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Space-group

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.

Figure1

Figure 1 Cubic space-groups (ZnTe F-43m selected).

The toolbar contains tool buttons to:

Figure2a Figure2b Figure2c

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:

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Silicon

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).

Non-centric space-group F d-3m

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.

Figure3a Figure3b Figure3c

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.

Figure4

Figure 4 Si F d-3m.


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Space-group *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!

Figure5a Figure5b Figure5c

Figure 5a General RPS code (*Fd-3m).

Figure 5b Special RPS code (with Wyckhoff letter).

Figure 5c Point group.

Figure6

Figure 6 Si *F d-3m.

Figure7

Figure 7 Si at (0,0,0) and *F d-3m space-group.

It sometimes advantageous (in order to increase calculation speed) to simplify the RPS code and to keep only the appropriate equivalent positions. This is possible using the RPS code.

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)


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Orthorhombic space-groups

Figure9

Figure 9 Convential (a ≤ b ≤ c) orthorhombic space-groups.

Figure10

Figure 9 Non-convential orthorhombic space-groups.