Molecular Ordering in Solid Hydrogen and Nitrogen in the Molecular-Field Approximation

A quantum-mechanical theory is given for the transition temperature and the nature of the ordered state for a system of molecules interacting through general quasiquadrupolar forces. The transition temperature and the ordered structure below it are found to be, respectively, the lowest eigenvalue and the corresponding eigenvector of an Hermitian matrix derived quantum mechanically, within the framework of the molecular-field approximation. It is further shown that under certain conditions the same structure is also the ground-state structure. Application is then made to the phase transitions in solid hydrogen and nitrogen. For solid ${\mathrm{H}}_2$ and ${\mathrm{D}}_2$ the calculations are performed over the range of densities for which the molecular phases are expected to ramain stable. For Para-${\mathrm{H}}_2$ and ortho-${\mathrm{D}}_2$ at high densities, transitions to phases in which the molecules are orientationally ordered are found. Using the intermolecular potential given by Nakamura for solid hydrogen, the transitions are found to be always to a $T_h^6$ structure. It is also shown that for para-${\mathrm{D}}_2$, the assumption that $J=1\mathrm{}$ is a good quantum number cannot be justified even at normal density. For solid nitrogen the intermolecular potential given by Kohin is found to be incapable of predicting a transition from the $T_h^6\alpha$ phase to the high-pressure $D_{4h}^{14}\gamma$ phase. Possible other quasiquadrupolar potentials which may account for this transition are suggested.