Electronic relaxation, equation

For all known cases of iron-sulfur proteins, J > 0, meaning that the system is antiferromagnetically coupled through the Fe-S-Fe moiety. Equation (4) produces a series of levels, each characterized by a total spin S, with an associated energy, which are populated according to the Boltzmann distribution. Note that for each S level there is in principle an electron relaxation time. For most purposes it is convenient to refer to an effective relaxation time for the whole cluster. 

It is clear from the above equations that numerous parameters (proton exchange rate, kcx = l/rm rotational correlation time, tr electronic relaxation times, 1 /rlj2e Gd proton distance, rGdH hydration number, q) all influence the inner-sphere proton relaxivity. Simulated proton relaxivity curves, like that in Figure 3, are often used to visualize better the effect of the... 

Electronic relaxation is a crucial and difficult issue in the analysis of proton relaxivity data. The difficulty resides, on the one hand, in the lack of a theory valid in all real conditions, and, on the other hand, by the technical problems of independent and direct determination of electronic relaxation parameters. At low fields (below 0.1 T), electronic relaxation is fast and dominates the correlation time tc in Eq. (3), however, at high fields its contribution vanishes. The basic theory of electron spin relaxation of Gdm complexes, proposed by Hudson and Lewis, uses a transient ZFS as the main relaxation mechanism (100). For complexes of cubic symmetry Bloembergen and Morgan developed an approximate theory, which led to the equations generally... 

Recapitulating, the SBM theory is based on two fundamental assumptions. The first one is that the electron relaxation (which is a motion in the electron spin space) is uncorrelated with molecular reorientation (which is a spatial motion infiuencing the dipole coupling). The second assumption is that the electron spin system is dominated hy the electronic Zeeman interaction. Other interactions lead to relaxation, which can be described in terms of the longitudinal and transverse relaxation times Tie and T g. This point will be elaborated on later. In this sense, one can call the modified Solomon Bloembergen equations a Zeeman-limit theory. The validity of both the above assumptions is questionable in many cases of practical importance. 

Tie are also expected to be field-dependent. Their field dependence can be described by two parameters the electron relaxation time at low fields Tso, and the correlation time for the electron relaxation mechanism Ty (see Eq. (14) of Chapter 2) (5). However, Tso usually depends on (see Eq. (52) of Chapter 2). Therefore, it is preferable to select two different parameters for describing the field dependence of electron relaxation. For S > 1/2 systems, in case the electron relaxation is due to modulation of a time dependent transient zero-field splitting, A, (pseudorotational model), the Bloembergen-Morgan equations are obtained 5,6) ... 

The same approach has been developed for systems in solutions (15,16), and was found relevant in some cases. The electron relaxation time is described in the cases of Orbach-type mechanism by the equation (17)... 

The NMRD profiles of water solution of Ti(H20)g" have been shown in Section I.C.7 and have been already discussed. We only add here that the best fit procedures provide a constant of contact interaction of 4.5 MHz (61), and a distance of the twelve water protons from the metal ion of 2.62 A. If a 10% outer-sphere contribution is subtracted from the data, the distance increases to 2.67 A, which is a reasonably good value. The increase at high fields in the i 2 values cannot in this case be ascribed to the non-dispersive term present in the contact relaxation equation, as in other cases, because longitudinal measurements do not indicate field dependence in the electron relaxation time. Therefore they were related to chemical exchange contributions (see Eq. (3) of Chapter 2) and indicate values for tm equal to 4.2 X 10 s and 1.2 X 10 s at 293 and 308 K, respectively. 

Another important parameter that influences the inner sphere relaxivity of the Gd(III)-based contrast agents is the electronic relaxation time. Both the longitudinal and transverse electron spin relaxation times contribute to the overall correlation times xa for the dipolar interaction and are usually interpreted in terms of a transient zero-field splitting (ZFS) interaction (22). The pertinent equations [Eqs. (6) and (7)] that describe the magnetic field dependence of 1/Tie and 1/T2e have been proposed by Bloembergen and Morgan and... 

If the paramagnetic center is part of a solid matrix, the nature of the fluctuations in the electron nuclear dipolar coupling change, and the relaxation dispersion profile depends on the nature of the paramagnetic center and the trajectory of the nuclear spin in the vicinity of the paramagnetic center that is permitted by the spatial constraints of the matrix. The paramagnetic contribution to the relaxation equation rate constant may be generally written as... 

Another fairly conservative reaction is the removal or attachment of a single electron from/to a molecule. As already discussed in Chapter 5, Koopmans theorem equates the energy of the HOMO with the negative of the IP. This approximation ignores the effect of electronic relaxation in the ionized product, i.e., the degree to which the remaining electrons redistribute themselves following the detachment of one from the HOMO. If we were to... 

Equation (17) provides a form for discussion of the fundamental rate processes of photochemistry. The situation is analogous to the use of the Eyring equation for discussion of the rates of thermal reactions. Whether or not eq. (17) will prove to be as useful as the Eyring equation is a matter for speculation. I propose that photochemists try the equation as a vehicle for correlation of their results. If some other, more attractive, formulation of the rates of electronic relaxation should appear, conversion of semi-ordered discussion to a new form will probably be relatively easy. Although the substance of my suggestion is completed, a brief comment concerning the variables in eq. (17) is probably in order. 

The sole presence of an electron spin causes nuclear relaxation. The correlation time for the electron nucleus interaction is presented as well as equations valid for dipolar and contact interaction. To do so, electron relaxation mechanisms need to be quickly reviewed. All the subtleties of nuclear relaxation enhancements are presented pictorially and quantitatively. 

When rc is significantly affected by rs, then it can be magnetic field dependent, since electron relaxation can often be described by equations similar to those for nuclear relaxation (see for example Eqs. (3.11) and (3.12)). Owing to this dependence, longitudinal (T e) and transverse (T ) electronic relaxation times may not be equal and, strictly speaking, when electron relaxation is the dominant correlation time, Eqs. (3.16) and (3.17) should be written as... 

Let us discuss first the case in which only the first term is present. In the Solomon and Bloembeigen equations for / , (i = 1, 2) there is the cos parameter at the denominator of a Lorentzian function. Up to now cos has been taken equal to that of the free electron. However, in the presence of orbital contributions, the Zeeman splitting of the Ms levels changes its value and cos equals xs / o or (g/h)pBBo- When g is anisotropic (see Fig. 1.16), the value of cos is different from that of the free electron and is orientation dependent. The principal consequence is that another parameter (at least) is needed, i.e. the 0 angle between the metal-nucleus vector and the z direction of the g tensor (see Section 1.4). A second consequence is that the cos fluctuations in solution must be taken into account when integrating over all the orientations. Appropriate equations for nuclear relaxation have been derived for both the cases in which rotation is faster [40,41] or slower [42,43] than the electronic relaxation time. In practical cases, the deviations from the Solomon profile are within 10-20% (see for example Fig. 3.14). 

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