Editing Nuclear Overhauser effect (section)

==Relaxation==
[[File:NOE energy levels 2.png|thumb|upright=1.3|Nuclear spin energy level diagram for two spin {{frac|1|2}} nuclei.<ref name="Claridge" />]]
[[File:Steady state noe ps.png|thumb|upright=1.3|Steady-state pulse sequence for <sup>1</sup>H NOE experiments]]

The NOE and nuclear [[spin-lattice relaxation]] are closely related phenomena. For a single spin-{{frac|1|2}} nucleus in a magnetic field there are two energy levels that are often labeled &alpha; and &beta;, which correspond to the two possible spin quantum states, +{{frac|1|2}} and -{{frac|1|2}}, respectively. At thermal equilibrium, the population of the two energy levels is determined by the [[Boltzmann distribution]] with spin populations given by ''P''<sub>α</sub> and ''P''<sub>β</sub>. If the spin populations are perturbed by an appropriate RF field at the transition energy frequency, the spin populations return to thermal equilibrium by a process called ''spin-lattice relaxation''. The rate of transitions from α to β is proportional to the population of state α, ''P''<sub>α</sub>, and is a first order process with rate constant ''W''. The condition where the spin populations are equalized by continuous RF irradiation (''P''<sub>α</sub> = ''P''<sub>β</sub>) is called ''saturation'' and the resonance disappears since transition probabilities depend on the population difference between the energy levels.

In the simplest case where the NOE is relevant, the resonances of two spin-{{frac|1|2}} nuclei, I and S, are chemically shifted but not [[J-coupling|J-coupled]]. The energy diagram for such a system has four energy levels that depend on the spin-states of I and S corresponding to αα, αβ, βα, and ββ, respectively. The ''W'''s are the probabilities per unit time that a transition will occur between the four energy levels, or in other terms the rate at which the corresponding spin flips occur. There are two single quantum transitions, ''W''<sub>1</sub><sup>I</sup>, corresponding to αα ➞ βα and αβ ➞ ββ; ''W''<sub>1</sub><sup>S</sup>, corresponding to αα ➞ αβ and βα ➞ ββ; a zero quantum transition, ''W''<sub>0</sub>, 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 rule]]s) 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 (''W''<sub>2</sub>) and between the αβ and the βα states (''W''<sub>0</sub>).

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 (<math>M_{S} = 0 </math>) is defined by the expression:
::: <math>\eta_{I}^{S} = \left(\frac{M_{I}^{S}-M_{0I}}{M_{0I}}\right)</math>
where <math>M_{0I}</math> is the magnetization (resonance intensity) of nucleus <math>I</math> 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
::: <math>\eta_{I}^{S}=\frac{M_{I}^{S}-M_{0I}}{M_{0I}} = \frac{\gamma_S }{\gamma_I }\frac{\sigma_{IS}}{\rho_{I}} = \frac{\gamma_S}{\gamma_I}\left(\frac{W_{2}-W_{0}}{2W_{1}^{I}+W_{0}+W_{2}}\right)</math>
where
::: <math>\rho_{I} = 2W_{1}^{I}+W_{0}+W_{2}</math> and <math>\sigma_{IS} = W_{2} - W_{0}</math>.
<math>\rho_{I}</math> is the total longitudinal dipolar relaxation rate (<math>1/T_{1}</math>) of spin ''I'' due to the presence of spin ''s'', <math>\sigma_{IS}</math> is referred to as the ''cross-relaxation'' rate, and <math>\gamma_{I}</math> and <math>\gamma_{S}</math> are the [[gyromagnetic ratio#For a nucleus|magnetogyric ratios]] characteristic of the <math>I</math> and <math>S</math> nuclei, respectively.

Saturation of the degenerate ''W''<sub>1</sub><sup>S</sup> transitions disturbs the equilibrium populations so that ''P''<sub>αα</sub> = ''P''<sub>αβ</sub> and ''P''<sub>βα</sub> = ''P''<sub>ββ</sub>. The system's relaxation pathways, however, remain active and act to re-establish an equilibrium, except that the ''W''<sub>1</sub><sup>''S''</sup> transitions are irrelevant because the population differences across these transitions are fixed by the RF irradiation while the population difference between the ''W''<sub>I</sub> transitions does not change from their equilibrium values. This means that if only the single quantum transitions were active as relaxation pathways, saturating the <math>S</math> resonance would not affect the intensity of the <math>I</math> resonance. Therefore to observe an NOE on the resonance intensity of I, the contribution of <math>W_{0}</math> and <math>W_{2}</math> 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 <math>\gamma_{I} = \gamma_{S}</math>, if <math>W_{2}</math> is the dominant relaxation pathway, then saturating <math>S</math> increases the intensity of the <math>I</math> resonance and the NOE is ''positive'', whereas if <math>W_0</math> is the dominant relaxation pathway, saturating <math>S</math> decreases the intensity of the <math>I</math> resonance and the NOE is ''negative''.