Reduced Spin Hamiltonian in High-Field NMR Spectroscopy

In high-field NMR the Zeeman term HZ dominates the spin Hamiltonian and in the absence of the high-frequency Brf. field, the energy levels of the total Hamiltonian can be determined using first-order perturbation theory, considering HZ as the unperturbed Hamiltonian H0 = HZ and the remaining interactions as the perturbation H1 = Ha + Hj + HD + Hq. In this case the different terms in the perturbation Hamiltonian H1 can be simplified, remaining only those parts that contribute in first-order perturbation theory thus commuting with H0, they are called the adiabatic or secular part of H1 (Ernst et al., 1992). To find the secular part of H1 we note that all the terms in H1 correspond to a contraction of two second-rank tensors, one a tensor operator derived from the spin operators 5!;-,lm = IitlIjim and another tensor Rfjlm dependent upon the interaction considered a:

Every second-rank tensor can be decomposed in a sum of three terms respectively a symmetric traceless tensor, an antisymmetric tensor and a scalar the trace:

I

where Si Im = f1 (%Im + %M) — 38imJ2l=x Sij,kk] is a symmetric traceless tensor, Si Im = [1 (Si;-,Im — SijtmI)] is an antisymmetric tensor and Sj = k=x Sij,kk is a scalar, the trace of SijiIm. With the

same decomposition applied to the tensor Ra Im, Hi,a becomes:

Considering without loss of generality the external magnetic induction Bo along the Z axis of the laboratory frame, B0 = B0ez and introducing the raising and lowering spin operators Ik + = Ikx+Hk,y and Ik, = Ik,x — iIky it is possible to show that in first-order perturbation theory the different contributions to Hi become:

 
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