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

ISOSUBGROUP

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**.

**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.

Space group preferences

Starting ISOSUBGROUP

ISOSUBGROUP: k vector

ISOSUBGROUP: irreducible representation

ISOSUBGROUP: table set up

ISOSUBGROUP: table

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*.

**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."

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)."

**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.