This problem proves the crystallographic restriction a third way

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Condensed Matter, Materials & Soft Matter · Crystal structure, bonding and diffraction

Solid State Physics, Ch Symmetry and the classification of crystals

Problem

This problem proves the crystallographic restriction a third way, using only the length of a shortest lattice vector. Suppose a Bravais lattice has a non-identity rotation RR by θ=360∘/n\theta=360^\circ/n, with integer n≥2n\ge2, about an axis through a lattice point OO. Among the non-zero lattice vectors perpendicular to the axis (they exist, since v−Rv\mathbf v-R\mathbf v is one for any lattice vector v\mathbf v off the axis), let u\mathbf u be a shortest one, of length aa. ((a)) Show that Ru−uR\mathbf u-\mathbf u is a lattice vector perpendicular to the axis, and find its length in terms of aa and θ\theta. Hence show that no lattice has a rotation axis of order n≥7n\ge7. ((b)) The argument of part (a) says nothing about n=5n=5. Find another combination of u\mathbf u and its rotated images that is a non-zero lattice vector shorter than aa when n=5n=5, and so exclude five-fold axes too.

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