Fundamental relationships

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, pp. 3-4 | 1 | 2 |

Section 1.1.3. Fundamental relationships

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. Fundamental relationships

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We now present a brief derivation and a summary of the most important relationships between the direct and the reciprocal bases. The usual conventions of vector algebra are observed and the results are presented in the conventional crystallographic notation. Equations (1.1.2.1) and (1.1.2.2) now become $[{\bf a}\cdot {\bf b}^{*} = {\bf a}\cdot {\bf c}^{*} = {\bf b}\cdot {\bf a}^{*} = {\bf b}\cdot {\bf c}^{*} = {\bf c}\cdot {\bf a}^{*} = {\bf c}\cdot {\bf b}^{*} = 0 \eqno(1.1.3.1)]$ and $[{\bf a}\cdot {\bf a}^{*} = {\bf b}\cdot {\bf b}^{*} = {\bf c}\cdot {\bf c}^{*} = 1, \eqno(1.1.3.2)]$ respectively, and the relationships are obtained as follows.

1.1.3.1. Basis vectors

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It is seen from (1.1.3.1) that $[{\bf a}^{*}]$ must be proportional to the vector product of b and c, $[{\bf a}^{*} = K ({\bf b} \times {\bf c}),]$ and, since $[{\bf a}\cdot {\bf a}^{*} = 1]$ , the proportionality constant K equals $[1/[{\bf a}\cdot ({\bf b} \times {\bf c})]]$ . The mixed product $[{\bf a}\cdot ({\bf b} \times {\bf c})]$ can be interpreted as the positive volume of the unit cell in the direct lattice only if a, b and c form a right-handed set. If the above condition is fulfilled, we obtain $[ {\bf a}^{*} = {{\bf b} \times {\bf c}\over {V}},\quad {\bf b}^{*} = {{\bf c} \times {\bf a}\over {V}},\quad {\bf c}^{*} = {{\bf a} \times {\bf b}\over {V}} \eqno(1.1.3.3)]$ and analogously $[ {\bf a} = {{\bf b^{*} \times c^{*}}\over {V}^{*}},\quad {\bf b} = {{\bf c^{*} \times a^{*}}\over {V}^{*}},\quad {\bf c} = {{\bf a^{*} \times b^{*}}\over {V}^{*}}, \eqno(1.1.3.4)]$ where V and $[ {V}^{*}]$ are the volumes of the unit cells in the associated direct and reciprocal lattices, respectively. Use has been made of the fact that the mixed product, say $[{\bf a}\cdot ({\bf b} \times {\bf c})]$ , remains unchanged under cyclic rearrangement of the vectors that appear in it.

1.1.3.2. Volumes

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The reciprocal relationship of V and $[ {V}^{*}]$ follows readily. We have from equations (1.1.3.2), (1.1.3.3) and (1.1.3.4) $[ {\bf c}\cdot {\bf c^{*}} = {({\bf a} \times {\bf b})\cdot ({\bf a}^{*} \times {\bf b}^{*})\over {VV}^{*}} = 1.]$ If we make use of the vector identity $[({\bf A} \times {\bf B})\cdot ({\bf C} \times {\bf D}) = ({\bf A}\cdot {\bf C}) ({\bf B}\cdot {\bf D}) - ({\bf A}\cdot {\bf D}) ({\bf B}\cdot {\bf C}), \eqno(1.1.3.5)]$ and equations (1.1.3.1) and (1.1.3.2), it is seen that $[ {V}^{*} = 1/{V}]$ .

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):

(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), 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)–(1.1.3.8). For example, $[\cos \alpha = {\cos \beta^{*} \cos\gamma^{*} - \cos \alpha^{*}\over \sin \beta^{*} \sin \gamma^{*}}.]$

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, pp. 3-4