Splitting of the phonon spectrum in 3 He- 4 He solutions

Abstract

The dynamic structure function for 4 He- 4 He density correlations S 44 ( k , ω) is calculated as a function of the wave vector k and frequency ω for a simple model of a 3 He- 4 He solution at various temperatures and small 3 He molar concentration x . The inputs to the model include the measured 4 He phonon spectrum and the zero-concentration 3 He quasiparticle energy (above the k=0 value), which is taken to be quadratic in k and to intersect the phonon spectrum at k c near the roton minimum. Taking into account the decay of a phonon into a quasiparticle-hole pair, which is effected in the model by a quantum hydrodynamic interaction, we find that the phonon spectrum is split into two distinct branches. At k c the splitting of the phonon branches depends on a coupling constant γ and the crossover parameter ε (k c )/ k c . If in 3 He- 4 He solutions γ is large enough and ε (k c )/ k c is small enough, then the phonon spectrum is split into two branches.