Internal and External Vibrations in Molecular Crystals

In order to simplify the problem, one identifies from the beginning the well-bound molecules or complex ions in the crystal and treats them as rigid bodies. These rigid bodies will execute not only translational oscillations but also rotational oscillations. Coupling is allowed for between all the various units in the crystal and the resulting vibrations are called external vibrations. The effects of nonrigidity of the molecules or complex ions may be examined separately, thereby treating the effects of the crystal 

It is now an easy excercise to determine the potential energy given by (4.135), 

c = cosa and ijj = exp(iqa). Now, the motion of a rigid molecule can always be described by specifying the coordinates (t) of its center of gravity and some suitable angles (tp) which specify its orientation. The displacement coordinates of the individual atoms of this molecule are then functions of the coordinates t and cp. From Fig.4.21 it is apparent that for small displacements t and rotations w, the following relations hold  

From (4.148) it is obvious that due the rigidity of the XY2 molecules, there are only three independent coordinates. Substituting (4.148) in (4.143) yields 

Using Lagrange equations, one obtains the equations of motion in the usual way and the eigenvalues are 

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