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# Inelastic neutron scattering

The ILL has firmly established itself as a pioneer in neutron science and technology. Neutron beams are used to carry out frontier research in diverse fields.

### Neutron techniques

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# Inelastic neutron scattering (INS)

In inelastic neutron scattering experiments, the intensity of the scattered neutrons is analysed with respect to the momentum ħq and the energy ħω exchanged between the neutrons and the scattering system. One measures the double differential scattering cross-section:

$$\frac{\partial \sigma}{\partial \Omega \partial \omega} = \frac{k_f}{k_i} [\sigma_{coh}\cdot S_{coh}(\overrightarrow{Q},\omega) + \sigma_{inc}\cdot S_{inc}(\overrightarrow{Q},\omega)]/4\pi$$

where Ω is the solid angle, σcoh and σinc are the coherent and incoherent cross-sections,  Scoh(q,ω) and Sinc(q,ω) are the coherent and incoherent scattering functions, respectively.

A hypothetical INS spectrum is shown below. One can recognize the different kinds of possible excitations across a wide frequency range.

#### Incoherent inelastic neutron scattering (IINS)

The vibrational interactions between a molecule and an electromagnetic radiation can be described by a simple harmonic oscillator model. In terms of vibrations, a molecule is equivalent to a system of weakly coupled harmonic oscillators. The decomposition of coupled harmonic oscillators into a collection of independent oscillators is known as a normal mode expansion and the independent oscillators are called normal modes. These normal modes are defined as modes of vibration where the respective atomic motions of the atoms all reach their maximum and minimum displacements at the same time. These normal modes can be expressed in terms of bond stretches and angle deformation. The vibration frequencies of these modes are characteristic of a molecule and this property is used in vibrational spectroscopies to characterize a sample.

Whereas coherent inelastic neutron scattering can be used for the study of collective excitations in single-crystals, such as phonons and magnons, incoherent inelastic neutron scattering probes the local environment of the scattering atom and is complementary to the optical techniques of infrared and Raman spectroscopies. This vibrational neutron spectroscopy yields information both on the local structure and on the sample dynamics, e.g. diffusion mechanisms. IINS resembles Raman scattering with photons replaced by neutrons but with no neutron selection rules.

#### References

J.L. Koenig, Spectroscopy of polymers, Elsevier, 1999, 2nd edition.
D.J. Bower and W.F. Maddams, The vibrational spectroscopy of polymers, Cambridge University Press, 1989.