The Nuclear Zeeman Interaction

If the interaction of the electron with an applied external magnetic field were the only effects detectable by EPR, then all spectra would consist of a single line and would be of little interest to chemists. However, the most useful chemical information that can be derived from an EPR spectrum usually results from nuclear 

Isotope Spin (%) Abundance Magnetogyric ratio, yxlO ()T ) ENDOR freq. (MHz at 0.350T) 

Some nuclei also possess spin, when the number of neutrons and protons are both uneven, and therefore have spin angular momentum. The spin of a nucleus is described by the spin quantum number, I. The angular momentum of a nucleus with spin I is given by  

Equation 1.14 are constants, they can all be replaced by another constant called the nuclear magneton Xn  

Equation 1.16 is analogous to Equation 1.3 for the electron, except now the negative sign is absent since the magnetic moment of the nucleus is collinear and parallel to the spin itself The simple spin Hamiltonian for a nuclear spin can thus be written 

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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)]. 

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... 

The physical origins of the effect can be illustrated from Figure 2, which shows the energy level scheme for an I = 1 nucleus, such as coupled to an S = 1/2 electron spin. In the case considered here (i.e., the electron-nuclear interaction, the nuclear Zeeman interaction, and the nuclear quadrupole interaction, all of the same order), microwaves can induce both allowed and semi-forbidden transitions between states in the Mj = 1/2 manifold (a) and the Ms = - 1/2 manifold (/ ). Simultaneous excitation of both kinds of transitions by the echo generating microwave pulses gives rise to interference effects, which manifest themselves as variations in the echo amplitude and thus cause the modulation of the echo envelope. Where a number of nuclei are coupled to the same electron spin, the level scheme becomes more complicated, but it is possible to factor out contributions due to coupling with each nucleus in the overall modulation pattern. If v(U l2>" n) is the modulation function due due to coupling with n nuclei, then... 

Figure 2. Energy level scheme for an I = 1 nucleus (e.g., l N) weakly coupled to an S = 1/2 electron spin. For the case shown here, the nuclear quadrupolar interaction and the nuclear Zeeman interaction are of the same order. Interference between allowed transitions...

Finally, [Eq. (6)] represents the interaction between the field and the nuclear magnetic moment. g is also a second-rank tensor that reflects anisotropies in the currents experienced by the shielded nuclei. However, the nuclear Zeeman interaction is, when included at all, usually treated as a scalar quantity since the magnitudes of these anisotropies are small compared to the frequencies and widths of the magnetic resonance spectra of triplet states. 

In the presence of quadrupole interactions the spin hamiltonian may be written as a sum of the nuclear Zeeman interaction and the quadrupole coupling term... 

The High-Field Approximation In most NMR experiments the nuclear Zeeman interaction with the static external magnetic field is much stronger than all other interactions of the nuclear spins. As a result of these differences in the size, it is usually possible to treat these interactions in first order perturbation theory, i.e. use only those terms which commute with the Zeeman Hamiltonian, the so called secular terms. This approximation is called the high field approximation. While the single particle interactions like CSA or quadrupolar interaction have a unique form, for bilinear interactions, one has to distinguish between a homonuclear and a hetero-nuclear case. The secular parts of Hamiltonians discussed in the previous section are collected in Table 1. 

ENDOR techniques work rather poorly if the hyperfine interaction and the nuclear Zeeman interaction are of the same order of magnitude. In this situation, electron and nuclear spin states are mixed and formally forbidden transitions, in which both the electron and nuclear spin flip, become partially allowed. Oscillations with the frequency of nuclear transitions then show up in simple electron spin echo experiments. Although such electron spin echo envelope modulation (ESEEM) experiments are not strictly double-resonance techniques, they are treated in this chapter (Section 5) because of their close relation and complementarity to ENDOR. The ESEEM experiments allow for extensive manipulations of the nuclear spins and thus for a more detailed separation of interactions. From the multitude of such experiments, we select here combination-peak ESEEM and hyperfine sublevel correlation spectroscopy (HYSCORE), which can separate the anisotropic dipole-dipole part of the hyperfine coupling from the isotropic Fermi contact interaction. 

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