Atomic Theory of the Two-Fluid Model of Liquid Helium

TLDR

The authors argue that excitations in liquid helium can be described by a wave function nearly of the form Σᵢ f(rᵢ) φ, where φ is the ground‑state wave function, and note that this view is insufficient to explain the λ‑transition and critical flow velocity, especially in dilute He‑3/He‑4 solutions. The thermodynamic and hydrodynamic equations of the two‑fluid model are discussed from this perspective. Using a variational principle, the trial function f(r)=exp(ik·r) minimizes the energy, yielding E(k)=ħ²k²/(2mS(k)), where S(k) is the liquid’s structure factor. The dispersion relation derived from the trial function exhibits a linear phonon branch at small k, a roton minimum whose energy Δ is predicted to be twice the experimental value, and an effective mass μ that agrees with Landau’s specific‑heat data; additionally, the He‑3 effective mass in a dilute He‑4 solution is calculated to be about six atomic mass units.

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.