Monoclinic (MCL)#
Pearson symbol: mP
Constructor: MCL()
It is defined by four parameter: \(a\), \(b\), \(c\) and \(\alpha\) with primitive and conventional cell:
with
Order of parameters: \(b \le c\), \(\alpha < 90^{\circ}\).
K-path#
\(\mathrm{\Gamma-Y-H-C-E-M_1-A-X-H_1\vert M-D-Z\vert Y-D}\)
Point |
\(\times\boldsymbol{b}_1\) |
\(\times\boldsymbol{b}_2\) |
\(\times\boldsymbol{b}_3\) |
|---|---|---|---|
\(\mathrm{\Gamma}\) |
\(0\) |
\(0\) |
\(0\) |
\(\mathrm{A}\) |
\(1/2\) |
\(1/2\) |
\(0\) |
\(\mathrm{C}\) |
\(0\) |
\(1/2\) |
\(1/2\) |
\(\mathrm{D}\) |
\(1/2\) |
\(0\) |
\(1/2\) |
\(\mathrm{D_1}\) |
\(1/2\) |
\(0\) |
\(-1/2\) |
\(\mathrm{E}\) |
\(1/2\) |
\(1/2\) |
\(1/2\) |
\(\mathrm{H}\) |
\(0\) |
\(\eta\) |
\(1-\nu\) |
\(\mathrm{H_1}\) |
\(0\) |
\(1-\eta\) |
\(\nu\) |
\(\mathrm{H_2}\) |
\(0\) |
\(\eta\) |
\(-\nu\) |
\(\mathrm{M}\) |
\(1/2\) |
\(\eta\) |
\(1-\nu\) |
\(\mathrm{M_1}\) |
\(1/2\) |
\(1-\eta\) |
\(\nu\) |
\(\mathrm{M_2}\) |
\(1/2\) |
\(\eta\) |
\(-\nu\) |
\(\mathrm{X}\) |
\(0\) |
\(1/2\) |
\(0\) |
\(\mathrm{Y}\) |
\(0\) |
\(0\) |
\(1/2\) |
\(\mathrm{Y_1}\) |
\(0\) |
\(0\) |
\(-1/2\) |
\(\mathrm{Z}\) |
\(1/2\) |
\(0\) |
\(0\) |
Variations#
There are no variations for monoclinic lattice. One example is predefined: mcl with
\(a = \pi\), \(b = 1.3 \pi\) \(c = 1.6 \pi\) and \(\alpha = 75^{\circ}\).
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("mcl_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("mcl_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("mcl_wigner-seitz.png")
# Interactive plot:
backend.show()
Cell standardization#
Conditions \(b \le c\) and \(\alpha < 90^{\circ}\) are checked directly. Matrix \(\boldsymbol{S}\) is constructed in three steps.
Step 1#
If \(\beta = \gamma = \frac{\pi}{2}\) and \(\alpha \ne \frac{\pi}{2}\) (i.e. \(\boldsymbol{a}_1\cdot\boldsymbol{a}_3 = \boldsymbol{a}_1\cdot\boldsymbol{a}_2 = 0\) and \(\boldsymbol{a}_2\cdot\boldsymbol{a}_3 \ne 0\), then
\[(\boldsymbol{a}_1^1, \boldsymbol{a}_2^1, \boldsymbol{a}_3^1) = (\boldsymbol{a}_1, \boldsymbol{a}_2, \boldsymbol{a}_3)\]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 \(\alpha = \gamma = \frac{\pi}{2}\) and \(\beta \ne \frac{\pi}{2}\) (i.e. \(\boldsymbol{a}_2\cdot\boldsymbol{a}_3 = \boldsymbol{a}_1\cdot\boldsymbol{a}_2 = 0\) and \(\boldsymbol{a}_1\cdot\boldsymbol{a}_3 \ne 0\), then
\[(\boldsymbol{a}_1^1, \boldsymbol{a}_2^1, \boldsymbol{a}_3^1) = (\boldsymbol{a}_2, \boldsymbol{a}_3, \boldsymbol{a}_1)\]and
\[\begin{split}\boldsymbol{S}_1 = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{pmatrix} \qquad \boldsymbol{S}_1^{-1} = \boldsymbol{S}_1^T = \begin{pmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix}\end{split}\]If \(\alpha = \beta = \frac{\pi}{2}\) and \(\gamma \ne \frac{\pi}{2}\) (i.e. \(\boldsymbol{a}_2\cdot\boldsymbol{a}_3 = \boldsymbol{a}_1\cdot\boldsymbol{a}_3 = 0\) and \(\boldsymbol{a}_1\cdot\boldsymbol{a}_2 \ne 0\), then
