Which cell?

Some confusion arrives with the term "cell" (unit? primitive? standardized? conventional?). In this page we explain how wulfric understands the term "cell" and it's variants.

Picture below summarize the types of the cells that are discussed. For the details on how transformation matrices \(\boldsymbol{C}\) and \(\boldsymbol{S}\) act see Transformation of the cell.

"The" cell

The main cell of the package ("primitive cell" in the picture above). Almost in all functions of wulfric the input cell is interpreted as primitive (i.e. in the context of the lattice or crystal). However, it might not be the actual primitive cell of the crystal structure or lattice that is considered by the user.

Standardized cells

Standardization in wulfric follows the Setyawan and Curtarolo [1] paper (SC paper), although in the future we may implement other standardization conventions (open a Feature request if you are interested). In the SC paper two types of cells are defined: conventional one and primitive one. We use transformation matrix \(\boldsymbol{S}\) to compute the standardized primitive cell

\[\boldsymbol{A^s} = \boldsymbol{S}^T \boldsymbol{A}\]

and transformation matrix \(\boldsymbol{C}\) to compute the standardized conventional cell

\[\boldsymbol{A^{cs}} = \boldsymbol{C}^T \boldsymbol{A^s}\]

Details on the construction of matrices \(\boldsymbol{S}\) and exact forms of matrices \(\boldsymbol{C}\) are provided in the individual pages for each of the 14 Bravais lattices. Note that \(\boldsymbol{C} = (\boldsymbol{P^{\prime}})^{-1}\) from Table 2 of [2]. An inverse is present as we define the transformation matrix \(\boldsymbol{C}\) as from primitive to conventional and transformation matrix \(\boldsymbol{P^{\prime}}\) in [2] is defined as from conventional to primitive. Standardized cells are written in the default orientation as in the SC paper in the individual pages for Bravais lattices.

Matrix \(\boldsymbol{S}\) is orthonormal for all Bravais lattices, except for the Base-centred monoclinic (MCLC). All matrices satisfy \(\det(\boldsymbol{S}) = 1\).

Primitive cell contains exactly 1 lattice point per cell, while conventional cell might include more than one lattice point.

Wulfric deals with arbitrary orientation of the the cell (consequently, the crystal) and automatically recomputes relative coordinates of the high symmetry k-points with respect to the orientation of the crystall (i.e. given cell). Standardization (cell.get_standardized()) may change the choice of the cell, but will not change the orientation of the crystal in the real space. Standardization (crystal.standardize()) may change relative coordinates of atoms, but will not change their position in the real space. It is not necessary to standardize the cell to obtain the K-points as defined in [1].

References

[1] (1,2)

Setyawan, W. and Curtarolo, S., 2010. High-throughput electronic band structure calculations: Challenges and tools. Computational materials science, 49(2), pp. 299-312.

[2] (1,2)

Hinuma, Y., Pizzi, G., Kumagai, Y., Oba, F. and Tanaka, I., 2017. Band structure diagram paths based on crystallography. Computational Materials Science, 128, pp.140-184.