Zero-temperature properties of matter and the quantum theorem of corresponding states: The liquid-to-crystal phase transition for Fermi and Bose systems

Abstract

The zero-temperature properties of matter with an interaction pair potential of the Lennard-Jones form are studied further within the context of the quantum theorem of corresponding states. In particular, the phase transition between the liquid and gaseous phases is studied for systems obeying either Bose-Einstein or Fermi-Dirac statistics. In contrast to well-known systems, the nature of this transition in these quantum systems depends on the statistics in a fundamental way. We find that it is illuminating to extend the usual thermodynamic variable space to include the corresponding-states quantum parameter $\ensuremath{\eta}=\frac{{\ensuremath{\hbar}}^{2}}{m\ensuremath{\epsilon}{\ensuremath{\sigma}}^{2}}$. It is shown that the phase transitions occur at zero temperature as $\ensuremath{\eta}$ is varied. For Bose systems it is found that a second-order liquid-to-gas transition occurs at a value of $\ensuremath{\eta}=0.456$. Thus, for Bose systems there is no coexistence region. In sharp contrast, for Fermi systems, there is a range of values of $\ensuremath{\eta}$ for which the liquid and gaseous phases can coexist. This coexistence region exists in the range $0.29\ensuremath{\le}\ensuremath{\eta}\ensuremath{\le}0.33$. The essential features of the behavior of both Bose and Fermi systems can be understood in terms of simple models. Detailed numerical results are presented for both cases.