E. Hitzer
Reciprocal space and crystal planes in geometric algebra
in Electronic Proceedings of AGACSE 2010, Amsterdam, The Netherlands, 14-16 June 2010.

Abstract: This contribution discusses the geometry of kD 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 R_{n+1} (see [2]) and in the conformal algebra R_{n+1,1}. The crystallographic notions of d-spacing, phase angle (in structure factors), extinction of Bragg re ections, and the interfacial angles of crystal planes are obtained in the same context.

[2] H. Grassmann, edited by F. Engel, Die Ausdehnungslehre von 1844 und die Geometrische Analysis, Vol. 1, Part 1, Teubner, Leipzig, 1894.

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