Orthorhombic (ORC)#
Pearson symbol: oP
Constructor: ORC()
It is defined by three parameter: \(a\), \(b\) and \(c\) with primitive and conventional cell:
with
Order of parameters: \(a < b < c\)
K-path#
\(\mathrm{\Gamma-X-S-Y-\Gamma-Z-U-R-T-Z\vert Y-T\vert U-X\vert S-R}\)
Point |
\(\times\boldsymbol{b}_1\) |
\(\times\boldsymbol{b}_2\) |
\(\times\boldsymbol{b}_3\) |
|---|---|---|---|
\(\mathrm{\Gamma}\) |
\(0\) |
\(0\) |
\(0\) |
\(\mathrm{R}\) |
\(1/2\) |
\(1/2\) |
\(1/2\) |
\(\mathrm{S}\) |
\(1/2\) |
\(1/2\) |
\(0\) |
\(\mathrm{T}\) |
\(0\) |
\(1/2\) |
\(1/2\) |
\(\mathrm{U}\) |
\(1/2\) |
\(0\) |
\(1/2\) |
\(\mathrm{X}\) |
\(1/2\) |
\(0\) |
\(0\) |
\(\mathrm{Y}\) |
\(0\) |
\(1/2\) |
\(0\) |
\(\mathrm{Z}\) |
\(0\) |
\(0\) |
\(1/2\) |
Variations#
There are no variations for orthorhombic lattice.
One example is predefined: orc with
\(a = \pi\), \(b = 1.5\pi\) and \(c = 2\pi\).
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("orc_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")
# Save an image:
backend.save("orc_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("orc_wigner-seitz.png")
# Interactive plot:
backend.show()
Cell standardization#
Lengths of the lattice vectors have to satisfy \(\vert\boldsymbol{a}_1^s\vert < \vert\boldsymbol{a}_2^s\vert < \vert\boldsymbol{a}_3^s\vert\) for the primitive cell in a standard form.
If \(\vert \boldsymbol{a}_3\vert > \vert \boldsymbol{a}_2\vert > \vert \boldsymbol{a}_1\vert\), 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 \(\vert \boldsymbol{a}_3\vert > \vert \boldsymbol{a}_1\vert > \vert \boldsymbol{a}_2\vert\), then
\[(\boldsymbol{a}_1^s, \boldsymbol{a}_2^s, \boldsymbol{a}_3^s) = (-\boldsymbol{a}_2, -\boldsymbol{a}_1, -\boldsymbol{a}_3)\]and
\[\begin{split}\boldsymbol{S} = \boldsymbol{S}^{-1} = \boldsymbol{S}^T = \begin{pmatrix} 0 & -1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & -1 \end{pmatrix}\end{split}\]If \(\vert \boldsymbol{a}_2\vert > \vert \boldsymbol{a}_3\vert > \vert \boldsymbol{a}_1\vert\), then
\[(\boldsymbol{a}_1^s, \boldsymbol{a}_2^s, \boldsymbol{a}_3^s) = (-\boldsymbol{a}_1, -\boldsymbol{a}_3, -\boldsymbol{a}_2)\]and
\[\begin{split}\boldsymbol{S} = \boldsymbol{S}^{-1} = \boldsymbol{S}^T = \begin{pmatrix} -1 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & -1 & 0 \end{pmatrix}\end{split}\]If \(\vert \boldsymbol{a}_2\vert > \vert \boldsymbol{a}_1\vert > \vert \boldsymbol{a}_3\vert\), 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 \(\vert \boldsymbol{a}_1\vert > \vert \boldsymbol{a}_3\vert > \vert \boldsymbol{a}_2\vert\), 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 \(\vert \boldsymbol{a}_1\vert > \vert \boldsymbol{a}_2\vert > \vert \boldsymbol{a}_3\vert\), then
\[(\boldsymbol{a}_1^s, \boldsymbol{a}_2^s, \boldsymbol{a}_3^s) = (-\boldsymbol{a}_3, -\boldsymbol{a}_2, -\boldsymbol{a}_1)\]and
\[\begin{split}\boldsymbol{S} = \boldsymbol{S}^{-1} = \boldsymbol{S}^T = \begin{pmatrix} 0 & 0 & -1 \\ 0 & -1 & 0 \\ -1 & 0 & 0 \end{pmatrix}\end{split}\]
Note
All six changes of the cell preserve handiness of the original one.
Edge cases#
If \(a = b \ne c\) or \(a = c \ne b\) or \(b = c \ne a\), then the lattice is Tetragonal (TET).
If \(a = b = c\), then the lattice is Cubic (CUB).