Bloembergen equation

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

The Bloembergen-Morgan equations, Eqs. (14) and (15), predict that the electron spin relaxation rates should disperse at around msTy = 1. This will make the correlation times for the dipolar and scalar interaction, %ci and respectively, in Eq. (11) dependent on the magnetic field. A combination of the modified Solomon-Bloembergen equations (12) and (13), for nuclear relaxation rates with the Bloembergen-Morgan equations for the field dependence... 

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. 

Besides these two main assumptions, the modified Solomon-Bloembergen equations, Eqs. (12) and (13), contain several additional approximations, the most important of which are the following ... 

Integration per partes of the Fokker-Planck equation for the quasidistribution (5=1) = P (the choice of a particular s is a question of taste only) allows us to write the appropriate equations for the cumulants. In what follows, we assume that damping is included only by way of coupling to the reservoir at zero temperature, that is, (n,) = 0. The first truncation (the cumulants higher than first-order vanish) leads to the classical limit. Then, from Eq. (80), we get the classical Bloembergen equations [102] [see Eqs. (1)] ... 

The s terms in Eq. (80) contribute only the term E,2 in Eq. (97). Thus, the term represents the quantum diffusional. v-terms in the Fokker-Planck equation. The other terms in Eqs. (93)-(100) originate in the drift terms of the Fokker-Planck equation. The terms B12 and C in Eqs. (93)-(94) play the role of feedback terms that pump quantum fluctuations into the classical Bloembergen equations. If the s terms in Eq. (80) do not appear (the classical case), the term in Eq. (97) does not appear, either. In this case the subset (95)—(100) with zero initial conditions has zero solutions and in consequence leads to the first truncation [171]. 

The enhancement of the longitudinal contact relaxation rate, after extension to the general case S > V2, is given by the Bloembergen equation [29,36]... 

The electronic term which is the first term in the Hamiltonian written in Eq. (3.13) and used to derive the Solomon and Bloembergen equations (Eqs. (3.16), (3.17), (3.19), (3.20), (3.26), (3.27)) may be inappropriate in many cases, since the electron energy levels may be strongly affected by the presence of ZFS or hyperfine coupling with the metal nucleus. Therefore, the electron static Hamiltonian to be solved to find the cos values, i.e. all electron energy transitions, and their probabilities, will be, in general,... 

Assuming that the relaxation of a given proton is due to the paramagnetism of the lanthanide ions then for the elements other than gadolinium the Solomon-Bloembergen equations for the water relaxation rates, 1/Tj or 1/T2, reduce to... 

The inner-sphere relaxation refers to the contribution from the water molecules that are directly bound to the gadolinium, and is expressed by the Solomon-Bloembergen equations (Equations 10.3-10.6) [15, 16],... 

It has been known since the earliest days of magnetic resonance spectroscopy that paramagnetic compounds, containing one or more unpaired spins, enhance the relaxation rates for the water protons in which they are dissolved." The extent of this enhancement, termed relaxivity, is highly dependent on the magnitude of the dipole-dipole interactions between the electron spin on the paramagnetic metal complex and the proton spin on the water molecule in question. These interactions are often quite complex. They can be treated on a formal (theoretical) level by the Solomon-Bloembergen equations." On a strictly practical level, however, the devel-... 

Using the Solomon-Bloembergen equations, together with the expression for paramagnetic contributions to nuclear spin-lattice and spin-spin relaxation rates (i ipara and / 2para) in a paramagnetic system are expressed by... 

The most important relaxation mechanism in this complex was, in accordance with Shulman, assumed to be modulation of the hyperfine interaction. The hyperfine coupling constant (A/h), was calculated from the Bloembergen equation [340]... 

The Tjjy[ value in the paramagnetic reagent-substrate adduct is determined by (i) the dipolar interaction between the observed nucleus and the unpaired electron(s) and (ii) the contact interaction between the magnetically active nucleus and the unpaired electron density localized in the position of the nucleus itself. An analytical form is given by the Solomon-Bloembergen equation (here for Tj an analogous expression is derived for T2) ... 

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