Monoclinic (MCL)#
Pearson symbol: mP
Constructor: MCL()
It is defined by four parameter: \(a\), \(b\), \(c\) and \(\alpha\) with primitive and conventional lattice:
Order of parameters: \(b \le c\), \(\alpha < 90^{\circ}\).
Cell standardization#
Angles between first and third lattice vectors (\(\beta\)) and between first and second lattice vectors (\(\gamma\)) have to be equal to \(90^{\circ}\).
Angle between second and third lattice vectors (\(\alpha\)) has to be less then \(90^{\circ}\).
Length of the second lattice vector has to be less or equal to the length of the third lattice vector.
If these conditions are not satisfied, then the lattice is transformed to the standard form:
First we ensure the \(\beta = 90^{\circ}\) and \(\gamma = 90^{\circ}\):
- If \(\beta \ne 90^{\circ}\):
- \[(\boldsymbol{a}_1, \boldsymbol{a}_2, \boldsymbol{a}_3) \rightarrow (\boldsymbol{a}_2, \boldsymbol{a}_3, \boldsymbol{a}_1)\]
- If \(\gamma \ne 90^{\circ}\):
- \[(\boldsymbol{a}_1, \boldsymbol{a}_2, \boldsymbol{a}_3) \rightarrow (\boldsymbol{a}_3, \boldsymbol{a}_1, \boldsymbol{a}_2)\]
Then we ensure the \(\alpha < 90^{\circ}\):
- If \(\alpha > 90^{\circ}\):
- \[(\boldsymbol{a}_1, \boldsymbol{a}_2, \boldsymbol{a}_3) \rightarrow (\boldsymbol{a}_1, \boldsymbol{a}_3, -\boldsymbol{a}_2)\]
Finally, we ensure the \(b \le c\):
- If \(b > c\):
- \[(\boldsymbol{a}_1, \boldsymbol{a}_2, \boldsymbol{a}_3) \rightarrow (-\boldsymbol{a}_1, \boldsymbol{a}_3, \boldsymbol{a}_2)\]
Note
First and second lattice vectors are multiplied by \(-1\) in some cases to preserve the handedness of the cell.
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 two variations for monoclinic lattice.
One example is predefined: mcl with
MCL(pi, 1.3 * pi, 1.6 * pi, alpha=75)
\(a = \pi\), \(b = 1.3 \pi\) \(c = 1.6 \pi\) and \(\alpha = 75^{\circ}\).
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("MCL")
backend = rad.PlotlyBackend()
backend.plot(l, kind="brillouin-kpath")
# Save an image:
backend.save("mcl_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("MCL")
backend = rad.PlotlyBackend()
backend.plot(l, kind="primitive")
# Save an image:
backend.save("mcl_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("MCL")
backend = rad.PlotlyBackend()
backend.plot(l, kind="wigner-seitz")
# Save an image:
backend.save("mcl_wigner-seitz.png")
# Interactive plot:
backend.show()
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).