# Motional narrowing

In physics and chemistry, motional narrowing is a phenomenon where a certain resonant frequency has a smaller linewidth than might be expected, due to motion in an inhomogeneous system.[1] The discovery of motional narrowing has been attributed to Nicolaas Bloembergen during his thesis work in the 1940s[2]

## Example: NMR spectroscopy

A common example is NMR.[1] In this process, the nuclear spin of an atom starts rotating, with the frequency of rotation proportional to the external magnetic field that the atom experiences. However, in an inhomogeneous medium, the magnetic field often varies from point to point (depending, for example, on the magnetic susceptibility of nearby atoms), so the frequency of nuclear spin rotation is different in different places. Therefore, when detecting the resonant rotation frequency, there is a linewidth (i.e., finite range of different frequencies) due to the variation in that resonant frequency from point to point. (This is called "inhomogeneous broadening".)

However, if the atoms are diffusing around the system, they will experience a higher magnetic field than average sometimes, and a lower magnetic field than average other times. Therefore, (in accordance with the central limit theorem), the time-averaged magnetic field experienced by an atom has less variation than the instantaneous magnetic field does. As a consequence, when detecting the resonant rotation frequency, the linewidth is smaller (narrower) than it would be if the atoms were stationary. This is the motional narrowing effect.

## Example: Vibrational spectroscopy

A similar phenomenon occurs in many other systems. Another example is vibrational modes in a liquid. Each molecule of the liquid has vibrational modes, and the vibrational frequency is influenced by the positions of nearby molecules. However, if the nearby molecules reorient and move around fast enough, the vibration will essentially occur at an averaged frequency, and therefore have a smaller linewidth. For example, simulations suggest that the OH stretch vibration linewidth in liquid water is 30% smaller than it would be without this motional narrowing effect.[3]

## References

1. ^ a b Solid state: nuclear methods by J. N. Mundy, section 6.2.1.1, page 441.
2. ^ Eli Yablonovitch (2017). "Obituary: Nicolaas Bloembergen (1920–2017)". Nature. 550 (7677): 458. Bibcode:2017Natur.550..458Y. doi:10.1038/550458a. PMID 29072260.
3. ^ "The Effects of Dissolved Halide Anions on Hydrogen Bonding in Liquid Water", J. D. Smith et al., doi:10.1021/ja071933z