corresponding to αα ➞ βα and αβ ➞ ββ; W1S, corresponding to αα ➞ αβ and βα ➞ ββ; a zero quantum transition, W0, corresponding to βα ➞ αβ, and a double quantum transition corresponding to αα ➞ ββ. While rf irradiation can only induce single-quantum transitions (due to so-called quantum mechanical selection rules) giving rise to observable spectral lines, dipolar relaxation may take place through any of the pathways. The dipolar mechanism is the only common relaxation mechanism that can cause transitions in which more than one spin flips. Specifically, the dipolar relaxation mechanism gives rise to transitions between the αα and ββ states (W2) and between the αβ and the βα states (W0). Expressed in terms of their bulk NMR magnetizations, the experimentally observed steady-state NOE for nucleus I when the resonance of nucleus S is saturated () is defined by the expression: where is the magnetization (resonance intensity) of nucleus at thermal equilibrium. An analytical expression for the NOE can be obtained by considering all the relaxation pathways and applying the Solomon equations to obtain where and . is the total longitudinal dipolar relaxation rate () of spin I due to the presence of spin s, is referred to as the cross-relaxation rate, and and are the magnetogyric ratios characteristic of the and nuclei, respectively. Saturation of the degenerate W1S transitions disturbs the equilibrium populations so that Pαα = Pαβ and Pβα = Pββ. The system's relaxation pathways, however, remain active and act to re-establish an equilibrium, except that the W1S transitions are irrelevant because the population differences across these transitions are fixed by the RF irradiation while the population difference between the WI transitions does not change from their equilibrium values. This means that if only the single quantum transitions were active as relaxation pathways, saturating the resonance would not affect the intensity of the resonance. Therefore to observe an NOE on the resonance intensity of I, the contribution of and must be important. These pathways, known as cross-relaxation pathways, only make a significant contribution to the spin-lattice relaxation when the relaxation is dominated by dipole-dipole or scalar coupling interactions, but the scalar interaction is rarely important and is assumed to be negligible. In the homonuclear case where , if is the dominant relaxation pathway, then saturating increases the intensity of the resonance and the NOE is positive, whereas if is the dominant relaxation pathway, saturating