The effect of an external magnetic field

Consideration of the spin-orbit interaction and the effect of an external magnetic field on the electronic ground state of an ion in a CF allows evaluation of the various terms in the spin-Hamiltonian of Equation (31). In addition, the interaction of the nucleus of the paramagnetic ion and ligand nuclei with the d-electron cloud must be considered. In this way the experimentally determined terms of the spin-Hamiltonian may be related to such parameters as the energy differences between levels of the ion in the CF and amount of charge transfer between d-electrons and ligands. 

Let us consider the effect of an external magnetic field on the angular momenta distribution at a level populated in the fluorescence process see Section 3.4, Fig. 3.14. In the presence of an external magnetic field the following polarization moments are created on the lower level J" via spontaneous transitions at weak excitation, x — 0 ... 

We shall see later how, with the appropriate expressions for the magnetic vector potentials, the effects of an external magnetic vector potentials, the effects of an external magnetic field can be introduced into the vibration rotation Hamiltonian. 

Before considering the effects of an external magnetic field, it is desirable to reformulate the expression for the total kinetic energy (8.83) in terms of the total angular momentum J and the total electronic orbital angular momentum L Equation (8.83) can be rewritten for a rigid molecule (va = 0) in the form... 

Since the transition from initial to final singlet states is spin-conserving, it has been postulated that y is larger the greater the number of pair states with singlet character, i.e., the greater the number of terms in (85) with 0 [193]. The effect of an external magnetic field on y (thus on ys) may be understood on this basis (see Sec. 2.5.3.1). 

Fig. 3. The NMR experiment. The effect of an external magnetic field upon the energy of a nucleus with spin / = J.

A number of techniques have been applied to probe the details of the ECL processes. Triplet intermediates resulting from the redox reaction have been identified by interception with fnms-stilbene and determination of the extent of trans- to cis-isomerization (Freed and Faulkner, 1971) as well as by observing the effect of an external magnetic field on the emission intensity (Faulkner et al., 1972 Tachikawa and Bard, 1974). In the latter studies, it was shown that a magnetic field decreases the rate of the triplet-triplet annihilation reaction, (112) or (114), and also the rate of quenching of triplets by radical ions [eqn (116)]. Since the rate of the radical ion redox... 

As shown in the previous section, several magnetic effects, such as the spin-orbit effect are included in the relativistic effect. To consider the effect of an external magnetic field, however, the vector potential. A, should be incorporated in the momentum operator (Jensen 2006),... 

A large part of the theory of the effect of an external magnetic field on lanthanide ions in crystals was developed in the context of paramagnetic resonance (Bleaney and Stevens 1953). It is that field that bequeathed optical spectroscopists the spin Hamiltonian and its various elaborations. This is not to say that such devices have ever been taken much advantage of. After all, the perturbation Hamiltonian m-(L + 2S), where P is the Bohr magneton and H the applied magnetic field, is particularly simple to evaluate, since the quantum number S and L are used in defining all lanthanide states. 

The fact that, for each line, a different set of axes has been set up complicates the analysis of the effect of an external magnetic field H on the nuclear moment p. The direct term, - i H, needs to be multiplied by a factor to allow for cross terms involving the Zeeman Hamiltonian and the hyperfine interaction given by (108), as was recognized first by Elliott (1957). But as soon as p is replaced by / (times a factor), we arrive at combinations such as that can be easily evaluated in the basis f, M > only if the coordinate frame defined by the principal axes is used. The required transformation of axes, which varies from line to line, has been described by Cone et al. (1984). 

The separate descriptions of the zero-field splitting and the effect of an external magnetic field on the atomic sublevels can now be combined into a unified description using the following spin Hamiltonian... 

Table 5. Electric-dipole probabilities for a- and tr-polarized Mjg Eg transitions in trigonal (P-systems under the effect of an external magnetic field that is aligned parallel to the molecular trigonal axis (cf. Ref. [32])...

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