A Contribution to the Theory of Bose-Einstein Liquids

Abstract

Smoothed potential models for liquid ${\mathrm{He}}^{4}$ and liquid ${\mathrm{He}}^{3}$ in which each are considered as ideal Bose-Einstein and Fermi-Dirac gases situated in potential wells of potential $\ensuremath{-}{{\ensuremath{\chi}}_{4}}^{0}$ and $\ensuremath{-}{{\ensuremath{\chi}}_{3}}^{0}$, respectively, are considered. It is shown that the degeneracy temperature, ${T}_{0}$ of pure liquid ${\mathrm{He}}^{4}$ on this model can be deduced and that the $\ensuremath{\lambda}$-transition temperatures, ${T}_{\ensuremath{\lambda}}$, of solutions of such liquids can be calculated. The calculated values of ${T}_{\ensuremath{\lambda}}$ as a function of concentration of ${\mathrm{He}}^{3}$ appear to be in satisfactory agreement with the observed values. From these considerations, predictions are made regarding the behavior of solutions of two Bose-Einstein liquids; e.g., ${\mathrm{He}}^{6}$ in ${\mathrm{He}}^{4}$. It is shown, moreover, that solutions of such model liquids obey the third law of thermodynamics.Detailed calculations have been made of the vapor pressures of such smoothed potential liquid models of ${\mathrm{He}}^{4}$ and ${\mathrm{He}}^{3}$ both in the pure state and in solution. The results for the vapor pressures of the solutions indicate that, in the temperature range above 1\ifmmode^\circ\else\textdegree\fi{}K for solutions of not too high concentration of ${\mathrm{He}}^{3}$, the total vapor pressure would be higher than that given by Raoult's law for temperatures both above and below the $\ensuremath{\lambda}$-temperature of the solution. In this way the experimental results for the vapor pressure of such solutions, first emphasized by Taconis, et al., can be explained, and good agreement between theory and experiment is evident. Explicit formulas are given for further numerical evaluation.Finally the results of calculations of the total vapor pressure of model solutions of ${\mathrm{He}}^{3}$ in ${\mathrm{He}}^{4}$ in which the ${\mathrm{He}}^{3}$ Fermi-Dirac liquid model is extremely degenerate are discussed. It would appear that at very low temperatures (0.5\ifmmode^\circ\else\textdegree\fi{}K) the partial vapor pressure of ${\mathrm{He}}^{3}$ in such solutions of ${\mathrm{He}}^{3}$ in ${\mathrm{He}}^{4}$ and the distribution coefficient, $\frac{{C}_{v}}{{C}_{L}}$ should become smaller than the values calculable from Raoult's law.