Space-Group Symmetry, Basic Tab
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Dialog Box Tab - called from: Symmetry dialog

Starting with V4.0 of ATOMS, space-group symmetry is obtained using licensed excerpts from the SGInfo program of Ralf Grosse-Kunstleve. This allows more complete selection of alternate orientations and origins than in previous versions of ATOMS, and also Shubnikov black-and-white symmetry.

You can specify the space group in any of three ways: 1) the Hermann-Maughin (H-M) or International symbol; 2) the Hall symbol (S.R. Hall: Acta Cryst., A37, 517, 1981); or 3) the number of the group in the International Tables for X-ray Crystallography.

International Tables Volume. You have the option of using the older version of the Tables (called the Second Edition: various revisions and reprints from 1952 to 1979) in which the symmetry information was in Volume I, or the newer version (1983 onwards) in which it is in Volume A. The principal difference between the two versions is that in the older one the unique axis of a monoclinic group is assumed to be the c-axis, whereas in the newer one it is assumed to be the b-axis. Thus entering the symbol P2/m gives two different orientations depending on the volume selected. You can always specify the orientation by entering the long form of the symbol, i.e. P 2/m 1 1, P 1 2/m 1 or P 1 1 2/m.

The H-M symbol can be typed into the edit box in either short form or long form, with or without spaces between positions. However, it is usually safer to select the symbol from the list box at the bottom, which gives the standard-form symbols for all the space-groups. Clicking the Select button copies the relevant data to the edit boxes at the top; it does not actually generate the symmetry. Symmetry generation is done after you click OK - this may take a few seconds. If the space group you select does not appear to be consistent with your choice of axes, a warning box appears, but in most cases consistency is not required. However, if you selected trigonal rhombohedral axes in the Title/Axes dialog, only a rhombohedral space group may be selected. This is done by adding :R to the end of the symbol in the case of H-M symbols, or asterisk (*) in the case of Hall symbols. This is a change from versions of ATOMS previous to V4.0, in which the orientation of the space group was automatically determined from the Title/Axes dialog. If you select the rhombohedral setting and then switch to the Custom symmetry option, the lattice type will be P; it will be R if the axes are hexagonal.

Origin of coordinates. In the International Tables, 24 space groups in the orthorhombic, tetragonal and cubic systems are given with a choice of origin: 1) not on a center of inversion or 2) on a center of inversion. These two origins are selected by adding :1 or :2 respectively to the end of the H-M symbol or the number. Although the origin on the center is second in the Tables and in the list, if the number is omitted this will be the default. This difference in origin is explicit in the Hall symbol, and other choices of origin may be specified for any space group with the Hall symbol.

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Shubnikov Symmetry. You can use Shubnikov black-and-white symmetry to show magnetic or other properties of crystals. If you choose this option you must enter at least some of the relevant parameters in the Shubnikov Tab. You enter the Shubnikov space-group symmetry using modified versions of either the H-M or Hall symbols. After entering the Shubnikov symmetry in the Basic Tab, you can check the Shubnikov lattice type and basis operations in the Shubnikov Tab after clicking OK or Apply. The H-M symbol is modified in the standard way, by a) adding a subscript (actually a postscript) to the lattice symbol; or b) adding a prime or apostrophe to the individual "positions" or basis operations in the symbol. To enter a Shubnikov symbol you must separate the lattice symbol and the individual positions by at least one space or underscore. The entries in the list box already have these separations, so it is advisable to select one of these and then edit it. The Hall symbol is modified in a similar way, but instead of using a prime you must use the "^" character. An initial minus sign on the Hall symbol signifies a center of inversion - if you want to assign Shubnikov inversion to this center, add "^" to the lattice symbol, not to the minus sign.

Standard H-M Shubnikov lattice symbols use both lower- and upper-case subscripts A, B, C for "color" face centering, and a, b, c for edge centering. In ATOMS a capital letter will always indicate face centering, and either lower-case a, b, c or x, y, z (or X, Y, Z) will indicate edge centering. Note that there is little checking for self-consistency, either for input through the symbol or explicit operators in the Shubnikov Tab. The user is responsible for entering a valid Shubnikov space group.

Shubnikov inversion is considered to apply to the spin of a magnetic atom, rather than directly to the vector which shows the magnetic direction. This means that improper operations, including a center of inversion, planes of symmetry and improper (bar) axes, result in inversion or reversal of the magnetic spin vector when the operation is not primed or Shubnikov, and no inversion when the operation is primed or Shubnikov. Of course, the resulting spin-vector orientation depends also on the orientation of the spin with respect to the symmetry operator - when the vector is parallel to an axis or plane the result is completely different from when it is perpendicular. ATOMS can apply Shubnikov symmetry in this way, or in certain other ways - see the Shubnikov Tab.

Magnetic or other Shubnikov symmetry normally involves entries in three different places:

1) The Space Group from Table symmetry option (this dialog), including the Shubnikov Tab;
2) The Atomic Vectors dialog (Input1 menu), to set the display parameters of the vectors; and
3) the Revise Atom dialog, Vector Tab for individual input atoms, to set the orientation of the vectors on the atoms.

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Molecules. It may sometimes be desirable to use the Space Group symmetry option for a molecule rather than the Point Group option because in some point groups you can select different orientations of the symmetry operators with respect to the coordinate axes by choosing the operations from different space groups. Space-group operations are applicable to point groups, provided there are no translations - that is, you cannot use space groups with screw axes, glide planes, or non-primitive lattices. For example, in number 111, P42m - D2d1 the mirror planes are 45 degrees from the structure axes, whereas in no. 115, P4m2 - D2d5, the mirror planes are parallel to the axes. If you choose the Point Group symmetry option and enter the point group 4m2 - D2d, the first of these orientations, as in number 111, will always be used. These two space groups actually differ in other ways than the 45 degree rotation, but this is irrelevant if the operations are used without lattice translations. The choice between the two space groups depends on the relative orientation of coordinate axes and symmetry elements. In this case, if an atom lies on a 2-fold axis it will belong to a set of four (rather than eight if it does not lie on any symmetry elements). For no. 111, such atoms will have coordinates like x,0,0, whereas for no. 115 they will be like x,x,0.
In the trigonal, hexagonal and tetragonal systems several space groups have alternate orientations at 30 or 45 degrees from each other as in the case of D2d. If the symmetry for a molecule is specified with the Point Group option, the "standard" orientation is generated. One may access the alternate orientation by the Space Group option and giving the alternate space group as follows (a B before a number indicates a "bar" or inversion axis:

  
Point Group
Standard Orientation
Alternate Orientation
42m - D2d
PB42M (no. 111)
PB4M2 (no. 115)
32 - D3
P321 (no. 150)
P312 (no. 149)
3m - C3v
P3M1 (no. 156)
P31M (no. 157)
32/m - D3d
PB3M1 (no. 164)
PB31M (no. 162)
6m2 - D3h
PB6M2 (no. 187)
PB62M (no. 189)


You can also use this symmetry option to select a non-standard setting for monoclinic or orthorhombic point groups as
discussed in the previous section. For example, you can cause the unique axis of groups 2 - C2, m - Cs or 2/m - C2h to be either a, b or c (the standard setting for ATOMS is unique axis b). You can also cause the 2-fold axis of mm2 - C2v to be parallel to any of the three structure axes. If you do choose this option, using space-group operations for a molecule, be sure that the space group has no screw axes or glide planes, and has a primitive Bravais lattice.