Bloembergen-Morgan theory

Fig. 2. Calculated relaxivities as a function of the water exchange rate for various proton Larmor frequencies and rotational correlation times, tr. The simulations have been performed by using the common Solomon-Bloembergen-Morgan theory of paramagnetic relaxation.

Fig. 4. Inner sphere contribution to the proton relaxivity as a function of the proton Larmor frequency. The curves were calculated on the basis of the Solomon-Bloembergen-Morgan theory for different values of the rotational correlation time, tr, and q — 1, kex — 10 x 106 s-1, tv = 20 ps, A2 = 0.1 x 102Os-2.

B. The modified SoIomon-BIoembergen equations and the Solomon-Bloembergen-Morgan theory... 

B. The Modified Solomon-Bloembergen Equations and the Solomon-Bloembergen-Morgan Theory... 

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... 

The electronic relaxation rates, as described by Bloembergen, Morgan and McLachlan [12], also depend on the magnetic field. For Gd(III) complexes they are usually interpreted in terms of zero field splitting interactions (ZFS). The electronic relaxation rates can be described by the Eqs. (14-16), often called as the Bloembergen-Morgan theory of paramagnetic electron spin relaxation ... 

The principle of IS relaxation is a chemical exchange in which one or more water molecules of the first coordination sphere of the paramagnetic center are replaced by others. This mechanism allows the propagation of the paramagnetic effect. The IS model is described by the Solomon-Bloembergen-Morgan theory. 

In Equations 4 and 5, A2 is the mean square ZFS energy and rv is the correlation time for the modulation of the ZFS, resulting from the transient distortions of the complex. The combination of Equations (3)—(5) constitutes a complete theory to relate the paramagnetic relaxation rate enhancement to microscopic properties (Solomon-Bloembergen Morgan (SBM) theory).15,16... 

Let us first examine the NMRD profiles of systems with correlation times Xci = Tie assuming the Solomon-Bloembergen-Morgan (SBM) theory (see Section II.B of Chapter 2) (2-5) is valid, and in the absence of contact relaxation. We report here the relevant equations for readers ... 

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... 

As relaxation is also field dependent, Bloembergen and Morgan developed a theory for the field dependence of Tie (Equation 10.9) that accounts for the discrepancies for ions with 5> 1/2 [17, 18]. 

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