Crystal structure, x-ray diffraction and oblique geometry

The main mathematical difficulty in crystal geometry and x-ray diffraction stems from an oblique system of coordinates as soon as we move on from the simple cubic to the body-centered cubic case without mentioning the non-cubic systems. We show in this work that there exists a simple way for dealing with oblique geometry for all crystal systems by combining direct and reciprocal space geometrical elements with the help of a small set of vector identities that are not usually taught in a general physics curriculum. This allows us to derive for all crystalline systems compact formulas describing directions, planes, moduli, angles and scalar products of vectors belonging to either direct or reciprocal lattices. Interplanar distances and angles are also derived with this approach. © 2019 European Physical Society.