decreases the intensity of the resonance and the NOE is negative. Molecular motion Whether the NOE is positive or negative depends sensitively on the degree of rotational molecular motion. The three dipolar relaxation pathways contribute to differing extents to the spin-lattice relaxation depending a number of factors. A key one is that the balance between ω2, ω1 and ω0 depends crucially on molecular rotational correlation time, , the time it takes a molecule to rotate one radian. NMR theory shows that the transition probabilities are related to and the Larmor precession frequencies, , by the relations: where is the distance separating two spin- nuclei. For relaxation to occur, the frequency of molecular tumbling must match the Larmor frequency of the nucleus. In mobile solvents, molecular tumbling motion is much faster than . The so-called extreme-narrowing limit where ). Under these conditions the double-quantum relaxation W2 is more effective than W1 or W0, because τc and 2ω0 match better than τc and ω1. When ω2 is the dominant relaxation process, a positive NOE results. This expression shows that for the homonuclear case where I = S, most notably for 1H NMR, the maximum NOE that can be observed is 1\2 irrespective of the proximity of the nuclei. In the heteronuclear case where I ≠ S, the maximum NOE is given by 1\2 (γS/γI), which, when observing heteronuclei under conditions of broadband proton decoupling, can produce major sensitivity improvements. The most important example in organic chemistry is observation of 13C while decoupling 1H, which also saturates the 1J resonances. The value of γS/γI is close to 4, which gives a maximum NOE enhancement of 200% yielding resonances 3 times as strong as they would be without NOE. In many cases, carbon atoms have an attached proton, which causes the relaxation to be dominated by dipolar relaxation and the NOE to be near maximum. For non-protonated carbon atoms the NOE enhancement is small while for carbons that relax by relaxation mechanisms by other than dipole-dipole interactions the NOE enhancement can be significantly reduced. This is one motivation for using deuteriated solvents (e.g. CDCl3) in 13C NMR. Since deuterium relaxes by the quadrupolar mechanism, there are no cross-relaxation pathways and NOE is non-existent. Another important case is 15N, an example where the value of its magnetogyric ratio is negative. Often 15N resonances are reduced or the NOE may actually null out the resonance when