Morgan equation

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],...

The standard least-squares approach provides an alternative to the Galerkin method in the development of finite element solution schemes for differential equations. However, it can also be shown to belong to the class of weighted residual techniques (Zienkiewicz and Morgan, 1983). In the least-squares finite element method the sum of the squares of the residuals, generated via the substitution of the unknown functions by finite element approximations, is formed and subsequently minimized to obtain the working equations of the scheme. The procedure can be illustrated by the following example, consider... 

Note These equations are from Doming, S. N. Morgan, S. L. Experimental Design A Chemometric Approach. Elsevier Amsterdam, 1987, and pseudo-three-dimensional plots of the response surfaces can be found in their figures 11.4, 11.5, and 11.14. The response surface for problem (a) also is shown in Color Plate 13. 

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... 

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... 

Unfortunately, phases of geochemical interest are not ideal. As well, aqueous species do not occur in a pure form, since their solubilities in water are limited, so a new choice for the standard state is required. For this reason, the chemical potentials of species in solution are expressed less directly (Stumm and Morgan, 1996, and Nordstrom and Munoz, 1994, e.g., give complete discussions), although the form of the ideal solution equation (Eqn. 3.4) is retained. 

Geochemists, however, seem to have reached a consensus (e.g., Karpov and Kaz min, 1972 Morel and Morgan, 1972 Crerar, 1975 Reed, 1982 Wolery, 1983) that Newton-Raphson iteration is the most powerful and reliable approach, especially in systems where mass is distributed over minerals as well as dissolved species. In this chapter, we consider the special difficulties posed by the nonlinear forms of the governing equations and discuss how the Newton-Raphson method can be used in geochemical modeling to solve the equations rapidly and reliably. 

Catalysis by such a mechanism can be accounted for in a kinetic rate equation by including as a factor the catalyzing mineral s surface area. Sung and Morgan (1981), for example, in studying the oxidation on Mn11 by dissolved dioxygen,... 

Various empirical relations are available for calculating individual ion activity coefficients [discussed by Stumm and Morgan (1996) for natural waters and Sposito (1984a, b), for soil solutions]. In the calculations in this book I used the Davies equation ... 

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 ... 

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... 

At 20.5 °C and between pH 6.0 and 7.5, k ranges from 1.28-1.83 10 min MPa L M. As the above equation indicates, the rate of oxidation increases one hundredfold per pH unit. In other words, oxidation is extremely slow below pH 6 and rises sharply above this value. It also increases tenfold for a 15°C increase in temperature. Increasing the ionic strength of the system retards oxidation (Sung Morgan, 1980 Millero et ak, 1987). The oxidation/hydrolysis process is accelerated by increasing the stirrer speed (Perez et al. 1998 Perez Umitsu 2000). 

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