Triclinic (TRI)#

Pearson symbol: aP

Constructor: TRI()

It is defined by six parameters: \(a\), \(b\), \(c\) and \(\alpha\), \(\beta\), \(\gamma\). with primitive and conventional cell:

\[\begin{split}\begin{matrix} \boldsymbol{a}_1 = (a, 0, 0)\\ \boldsymbol{a}_2 = (b\cos\gamma, b\sin\gamma, 0)\\ \boldsymbol{a}_3 = (c\cos\beta, \dfrac{c(\cos\alpha - \cos\beta\cos\gamma)}{\sin\gamma}, \dfrac{c}{\sin\gamma}\sqrt{\sin^2\gamma - \cos^2\alpha - \cos^2\beta + 2\cos\alpha\cos\beta\cos\gamma}) \end{matrix}\end{split}\]

with

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

K-path#

TRI_1a#

\(\mathrm{X-\Gamma-Y\vert L-\Gamma-Z\vert N-\Gamma-M\vert R-\Gamma}\)

Point	\(\times\boldsymbol{b}_1\)	\(\times\boldsymbol{b}_2\)	\(\times\boldsymbol{b}_3\)
\(\mathrm{\Gamma}\)	\(0\)	\(0\)	\(0\)
\(\mathrm{L}\)	\(1/2\)	\(1/2\)	\(0\)
\(\mathrm{M}\)	\(0\)	\(1/2\)	\(1/2\)
\(\mathrm{N}\)	\(1/2\)	\(0\)	\(1/2\)
\(\mathrm{R}\)	\(1/2\)	\(1/2\)	\(1/2\)
\(\mathrm{X}\)	\(1/2\)	\(0\)	\(0\)
\(\mathrm{Y}\)	\(0\)	\(1/2\)	\(0\)
\(\mathrm{Z}\)	\(0\)	\(0\)	\(1/2\)

TRI_2a#

\(\mathrm{X-\Gamma-Y\vert L-\Gamma-Z\vert N-\Gamma-M\vert R-\Gamma}\)

Point	\(\times\boldsymbol{b}_1\)	\(\times\boldsymbol{b}_2\)	\(\times\boldsymbol{b}_3\)
\(\mathrm{\Gamma}\)	\(0\)	\(0\)	\(0\)
\(\mathrm{L}\)	\(1/2\)	\(1/2\)	\(0\)
\(\mathrm{M}\)	\(0\)	\(1/2\)	\(1/2\)
\(\mathrm{N}\)	\(1/2\)	\(0\)	\(1/2\)
\(\mathrm{R}\)	\(1/2\)	\(1/2\)	\(1/2\)
\(\mathrm{X}\)	\(1/2\)	\(0\)	\(0\)
\(\mathrm{Y}\)	\(0\)	\(1/2\)	\(0\)
\(\mathrm{Z}\)	\(0\)	\(0\)	\(1/2\)

TRI_1b#

\(\mathrm{X-\Gamma-Y\vert L-\Gamma-Z\vert N-\Gamma-M\vert R-\Gamma}\)

Point	\(\times\boldsymbol{b}_1\)	\(\times\boldsymbol{b}_2\)	\(\times\boldsymbol{b}_3\)
\(\mathrm{\Gamma}\)	\(0\)	\(0\)	\(0\)
\(\mathrm{L}\)	\(1/2\)	\(-1/2\)	\(0\)
\(\mathrm{M}\)	\(0\)	\(0\)	\(1/2\)
\(\mathrm{N}\)	\(-1/2\)	\(-1/2\)	\(1/2\)
\(\mathrm{R}\)	\(0\)	\(-1/2\)	\(1/2\)
\(\mathrm{X}\)	\(0\)	\(-1/2\)	\(0\)
\(\mathrm{Y}\)	\(1/2\)	\(0\)	\(0\)
\(\mathrm{Z}\)	\(-1/2\)	\(0\)	\(1/2\)

TRI_2b#

\(\mathrm{X-\Gamma-Y\vert L-\Gamma-Z\vert N-\Gamma-M\vert R-\Gamma}\)

Point	\(\times\boldsymbol{b}_1\)	\(\times\boldsymbol{b}_2\)	\(\times\boldsymbol{b}_3\)
\(\mathrm{\Gamma}\)	\(0\)	\(0\)	\(0\)
\(\mathrm{L}\)	\(1/2\)	\(-1/2\)	\(0\)
\(\mathrm{M}\)	\(0\)	\(0\)	\(1/2\)
\(\mathrm{N}\)	\(-1/2\)	\(-1/2\)	\(1/2\)
\(\mathrm{R}\)	\(0\)	\(-1/2\)	\(1/2\)
\(\mathrm{X}\)	\(0\)	\(-1/2\)	\(0\)
\(\mathrm{Y}\)	\(1/2\)	\(0\)	\(0\)
\(\mathrm{Z}\)	\(-1/2\)	\(0\)	\(1/2\)

Variations#

There are four variations of triclinic lattice.

