Neutron diffraction: a primer

1.1 Provision of neutron beams

Neutrons are abundant in nature because, together with protons, they make up atomic nuclei. With the exception of hydrogen, light atoms contain an approximately equal number of protons and neutrons, while the heavier atoms have more neutrons than protons. That being said, it does not surprise that about 40 % of the mass of your body is made up of neutrons, tightly stored in the nuclei. Neutron diffraction requires large machines, however, simply because of the strong force of particle physics which must be overcome to release neutrons from the nuclei. There are three major fundamental processes that underlie neutron scattering facilities, as depicted in Figure 1: fission, spallation, and low-energy nuclear reactions induced by charged particles. ¹

Figure 1:

Illustration of the three major fundamental processes providing neutrons for scattering instruments: fission (top), spallation (middle) and low energy nuclear reactions (bottom). Adapted from ref 2 by kind permission of Wiley-VCH.

To begin with, fission takes place as a chain reaction in research reactors when a fissile nucleus, usually ²³⁵U, captures a neutron of the right speed. The nucleus then splits into two fragments of almost equal mass, releasing on average about 2.5 neutrons in the process. One neutron is needed to sustain the chain reaction after it has been slowed down (or moderated, another way of saying it) because fission neutrons have a large average energy of 2 MeV.

Second, there are accelerator-driven spallation neutron sources in which heavy metal targets (typically U, W, Ta, Pb, Hg) are bombarded with protons at huge energies around or above 1 GeV. After collision, the highly excited nuclei emit neutrons, protons and also pions, which collide again to produce more excited nuclei from which particles (mainly neutrons) are emitted. Typically, 10 to 30 neutrons per proton are released in the entire process. Most of these neutrons are so-called “evaporation neutrons” with average energies of about 2 MeV but there is also a fraction of high-energy cascade neutrons with GeV energies up to the incident proton energy. While spallation is the most efficient process for releasing neutrons, much heavier shielding is required due to the high-energy particles and, secondary, the high peak flux.

Third, nuclear reactions at accelerator-driven neutron sources at energies below 100 MeV have rather low neutron yields of about 10⁻² to a few 10⁻¹ neutrons per collision. At energies below 30 MeV – the domain of so-called Compact Accelerator-driven Neutron Sources (CANS) – light metal targets (Li, Be) are most suitable whereas at higher energies, proton reactions in heavy elements are more efficient. ³ Thanks to recent technological developments, such High-Current Accelerator-driven Neutron Sources (HiCANS) can provide performance on par with fission or spallation sources. ⁴ Accelerator-driven sources have a decisive advantage to reactor sources because neutrons can be produced on demand. With the time structure of a pulsed source, and the relation between wavelength and time-of-flight of the neutrons, one can use the entire spectrum instead of a monochromatic beam. It is then the peak flux rather than only the time average flux which matters. More accurately, rather than flux we may consider the (peak) brilliance , with a unit of n s⁻¹ cm⁻² sr⁻¹ Å⁻¹, quantifying how many neutrons are created per time interval and wavelength band and passing through an area and solid angle element, which ultimately determines the useful neutrons that can be scattered by the sample.

With neutron diffraction we want to measure the distances between atoms, which requires a “ruler” of similar size. This means that the neutron wavelength should be comparable to interatomic distances of about 0.1 nm (=10⁻¹⁰ m = 1 Å, the so-called Ångström crystallographic unit which comes in handy). With the symbol λ for the neutron wavelength, p for the neutron momentum, m for the neutron mass, v for the neutron velocity and h for the Planck constant, the neutron kinetic energy E = p 2 2 m can be related to the neutron wavelength using the De Broglie relation λ = h p = h m v . In practical units this reads

Thus, neutrons of wavelength 1 Å have a kinetic energy of about 82 meV. This fits well with typical excitation energies in solids, making neutrons also a perfect probe for studying elementary excitations in solids.

But now there is a problem: all of the above processes release neutrons with energies in the MeV range, while for neutron diffraction we need neutrons with kinetic energies about 7–8 orders of magnitude smaller. That is to say that the neutrons from the primary reactions have to be slowed down to these energies, which is usually achieved by collisions with light elements, mainly hydrogen, in so-called moderators, as already said before. The thermal equilibrium between the neutrons and the nuclei of the moderator material produces a Maxwellian spectrum with an average neutron energy determined by the temperature of the moderator. Depending on the temperature of the moderator, one speaks of “hot”, “thermal” and “cold” neutrons, as summarized in Table 1 and illustrated in Figure 2. These terms refer to the temperature equivalent of the neutrons’ kinetic energy according to the Boltzmann conversion E _kin = k _B∙T; note that 1 meV corresponds to about 11.6 K. The moderator is surrounded by a reflector, e.g., beryllium or lead, which effectively scatters the escaping higher energy neutrons back into the moderator for further thermalization.

Figure 2:

Maxwellian distribution of neutrons for several temperatures of the medium, which the beam from the reactor core passes through (moderator); grey shaded area is an example of a neutron wavelength band selected by a monochromator.

Both the extraction and transport of neutrons from the moderators is done through evacuated beam tubes, which allow the neutrons to leave the moderator and pass through the biological shielding to the neutron instrument.

1.2 Neutron–matter interaction

Somewhat simplified, X-rays, electrons, and neutrons are the main probes for studying matter at atomic length scales. Because X-rays and electrons interact with matter mainly through the strong electrostatic Coulomb interaction, this leads to a limited penetration depth simply due to scattering from the electron cloud of the atoms. For neutrons, being neutral particles with no electrostatic Coulomb interaction, however, condensed matter appears quite empty, as the main interaction with atoms is mediated through the short-range strong force with atomic nuclei, and they are about five orders of magnitude smaller than the size of an atom’s electron cloud (= the size of the atom). This explains why most materials are fairly transparent to neutrons.

