Crystal planes and reciprocal space in Clifford geometric algebra

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This paper discusses the geometry of kD crystal cells given by (k+1) points in a projective space 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(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(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