The Debye-Waller factor for small cubes and thin films

The dependence of the Debye-Waller factor on the size of cubes and plates is calculated by assuming the Debye theory, but using the frequency distribution which incorporates a term that is proportional to the surface area. The ratio of B for small particles to B for large particles is less than 0.01 for cubes with 10⁹ atoms or more and for plates which are more than 330 atoms thick. The inclusion or exclusion of the zero point energy terms has a large influence on the size correction which has to be applied to small particles and thin films.