. Lett. 47 (1980) 789.

[3] C. Hu and T.E. Ham, Phys. Rev. 24B (1981) 2478. [4] W.F. Brinkman and H. Smith, Phys. Lett. 51A (1975)

449.

204