RHL2

Rhombohedral cell is defined by two parameters \(a\) and \(\alpha\).

RHL lattice has variation RHL2 when \(\alpha > 90^{\circ}\).

Cell constructor

To get an example of the cell use wulfric.cell.SC_RHL().

wulfric.cell.sc_get_example() returns an example where \(a = \pi\) and \(\alpha = 110^{\circ}\).

import wulfric

cell = wulfric.cell.sc_get_example("RHL2")
atoms = dict(positions=[[0, 0, 0]], spglib_types=[1])

# To avoid multiple calls to spglib one can do it once and then pass spglib_data
# to the functions where it is needed
spglib_data = wulfric.get_spglib_data(cell=cell, atoms=atoms)

kp = wulfric.Kpoints.from_crystal(cell=cell, atoms=atoms, convention="SC")

conv_cell, conv_atoms = wulfric.crystal.get_conventional(
    cell=cell, atoms=atoms, convention="SC", spglib_data=spglib_data
)

prim_cell, prim_atoms = wulfric.crystal.get_primitive(
    cell=cell, atoms=atoms, convention="SC", spglib_data=spglib_data
)

variation = wulfric.crystal.sc_get_variation(
    cell=cell, atoms=atoms, spglib_data=spglib_data
)

assert variation == "RHL2"

print(variation)

RHL2

K-path

print(kp.path_string)

GAMMA-P-Z-Q-GAMMA-F-P1-Q1-L-Z

High-symmetry points

print(kp.hs_table(decimals=4))

Name    rel_b1  rel_b2  rel_b3      k_x     k_y     k_z
GAMMA   0.0000  0.0000  0.0000   0.0000  0.0000  0.0000
F      -0.0000  0.5000 -0.5000   0.8717  0.6104 -0.5982
L       0.0000  0.5000  0.0000   0.8717  0.6104  0.6475
P       0.3726  0.3726 -0.6274   1.2991 -0.0000 -0.5982
P1     -0.3726  0.6274 -0.3726   0.4443  1.2208 -0.5982
Q       0.2451  0.2451  0.2451   0.8548 -0.0000  1.2457
Q1     -0.2451  0.7549 -0.2451   0.8887  1.2208  0.0493
Z       0.5000  0.5000 -0.5000   1.7434 -0.0000  0.0493

Brillouin zone and default k-path

pe = wulfric.PlotlyEngine(_sphinx_gallery_fix=True)

pe.plot_brillouin_zone(
    cell=prim_cell, color="red", legend_label="Brillouin zone of the primitive cell"
)
pe.plot_brillouin_zone(
    cell=cell, color="chocolate", legend_label="Brillouin zone of the original cell"
)
pe.plot_kpath(kp=kp)
pe.plot_kpoints(kp=kp, only_from_kpath=True)

pe.show(axes_visible=False)

Cells of real space

pe = wulfric.PlotlyEngine(_sphinx_gallery_fix=True)

pe.plot_cell(cell=cell, legend_label="Original cell", color="Chocolate")
pe.plot_cell(cell=prim_cell, legend_label="Primitive cell", color="Black")
pe.plot_cell(cell=conv_cell, legend_label="Conventional cell", color="Blue")
pe.plot_wigner_seitz_cell(
    cell=prim_cell, legend_label="Wigner-Seitz cell", color="green"
)

pe.show(axes_visible=False)

Edge cases

In rhombohedral lattice \(a = b = c\) and \(\alpha = \beta = \gamma\), thus three edge cases exist:

If \(\alpha = 60^{\circ}\), then the lattice is FCC.

If \(\alpha \approx 109.47122063^{\circ}\) (\(\cos(\alpha) = -1/3\)), then the lattice is BCC.

If \(\alpha = 90^{\circ}\), then the lattice is CUB.