The Formalism for Electronic Spins

The effective spin Hamiltonian for energetically well isolated and orbitally nongene-rate ground states of transition metal ions with spin S is generally given by [1, 41, 103]  

If D is taken as a traceless tensor, Tr(/)) = Da = 0, there remain only two independent components for D (neglecting the three Euler angles for orientation in a general coordinate system). Usually, these are the parameters D and E, for the axial and rhombic contribution to the ZFS  

The ratio 7Z) is the so-called rhombicity parameter. With some algebra, and 

Although D is in general not necessarily traceless,(4.71) holds for the present 

The values of the rhombicity parameters are conventionally limited to the range 0 EjD 1/3 without loss of generality. This corresponds to the choice of a proper coordinate system, for which /)zz (in absolute values) is the largest component of the D tensor, and /) is smaller than Dyy. Any value of rhombicity outside the proper interval, obtained from a simulation for instance, can be projected back to 0 EID 1/3 by appropriate 90°-rotations of the reference frame, that is, by permutations of the diagonal elements of D. To this end, the set of nonconventional parameters D and EID has to be converted to the components of a traceless 3x3 tensor D using the relationships 

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