Special directions in the Brillouin zone

TL;DR

This paper presents rules for identifying special directions in the Brillouin zone of various crystal lattices to optimize the description of physical quantities with Gamma1 symmetry, aiding experimental and theoretical analysis.

Contribution

It introduces a systematic method for selecting special directions in the Brillouin zone applicable to multiple lattice types, improving the reconstruction of physical quantities from limited data.

Findings

Rules enable construction of symmetric quantities from limited directional data

Applicable to Fermi surface, momentum density, and experimental projections

Facilitates accurate electron momentum density reconstruction

Abstract

Rules are given for determining special directions in the Brillouin zone which optimize the descrip-tion of various physical quantities with Gamma1 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 spe-cial 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 recon-struct 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.