Harold T. Stokes and Branton J. Campbell, Department of Physics and Astronomy, Brigham Young University, Provo, Utah 84602, USA, branton_campbell@byu.edu
ISOSPACEGROUP is a utility for displaying information about crystallographic space groups. This information includes centering, non-lattice generators, non-lattice operators, and Wyckoff positions, and can be displayed in various settings.
3-Dimensional Nonmagnetic Space Groups. These are the space groups given in International Tables of Crystallography, Vol. A (ITA). The user can select the space group from the drop-down list or enter the space-group number in the text field. If a space group is selected from the drop-down list, the text field is ignored.
3-Dimensional Magnetic Space Groups. These space groups are based on the tables of Litvin and of Bradley and Cracknell. See ISO-MAG for more details. The user can select the space group from the drop-down lists or enter the space-group number in the text field. One list contains the space group in the order of the BNS setting, and the other list in order of the OG setting. Selecting a space group from one list or the other does not determine the setting of the displayed space group. The actual setting displayed must be selected in Magnetic space group preferences below. If a space group is selected from the BNS-ordered drop-down list, then the other drop-down list is ignored. If a space group is selected from either drop-down list, the text field is ignored. The space-group number entered in the text field must be consistent with the Magnetic space group preferences selected. If the BNS setting is selected, then the number entered must be a number for the BNS setting. If the OG setting is selected, then the number entered must be a number for the OG setting.
(3+d)-Dimensional Nonmagnetic Superspace groups, d=1,2,3. These space groups are based on the tables of Janner, Janssen, and de Wolff and of Stokes, Campbell, and van Smaalen. See ISO-(3+d)D for more details. The user can select the space group from the drop-down list or enter the space-group number in the text field. The drop-down list contains only the (3+1)-dimensional superspace groups. To select superspace groups for d=2,3, the user must enter the space-group number in the text field. If a space group is selected from the drop-down list, the text field is ignored.
(3+d)-Dimensional Magnetic Superspace groups, d=1,2,3. These space groups are based on the tables of Stokes and Campbell. See ISO-(3+d)D for more details. The user enters the space-group number in the text field.
Wild Cards. Wild cards can be used in the space-group numbers entered in the text fields (except for nonmagnetic 3-dimensional space groups). There are two ways to do this: (1) use an asterisk (*) in place of one of the components or (2) simply omit components at the end of the number. (By components, we mean the numbers separated by periods in the space-group number.) For example, 136.*.19 in the text field for nonmagnetic superspace groups would select all groups, d=1,2,3, based on the basic space group 136 and Bravais class 19. As another example, 136 in the text field for 3-dimensional magnetic space groups would select all groups based on the nonmagnetic space group 136.
Display Options.
(1) The notation for generators and operators. Select the
x,y,z,... notation encountered in International Tables of
Crystallography, Vol. A (ITA), or select the Seitz notation. In
the Seitz notation, the point-group operator is displayed followed by
the axis of rotation (if any) in square brackets.
(2) If the ITA notation is selected above, the notation for
generators and operators in the superspace groups can be further
selected as x,y,z,t,... or x1,x2,x3,x4,..., or
xs1,xs2,xs3,xs4,....
(3) Show Wyckoff positions.
(4) Show supercentered setting for superspace groups. This applies
only if the standard setting is selected below in the Superspace group
preferences.
Space group preferences. The International Tables of
Crystallography, Vol. A (ITA) gives more than one setting for many
of the space groups. The selections below affect both nonmagnetic and
magnetic 3-dimensional space groups. They also affect both the
nonmagnetic and magnetic (3+d)-dimensional superspace groups, but only
if the basic setting is selected below in the Superspace group
preferences
(1) Monoclinic space groups have settings for six different
orientations of the axes: a(b)c, c(-b)a, ab(c),
ba(-c), (a)bc, or (-a)cb. Unique axes are in
parentheses. See Table 4.3.1 in ITA for more details.
(2) Most monoclinic space groups also have settings for different
cell choices: 1, 2, or 3.
(3) Orthorhombic space groups have six different choices for the
orientation of axes: abc, ba-c, cab, -cba,
bca, or a-cb. See Table 4.3.1 in ITA for more details.
(4) Trigonal space groups (for example, #146, R3) have settings
using hexagonal axes and rhombohedral axes.
(5) Some orthorhombic, tetragonal, and cubic space groups (for
example, #227 Fd-3m) have two choices for the position of the origin,
one of which (origin choice 2) in located at a point of
inversion.
Magnetic space group preference. Select either the BNS (Belov-Neronova-Smirnova) or OG (Opechowski-Guccione) setting. See ISO-MAG for more details about these two settings. Note that the OG setting is not yet implemented for magnetic superspace groups.
Superspace group preference. Select either the standard or the basic setting. The standard setting is given in the tables at ISO-(3+d)D. The basic setting uses the Space group preferences selected above. The supercentered setting is displayed only when the standard setting is selected here.
Use Alternate Setting. Enter the transformation from the standard setting to the desired alternate (and possibly non-standard) setting of the group. Separate field sets are provided for commensurate and incommensurate groups. As visually organized on the page, these fields comprise S-1t, the inversed transpose of the matrix S that transforms an atomic coordinate or superspace coordinate x from the standard setting to the alternate setting, i.e. x' = S x. The structure of this matrix for an incommensurate group is described in detail HERE.