Section 9.8.3.6. Ambiguities in the notation

The invariant part of the translation part of a (3 + 1)-dimensional superspace-group element is uniquely determined by (9.8.3.5). This does not imply that for each element of the point group there is a translation for which the invariant part is unique up to lattice vectors. The reason is that, for a given element R of the point group and given origin, may be changed when R is combined with a three-dimensional lattice translation . This situation is well known in ordinary three-dimensional crystallography. For example, the twofold rotation in the space group has according to Volume A of International Tables for Crystallography a translation part . Its invariant part is . However, when the translation part is equivalently taken as , the invariant part vanishes. Therefore, in the symbol for that space group, the corresponding generator is given as the rotation `2' and not as the screw axis `'.

The same situation may occur in 3 + 1 dimensions. This can be seen very clearly from the definition of τ [equation (9.8.3.8)]. Since v is only determined modulo a lattice vector, one may add to it a lattice vector that has a non-vanishing product with q^r. This results in a change for τ. For example, the (3 + 1)-dimensional space group has a mirror perpendicular to the a axis with associated value τ = 0. The parallel mirror at a distance a/2 has v = a and consequently . Hence, the symbols and indicate the same group. This non-uniqueness in the symbol, however, does not have serious practical consequences.

Another source of ambiguity is the fact that the assignment of a satellite to a main reflection is not unique. For example, the reflection conditions for the group are h + k + l = even because of the centring and l + m = even and h + m = even for h0lm because of the two glide planes perpendicular to the b axis. When one takes for the modulation vector q = γ′c* = (1 − γ)c*, the new indices are h, k, l′, and m′ with l′ = l + m and m′ = −m. Then the reflection conditions become l′ = even and h + m = even for . The first of these conditions implies the symbol for the group considered. This, however, is the symbol for the nonequivalent group with condition h = even for h0lm. This difficulty may be avoided by sometimes using a non-standard setting of the three-dimensional space group (see Yamamoto et al., 1985). In this case, the setting I2ab instead of I2cb avoids the problem.