Crystal planes and reciprocal space in Clifford geometric algebra

✍ Scribed by E. Hitzer

• Publisher: John Wiley and Sons
• Year: 2011
• Tongue: English
• Weight: 190 KB
• Volume: 34
• Category: Article
• ISSN: 0170-4214
• DOI: 10.1002/mma.1442

✦ Synopsis

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 (Die Ausdehnungslehre von 1844 und die Geom. Anal. Teubner: Leipzig, 1894)) 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.