3.14 Space groups in GSAS-II

In GSAS-II, to simplify the listings of the symmetry operations, they are shown in a manner that minimizes the number of operators that must be provided. This is done by providing only the unique operators, if there is a center of symmetry and lists the centering operations. The full set of operations will be created by applying the center of symmetry and centering operations to the unique operators. An example will show how this works. For the space group, F 2/c (which is a non-standard setting of the monoclinic space group P 2/c) there are 16 symmetry operations,

GSAS-II displays the window in Fig. 3.3 for this space group.

Note that at the bottom of this window, two unique symmetry operations are given: x,y,z and -x,y,½-z, but as noted the multiplicity of a general site is 16, meaning there must be a total of 16 symmetry operations. The two initially supplied symmetry operations are doubled because this space group is noted as centrosymmetric. Thus, the center of symmetry operation, -x,-y,-z, is applied to each of the two initial symmetry operations, generating two additional operations, -x,-y,-z and x,-y, ½+z. These four symmetry operations are also operated upon by the four centering operations, listed as (0,0,0)+, (0,½,½)+, (½,0,½)+, (½,½,0)+. The first centering operation gives us back the four operators already listed, but adding (0,½,½) to each of these the four operators we already have provides us with four new operations: x,½+y,½+z, -x,½+y,-z, -x,½-y,½-z and x,½-y,z. Likewise, applying the (½,0,½)+, (½,½,0)+ operations will generate the other 8 symmetry operations. Thus, we generate all 16 of the operators listed above. Pressing the “Print Ops” button causes these 16 operations to be displayed in the GSAS-II console window.

The use of these grouped operators allows the most complex space groups (which have 192 operators) to be shown in GSAS-II with only 24 basic operators, where these 24 operators will be doubled to 48 by a center of symmetry, and then each of the three additional centering operators will provide another 48 operators giving us a total of 192 (=24*2*4).

Space groups are input to GSAS-II with a space between each set of symmetry operations, so the space group P6₃/mmc will be specified as P 63/m m c, though for standard settings, space group symbols will be recognized even if the spaces have not been specified correctly.

One place where GSAS-II is quite restrictive is with centrosymmetric space groups where the highest symmetry location in the cell is not a center of symmetry. Such space groups are documented in the International Tables with two choices for origin. In the first origin setting (“Origin 1”), the highest symmetry position is selected as (0,0,0), while “Origin 2” places a center of symmetry at (0,0,0). Origin 2 is computationally much simpler (since reflection phases are real). GSAS-II only accepts structures in Origin 2 settings in those space groups where there is a choice. When a structure is imported into GSAS-II where this choice exists, if symmetry operators are in the file (currently only in some CIFs), and an Origin 1 setting is detected, GSAS-II will perform the necessary translation to place the structure into Origin 2. Without symmetry operators, the origin can not be determined automatically, but atom multiplicities are computed using both origin settings and usually it is clear which origin setting will provide the correct stoichiometry. As a last resort, plotting of the structure will almost always make it very clear if a reasonable bonding geometry is being generated. If the origin setting is wrong, the generated diffraction pattern will also be wrong.

Finally, in rhombohedral space groups, the cell is assumed to be hexagonal. To specify that the primitive rhombohedric cell should be used, add a final letter “r” to the space group name. Thus, “R 3” will provide symmetry for a hexagonal cell, but “R 3 r” will provide the symmetry for a rhombohedric cell. Usually, refinements are more stable with the hexagonal cell.