List of matrices and vectors of a specific space group
op_xyz_list_to_matrix_list.RdReturns \(3\times 3\) matrices and \(3\times 1\) vectors corresponding to point group operations, group translations and cell centring of a given space group.
Arguments
- op_xyz_list
A named list made of two vectors. The first vector, SYMOP, contains strings describing the symmetry operators. The second vector, CENOP, contains strings describing the centring of the unit cell.
Value
mat_ops_list A named list consisting of 3 lists. The first list, PG, contains \(3\times 3\) point group matrices; the second list, T, contains the same number of \(3\times 1\) translation vectors. The first matrix is always the identity matrix, the first translation vector is always the null vector. The third list, C, consists of centering vectors; the first centering vector is always the null vector. To summarize, the output looks like the following:
[[ [[I,M2,M3,...,Mn]] , [[O,V2,V3,...,Vn]] , [[O,C2,C3,...,Cm]] ]] where: I = identity 3X3 matrix 0 = null 3X1 vector M2,M3,...,Mn = point group 3X3 matrices V2,V3,...,Cn = translation 3X1 vectors C2,C3,...,Cm = centering 3X1 vectors
Details
A crystallographic space group consists of a series of transformations on a point \((x_f,y_f,z_f)\) in space that are mathematically implemented as the product of a \(3\times 3\) point-group matrix and the point fractional coordinates, \((x_f,y_f,z_f)\), followed by a sum with a \(3\times 1\) translation vector. The complete set of points thus produced can be cloned into a new and shifted set translated of an amount represented by a \(3\times 1\) centring vector.
Examples
# Symmetry operators for space group number 3, P 1 2 1
SG <- "P 1 2 1"
op_xyz_list <- syminfo_to_op_xyz_list(SG)
mat_ops_list <- op_xyz_list_to_matrix_list(op_xyz_list)
names(mat_ops_list)
#> [1] "PG" "T" "C"