CARAT Introduction / Examples / How do the Z-classes in the Q-class of G distribute into their Bravais flocks?
Author: Oliver Heidbüchel
Description: CARAT Introduction: Example 2 - How do the Z-classes in the Q-class of G distribute into their Bravais flocks?
Keywords: CARAT, crystallography, spacegroups, space groups, space group, spacegroup, bravaisgroup, bravaisgroups, bravais groups, bravais group, mathematics, algorithms, tables, algorithm, table, examples, example, Bravais_catalog, Bravais_type, Is_finite, Order, Symbol, QtoZ, Z-class, Q-class, Z class, Q class, arithmetic class, geometric class, arithmetic, geometric, class, bravais flock, flock, bravais, bravais flocks, flocks

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### Example 2

Let G be the group generated by the two matrices below. How do the Z-classes in the Q-class of G distribute into their Bravais flocks? In particular, how many Bravais flocks are involved?

    6
    0 1 0 0 0 0
    1 0 0 0 0 0
    0 0 0 1 0 0
    0 0 1 0 0 0
    0 0 0 0 1 0
    0 0 0 0 0 1
    6
    0 0 1 0 0 0
    0 0 0 1 0 0
    0 0 0 0 1 0
    0 0 0 0 0 1
    1 0 0 0 0 0
    0 1 0 0 0 0

### Used Programs

Bravais_catalog/Datei, Bravais_type, Is_finite, Order, Symbol, QtoZ

### Solution

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

         #g2

   or

         #2

   as top row.
2. Call

          Symbol G

   to compute the family symbol of the group in G.
   It turns out to be 3;2-2;1 .
3. Call

          Bravais_catalog

   and input the symbol '3;2-2;1'. It says that there are 3 homogeneously decomposable Bravais groups in the family and that there are altogether 6 + 8 + 4 Z-classes of Bravais groups. Don't calculate the Bravais groups.
4. Call

          Order -o G >> G

   It adds a bootom line to the file 'G' containing the order of the group G in factorized form to prepare the file as input for QtoZ.

   Alternatives:

   1. Is_finite -o G >> G has the same effect, but works faster.
   2. Order -O G produces a slightly more complete file, containing all the necessary information.
5. Call

          QtoZ -D G

   Here the Q-class of the group G is split up into Z-classes. It turns out that there are 18 Z-classes. Because of the -D-option the representative groups are put into the following files

   G.1.1 | G.1.2 | G.1.3 | G.1.4 | G.1.5 | G.1.6 | G.1.7 | G.1.8
   G.2.1 | G.2.2 | G.2.3 | G.2.4 | G.2.5 | G.2.6
   G.3.1 | G.3.2 | G.3.3 | G.3.4

6. Call

          Bravais_type G.1.1 > type.G.1.1

   etc. to

          Bravais_type G.3.4 > type.G.3.4

   to compute the Bravais group in the Bravais catalog which is Z-equivalent to the Bravais group of the group in G.1.1, etc. to G.3.4:

   type.G.1.1 | type.G.1.2 | type.G.1.3 | type.G.1.4 | type.G.1.5 | type.G.1.6 | type.G.1.7 | type.G.1.8
   type.G.2.1 | type.G.2.2 | type.G.2.3 | type.G.2.4 | type.G.2.5 | type.G.2.6
   type.G.3.1 | type.G.3.2 | type.G.3.3 | type.G.3.4

   It turns out that each Bravais type in the family is represented with multiplicity one.

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last change: 11.09.2000