Auto-relaxation rates

The coupling term, traditionally denoted by cr B (which has however nothing to do with the screening coefficient of Section 2.2), is the so-called cross-relaxation rate and is a relaxation parameters which depends exclusively on the dipolar interaction between nuclei A and B, contrary to auto-relaxation rates which are compounds of several contributions. For instance, if A is a carbon-13, the auto-relaxation rate can always be written as... 

When r s, one has interconversion between operators Br and Bs, and Rrs is a cross-relaxation rate. Note that the cross-relaxation may or may not contain interference effects depending on the indices l and /, which keep track of interactions Cyj and C,. Cross-correlation rates and cross-relaxation rates have not been fully utilized in LC. However, there is a recent report41 on this subject using both the 13C chemical shielding anisotropy and C-H dipolar coupling relaxation mechanisms to study a nematic, and this may be a fruitful arena in gaining dynamic information for LC. We summarize below some well known (auto-)relaxation rates for various spin interactions commonly encountered in LC studies. 

Now the expectation (mean) value of any physical observable (A(t)) = Yv Ap(t) can be calculated using Eq. (22) for the auto-correlation case (/ = /). For instance, A can be one of the relaxation observables for a spin system. Thus, the relaxation rate can be written as a linear combination of irreducible spectral densities and the coefficients of expansion are obtained by evaluating the double commutators for a specific spin-lattice interaction X in the auto-correlation case. In working out Gm x) [e.g., Eq. (21)], one can use successive transformations from the PAS to the (X, Y, Z) frame, and the closure property of the rotation group to rewrite D2mG(Qp ) so as to include the effects of local segmental, molecular, and/or collective motions for molecules in LC. The calculated irreducible spectral densities contain, therefore, all the frequency and orientational information pertaining to the studied molecular system. 

The symbols Rauto and Rcross within the relaxation matrix are the auto- and cross-relaxation rates, respectively. and (l2Z) are the longitudinal magnetizations of spin 1 and 2, respectively, and the brackets indicate averaging over the whole ensemble of spins. Rcross in terms of the spectral densities is given by... 

Fig. 2 HN(CO)CA-derived experiment for the measurement of /nhc dipole-CSA CCR-rates. Various ratios of NBD peptide and NEMO were used. A 200-fold excess of ligand optimal conditions. B 100-fold excess auto relaxation and cross relaxation are too fast. The resonance assignment of the peptide is indicated...

A suitable CCR-rate to determine the backbone torsion angle 0 by CCR is the /NHCHcf dipole-dipole CCR-rate that conveniently can be measured by an HNCA-derived experiment [44]. Alternatively, like for the torsion angle 0, the FcuaCii-i) dipole-CSA CCR can be measured by a triple-resonance experiment that is derived from a combination of HNCA and HNCO experiments [45]. Also, CCR experiments for which the rate depends on 0 and

dipole-dipole CCR experiment can be used [46]. Unfortunately for the peptide under investigation, we were not able to successfully record any of these spectra, possibly due to the relatively strong auto relaxation. 

It is possible to extract quantitative data from a 2D display. For example, in the simplest case of a two-site exchange with equal populations and equal spin-lattice relaxation times, the exchange rate k can be obtained from the ratio of the intensity of an auto-peak to that of a cross-peak (eq. 1) (12). 

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