Phonon Scattering by Isotopes in Solid Helium

Scattering of phonons by isotopes in solid helium is stronger than predicted by the mass difference alone, since the change in zero-point energy causes local distortion, which scatters phonons owing to the anharmonicity of the solid. Earlier theories of this effect disregarded the change of the local bulk modulus due to the changed zero-point energy. This change is estimated by comparing the velocity of sound in ${\mathrm{He}}^3$ and ${\mathrm{He}}^4$ at the same atomic volume. It is shown that this change compensates the effect of distortion on the bulk modulus, but not on the shear modulus. Longitudinal phonons are thus scattered by isotopes less strongly than are transverse phonons. A generalization of the Callaway theory is indicated to take account of this difference.