Multiple Scattering and Many-Body Theory: Free Energy of Electrons in Helium

An electron in helium vapor at 4°K is characterized by its $s$-wave scattering length off helium atoms. This scattering length is small compared with the average interparticle spacing in the helium vapor. Consequently, we have taken as a model a particle interacting via hard-core repulsion with an ideal gas. This establishes a connection with the hard-sphere Bose problem of more general interest. The model described above is simpler (a) because it has only Boltzmann statistics and (b) because the electron is very light compared with the helium atoms. For this model, we have calculated the interaction free energy of the electron assuming that it is in statistical equilibrium with the helium gas. In the $s$-wave approximation, it is shown that this interaction free energy is rigorously $2\pi \rho a\left(\frac{{\hslash }^2}{m}\right)$ due to single scattering, all higher order multiple-scattering effects being zero. Here, $\rho$ is the average density of helium atoms, $d$ is the scattering length, and $m$, is the electron mass. Since the $p$-wave approximation contributes a term of order $\rho a^3$, it is evident that the term $2\pi \rho a\left(\frac{{\hslash }^2}{m}\right)$ is good to higher densities than might previously have been supposed. This provides partial justification for the "bubble" model of the electron mobility since the term $2\pi \rho a\left(\frac{{\hslash }^2}{m}\right)$ is certainly good up to densities at which the free energy of the "bubble" configuration becomes smaller.