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# Introduction

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

#### Questions?

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# Some introductory elements

Neutron diffraction (as well as X-ray diffraction) is a standard method to study the crystal structure of a solid, or, in a broader manner, to have insights about the structure of a material. Every type of materials can be investigated with this technique, ranging from crystals to liquids via disordered polymers or colloids in solution. As matter is never completely disordered, these techniques yield at least correlation distances exhibited by the sample when it is disordered, diffraction peaks corresponding to a particular crystal structure when it is crystalline.

One generally distinguishes between cold neutrons (energy from 5·10-5 eV to 0.025eV, wavelength between roughly 2 and 40Å), thermal neutrons (energy of about 0.025eV, wavelength of about 1.8 Å) and hot neutrons (energy of about 0.2eV, wavelength of about 0.6Å). Neutrons used for scattering studies have wavelengths of the same order of magnitude of interatomic distances, which explains why they are such a useful tool for investigating the structure of materials.

A typical neutron scattering experiment: an incident plane wave of flux J0  irradiates a sample, from which the scattered spherical wave emanates in all directions.

#### Neutron cross-section and scattering length

The task of the scientist is to measure the flux J of the scattered rays as a function of the scattering direction. The ratio J/J0 is referred to as the differential cross section. Integrating the differential cross section throughout the solid angle Ω gives the total scattering cross section which is:

$$\sigma_{tot} = \frac{total\ number\ of\ particles\ scattered\ in\ all\ directions\ per\ second}{flux\ of\ the\ incident\ beam}$$

The diffraction of neutrons by matter results from the combination of two different phenomena:
1-    scattering of neutrons by individual atomic nuclei in it,
2-    interference among the waves scattered by these primary events.
Strictly speaking the term diffraction refers to combination of 1- and 2-, whereas scattering refers only to phenomenon 1-.
Neutrons are scattered by the nuclei in matter. The efficacy of neutron scattering by a nucleus is expressed by its scattering length b:

$$b^2 = \frac{d\sigma}{d\Omega}$$

whose value is dependent of the incident neutron wavelength. The strength of the nucleus-neutron interaction depends on the details of the nuclear structure, which is not related to the atomic number in any simple way. Therefore, the value of b can vary greatly between elements of similar atomic number and even between isotopes of the same element. For a particular nucleus concerned, it depends on the spin state of the nucleus-neutron system. If the nucleus has nonzero spin i, the spin of the nucleus-neutron system is either i + 1/2 or i - 1/2 and the associated scattering length is either b+ or b-, respectively. In case of scattering by an assembly of nuclei, the probable presence of isotopes and if the spin of the nuclei is nonzero, the value of b will vary randomly from nucleus to nucleus. The consequence is that the scattered intensity contains not only a component that reflects the structure as usual, but also another component that arises simply from this randomness and has nothing to do with the structure. The first component is associated to a coherent scattering length bcoh and the corresponding coherent cross section σcoh whereas the second one to an incoherent scattering length binc and the corresponding incoherent cross section σinc.

#### Complementarity between neutron and X-ray diffractions

The following table gives the values of the thermal neutron coherent and incoherent cross-sections and X-ray cross sections at 3.3 keV (corresponding to the energy of the Kα line) for some common elements and isotopes, in barns (= 10-24 cm2).

Element
Thermal neutronsX rays (3.3 keV)

σcoh (barn)
σinc (barn)σ (barn)
1H
1.76
79.9
0.79
2D
5.59
2.04

C
5.55
0.001
1347
N
11.01
0.49
2600
O
4.23
0.0008
4394

If X-ray and neutron diffractions have a lot of features in common, they also exhibit essential differences that we would like to emphasize here.

-    Energy: of the order of 10 keV for X-rays, of the order of 10 meV for neutrons. This will have an important consequence for the study of atomic motions.
-    Cross sections: whereas the X-ray atomic scattering factor increases smoothly with atomic number, the neutron cross section varies seemingly randomly among elements. Thus hydrogen is almost invisible to X-rays and scattering from heavy atoms can often overwhelm the scattering from the rest of the material. This difficulty can be advantageously avoided with neutron diffraction.
-    Magnetic moment: in contrast to X-rays, neutrons have a magnetic moment, allowing the use of neutrons to study magnetic structure of materials.

The recourse to neutron scattering will be especially important in polymer science, and this technique is still much more powerful when used in conjunction with deuterium labeling. In this technique, selected hydrogen atoms in polymer molecules are replaced by deuterium. As the coherent scattering lengths of hydrogen and deuterium are very different, the cross section for scattering neutrons is then greatly modified but all the other physical properties of the molecules remain essentially unaltered. Selective deuterium labeling thus allows one to make some of the molecules or only one part of the molecules "visible".
Note also that if the large incoherent neutron scattering cross section by hydrogen is a complication for structure studies - that can be overcome by sample deuteration - this is a blessing to study the dynamics of hydrogenous materials.

#### References

R.-J. Roe, Methods of X-Ray and Neutron Scattering in Polymer Science, Oxford University Press, 2000.
B.T.M. Willis and C.J. Carlile, Experimental Neutron Scattering, Oxford University Press, 2009.

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