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

This program finds the standard setting for a (3+*d*)-dimensional
superspace group (*d*=1,2,3), given a list of generators in any setting
of the user's choice. The generators of the centering translation group
and the non-lattice symmetry group are entered separately. After identifying
the superspace group, this utility displays detailed information about the standard setting
of the equivalent group and
presents the affine transformation matrix that takes operators from the
user's input setting to the standard setting.

See the Example Page, which contains default input. Simply click the "OK" button obtain the corresponding output.

Spaces are ignored by the program. Fractions should be entered as a ratio of two integers (for example, 1/2, not 0.5). Operators may be enclosed in parentheses, but do not need to be. Multiple operators can appear on the same line if they are delimited by either semicolons or parentheses.

**(1) Centering positions**

You only need to enter enough operators to generate the full list. For example, if you enter (1/2,1/2,0,0) and (1/2,0,1/2,0), the program will complete the list by adding (0,0,0,0) and (0,1/2,1/2,0). However, additional operators can also be included beyond those in the generating set. Components of each centering translation may be separated by commas if you wish, though (1/2,1/2,0,0) can also be entered as (1/21/200).

Examples:

(1/2,1/2,0,0)(1/2,0,1/2,0)

1/2,1/2,0,0; 1/2,0,1/2,0

1/21/200;1/201/20

**(2) Symmetry operators**

You only need to enter enough operators to generate the full list. For example, if you enter (-y,x,z,t+1/4), the program will complete the list by adding (x,y,z,t), (-x,-y,z,t+1/2), and (y,-x,z,t+3/4). However, additional operators can also be included beyond those in the generating set. Components of each operator should be separated by commas.

When the "magnetic" box near the bottom of the page is checked, adding either a prime (') symbol or -1 to the end of a representative point operator or generator causes it to be time reversed. If a +1 is added, or if nothing is added, the operator is not time reversed. Note that the centering vectors are actual lattice translations and cannot therefore be time reversed. To add a time-reversed translation (or pure time reversal), add it to the list of representative point operators in the second input box.

Examples:

(x,y,-z,-t,u)(1/2+x,-y,z,1/2+t,u)

x,y,-z,-t,u; 1/2+x,-y,z,1/2+t,u

x1,x2,-x3,-x4,x5; 1/2+x1,-x2,x3,1/2+x4,x5

x1s,x2s,-x3s,-x4s,x5s; 1/2+x1s,-x2s,x3s,1/2+x4s,x5s

x,y,-z,-t,u; 1/2+x,-y,z,1/2+t,u' (with magnetic boxed checked)

x,y,-z,-t,u,+1; 1/2+x,-y,z,1/2+t,u,-1 (with magnetic boxed checked)

The output notation for superspace-group operators will match that of
the input: There are three choices: (*x,y,z,t,u,v*),
(*x _{1},x_{2},x_{3},x_{4},x_{5},x_{6}*),
and
(

**(1) Input setting**

Reproduces the centering translations and symmetry operators entered by the user, in addition to all other operations generated by them.

**(2) Standard setting**

Includes detailed information about the standard setting of the superspace group. See about these tables for an explanation of the information displayed.

**(3) Affine transformation to standard setting**

Presents the affine transformation (and its inverse) that transforms symmetry operators from the user-input setting into the standard setting, and summarizes the resulting relationships between the basis vectors and modulation vectors of the input and standard settings. See a Detailed Explanation of the matrix manipulations involved.