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Matrix-column pairs

As was mentioned already in Section 2.2, an affine mapping A is described by a matrix A and a column a, see equations (2.2.1) and (2.2.5) on p. . Crystallographic symmetry operations are special affine mappings. They will be designated by the letter W and described by the matrix W and the column w. Their description is analogous to equation (2.2.1):

$$\begin{array}{rcl} \tilde{x} & = & W_{11}x+W_{12}y+W_{13}z... ... \\ \tilde{z} & = & W_{31}x+W_{32}y+W_{33}z+w_3 \end{array}$$ (4.1.1)

There are different ways of simplifying this array. One of them leads to the description with $\Sigma$ sign and indices in analogy to that for mappings, see equations (2.4.1) and (2.4.2). It will not be followed here. Another one is the symbolic description introduced in Section 2.3. It will be treated now in more detail.

Step 1 One writes the system of equations in the form

$$\left( \begin{array}{r} \tilde{x} \\ \tilde{y} \\ \tilde{z}... ...\left( \begin{array}{r} w_1 \\ w_2 \\ w_3 \end{array} \right).$$ (4.1.2)

The form 4.1.2 has the advantage that the coordinates and the coefficients which describe the mapping are no longer intimately mixed but are more separated in the equation. For actual calculations with concrete mappings this form is most appropriate, applying the definitions (D 2.4.3) and (D 2.4.2). For the derivation of general formulae, a further abstraction is advantageous.

Step 2 Denoting the coordinate columns by $\tilde{\mbox{\textit{\textbf{x}}}}$ and x, the ($3\times3$) matrix by W, and the column by w, one obtains in analogy to equation (2.2.5)

$$\tilde{\mbox{\textit{\textbf{x}}}} = \mbox{\textit{\textbf{W}}}\,\mbox{\textit{\textbf{x}}} + \mbox{\textit{\textbf{w}}}.$$ (4.1.3)

Step 3 Still the coordinate part and the mapping part are not completely separated. Therefore, one writes

$$\tilde{\mbox{\textit{\textbf{x}}}} = (\mbox{\textit{\textbf{... ...vert\,\mbox{\textit{\textbf{w}}})\,\mbox{\textit{\textbf{x}}}.$$ (4.1.4)

The latter form is called the SEITZ notation.

Note that the forms (4.1.1) to (4.1.4) of the equations are only different ways of describing the same mapping W. The matrix-column pairs (W,w) or (W$\vert$w) are suitable in particular for general considerations; they present the pure description of the mapping, and the coordinates are completely eliminated. Therefore, in Section 4.2 the pairs are used for the formulation of the combination VU of 2 symmetry operations V and U and of the inverse W$^{-1}$ of a symmetry operation W. However, if one wants to provide a list of specific mappings, then there is no way to avoid the explicit description by the formulae 4.1.1 or 4.1.2, see Section 4.6.

With the matrix-column pairs one can replace geometric considerations by analytical calculations. To do this one first determines those matrix-column pairs which describe the symmetry operations to be studied. This will be done in Section 5.1. Then one performs the necessary procedures with the matrix-column pairs, e.g. combination or reversion, see Section 4.2. Finally, one has to extract the geometric meaning from the resulting matrix-column pairs. This last step is shown in Section 5.2.

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