Body-centred tetragonal (BCT)

Pearson symbol: tI

Constructor: BCT()

It is defined by two parameters \(a\) and \(c\) with \(a \ne c\). Standardized primitive and conventional cells in the default orientation are

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

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

Transformation matrix from standardized primitive cell to standardized conventional cell is

\[\begin{split}\boldsymbol{C} = \begin{pmatrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \\ \end{pmatrix} \qquad \boldsymbol{C}^{-1} = \begin{pmatrix} -0.5 & 0.5 & 0.5 \\ 0.5 & -0.5 & 0.5 \\ 0.5 & 0.5 & -0.5 \\ \end{pmatrix}\end{split}\]

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

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

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

import wulfric as wulf

cell = wulf.cell.get_cell_example("BCT1")
backend = wulf.visualization.PlotlyBackend()
backend.plot(cell, kind="brillouin-kpath")
# Save an image:
backend.save("bct1_reciprocal.png")
# Interactive plot:
backend.show()

Primitive, Wigner-Seitz and conventional cells

Click on the legend to hide a cell.

import wulfric as wulf

cell = wulf.cell.get_cell_example("BCT1")
backend = wulf.visualization.PlotlyBackend()
backend.plot(cell, kind="primitive", label="primitive", color="black")
backend.plot(cell, kind="conventional", label="conventional", color="blue")
backend.plot(cell, kind="wigner-seitz", label="wigner-seitz", color="green")
# Save an image:
backend.save("bct1_real.png")
# Interactive plot:
backend.show()

BCT₂

Brillouin zone and default kpath

import wulfric as wulf

cell = wulf.cell.get_cell_example("BCT2")
backend = wulf.visualization.PlotlyBackend()
backend.plot(cell, kind="brillouin-kpath")
# Save an image:
backend.save("bct2_reciprocal.png")
# Interactive plot:
backend.show()

Primitive, Wigner-Seitz and conventional cells

Click on the legend to hide a cell.

import wulfric as wulf

cell = wulf.cell.get_cell_example("BCT2")
backend = wulf.visualization.PlotlyBackend()
backend.plot(cell, kind="primitive", label="primitive", color="black")
backend.plot(cell, kind="conventional", label="conventional", color="blue")
backend.plot(cell, kind="wigner-seitz", label="wigner-seitz", color="green")
# Save an image:
backend.save("bct2_real.png")
# Interactive plot:
backend.show()

Cell standardization

Condition \(a \ne c\) result in the condition \(\alpha^s = \beta^s \ne \gamma^s\) for the primitive cell in a standard form. Therefore, wulfric uses angles of the primitive cell for standardization.

• If \(\alpha = \beta \ne \gamma\) then

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

  and

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

• If \(\beta = \gamma \ne \alpha\) then

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

  and

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

• If \(\alpha = \gamma \ne \beta\) then

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

  and

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

Edge cases

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