Base-centred orthorhombic (ORCC)#
Pearson symbol: oS
Constructor: ORCC()
It is defined by three parameter: \(a\), \(b\) and \(c\) with conventional cell:
And primitive cell:
with
Order of parameters: \(a < b\)
K-path#
\(\mathrm{\Gamma-X-S-R-A-Z-\Gamma-Y-X_1-A_1-T-Y\vert Z-T}\)
Point |
\(\times\boldsymbol{b}_1\) |
\(\times\boldsymbol{b}_2\) |
\(\times\boldsymbol{b}_3\) |
|---|---|---|---|
\(\mathrm{\Gamma}\) |
\(0\) |
\(0\) |
\(0\) |
\(\mathrm{A}\) |
\(\zeta\) |
\(\zeta\) |
\(1/2\) |
\(\mathrm{A_1}\) |
\(-\zeta\) |
\(1-\zeta\) |
\(1/2\) |
\(\mathrm{R}\) |
\(0\) |
\(1/2\) |
\(1/2\) |
\(\mathrm{S}\) |
\(0\) |
\(1/2\) |
\(0\) |
\(\mathrm{T}\) |
\(-1/2\) |
\(1/2\) |
\(1/2\) |
\(\mathrm{X}\) |
\(\zeta\) |
\(\zeta\) |
\(0\) |
\(\mathrm{X_1}\) |
\(-\zeta\) |
\(1-\zeta\) |
\(0\) |
\(\mathrm{Y}\) |
\(-1/2\) |
\(1/2\) |
\(0\) |
\(\mathrm{Z}\) |
\(0\) |
\(0\) |
\(1/2\) |
Variations#
There are no variations for base-centered orthorombic.
One example is predefined: orcc with
\(a = \pi\), \(b = 1.3\pi\) and \(c = 1.7\pi\).
Examples#
Brillouin zone and default kpath#
# Wulfric - Crystal, Lattice, Atoms, K-path.
# Copyright (C) 2023-2024 Andrey Rybakov
#
# e-mail: anry@uv.es, web: adrybakov.com
#
# This program is free software: you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation, either version 3 of the License, or
# (at your option) any later version.
#
# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU General Public License for more details.
#
# You should have received a copy of the GNU General Public License
# along with this program. If not, see <https://www.gnu.org/licenses/>.
import wulfric as wulf
l = wulf.lattice_example("{name}")
# Standardization is explicit since 0.3
l.standardize()
backend = wulf.PlotlyBackend()
backend.plot(l, kind="brillouin-kpath")
# Save an image:
backend.save("orcc_brillouin.png")
# Interactive plot:
backend.show()
Primitive and conventional cell#
# Wulfric - Crystal, Lattice, Atoms, K-path.
# Copyright (C) 2023-2024 Andrey Rybakov
#
# e-mail: anry@uv.es, web: adrybakov.com
#
# This program is free software: you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation, either version 3 of the License, or
# (at your option) any later version.
#
# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU General Public License for more details.
#
# You should have received a copy of the GNU General Public License
# along with this program. If not, see <https://www.gnu.org/licenses/>.
import wulfric as wulf
l = wulf.lattice_example("{name}")
# Standardization is explicit since 0.3
l.standardize()
backend = wulf.PlotlyBackend()
backend.plot(l, kind="primitive", label="primitive")
backend.plot(l, kind="conventional", label="conventional", color="black")
# Save an image:
backend.save("orcc_real.png")
# Interactive plot:
backend.show()
Wigner-Seitz cell#
# Wulfric - Crystal, Lattice, Atoms, K-path.
# Copyright (C) 2023-2024 Andrey Rybakov
#
# e-mail: anry@uv.es, web: adrybakov.com
#
# This program is free software: you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation, either version 3 of the License, or
# (at your option) any later version.
#
# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU General Public License for more details.
#
# You should have received a copy of the GNU General Public License
# along with this program. If not, see <https://www.gnu.org/licenses/>.
import wulfric as wulf
l = wulf.lattice_example("{name}")
# Standardization is explicit since 0.3
l.standardize()
backend = wulf.PlotlyBackend()
backend.plot(l, kind="wigner-seitz")
# Save an image:
backend.save("orcc_wigner-seitz.png")
# Interactive plot:
backend.show()
Cell standardization#
Length of third vector of the primitive cell has to be different from the lengths of the first two vectors of the primitive cell. Together with the \(a < b\) we arrive at the following condition of the angles of the primitive cell in a standard form: \(\alpha = \beta = 90^{\circ}\) and \(\gamma > 90^{\circ}\). In practice this condition simplifies to \(\boldsymbol{a}_2 \cdot \boldsymbol{a}_3 = \boldsymbol{a}_1 \cdot \boldsymbol{a}_3 = 0\) and \(\boldsymbol{a}_1 \cdot \boldsymbol{a}_2 < 0\). We use angles of the primitive cell for standardization.
