QENS refers to those inelastic processes in neutron scattering that are almost elastic. The term is usually considered to mean a broadening of the elastic line in the energy spectrum rather than the appearance of discrete peaks representing inelastic events. As already mentioned, neutron scattering - and thus also QENS - contains both coherent and incoherent components. The coherent component yields information about interference phenomena between atoms, such as lattice distortions or short-range order. Incoherent scattering relates to scattering by individual atoms: if the atoms or molecules undergo stochastic motions (translational or rotational diffusion) during the scattering event, this single-particle scattering is accompanied by the transfer of energy to or from the neutrons. As the motions are not quantized, this gives a broadening of the sharp line arising from elastically scattered neutrons. Thus QENS can be used to study diffusion in solids, where an individual particle performs a random walk over the crystal lattice. In this case, the incoherent component of scattering gives the jump rates, jump lengths and jump directions of the diffusion particle.

In general, for an atom which diffuses within a fixed volume, the incoherent scattering function S_{inc}(**q**,ω) in the elastic region is separable into a purely elastic component, A_{0}(**q**)δ(ω), and a quasi-elastic component centered on ω = 0, A_{1}(**q**)L(ω), where δ(ω) is the delta function and L(ω) is a Lorentzian function. The first term contains information on the geometry of the diffusing entity and the Lorentzian term information on the time scale of its diffusion. The fall in the elastic intensity as a function of q can be represented by a structure factor called the elastic incoherent structure factor (EISF), which is defined as:

For purely translational diffusion, the EISF is everywhere zero except at q = 0. For the rotational diffusion of a particle, the EISF is unity at q = 0 and falls to a minimum at a q value which is inversely related to the radius of gyration of the rotating particle and is thereafter oscillatory in character. The precise shape of EISF is indicative of the geometry of the diffusional process and is particularly sensitive to the model in the q region beyond the first minimum.

Another feature of the incoherent scattering function, namely the q-dependence of the quasielastic line width f(q) of the central component, which is due to translational long-range diffusion, can be also used to validate a diffusion model. For some cases in which the EISF is the same for two different diffusion processes, it allows one to determine which one is the faster or slower, respectively.

#### References

B.T.M. Willis and C.J. Carlile, Experimental Neutron Scattering, Oxford University Press, 2009.

R.E. Lechner, Neutron investigations of superprotonic conductors, Ferroelectrics, 167 (1995) 83-98.