Considering the elementary excitation spectrum of ${\mathrm{He}}^3$-${\mathrm{He}}^4$ mixtures to be of the form proposed by Landau and Pomeranchuk, we calculate the scattering cross section for roton and ${\mathrm{He}}^3$ quasiparticle collisions, taking final-state interactions into account. Our theory demonstrates the importance of final-state interactions in renormalizing the roton energy and lifetime. Previous theories based on the Born approximation are shown to give unreliable results for the change of the energy and lifetime for rotons in dilute ${\mathrm{He}}^3$-${\mathrm{He}}^4$ mixtures owing to roton-${\mathrm{He}}^3$ scattering. We obtain upper bounds on the changes in the energy and lifetime of a roton as a function of the roton-${\mathrm{He}}^3$ coupling strength, using a simplified model for the coupling. These bounds give an insignificant change of the roton energy with the ${\mathrm{He}}^3$ concentration and thus explain recent neutron-scattering and Raman data on the mixtures. Effects of level repulsion between rotons and the ${\mathrm{He}}^3$ quasiparticle-hole continuum are calculated, and estimated to be small on the basis of recent Raman data. However, decay of a roton into a ${\mathrm{He}}^3$ quasiparticle-hole pair may give rise to an interesting concentration dependence of the roton linewidth. Further experimental studies of the mixtures are suggested, which may check the detailed predictions of the theory and provide insight into the momentum dependence of the coupling parameters. The present analysis represents an essential link between microscopic theories of the quasiparticle coupling and related experiments on dilute ${\mathrm{He}}^3$-${\mathrm{He}}^4$ mixtures.

• Received 19 March 1973

DOI:https://doi.org/10.1103/PhysRevA.8.1973