ISO(3+d)D
Tables of (3+d)-Dimensional Superspace Groups

Harold T. Stokes and Branton J. Campbell, Brigham Young University, Provo, Utah, USA
Sander van Smaalen, University of Bayreuth, Bayreuth, Germany

Comparison with tables of Orlov and Chapuis

OC: I. P. Orlov and G. Chapuis, "List of (3+1) Dimensional Superspace Groups," superspace.epfl.ch/groups/ (2005)

ITC-C: T. Janssen, A. Janner, A. Looijenga-Vos, and P. M. de Wolff, "Incommensurate and commensurate modulated structures," International Tables for Crystallography, Vol. C, pp. 907-955. Edited by E. Prince (Kluwer, Dordrecht, 2004).

The OC table lists group operators and reflection conditions for each (3+1)-dimensional space group. Our (3+1)D superspace group operators are consistent with those in the OC table for all but two groups, 100.5 P4bm(1/2,1/2,g)qq0 and 100.6 P4bm(1/2,1/2,g)qqs. Our operators for these groups are consistent with the reflection conditions listed in the ITC-C and OC tables, while the OC operators are not. The operators of these two groups differ only slightly and appear to have be reversed in the OC table.

Both groups belong to Bravais class 20 P4/mmm(1/2,1/2,g). From ITC-C, Table 9.8.3.2(a), we find that the conventional basis (a_c^*, b_c^*, c_c^*, qⁱ) of this class is given by (1/2,1/2,0), (-1/2,1/2,0), (0,0,1), (0,0,g). Since a_4c^*=qⁱ=(0,0,g) and a₄^*=q=(1/2,1/2,g), we can write

a_1c^* = 1/2 a₁^* + 1/2 a₂^*
a_2c^* = -1/2 a₁^* + 1/2 a₂^*
a_3c^* = a₃^*
a_4c^* = -1/2 a₁^* -1/2 a₂^* + a₄^*

Using the identities,

a_k^*.a_n = 1 if k=n and = 0 otherwise,
a_kc^*.a_nc = 1 if k=n and = 0 otherwise,

we obtain a relation between the basis of the direct lattice in real space in the basic-spacegroup setting and in the supercentered setting,

a_1c = a₁ + a₂ + a₄
a_2c = -a₁ + a₂
a_3c = a₃
a_4c = a₄

Now, consider the diagonal mirror reflection y+1/2,x+1/2,z in the basic space group P4bm. The OC table lists this operator as y+1/2,x+1/2,z,t+1/2 in 100.5 and as y+1/2,x+1/2,z,t in 100.6.

In the supercentered setting, y+1/2,x+1/2,z,t+1/2 becomes X+1/2,-Y,Z,T which is equivalent to X,-Y+1/2,Z,T+1/2 because of the centering translation (1/2,1/2,0,1/2). For the reflections of the H0Lm class, we obtain the condition m=2n, which is a reflection condition for 100.6, not 100.5.

In the supercentered setting, y+1/2,x+1/2,z,t becomes X+1/2,-Y,Z,T-1/2 which is equivalent to X,-Y+1/2,Z,T because of the centering translation (1/2,1/2,0,1/2). This produces no conditions on reflections of the H0Lm class, consistent with the reflection conditions listed for 100.5.

Thus, based on the reflection conditions listed in ITC-C, we conclude that the OC operators for 100.5 belong to 100.6, and the OC operators for 100.6 belong to 100.5.