High-order operators

In systems with electron spin S>1/2, higher-order terms have to be included in the spin Hamiltonian in order to accurately describe EPR spectra. The most common parameters are the second-order zero-field splitting parameters D and E.

In addition to the terms of D and E, a commonly used complete set of high-order operators are the extended Stevens operators (ESO) for and .

The high-order part of the spin Hamiltonian for one electron spin written in terms of ESOs is

where each of the operators is hermitian, and the associated coefficients are always real-valued. Orders with k>6 are practically irrelevant.

In contrast to spherical tensor operators (analogous to p₊, p₀, and p_- orbitals), these operators are tesseral (analogous to p_x, p_y and p_z). Positive k corresponds to cosine tesseral components, and negative k corresponds to sine tesseral components.

Be careful when using literature values for the coefficients, since conventions for the normalization and the phase of these operators vary widely. Also, many papers contain a significant amount of typos.

In EasySpin, matrix representations of the extended Stevens operators are provided by the function stev and can be used in most simulation functions.

The commonly used second-order zero-field splitting parameters D and E are related to the coefficients by

The associated second-order Hamiltonian terms have the forms

The traditional cubic and axial fourth-order parameters a and F are related to the coefficients via

The associated fourth-order Hamiltonian terms have the forms

(see Abragam/Bleaney, p. 142, 437; Pilbrow, p. 125, 129). The axes x, y, and z are the fourfold symmetry axes of the cubic part of the crystal field.

The following table lists all extended Stevens operators for k=2, 4 and 6 in terms of polynomials in S₊, S_- and S_z. A more general list of extended Stevens operators (including odd k) can be found in Altshuler/Kozyrev. Abragam/Bleaney contains only a partial, but compatible list. Ryabov (1999) gives a formula for computing Stevens operator polynomials for arbitrary k and q.

Here are the some useful references for extended Stevens operators:

• S. A. Altshuler, B. M. Kozyrev, Electron Paramagnetic Resonance in Compounds of Transition Elements, 2nd edn., Wiley (1974), Appendix V, p.512.
• A. Abragam, B. Bleaney, Electron Paramagnetic Resonance of Transition Ions, Dover (1986), Appendix B, Table 16, p. 863.
• I. D. Ryabov, J. Magn. Reson. 140, 141-145 (1999)
• C. Rudowicz, C. Y. Chung, J. Phys.: Condens. Matter 16, 1-23 (2004)

Note that there are typos in Altshuler/Kozyrev and in Abragam/Bleaney, as discussed by Rudowicz (2004).