Crystal planes and reciprocal space in Clifford geometric algebra

TL;DR

This paper explores the geometric and algebraic structures underlying crystallography using Clifford geometric algebra, providing a unified framework for concepts like reciprocal space, $d$-spacing, and Bragg reflections.

Contribution

It introduces a novel geometric algebra approach to crystallography, linking projective and conformal algebra representations of crystal geometry and reciprocal space.

Findings

Unified geometric interpretation of reciprocal vectors and crystal planes

Derivation of crystallographic concepts within Clifford algebra framework

Enhanced understanding of Bragg reflections and interfacial angles

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