Crystal planes and reciprocal space in Clifford geometric algebra

Abstract

This paper discusses the geometry of $k$D crystal cells given by $(k+1)$ points in a projective space $\R^{n+1}$. We show how the concepts of barycentric and fractional (crystallographic) coordinates, reciprocal vectors and dual representation are related (and geometrically interpreted) in the projective geometric algebra $Cl(\R^{n+1})$ (see Grassmann H., edited by Engel F., Die Ausdehnungslehre von 1844 und die Geom. Anal., vol. 1, part 1, Teubner: Leipzig, 1894.) and in the conformal algebra $Cl(\R^{n+1,1})$. The crystallographic notions of $d$-spacing, phase angle, structure factors, conditions for Bragg reflections, and the interfacial angles of crystal planes are obtained in the same context. Keywords: Clifford geometric algebra, crystallography, reciprocal space, $d$-spacing, phase angle, structure factors, Bragg reflections, interfacial angles