International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

International Tables for Crystallography (2006). Vol. C, ch. 9.8, pp. 936-937

Section 9.8.3.6. Ambiguities in the notation

T. Janssen,a A. Janner,a A. Looijenga-Vosb and P. M. de Wolffc

aInstitute for Theoretical Physics, University of Nijmegen, Toernooiveld, NL-6525 ED Nijmegen, The Netherlands,bRoland Holstlaan 908, NL-2624 JK Delft, The Netherlands, and cMeermanstraat 126, 2614 AM, Delft, The Netherlands

9.8.3.6. Ambiguities in the notation

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The invariant part [v^o_s] of the translation part [v_s] of a (3 + 1)-dimensional superspace-group element is uniquely determined by (9.8.3.5)[link]. 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, [v_s] may be changed when R is combined with a three-dimensional lattice translation [w_s=({\bf w},0)]. This situation is well known in ordinary three-dimensional crystallography. For example, the twofold rotation [(x,y,z)\rightarrow(-x,z,y)] in the space group [P4_132] has according to Volume A of International Tables for Crystallography[link] a translation part [(\,{1\over4},{3\over4},{1\over4}\,)]. Its invariant part is [(0,{1\over2},{1\over2}\,)]. However, when the translation part is equivalently taken as [(\,{1\over4},{3\over4},-{3\over4}\,)], 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 `[2_1]'.

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)[link]]. Since v is only determined modulo a lattice vector, one may add to it a lattice vector that has a non-vanishing product with qr. This results in a change for τ. For example, the (3 + 1)-dimensional space group [Pmmm(\,{1\over2}0\gamma)000=A^{Pm m\,m}_{\hskip 4.5pt 1\ 1\ \bar1}] 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 [\tau={1\over2}]. Hence, the symbols [Pmmm(\,{1\over2}0\gamma)000] and [Pmmm(\,{1\over2}0\gamma)s00] 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 [I2cb(00\gamma)0s0=P\kern.5pt^{I2cb}_{\kern3pt\bar1s\bar1}] 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 [h0l'm']. The first of these conditions implies the symbol [I2cb(00\gamma)000=P\kern.5pt^{I2 c b}_{\kern 3pt\bar11\bar1}] 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[link]). In this case, the setting I2ab instead of I2cb avoids the problem.

References

International Tables for Crystallography (2005). Vol. A, edited by Th. Hahn, fifth ed. Heidelberg: Springer.
Yamamoto, A., Janssen, T., Janner, A. & de Wolff, P. M. (1985). A note on the superspace groups for one-dimensionally modulated structures. Acta Cryst. A41, 528–530.








































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