Seth Van Orden, Harold T. Stokes, and Branton J. Campbell, Department of Physics and Astronomy, Brigham Young University, Provo, Utah 84602, USA,


ISOSUBGROUP is a utility for listing isotropy subgroups associated with irreducible representations (IRs) of a parent space group. These include (1) nonmagnetic IRs which result in subgroups with one of the 230 crytallographic sapce-group symmetries, (2) magnetic IRs which result in subgroups with one of the 1651 magnetic space-group symmetries and (3) IRs which result in incommensurate subgroups with superspace-group symmetry, both nonmagnetic and magnetic.


International Tables. International Tables for Crystallography, Vol. A, Edited by Theo Hahn (Kluwer Academic, Dordrecht).

Stokes and Hatch. Tables of isotropy subgroups found in H. T. Stokes and D. M. Hatch, Isotropy Subgroups of the 230 Crystallographic Space Groups (World Scientific, Singapore, 1988). These tables are limited to special k points.

Quick links

Space group preferences
ISOSUBGROUP: irreducible representation
ISOSUBGROUP: table set up

Space-group preferences

The International Tables gives more than one setting for some space groups. A desired setting can be specified when creating or modifying a SUBGROUP.

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 International Tables for more details.

Most monoclinic space groups also have settings for different cell choices: 1, 2, or 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 International Tables for more details.

Trigonal space groups (for example, #146, R3) have settings using hexagonal axes and rhombohedral axes.

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. Choose orgin choice 1 or 2.

For (3+d)-dimensional superspace groups, choose either (1) the standard setting (listed in Vol. C of International Tables for d=1 and in the ISO(3+d)D tables for d=1,2,3) or (2) the setting of the basic space group as given by the above choices in Vol. A of International Tables.

IR Matrices

An IR maps each space-group operator onto a matrix. ISOSUBGROUP uses the matrices listed at the ISO-IR website. These matrices were chosen to have a specific "block" form so that the contributions from different modulation k vectors would appear separated from each other in the order parameters. For example, consider a k point (a,0,0) in a cubic space group. The space group operators generate three modulation vectors from this k point: (a,0,0), (0,a,0), (0,0,a). Suppose that one of the IRs at this k point is six-dimensional. Then two of the dimensions would be associated with each modulation. An order parameter (a,b,0,0,0,0) would generate a distortion with a modulation vector (a,0,0), an order parameter (0,0,a,b,0,0) would generate a distortion with a modulation vector (0,a,0), an order parameter (a,b,c,d,0,0) would generate a distortion with two superposed modulation vectors, (a,0,0) and (0,a,0), etc. You would be able to see by inspection from the order parameter direction which modulation vectors were involved in the distortion. To accomplish this, IR matrices must be chosen to have a certain form. In addition to putting these matrices into "block" form, we also chose the matrices so that those representing pure translations would have a specific form, and we chose the matrices for types 2 and 3 IRs so that a simple transformation would bring them into complex block-diagonal form.


Parent space group. Enter the parent space group symmetry. You may either choose the space group from the drop-down menu on the left or enter the space group number in the box on the right. Each line in the drop-down menu contains (1) the space-group number from International Tables, (2) the short Hermann-Mauguin symbol, and (3) the Schoenflies symbol. The Hermann-Mauguin symbols in the drop-down menu are generic and do not influence the space-group preferences. If any character is entered into the box on the right, the drop-down menu selection will be ignored.

Number of superposed irreducible representations. If you want to superpose distortions from more than one primary order parameter, you need to couple two or more IRs. Enter the number of superposed IRs. When you click ok, this will take you to a page where you will choose a k point for each of the superposed IRs. After making your initial selections, you will see two additional pages to select an IR and an order-parameter direction.

Space group preferences. If the parent space group symmetry you selected has more than one setting in International Tables, then you should select the desired setting. The same settings are available to both non-magnetic and magnetic space groups.

Clicking on "OK" takes you to a page, "ISOSUBGROUP: k vector."

K Vector

Choose a k point in the first Brillouin zone. This choice affects the possible superlattices which can result from the phase transition. Each line in the drop-down menu contains (1) the label of the k point using the notation of Miller and Love, (2) the label of the k point using the notation of Kovalev (only included for special k points), and (3) the coordinates of the point in terms of the basis vectors of the reciprocal lattice of the conventional lattice defined in International Tables. Some points contain one or more of the parameters a, b, or g (for example, a,0,0). You must enter the values of the parameters needed for fully specifying the position of the point. If no parameters are needed (for example, the k point 0,0,0), you do not need to enter any values. You must enter all parameters as rational numbers (for example, 1/2 instead of 0.5).

Incommensurate modulations. Incommensurate k-points are points with one or more irrational components. If you want to explore an incommensurate modulation at a given k-point, choose the # of incommensurate modulations from the drop-down menu. You don't need to enter any values for the parameters a,b,g.

Magnetic. Use this checkbox to select which of your k points you would like to be magnetic.

Clicking on "OK" takes you to the page, "ISOSUBGROUP: irreducible representaions (IR)."

Irreducible Representaion (IR)

IR. Choose an irreducible representation (IR). The list in the drop-down menu contains IRs associated with the k point you selected. Each line in the drop-down menu contains the label of the IR using the notation of (1) Miller and Love and (2) Kovalev (only included for IRs associated with special k points). Type-2 and type-3 IRs are complex. We want real IRs since distortions induced by the IR must be real. In these cases, we obtain the physical IR from the direct sum of the IR and its complex conjugate. These are indicated in the notation by a pair of IR symbols (for example, P1P1, where P1 is a type-2 IR which is equivalent to its own complex conjugate, and A2A3, where A2 and A3 are type-3 IRs which are complex conjugates of each other). Note that physical IRs are reducible with respect to complex numbers but irreducible with respect to real numbers. When dealing with magnetic distortions, IRs that produce magnetic moments have an "m" prepended to their labels.

Table set-up. This page also allows you to select the information you would like to display in the generated table on the next page. Each checkbox represents data that will take the form of columns in the generated table. Check all the boxes for the data you wish to display.

Clicking on "OK" takes you to the page, "ISOSUBGROUP: table."


Real-time calculations. ISOSUBGROUP uses precomputed data tables containing the isotropy subgroups for single IRs at special k points. For any other case, the isotropy subgroups must be generated on demand and saved to a temporary file on the server. The generation of isotropy subgroups may take anywhere from a few seconds to many hours. Be prepared to wait while they are being generated. Factors that increase the time required include a high-symmetry parent, a low-symmetry distortion, or the coupling of multiple IRs. Couple more than three IRs of a cubic parent with caution. Calculations on the server are automatically killed if they have not run to completion within one hour, and all temporary files on the server are automatically deleted once a week. Contact us if you need help with a special case that warrants an exception to these policies.