Select here the parameters to build the reciprocal lattice of the Cell lattice.
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 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.
Lattice
Select here if the reciprocal lattice of the primitive lattice related
with the Cell lattice should also be constucted.
Planes
Select here if Plane objects (described by cross patterns, as in
stereographic projections) should be added to the reciprocal lattice nodes.
Constant
The reciprocal lattice constant
K is often set to
1.0 in crystallography,
while in solid state physics the value commonly used is
2 PI, the default used
here (it is easier to change it to
1 than the other way).
Bravais
Users can also choose the so-called Bravais polar lattice, where the constant 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.