CARAT Introduction / Examples / Do given matrices generate a space group? If so, find its name and normal representation.
Author: Oliver Heidbüchel
Description: CARAT Introduction: Example 5 - Do given matrices generate a space group? If so, find its name and normal representation.
Keywords: CARAT, crystallography, spacegroups, space groups, space group, spacegroup, mathematics, algorithms, algorithm, examples, example, Extract, Is_finite, Presentation, Standard_affine_form, matrix, matrices, generator, generators, Name

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### Example 5

Do the following given matrices generate a space group? If so, find its name and normal representation.

A. |
    7
    0 1 0 0 0 0 1
    1 0 0 0 0 0 -1
    0 0 0 1 0 0 0
    0 0 1 0 0 0 0
    0 0 0 0 1 0 0
    0 0 0 0 0 1 0
    0 0 0 0 0 0 1
    7
    0 0 1 0 0 0 0
    0 0 0 1 0 0 0
    0 0 0 0 1 0 0
    0 0 0 0 0 1 0
    1 0 0 0 0 0 0
    0 1 0 0 0 0 0
    0 0 0 0 0 0 1
| B. |
    6       /2
     -1  1 0  0 -1 -2
     -3  1 2  0 -1 -2
      3  1 0  0  1 -4
      3 -1 2  0  1 -2
      0  0 2 -2  2  0
      0  0 0  0  0  2
    6       /2
     -2  0 0 0 -1 -1
     -4  0 2 0 -1 -1
      2 -2 0 2 -1 -1
      2 -2 2 0  1 -1
      2  0 2 0  0  0
      0  0 0 0  0  2

### Used Programs

Extract, Is_finite, Name, Order, Presentation, Standard_affine_form

### Solution

1. 1. Write the two matrices in a file 'group' and add

            #g2

      or

            #2

      as top row.
   2. Call

            Extract group > linear_part

      to write the linear part of the affine group, whose generators are in the file 'group', to the file 'linear_part' in bravais_TYP as format. At the same time invariant forms are calculated.
   3. Call

             Is_finite linear_part

      to find that the potential point group has finite order, 12 in this case. The output is

          The order of the group is: 12

   4. Call

             Presentation linear_part > pres

      to get a presentation of the group in file 'linear_part' and to write this to the file 'pres' in matrix_TYP as format.
   5. Call

             Standard_affine_form group pres

      to find that the rank of the translation lattice of the affine group in 'g' is not full, i. e. 6, but only 3. Hence one has a subperiodic group and not a space group. The output is

          The rank of the translation lattice is 3

2. 1. Write the two matrices in a file 'group' and add

            #g2

      or

            #2

      as top row.
   2. Call

            Extract group > linear_part

      to write the linear part of the affine group, whose generators are in the file 'group', to the file 'linear_part' in bravais_TYP as format. At the same time invariant forms are calculated.
   3. Call

             Order linear_part

      to find that the potential point group has finite order, 720 in this case (Here we use "Order" in contrast to part A. because "Is_finite" only works for integral matrices). The output is

          The order of the group is 720

   4. Call

             Presentation linear_part > pres

      to get a presentation of the group in file 'linear_part' and to write this to the file 'pres' in matrix_TYP as format.
   5. Call

             Standard_affine_form group pres > S

      to get the space group in standard form in file 'S'. So the given matrices really generate a space group.
   6. Call

             Name -o S

      to get the CARAT name and the CARAT representative for this affine class:

             #g2 % standard group for S
             6       /2              % generator
               0 2  0 0 -2 -2
               0 0  2 0  0  0
               2 0  0 0  2 -1
              -2 2 -2 0 -2  0
               0 0  2 2  2  0
               0 0  0 0  0  2
             6       /2              % generator
              -2  0  0  0 -2 -1
               0  0  2  0  0  0
               2 -2  0 -2  0  0
              -2  2 -2  0 -2  0
               2  0  2  0  0  0
               0  0  0  0  0  2
             % order of the group unknown
             qname: group.1040 zname: 3 1 aff_name: 1

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last change: 22.09.2000