Body-centred tetragonal (BCT)#

Pearson symbol: tI

Constructor: BCT()

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

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

And primitive lattice:

\[\begin{split}\begin{matrix} \boldsymbol{a}_1 &=& (-a/2, &a/2, &c/2)\\ \boldsymbol{a}_2 &=& (a/2, &-a/2, &c/2)\\ \boldsymbol{a}_3 &=& (a/2, &a/2, &-c/2) \end{matrix}\end{split}\]

Cell standardization#

Length of third lattice vector of the conventional cell 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#

BCT₁#

\(\mathrm{\Gamma-X-M-\Gamma-Z-P-N-Z_1-M\vert X-P}\)

\[\eta = \dfrac{1 + c^2/a^2}{4}\]

Point	\(\times\boldsymbol{b}_1\)	\(\times\boldsymbol{b}_2\)	\(\times\boldsymbol{b}_3\)
\(\mathrm{\Gamma}\)	\(0\)	\(0\)	\(0\)
\(\mathrm{M}\)	\(-1/2\)	\(1/2\)	\(1/2\)
\(\mathrm{N}\)	\(0\)	\(1/2\)	\(0\)
\(\mathrm{P}\)	\(1/4\)	\(1/4\)	\(1/4\)
\(\mathrm{X}\)	\(0\)	\(0\)	\(1/2\)
\(\mathrm{Z}\)	\(\eta\)	\(\eta\)	\(-\eta\)
\(\mathrm{Z}_1\)	\(-\eta\)	\(1-\eta\)	\(\eta\)

BCT₂#

\(\mathrm{\Gamma-X-Y-\Sigma-\Gamma-Z-\Sigma_1-N-P-Y_1-Z\vert X-P}\)

\[\begin{matrix} \eta = \dfrac{1 + a^2/c^2}{4} & \zeta = \dfrac{a^2}{2c^2} \end{matrix}\]

Point	\(\times\boldsymbol{b}_1\)	\(\times\boldsymbol{b}_2\)	\(\times\boldsymbol{b}_3\)
\(\mathrm{\Gamma}\)	\(0\)	\(0\)	\(0\)
\(\mathrm{N}\)	\(0\)	\(1/2\)	\(0\)
\(\mathrm{P}\)	\(1/4\)	\(1/4\)	\(1/4\)
\(\mathrm{\Sigma}\)	\(-\eta\)	\(\eta\)	\(\eta\)
\(\mathrm{\Sigma_1}\)	\(\eta\)	\(1-\eta\)	\(-\eta\)
\(\mathrm{X}\)	\(0\)	\(0\)	\(1/2\)
\(\mathrm{Y}\)	\(-\zeta\)	\(\zeta\)	\(1/2\)
\(\mathrm{Y}_1\)	\(1/2\)	\(1/2\)	\(-\zeta\)
\(\mathrm{Z}\)	\(1/2\)	\(1/2\)	\(-1/2\)

Variations#

There are two variations of body-centered tetragonal lattice.

BCT₁#

\(c < a\).

Predefined example: bct1 with \(a = 1.5\pi\) and \(c = \pi\).

BCT₂#

\(c > a\).

Predefined example: bct2 with \(a = \pi\) and \(c = 1.5\pi\).

Examples#

BCT₁#

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("BCT1")
backend = rad.PlotlyBackend()
backend.plot(l, kind="brillouin-kpath")
# Save an image:
backend.save("bct1_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("BCT1")
backend = rad.PlotlyBackend()
backend.plot(l, kind="primitive", label="primitive")
backend.plot(l, kind="conventional", label="conventional", color="black")
# Save an image:
backend.save("bct1_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("BCT1")
backend = rad.PlotlyBackend()
backend.plot(l, kind="wigner-seitz")
# Save an image:
backend.save("bct1_wigner-seitz.png")
# Interactive plot:
backend.show()

BCT₂#

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("BCT2")
backend = rad.PlotlyBackend()
backend.plot(l, kind="brillouin-kpath")
# Save an image:
backend.save("bct2_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("BCT2")
backend = rad.PlotlyBackend()
backend.plot(l, kind="primitive", label="primitive")
backend.plot(l, kind="conventional", label="conventional", color="black")
# Save an image:
backend.save("bct2_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("BCT2")
backend = rad.PlotlyBackend()
backend.plot(l, kind="wigner-seitz")
# Save an image:
backend.save("bct2_wigner-seitz.png")
# Interactive plot:
backend.show()

Edge cases#

If \(a = c\) then the lattice is Body-centered cubic (BCC).

Body-centred tetragonal (BCT)#

Cell standardization#

K-path#

BCT1#

BCT2#

Variations#

BCT1#

BCT2#

Examples#

BCT1#

Brillouin zone and default kpath#

Primitive and conventional cell#

Wigner-Seitz cell#

BCT2#

Brillouin zone and default kpath#

Primitive and conventional cell#

Wigner-Seitz cell#

Edge cases#

This Page

BCT₁#

BCT₂#

BCT₁#

BCT₂#

BCT₁#

BCT₂#