Spin relaxation rate

B1.13.3.1 SPIN-LATTICE AND SPIN-SPIN RELAXATION RATES... 

Figure 3 Effect of the water exchange rate, kex, and the rotational correlation time, rR, on inner-sphere proton relaxivity. The plot was simulated for a particular value of the longitudinal electron spin relaxation rate, 1/Tie — 5.28xlOss 1. The marketed contrast agents all have relaxivities around 4—5mM 1s 1 in contrast to the theoretically attainable values over lOOrnM-1 s 1, and this is mainly due to their fast rotation...

Outer sphere relaxation arises from the dipolar intermolecular interaction between the water proton nuclear spins and the gadolinium electron spin whose fluctuations are governed by random translational motion of the molecules (106). The outer sphere relaxation rate depends on several parameters, such as the closest approach of the solvent water protons and the Gdm complex, their relative diffusion coefficient, and the electron spin relaxation rate (107-109). Freed and others (110-112) developed an analytical expression for the outer sphere longitudinal relaxation rate, (l/Ti)os, for the simplest case of a force-free model. The force-free model is only a rough approximation for the interaction of outer sphere water molecules with Gdm complexes. 

Up to now, no direct measurements of diffusion coefficients have been reported for any system that show the low-temperature upturn just described, and it may well be that for most systems involving hydrogen such effects would occur only at ultra-low temperatures and minuscule diffusion rates. Also, the impurities and imperfections always present in real materials might well trap nearly all the diffusant atoms at the low temperatures at which coherent transport might be expected in ideal material. However, a recent measurement by Kiefl et al. (1989) of the (electronic) spin relaxation rate of muonium in potassium chloride over a range of temperatures gives spectacular support to the concept of coherent tunneling at low temperatures. (See Fig. 6 of Chapter 15 and the associated discussion.)... 

Fig. 14 An Arrhenius plot of the high-spin to low-spin relaxation rate obtained from the fits of the Mossbauer spectra of [Fe(HC(pz)3)2](BF4)2 shown in Fig. 13. Data obtained from [46]... 

The most important advantage of ESE-ENDOR lies in the fact that it does not critically depend on spin relaxation rates like the conventional cw-ENDOR technique, i.e. ESE-ENDOR might be more sensitive than the latter one. In ESE-ENDOR changes in the echo intensities up to 100% have been reported109). 

In addition to the use of quadrupolar splitting, the spin relaxation rate can also be used to calculate the specific surface area ratios for pulps beaten to different degrees and the results for an unbleached pulp agree closely and confirm the 250% increase in surface area measured by isotherm data (Table 5.4). 

Several approaches to determination of the overall rotational diffusion tensor from 15N relaxation data were suggested in the literature [15, 47, 49, 51-53]. The approach described here uses the orientational dependence of the ratio of spin-relaxation rates [49]... 

A common assumption in the relaxation theory is that the time-correlation function decays exponentially, with the above-mentioned correlation time as the time constant (this assumption can be rigorously derived for certain limiting situations (18)). The spectral density function is then Lorentzian and the nuclear spin relaxation rate of Eq. (7) becomes ... 

The MSB-equations were first presented by Connick and Mat (23) and by Reuben et al. (24). A formal derivation of these equations can be found, in a somewhat sketchy form, in the article by Gueron (25) and in a more stringent version in an article by Benetis et al. (26). Since oos 658m/ if / is a proton, and even more if it represents another nuclear spin, the first and third term of the DD part of Eq. (12) can safely be combined into a seven term , dispersing at msXc2 = 1 while the three term disperses at m/Td = 1. Similar equations can also be derived for the nuclear spin-spin relaxation rate in a paramagnetic complex ... 

The relaxation rates in Eqs. (12) and (13) depend now on the magnetic field in a more complicated way. Not only are the Larmor frequencies in the denominators of the Lorentzians proportional to the field, the electron spin relaxation rates are, in principle, also field-dependent. 

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... 

We wish to mention that a modified, parametrized, form of Eq. (15), for effective electron spin spin relaxation rate in S = 7/2 complexes of Gd(III) and Eu(II), was used in some articles from the Merhach group (28-30). 

Assuming that the lattice can, on the time scale relevant for the evolution of the nuclear spin density operator, be considered to remain in thermal equilibrium, a = a, and applying the Redfield theory to the nuclear spin sub-system allows us to obtain the following expressions for nuclear spin-lattice and spin spin relaxation rates ... 

The assumption of a single electron spin and a single T2 holds usually for S = 1/2 and for S > 1 in certain limits. Let us assume that the instantaneous distortions of the solvation sphere of the ion result in a transient ZFS and that the time-dependence of the transient ZFS can be described by the pseudorotation model, with the magnitude of the transient ZFS equal to At and the correlation time t . The simple picture of electron relaxation for S = 1 is valid if the Redfield condition (Att <5c 1) applies. Under the extreme narrowing conditions ((Os v 1), the longitudinal and transverse electron spin relaxation rates are equal to each other and to the low-field limit rate Tgo, occurring in Eqs. (14) and (15). The low field-limit rate is then given by (27,86) ... 

Using the case of S = 5/2 as an illustrative example, he demonstrated that it was possible to derive closed-form analytical expressions for the PRE of the form of the SBM equations times (1 + correction term). For typical parameter values, the effect of the correction term was to increase the prediction of the SBM theory by 5-7%. A similar approach was also applied to the S = 7/2 system, such as Gd(III) (101), where the correction terms could be larger. For that case, the estimations of the electron spin relaxations rates, obtained in the solution for PRE, were also used for simulations of ESR lineshapes. 

I J XgJ, Xg2- In this case, the two metal ions can be considered to have a single set of electron spin relaxation rates. If no additional relaxation mechanisms are established, such common relaxation rates are about equal to the fastest relaxation rates of the uncoupled spins. Actually, calculations indicate the presence of different electron relaxation rates for each level and for each transition. The electron relaxation rates for the pair are the sum of the rates of the two spins, weighted by coefficients depending on the transition 108). 

Kruk and co-workers formulated, in analogy with the inner-sphere case, the expression for the nuclear spin relaxation rate of solvent nuclei, as... 

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