# Neutron reflectometry

Neutron reflectometry is a neutron diffraction technique for measuring the structure of thin films, similar to the often complementary techniques of X-ray reflectivity and ellipsometry. The technique provides valuable information over a wide variety of scientific and technological applications including chemical aggregation, polymer and surfactant adsorption, structure of thin film magnetic systems, biological membranes, etc.

## Technical details

The technique involves shining a highly collimated beam of neutrons onto an extremely flat surface and measuring the intensity of reflected radiation as a function of angle or neutron wavelength. The exact shape of the reflectivity profile provides detailed information about the structure of the surface, including the thickness, density, and roughness of any thin films layered on the substrate.

Neutron reflectometry is a specular reflection technique, where the angle of the incident beam is equal to the angle of the reflected beam. The reflection is usually described in terms of a momentum transfer vector, denoted ${\displaystyle q_{z}}$, which describes the change in momentum of a neutron after reflecting from the material. Conventionally the ${\displaystyle z}$ direction is defined to be the direction normal to the surface, and for specular reflection, the scattering vector has only a ${\displaystyle z}$-component. A typical neutron reflectometry plot displays the reflected intensity (relative to the incident beam) as a function of the scattering vector:

${\displaystyle q_{z}={\frac {4\pi }{\lambda }}\sin(\theta )}$

where ${\displaystyle \lambda }$ is the neutron wavelength, and ${\displaystyle \theta }$ is the angle of incidence. The Abeles matrix formalism or the Parratt recursion can be used to calculate the specular signal arising from the interface.

The wavelength of the neutrons used for reflectivity are typically on the order of 0.2 to 1 nm (2 to 10 Å). This technique requires a neutron source, which may be either a research reactor or a spallation source (based on a particle accelerator). Like all neutron scattering techniques, neutron reflectometry is sensitive to contrast arising from different nuclei (as compared to electron density, which is measured in x-ray scattering). This allows the technique to differentiate between various isotopes of elements. Neutron reflectometry measures the neutron scattering length density (SLD) and can be used to accurately calculate material density if the atomic composition is known.

## Neutron scattering lengths

A neutron may pass by a nucleus with a probability determined by the nuclear interaction distance, or be absorbed, or undergo scattering that may be either coherent or incoherent.[1] The scattering length of neutrons varies by element and isotope in a way that appears random, whereas the scattering of X-rays generally increases with the atomic number.[1][2]

The scattering length may be either positive or negative. The scattering cross-section is equal to the square of the scattering length multiplied by 4π,[3] i.e. the area of a circle with radius twice the scattering length. In some cases, as with titanium and niobium, it is possible to mix isotopes of an element whose lengths are of opposite signs to give a net scattering length of zero, in which case coherent scattering will not occur at all. However, neutrons will still undergo strong incoherent scattering in these materials.[1]

There is a large difference in scattering length between protium (-0.374) and deuterium (0.667). This means that protium, the most common isotope of hydrogen, is poorly imaged due to its smaller absolute magnitude and because its scattering density tends to cancel that of adjacent carbon, nitrogen, or oxygen, which have positive scattering lengths. Because of these factors, as well as the much larger incoherent scattering cross section of protium, ordinary proteins cannot be imaged as well as those that are fully deuterated. Specific exchangeable hydrogens in a non-deuterated protein may be imaged if it is exposed to heavy water.[4]

element protons isotope X-ray scattering
1012bX/cm
neutron scattering
1012bcoh/cm
coherent
cross-section
σcoh (barn)
incoherent
cross-section
σinc (barn)
absorption
cross-section
σa (barn)
Hydrogen 1 1 0.282[2][5] -0.374[1][2][5][6] 1.758[1] 79.7,[6] 80.27[1] 0.33,[6] 0.383[1]
Hydrogen 1 2 0.282[2][5] 0.667[1][2][5][6] 5.592[1] 2.0,[6] 2.05[1] 0.0005[1][6]
Boron 5 natural 0.530[1] 3.54[1] 1.70[1] 767.0[1]
Carbon 6 12 1.69[2][5] 0.665[1][2][5][6] 5.550[1] 0.0,[6] 0.001[1] 0.0035,[6] 0.004[1]
Nitrogen 7 14 1.97[2][5] 0.936,[1] 0.940,[2] 0.94[5][6] 11.01[1] 0.3,[6] 0.5[1] 1.9[1][6]
Oxygen 8 16 2.16,[2] 2.26[5] 0.580,[2] 0.58[1][5][6] 4.232[1] 0.0,[6] 0.000[1] 0.00019,[6] 0.0002[1]
Aluminum 13 natural 0.345,[1] 0.35[6] 1.495[1] 0.0,[6] 0.008[1] 0.23,[6] 0.231[1]
Silicon 14 natural 0.42[6][7] 0.0[6] 0.17[6]
Phosphorus 15 30 3.23[2] 0.510[2]
Sulfur 16 32 4.51[2][5] 0.280,[2] 0.28[5]
Titanium 22 natural -0.344,[1] -0.34[6][7] 1.485[1] 2.87,[1] 3.0[6] 6.09,[1] 6.1[6]
Vanadium 23 natural -0.038[1][1] 0.018[1] 5.07[1] 5.08[1]
Chromium 24 natural 0.364[1] 1.66[1] 1.83[1] 3.05[1]
Manganese 25 55 (natural) -0.373[1] 1.75[1] 0.4[1] 13.3[1]
Iron 26 natural 0.945,[1] 0.95[6] 11.22[1] 0.4[1][6] 2.56,[1] 2.6[6]
Nickel 28 natural 1.03[1] 13.3[1] 5.2[1] 4.49[1]
Copper 29 natural 0.772[1] 7.485[1] 0.55[1] 3.78[1]
Zirconium 40 natural 0.716,[1] 0.72[6] 6.44[1] 0.02,[1] 0.3[6] 0.18,[6] 0.185[1]
Niobium 41 93 (natural) 0.7054[1] 6.253[1] 0.0024[1] 1.15[1]
Molybdenum 42 natural 0.672[1] 5.67[1] 0.04[1] 2.48[1]
Cadmium 48 natural 0.487[1] 3.04[1] 3.46[1] 2520[1]
Tin 50 natural 0.623[1] 4.87[1] 0.022[1] 0.626[1]
Cerium 58 natural 0.48[6] 0.0[6] 0.63[6]
Gadolinium 64 natural 0.65[1] 29.3[1] 151[1] 49700[1]
Tantalum 73 natural 0.691[1] 6.00[1] 0.01[1] 20.6[1]
Tungsten 74 natural 0.486[1] 2.97[1] 1.63[1] 18.3[1]
Gold 79 197 22.3[2] 0.760[2]
Lead 82 natural 0.941[1] 11.115[1] 0.003[1] 0.171[1]
Thorium 90 232 (natural) 0.98[6] 0.00[6] 7.4[6]
Uranium 92 natural 0.842[1][6] 8.903[1] 0.00,[6] 0.005[1] 7.5,[6] 7.57[1]

More comprehensive data is available from NIST[8] and Atominstitut of Vienna.[9]

## Comparison to other reflectometry techniques

Although other reflectivity techniques (in particular optical reflectivity, x-ray reflectometry) operate using the same general principles, neutron measurements are advantageous in a few significant ways. Most notably, since the technique probes nuclear contrast, rather than electron density, it is more sensitive for measuring some elements, especially lighter elements (hydrogen, carbon, nitrogen, oxygen, etc.). Sensitivity to isotopes also allows contrast to be greatly (and selectively) enhanced for some systems of interest using isotopic substitution, and multiple experiments that differ only by isotopic substitution can be used to resolve the phase problem that is general to scattering techniques. Finally, neutrons are highly penetrating and typically non-perturbing: which allows for great flexibility in sample environments, and the use of delicate sample materials (e.g., biological specimens). By contrast x-ray exposure may damage some materials, and laser light can modify some materials (e.g. photoresists). Also, optical techniques may include ambiguity due to optical anisotropy (birefringence), which complementary neutron measurements can resolve. Dual polarisation interferometry is one optical method which provides analogous results to neutron reflectometry at comparable resolution although the underpinning mathematical model is somewhat simpler, i.e. it can only derive a thickness (or birefringence) for a uniform layer density.

Disadvantages of neutron reflectometry include the higher cost of the required infrastructure, the fact that some materials may become radioactive upon exposure to the beam, and insensitivity to the chemical state of constituent atoms. Moreover, the relatively lower flux and higher background of the technique (when compared to x-ray reflectivity) limit the maximum value of ${\displaystyle q_{z}}$ that can be probed (and hence the measurement resolution).

## Partial list of neutron reflectometers

A catalogue of neutron reflectometers is available at www.reflectometry.net.

## Partial list of neutron reflectometry software

Other neutron reflection analysis software is listed at www.reflectometry.net.

## References

1. M.T. Hutchings; P.J. Withers; T.M. Holden; Torben Lorentzen (Feb 28, 2005). Introduction to the Characterization of Residual Stress by Neutron Diffraction. CRC Press.
2. Dmitri I. Svergun; Michel H. J. Koch; Peter A. Timmins; Roland P. May (Aug 8, 2013). Small Angle X-Ray and Neutron Scattering from Solutions of Biological Macromolecules. OUP Oxford.
3. ^ Amparo Lopez-Rubio & Elliot Paul Gilbert (2009). "Neutron scattering: a natural tool for food science and technology research" (PDF). Trends in Food Science & Technology: 1–11.
4. ^ Fong Shu; Venki Ramakrishnan & Benno P. Schoenborn. "Enhanced visibility of hydrogen atoms by neutron crystallography on fully deuterated myoglobin". PNAS. 97 (8): 3872–3877. Bibcode:2000PNAS...97.3872S. doi:10.1073/pnas.060024697.
5. Oliver C. Mullins, Eric Y. Sheu, eds. (Nov 11, 2013). Structures and Dynamics of Asphaltenes. Springer Science & Business Media. p. 161.
6. N.K. Kanellopoulos, ed. (Sep 26, 2000). Recent Advances in Gas Separation by Microporous Ceramic Membranes.
7. ^ a b F. Rodríguez-Reinoso, Jean Rouquerol, KK Unger, Kenneth S.W. Sing, eds. (Aug 26, 1994). Characterization of Porous Solids III. Elsevier.
8. ^
9. ^