The Lifetime of Interacting Ripplons on the Surface of Superfluid-Helium

Ripplons, or quantised surface waves, are known to exist on the surface of superfluid He-4. We follow the work of previous authors and describe these excitations by quantising the classical expressions for the energy of such surface waves. The resulting quantum mechanical Hamiltonian consists of a term describing free ripplons, together with an additional term (cubic in the ripplon variables) which induces the mutual scattering of ripplons and has the effect of causing any one ripplon to have a finite lifetime. We present here a many body calculation which investigates the lifetimes of ripplons at a temperature T = 0 and the lifetimes of acoustic ripplons at a finite temperature. In these two limits our calculation agrees with separate calculations performed by previous authors carried out at the two limits; however our calculations readily allow a sensible interpolation between these limits. We report hitherto unsuspected divergences in the calculations at finite temperature and we comment on the relevance of such calculations to the results of recent experiments which yield information on the ripplon lifetimes.