The Rotational Motion of Molecules in Crystals

It is shown by the discussion of the wave equation for a diatomic molecule in a crystal that the motion of the molecule in its dependence on the polar angles $\theta$ and $\phi$ may approach either one of two limiting cases, oscillation and rotation. If the intermolecular forces are large and the moment of inertia of the molecule is large (as in ${\mathrm{I}}_2$, for example), the eigenfunctions and energy levels approach those corresponding to oscillation about certain equilibrium orientations; if they are small (as in ${\mathrm{H}}_2$), the eigenfunctions and energy levels may approximate those for the free molecule, even in the lowest quantum state.

It is found in this way that crystalline hydrogen at temperatures somewhat below the melting point is a nearly perfect solid solution of symmetric and antisymmetric molecules, the latter retaining the quantum weight 3 for the state with $j=1\mathrm{}$ as well as the spin quantum weight 3. This leads to the expression $S=-n_{A}Rlog{n}_A-(1-n_A)Rlog(1-n_A)+n_{A}Rlog9+S_{\mathrm{tr}},$ in which $S_{\mathrm{tr}}$ is the translational entropy, for the entropy of the solid at these temperatures. At lower temperatures (around 5°K) the solid solution becomes unstable relative to phases of definite composition, and the entropy falls to $S=n_{A}Rlog3+S_{\mathrm{tr}},$ the entropy of mixing and of the quantum weight 3 for $j=1\mathrm{}$ being lost at the same time. Only at temperatures of about 0.001°K will the spin quantum weight entropy be lost.

Gradual transitions covering a range of temperatures and often unaccompanied by a change in crystal structure, reported for C${\mathrm{H}}_4$, HCl, the ammonium halides, and other substances, are interpreted as changes from the state in which most of the molecules are oscillating to that in which most of them are rotating. The significance of molecular rotation in the interpretation of other phenomena is also discussed.