page_title: Nuclear Overhauser effect
section_title: Relaxation
hierarchical_section_title: Nuclear Overhauser effect / Relaxation
context_page_description: The nuclear Overhauser effect is the transfer of nuclear spin polarization from one population of spin-active nuclei to another via cross-relaxation. A phenomenological definition of the NOE in nuclear magnetic resonance spectroscopy is the change in the integrated intensity of one NMR resonance that occurs when another is saturated by irradiation with an RF field. The change in resonance intensity of a nucleus is a consequence of the nucleus being close in space to those directly affected by the RF perturbation.
  The NOE is particularly important in the assignment of NMR resonances, and the elucidation and confirmation of the structures or configurations of organic and biological molecules. The two-dimensional NOE experiment is an important tool to identify stereochemistry of proteins and other biomolecules in solution, whereas in solid form crystal x-ray diffraction must be used to identify the stereochemistry.
context_section_description: The NOE and nuclear spin-lattice relaxation are closely related phenomena. For a single spin-​¹⁄₂ nucleus in a magnetic field there are two energy levels that are often labeled α and β, which correspond to the two possible spin quantum states, +​¹⁄₂ and -​¹⁄₂, respectively.  At thermal equilibrium, the population of the two energy levels is determined by the Boltzmann distribution with spin populations given by Pα and Pβ. 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, Pα, and is a first order process with rate constant W.  The condition where the spin populations are equalized by continuous RF irradiation (Pα = Pβ) 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-​¹⁄₂ nuclei, I and S, are chemically shifted but not 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, W₁ᴵ, corresponding to αα ➞ αβ and βα ➞ ββ; W₁S, corresponding to αα ➞ βα 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, satur
caption: Nuclear spin energy level diagram for two spin ​1⁄2 nuclei. [3]