Body-centered cubic (BCC)

Pearson symbol: cI

Constructor: BCC()

It is defined by one parameter: \(a\) with conventional cell:

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

And primitive cell:

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

with

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

K-path

\(\mathrm{\Gamma-H-N-\Gamma-P-H\vert P-N}\)

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

Variations

There are no variations for body-centered cubic lattice. One example is predefined: bcc with \(a = \pi\).

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

Cell standardization

No standardization required.

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