Analysis of the reciprocal lattice of crystals with non-primitive Bravais cells

Abstract

The aim of the work was proof the independence of the reciprocal lattice of a crystal from the choice of a cell in its direct lattice. The object of analysis was the dependence of the position of reciprocal lattice cell sites and direct lattice parameters based on quadratic formulas, as well as the structural amplitudes («weights» of reciprocal lattice sites) for crystals with non-primitive cells. The following tasks were solved: choice of a Bravais rapper with a P-type cell in a crystal with I-, F- and C-type cells; construction of direct and reciprocal metric tensors for centered cells of a direct lattice and for P-type cells of crystal lattices; establishment of the transition rule from the HKL indices of the nodes of the reciprocal lattice with the selected P-cell to the hkl indices of the same nodes for non-primitive cells based on quadratic formulas; determining a connection between HKL and hkl indices using metric tensors; analysis of the results obtained by two different methods (based on quadratic formulas and using metric tensors); proof, that in the reciprocal lattice of crystals with non-primitive cells, the nodes that correspond to «fading» reflections are actually absent in the reciprocal lattice. The research results are addressed to specialists in the field of crystal physics and represent a rigorous proof that fills a gap in the mathematical apparatus of the kinetic theory of scattering by crystals with non-primitive Bravais cells. It allows eliminating redundant statements about the «extinction» of part of the reflexes that are not recorded on X-rays due to their real absence from scientific circulation.