Body-centred orthorhombic (ORCI)#
Pearson symbol: oI
Constructor: ORCI()
It is defined by three parameters \(a\), \(b\) and \(c\) with \(a < b < c\). Standardized primitive and conventional cells in the default orientation are
Transformation matrix from standardized primitive cell to standardized conventional cell is
K-path#
\(\mathrm{\Gamma-X-L-T-W-R-X_1-Z-\Gamma-Y-S-W\vert L_1-Y\vert Y_1-Z}\)
Point |
\(\times\boldsymbol{b}_1^s\) |
\(\times\boldsymbol{b}_2^s\) |
\(\times\boldsymbol{b}_3^s\) |
|---|---|---|---|
\(\mathrm{\Gamma}\) |
\(0\) |
\(0\) |
\(0\) |
\(\mathrm{L}\) |
\(-\mu\) |
\(\mu\) |
\(1/2 - \delta\) |
\(\mathrm{L_1}\) |
\(\mu\) |
\(-\mu\) |
\(1/2 + \delta\) |
\(\mathrm{L_2}\) |
\(1/2-\delta\) |
\(1/2+\delta\) |
\(-\mu\) |
\(\mathrm{R}\) |
\(0\) |
\(1/2\) |
\(0\) |
\(\mathrm{S}\) |
\(1/2\) |
\(0\) |
\(0\) |
\(\mathrm{T}\) |
\(0\) |
\(0\) |
\(1/2\) |
\(\mathrm{W}\) |
\(1/4\) |
\(1/4\) |
\(1/4\) |
\(\mathrm{X}\) |
\(-\zeta\) |
\(\zeta\) |
\(\zeta\) |
\(\mathrm{X_1}\) |
\(\zeta\) |
\(1-\zeta\) |
\(-\zeta\) |
\(\mathrm{Y}\) |
\(\eta\) |
\(-\eta\) |
\(\eta\) |
\(\mathrm{Y_1}\) |
\(1-\eta\) |
\(\eta\) |
\(-\eta\) |
\(\mathrm{Z}\) |
\(1/2\) |
\(1/2\) |
\(-1/2\) |
Variations#
There are no variations for body-centered orthorombic.
One example is predefined: orci with
\(a = \pi\), \(b = 1.3\pi\) and \(c = 1.7\pi\).
Examples#
Brillouin zone and default kpath#
import wulfric as wulf
cell = wulf.cell.get_cell_example("ORCI")
backend = wulf.visualization.PlotlyBackend()
backend.plot(cell, kind="brillouin-kpath")
# Save an image:
backend.save("orci_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("ORCI")
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("orci_real.png")
# Interactive plot:
backend.show()
Cell standardization#
Condition \(a < b < c\) implies \(\gamma^s < \beta^s < \alpha^s\) for the primitive cell in a standard form. Therefore, wulfric uses angles of the primitive cell for standardization.
If \(\gamma < \beta < \alpha\), 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 \(\gamma < \alpha < \beta\), then
\[(\boldsymbol{a}_1^s, \boldsymbol{a}_2^s, \boldsymbol{a}_3^s) = (-\boldsymbol{a}_2, -\boldsymbol{a}_1, -\boldsymbol{a}_3)\]and
\[\begin{split}\boldsymbol{S} = \begin{pmatrix} 0 & -1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & -1 \end{pmatrix} \qquad \boldsymbol{S}^{-1} = \begin{pmatrix} 0 & -1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & -1 \end{pmatrix}\end{split}\]If \(\beta < \gamma < \alpha\), then
\[(\boldsymbol{a}_1^s, \boldsymbol{a}_2^s, \boldsymbol{a}_3^s) = (-\boldsymbol{a}_1, -\boldsymbol{a}_3, -\boldsymbol{a}_2)\]and
\[\begin{split}\boldsymbol{S} = \begin{pmatrix} -1 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & -1 & 0 \end{pmatrix} \qquad \boldsymbol{S}^{-1} = \begin{pmatrix} -1 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & -1 & 0 \end{pmatrix}\end{split}\]If \(\beta < \alpha < \gamma\), then
\[(\boldsymbol{a}_1^s, \boldsymbol{a}_2^s, \boldsymbol{a}_3^s) = (\boldsymbol{a}_3, \boldsymbol{a}_1, \boldsymbol{a}_2)\]and
\[\begin{split}\boldsymbol{S} = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{pmatrix} \qquad \boldsymbol{S}^{-1} = \begin{pmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix}\end{split}\]If \(\alpha < \gamma < \beta\), 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 \(\alpha < \beta < \gamma\), then
\[(\boldsymbol{a}_1^s, \boldsymbol{a}_2^s, \boldsymbol{a}_3^s) = (-\boldsymbol{a}_3, -\boldsymbol{a}_2, -\boldsymbol{a}_1)\]and
\[\begin{split}\boldsymbol{S} = \begin{pmatrix} 0 & 0 & -1 \\ 0 & -1 & 0 \\ -1 & 0 & 0 \end{pmatrix} = \boldsymbol{S}^{-1} = \begin{pmatrix} 0 & 0 & -1 \\ 0 & -1 & 0 \\ -1 & 0 & 0 \end{pmatrix}\end{split}\]
Note
All six changes of the cell preserve handiness of the original one.
Edge cases#
If \(a = b \ne c\) or \(a = c \ne b\) or \(b = c \ne a\), then the lattice is Body-centred tetragonal (BCT).
If \(a = b = c\), then the lattice is Body-centered cubic (BCC).