TRANSFORMSSG help

Harold T. Stokes and Branton J. Campbell, Department of Physics and Astronomy, Brigham Young University, Provo, Utah, 84602, USA, branton_campbell@byu.edu
Sander van Smaalen, University of Bayreuth, Bayreuth, Germany, smash@uni-bayreuth.de

Given the operators and modulation vectors of a (3+d)-dimensional superspace group (d=1,2,3), and the components of a superspace transformation matrix, this utility transforms the operators and modulation vectors to the new superspace setting. The generators of the centering translation group and the non-lattice symmetry group are entered separately. Detailed information about the transformed basis vectors, modulation vectors and origin are displayed. See a Detailed Explanation of the matrix manipulations involved.

Input

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

(1) Centering positions

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.  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).

(2) Symmetry operators

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.  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 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.

The output notation for superspace-group operators will match that of the input: There are three choices: (x,y,z,t,u,v), (x1,x2,x3,x4,x5,x6), and (xs1,xs2,xs3,xs4,xs5,xs6). See the ISO(3+d)D help page for more information about these notations.

(3) Old q vectors

Enter the components of the modulation vectors (i.e. the σ matrix) in the old setting as either rational (e.g. 1/2) or decimal (e.g. 0.5) fractions. They are not required and can be left blank if you wish. If entered, they will be transformed along with the operators.

(4) New reciprocal lattice basis vectors and q vectors in terms of the old ones

The new 3D reciprocal-space vectors are related to the old ones via the SR portion of affine transformation matrix S, whereas the new q vectors depend on both the old q vectors (via SM) and the old 3D reciprocal-space vectors (via Sε). These components must be entered as integers or rational fractions (e.g. 0 or 1/2, but not 0.5). Any fields left blank will be assumed to be zero. Take care that the determinant of the resulting transformation matrix is not singular.

(5) Old superspace origin in the new setting

The superspace origin of the old setting must be specified in the superspace coordinates of the new setting. Its components are the coefficients of the new superspace (not external space) basis vectors. These components must be entered as integers or rational fractions (e.g. 0 or 1/2, but not 0.5). Any fields left blank will be assumed to be zero.

Output

(1) Input setting

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

(2) New setting

Includes detailed information about the new setting of the superspace group.

(3) Affine transformation

Summarizes the affine transformation (and its inverse) specified by the user and the resulting relationships between the input and new settings, including the basis vectors (direct and reciprocal), the q-vectors (vector and component representations), and the origins.