Example

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. 8 | 1 | 2 |

Section 1.1.5.2. Example

U. Shmueli^a ^*

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

1.1.5.2. Example

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This example deals with the construction of a Cartesian system in a crystal with given basis vectors of its direct lattice. We shall also require that the Cartesian system bears a clear relationship to at least one direction in each of the direct and reciprocal lattices of the crystal; this may be useful in interpreting a physical property which has been measured along a given lattice vector or which is associated with a given lattice plane. For a better consistency of notation, the Cartesian components will be denoted as contravariant.

The appropriate version of equations (1.1.5.1) and (1.1.5.2) is now $[{\bf r} = x^{i}{\bf a}_{i} \eqno(1.1.5.11)]$ and $[{\bf r} = X^{k}{\bf e}_{k}, \eqno(1.1.5.12)]$ where the Cartesian basis vectors are: $[{\bf e}_{1} = {\bf r}_{L}/|{\bf r}_{L}|]$ , $[{\bf e}_{2} = {\bf r}^{*}/|{\bf r}^{*}|]$ and $[{\bf e}_{3} = {\bf e}_{1} \times {\bf e}_{2}]$ , and the vectors $[{\bf r}_{L}]$ and $[{\bf r}^{*}]$ are given by $[{\bf r}_{L} = u^{i}{\bf a}_{i} \hbox{ and } {\bf r}^{*} = h_{k}{\bf a}^{k},]$ where $[u^{i}]$ and $[h_{k}]$ , i, k = 1, 2, 3, are arbitrary integers. The vectors $[{\bf r}_{L}]$ and $[{\bf r}^{*}]$ must be mutually perpendicular, $[{\bf r}_{L} \cdot {\bf r}^{*} = u^{i}h_{i} = 0]$ . The $[X^{1}(X)]$ axis of the Cartesian system thus coincides with a direct-lattice vector, and the $[X^{2}(Y)]$ axis is parallel to a vector in the reciprocal lattice.

Since the basis in (1.1.5.12) is a Cartesian one, the required transformations are given by equations (1.1.5.10) as $[x^{i} = X^{k}(T^{-1})_{k}^{i} \hbox{ and } X^{i} = x^{k}T_{k}^{i}, \eqno(1.1.5.13)]$ where $[T_{k}^{i} = {\bf a}_{k} \cdot {\bf e}_{i}]$ , k, i = 1, 2, 3, form the matrix of the scalar products. If we make use of the relationships between covariant and contravariant basis vectors, and the tensor formulation of the vector product, given in Section 1.1.4 above (see also Chapter 3.1 ), we obtain $[\eqalignno{ T_{k}^{1} &= {1\over |{\bf r}_{L}|} g_{ki}u^{i}\cr T_{k}^{2} &= {1\over |{\bf r}^{*}|} h_{k} &(1.1.5.14)\cr T_{k}^{3} &= {V\over |{\bf r}_{L}||{\bf r}^{*}|} e_{kip}u^{i}g^{pl}h_{l}.}]$

Note that the other convenient choice, $[{\bf e}_{1}\propto {\bf r}^{*}]$ and $[{\bf e}_{2}\propto {\bf r}_{L}]$ , interchanges the first two columns of the matrix T in (1.1.5.14) and leads to a change of the signs of the elements in the third column. This can be done by writing $[e_{kpi}]$ instead of $[e_{kip}]$ , while leaving the rest of $[T_{k}^{3}]$ unchanged.

References

International Tables for Crystallography (2006). Vol. B. ch. 1.1, p. 8