Solomon equations solution

Fig. 5.36. Water H NMRD profiles for an aqueous solution of Cu(OH2)j+ at 298 K. The solid line represents the best-fit curve obtained using the Solomon equation (Eq. (3.16)), with a Cu-H distance of 2.7 A and xc = 2.6 x 10-11 s.

Using a simple kinetic model, Solomon demonstrated that the spin-lattice relaxation of the I and S spins was described by a system of coupled differential equations, with bi-exponential functions as general solutions. A single exponential relaxation for the I spin, corresponding to a well-defined Tu, could only be obtained in certain limiting situations, e.g., if the other spin, S, was different from I and had an independent and highly efficient relaxation pathway. This limit is normally fulfilled if S represents an electron spin. The spin-lattice relaxation rate, for the nuclear spin, I, is in such a situation given by ... 

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).

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). 

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],...

Tisp, and relative viscosity, riiei, at only one low concentration. Chuali et al. [37] examined die application of several single-point equations for PTT. They found tiiat when tire solution concenttation is <0.005 g/dL, Psp can be approximated to [p] witiiin 3 %. The single-point equation used in tiiis author s laboratoiy is from Solomon and Ciuta [38], as follows ... 

Materials. Film samples of PET were obtained from the Specialty Films Division of 3M Company, St.Paul, Minnesota 551, U.S.A. Two samples of thicknesses 6.3 u and 12.5 u contained no additives. All of the other samples contained varying amounts of a substituted 2,U-dihydroxybenzophenone type (9%10) ultraviolet light absorber. The densities of the film samples were in the range 1.31 to 1.32 g/ml. Relative viscosities of 1 1 phenolrtetrachloroethylene solutions of PET were determined using a Ubbelohde viscometer at 25+0.1 C. Intrinsic viscosities were calculated using the equation of Solomon... 

For moderately concentrated solutions, another equation proposed Solomon and Ciuta... 

The measurement of intrinsic viscosity using capillary viscometers can be a labour intensive and time-consuming exercise. However, polymer chemists undertaking characterisation studies in this way have been spared a significant amount practical work as a result of the development of so-called single point equations . These provide a method by which intrinsic viscosity can be determined when the flow time for the polymer solution is determined at only one concentration and compared to the flow time for that of the solvent alone. Solomon and Ciuta [23] proposed the following equation for use ... 

On selective saturation by RE irradiation at the resonance frequency of one of the spins (say S), it is generally assumed that the magnetization of the irradiated spin is fully saturated at all times, that is S (t) = 0, for all values of L The steady state solution of the Solomon s equations is then obtained by making the time derivatives on the left-hand-side of Equations [1] and [2] as equal to zero, yielding the steady state value of from Equation [1] as... 

The advantage of the transient NOE experiment is that the transport of magnetization takes place in the absence of RE irradiation and also the dynamics of Solomon s equations are identical to the 2D NOE experiment, described in the next section. The driven experiment has no 2D analogue and the solution given earlier in Equation [4] has the limitation that the details of saturation are not included. In fact if one uses Equation [2] instead of Equation [1], for the steady-state solution, by substituting dSJ t)lAt = 0 S (t) = 0, one obtains a wrong result... 

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