Vibration Spectrum and Heat Capacity of a Chain Polymer Crystal

The Born‐von Karman model for a chain polymer crystal previously studied by Stockmayer and Hecht is re‐examined using only analytic approximations rather than numerical methods. The analytic approach brings out many peculiar properties of the chain polymer model that did not appear in the numerical treatment. One branch of the frequency distribution g(v) is shown to be proportional to v² for the smallest values of v, approximately proportional to $v^{\frac{3}{2}}$ for slightly larger v, to v^½ for still larger v and to v^—½ in still a fourth range of small values of v. An accurate graph of g(v) is constructed for the entire range of frequencies using values of the force constants suggested by Stockmayer and Hecht. g(v) is shown to have approximate singularities of a type not anticipated by van Hove in his broad treatment of singularities for general systems. This anomalous behavior results from having strong valence forces resisting the bending of bond angles. A classification and description of various kinds of singularities that may arise for systems with strong valence forces are given in an appendix.