Tetragonal (TET)

Pearson symbol: tP

Constructor: TET()

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

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

Transformation matrix from standardized primitive cell to standardized conventional cell is

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

K-path

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

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

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

import wulfric as wulf

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

Primitive and Wigner-Seitz cells

Click on the legend to hide a cell.

import wulfric as wulf

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

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 \(\vert\boldsymbol{a}_1\vert = \vert\boldsymbol{a}_2\vert \ne \vert\boldsymbol{a}_3\vert\), 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 \(\vert\boldsymbol{a}_2\vert = \vert\boldsymbol{a}_3\vert \ne \vert\boldsymbol{a}_1\vert\), 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 \(\vert\boldsymbol{a}_1\vert = \vert\boldsymbol{a}_3\vert \ne \vert\boldsymbol{a}_2\vert\), 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 Cubic (CUB).