\[(\boldsymbol{a}_1^1, \boldsymbol{a}_2^1, \boldsymbol{a}_3^1) = (\boldsymbol{a}_3, \boldsymbol{a}_1, \boldsymbol{a}_2)\]and
\[\begin{split}\boldsymbol{S}_1 = \begin{pmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix} \qquad \boldsymbol{S}_1^{-1} = \boldsymbol{S}_1^T = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{pmatrix}\end{split}\]
Step 2#
If \(\vert \boldsymbol{a}_2^1 \vert \le \vert \boldsymbol{a}_3^1 \vert\), then
\[(\boldsymbol{a}_1^2, \boldsymbol{a}_2^2, \boldsymbol{a}_3^2) = (\boldsymbol{a}_1^1, \boldsymbol{a}_2^1, \boldsymbol{a}_3^1)\]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 \(\vert \boldsymbol{a}_2^1 \vert > \vert \boldsymbol{a}_3^1 \vert\), then
\[(\boldsymbol{a}_1^2, \boldsymbol{a}_2^2, \boldsymbol{a}_3^2) = (-\boldsymbol{a}_1^1, \boldsymbol{a}_3^1, \boldsymbol{a}_2^1)\]and
\[\begin{split}\boldsymbol{S}_2 = \begin{pmatrix} -1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix} \qquad \boldsymbol{S}_2^{-1} = \boldsymbol{S}_2^T = \begin{pmatrix} -1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix}\end{split}\]
Step 3#
If \(\alpha^2 < \frac{\pi}{2}\) (i.e \(\boldsymbol{a}_2 \cdot \boldsymbol{a}_3 > 0\)), then
\[(\boldsymbol{a}_1^s, \boldsymbol{a}_2^s, \boldsymbol{a}_3^s) = (\boldsymbol{a}_1^2, \boldsymbol{a}_2^2, \boldsymbol{a}_3^2)\]and
\[\begin{split}\boldsymbol{S}_3 = \boldsymbol{S}_3^{-1} = \boldsymbol{S}_3^T = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}\end{split}\]If \(\alpha^2 > \frac{\pi}{2}\) (i.e \(\boldsymbol{a}_2 \cdot \boldsymbol{a}_3 < 0\)), then
\[(\boldsymbol{a}_1^s, \boldsymbol{a}_2^s, \boldsymbol{a}_3^s) = (-\boldsymbol{a}_1^2, -\boldsymbol{a}_2^2, \boldsymbol{a}_3^2)\]and
\[\begin{split}\boldsymbol{S}_3 = \begin{pmatrix} -1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \qquad \boldsymbol{S}_3^{-1} = \boldsymbol{S}_3^T = \begin{pmatrix} -1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{pmatrix}\end{split}\]
Finally#
Note
All changes of the cell preserve handiness of the original one.
Edge cases#
If (\(\alpha = 60^{\circ}\) or \(\alpha = 120^{\circ}\)) and \(b = c\), then the lattice is Hexagonal (HEX).
If (\(\alpha = 30^{\circ}\) or \(\alpha = 150^{\circ}\) or \(\alpha = 45^{\circ}\) or \(\alpha = 145^{\circ}\)) and \(b = c\), then the lattice is Base-centred orthorhombic (ORCC).
If (\(\alpha = 60^{\circ}\) or \(\alpha = 120^{\circ}\)) and \(a \ne b = c/2\), then the lattice is Orthorhombic (ORC).
If \(a \ne b \ne c\) and \(\alpha = 90^{\circ}\), then the lattice is Orthorhombic (ORC).
If (\(\alpha = 60^{\circ}\) or \(\alpha = 120^{\circ}\)) and \(a = b = c/2\), then the lattice is Tetragonal (TET).
If (\(a = b \ne c\) or \(a = c \ne b\) or \(b = c \ne a\)) and \(\alpha = 90^{\circ}\), then the lattice is Tetragonal (TET).
If \(a = b = c\) and \(\alpha = 90^{\circ}\), then the lattice is Cubic (CUB).