Bloembergen-Morgan equations

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... 

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) ... 

Fig. 7. Paramagnetic enhancements to solvent NMRD profiles for Fe(H20)g" " solutions at 298 K with (A) pure water and ( ) 60% glycerol. The lines represent the best fit curves using the Solomon-Bloembergen-Morgan equations [Eqs. (l)-(6)] 36).

Fig. 5.3. Water proton longitudinal relaxivity as a function of proton Larmor frequency ( H NMRD profiles) for solutions of Fe(OH2) + at ( ) 278 K, ( ) 288 K, (A) 298 K, ( ) 308 K. High field transverse relaxivity data at 308 K >) are also shown. The lines represent the best fit curves using the Solomon-Bloembergen-Morgan equations (Eqs. (3.11), (3.12), (3.16), (3.17), (3.26) and (3.27)) [4],...

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... 

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

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

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... 

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... 

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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