Face-centred orthorhombic (ORCF)#
Pearson symbol: oF
Constructor: ORCF()
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#
ORCF1#
\(\mathrm{\Gamma-Y-T-Z-\Gamma-X-A_1-Y\vert T-X_1\vert X-A-Z\vert L-\Gamma}\)
Point |
\(\times\boldsymbol{b}_1^s\) |
\(\times\boldsymbol{b}_2^s\) |
\(\times\boldsymbol{b}_3^s\) |
|---|---|---|---|
\(\mathrm{\Gamma}\) |
\(0\) |
\(0\) |
\(0\) |
\(\mathrm{A}\) |
\(1/2\) |
\(1/2 + \zeta\) |
\(\zeta\) |
\(\mathrm{A_1}\) |
\(1/2\) |
\(1/2 - \zeta\) |
\(1-\zeta\) |
\(\mathrm{L}\) |
\(1/2\) |
\(1/2\) |
\(1/2\) |
\(\mathrm{T}\) |
\(1\) |
\(1/2\) |
\(1/2\) |
\(\mathrm{X}\) |
\(0\) |
\(\eta\) |
\(\eta\) |
\(\mathrm{X_1}\) |
\(1\) |
\(1-\eta\) |
\(1-\eta\) |
\(\mathrm{Y}\) |
\(1/2\) |
\(0\) |
\(1/2\) |
\(\mathrm{Z}\) |
\(1/2\) |
\(1/2\) |
\(0\) |
ORCF2#
\(\mathrm{\Gamma-Y-C-D-X-\Gamma-Z-D_1-H-C\vert C_1-Z\vert X-H_1\vert H-Y\vert L-\Gamma}\)
Point |
\(\times\boldsymbol{b}_1^s\) |
\(\times\boldsymbol{b}_2^s\) |
\(\times\boldsymbol{b}_3^s\) |
|---|---|---|---|
\(\mathrm{\Gamma}\) |
\(0\) |
\(0\) |
\(0\) |
\(\mathrm{C}\) |
\(1/2\) |
\(1/2 - \eta\) |
\(1 - \eta\) |
\(\mathrm{C_1}\) |
\(1/2\) |
\(1/2 + \eta\) |
\(\eta\) |
\(\mathrm{D}\) |
\(1/2-\delta\) |
\(1/2\) |
\(1 - \delta\) |
\(\mathrm{D_1}\) |
\(1/2+\delta\) |
\(1/2\) |
\(\delta\) |
\(\mathrm{L}\) |
\(1/2\) |
\(1/2\) |
\(1/2\) |
\(\mathrm{H}\) |
\(1 - \phi\) |
\(1/2 - \phi\) |
\(1/2\) |
\(\mathrm{H_1}\) |
\(\phi\) |
\(1/2 + \phi\) |
\(1/2\) |
\(\mathrm{X}\) |
\(0\) |
\(1/2\) |
\(1/2\) |
\(\mathrm{Y}\) |
\(1/2\) |
\(0\) |
\(1/2\) |
\(\mathrm{Z}\) |
\(1/2\) |
\(1/2\) |
\(0\) |
ORCF3#
\(\mathrm{\Gamma-Y-T-Z-\Gamma-X-A_1-Y\vert X-A-Z\vert L-\Gamma}\)
Point |
\(\times\boldsymbol{b}_1^s\) |
\(\times\boldsymbol{b}_2^s\) |
\(\times\boldsymbol{b}_3^s\) |
|---|---|---|---|
\(\mathrm{\Gamma}\) |
\(0\) |
\(0\) |
\(0\) |
\(\mathrm{A}\) |
\(1/2\) |
\(1/2 + \zeta\) |
\(\zeta\) |
\(\mathrm{A_1}\) |
\(1/2\) |
\(1/2 - \zeta\) |
\(1-\zeta\) |
\(\mathrm{L}\) |
\(1/2\) |
\(1/2\) |
\(1/2\) |
\(\mathrm{T}\) |
\(1\) |
\(1/2\) |
\(1/2\) |
\(\mathrm{X}\) |
\(0\) |
\(\eta\) |
\(\eta\) |
\(\mathrm{X_1}\) |
\(1\) |
\(1-\eta\) |
\(1-\eta\) |
\(\mathrm{Y}\) |
\(1/2\) |
\(0\) |
\(1/2\) |
\(\mathrm{Z}\) |
\(1/2\) |
\(1/2\) |
\(0\) |
Variations#
There are three variations of face-centered orthorombic lattice.
