Hyperfine couplings spin = 1/2 systems

The most straightforward way to understand the origin of ESEEM and the physical chemistry behind its detection and analysis is to step back ft om this Cu(II) center and focns on an 5 = 1 /2, 7 = 1/2 coupled spin system. The spin Hamiltonian for this system consists of electronic Zeeman, nuclear Zeeman, and electron-nnclear hyperfine interaction terms. For the case of an isotropic electron -matrix and an axial hyperfine interaction, this Hamiltonian can be conveniently written in the laboratory reference Ifame,... 

ESR line widths are also sensitive to processes that modulate the g-value or hyperfine coupling constants or limit the lifetime of the electron spin state. The effects are closely analogous to those observed in NMR spectra of dynamical systems. However, since ESR line widths are typically on the order of 0.1-10 G... 

In the nuclear spin quantization axis system, the last term has the form, KSZ IZ" where K is the effective hyperfine coupling for the particular orientation. Thus ... 

The method presented here for evaluating energy levels from the spin Hamiltonian and then determining the allowed transitions is quite general and can be applied to more complex systems by using the appropriate spin Hamiltonian. Of particular interest in surface studies are molecules for which the g values, as well as the hyperfine coupling constants, are not isotropic. These cases will be discussed in the next two sections. 

Arnold s scale is derived for the action of a single substituent on the benzylic 7c-system. It cannot be used to estimate the influence of several substituents on the system under consideration. In this way it is, therefore, not possible to gain insight into the problem of captodative stabilization of a radical centre. The investigation of the spin-density distribution in benzylic radicals has been extended (Korth et al., 1987) to include multiple substitution patterns. Three types of benzylic radicals were considered a,p-disubsti-tuted a-methylbenzyl radicals [17], a-substituted p-methylbenzyl radicals [18] and a-substituted benzyl radicals [19]. In [17] and [18] the hyperfine coupling constants of the methyl hydrogens were used to determine the spin-density... 

We now come back to the simplest possible nuclear spin system, containing only one kind of nuclei 7, hyperfine-coupled to electron spin S. In the Solomon-Bloembergen-Morgan theory, both spins constitute the spin system with the unperturbed Hamiltonian containing the two Zeeman interactions. The dipole-dipole interaction and the interactions leading to the electron spin relaxation constitute the perturbation, treated by means of the Redfield theory. In this section, we deal with a situation where the electron spin is allowed to be so strongly coupled to the other degrees of freedom that the Redfield treatment of the combined IS spin system is not possible. In Section V, we will be faced with a situation where the electron spin is in... 

Fig. 4. Effect of (A) axial zero field splitting for the spin systems S = 1,3/2,2, and 5/2 (with Bo applied along the z direction of the ZFS tensor), and (B) isotropic hyperfine coupling with the metal nucleus for systems with I = 1/2, S = 1/2 and I = 3/2, S = 1/2.

In general, fluctuations in any electron Hamiltonian terms, due to Brownian motions, can induce relaxation. Fluctuations of anisotropic g, ZFS, or anisotropic A tensors may provide relaxation mechanisms. The g tensor is in fact introduced to describe the interaction energy between the magnetic field and the electron spin, in the presence of spin orbit coupling, which also causes static ZFS in S > 1/2 systems. The A tensor describes the hyperfine coupling of the unpaired electron(s) with the metal nuclear-spin. Stochastic fluctuations can arise from molecular reorientation (with correlation time Tji) and/or from molecular distortions, e.g., due to collisions (with correlation time t ) (18), the latter mechanism being usually dominant. The electron relaxation time is obtained (15) as a function of the squared anisotropies of the tensors and of the correlation time, with a field dependence due to the term x /(l + x ). 

In general the NMRD profiles are affected also by other parameters characterizing the electron spin system such as the g -factor, the hyperfine coupling with the metal nuclear spin (for I > 1/2 systems) and the ZFS (for S > 1/2 systems). 

The essential requirement for the effect to occur is a coupling of the nuclear spins with the electronic spins so that the predominant nuclear spin relaxation mechanism is via the electron spin system. In metals this coupling is via the hyperfine interaction. Another source of coupling is via the dipole-dipole interaction between nuclear and electronic spins. 

In addition to g tensor anisotropy, EPR spectra are often strongly affected by hyperfine interactions between the nuclear spin I and the electron spin S. These interactions take the form T A S, where A is the hyperfine coupling tensor. Like the g tensor, the A tensor is a second-order third-rank tensor that expresses orientation dependence, in this case, of the hyperfine coupling. The A and g tensors need not be colinear in other words, A is not necessarily diagonal in the coordinate systems which diagonalize g. 

Once a description of the electronic structure has been obtained in these terms, it is possible to proceed with the evaluation of spectroscopic properties. Specifically, the hyperfine coupling constants for oligonuclear systems can be calculated through spin projection of site-specific expectation values. A full derivation of the method has been reported recently (105) and a general outline will only be presented here. For the calculation of the hyperfine coupling constants, the total system of IV transition metal centers is viewed as composed of IV subsystems, each of which is assumed to have definite properties. Here the isotropic hyperfine is considered, but similar considerations apply for the anisotropic hyperfine coupling constants. For the nucleus in subsystem A, it can be... 

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