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. 915-916
Section 9.8.3.1. Tables of Bravais lattices |
The (3 + 1)-dimensional lattice is determined by the three-dimensional vectors a*, b*, c* and the modulation vector q. The former three vectors give by duality a, b, and c, the external components of lattice basis vectors, and the products , , and the corresponding internal components. Therefore, it is sufficient to give the arithmetic crystal class of the group and the components σj (σ1 = α, σ2 = β, and σ3 = γ) of the modulation vector q with respect to a conventional basis a*, b*, c*. The arithmetic crystal class is denoted by a modification of the symbol of the three-dimensional symmorphic space group of this class (see Chapter 1.4 ) plus an indication for the row matrix σ (having entries ). In this way, one obtains the so-called one-line symbols used in Tables 9.8.3.1 and 9.8.3.2.
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As an example, the symbol denotes a Bravais class for which the main reflections belong to a B-centred monoclinic lattice (unique axis c) and the satellite positions are generated by the point-group transforms of . Then the matrix σ becomes . It has as irrational part and as rational part . The external part of the (3 + 1)-dimensional point group of the Bravais lattice is 2/m. By use of the relation [cf. (9.8.2.4)] we see that the operations 2 and m are associated with the internal space transformations ɛ = 1 and ɛ = −1, respectively. This is denoted by the one-line symbol for the (3 + 1)-dimensional point group of the Bravais lattice. In direct space, the symmetry operation {R, ɛ(R)} is represented by the matrix Γ(R) which transforms the components , of a vector to: The operations (2, 1) and are represented by the matrices: The 3 × 3 part of each matrix is obtained by considering the action of R on the external part v of . The 1 × 1 part is the value of the ɛ associated with R and the remaining part follows from the relation
Bravais classes can be denoted in an alternative way by two-line symbols. In the two-line symbol, the Bravais class is given by specifying the arithmetic crystal class of the external symmetry by the symbol of its symmorphic space group, the associated elements by putting their symbol under the corresponding symbols of , and by the rational part indicated by a prefix. In the following table, this prefix is given for the components of that play a role in the classification. Note that the integers appearing here are not equivalent to zero because they express components with respect to a conventional lattice basis (and not a primitive one). For the Bravais class mentioned above, the two-line symbol is . This symbol has the advantage that the internal transformation (the value of ɛ) is explicitly given for the corresponding generators. It has, however, certain typographical drawbacks. It is rare for the printer to put the symbol together in the correct manner: .
In Tables 9.8.3.1 and 9.8.3.2 the symbols for the (2 + d)- and (3 + 1)-dimensional Bravais classes are given in the one-line form. It is, however, easy to derive from each one-line symbol the corresponding two-line symbol because the bottom line for the two-line symbol appears in the tables as the internal part of the point-group symbol.
The number of symbols in the bottom line of the two-line symbol should be equal to that of the generators given in the top line. A symbol `1' is used in the bottom line if the corresponding is the unit transformation. If necessary, a mirror perpendicular to a crystal axis is indicated by and one that is not by . This situation only occurs for . So the (2 + 2)-dimensional class is actually and is different from the class . In a one-line symbol, their difference is apparent, the first being 4mp(α0), whereas the second is 4mp(αα).
References
Brown, H., Bülow, H., Neubüser, J., Wondratschek, H. & Zassenhaus, H. (1978). Crystallographic groups of four-dimensional space. New York: John Wiley.Google ScholarJanssen, T. (1969). Crystallographic groups in space and time. III. Four-dimensional Euclidean crystal classes. Physica (Utrecht), 42, 71–92.Google Scholar