For the examples of variations \(a\) is set to \(1\); \(b\) and \(c\) fulfil the conditions:
\(b = \dfrac{c}{\sqrt{c^2 - 1}}\)
\(c > \sqrt{2}\)
First condition defines in ORCF3 lattice and ensures ordering of lattice parameters \(b > a\). Ordering \(c > b\) is forced by second condition.
For ORCF1 and ORCF2 lattices \(a < 1\) and \(a > 1\) is chosen. While \(b\) and \(c\) are the same as for ORCF3 lattice.
At the end all three parameters are multiplied by \(\pi\).
ORCF1#
\(\dfrac{1}{a^2} > \dfrac{1}{b^2} + \dfrac{1}{c^2}\).
Predefined example: orcf1 with
\(a = 0.7\pi\), \(b = 5\pi/4\) and \(c = 5\pi/3\).
ORCF2#
\(\dfrac{1}{a^2} < \dfrac{1}{b^2} + \dfrac{1}{c^2}\).
Predefined example: orcf2 with
\(a = 1.2\pi\), \(b = 5\pi/4\) and \(c = 5\pi/3\).
ORCF3#
\(\dfrac{1}{a^2} = \dfrac{1}{b^2} + \dfrac{1}{c^2}\).
Predefined example: orcf3 with
\(a = \pi\), \(b = 5\pi/4\) and \(c = 5\pi/3\).
Examples#
ORCF1#
Brillouin zone and default kpath#
import wulfric as wulf
cell = wulf.cell.get_cell_example("ORCF1")
backend = wulf.visualization.PlotlyBackend()
backend.plot(cell, kind="brillouin-kpath")
# Save an image:
backend.save("orcf1_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("ORCF1")
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("orcf1_real.png")
# Interactive plot:
backend.show()
ORCF2#
Brillouin zone and default kpath#
import wulfric as wulf
cell = wulf.cell.get_cell_example("ORCF2")
backend = wulf.visualization.PlotlyBackend()
backend.plot(cell, kind="brillouin-kpath")
# Save an image:
backend.save("orcf2_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("ORCF2")
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("orcf2_real.png")
# Interactive plot:
backend.show()
ORCF3#
Brillouin zone and default kpath#
import wulfric as wulf
cell = wulf.cell.get_cell_example("ORCF3")
backend = wulf.visualization.PlotlyBackend()
backend.plot(cell, kind="brillouin-kpath")
# Save an image:
backend.save("orcf3_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("ORCF3")
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("orcf3_real.png")
# Interactive plot:
backend.show()
Cell standardization#
Condition \(a < b < c\) implies \(\vert\boldsymbol{a}_1^s\vert > \vert\boldsymbol{a}_2^s\vert > \vert\boldsymbol{a}_3^s\vert\) for the lattice vectors of the primitive cell in a standard form. Therefore, wulfric uses the primitive lattice vectors for the standardization:
If \(\vert \boldsymbol{a}_3\vert < \vert \boldsymbol{a}_2\vert < \vert \boldsymbol{a}_1\vert\), 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 \(\vert \boldsymbol{a}_3\vert < \vert \boldsymbol{a}_1\vert < \vert \boldsymbol{a}_2\vert\), 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 \(\vert \boldsymbol{a}_2\vert < \vert \boldsymbol{a}_3\vert < \vert \boldsymbol{a}_1\vert\), 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 \(\vert \boldsymbol{a}_2\vert < \vert \boldsymbol{a}_1\vert < \vert \boldsymbol{a}_3\vert\), then
\[(\boldsymbol{a}_1^s, \boldsymbol{a}_2^s, \boldsymbol{a}_3^s) = (\boldsymbol{a}_3, \boldsymbol{a}_1, \boldsymbol{a}_2)\]and
\[\begin{split}\boldsymbol{S} = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{pmatrix} \qquad \boldsymbol{S}^{-1} = \begin{pmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix}\end{split}\]If \(\vert \boldsymbol{a}_1\vert < \vert \boldsymbol{a}_3\vert < \vert \boldsymbol{a}_2\vert\), 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 \(\vert \boldsymbol{a}_1\vert < \vert \boldsymbol{a}_2\vert < \vert \boldsymbol{a}_3\vert\), 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 Face-centred cubic (FCC).