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The equivalent position generators

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The lattice associated to a unit cell is defined from one of the seven crystalline systems, six unit cell parameters and one equivalent position generator which can be one of the 14th Bravais lattices or one of the 230 standard space groups.

The lattice is defined implicitly from the equivalent position generator associated to the unit cell and the list of lattice points of the lattice is created automatically from the lattice translations of the selected equivalent position generator.

The equivalent position generators represent the symmetries groups (or space groups). The space group is the combination of all the possible transformations of symmetry in a crystalline structure. 230 space groups can be obtained. Each space group is a group in the mathematical sense of the word.

Creating the unit cell of diamond

CaRIne offers three ways to create the unit cell of diamond :

• without any equivalent position generator,
• using a Bravais lattice,
• using a space group.

For the first solution, the height atoms of carbon must be given (0,0,0 0,1/2,1/2 1/2,0,1/2 ... 3/4,3/4,1/4), then the unit cell parameters must be set to : 3.567 Å for the three lengths(a, b, c) and 90° for the three angles (alpha, beta, gamma).

The FCC Bravais lattice must be selected for the second solution. Two atoms of carbon must be entered (0,0,0 and 1/4,1/4,1/4), and only a must be set to 3.567 Å.

For the last solution, the FD-3M space group (227) must be selected and only one atom of carbon has to be given (0,0,0). The FD-3M space group will generate seven equivalent positions.

Space group origin

The (xyz) coordinates of all the atoms defined in a unit cell must be given regarding to the space group origin.

Example : The two solutions to create the unit cell of diamond from the FD-3M space group

After the selection of the FD-3M space group (227) has been completed, an atom of carbon can be added to the asymetric unit at the (0,0,0) position if the space group origin is set to (1/8,1/8,1/8), or added at the (1/8,1/8,1/8) position if the space group origin is set to (0,0,0). The first solution gives the conventional visualisation of the diamond (atoms of carbon are placed at the corners of the unit cell).

 C (0,0,0) C (1/8,1/8,1/8) Space group origin (1/8,1/8,1/8) Space group origin (0,0,0)