Nuclear Zeeman interactions

The leading term in T nuc is usually the magnetic hyperfine coupling IAS which connects the electron spin S and the nuclear spin 1. It is parameterized by the hyperfine coupling tensor A. The /-dependent nuclear Zeeman interaction and the electric quadrupole interaction are included as 2nd and 3rd terms. Their detailed description for Fe is provided in Sects. 4.3 and 4.4. The total spin Hamiltonian for electronic and nuclear spin variables is then ... 

In Equation (6) ge is the electronic g tensor, yn is the nuclear g factor (dimensionless), fln is the nuclear magneton in erg/G (or J/T), In is the nuclear spin angular momentum operator, An is the electron-nuclear hyperfine tensor in Hz, and Qn (non-zero for fn > 1) is the quadrupole interaction tensor in Hz. The first two terms in the Hamiltonian are the electron and nuclear Zeeman interactions, respectively the third term is the electron-nuclear hyperfine interaction and the last term is the nuclear quadrupole interaction. For the usual systems with an odd number of unpaired electrons, the transition moment is finite only for a magnetic dipole moment operator oriented perpendicular to the static magnetic field direction. In an ESR resonator in which the sample is placed, the microwave magnetic field must be therefore perpendicular to the external static magnetic field. The selection rules for the electron spin transitions are given in Equation (7)... 

An exception to this rule arises in the ESR spectra of radicals with small hyperfine parameters in solids. In that case the interplay between the Zeeman and anisotropic hyperfine interaction may give rise to satellite peaks for some radical orientations (S. M. Blinder, J. Chem. Phys., 1960, 33, 748 H. Sternlicht,./. Chem. Phys., 1960, 33, 1128). Such effects have been observed in organic free radicals (H. M. McConnell, C. Heller, T. Cole and R. W. Fessenden, J. Am. Chem. Soc., 1959, 82, 766) but are assumed to be negligible for the analysis of powder spectra (see Chapter 4) where A is often large or the resolution is insufficient to reveal subtle spectral features. The nuclear Zeeman interaction does, however, play a central role in electron-nuclear double resonance experiments and related methods [Appendix 2 and Section 2.6 (Chapter 2)]. 

HypB protein, 47 289 HypC protein, 47 289 Hyperfine coupling, 13 149-178 anisotropic, 13 150-161 Hyperfine coupling anisotropic dipolar, 13 150-154 nuclear Zeeman interaction, 13 155 quadrupole interaction, 13 154, 155 factors affecting magnitude of metal influence of charge on metal, 13 169-170 isotropic and anisotropic, 13 166-170 libration, 13 170... 

The first and second terms describe the electron and nuclear Zeeman interactions, where ys- and y are the gyromagnetic ratios of the electron and nucleus, respectively, and B0 is the externally applied magnetic field. This description of the electron Zeeman interaction is appropriate for a free electron or organic radical, but for metal ions or semiconductors it should be rewritten as gjuB(S B0) where g is the y-f actor of the unpaired electron and juB is the Bohr magnetron. The terms Hs and Ho in... 

There remain two other important magnetic interactions involving the nuclear spin magnetic moments, which cannot be derived from the present analysis, although their presence is reasonably self evident by analogy with corresponding electron spin terms which we have derived earlier. They are the nuclear Zeeman interaction,... 

We have finally reached our goal. The first term in (8.15 5) describes the nuclear Zeeman interaction, and was introduced in equation (8.4). cr is called the shielding ,... 

Earlier in this chapter when dealing with the nuclear Zeeman interaction we calculated the behaviour of the nuclear spin levels of H2 ignoring the effects of nuclear shielding. We now return to this question in more detail. We have shown that the Zeeman interaction for a nucleus of spin I should be written in the form... 

We treat the nuclear Zeeman interaction by remaining in the space-fixed axis system, and obtain... 

The second term here is the nuclear Zeeman interaction where gN is the nuclear g-factor, N the nuclear Bohr magneton, and ][ the nuclear spin. 

The isotropic form of the nuclear Zeeman interaction was discussed in detail in Section 1.2.3.1. This interaction is observed in isotropic media, and also in cases where the molecular orbital hosting the unpaired electron has substantial s character. The resulting isotropic hyperfrne couphng is related to the finite probability of the unpaired electron being at the nucleus. The spherical symmetry of the s orbital explains the isotropic nature of the interaction which is given by ... 

Fourthly, two unpaired electrons interact because of the overlap of their electronic orbitals. This gives rise to the so-called exchange energy, which again changes the resonance frequency of the individual electrons compared to that of the free electron. In Table 3 the four interactions are tabulated, together with their mathematical expressions. We have neglected the nuclear Zeeman interaction as this is more than six hundred times smaller than the electronic Zeeman interaction and, to first order, does not influence the EPR resonance. 

Consider the expression for the energy of a one-electron (S = -j, "is = one-proton (/ = mj = y) system, now including the nuclear Zeeman interaction (whose sign is opposite that of the electron Zeeman interaction) ... 

One key aspect of ENDOR spectroscopy is the nuclear relaxation time, which is generally governed by the dipolar coupling between nucleus and electron. Another key aspect is the ENDOR enhancement factor, as discussed by Geschwind [294]. The radiofrequency frequency field as experienced by the nucleus is enhanced by the ratio of the nuclear hyperfine field to the nuclear Zeeman interaction. Still another point is the selection of orientation concept introduced by Rist and Hyde [276]. In ENDOR of unordered solids, the ESR resonance condition selects molecules in a particular orientation, leading to single crystal type ENDOR. Triple resonance is also possible, irradiating simultaneously two nuclear transitions, as shown by Mobius et al. [295]. 

The Nuclear Zeeman Interaction The nuclear Zeeman term represents the direct interaction between the external magnetic field and the nuclear magnetic moment. This is usually neglected since it cancels for transitions between states with identical values of m/. When forbidden transitions are being considered, however, it is sometimes necessary to take account of this effect. It is applicable to both anisotropic and isotropic spectra. 

In Eq. (1) the first term is the nuclear Zeeman interaction, which is of the form... 

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