International
Tables for Crystallography Volume C Mathematical, physical and chemical tables Edited by E. Prince © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. C. ch. 9.8, pp. 938-939
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Definition 1. The Laue point group of the diffraction pattern is the point group in three dimensions that transforms every diffraction peak into a peak of the same intensity.2
Because all diffraction vectors are of the form (9.8.4.5), the action of an element R of the Laue group is given by The (3 + d) × (3 + d) matrices form a finite group of integral matrices for K equal to or to one of its subgroups. A well known theorem in algebra states that then there is a basis in 3 + d dimensions such that the matrices on that basis are orthogonal and represent (3 + d)-dimensional orthogonal transformations . The corresponding group is a (3 + d)-dimensional crystallographic group denoted by . Because R is already an orthogonal transformation on V, is reducible and can be expressed as a pair of orthogonal transformations, in 3 and d dimensions, respectively. The basis on which acts according to is denoted by . It spans a lattice that is the reciprocal of the lattice with basis elements . The pairs , sometimes also noted , leave invariant: where Γ(R) is the transpose of .
In many cases, one can distinguish a lattice of main reflections, the remaining reflections being called satellites. The main reflections are generally more intense. Therefore, main reflections are transformed into main reflections by elements of the Laue group. On a standard lattice basis (9.8.4.13), the matrices Γ(R) take the special form The transformation of main reflections and satellites is then given by as in (9.8.4.15), the relation with Γ(R) being (as already said) where the tilde indicates transposition. Accordingly, on a standard basis one has The set of matrices for R elements of K forms a crystallographic point group in three dimensions, denoted , having elements R of O(3), and the corresponding set of matrices forms one in d dimensions denoted by with elements of O(d).
For a modulated crystal, one can choose the (i = 1, 2, 3) of a standard basis. These span the (reciprocal) lattice of the basic structure. One can then express the additional vectors (which are modulation wavevectors) in terms of the basis of the lattice of main reflections: The three components of the jth row of the (d × 3)-dimensional matrix σ are just the three components of the jth modulation wavevector with respect to the basis . It is easy to show that the internal components (i = 1, 2, 3) of the corresponding dual standard basis can be expressed as This follows directly from (9.8.4.19) and the definition of the reciprocal standard basis (9.8.4.13). From (9.8.4.16) and (9.8.4.17), a simple relation can be deduced between σ and the three constituents , , and of the matrix Γ(R): Notice that the elements of are integers, whereas σ has, in general, irrational entries. This requires that the irrational part of σ gives zero when inserted in the left-hand side of equation (9.8.4.21). It is therefore possible to decompose formally σ into parts and as follows. where the sum is over all elements of the Laue group of order N. It follows from this definition that This implies The matrix has rational entries and is called the rational part of σ. The part is called the irrational (or invariant) part.
The above equations simplify for the case d = 1. The elements are the three components of the wavevector q, the row matrix has the components of and = ɛ = ±1 since, for d = 1, q can only be transformed into ±q. One has the corresponding relations and The reciprocal-lattice vector that gives the difference between and has as components the elements of the row matrix .