Relaxation Mechanisms and Correlation Times

The complete microscopic details of how longitudinal (spin-lattice, T]) and transverse (spin-spin, T- relaxation occur is beyond the scope of this book. But a little further discussion might be profitable in order to provide us with at least a qualitative understanding of the subject. 

The above dipole-dipole mechanism for spin-lattice relaxation depends on the interaction of the target nucleus with the magnetic field B/ of a lattice nucleus with magnetic moment p . The magnitude of B, is governed by the equation 

In addition to the direct interaction of magnetic dipoles, spin-lattice relaxation can proceed by way of interactions between the magnetic dipole of the target nucleus and fluctuating electric fields in the lattice. This is why neighboring quadrupolar nuclei (those with / 5, Section 2.1) can bring about very efficient spin-lattice relaxation (short values). 

As mentioned above, the frequency of rotational or translational motion of magnetic and electric fields in the lattice nuclei is critical to the effectiveness of spin-lattice relaxation. It cannot be too fast or too slow. It is common to express the frequencies of these types of molecular motion in terms of a so-called correlation time x,.. If the angular rotation frequency is CO (in radians per second), the rotational correlation time is 1/co, the time required for a molecule (or part of a molecule) to rotate 1 rad. Similarly, the translational correla- 

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As stated in Section II.B of Chapter 2, the actual correlation time for electron-nuclear dipole-dipole relaxation, is dominated by the fastest process among proton exchange, rotation, and electron spin relaxation. It follows that if electron relaxation is the fastest process, the proton correlation time Xc is given by electron-spin relaxation times Tie, and the field dependence of proton relaxation rates allows us to obtain the electron relaxation times and their field dependence, thus providing information on electron relaxation mechanisms. If motions faster than electron relaxation dominate Xc, it is only possible to set lower limits for the electron relaxation time, but we learn about some aspects on the dynamics of the system. In the remainder of this section we will deal with systems where electron relaxation determines the correlation time. 

Figure 10.4 Schematic diagram of the three types of relaxation mechanisms and the major parameters in the relaxation, which are the number of coordinated water molecules, water exchange rate (k ), and reorientational correlation time (xr).

Hirschinger et al [95] also used the anisotropy of spin-lattice relaxation to distinguish the two mechanisms, in this case the relaxation time associated with quadrupolar order, T q. A pulse sequence similar to Figure 8.2(b) was used. Quadrupolar order for the amorphous regions was shown to decay quickly, and for longer values of DE, only crystalline quadrupole order remained. The spectra could be closely simulated by a diffusion model and correlation time of 100 ps. Calculated jump lineshapes possess a central peak that is not present in the experimental data. 

In order to implement the relaxation calculations some additional definitions should be provided i) the nuclear spin interactions acting as relaxation mechanisms and ii) a molecular model motion, including iii) the distribution of correlation times at which the motion is occurring. These features will depend on aspects such as the temperature, the physical state of the sample, and the magnitude of the external applied magnetic field, among others. A more detailed analysis of such relaxation mechanisms can be found in References [4,8,25,26]. 

It should be noted that chemical exchange of a quadrupolar nucleus to and from a macromolecular binding site also produces fluctuations in the electric field gradient sensed by the nucleus and thus provides a second mechanism for quadrupole relaxation. The effective correlation time will now be given by eq. 27... 

The measurement of correlation times in molten salts and ionic liquids has recently been reviewed [11] (for more recent references refer to Carper et al. [12]). We have measured the spin-lattice relaxation rates l/Tj and nuclear Overhauser factors p in temperature ranges in and outside the extreme narrowing region for the neat ionic liquid [BMIM][PFg], in order to observe the temperature dependence of the spectral density. Subsequently, the models for the description of the reorientation-al dynamics introduced in the theoretical section (Section 4.5.3) were fitted to the experimental relaxation data. The nuclei of the aliphatic chains can be assumed to relax only through the dipolar mechanism. This is in contrast to the aromatic nuclei, which can also relax to some extent through the chemical-shift anisotropy mechanism. The latter mechanism has to be taken into account to fit the models to the experimental relaxation data (cf [1] or [3] for more details). Preliminary results are shown in Figures 4.5-1 and 4.5-2, together with the curves for the fitted functions. 

Up to now it has been tacitly assumed that each molecular motion can be described by a single correlation time. On the other hand, it is well-known, e.g., from dielectric and mechanical relaxation studies as well as from photon correlation spectroscopy and NMR relaxation times that in polymers one often deals with a distribution of correlation times60 65), in particular in glassy systems. Although the phenomenon as such is well established, little is known about the nature of this distribution. In particular, most techniques employed in this area do not allow a distinction of a heterogeneous distribution, where spatially separed groups move with different time constants and a homogeneous distribution, where each monomer unit shows essentially the same non-exponential relaxation. Even worse, relaxation... 

Polycarbonate (PC) serves as a convenient example for both, the direct determination of the distribution of correlation times and the close connection of localized motions and mechanical properties. This material shows a pronounced P-relaxation in the glassy state, but the nature of the corresponding motional mechanism was not clear 76 80> before the advent of advanced NMR techniques. Meanwhile it has been shown both from 2H NMR 17) and later from 13C NMRSI) that only the phenyl groups exhibit major mobility, consisting in 180° flips augmented by substantial small angle fluctuations about the same axis, reaching an rms amplitude of 35° at 380 K, for details see Ref. 17). 

The Mossbauer spectra of the complex [Fe(acpa)2]PF6 shown in Fig. 26 have also been interpreted on the basis of a relaxation mechanism [168]. For the calculations, the formalism using the modified Bloch equations again was employed. The resulting correlation times x = XlXh/(tl + Xh) are temperature dependent and span the range between 1.9 x 10 s at 110 K and 0.34 x 10 s at 285 K. Again the correlation times are reasonable only at low temperatures, whereas around 200 K increase of the population of the state contributes to... 

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... 

Their Gdm complexes were found to show significantly increased relaxivity upon interaction with Zn11, Ca11, and Mg11 ions. The relaxivity increase was related to an increase of the rotational correlation time, and the mechanism was ascribed to the formation of coordination oligomers or polymers that are typical for bisphosphonate complexes. 

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