Cross-relaxation rate constants

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. 

Hence a plot of 77 against mixing time will give a straight line of slope cr/5 this is a method used for measuring the cross-relaxation rate constant. A single experiment for one value of the mixing time will reveal the presence of NOE enhancements. 

It is important to realise that the value of the steady-state NOE enhancement depends on the ratio of cross-relaxation rate constant to the self relaxation rate constant for the spin which is receiving the enhancement. If this spin is relaxing quickly, for example as a result of interaction with many other spins, the size of the NOE enhancement will be reduced. So, although the size of the enhancement does depend on the cross-relaxation rate constant the size of the enhancement cannot be interpreted in terms of this rate constant alone. This point is illustrated by the example in the margin. 

We see that the time dependence and size of the NOE enhancement depends on the relative sizes of the cross-relaxation rate constant oIS and the self relaxation rate constants R, and Rs. It turns out that these self-rates are always positive, but the cross-relaxation rate constant can be positive or negative. The reason for this is that <7IS = (W2 - W(j) and it is quite possible for W(j to be greater or less than W2. 

A positive cross-relaxation rate constant means that if spin S deviates from equilibrium cross-relaxation will increase the magnetization on spin I. This leads to an increase in the signal from I, and hence a positive NOE enhancement. This situation is typical for small molecules is non-viscous solvents. 

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) ... 

The complete set of pairwise cross-relaxation rate constants forms the off-diagonal elements of the full relaxation matrix, L. The diagonal elements are the autorelaxation rates of each of the protons. The goal is to determine... 

Wi), As = (2Ws) aMRis = (2W/ + 2Ws). The term pi/s is known as the autorelaxation rate constant (sometimes referred to as self relaxation rate constant) which describes the I and 5 spin lattice relaxation, ajs is known as the cross-relaxation rate constant and describes the rate at which magnetisation is transferred between the two spins. Ai/s describes the transfer of the Iz(t)Sz(t) magnetisation to / and S spin magnetisation respectively. The term Ris is the self relaxation rate constant of 2fSz. Unfortunately, like the Bloch equations, the first-order rate constants are treated as parameters without taking into account the explicit nature of the relaxation mechanism. 

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