E. Hitzer
Crystal planes and reciprocal space in Clifford geometric algebra
Math. Methods in the Applied Sciences, Vol. 34, Iss. 12, pp. 1421-1429, August 2011, Article first published online: 15 Feb. 2011, (2011), DOI: 10.1002/mma.1442.

Abstract: This paper 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 Cl(R^{n+1}) (see [8]) 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.

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

Keywords: Clfford geometric algebra, crystallography, reciprocal space, d-spacing, phase angle, structure factors, Bragg reflections, interfacial angles.

AMS Subject Classification: 74E15, 15A66.


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