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

The zero-temperature properties of matter with an interaction pair potential of the Lennard-Jones form are studied in the context of the quantum theorem of corresponding states. In particular, the phase transition between the fluid and crystalline phases is studied for systems obeying either Fermi-Dirac or Bose-Einstein statistics. It is found that the solidification pressure of a Fermi system is much lower than that of a Bose system with the same mass and pair potential. We find that it is illuminating to extend the usual thermodynamic variable space to include the corresponding-states quantum parameter $\eta \equiv \frac{{\hslash }^2}{m\epsilon {\sigma }^2}$ which is defined in the text. It is shown that phase transitions occur at zero temperature as $\eta$ is varied; in particular, a first-order liquid-solid transition is described in detail and a model is discussed in which a second-order magnetic transition occurs at a critical value of $\eta$. It is suggested that the full complexity of the phase-transition behavior, which is observed at finite temperatures arising from various properties of the potential, will also be observed at zero temperature mainly as a fundamental consequence of the quantum-mechanical zero-point kinetic energy.