Body-centred tetragonal (BCT)#

Pearson symbol: tI

Constructor: BCT()

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

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

And primitive cell:

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

with

\[\begin{split}\boldsymbol{C} = \dfrac{1}{2} \begin{pmatrix} -1 & 1 & 1 \\ 1 & -1 & 1 \\ 1 & 1 & -1 \end{pmatrix} \qquad \boldsymbol{C}^{-1} = \begin{pmatrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{pmatrix}\end{split}\]

Order of parameters: \(a \ne c\)

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#

# 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("bct1_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", 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#

# 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("bct1_wigner-seitz.png")
# Interactive plot:
backend.show()

BCT₂#

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("bct2_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", 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#

# 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("bct2_wigner-seitz.png")
# Interactive plot:
backend.show()

Cell standardization#

Condition \(a \ne c\) result in the condition \(\alpha = \beta \ne \gamma\) for the primitive cell in a standard form, in practice this condition simplifies to \(\boldsymbol{a}_2^s\cdot\boldsymbol{a}_3^s = \boldsymbol{a}_1^s\cdot\boldsymbol{a}_3^s \ne \boldsymbol{a}_1^s\cdot\boldsymbol{a}_2^s\) for the primitive cell in a standard form. We use angles of the primitive cell for standardization.

If \(\alpha = \beta \ne \gamma\) (i.e. \(\boldsymbol{a}_2\cdot\boldsymbol{a}_3 = \boldsymbol{a}_1\cdot\boldsymbol{a}_3 \ne \boldsymbol{a}_1\cdot\boldsymbol{a}_2\)), 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 \(\beta = \gamma \ne \alpha\) (i.e. \(\boldsymbol{a}_1\cdot\boldsymbol{a}_3 = \boldsymbol{a}_1\cdot\boldsymbol{a}_2 \ne \boldsymbol{a}_2\cdot\boldsymbol{a}_3\)), 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 \(\alpha = \gamma \ne \beta\) (i.e. \(\boldsymbol{a}_2\cdot\boldsymbol{a}_3 = \boldsymbol{a}_1\cdot\boldsymbol{a}_2 \ne \boldsymbol{a}_1\cdot\boldsymbol{a}_3\)), then

\[(\boldsymbol{a}_1^s, \boldsymbol{a}_2^s, \boldsymbol{a}_3^s) = (\boldsymbol{a}_3, \boldsymbol{a}_1, \boldsymbol{a}_2)\]

and

\[\begin{split}\boldsymbol{S} = \begin{pmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix} \qquad \boldsymbol{S}^{-1} = \boldsymbol{S}^T = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{pmatrix}\end{split}\]

Edge cases#

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

Body-centred tetragonal (BCT)#

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#

Cell standardization#

Edge cases#

This Page

BCT₁#

BCT₂#

BCT₁#

BCT₂#

BCT₁#

BCT₂#