TRI_1a#

\(k_{\alpha} > 90^{\circ}, k_{\beta} > 90^{\circ}, k_{\gamma} > 90^{\circ}, k_{\gamma} = \min(k_{\alpha}, k_{\beta}, k_{\gamma})\)

TRI_2a#

\(k_{\alpha} > 90^{\circ}, k_{\beta} > 90^{\circ}, k_{\gamma} = 90^{\circ}\)

TRI_1b#

\(k_{\alpha} < 90^{\circ}, k_{\beta} < 90^{\circ}, k_{\gamma} < 90^{\circ}, k_{\gamma} = \max(k_{\alpha}, k_{\beta}, k_{\gamma})\)

TRI_2b#

\(k_{\alpha} < 90^{\circ}, k_{\beta} < 90^{\circ}, k_{\gamma} = 90^{\circ}\)

In definition of the examples we cheated and defined them through reciprocal lattice parameters.

Examples#

TRI_1a#

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("tri1a_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("tri1a_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("tri1a_wigner-seitz.png")
# Interactive plot:
backend.show()

TRI_2a#

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("tri2a_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("tri2a_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("tri2a_wigner-seitz.png")
# Interactive plot:
backend.show()

TRI_1b#

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("tri1b_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("tri1b_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("tri1b_wigner-seitz.png")
# Interactive plot:
backend.show()

TRI_2b#

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("tri2b_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("tri2b_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("tri2b_wigner-seitz.png")
# Interactive plot:
backend.show()

Cell standardization#

Triclinic cell is unique, as standardization is performed based on the reciprocal primitive cell. As all transformations involved are described by orthonormal matrices, the reciprocal and real-space cells are transformed in a simplified manner (Note: \(\boldsymbol{S}^T = \boldsymbol{S}^{-1}\)):

\[\begin{split}(\boldsymbol{a}_1, \boldsymbol{a}_2, \boldsymbol{a}_3) &= (\boldsymbol{a}_1^s, \boldsymbol{a}_2^s, \boldsymbol{a}_3^s) \boldsymbol{S}\\ (\boldsymbol{b}_1, \boldsymbol{b}_2, \boldsymbol{b}_3) &= (\boldsymbol{b}_1^s, \boldsymbol{b}_2^s, \boldsymbol{b}_3^s) \boldsymbol{S}\end{split}\]

One of the four conditions have to be met:

\(k_{\gamma} = 90^{\circ}\) and other two angles are \(> 90^{\circ}\).
\(k_{\gamma} = 90^{\circ}\) and other two angles are \(< 90^{\circ}\).
All reciprocal cell angles (\(k_{\alpha}\), \(k_{\beta}\), \(k_{\gamma}\)) are \(> 90^{\circ}\) and \(k_{\gamma} = \min(k_{\alpha}, k_{\beta}, k_{\gamma})\).
All reciprocal cell angles (\(k_{\alpha}\), \(k_{\beta}\), \(k_{\gamma}\)) are \(< 90^{\circ}\) and \(k_{\gamma} = \max(k_{\alpha}, k_{\beta}, k_{\gamma})\).

The standardization matrix is constructed in two steps

Step 1#

In this step we ensure either of the two conditions: all angles are \(\le 90^{\circ}\) or all angles are \(\ge 90^{\circ}\).

If \(k_{\alpha} \ge \frac{\pi}{2}\) and \(k_{\beta} \ge \frac{\pi}{2}\) and \(k_{\gamma} \ge \frac{\pi}{2}\) or \(k_{\alpha} \le \frac{\pi}{2}\) and \(k_{\beta} \le \frac{\pi}{2}\) and \(k_{\gamma} \le \frac{\pi}{2}\) (i.e \(\boldsymbol{b}_2\cdot\boldsymbol{b}_3 \le 0\) and \(\boldsymbol{b}_1\cdot\boldsymbol{b}_3 \le 0\) and \(\boldsymbol{b}_1\cdot\boldsymbol{b}_2 \le 0\) or \(\boldsymbol{b}_2\cdot\boldsymbol{b}_3 \ge 0\) and \(\boldsymbol{b}_1\cdot\boldsymbol{b}_3 \ge 0\) and \(\boldsymbol{b}_1\cdot\boldsymbol{b}_2 \ge 0\), then

\[(\boldsymbol{b}_1^1, \boldsymbol{b}_2^1, \boldsymbol{b}_3^1) = (\boldsymbol{b}_1, \boldsymbol{b}_2, \boldsymbol{b}_3)\]

