Tetragonal (TET)

Pearson symbol: tP

Constructor: TET()

It is defined by two parameters: \(a\) and \(c\) with primitive and conventional lattice:

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

Cell standardization

Length of third lattice vector has to be different from the first two. If this condition is not satisfied, then the lattice is transformed to the standard form:

• If \(\vert\boldsymbol{a}_1\vert = \vert\boldsymbol{a}_3\vert \ne \vert\boldsymbol{a}_2\vert\)

  \[(\boldsymbol{a}_1, \boldsymbol{a}_2, \boldsymbol{a}_3) \rightarrow (\boldsymbol{a}_3, \boldsymbol{a}_1, \boldsymbol{a}_2)\]

• If \(\vert\boldsymbol{a}_2\vert = \vert\boldsymbol{a}_3\vert \ne \vert\boldsymbol{a}_1\vert\)

  \[(\boldsymbol{a}_1, \boldsymbol{a}_2, \boldsymbol{a}_3) \rightarrow (\boldsymbol{a}_2, \boldsymbol{a}_3, \boldsymbol{a}_1)\]

K-path

\(\mathrm{\Gamma-X-M-\Gamma-Z-R-A-Z\vert X-R\vert M-A}\)

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

Variations

There are no variations for tetragonal lattice. One example is predefined: tet with \(a = \pi\) and \(c = 1.5\pi\)

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("TET")
backend = rad.PlotlyBackend()
backend.plot(l, kind="brillouin-kpath")
# Save an image:
backend.save("tet_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("TET")
backend = rad.PlotlyBackend()
backend.plot(l, kind="primitive")
# Save an image:
backend.save("tet_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("TET")
backend = rad.PlotlyBackend()
backend.plot(l, kind="wigner-seitz")
# Save an image:
backend.save("tet_wigner-seitz.png")
# Interactive plot:
backend.show()

Edge cases

If \(a = c\), then the lattice is Cubic (CUB).