The crystallographic restriction was proved for rotations

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Group theory and symmetry · Group theory and symmetry

Invariance, Ch Point groups and crystallographic symmetry

Problem

The crystallographic restriction was proved for rotations. A lattice can also be mapped onto itself by an improper operation, such as a mirror plane or the inversion through a lattice point. ((a)) Show that an improper operation that maps a three-dimensional lattice onto itself has a matrix of whole numbers in a primitive basis. Use its trace in Cartesian axes, −1+2cos⁡θ-1+2\cos\theta with θ\theta between 00 and π\pi, to find every angle θ\theta it can have, and name the operation SnS_n that goes with each angle, with S1=σS_1=\sigma and S2=iS_2=i. ((b)) In the Hermann-Mauguin notation, nˉ\bar n is an nn-fold rotation followed by the inversion, so its matrix is −Rz(2π/n)-\mathsf{R}_z(2\pi/n). For n=1,2,3,4,6n=1,2,3,4,6 find the trace and determinant of nˉ\bar n, and use them to identify each nˉ\bar n with one of the operations found in (a).

Browse the problem library