For historical reasons, crystal field parameters (effectively the radial matrix elements of the crystal field interactions) may be expressed in two different "normalisation", which we shall call Stevens and Wybourne. Stevens [53,25] initially expressed the radial parts of the crystal field interaction in terms of angular momentum operators , , . He did this by taking the Cartesian expressions for the tesseral harmonic functions (see Appendix F), and replacing all instances of the coordinates , , and with , and and allowing for the commutation relations of the angular momentum operators, but without considering the normalisation condition of these functions and hence are missing the prefactors before the square brackets in the expressions in Appendix F. We denote these prefactors . The Stevens crystal field Hamiltonian is thus
where the product is commonly taken in the literature as the crystal field parameter, because the factorisation into an intrinsic parameter and the expectation value of the radial wavefunction is derived from the point charge model and is not generally valid. Alternatively, the product is also commonly used, particularly in the neutron scattering literature. are the Stevens factors: for these correspond to the number of electrons in the unfilled shell , respectively.
Wybourne [43] and subsequent coauthors on the other hand chose to use the tensor operators which transform in the same way as the functions , where are the spherical harmonic functions, to describe the crystal field. Thus the angulardependent part of the crystal field matrix elements used by Wybourne differed from that of Stevens by the factor and for . The crystal field Hamiltonian used by Wybourne is thus (in our notation)
The disadvantage of the Wybourne approach is that one requires imaginary crystal field parameters, because the tensor operators are not Hermitian. In McPhase, we have instead chosen to use slightly different tensor operators , which are the Hermitian combinations of the ,
giving the Hamiltonian
Our parameters therefore have the same normalisation as the Wybourne parameters but will be real.
In summary:
