A symptotic formulae relating to the physical theory of crystals

Abstract

In the physical theory of crystals great formal difficulties are encountered when the exact shape of the crystal is to be taken into account. Certain methods of approximation have therefore been developed and successfully used by several authors. However, as the validity of these methods was recently questioned by Sir C. V. Raman, a more rigorous examination of the problem had to be undertaken. It is found that the old procedure is fully justified provided the number of boundary particles is small compared with the total number of particles in the crystal. In particular, it is shown that lattice sums may in general be replaced by the corresponding infinite series, and that the distribution of frequencies follows with sufficient accuracy Born's law for cyclic crystals. Upper bounds are obtained for the errors caused by these approximations.