International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

© International Union of Crystallography 2006
International Tables for Crystallography (2006). Vol. B. ch. 1.1, p. 4   | 1 | 2 |

Section 1.1.3.3. Angular relationships

U. Shmuelia*

aSchool of Chemistry, Tel Aviv University, 69 978 Tel Aviv, Israel
Correspondence e-mail: ushmueli@post.tau.ac.il

1.1.3.3. Angular relationships

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The relationships of the angles [\alpha, \beta, \gamma] between the pairs of vectors (b, c), (c, a) and (a, b), respectively, and the angles [\alpha^{*}, \beta^{*}, \gamma^{*}] between the corresponding pairs of reciprocal basis vectors, can be obtained by simple vector algebra. For example, we have from (1.1.3.3)[link]:

(i) [{\bf b}^{*}\cdot {\bf c}^{*} = b^{*} c^{*} \cos \alpha^{*}], with [b^{*} = {ca \sin \beta\over V}\quad \hbox{and}\quad c^{*} = {ab \sin \gamma\over V}] and (ii)

[{\bf b}^{*}\cdot {\bf c}^{*} = {({\bf c} \times {\bf a})\cdot ({\bf a} \times {\bf b})\over {V}^{2}}.] If we make use of the identity (1.1.3.5)[link], and compare the two expressions for [{\bf b}^{*}\cdot {\bf c}^{*}], we readily obtain [\cos \alpha^{*} = {\cos \beta \cos \gamma - \cos \alpha\over \sin \beta \sin \gamma}. \eqno(1.1.3.6)] Similarly, [\cos \beta^{*} = {\cos \gamma \cos \alpha - \cos \beta\over \sin \gamma \sin \alpha} \eqno(1.1.3.7)] and [\cos \gamma^{*} = {\cos \alpha \cos \beta - \cos \gamma\over \sin \alpha \sin \beta}. \eqno(1.1.3.8)] The expressions for the cosines of the direct angles in terms of those of the reciprocal ones are analogous to (1.1.3.6)[link]–(1.1.3.8)[link]. For example, [\cos \alpha = {\cos \beta^{*} \cos\gamma^{*} - \cos \alpha^{*}\over \sin \beta^{*} \sin \gamma^{*}}.]








































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