Rotating cross-relaxation rate

In Equation (15), R others encompasses all secondary interactions which are not included in the first two terms (for instance the interaction with an unpaired electron, the spin-rotation interaction,...). By contrast, the expression of the cross-relaxation rate is simply... 

The cross-relaxation rates between two spins can be experimentally measured in the laboratory (rotating frame They depend on interspin distance r and correlation time that modulates the dipole-dipole interaction [4] ... 

For macromolecules (or small molecules in viscous solvents at a low temperature) in a high magnetic field, wqTc 3> 1. At this spin-diffusion limit the rotating-frame cross-relaxation rate is twice as fast as in the laboratory frame, and the rates are of the opposite sign, 5 = —1/2 (fig. 1, top). 

Fig. 1. Contour plot of the dependence of the cross-relaxation rate in the laboratory frame, a" (solid line), and rotating frame, cr (dashed line), on the interproton distance, r, and correlation time, Tc. Rigid body isotropic motion is assumed (eqs. (1) and (2)). Top panel shows the dependence of the ratio of the two cross-relaxation rates on correlation time, a is always positive, whereas cr" is positive for tuoTc < v/5/2 and negative for woTi < /5/2.

Measurement of the cross-relaxation rate in either the laboratory or the rotating frame suffices. This assumes that the respective correlation time function, /(tc), is the same for all spin pairs. If the calibration of the crossrelaxation rate is not feasible (e.g., no suitable spin pair, spectral overlap, motion not isotropic, molecule has internal mobility), the value of /(tc) for each spin pair (or for a group of selected pairs), i.e., the correlation time, must be explicitly determined. 

In order to test the validity of equation (7) for the EHB system, we plotted I I as a function of t T, an example of which is shown in Fig. 7. The linear behaviour of the plot indicates good agreement with equation (7). Analogous plots for A-F, B-E and B-F relaxation rates as a function of tj/T also showed good linearity. The linearity of the i versus tj/T plots implies that overall rotation is extremely important in determining intramolecular cross-relaxation rates and that reorientation of each of the relaxation vectors is adequately described by one correlation time. 

Motions slower than the overall rotational correlation time, including conformational exchange on the ps-ms time scale, can lead to effects similar to cross-correlated relaxation in multiple-quantum coherences. These slow motions can only modulate isotropic spin interactions, such as J-couplings and isotropic chemical shifts, as the anisotropic interactions are already averaged out by the fast dynamics. The cross-correlated modulation of the isotropic chemical shifts (CSM) of two nuclei has the same effect as their CSA/CSA cross-correlation and can be measured from the difference in relaxation rates of ZQ and DQ coherences of the two nuclei. Two publications presented different schemes for the measurement of the CSM/CSM cross-correlated relaxation. Majumdar and Ghose proposed to evaluate the cross-relaxation rates from the conversion... 

Anisotropic and Rapid Internal Motions. The cross-relaxation rate constants depend not only on the intemuclear separation but also the correlation time. Even for a spherically symmetric rotating body, each cross-relaxation rate constant depends on two parameters. However, for a rigid spherically symmetric rotor, there is a single unique correlation time, that can be determined by relaxation methods on X nuclei, by cross-relaxation between protons that have a Imown fixed separation or by non-NMR methods based on rotational diffusion. 

Nucleic acids > ca. 10 bp long are not spherically symmetric. To a good approximation they are equivalent to circular cylinders with a hydrodynamic diameter of 20-23 A for DNA (33-35) and 25 A for RNA (35). The correlation function for such symmetric top molecules consist of three exponentials, whose arguments are combinations only of the correlation time for end over end tumbling (tl) and for rotation about the principal symmetiy axis (ts). Thus for anisotropic motion, two independent correlation times are needed to describe the rotational diffusion. The spectral density function also depends on the angle (0) the interproton vector makes with the principal axis. J(0), and hence the cross-relaxation rate constant, varies as a function of this angle according to (.16) ... 

A = cross peak intensity B = magnetic flux density / = coupling constant = population of rotamer i R y= relaxation rate between spins i and / r = internuclear distance 0 = probability of double- and zero-quantum transitions, respectively, in the rotating frame W2, Wq = transition probability for double- and zero-quantum transitions, respectively y = gyromagnetic ratio, o = cross-relaxation rate r, t2 = correlation times = correlation time tjjj = mixing time (j) ip a) = peptide backbone angles X = bond angles of peptide side-chains (Oq = Larmor frequency. 

Since the right-hand-side is independent of n, we can remove this index from J. Considerable complication is introduced if the overall rotational motion is not isotropic, since then the orientation of the intemuclear vector with respect to the rotational axes can have an important effect on cross-relaxation rates. This dependence can be an important source of structural information, but such considerations are beyond the scope of the current article. 

In general, for transitions that are slow relative to the rotational tumbling time, cross-relaxation rates are Just the weighted averages of the rates for the various conformers. 

The fact that mutually different cross sectional patterns were observed in the SC-2D NMR spectrum where 7 was chosen to be 0 ms indicates that both inter- and intramolecular cross relaxation rates and spin flip-flop rates between interacting pairs of protons are relatively slow. This can be understood if one considers that dipolar interactions are partially averaged out by fast translational and rotational molecular motions in the liquid crystalline phase in contrast to the solid phase. 

The value of the magnetic hyperfine interaction constant C = 22.00 kHz is supposed to be reliably measured in the molecular beam method [71]. Experimental data for 15N2 are shown in Fig. 1.24, which depicts the density-dependence of T2 = (27tAv1/2)-1 at several temperatures. The fact that the dependences T2(p) are linear until 200 amagat proves that binary estimation of the rotational relaxation rate is valid within these limits and that Eq. (1.124) may be used to estimate cross-section oj from... 

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