and

\[\begin{split}\boldsymbol{S}_1 = \boldsymbol{S}_1^{-1} = \boldsymbol{S}_1^T = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}\end{split}\]
If \(k_{\alpha} \ge \frac{\pi}{2}\) and \(k_{\beta} \ge \frac{\pi}{2}\) and \(k_{\gamma} \le \frac{\pi}{2}\) or \(k_{\alpha} \le \frac{\pi}{2}\) and \(k_{\beta} \le \frac{\pi}{2}\) and \(k_{\gamma} \ge \frac{\pi}{2}\) (i.e \(\boldsymbol{b}_2\cdot\boldsymbol{b}_3 \le 0\) and \(\boldsymbol{b}_1\cdot\boldsymbol{b}_3 \le 0\) and \(\boldsymbol{b}_1\cdot\boldsymbol{b}_2 \ge 0\) or \(\boldsymbol{b}_2\cdot\boldsymbol{b}_3 \ge 0\) and \(\boldsymbol{b}_1\cdot\boldsymbol{b}_3 \ge 0\) and \(\boldsymbol{b}_1\cdot\boldsymbol{b}_2 \le 0\), then

\[(\boldsymbol{b}_1^1, \boldsymbol{b}_2^1, \boldsymbol{b}_3^1) = (-\boldsymbol{b}_1, -\boldsymbol{b}_2, \boldsymbol{b}_3)\]

and

\[\begin{split}\boldsymbol{S}_1 = \boldsymbol{S}_1^{-1} = \boldsymbol{S}_1^T = \begin{pmatrix} -1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{pmatrix}\end{split}\]
If \(k_{\alpha} \ge \frac{\pi}{2}\) and \(k_{\beta} \le \frac{\pi}{2}\) and \(k_{\gamma} \ge \frac{\pi}{2}\) or \(k_{\alpha} \le \frac{\pi}{2}\) and \(k_{\beta} \ge \frac{\pi}{2}\) and \(k_{\gamma} \le \frac{\pi}{2}\) (i.e \(\boldsymbol{b}_2\cdot\boldsymbol{b}_3 \le 0\) and \(\boldsymbol{b}_1\cdot\boldsymbol{b}_3 \ge 0\) and \(\boldsymbol{b}_1\cdot\boldsymbol{b}_2 \le 0\) or \(\boldsymbol{b}_2\cdot\boldsymbol{b}_3 \ge 0\) and \(\boldsymbol{b}_1\cdot\boldsymbol{b}_3 \le 0\) and \(\boldsymbol{b}_1\cdot\boldsymbol{b}_2 \ge 0\), then

\[(\boldsymbol{b}_1^1, \boldsymbol{b}_2^1, \boldsymbol{b}_3^1) = (-\boldsymbol{b}_1, \boldsymbol{b}_2, -\boldsymbol{b}_3)\]

and

\[\begin{split}\boldsymbol{S}_1 = \boldsymbol{S}_1^{-1} = \boldsymbol{S}_1^T = \begin{pmatrix} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{pmatrix}\end{split}\]
If \(k_{\alpha} \le \frac{\pi}{2}\) and \(k_{\beta} \ge \frac{\pi}{2}\) and \(k_{\gamma} \ge \frac{\pi}{2}\) or \(k_{\alpha} \ge \frac{\pi}{2}\) and \(k_{\beta} \le \frac{\pi}{2}\) and \(k_{\gamma} \le \frac{\pi}{2}\) (i.e \(\boldsymbol{b}_2\cdot\boldsymbol{b}_3 \ge 0\) and \(\boldsymbol{b}_1\cdot\boldsymbol{b}_3 \le 0\) and \(\boldsymbol{b}_1\cdot\boldsymbol{b}_2 \le 0\) or \(\boldsymbol{b}_2\cdot\boldsymbol{b}_3 \le 0\) and \(\boldsymbol{b}_1\cdot\boldsymbol{b}_3 \ge 0\) and \(\boldsymbol{b}_1\cdot\boldsymbol{b}_2 \ge 0\), then

\[(\boldsymbol{b}_1^1, \boldsymbol{b}_2^1, \boldsymbol{b}_3^1) = (\boldsymbol{b}_1, -\boldsymbol{b}_2, -\boldsymbol{b}_3)\]

and

\[\begin{split}\boldsymbol{S}_1 = \boldsymbol{S}_1^{-1} = \boldsymbol{S}_1^T = \begin{pmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{pmatrix}\end{split}\]

Step 2#

At this step we ensure that \(k_{\gamma}\) is an appropriate extremum.

