GAMGI Interfaces: Cell Measure

Cell Measure
Cell->Measure analyses cell properties. Currently only the reciprocal lattice is available.

### Cell

The name and number of the cell to analyse should be written here.

An easy technique to identify a cell is to click the mouse over its visual representation: if GAMGI recognises the cell and the Cell entry is empty (or filled with separator characters, as space, tab, newline), the cell is analysed, otherwise the mouse action is ignored.

### Reciprocal

The reciprocal lattice for the selected cell is calculated and attributed to a new cell object, in a new layer. The primitive vectors for the old and new lattices, a1,a2,a3 and b1,b2,b3 respectively, are related by the equations:

    b1 = K (a2 x a3) / (a1 . a2 x a3)
    b2 = K (a3 x a1) / (a1 . a2 x a3)
    b3 = K (a1 x a2) / (a1 . a2 x a3)

The reciprocal lattice constant K is often set to 1.0 in Crystallography, while in Solid State Physics the value mostly used is 2PI, the default used here, as it is easier to change it to 1 than the reverse. Users can also use the value calculated for the so-called Bravais polar lattice, where K is made equal to V**2/3, where V is the primitive cell volume, so direct and reciprocal lattices have exactly the same volume per node.

The new layer has the default layer properties (except no visibility to outside) and the new cell has the initial cell properties (except those related with the reciprocal lattice transformation). Space group information is discarded for the new cell.

For all Bravais lattices, the reciprocal lattice transformation follows these rules: 1) the crystalographic system remains the same; 2) P, C, R lattices are transformed into P, C, R lattices, while I, F lattices are transformed into F, I lattices, respectively; 3) the reciprocal lattice of the reciprocal lattice is the initial lattice.

When Cubic, Tetragonal, Orthorhombic or Hexagonal R lattices are transformed, the orientation of the conventional cell vectors is preserved.

The reciprocal lattice of a Tetragonal I lattice is a Tetragonal F lattice, which is usually not represented, because it is equivalent to a Tetragonal I lattice, rotated 45 degrees and scaled by (1/sqrt(2), 1/sqrt(2), 0). GAMGI represents the reciprocal lattice of a Tetragonal I lattice as a new Tetragonal I lattice, rotated and scaled by those values, so orientation and lengths are fully preserved. In particular, the reciprocal of the reciprocal of a Tetragonal I lattice is represented as the initial cell rotated by 90 degrees around the origin, which still represents the same lattice, due to the rotation symmetry of the Tetragonal system.

Hexagonal P lattices are rotated 30 degrees, so the reciprocal of the reciprocal of a Hexagonal P lattice is represented as the initial lattice rotated by 60 degrees around the origin, which still represents the same lattice, due to the rotation symmetry of the Hexagonal system.

For Monoclinic P, C lattices, the first conventional vector (aligned along the x axis) becomes a vector on the plane xz, the second vector (aligned along the y axis) continues aligned along the y axis, and the third vector (a vector on the xz plane) becomes aligned along the z axis. For the reciprocal of the reciprocal lattice, the conventional vectors become exactly the same as for the initial lattice.

For Triclinic P lattices, the first conventional vector (aligned along the x axis) becomes a generic vector pointing down, the second vector (aligned on the xy plane) becomes aligned on the yz plane, and the third vector (a generic vector pointing up) becomes aligned along the z axis. For the reciprocal of the reciprocal lattice, the conventional vectors become exactly the same as for the initial lattice.
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