Elementary Theory of Liquid Helium: Refinement of the Theory and Comparison with Feynman's Theory

It has been suggested that the excitations of liquid ${\mathrm{He}}^3$ can be represented approximately as excitations of pairs of helium atoms acting as hindered plane rotators. For the low excitations $N$ helium atoms may be considered as $\frac{N}{2}$ pairs; for the higher excitations the possibility of random pairing must be taken into account, a situation which is discussed in some detail. Because of fluctuations in the hindering potential, and because of interactions between localized excitations, the energy levels of the plane rotators are broadened into energy bands, which can give a good account of the specific-heat curve. The multiplicity of these levels is related on the one hand to the statistics and spin of ${\mathrm{He}}^3$ atoms, and on the other to the vibrational modes of a quasi-lattice approximating the liquid, and is also concerned with the development of communal entropy in a quantum liquid. In ${\mathrm{He}}^4$ the low-lying energy levels are excluded because of the spin and the statistics. Vibrations of single atoms, pair rotators, and double pairs, rotating or oscillating like meshed gears, have excitation energies close to that of a roton (the energies of the latter two types are estimated by comparison with ${\mathrm{He}}^3$, allowing for the effect of the increased density on the hindering potential). These types of excitation are not all independent, however, and there is moderate difficulty in accounting for the multiplicity of the excitations as deduced empirically from the number of rotons appearing at different temperatures, which is obtained reasonably accurately from the specific heats. The picture presented here is compared with Feynman's theory; it concluded that there is good correspondence—even the difficulty concerning the multiplicity of the excitations being present in both cases. The relation of a broadened band of energy levels to the idea of a gas of excitations is considered, and it is concluded that such a gas should obey Fermi-Dirac statistics.