Face-centred cubic (FCC)

Pearson symbol: cF

Constructor: FCC()

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

\[\begin{split}\begin{matrix} \boldsymbol{a}_1^s &=& (0, &a/2, &a/2)\\ \boldsymbol{a}_2^s &=& (a/2, &0, &a/2)\\ \boldsymbol{a}_3^s &=& (a/2, &a/2, &0) \end{matrix}\end{split}\]

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

Transformation matrix from standardized primitive cell to standardized conventional cell is

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

K-path

\(\mathrm{\Gamma-X-W-K-\Gamma-L-U-W-L-K\vert U-X}\)

Point	\(\times\boldsymbol{b}_1^s\)	\(\times\boldsymbol{b}_2^s\)	\(\times\boldsymbol{b}_3^s\)
\(\mathrm{\Gamma}\)	\(0\)	\(0\)	\(0\)
\(\mathrm{K}\)	\(3/8\)	\(3/8\)	\(3/4\)
\(\mathrm{L}\)	\(1/2\)	\(1/2\)	\(1/2\)
\(\mathrm{U}\)	\(5/8\)	\(1/4\)	\(5/8\)
\(\mathrm{W}\)	\(1/2\)	\(1/4\)	\(3/4\)
\(\mathrm{X}\)	\(1/2\)	\(0\)	\(1/2\)

Variations

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

Examples

Brillouin zone and default kpath

import wulfric as wulf

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

Cell standardization

No standardization required.

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