Zero field splitting describes various interactions of the energy levels of a molecule or ion resulting from the presence of more than one unpaired electron. The unpaired electrons mutually interact to give rise to two or more energy states. It is well known that degeneracy is lifted in the presence of a magnetic field, but zero field splitting occurs even in the absence of a magnetic field. ZFS is responsible for many effects in related to the magnetic properties of materials, as manifested in their electron spin resonance spectra and magnetism.
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The classic case for ZFS is the spin triplet, i.e., the S=1 spin system. In the presence of a magnetic field, the levels with different values of magnetic spin quantum number (MS=0,±1) are separated and the Zeeman splitting dictates their separation. In the absence of magnetic field, the 3 levels of the triplet are isoenergetic to the first order. However, when the effect interelectron repulsions are considered, the energy of the three sublevels of the triplet are separated. This effect is ZFS. The degree of separation depends on the symmetry of the system. The effects of ZFS are often most dramatically manifested in EPR spectra.
Quantum mechanical description
The corresponding Hamiltonian can be written as:
Where S is the total Spin quantum number, and
Algebraic derivation
The start is the corresponding Hamiltonian
with
The key is to express
To find the value for the deviation
By inserting (4) and (3) into (2) the result reads as:
Note, that in the second line in (5)
By defining D and E parameters equation (6) becomes to:
with