Hexagonal (HEX)#
Pearson symbol: hP
Constructor: HEX()
It is defined by two parameter: \(a\) and \(c\) with primitive and conventional cell:
with
K-path#
\(\mathrm{\Gamma-M-K-\Gamma-A-L-H-A\vert L-M\vert K-H}\)
Point |
\(\times\boldsymbol{b}_1\) |
\(\times\boldsymbol{b}_2\) |
\(\times\boldsymbol{b}_3\) |
|---|---|---|---|
\(\mathrm{\Gamma}\) |
\(0\) |
\(0\) |
\(0\) |
\(\mathrm{A}\) |
\(0\) |
\(0\) |
\(1/2\) |
\(\mathrm{H}\) |
\(1/3\) |
\(1/3\) |
\(1/2\) |
\(\mathrm{K}\) |
\(1/3\) |
\(1/3\) |
\(0\) |
\(\mathrm{L}\) |
\(1/2\) |
\(0\) |
\(1/2\) |
\(\mathrm{M}\) |
\(1/2\) |
\(0\) |
\(0\) |
Variations#
There are no variations for hexagonal lattice.
One example is predefined: hex with \(a = \pi\) 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("hex_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("hex_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("hex_wigner-seitz.png")
# Interactive plot:
backend.show()
Cell standardization#
Since parameters \(a\) and \(c\) are not restricted (i.e. \(a = c\) is allowed), we use angles \(\alpha\), \(\beta\) and \(\gamma\) to determine the standard form of the cell. For the primitive cell in a standard form \(\alpha = \beta = 90^{\circ}\) and \(\gamma = 120^{\circ}\). In practice these conditions are equivalent to \(\boldsymbol{a}_2 \cdot \boldsymbol{a}_3 = \boldsymbol{a}_1 \cdot \boldsymbol{a}_3 = 0\) and \(\boldsymbol{a}_1 \cdot \boldsymbol{a}_2 < 0\).
If \(\alpha = \beta = \pi\) and \(\gamma = \frac{2\pi}{3}\) (i.e. \(\boldsymbol{a}_2 \cdot \boldsymbol{a}_3 = \boldsymbol{a}_1 \cdot \boldsymbol{a}_3 = 0\) and \(\boldsymbol{a}_1 \cdot \boldsymbol{a}_2 < 0\)), 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 \(\beta = \gamma = \pi\) and \(\alpha = \frac{2\pi}{3}\) (i.e. \(\boldsymbol{a}_1 \cdot \boldsymbol{a}_3 = \boldsymbol{a}_1 \cdot \boldsymbol{a}_2 = 0\) and \(\boldsymbol{a}_2 \cdot \boldsymbol{a}_3 < 0\)), 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 \(\alpha = \gamma = \pi\) and \(\beta = \frac{2\pi}{3}\) (i.e. \(\boldsymbol{a}_2 \cdot \boldsymbol{a}_3 = \boldsymbol{a}_1 \cdot \boldsymbol{a}_2 = 0\) and \(\boldsymbol{a}_1 \cdot \boldsymbol{a}_3 < 0\)), then
\[(\boldsymbol{a}_1^s, \boldsymbol{a}_2^s, \boldsymbol{a}_3^s) = (\boldsymbol{a}_3, \boldsymbol{a}_1, \boldsymbol{a}_2)\]and
\[\begin{split}\boldsymbol{S} = \begin{pmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix} \qquad \boldsymbol{S}^{-1} = \boldsymbol{S}^T = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{pmatrix}\end{split}\]