Special directions in the Brillouin zone

Rules are given for determining special directions in the Brillouin zone which optimize the description of various physical quantities with Γ1-type symmetry. We consider the cubic, hexagonal, tetragonal and trigonal (e.g. Bi) lattice. These rules allow us to construct, in all of momentum space, quantities which have the full symmetry of the Brillouin zone such as the Fermi surface, momentum density and others, from the knowledge of these quantities for a limited number of special directions. These results can also be used for determining the projections which should be measured either in Compton scattering or in positron-annihilation experiments in order to reconstruct properly the electron-momentum density. Such a treatment is approximately equivalent to applying Gaussian quadrature in calculations of the radial functions that occur in the expansion of data into lattice harmonics.