The Electron Zeeman Term

z coordinate system is defined by the g-matrix principal axes  

The magnetic field vector in that coordinate system is  

The transformation from the x,y,z axes to x, y, z (the axis system in which S is quantized) must then take this form to  

Note that the angle x is left indeterminant by this transformation. This amounts to saying that Sx and Sy are not fixed in space by the quantization of S along gB. The above result was used in deriving eqn (4.4) in Chapter 4. 

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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. 

Before we develop the resonance conditions for systems with hyperhne and with zero-held interactions, we return to the electronic Zeeman term S B as an example interaction to discuss a hitherto ignored complexity that is key to the usefulness of EPR spectroscopy in (bio)chemistry, namely anisotropy the fact that all interactions... 

Suppose we have an isolated system with a single unpaired electron and no hyper-fine interaction. Mononuclear low-spin Fe111 and many iron-sulfur clusters fall in this category (cf. Table 4.2). The only relevant interaction is the electronic Zeeman term, so the spin Hamiltonian is... 

As mentioned in Sect. 3.3, the electron Zeeman term contributes to Wba if first order base functions. Explicit first order expressions of Wba for l.h. and r.h. rotating fields, I = 1/2 and an isotropic or purely dipolar hfs tensor are given in Table 2. 

For a hydrogen atom in an external field of 10,000 G, draw a figure that shows the effect on the original 1 s energy level of including first the electron Zeeman term, then the nuclear Zeeman term, and finally the hyperfine coupling term in the Hamiltonian. 

In Eq. (3-2), / is the value of the exchange integral between two electron spins (Si and S2) and the 1/2 term is put for convenience although it is not used in many textbooks. Eq. (3-3) is just the sum of spin Hamiltonian of one radical given by Eq. (2-22), but the nuclear Zeeman terms are omitted in Eq. (3-3) because their magnitude is much smaller than those of the electron Zeeman term and the HFC one. In Eq. (3-3), ga and gb are the isotropic g-values of two component radicals (radicals a and b) in a radical pair, respectively, and A, and A are the isotropic HFC constants with nuclear spins (/, and 4) in radicals a and b, respectively. 

The first-order perturbation correction to the energy is given by the electronic Zeeman term as follows... 

Zero-field splittings occur in ESR spectra of paramagnetic species with electron spin S > 2. The energy for the transition (ms -<-> ms - 1) is AE = gHBB+1 F(ms-l/2) provided that the electron Zeeman term is the dominant one, gfXBB >> F. The ESR spectrum then contains 2 S lines, separated by 3 F. With the field in the xy-plane one has ... 

ENDOR signals due to nuclei with arbitrary values of the nuclear spin I can be analyzed by this unconventional perturbation method, where the electronic Zeeman term is the dominating one, while the hyperfine, quadrupole and nuclear Zeeman... 

The formulae by Iwasaki [19] reproduced in Appendix A3.2.2 are applicable also for S > Vi, when the electron Zeeman term dominates. Procedures based on diagonalisation of the spin Hamiltonian [59] apply generally. 

Theoretical aspects of NMR of free radicals containing spin-1/2 nuclei have been thoroughly discussed in the literature.29 Extension of these studies to free radicals containing quadrupolar nuclei (i.e., nuclei with nuclear spin 1) is relatively straightforward, especially in the solution phase with which we are mainly concerned here. The quadrupolar interaction, however, does lead to fast relaxation and hence broadens the NMR signals. We make a further assumption that the electronic Zeeman term, giSH, is much... 

The spin projection factors 7/3 and —4/3 relate the tensors of the coupled system to the local tensors and reflect the orientation of the local spins relative to the system spin, S = Sa -f Sb- Since gjb > 2.0 (y = x, y, z) the (-4/3) factor of gb leads to g values below g = 2.00. " For applied magnetic fields B > 0.05 T the electronic Zeeman interaction in Equation (5) is at least 20 times larger than the hyperfine interactions. Consequently, the expectation value of the electronic spin, , is determined by the electronic Zeeman term, allowing us to replace the spin operator S... 

The difficulty of this definition (or detection) of interactions is one of scale the perturbations that one is trying to observe are orders of magnitude smaller than actual measurements. Computationally, a better approach is to attempt evaluation of the individual perturbations directly, and then define the total interaction energy as a sum of the individual perturbation energies. In the case of EMR spectroscopy, this is exactly what we are doing by using ENDOR or ESEEM. We know that the effects that we expect to see will become manifest in the nuclear hyperfine terms, so rather than try to measure these from differences in the EMR spectrum, which includes the electron Zeeman term, we turn instead to the ENDOR and ESEEM, which detect the nuclear hyperfine interactions. 

In effect, ENDOR and ESEEM spectra permit one to conceptually drop the electronic Zeeman term fi-om the spin Hamiltonian and work on an energetic scale that is comparable to NMR spectroscopy. Both techniques can be used to obtain specific terms of the spin Hamiltonian provided that one has the means to experimentally deconvolute the spectroscopic transition energies. The primary difference between ENDOR and ESEEM resides with the manner in which the electronic... 

The first term of Eq. 4 represents the electronic Zeeman term, Si and 2 the spin operators for tvro triplet carbenes, and the second term represents the electronic exchange interaction. If Si and 2 couple to a quintet state (S = 2), ( 1 + 2) = S = S(S + 1) = 6. If they couple to a singlet, S = 0. Therefore, this term directly results in the energy level scheme, indicated in the inset of Fig. 9.13. The pure singlet and the pure quintet states are split by Assq which turns out to be the characteristic property of each dicarbene. The third term of Eq. 4 represents the magnetic dipole-dipole coupling of the two triplet carbenes ... 

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