Effect of dipole interaction on collective modes in³He-A

Abstract

A general theory for the correlation functions of superfluid³He which takes into account rigorously the magnetic dipole interaction is developed. The resulting equations are solved for the Anderson-Brinkman-Morel (ABM) state and for wave vectorsq oriented parallel to the energy gap axis. Then the dispersion relations of low-frequency modes, including Fermi liquid corrections and damping due to pair breaking, are calculated in the zero-temperature and zero-field limit. There are two real frequency modes arising from each of the longitudinal and transverse spin density correlation functions: a spin wave and an orbit wave,both exhibiting a frequency gap where that of the spin wave is somewhat modified in comparison to the unperturbed longitudinal nuclear magnetic resonance frequency Ω ^/ABM _L .The orbit wave is damped much more strongly than the spin wave. Further, there are two real frequency modes arising from the density correlation function: the sound wave, having a frequency gap of the order Ω ^/ABM _L , and an orbit wave, exhibiting a gap in wave number of order Ω ^/ABM _L /v _F.—The NMR frequency undergoes a small splitting, which is the result of the splitting of the energy gap due to the dipole interaction. One of the two gaps still has nodes.—In addition to these low-frequency modes our equations yield resonances at frequencies of the order of the gap frequency Δ ₀ /ħ, i.e., at ω=1.22 Δ ₀ /ħ and at ω=1.58 Δ ₀ /ħ. The damping and the oscillator strengths of these resonances are calculated.