The rigid-body model for molecular crystals

In molecular crystals, the separation between internal and external modes is of importance. Except for torsional oscillations in some types of molecules, the internal modes have much higher frequencies than the external modes. According to expressions such as Eqs. (2.51) and (2.58), the latter are then the dominant 

The most general motions of a rigid body consist of rotations about three axes, coupled with translations parallel to each of the axes. Such motions correspond to screw rotations. A libration around a vector A (Ai,A2, A3), with length corresponding to the magnitude of the rotation, results in a displacement 5r, such that 

When a body undergoes vibrations, the displacements vary with time, so time averages must be taken to derive the mean-square displacements, as we did to obtain the lattice-dynamical expression of Eq. (2.58). If the librational and translational motions are independent, the cross products between the two terms in Eq. (2.69) average to zero, and the elements of the mean-square displacement tensor of atom n, U j, are given by 

If a rotation axis is correctly oriented, but incorrectly positioned, an additional translation component, perpendicular to the rotation axes, is introduced. The 

The quadratic correlation between librational and translational motions can be allowed for by including in Eq. (2.70) cross terms of the type Dikt ), or 

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