If \(k_{\gamma} = min(k_{\alpha}, k_{\beta}, k_{\gamma})\) and \(k_{\gamma} \ge \frac{\pi}{2}\) or \(k_{\gamma} = max(k_{\alpha}, k_{\beta}, k_{\gamma})\) and \(k_{\gamma} \le \frac{\pi}{2}\) (i.e. \(k_{\gamma} = min(k_{\alpha}, k_{\beta}, k_{\gamma})\) and \(\boldsymbol{b}_1\cdot\boldsymbol{b}_2 \le 0\) or \(k_{\gamma} = max(k_{\alpha}, k_{\beta}, k_{\gamma})\) and \(\boldsymbol{b}_1\cdot\boldsymbol{b}_2 \ge 0\), then

\[(\boldsymbol{b}_1^s, \boldsymbol{b}_2^s, \boldsymbol{b}_3^s) = (\boldsymbol{b}_1^1, \boldsymbol{b}_2^1, \boldsymbol{b}_3^1)\]

and

\[\begin{split}\boldsymbol{S}_2 = \boldsymbol{S}_2^{-1} = \boldsymbol{S}_2^T = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}\end{split}\]
If \(k_{\beta} = min(k_{\alpha}, k_{\beta}, k_{\gamma})\) and \(k_{\beta} \ge \frac{\pi}{2}\) or \(k_{\beta} = max(k_{\alpha}, k_{\beta}, k_{\gamma})\) and \(k_{\beta} \le \frac{\pi}{2}\) (i.e. \(k_{\beta} = min(k_{\alpha}, k_{\beta}, k_{\gamma})\) and \(\boldsymbol{b}_1\cdot\boldsymbol{b}_3 \le 0\) or \(k_{\beta} = max(k_{\alpha}, k_{\beta}, k_{\gamma})\) and \(\boldsymbol{b}_1\cdot\boldsymbol{b}_3 \ge 0\), then

\[(\boldsymbol{b}_1^s, \boldsymbol{b}_2^s, \boldsymbol{b}_3^s) = (\boldsymbol{b}_3^1, \boldsymbol{b}_1^1, \boldsymbol{b}_2^1)\]

and

\[\begin{split}\boldsymbol{S}_2 = \begin{pmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix} \qquad \boldsymbol{S}_2^{-1} = \boldsymbol{S}_2^T = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{pmatrix}\end{split}\]
If \(k_{\alpha} = min(k_{\alpha}, k_{\beta}, k_{\gamma})\) and \(k_{\alpha} \ge \frac{\pi}{2}\) or \(k_{\alpha} = max(k_{\alpha}, k_{\beta}, k_{\gamma})\) and \(k_{\alpha} \le \frac{\pi}{2}\) (i.e. \(k_{\alpha} = min(k_{\alpha}, k_{\beta}, k_{\gamma})\) and \(\boldsymbol{b}_2\cdot\boldsymbol{b}_3 \le 0\) or \(k_{\alpha} = max(k_{\alpha}, k_{\beta}, k_{\gamma})\) and \(\boldsymbol{b}_2\cdot\boldsymbol{b}_3 \ge 0\), then

\[(\boldsymbol{b}_1^s, \boldsymbol{b}_2^s, \boldsymbol{b}_3^s) = (\boldsymbol{b}_2^1, \boldsymbol{b}_3^1, \boldsymbol{b}_1^1)\]

and

\[\begin{split}\boldsymbol{S}_2 = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{pmatrix} \qquad \boldsymbol{S}_2^{-1} = \boldsymbol{S}_2^T = \begin{pmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix}\end{split}\]

Finally#

\[\boldsymbol{S} = \boldsymbol{S}_2 \boldsymbol{S}_1 \qquad \boldsymbol{S}^{-1} = \boldsymbol{S}_1^{-1} \boldsymbol{S}_2^{-1}\]

Triclinic (TRI)#

K-path#

TRI1a#

TRI2a#

TRI1b#

TRI2b#

Variations#

TRI1a#

TRI2a#

TRI1b#

TRI2b#

Examples#

TRI1a#

Brillouin zone and default kpath#

Primitive and conventional cell#

Wigner-Seitz cell#

TRI2a#

Brillouin zone and default kpath#

Primitive and conventional cell#

Wigner-Seitz cell#

TRI1b#

Brillouin zone and default kpath#

Primitive and conventional cell#

Wigner-Seitz cell#

TRI2b#

Brillouin zone and default kpath#

Primitive and conventional cell#

Wigner-Seitz cell#

Cell standardization#

Step 1#

Step 2#

Finally#

This Page

TRI_1a#

TRI_2a#

TRI_1b#

TRI_2b#

TRI_1a#

TRI_2a#

TRI_1b#

TRI_2b#

TRI_1a#

TRI_2a#

TRI_1b#

TRI_2b#