Rhombohedral (RHL)

Pearson symbol: hR

Constructor: RHL()

It is defined by two parameters \(a\) and \(\alpha\). Standardized primitive and conventional cells in the default orientation are

\[\begin{split}\begin{matrix} \boldsymbol{a}_1^s &=& \boldsymbol{a}_1^{cs} &=& (a\cos(\alpha / 2), &-a\sin(\alpha/2), &0)\\ \boldsymbol{a}_2^s &=& \boldsymbol{a}_2^{cs} &=& (a\cos(\alpha / 2), &a\sin(\alpha/2), &0)\\ \boldsymbol{a}_3^s &=& \boldsymbol{a}_3^{cs} &=& \left(\dfrac{\cos\alpha}{\cos(\alpha/2)}\right., &0, &\left.a\sqrt{1 - \dfrac{\cos^2\alpha}{\cos^2(\alpha/2)}}\right) \end{matrix}\end{split}\]

Transformation matrix from standardized primitive cell to standardized conventional cell is

\[\begin{split}\boldsymbol{C} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{pmatrix} \qquad \boldsymbol{C}^{-1} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{pmatrix}\end{split}\]

K-path

RHL₁

\(\mathrm{\Gamma-L-B_1\vert B-Z-\Gamma-X\vert Q-F-P_1-Z\vert L-P}\)

\[\begin{matrix} \eta = \dfrac{1 + 4\cos\alpha}{2 + 4\cos\alpha} & \nu = \dfrac{3-2\eta}{4} \end{matrix}\]

Point	\(\times\boldsymbol{b}_1^s\)	\(\times\boldsymbol{b}_2^s\)	\(\times\boldsymbol{b}_3^s\)
\(\mathrm{\Gamma}\)	\(0\)	\(0\)	\(0\)
\(\mathrm{B}\)	\(\eta\)	\(1/2\)	\(1 - \eta\)
\(\mathrm{B_1}\)	\(1/2\)	\(1-\eta\)	\(\eta - 1\)
\(\mathrm{F}\)	\(1/2\)	\(1/2\)	\(0\)
\(\mathrm{L}\)	\(1/2\)	\(0\)	\(0\)
\(\mathrm{L_1}\)	\(0\)	\(0\)	\(-1/2\)
\(\mathrm{P}\)	\(\eta\)	\(\nu\)	\(\nu\)
\(\mathrm{P_1}\)	\(1-\nu\)	\(1-\nu\)	\(1-\eta\)
\(\mathrm{P_2}\)	\(\nu\)	\(\nu\)	\(\eta - 1\)
\(\mathrm{Q}\)	\(1-\nu\)	\(\nu\)	\(0\)
\(\mathrm{X}\)	\(\nu\)	\(0\)	\(-\nu\)
\(\mathrm{Z}\)	\(1/2\)	\(1/2\)	\(1/2\)

RHL₂

\(\mathrm{\Gamma-P-Z-Q-\Gamma-F-P_1-Q_1-L-Z}\)

\[\begin{matrix} \eta = \dfrac{1}{2\tan^2(\alpha/2)} & \nu = \dfrac{3-2\eta}{4} \end{matrix}\]

Point	\(\times\boldsymbol{b}_1\)	\(\times\boldsymbol{b}_2\)	\(\times\boldsymbol{b}_3\)
\(\mathrm{\Gamma}\)	\(0\)	\(0\)	\(0\)
\(\mathrm{F}\)	\(1/2\)	\(-1/2\)	\(0\)
\(\mathrm{L}\)	\(1/2\)	\(0\)	\(0\)
\(\mathrm{P}\)	\(1-\nu\)	\(-\nu\)	\(1-\nu\)
\(\mathrm{P_1}\)	\(\nu\)	\(\nu-1\)	\(\nu-1\)
\(\mathrm{Q}\)	\(\eta\)	\(\eta\)	\(\eta\)
\(\mathrm{Q_1}\)	\(1-\eta\)	\(-\eta\)	\(-\eta\)
\(\mathrm{Z}\)	\(1/2\)	\(-1/2\)	\(1/2\)

Variations

There are two variations for rhombohedral lattice.

RHL₁

\(\alpha < 90^{\circ}\).

Predefined example: rhl1 with \(a = \pi\) and \(\alpha = 70^{\circ}\)

RHL₂

\(\alpha > 90^{\circ}\).

Predefined example: rhl2 with \(a = \pi\) and \(\alpha = 110^{\circ}\)

Examples

RHL₁

Brillouin zone and default kpath

import wulfric as wulf

cell = wulf.cell.get_cell_example("RHL1")
backend = wulf.visualization.PlotlyBackend()
backend.plot(cell, kind="brillouin-kpath")
# Save an image:
backend.save("rhl1_reciprocal.png")
# Interactive plot:
backend.show()

Primitive and Wigner-Seitz cells

Click on the legend to hide a cell.

import wulfric as wulf

cell = wulf.cell.get_cell_example("RHL1")
backend = wulf.visualization.PlotlyBackend()
backend.plot(cell, kind="primitive", label="primitive", color="black")
backend.plot(cell, kind="wigner-seitz", label="wigner-seitz", color="green")
# Save an image:
backend.save("rhl1_real.png")
# Interactive plot:
backend.show()

RHL₂

Brillouin zone and default kpath

import wulfric as wulf

cell = wulf.cell.get_cell_example("RHL2")
backend = wulf.visualization.PlotlyBackend()
backend.plot(cell, kind="brillouin-kpath")
# Save an image:
backend.save("rhl2_reciprocal.png")
# Interactive plot:
backend.show()

Primitive and Wigner-Seitz cells

Click on the legend to hide a cell.

import wulfric as wulf

cell = wulf.cell.get_cell_example("RHL2")
backend = wulf.visualization.PlotlyBackend()
backend.plot(cell, kind="primitive", label="primitive", color="black")
backend.plot(cell, kind="wigner-seitz", label="wigner-seitz", color="green")
# Save an image:
backend.save("rhl2_real.png")
# Interactive plot:
backend.show()

Cell standardization

No standardization required.

\[\begin{split}\boldsymbol{S} = \boldsymbol{S}^{-1} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}\end{split}\]

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 Face-centred cubic (FCC)

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

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