Group-Theoretical Selection Rules in Inelastic Neutron Scattering within the Rigid-Molecule Model

The model-independent technique of Elliott and Thorpe (ET) is extended to apply to a class of models in which molecular units undergo translational vibrations with respect to each other and also librations, but in which the internal vibronics of the molecules are neglected. Within the model we find that the ET "structure function" $F^{\mathrm{}\left(r\right)\mathrm{}}\left(\overrightarrow{\mathrm{k}}\right)$, associated with irreducible representation $r$ and momentum transfer $\overrightarrow{\mathrm{k}}$, can be written in the form $F^{\mathrm{}\left(r\right)\mathrm{}}\left(\overrightarrow{\mathrm{k}}\right)=F^{\mathrm{}\left(r\right)\mathrm{}}\left(\overrightarrow{\mathrm{k}}\mathrm{}\right|R)+F^{\mathrm{}\left(r\right)\mathrm{}}(\overrightarrow{\mathrm{k}}\left|\theta \right)$, where $R$ and $\theta$ signify translational and rotational oscillations. Moreover, the translational part is identical to that of ET except that the atomic scattering lengths $a_n$ which appear in their result are to be replaced by $\overrightarrow{\mathrm{k}}$-dependent molecular form factors $a_n\left({\overrightarrow{\mathrm{k}}}^{\prime }\right)$. $F^{\mathrm{}\left(r\right)\mathrm{}}\left(\overrightarrow{\mathrm{k}}\right|\theta )$ contains a vector form factor equal to $i{\nabla }_{{\overrightarrow{\mathrm{k}}}^{\prime }}{a}_n\left({\overrightarrow{\mathrm{k}}}^{\prime }\right)$, where ${\overrightarrow{\mathrm{k}}}^{\prime }$ is related to $\overrightarrow{\mathrm{k}}$ via a rotation. Mathematically, their result is contained in ours as a special case. Physically, we indicate how to use both procedures in concert, thereby aiding in the identification of $r$ as well as in separating the internal from external vibrations and among the latter, the translational and rotational parts thereof. At the Brillouin-zone boundary we employ the so-called multiplier representations, thereby achieving a simplification both of our results and theirs. By significantly reducing the number of phonon modes to be considered in complex molecular crystals, we have likewise increased the diagnostic power of this method which requires no detailed knowledge of force constants. It is hoped that our results will receive wide application in the identification of phonons in such crystals.