Theory of Phase Transitions in Solid Heavy Methane

It is the purpose of this paper to explain the existence of three solid phases in CD4, and to predict the molecular ordering in each phase, by straightforward deduction from a reasonable assumption concerning the orientational coupling of the molecules. Molecular and lattice vibrations are neglected, the crystal being treated as an f.c.c. array of spherical rotators carrying distributions of charge with tetrahedral symmetry. It is assumed that the dominant term in the orientational coupling of the molecules is their electrostatic interaction, of which only the octopole-octopole interaction of next neighbors is retained in the calculation. The magnitude of the effective octopole moment is the only disposable constant in the theory. The statistical calculation is based on a classical version of the self-consistent field idea; neglect of quantum effects makes the results inapplicable to CH4, which shows large isotope differences from CD4. The conditions for self-consistency in the theory appear as a family of integro-functional equations, one for each molecule in the crystal. These are brought into convenient form by introduction of tetrahedral harmonics and associated tetrahedral rotator functions. The consistency equations are first solved for the cases in which the orientational distribution function is the same for all molecules in the crystal. There exist three distinct solutions of this type, in which all molecules are subject to identical orienting fields with symmetries T, C2, or D3, respectively. More general solutions are then found, for which the molecules are not all equivalent, but may be freely rotating, or in orienting fields of symmetry T, C2, or D3. The consistency relations state restrictions on the distribution of such molecules in the crystal lattice, and fix the strengths of the orienting fields; they can be satisfied only for small numbers of molecules in a crystal cell. An apparently exhaustive tabulation is made of all solutions with low free energy. Three of these minimize F in some temperature range, and describe stable phases. At the lowest temperatures the stable phase has tetragonal symmetry Vd, with all molecules oscillating about equivalent equilibrium orientations. As T rises this undergoes a first-order transformation to a phase with octahedral symmetry O, in which one molecule in four rotates freely, surrounded by a shell of oscillating next neighbors. At higher temperatures this undergoes a transformation (described by the theory as of second order) to an orientationally disordered phase. If the molecular octopole moment is assigned the magnitude 0.504×10—24 electron cm3 needed to make the predicted upper transition temperature agree with the 27.1°K observed in CD4, the predicted lower transition temperature is 24.4°K, as against the observed 22.2°K. Predictions of the theory are also in satisfactory agreement with observations on integrated heats of transition, zero-point entropy, and the optical properties of the phases.