The variational theory of Gasparini, Bhattacharyya, and DiPirro for the binding of a single ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ atom to a ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ film has been modified so as to remove the adjustable constants. In contrast with the older version, the new theory predicts that the ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ binding energy (the energy to evaporate a ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ atom from its lowest state in the film into the vacuum) increases as the film thickness decreases. The results for the binding energy and the excitation energy (the energy to raise a ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ from the ground state to the first excited state) are in good agreement with experiment. The same theory has been used to predict the elastic scattering of an atomic beam of ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ by a ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ film. The problem of calculating the ${}_{}{}^3{}_{}{}^{}\mathrm{-}_{}^3{}_{}{}^{}$He effective interaction $V^s$(q) is discussed in the context of the variational theory. It is found that, at constant ${\mu }_4$, the film is thicker in the neighborhood of a ${}_{}{}^3{}_{}{}^{}\mathrm{He}$. The combined thickening due to two ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ of opposite spin gives a contribution to the effective interaction which is attractive.

• Received 29 May 1984

DOI:https://doi.org/10.1103/PhysRevB.31.1338