A rigorous proof on the crystallographic restriction theorem

Abstract

With respect to the crystallographic restriction theorem, although there is a famous proof which is concurred and used by a lot of books in solid state physics, this paper analyses some incorrect or inappropriate arguments in the proof, for example, the possible value of m=1from the proof is in contradiction with eq.⑴; the argument of that the length of B’A’ must be the integral multiples of that of AB is not clear for real lattices of solids, and is required to be proved for various crystal lattices. On the basis of the viewpoint of that the inexistence of C5 axis of symmetry is equivalent of that pentagons are impossible to fill all the space with a connected array of pentagons, using a purely mathematical approach the paper rigorously proves the crystallographic restriction theorem.