If \(\alpha = \beta = \frac{\pi}{2}\) and \(\gamma > \frac{\pi}{2}\) (i.e. \(\boldsymbol{a}_2 \cdot \boldsymbol{a}_3 = \boldsymbol{a}_1 \cdot \boldsymbol{a}_3 = 0\) and \(\boldsymbol{a}_1 \cdot \boldsymbol{a}_2 < 0\)), then
\[(\boldsymbol{a}_1^s, \boldsymbol{a}_2^s, \boldsymbol{a}_3^s) = (\boldsymbol{a}_1, \boldsymbol{a}_2, \boldsymbol{a}_3)\]and
\[\begin{split}\boldsymbol{S} = \boldsymbol{S}^{-1} = \boldsymbol{S}^T = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}\end{split}\]If \(\alpha = \beta = \frac{\pi}{2}\) and \(\gamma < \frac{\pi}{2}\) (i.e. \(\boldsymbol{a}_2 \cdot \boldsymbol{a}_3 = \boldsymbol{a}_1 \cdot \boldsymbol{a}_3 = 0\) and \(\boldsymbol{a}_1 \cdot \boldsymbol{a}_2 > 0\)), then
\[(\boldsymbol{a}_1^s, \boldsymbol{a}_2^s, \boldsymbol{a}_3^s) = (\boldsymbol{a}_2, -\boldsymbol{a}_1, \boldsymbol{a}_3)\]and
\[\begin{split}\boldsymbol{S} = \begin{pmatrix} 0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix} \qquad \boldsymbol{S}^{-1} = \boldsymbol{S}^T = \begin{pmatrix} 0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}\end{split}\]If \(\beta = \gamma = \frac{\pi}{2}\) and \(\alpha > \frac{\pi}{2}\) (i.e. \(\boldsymbol{a}_1 \cdot \boldsymbol{a}_3 = \boldsymbol{a}_1 \cdot \boldsymbol{a}_2 = 0\) and \(\boldsymbol{a}_2 \cdot \boldsymbol{a}_3 < 0\)), then
\[(\boldsymbol{a}_1^s, \boldsymbol{a}_2^s, \boldsymbol{a}_3^s) = (\boldsymbol{a}_2, \boldsymbol{a}_3, \boldsymbol{a}_1)\]and
\[\begin{split}\boldsymbol{S} = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{pmatrix} \qquad \boldsymbol{S}^{-1} = \boldsymbol{S}^T = \begin{pmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix}\end{split}\]If \(\beta = \gamma = \frac{\pi}{2}\) and \(\alpha < \frac{\pi}{2}\) (i.e. \(\boldsymbol{a}_1 \cdot \boldsymbol{a}_3 = \boldsymbol{a}_1 \cdot \boldsymbol{a}_2 = 0\) and \(\boldsymbol{a}_2 \cdot \boldsymbol{a}_3 > 0\)), then
\[(\boldsymbol{a}_1^s, \boldsymbol{a}_2^s, \boldsymbol{a}_3^s) = (\boldsymbol{a}_3, -\boldsymbol{a}_2, \boldsymbol{a}_1)\]and
\[\begin{split}\boldsymbol{S} = \boldsymbol{S}^{-1} = \boldsymbol{S}^T = \begin{pmatrix} 0 & 0 & 1 \\ 0 & -1 & 0 \\ 1 & 0 & 0 \end{pmatrix}\end{split}\]If \(\alpha = \gamma = \frac{\pi}{2}\) and \(\beta > \frac{\pi}{2}\) (i.e. \(\boldsymbol{a}_2 \cdot \boldsymbol{a}_3 = \boldsymbol{a}_1 \cdot \boldsymbol{a}_2 = 0\) and \(\boldsymbol{a}_1 \cdot \boldsymbol{a}_3 < 0\)), then
\[(\boldsymbol{a}_1^s, \boldsymbol{a}_2^s, \boldsymbol{a}_3^s) = (\boldsymbol{a}_3, \boldsymbol{a}_1, \boldsymbol{a}_2)\]and
\[\begin{split}\boldsymbol{S} = \begin{pmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix} \qquad \boldsymbol{S}^{-1} = \boldsymbol{S}^T = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{pmatrix}\end{split}\]If \(\alpha = \gamma = \frac{\pi}{2}\) and \(\beta < \frac{\pi}{2}\) (i.e. \(\boldsymbol{a}_2 \cdot \boldsymbol{a}_3 = \boldsymbol{a}_1 \cdot \boldsymbol{a}_2 = 0\) and \(\boldsymbol{a}_1 \cdot \boldsymbol{a}_3 > 0\)), then
\[(\boldsymbol{a}_1^s, \boldsymbol{a}_2^s, \boldsymbol{a}_3^s) = (\boldsymbol{a}_1, -\boldsymbol{a}_3, \boldsymbol{a}_2)\]and
\[\begin{split}\boldsymbol{S} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0 \end{pmatrix} \qquad \boldsymbol{S}^{-1} = \boldsymbol{S}^T = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & -1 & 0 \end{pmatrix}\end{split}\]
Note
All six changes of the cell preserve handiness of the original one.
Edge cases#
If \(a = b\), then the lattice is Tetragonal (TET).
If \(a = b = \sqrt{2} c\), then the lattice is Cubic (CUB).