The Rotation of Molecules in Crystals

In the previous sections we have discussed the rotation and vibration of free molecules, that is, of molecules in the gas phase. There is strong evidence2 that molecules and parts of molecules in many crystals can rotate if the temperature is sufficiently high. The application2 3 of quantum mechanics to this problem has led to a clarification of the nature of the motion of a molecule within a crystal which is of some interest. The problem is closely related to that dealing with the rotation of one part of a molecule relative to the other parts, such as the rotation of methyl groups in hydrocarbon molecules.4 

The wave equation for a diatomic molecule in a crystal, considered as a rigid rotator, obtained by introducing V into the equation for the free rotator given in a footnote of Section 35c, is 

If the molecule being studied is made up of like atoms, such as is O2 or H2, then a reasonable form to assume for V is 

The wave equation 38-1 with the above form for V has been studied by Stern,1 who used the mathematical treatment given by A. H. Wilson.2 We shall not reproduce their work, although the method of solution is of some interest. The first steps are exactly the same as in the solution of the equation discussed in Section 18c except that a three-term recursion formula is obtained. The method of obtaining the energy levels from this three-term formula is then similar to the one which is discussed in Section 42c, where a similar situation is encountered. 

We have referred to the case of free rotation of methyl groups at the end of Section 36a. 

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