Solomon equation

We begm tliis section by looking at the Solomon equations, which are the simplest fomuilation of the essential aspects of relaxation as studied by NMR spectroscopy of today. A more general Redfield theory is introduced in the next section, followed by the discussion of the coimections between the relaxation and molecular motions and of physical mechanisms behind the nuclear relaxation. 

A good introductory textbook, includes a nice and detailed presentation of relaxation theory at the level of Solomon equations. 

CROSS-RELAXATION (AND CROSS-CORRELATION) SOLOMON EQUATIONS... 

Vi A and B. This interaction is responsible for a bi-exponential evolution of their polarization which is accounted for by two simultaneous differential equations called Solomon equations... 

Longitudinal spin order Extended Solomon equations... 

These equations involve additional parameters with respect to the classical Solomon equations (see (13)) which is the longitudinal order specific... 

The last point is about equivalent spins (or like spins as the two protons of the water molecule). Referring to Solomon equations (see (13)), we can notice that, because of this equivalence, the effective longitudinal relaxation rate is obtained by adding the cross-relaxation rate to the specific longitudinal relaxation rate ... 

For a spin-1/2 nucleus, such as carbon-13, the relaxation is often dominated by the dipole-dipole interaction with directly bonded proton(s). As mentioned in the theory section, the longitudinal relaxation in such a system deviates in general from the simple description based on Bloch equations. The complication - the transfer of magnetization from one spin to another - is usually referred to as cross-relaxation. The cross-relaxation process is conveniently described within the framework of the extended Solomon equations. If cross-correlation effects can be neglected or suitably eliminated, the longitudinal dipole-dipole relaxation of two coupled spins, such... 

The water proton NMRD of Co(OH2)g+ is reported in Fig. 5.33. The pseudooctahedral cobalt(II) complex provides almost field-independent water proton R values in the 0.01-60 MHz region [81]. By assuming the validity of the Solomon equation (Eq. (3.16)), both the cosxc = 1 and (ojtc = 1 dispersions can be placed at fields higher than 60 MHz, and therefore an upper limit for rc equal to 10 12 s can be set. Since the rotational correlation time, xr, is likely to be very similar... 

A more quantitative analysis of the data would require consideration of the possible effects of zero field splitting, which is known to be sizeable in cobalt(II) complexes. Such effects have been taken into account to explain the smoother shape of the dispersion curve with respect to what is predicted by the Solomon equation, and to obtain more accurate values of rs (see Section 3.7.1) [83]. 

The water proton NMRD profile of Cu(II) aqua ion at 298 K [108] (Fig. 5.36) is in excellent accordance with what expected from the dipole-dipole relaxation theory, as described by the Solomon equation (Eq. (3.16)). The best fitting procedure applied to a configuration of 12 water protons bound to the metal ion provides a distance between water protons and the paramagnetic center equal to 2.7 A, and a correlation time equal to 2.6 x 10 11 s, which defines the position of the cos dispersion. The correlation time is determined by rotation as expected from the Stokes-Einstein equation (Eq. (3.8)). The electron relaxation time is in fact expected to be one order of magnitude longer (see Table 5.6). This also ensures... 

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.

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