On the Relationship between Thermal Vibrations and X-Ray Diffuse Scatterings by Polymer Crystals. I. Vibrational Modes and Thermal Fluctuations in a Helical Polymer Chain

Thermal vibrations which are excited in a helical polymer chain are theoretically dealt with. Dispersion relations and thermal fluctuations are given in terms of cylindrical coordinates in order to apply for X-ray diffractions of polymer crystals. It is found from a viewpoint of lattice dynamics that there is the mode ( E (θ)-mode) in a helical polymer chain which takes zero's frequency when the wave number k ζ =1/ P ( P : pitch of helix). The mean square amplitude < R 2 k ζ =1/ P > of E (θ)-mode, which is calculated concretely, makes a divergence when k ζ =1/ P .