shgpy.tensor_definitions module

This module provides a collection of susceptibility tensors (both dipole and quadrupole) which have each been simplified using the symmetry constraints implied by different crystallographic point groups. The tensors are contained in dictionaries of numpy ndarray s, where the keys of the dictionaries are strings containing the name of the relevant point group. For example, to get the dipole tensor implied by the crystallographic point group C_4v, one would use

>>> import shgpy.tensor_definitions as td
>>> t1 = td.dipole['C_4v']

In addition to the dipole dictionary, this module also provides a quadrupole dictionary with the rank-4 susceptibility tensors needed to compute quadrupole SHG, as well as a special surface dictionary (which is a duplicate of dipole but with the variables renamed e.g. from xxx -> sxxx. This is useful when analyzing SHG signal involving both bulk and surface dipole sources.

All of the variables defined in this module are sympy.Symbol objects and are defined in the shg_symbols module.

The crystallographic point groups which are defined in this module are

• 'S_2'

• 'C_2h'

• 'D_2h'

• 'C_4h'

• 'D_4h'

• 'T_h'

• 'O_h'

• 'S_6'

• 'D_3d'

• 'C_6h'

• 'D_6h'

• 'C_2'

• 'C_1h'

• 'D_2'

• 'C_2v'

• 'C_4'

• 'S_4'

• 'D_4'

• 'C_4v'

• 'D_2d'

• 'O'

• 'T_d'

• 'T'

• 'D_3'

• 'C_3'

• 'C_3v'

• 'C_6'

• 'C_3h'

• 'D_6'

• 'C_6v'

• 'D_3h'

• 'C_1'

In quadrupole, there is an additional key 'Isotropic'.

Please note that an additional constraint on typical SHG tensors which is not captured by the definitions in this module is that, since the SHG response function is symmetric in exchange of the electric field vectors, SHG tensors should be symmetric in their last two indices. This constraint is not captured in these definitions because one could imagine a broader use case in which this symmetry is not necessarily valid. In scripts, one should use particularize() to implement this constraint.