Body-centred orthorhombic (ORCI)#

Pearson symbol: oI

Constructor: ORCI()

It is defined by three parameter: \(a\), \(b\) and \(c\) with conventional lattice:

\[\begin{split}\begin{matrix} \boldsymbol{a}_1 &=& (a, &0, &0)\\ \boldsymbol{a}_2 &=& (0, &b, &0)\\ \boldsymbol{a}_3 &=& (0, &0, &c) \end{matrix}\end{split}\]

And primitive lattice:

\[\begin{split}\begin{matrix} \boldsymbol{a}_1 &=& (-a/2, &b/2, &c/2)\\ \boldsymbol{a}_2 &=& (a/2, &-b/2, &c/2)\\ \boldsymbol{a}_3 &=& (a/2, &b/2, &-c/2) \end{matrix}\end{split}\]

Order of parameters: \(a < b < c\)

Cell standardization#

Lengths of the lattice vectors of the conventional cell have to satisfy \(\vert\boldsymbol{a}_1\vert < \vert\boldsymbol{a}_2\vert < \vert\boldsymbol{a}_3\vert\). All vectors of the primitive cell have the same length, therefore we use conventional lattice vectors for the standardization.

If this condition is not satisfied, then the lattice is transformed to the standard form:

First we order first two vectors by length:

If \(\vert\boldsymbol{a}_1\vert > \vert\boldsymbol{a}_2\vert\)
\[(\boldsymbol{a}_1, \boldsymbol{a}_2, \boldsymbol{a}_3) \rightarrow (\boldsymbol{a}_2, \boldsymbol{a}_1, -\boldsymbol{a}_3)\]

Then we find a correct place for the third vector:

If \(\vert\boldsymbol{a}_1\vert > \vert\boldsymbol{a}_3\vert\)
\[(\boldsymbol{a}_1, \boldsymbol{a}_2, \boldsymbol{a}_3) \rightarrow (\boldsymbol{a}_3, \boldsymbol{a}_1, \boldsymbol{a}_2)\]
If \(\vert\boldsymbol{a}_2\vert > \vert\boldsymbol{a}_3\vert\)
\[(\boldsymbol{a}_1, \boldsymbol{a}_2, \boldsymbol{a}_3) \rightarrow (\boldsymbol{a}_1, -\boldsymbol{a}_3, \boldsymbol{a}_2)\]

Note

The third lattice vector is multiplied by \(-1\) in some cases to preserve the handedness of the cell.

K-path#

\(\mathrm{\Gamma-X-L-T-W-R-X_1-Z-\Gamma-Y-S-W\vert L_1-Y\vert Y_1-Z}\)

\[\begin{matrix} \zeta = \dfrac{1 + a^2/c^2}{4} & \eta = \dfrac{1 + b^2/c^2}{4} & \delta = \dfrac{b^2 - a^2}{4c^2} & \mu = \dfrac{a^2 + b^2}{4c^2} \end{matrix}\]

Point	\(\times\boldsymbol{b}_1\)	\(\times\boldsymbol{b}_2\)	\(\times\boldsymbol{b}_3\)
\(\mathrm{\Gamma}\)	\(0\)	\(0\)	\(0\)
\(\mathrm{L}\)	\(-\mu\)	\(\mu\)	\(1/2 - \delta\)
\(\mathrm{L_1}\)	\(\mu\)	\(-\mu\)	\(1/2 + \delta\)
\(\mathrm{L_2}\)	\(1/2-\delta\)	\(1/2+\delta\)	\(-\mu\)
\(\mathrm{R}\)	\(0\)	\(1/2\)	\(0\)
\(\mathrm{S}\)	\(1/2\)	\(0\)	\(0\)
\(\mathrm{T}\)	\(0\)	\(0\)	\(1/2\)
\(\mathrm{W}\)	\(1/4\)	\(1/4\)	\(1/4\)
\(\mathrm{X}\)	\(-\zeta\)	\(\zeta\)	\(\zeta\)
\(\mathrm{X_1}\)	\(\zeta\)	\(1-\zeta\)	\(-\zeta\)
\(\mathrm{Y}\)	\(\eta\)	\(-\eta\)	\(\eta\)
\(\mathrm{Y_1}\)	\(1-\eta\)	\(\eta\)	\(-\eta\)
\(\mathrm{Z}\)	\(1/2\)	\(1/2\)	\(-1/2\)

Variations#

There are no variations for body-centered orthorombic. One example is predefined: orci with \(a = \pi\), \(b = 1.3\pi\) and \(c = 1.7\pi\).

Examples#

Brillouin zone and default kpath#

# RAD-tools - Sandbox (mainly condense matter plotting).
# Copyright (C) 2022-2024  Andrey Rybakov
#
# e-mail: anry@uv.es, web: rad-tools.org
#
# 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 radtools as rad

l = rad.lattice_example("ORCI")
backend = rad.PlotlyBackend()
backend.plot(l, kind="brillouin-kpath")
# Save an image:
backend.save("orci_brillouin.png")
# Interactive plot:
backend.show()

Primitive and conventional cell#

# RAD-tools - Sandbox (mainly condense matter plotting).
# Copyright (C) 2022-2024  Andrey Rybakov
#
# e-mail: anry@uv.es, web: rad-tools.org
#
# 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 radtools as rad

l = rad.lattice_example("ORCI")
backend = rad.PlotlyBackend()
backend.plot(l, kind="primitive", label="primitive")
backend.plot(l, kind="conventional", label="conventional", color="black")
# Save an image:
backend.save("orci_real.png")
# Interactive plot:
backend.show()

Wigner-Seitz cell#

# RAD-tools - Sandbox (mainly condense matter plotting).
# Copyright (C) 2022-2024  Andrey Rybakov
#
# e-mail: anry@uv.es, web: rad-tools.org
#
# 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 radtools as rad

l = rad.lattice_example("ORCI")
backend = rad.PlotlyBackend()
backend.plot(l, kind="wigner-seitz")
# Save an image:
backend.save("orci_wigner-seitz.png")
# Interactive plot:
backend.show()

Edge cases#

If \(a = b \ne c\) or \(a = c \ne b\) or \(b = c \ne a\), then the lattice is Body-centred tetragonal (BCT).

If \(a = b = c\), then the lattice is Body-centered cubic (BCC).