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 *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 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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