# Introduction

Neutron diffraction, or elastic neutron scattering, can be used to analyze a sample's crystal structure, as well as atomic and magnetic properties. The measurement is very similar to X-ray diffraction, as Bragg's law is used to determine the distance between atomic planes in crystalline and polycrystalline material (Figure 1). The difference with neutron scattering when compared to X-ray scattering is that neutrons can penetrate much deeper into the sample, even as deep as 100 mm (for aluminum)^{[1]}. Neutrons interact directly with the nucleus, making even low atomic number elements show a high intensity. This also makes it possible to measure magnetic properties in the sample, as neutrons have a spin of 1/2, thus having a magnetic moment.

Figure 1. Diffraction within a crystal structure. (Figure: Jarkko Larkio)

# Principle of neutron diffraction

High energy neutrons are produced either by fission in a nuclear reactor, or in a spallation source where target nuclei are bombarded with high energy protons. The resulting neutrons are brought into useful thermal range by collisions near ambient temperature with a material termed "moderator", usually water or liquid methane. Neutrons emerging from the moderator are filtered for wavelenght and direction with a single crystal of time-of-flight through a known distance.^{[2, p.13-16, 195-198]}

Resulting neutron beam is targeted at the sample where it undergoes diffraction. Scattered neutron waves from the sample add up or interfere with each other, producing sharp diffraction peaks at defined angles. Using Bragg's law, these angles can be used to calculate Miller indices and lattice planes, which in turn can be used to measure eg. residual stress.^{[2]}

Where,

*d* = spacing between lattice planes

*θ* = Bragg angle

*λ* = wavelenght of the coming neutrons

*h*, *k* and *l* are the Miller indeces which specify the planes of the atomic lattice

Width and height of the coming beam is controlled using slits (cadmium masks in Figure 2), and the overlap of incident and diffracted beam is called a **gage volume**. Sample is moved in three dimensions and rotated to measure an average of the lattice spacing within the gage volume.

Figure 2. Incident and diffracted beams on the sample. Gage volume is the intersection of the beams defined by the cadmium masks. (Figure: Jarkko Larkio)

# Calculating residual stress

Almost all manufacturing processes create residual stress, which can be defined as stresses that exist in materials and structures independent of any external loads. In the whole material or structure they sum up to create zero force and moment resultants, and thus may they may not be readily apparent. However they are still stresses and must be considered that way with external loading involved. Main effect of residual stresses is an addition to loading stresses. This addition can be beneficial or harmful, depending on the location and direction of the stress. ^{[2]}

When using a steady state reactor neutron source, strain is measured in the direction of the scattering vector Q (Figure 2). If the sample is strain free, we get the lattice spacing for a strain free sample, d_{0,hkl}. In a strained sample the lattice spacing are shifted and the elastic strain can be calculated by:

\( \epsilon_{hkl} = \frac{d_{hkl}-d_{0,hkl}}{d_{0,hkl}}= \frac{\Delta d_{hkl}}{d_{0,hkl}}= \frac{\sin\theta_{0,hkl}}{\sin\theta_{hkl}}-l \)

Where θ_{0,hkl} is the angle at which Bragg peak is observed in the strain free reference.^{[3]}

When using a pulsed neutron source, also known as time-of-flight instrument, elastic strain for a fixed angle can be calculated very similarly to the example above:

\( \epsilon_{hkl} = \frac{\Delta d_{hkl}}{d_{0,hkl}}= \frac{t_{hkl}-t_{0,hkl}}{t_{0,hkl}} \)

Here t_{0,hkl} is the time of flight for the strain free sample. ^{[3]}

# Analysis of magnetic structure

As neutrons have a spin of 1/2 and thus a magnetic moment, they are scattered by magnetic moments in the sample. This allows the determination of magnetic structures in much the same way as crystal structures are determined from the intensities of diffraction peaks from nuclear scattering. Neutrons interacting with different magnetic structures (Figure 3) show different scattering peaks due to the combination of relative orientation of the neutron magnetic moment, the atomic magnetic moment and the scattering vector.^{[4]}^{[5]}

Figure 3. Magnetic moments of ferromagnetic and antiferromagnetic materials. (Figure:Jarkko Larkio)

# Practical uses

Neutron diffraction is a very useful tool for crystallography. Even though it is similar in concept to X-ray diffraction, neutron diffraction works more as a complimentary analysis tool for X-ray diffraction. With X-rays, only the surface layer can be analyzed. With neutrons, having high penetration depth, bulk samples can be analyzed.

One practical application for neutron diffraction is that lattice constants for metals and crystalline materials can be measured very precisely. Mapping the lattice constants in the material can be used for measuring the residual stress in said material. Neutron diffraction has thus seen use in eg. automotive and aerospace industries to measure stress in high performance components. Due to the high penetration depth, this method can be used for bulk components such as pistons, crankshafts and gears. ^{[3]}

Another useful application for neutron diffraction comes from its ability to distinguish elements with differing atomic masses, or Z numbers. This makes it ideal for locating light atoms in the presence of heavy atoms, especially metal hydrides or complexes with agnostic bonds. Transition metal hydrides come in various different coordinations, from 1-coordinate terminal hydrides to 6-coordinate interstitial hydrides. The precise measurement of the hydride ligands to the metal centers cannot be done reliably with X-ray diffraction. ^{[6]}

# Current advances in neutron diffraction

A special technique and data analysis tool has been made for Laue three dimensional neutron diffraction (Laue3DND), which allows 3D mapping of grains for oligocrystalline samples. The method was tested on cubic *α*-Fe and tetragonal YBaCuFeO_{5.} They were able to determine the position and and orientation of 97 out of 100 grains from a synthetic dataset of *α*-Fe, as well as 24 (*α*-Fe) and 9 (YBaCuFeO_{5}) grains from the sample measured at FALCON. The precision obtained for the position and orientation was 430 µm and 1°. ^{[7]}

# Strengths and challenges of neutron diffraction methods

## Strengths

With neutron diffraction, isotopes and neighbors can be discriminated from each other, even if they have a similar Z number. Magnetic structure can be analyzed and measurement of light elements such as hydrogen is possible. Neutron beam has a high depth penetration and is almost completely non-damaging to the sample.

## Challenges

Some metals like vanadium have no diffraction peaks, making stress testing with neutrons impossible. Also some titanium alloys have low coherent scattering and absorb neutrons, resulting in a sub-optimal peak-to-backround ratio. ^{[2, p.198]} Systematic errors might come from eg. partly filled gage volume, large grain size, incorrect use of slits and intergranular effects. A strong neutron flux is needed, so currently measurements are only possible in large scale facilities.

# European research centers with neutron diffraction facilities

- Institut Laue-Langevin (ILL) and the Laboratoire Léon Brillouin (LLB) in France (Figure 4)
- ISIS Pulsed Neutron Facility in the UK
- Forschungs-Neutronenquelle Heinz Maier-Leibnitz (FRM II) and the Helmholtz-Zentrum Berlin (HBZ) in Germany,
- Frank Laboratory of Neutron Physics and Petersburg Nuclear Physics Institute in Russia,
- Nuclear Physics Institute (NPI) in the Czech Republic,
- Paul Scherrer Institute (PSI) in Switzerland,
- Institute for Energy Technology (IFE) in Norway and
- Budapest Neutron Centre (BNC) in Hungary

Figure 4. Institut Laue-Langevin in Grenobe. Figure from Wikipedia. License: CC BY-SA 3.0.

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