Matrices of metric tensors

Shmueli, U.

doi:10.1107/97809553602060000549

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

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

Section 1.1.3.4. Matrices of metric tensors

U. Shmueli^a ^*

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

1.1.3.4. Matrices of metric tensors

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Various computational and algebraic aspects of mutually reciprocal bases are most conveniently expressed in terms of the metric tensors of these bases. The tensors will be treated in some detail in the next section, and only the definitions of their matrices are given and interpreted below.

Consider the length of the vector $[{\bf r} = x{\bf a} + y{\bf b} + z{\bf c}]$ . This is given by $[|{\bf r}| = [(x{\bf a} + y{\bf b} + z{\bf c})\cdot (x{\bf a} + y{\bf b} + z{\bf c})]^{1/2} \eqno(1.1.3.9)]$ and can be written in matrix form as $[ |{\bf r}| = [{\bi x}^{T} {\bi Gx}]^{1/2}, \eqno(1.1.3.10)]$ where $[ {\bi x} = \pmatrix{x\cr y\cr z\cr},\quad {\bi x}^{T} = (xyz)]$ and $[ \eqalignno{ {\bi G} &= \pmatrix{{\bf a\cdot a} &{\bf a\cdot b} &{\bf a\cdot c}\cr {\bf b\cdot a} &{\bf b\cdot b} &{\bf b\cdot c}\cr {\bf c\cdot a} &{\bf c\cdot b} &{\bf c\cdot c}\cr} &(1.1.3.11)\cr &= \pmatrix{a^{2} &ab \cos \gamma &ac \cos \beta\cr ba \cos \gamma &b^{2} &bc \cos \alpha\cr ca \cos \beta &cb \cos \alpha &c^{2}\cr}. &(1.1.3.12)}%(1.1.3.12)]$ This is the matrix of the metric tensor of the direct basis, or briefly the direct metric . The corresponding reciprocal metric is given by $[\eqalignno{ {\bi G^{*}} &= \pmatrix{{\bf a^{*}\cdot a^{*}} &{\bf a^{*}\cdot b^{*}} &{\bf a^{*}\cdot c^{*}}\cr {\bf b^{*}\cdot a^{*}} &{\bf b^{*}\cdot b^{*}} &{\bf b^{*}\cdot c^{*}}\cr {\bf c^{*}\cdot a^{*}} &{\bf c^{*}\cdot b^{*}} &{\bf c^{*}\cdot c^{*}}\cr} &(1.1.3.13)\cr &= \pmatrix{a^{*2} &a^{*}b^{*} \cos \gamma^{*} &a^{*}c^{*} \cos \beta^{*}\cr b^{*}a^{*} \cos \gamma^{*} &b^{*2} &b^{*}c^{*} \cos \alpha^{*}\cr c^{*}a^{*} \cos \beta^{*} &c^{*}b^{*} \cos \alpha^{*} &c^{*2}\cr}. &(1.1.3.14)}%(1.1.3.14)]$ The matrices G and $[ {\bi G}^{*}]$ are of fundamental importance in crystallographic computations and transformations of basis vectors and coordinates from direct to reciprocal space and vice versa. Examples of applications are presented in Part 3 of this volume and in the remaining sections of this chapter.

It can be shown (e.g. Buerger, 1941) that the determinants of G and $[ {\bi G}^{*}]$ equal the squared volumes of the direct and reciprocal unit cells, respectively. Thus, $[ \hbox{det } ({\bi G}) = [{\bf a}\cdot ({\bf b} \times {\bf c})]^{2} = V^{2} \eqno(1.1.3.15)]$ and $[ \hbox{det } ({\bi G^{*}}) = [{\bf a}^{*}\cdot ({\bf b}^{*} \times {\bf c}^{*})]^{2} = V^{*2}, \eqno(1.1.3.16)]$ and a direct expansion of the determinants, from (1.1.3.12) and (1.1.3.14), leads to $[\eqalignno{ V &= abc (1 - \cos^{2} \alpha - \cos^{2} \beta - \cos^{2} \gamma \cr &\quad + 2 \cos \alpha \cos \beta \cos \gamma)^{1/2} &(1.1.3.17)}]$ and $[\eqalignno{ V^{*} &= a^{*}b^{*}c^{*} (1 - \cos^{2} \alpha^{*} - \cos^{2} \beta^{*} - \cos^{2} \gamma^{*} \cr &\quad + 2 \cos \alpha^{*} \cos \beta^{*} \cos \gamma^{*})^{1/2}. &(1.1.3.18)}]$ The following algorithm has been found useful in computational applications of the above relationships to calculations in reciprocal space (e.g. data reduction) and in direct space (e.g. crystal geometry).

(1) Input the direct unit-cell parameters and construct the matrix of the metric tensor [cf. equation (1.1.3.12)].
(2) Compute the determinant of the matrix G and find the inverse matrix, $[ {\bi G}^{-1}]$ ; this inverse matrix is just $[ {\bi G}^{*}]$ , the matrix of the metric tensor of the reciprocal basis (see also Section 1.1.4 below).
(3) Use the elements of $[ {\bi G}^{*}]$ , and equation (1.1.3.14), to obtain the parameters of the reciprocal unit cell.

The direct and reciprocal sets of unit-cell parameters , as well as the corresponding metric tensors, are now available for further calculations.

Explicit relations between direct- and reciprocal-lattice parameters, valid for the various crystal systems, are given in most textbooks on crystallography [see also Chapters 1.1 and 1.2 of Volume C (Koch, 2004)].

References

Buerger, M. J. (1941). X-ray crystallography. New York: John Wiley.Google Scholar

Koch, E. (2004). In International tables for crystallography, Vol. C. Mathematical, physical and chemical tables, edited by E. Prince, Chapters 1.1 and 1.2. Dordrecht: Kluwer Academic Publishers.Google Scholar

International Tables for Crystallography (2006). Vol. B. ch. 1.1, p. 4