CARAT Introduction / Examples / Do given matrices generate a space group? If so, find its name and normal representation.

Example 5

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

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

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. 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
```
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