a computer program for determination of the relations of Wyckoff positions
for a group - subgroup pair
Consider group-subgroup related space groups G > H. Atoms which are symmetrically
equivalent under G, i.e. belong to the same orbit of G, may become non
equivalent under H, (i.e. the orbit splits) and/or their site-symmetries
can be reduced. The structural relation between the group-subgroup related
phases follows from the relations of the occupied orbits, and this behavior
is the same for all orbits belonging to a Wyckoff position. The aim of
the program is the determination of the relations of the Wyckoff positions
(known as splitting of Wyckoff positions) for a group-subgroup symmetry
break G > H.
1. Specification of the group-subgroup chain G > H
(i) The groups G and H specified either by their sequential numbers,
as listed in International Tables for Crystallography, vol.A.
You can specify the origin choice, rhombohedral or hexagonal axes and the
Default values are:
If your group or subgroup data are in unconventional settings, you can
give the transformation matrices to the corresponding conventional
settings by clicking over "Unconventional settings".
- origin choice 2;
- hexagonal axes;
- unique axis b;
(ii) A transformation matrix relating the bases of the group and the subgroup.
The elements of the matrix can be given either as decimals or fractions.
If you don't know the transformation matrix click here
for more information about "How to find the correct transformation?"
By clicking over the button "Show Group-Subgroup Data" you
At this step the transformation matrix is checked. If it does not correspond
to the chosen group-subgroup chain, then the user is given the possibility
to correct it.
- Name and list of the Wyckoff positions of the group G.
- Name and list of the Wyckoff positions of the subgroup H.
- Characteristics of the group-subgroup chain -- the index of the subgroup
H in the group G and the transformation matrix.
2. Specification of the orbit of G
The orbits of G are selected either from a list
with Wyckoff positions of G, or by the coordinate triplet of any point
of the orbit of G.
To study the behavior of some of the Wyckoff positions of G:
If you want to obtain the behavior of a particular orbit:
check these Wyckoff positions from the list;
Give the coordinates (x, y, z) of any point from the orbit.
To view the behavior of the chosen orbit of G for the symmetry break G > H
- click over the button "Splitting"
The output file contains the following data:
Group Subgroup Data
Characteristics of the group-subgroup chain:
Splitting of the chosen orbit of G (or the corresponding Wyckoff position)
- the index of the subgroup H in the group G;
- the transformation matrix relating the bases of the group and the subgroup;
The result from the splitting is shown in table-form with the following columns:
The distribution of the representatives of the G-orbit into H-orbits
- group Wyckoff position letter and multiplicity;
- corresponding subgroup Wyckoff position(s) letter and multiplicity;
There is a link to the file with the full information for the splitting. This file contains:
Coordinate triplets of the G-orbit representatives (with respect to the
unit cell of H) referred to the bases of G and H. These are shown in a
table-form with the following columns:
- sequential number of the representative,
- coordinates of the representative referred to the group G basis,
- coordinates of the representative referred to the subgroup H basis.
The splitting of the Wyckoff position of G into the subgroup Wyckoff position(s);
The distribution of the representatives of the G-orbit into H-orbits is
shown in a table-form with the following columns:
- the Wyckoff letter of the subgroup Wyckoff position;
- sequential number of the group Wyckoff position's representative -- the
numbers correspond to those in the table above (the table with the list
of G orbit representatives, with respect to the unit cell of H, referred
to the basis of G and H);
- representative of the subgroup Wyckoff position, corresponding to the group
G representative with the sequential number given in the previous
There is a possibility to specify another orbit of the group G, by clicking over the button "Group-subgroup data".