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Figure 5.20

5.12 THE HOMONUCLEAR NUCLEAR OVERHAUSER EFFECT (NOE)

In the previous chapter the NOE was introduced but not explained in detail. Why should equalization of the populations of one spin cause an enhancement of the population difference of a nearby spin? The key to understand this effect is to take a close look at the longitudinal (Ti) relaxation process, now in the context of two nearby protons within one molecule. Consider two nonequivalent protons Ha and Hb in an organic molecule, with a distance between them of less than 5 A. If we saturate nucleus Ha by selective low-power irradiation, it will be out of equilibrium because the populations in the a (lower) and ft (upper) energy levels will be equal. From the point of view of Ha, Hb will be rotating around it at the tumbling rate of the molecule. For those molecules that are tumbling at a rate very close to the Larmor frequency (e.g., 300 MHz in a 7.05 T magnet), Ha will experience an oscillating magnetic field that will stimulate it to drop from the ft level to the a level. This is the major mechanism for Ti relaxation; if it were the only mechanism, there would be no nuclear Overhauser effect.

How can relaxation lead to an NOE? There is another pathway for relaxation that involves both Ha and Hb changing spin state simultaneously. For example, if both Ha and Hb are in the ft state (overall state ftft), both can flip simultaneously to the a state (overall state aa). This is called a double-quantum transition, and it is sensitive to magnetic fields oscillating at twice the Larmor frequency (600 MHz in our case). Small organic molecules (MW < 1000) tumble rapidly and have significant populations tumbling at both the Larmor frequency and twice the Larmor frequency (Fig. 5.13), so this "cross-relaxation" (simultaneous spin flip) is a significant relaxation pathway. Consider an extreme case where double-quantum relaxation is the only pathway available: after saturation of Ha, every time an Ha spin drops down from the ft state to the a state it drags an Hb spin (in the same molecule) down with it. This is fine for the Ha spins—they are reestablishing the Boltzmann distribution by increasing

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