Nature of Vibrational Modes in Ionic Crystals

The principal purpose of this paper is to discuss the nature of the modes of vibration in a solid. The following is found: (1) Waves in lattices are in general neither transverse nor longitudinal; in particular, they need not be transverse or longitudinal when the propagation vector k is very small. (2) The relationship $\frac{{\omega }_l}{{\omega }_t}={\left(\frac{{\epsilon }_0}{{\epsilon }_{\infty }}\right)}^{\frac{1}{2}}$ for "longitudinal" and "transverse" modes in ionic crystals applies, if at all, in a region of small, but nonzero, wave vector k. (3) The derivation of this relationship is based, at least implicitly, on the use of cyclic boundary conditions. (4) The use of cyclic boundary conditions is valid in statistical problems for crystals without long-range forces, but has never been justified for systems with Coulomb forces. (5) If cyclic boundary conditions are nonetheless used, it can be simply shown that for k=0, $\frac{{\omega }_l}{{\omega }_t}=1\mathrm{}$.