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Example 4

Find all space groups with point group G generated by the following two matrices and decide for which superlattices each extension splits.
4d1
-1 1 1 1

4
0 0 1 0
0 0 0 1
0 1 0 0
1 0 0 0

Used Programs

Conj_bravais, Extract, Normalizer, Sublattices/ZZprog, Tr, Tr_bravais, Vector_systems/Extensions

Solution

  1. Write the two matrices in a file 'g' and add
             #g2
          
    as top line. Now 'g' has bravais_TYP as format.
  2. Call
             Normalizer g > gn
          
    to produce a file 'gn' containing G, a basis of the space of invariant quadratic forms of G, generators for the normalizer of G in GL4(Z), and the order of G. Note 'gn' has bravais_TYP as format.
  3. Call
             Vector_systems gn > gout
          
    to write a set of representatives of the vector systems under the normalizer action into the file 'out'. It turs out that the order of the extension group is 8 and that there are 6 orbits, i. e. one has 6 affine classes going with the Z-class of G.
  4. For each of the 6 cozycles write it into a file 'coz' and call
             Extract -r gn coz > g.i
          
    to construct the affine classes in the Z-class of G in standard form (i.e. the translation lattice is Zn):
    g.1 g.2 g.3 g.4 g.5 g.6
  5. Call
             Tr_bravais gn > gn_tr
          
    to get the transposed group 'gn_tr of 'g'.
  6. Call
             Sublattices -b gn_tr > gl
          
    to write the matrices representing the bases of the Gtr-sublattices of Z4 of 2-power index into file 'gl'. (Note G and Gtr have order 26.) These sublattices are ordered by inclusion and form a chain:

    L1 = Z4 > L2 > L3 > L4 > 2 L1

  7. For each inverse write it into a file 'inv' call
             Tr inv > l.i
          
    to transpose it. So we get the G-invariant superlattices of Zn:

    l.1 = Zn < l.2 < l.3 < l.4 < 1/2 Zn

  8. The following is trivial: So we only have to consider 'g.2' to 'g.6' with the lattices 'l.2' to 'l.4'.
  9. Enlarge the matrices in the files 'l.2', 'l.3' and 'l.4', i.e. change
             4       /2              % tranposed of 1-th matrix of inv
              2  0  0  1
              0 -2  0 -1
              0  0 -2 -1
              0  0  0 -1
          
    to
              5       /2              % tranposed of 1-th matrix of inv
               2  0  0  1 0
               0 -2  0 -1 0
               0  0 -2 -1 0
               0  0  0 -1 0
               0  0  0  0 2
          
  10. To decide for which superlattice each extension splits, one has to add the translations of a superlattice 'l.j' to a spacegroup 'g.i' and transform this group such that the translation lattice is Zn (Cf. Standard_affine_form). This can be done in one step by calling:
             for i in 2 3 4 5 6 ; do
                for j in 2 3 4 ; do
                   Conj_bravais -i g.i l.j > g.i.j ;
                done ;
             done
          
    We get the following spacegroups:
    g.2.2 g.2.3 g.2.4
    g.3.2 g.3.4 g.3.4
    g.4.2 g.4.3 g.4.4
    g.5.2 g.5.3 g.5.4
    g.6.2 g.6.3 g.6.4
  11. For each of these groups, extract the cozycle and the point group:
             for i in 2 3 4 5 6 ; do
                for j in 2 3 4 ; do
                   Extract -p g.i.j > pg.i.j ;
                   Extract -c g.i.j > cg.i.j ;
                done ;
             done
          
    We get the following files:
    pg.2.2 pg.2.3 pg.2.4
    pg.3.2 pg.3.4 pg.3.4
    pg.4.2 pg.4.3 pg.4.4
    pg.5.2 pg.5.3 pg.5.4
    pg.6.2 pg.6.3 pg.6.4
    cg.2.2 cg.2.3 cg.2.4
    cg.3.2 cg.3.4 cg.3.4
    cg.4.2 cg.4.3 cg.4.4
    cg.5.2 cg.5.3 cg.5.4
    cg.6.2 cg.6.3 cg.6.4
  12. Call
             for i in 2 3 4 5 6 ; do
                for j in 2 3 4 ; do
                   Vector_systems -i pg.$i.$j cg.$i.$j ;
                done ;
             done
          
    to identify the spacegroups in the files 'g.i.j'. The name is 0 iff the extension splits. We get the following output:
             Name for the 1-th extension in pg.2.2: 1
             Name for the 1-th extension in pg.2.3: 1
             Name for the 1-th extension in pg.2.4: 0
             Name for the 1-th extension in pg.3.2: 2
             Name for the 1-th extension in pg.3.3: 0
             Name for the 1-th extension in pg.3.4: 0
             Name for the 1-th extension in pg.4.2: 2
             Name for the 1-th extension in pg.4.3: 0
             Name for the 1-th extension in pg.4.4: 0
             Name for the 1-th extension in pg.5.2: 3
             Name for the 1-th extension in pg.5.3: 1
             Name for the 1-th extension in pg.5.4: 0
             Name for the 1-th extension in pg.6.2: 0
             Name for the 1-th extension in pg.6.3: 0
             Name for the 1-th extension in pg.6.4: 0
          

Remarks


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last change: 07.01.2003