Orientational dynamics of superfluid 3He: A “two-fluid” model. I. Spin dynamics with relaxation

Abstract

This is the first of two papers devoted to the dynamics of the orientation of the Cooperpair wavefunction in superfluid 3He. The present paper deals with the dynamics of the spin orientation, including relaxation; a second paper will discuss the dynamics of the l -vector (orbital dynamics) in the A or A 1 phase. The theory developed here is claimed to be approximately valid in both the A and B phases for spatially homogeneous situations at all frequencies much less than the “gap frequency” Δ h ̷ ; it embraces nonlinear as well as linear phenomena. It is believed (but not proved) that it is also valid for the A 1 phase provided that the deviation from equilibrium is not too large. The output of the theory is three simple equations of spin dynamics (Eqs. (4.20)–(4.22)) which generalize the earlier “hydrodynamic” results of one of us and may be solved for a great many cases of practical interest. The theory used is a “two-fluid” model, in which the changes in macroscopic physical quantities arising from variation of the Cooper-pair wavefunction are associated with the “superfluid” component, while those arising from changes in the statistical occupation factors are attributed to the “normal” components. We calculate the “susceptibilities” associated with the two components separately, and show that the superfluid susceptibility is proportional to (T c − T) 1 2 in the limit T → T c . The physics of the model is illustrated by considering the longitudinal dipole resonance (internal Josephson effect): We observe that in the “tunneling” between the up- and down-spin bands it is only the superfluid component which is involved, so that the result of the tunneling is to produce an imbalance between the superfluid and normal components in the two bands separately; this imbalance simultaneously relaxes by spin-conserving collisions, with some relaxation time τ( T) which is of the order of the quasi-particle lifetime. For ωτ ⪡ 1 the theory reduces to the hydrodynamic theory, but for ωτ ⪢ 1 the hydrodynamic susceptibility χ is replaced by the superfluid susceptibility χ p ; for ωτ ∼ 1 the relaxation between the superfluid and normal components gives rise to damping of the NMR. After explaining the physical picture we derive general equations of motion with the help of operator definitions of the superfluid spin, etc. A comparison of our theory with the results of microscopic calculations shows that exact agreement is obtained for longitudinal motion, linear or nonlinear, at arbitrary frequencies. For transverse motion our theory is valid to first order in ωτ; it is not exact for ωτ ≳ 1 but gives surprisingly good agreement with the microscopic results when explicit comparison is possible. It is pointed out that a simple single-relaxation-time approximation to the solution of the kinetic equation may ignore important effects, and in particular in the region of moderate magnetic fields near T c probably actually gives worse results than our phenomenological theory. Applications considered include cw resonance linewidths, recovery of the magnetization following a field step, recovery after a large-angle pulse, the ringing-down of the B-phase “wall-pinned” mode, and the general theory of nonlinear longitudinal resonance. A useful general equation for energy dissipation in the limit ωτ ⪡ 1 is also given. An Appendix discusses the “anomalous” case where susceptibility anisotropy is important.