Dipolar relaxation rate constant

All other factors are constants. The angle 9 is the projection angle between bond vectors and is the sole unknown in Eq. 1, and can be readily and precisely determined by measurement of the cross-correlated dipolar relaxation rate. It should be emphasized that 9 is measured directly, without the need of experimental calibration as in the Karplus curve for J-coupling constants for example. 

Finally, we mention that Li T measurements have been used to study the rate of interconversion between tight and loose ion pairs of lithium fluorenide [44]. The quadrupolar relaxation contribution was derived from Li and Li T data, and a change of the Li quadrupole splitting constant, QSC, obtained via Li quadrupolar and dipolar relaxation rates (see Section 4) with ion pair structure, was indicated. 

Plainly, there will be no such effect unless relaxation pathways. It will be seen later on that such pathways are only present when there is dipolar relaxation between the two spins and that the resulting cross-relaxation rate constants have a strong dependence on the distance between the two spins. The observation of a nuclear Overhauser effect is therefore diagnostic of dipolar relaxation and hence the proximity of pairs of spins. The effect is of enormous value, therefore, in structure determination by NMR. 

As the size of the dipolar interaction depends on the product of the gyromagnetic ratios of the two nuclei involved, and the resulting relaxation rate constants depends on the square of this. Thus, pairs of nuclei with high gyromagnetic ratios are most efficient at promoting relaxation. For example, every thing else being equal, a proton-proton pair will relax 16 times faster than a carbon-13 proton pair. 

To measure distances in the wider temperature range, this procedure was modified. Relaxation of the carotenoid occurs through several different mechanisms including the dipolar-dipolar interaction. Assuming that kAA is the rate constant of the dipolar-dipolar interaction and K=(k,l + k2 + k3 +. ..) is the sum of the rate constants of all other relaxation pathways, we can extract kAA from the following equation ... 

As we shall see, all relaxation rates are expressed as linear combinations of spectral densities. We shall retain the two relaxation mechanisms which are involved in the present study the dipolar interaction and the so-called chemical shift anisotropy (csa) which can be important for carbon-13 relaxation. We shall disregard all other mechanisms because it is very likely that they will not affect carbon-13 relaxation. Let us denote by 1 the inverse of Tt. Rt governs the recovery of the longitudinal component of polarization, Iz, and, of course, the usual nuclear magnetization which is simply the nuclear polarization times the gyromagnetic constant A. The relevant evolution equation is one of the famous Bloch equations,1 valid, in principle, for a single spin but which, in many cases, can be used as a first approximation. 

As an example of the measurement of cross-correlated relaxation between CSA and dipolar couplings, we choose the J-resolved constant time experiment [30] (Fig. 7.26 a) that measures the cross-correlated relaxation of 1H,13C-dipolar coupling and 31P-chemical shift anisotropy to determine the phosphodiester backbone angles a and in RNA. Since 31P is not bound to NMR-active nuclei, NOE information for the backbone of RNA is sparse, and vicinal scalar coupling constants cannot be exploited. The cross-correlated relaxation rates can be obtained from the relative scaling (shown schematically in Fig. 7.19d) of the two submultiplet intensities derived from an H-coupled constant time spectrum of 13C,31P double- and zero-quantum coherence [DQC (double-quantum coherence) and ZQC (zero-quantum coherence), respectively]. These traces are shown in Fig. 7.26c. The desired cross-correlated relaxation rate can be extracted from the intensities of the cross peaks according to ... 

Table II reports the contact coupling constant for different aqua ion systems at room temperature. The contact coupling constant is a measure of the unpaired spin density delocalized at the coordinated protons. The values were calculated from the analysis of the contact contribution to the paramagnetic enhancements of relaxation rates in all cases where the correlation time for dipolar relaxation is dominated by x and Tig > x. In fact, in such cases the dispersion due to contact relaxation occurs earlier in frequency than the dispersion due to dipolar relaxation. In metalloproteins the contact contribution is usually negligible, even for metal ions characterized by a large contact contribution in aqua ion systems. This is due to the fact that the dipolar contribution is much larger because the correlation time increases by orders of magnitude, and x becomes longer than Tig. Under...

If the paramagnetic center is part of a solid matrix, the nature of the fluctuations in the electron nuclear dipolar coupling change, and the relaxation dispersion profile depends on the nature of the paramagnetic center and the trajectory of the nuclear spin in the vicinity of the paramagnetic center that is permitted by the spatial constraints of the matrix. The paramagnetic contribution to the relaxation equation rate constant may be generally written as... 

The contribution of the hyperfine interactions to the relaxation rates of the radical depends on whether the dominant contribution comes from the anisotropic (dipolar) or the isotropic (scalar) part of the hyperfine interaction. Usually, the anisotropic contribution predominates because this interaction can be readily modulated by the tumbling motion of the molecule. However, in radicals and radical anions such as the trifluoroaceto-phenone (115,116), the rotation of the CF3 group may modulate the isotropic part of the hyperfine interaction and the scalar relaxation W0 could dominate the dipolar transition W2. In such a case, the authors have pointed out that the sign of the resulting CIDNP will be independent of the sign of the isotropic hyperfine coupling constants. 

Another way of investigating the depth in energy to which charges are trapped (a measure of the stability towards thermal decay) may be obtained by thermally stimulated currents (TSC). In this technique, electrodes on the two sides of the electret are connected via a sensitive current meter and the specimen is then heated at a constant, slow rate (1 Cmin-1, say). Discrete current peaks are observed as a function of temperature as successively more deeply trapped charges are released (Fig. 7.22). Dipolar relaxation may also give peaks in the TSC spectrum (van Turnhout, 1975). 

Observed linewidths of NMR signals in paramagnetic systems vary enormously and the conditions that govern the observed widths are considerably more complex than in diamagnetic systems. Swift (30) reviewed the problem some years ago. Relaxation times of spin-j nuclei are governed by dipolar and hyperfine exchange (Fermi contact) relaxation processes. The dipolar interaction is normally dominant except in some delocalized systems in which considerable unpaired spin density exists on nuclei far removed from the metal ions (e.g. Ti-radicals). Distinction between the two processes can be made by consideration of the different mathematical expressions involved. For dipolar relaxation when o)fx 1 (t = rate constant for rotation of the species containing the coupled pair and to, = nuclear resonance frequency) ... 

Determination of ET rate constants. The proton relaxation of the cyclopentadienyl (cp) ring of the present mixed-valence complex is caused by the magnetic dipolar interaction with the electron spin on the Fe(III) ion of the fetiicenium unit (S=l/2). Then the proton spin-lattice relaxation rate of bfc+, 1/Ti, is represented by ... 

By the same experimental technique, the temperature dependence of the nuclear spin relaxation rates was investigated for the radical cations of dimethoxy- and trimethoxybenzenes [89], The rates of these processes do not appear to be accessible by other methods. As was shown, l/Tfd of an aromatic proton in these radicals is proportional to the square of its hyperfine coupling constant. This result could be explained qualitatively by a simple MO model. Relaxation predominantly occurs by the dipolar interaction between the proton and the unpaired spin density in the pz orbital of the carbon atom the proton is attached to. Calculations on the basis of this model were performed with the density matrix formalism of MO theory and gave an agreement of experimental and predicted relaxation rates within a factor of 2. 

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