The toolbar contains tool buttons to:
The space-group generates the Regular Point System codes (RPS orbits) using its Hermann-Mauguin short symbol. The window shown on Figure 1 displays the cubic space-groups. Several of these space-groups have 2 possible settings, centric or non-centric.
The * in front of the Hermann-Mauguin symbol indicates that the space-group is centric, i.e that it has a center of symmetry at the origin (000).
Several space-groups have a non-centric alternatice setting. For example *F d-3m, centric has the F d-3m non-centric equivalent.
The toolbar contains tool buttons to:
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Figure 2a General RPS code (ZnTe F-43m). |
Figure 2b Special RPS code (with Wyckhoff letter). |
Figure 2c Point group. |
The information provided by these panes is as follow:
Structures can be described by different space-groups, in particular centric structure where the symmetry center is at (0,0,0). As an example the structure of Si can be defined using the centric space-group *F d-3m with one Si atoms at (1/8, 1/8, 1/8), i.e. at the symmetry center, or the alternate non-centric space-group F d-3m with one Si atom at (0, 0, 0).
The General>pane of space-group F d-3m contains 48 elements (Fig. 3a) and shows that this space-group has no symmetry center. The 48 RPS codes with the FCC translations give the 192 orbit size for an atom at a general (x, y, z) position.
The Special pane provide the orbits (Wyckoff letter) of the symmetry elements. One Si atom at (0,0,0) will provide an orbit of 8 equivalent atoms (Fig. 3b). The point-group is shown in Fig. 3c.
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Figure 3a General RPS code (Fd-3m). |
Figure 3b Special RPS code (with Wyckhoff letter). |
Figure 3c Point group. |
Space-group F d-3 m is non-centric and the Silicon structure is entirely defined with one Si atom at (0, 0, 0) (Fig. 4). The Silicon structure given in this figure is the conventional one with 8 Si at the vertices each counting for 1/8, 6 at the center of the faces each counting for 1/2 and 4 inside the unit cell making 1 + 3 + 4 = 8 atoms per unit cell.
Figure 4 Si F d-3m.
It is possible to define Silicon with the centric *F d-3m space-group.
The symmetry center of this space-group is set at (1/8, 1/8, 1/8). The orbit of Si at (1/8, 1/8, 1/8) has 8 equivalent atoms (Figures 5a, 5b, 5c). All the atoms being inside the unit cell, the 3-D view of the Silicon structure is non-conventional (Fig. 6).
The equivalence the Si structures generated with *F d-3m or F d-3m can be check by comparing, for example, their (h,k,l) structure factors.
Setting the Si atom at (0,0,0) with space-group *F d-3m will result in a very curious Silicon structure (Fig. 7).
Checking the space-group or the atoms position of a new structure is always a pretty good idea!
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Figure 5a General RPS code (*Fd-3m). |
Figure 5b Special RPS code (with Wyckhoff letter). |
Figure 5c Point group. |
The space-group editor recognizes the alternate settings of orthorhombic structures where quite often the lattice parameters a,b,c are permutated (Figures 9, 10)