Reciprocal Space and Crystal Planes in Geometric Algebra

TL;DR

This paper explores the geometric relationships of crystal structures using projective and conformal geometric algebra, providing a unified interpretation of reciprocal space, crystal planes, and related crystallographic concepts.

Contribution

It introduces a geometric algebra framework to interpret reciprocal vectors, crystal planes, and crystallographic parameters, unifying these concepts in projective and conformal algebra.

Findings

Geometric interpretation of reciprocal vectors and crystal planes.

Unified algebraic framework for crystallographic concepts.

Application of geometric algebra to crystallography.

Abstract

This contribution 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 representati on are related (and geometrically interpreted) in the projective geometric algebra $\R_{n+1}$ (see H. Grassmann, edited by F. Engel, Sie Ausdehnungslehre von 1844 und die Geom. Anal., vol. 1, part 1, Teubner, Leipzig, 1894.) and in the conformal algebra $\R_{n+1,1}$. The crystallographic notions of $d$-spacing, phase angle (in structure factors), extinction of Bragg reflections, and the interfacial angles of crystal planes are obtained in the same context.