The Zeeman term

The matrix elements of the Zeeman term become expressed as 

When the g-tensor components are distinguished, one can use the fact that the scalar product of two vectors can be written in terms of the spherical tensor components, hence 

Now the Wigner-Eckart theorem, applied to the spherical tensor operators, implies 

For the SA = SB= 1/2 dimer the Zeeman matrix in the basis set of local (uncoupled) functions is 

In the case of isotropic exchange one can handle only the z-component of the Zeeman term the corresponding matrix is diagonal. 

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When a magnetic field, B, is applied to a ferromagnetic or ferrimagnetic particle with magnetic moment, p, the magnetic energy has a contribution from the Zeeman term... 

The g tensor can be easily related to more fundamental properties of the spin system by comparing the energy from the Zeeman term in the spin... 

We have seen in Chapter 2 that the electronic Zeeman term, the interaction between unpaired electrons in molecules and an external magnetic field, is the basis of EPR, but we have also discussed in Chapter 4 the fact that if a system has more than one unpaired electron, their spins can mutually interact even in the absence of an external field, and we have alluded to the fact that this zero-field interaction affords EPR spectra that are quite different from those caused by the Zeeman term alone. Let us now broaden our view to include many more possible interactions, but at the same time let us be systematic and realize that this plethora of possibilities is eventually reducible to five basic types only, two of which are usually so weak that they can be ignored. 

What would happen if we were to lower the microwave frequency from X-band to L-band (1 GHz). The Zeeman term for g 2.2 (an average value for copper) would correspond to a held of circa 325 gauss at 1 GHz, and so the two interactions S B and S I would be of comparable magnitude. In such situations the perturbation expressions become extremely complicated and lose all practical significance. 

The total Hamilton operator for solid state NMR, including the Zeeman term (Hq), is given in equation (1) ... 

The EPR spectra have always been interpreted2994 using an effective S = 2 spin Hamiltonian including the Zeeman term, /iBB g-S, and the hyperfine term, ICa-A-S, which describes the interaction of the unpaired electrons with the copper nucleus (7Cu = I). The spectra are very sensitive to the ratio between the isotropic coupling constant J and the local zero field splitting of nickel(II), Z)Ni.2982 In the limit J DNi it can easily be shown that the following relations hold ... 

Rapid molecular motions in solutions average to zero the dipolar and quadrupolar Hamiltonian terms. Hence, weak interactions (chemical shift and electron-coupled spin-spin couplings) are the main contributions to the Zeeman term. The chemical shift term (Hs) arises from the shielding effect of the fields produced by surrounding electrons on the nucleus ... 

In the low field limit the ZFS term (the second in the above equation) is dominant, while in the high field limit the Zeeman term (the first in the above equation) is dominant. 

There is no need to pass from the basis set of the ATs L,Mi,S,Ms) to the basis set of the atomic multiplets (LS),J,Mj) since such a unitary transformation does not lead to a gain in the computational effort. In the basis set of the L,Mi, S, Ms) functions, the operator Vax is diagonal but the operator Hso has off-diagonal matrix elements. In contrast, in the basis set of the (LS),J,Mj) kets, the operator Hso is diagonal, but the operator Vax has off-diagonal matrix elements. Therefore, none of these basis sets is appropriate for considering the Zeeman term as a small perturbation. 

When S > 1/2 and the metal site symmetry is lower than Oh or Td, there is a term in the spin Hamiltonian, in addition to the Zeeman term (gfill), that will split the (2S + 1 )MS spin degeneracy even in the absence of a magnetic field.32 This is shown in Equation 1.8, where D in the first term describes the effect of an axial distortion of the ligand field (z / x y) and E in the second term accounts for the presence of a rhombic ligand field y). 

An applied field in the xy-plane can tune the tunnel splittings Amm. via the Sx and Sy spin operators of the Zeeman terms that do not commute with the spin Hamiltonian. This effect can be demonstrated by using the Landau-Zener method (Section 3.1). Fig. 8 presents a detailed study of the tunnel splitting A 10 at the tunnel transition between m - +10, as a function of transverse fields applied at different angles (p, defined as the azimuth angle between the anisotropy hard axis and the transverse field (Fig. 4). 

Equation (9.106) is a field-free Hamiltonian, to which must be added the Zeeman terms the complete effective Hamiltonian for the problem is therefore... 

The Zeeman interaction and nuclear hyperfrne terms have to be added to this effective Hamiltonian. The Zeeman terms are, as in previous examples,... 

The Zeeman term is considered first and the hyperfine interaction is discussed in Section 4.2.2. The free electron g-value of = 2.0023 is shifted when the electron is surrounded by material, because of the spin-orbit coupling to the other electron states. The shift is... 

M. From Eqs (3-11) and (3-13), the S-T conversion of radical pars was found to be influenced by the following terms (a) the Zeeman term which is characterized by AgjUgB, (b) the hyperfine coupling (HFC) terms which are characterized by i4, and At, and (c) the exchange term which is characterized by J. Thus, the MFEs on chemical reactions through radical pairs can be classified by the following typical mechanisms ... 

Among the off-diagonal elements for the Zeeman terms (Hz) in the eigenbase of the zero field Hamiltonian, the following ones are only non-vanishing ... 

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