Atomic Theory of the Two-Fluid Model of Liquid Helium

Abstract

It is argued that the wave function representing an excitation in liquid helium should be nearly of the form $\ensuremath{\Sigma}{i}^{}f({\mathrm{r}}_{i})\ensuremath{\varphi}$, where $\ensuremath{\varphi}$ is the ground-state wave function, $f(\mathrm{r})$ is some function of position, and the sum is taken over each atom $i$. In the variational principle this trial function minimizes the energy if $f(\mathrm{r})=\mathrm{exp}(i\mathrm{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathrm{r})$, the energy value being $E(k)=\frac{{\ensuremath{\hbar}}^{2}{k}^{2}}{2mS(k)}$, where $S(k)$ is the structure factor of the liquid for neutron scattering. For small $k$, $E$ rises linearly (phonons). For larger $k$, $S(k)$ has a maximum which makes a ring in the diffraction pattern and a minimum in the $E(k)$ vs $k$ curve. Near the minimum, $E(k)$ behaves as $\ensuremath{\Delta}+\frac{{\ensuremath{\hbar}}^{2}{(k\ensuremath{-}{k}_{0})}^{2}}{2\ensuremath{\mu}}$, which form Landau found agrees with the data on specific heat. The theoretical value of $\ensuremath{\Delta}$ is twice too high, however, indicating need of a better trial function.Excitations near the minimum are shown to behave in all essential ways like the rotons postulated by Landau. The thermodynamic and hydrodynamic equations of the two-fluid model are discussed from this view. The view is not adequate to deal with the details of the $\ensuremath{\lambda}$ transition and with problems of critical flow velocity.In a dilute solution of ${\mathrm{He}}^{3}$ atoms in ${\mathrm{He}}^{4}$, the ${\mathrm{He}}^{3}$ should move essentially as free particles but of higher effective mass. This mass is calculated, in an appendix, to be about six atomic mass units.