The g Tensor Origin and Significance

In Sechons 1.2.1- 1.2.3, the basic theory and analysis of EPR spectra in fluid solution phase were examined. However, the theory and analysis of the spectra in the solid state, such as a heterogeneous catalyst, is more complex owing to anisotropies in the simple spin Hamiltonian introduced earlier in Equation 1.20, which only considered isotropic or averaged contributions from g and a. A more appropriate Hamiltonian for the solid state, which takes into account these anisotropies, is given by  

The isotropic g and a values are now replaced by two 3x3 matrices representing the g and A tensors and which arise from the anisotropic electron Zeeman and hyperfine interaction. Other energy terms may also be included in the spin Hamiltonian, including the anisotropic fine term D, for electron-electron interactions, and the anisotropic nuclear quadrupolar interaction Q, depending on the nucleus. Usually the quadrupolar interachons are very small, compared to A and D, are generally less than the inherent linewidth of the EPR signal and are therefore invisible by EPR. They are readily detected in hyperfine techniques such as ENDOR and HYSCORE. All these terms (g. A, D) are anisotropic in the solid state, and must therefore be defined in terms of a tensor, which will be explained in this section. 

According to the basic EPR resonance Equahon 1.9, the frequency required for the EPR transition depends only on B and Xb since the g value in this equation is isotropic. However, since the EPR spectrum in the solid state will depend on the relative orientation of the applied field with respect to the paramagnetic species in the powder. Equation 1.9 must be modified to include this angular dependence  

Because g now depends on the angles (6,( )) it should be described using the following electron Zeeman Hamiltonian  

These angular variations are responsible for the different g values found in the FPR spectrum (i.e. qualitatively they depend on the symmetry of the electronic wave function). However these deviations from ge actually arise from the admixture of orbital angular momentum into the spin ground state via spin orbit coupling. The extent of this admixing depends on which orbital contributes to the spin ground state (p, d or f). The real components of the g matrix are then given by  

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