The strong interaction between neutrons and nuclei is an extremely complex particle physics problem. Nonetheless, Nobel laureate Enrico Fermi took advantage of the point-like nature of nuclei being so much smaller than the wavelength of thermal neutrons to describe the interaction, and this led to the so-called Fermi pseudopotential for nuclear scattering: ⁵

Here, r is the distance between the neutron and the nucleus, m is the mass of the neutron, and δ is the delta-function reflecting the effectively vanishing interaction range. The pre-factor in (2) is chosen such that the total cross section σ _tot, i.e., the total number of neutrons scattered per second normalized to the flux of incident neutrons, is given by:

Since a point-like scatterer has no internal structure, the scattering from it is isotropic, i.e., nuclear scattering has no angular dependence. The important quantity in (2) and (3) is the bound scattering length b, which is generally a complex number in need of explanation. While the real part is a measure of the strength of the interaction, the imaginary part measures absorption. Moreover, b is a phenomenological parameter that varies rather randomly throughout the periodic table, so it must be determined experimentally, for values see ref 6. The real part of b is positive for a repulsive pseudopotential, the normal case. In rare but important cases it can be negative, however, corresponding to an attractive potential. In addition, b is isotope specific and depends on nuclear spin states: b ₊ when the neutron (with spin = ½) and nuclear spin (I) are aligned (total spin = I+½), b _– for total spin = I−½.

The fact that the distribution of isotopes and nuclear spin orientations is usually random has the important consequence that scattering can be separated into a coherent and incoherent part. For coherent scattering, the scattered waves from different scattering centers interfere, i.e., a regular superposition of the relative phases occurs. In accordance with (3), coherent scattering depends on the average value b _coh = 〈b〉 of the scattering length of the nuclei:

In a crystal with lattice periodicity (that is, a regular crystal with crystallographic symmetry), the coherent scattering length is responsible for the Bragg scattering, just like for the typical X-ray case. Incoherent scattering generates noise, however, because it arises from the random distribution of the deviations of the scattering length from their mean value and does not show interference effects:

An important and real-world example is given by elemental hydrogen ¹ ₁H, which has a nuclear spin of ½. With b ₊ = 10.82 fm and b ₋ = −47.42 fm one obtains b coh = 1 4 b − + 3 4 b + = − 3.74  fm and σ _coh = 1.76 barn, σ inc = 4 π ( 1 4 b − 2 + 3 4 b + 2 ) = 80.27  barn (1 barn = 100 fm² = 10⁻²⁸ m² = 10⁻²⁴ cm²). So, the most abundant hydrogen ¹ ₁H has a small coherent but a huge incoherent cross section, and this leads to a high flat background for scattering from samples containing hydrogen. Fortunately enough, deuterium ² ₁H ≡ D, on the other hand, has cross sections of σ _coh = 5.59 barn, σ _inc = 2.05 barn. Hence, the much smaller incoherent scattering cross section of D can be used to minimize background otherwise present due to hydrogen’s incoherent scattering and allows for contrast variation by isotope substitution.

Figure 3 displays neutron cross sections for selected elements. While the cross section for high-energy X-rays is (to first approximation) proportional to the number of electrons (or the atomic number Z) squared and varies continuously throughout the periodic table, this is very different for neutrons. Hydrogen is virtually invisible for X-rays in the presence of heavier elements, and neighboring elements like Mn, Fe or Ni are hardly distinguishable for high-energy X-rays. Not so for neutron diffraction, thereby offering a clear advantage of neutrons.

Figure 3:

Coherent scattering cross sections of selected elements throughout the periodic table for neutrons. The area of the circle is proportional to the cross section. The colors blue and green indicate the sign of the coherent scattering length (positive and negative, respectively). For comparison, the X-ray Thomson scattering cross section for a single electron is 0.665 barn. The classical coherent X-ray cross section at high energies varies smoothly with the square of the number of electrons throughout the periodic table.

For magnetic materials, an additional scattering mechanism besides nuclear scattering must be considered. Magnetic scattering is due to the electromagnetic interaction with the internal magnetic field, resulting from the magnetic moment associated with the neutron’s nuclear spin. Fortunately, magnetic scattering is comparable in magnitude to nuclear scattering: the magnetic scattering length of neutrons from a single unpaired electron amounts to − 1 2 γ n ∙ r 0 ≈ − 2.67 fm, where γ _n denotes the magnetic dipole moment of the neutron expressed in nuclear magnetons and r ₀ is the classical electron radius. Since magnetism is a vector quantity, magnetic neutron scattering is strongly dependent on the direction of the magnetic moment and the polarization of the neutron beam. ⁵ Also, because scattering occurs from unpaired electrons of the atom they belong to, magnetic neutron scattering exhibits a form factor dependence similar to the form factor of X-rays. Unlike nuclear scattering, which is independent of the scattering angle (see above), the intensity of magnetic scattering from an atom decreases rapidly with increasing scattering angle. In fact, it decreases even faster than for X-rays because the unpaired electrons are in the atom’s outer region, so magnetic Bragg peaks are found for small diffraction angles, in the “early” region of the standard diffraction pattern. Magnetic form factors are tabulated in refs 6 or 7.

Let’s turn to the imaginary part of the scattering length, connected with neutron absorption. Most elements have a very small neutron absorption cross section which has been tabulated, ⁶ proportional to the neutron wavelength:

It can be argued that the wavelength λ and the neutron velocity v are inversely proportional, so that for longer wavelengths, i.e., lower velocities, the neutron stays near the nucleus for a correspondingly longer time, leading to a higher absorption cross section. There are some elements, however, that exhibit strong neutron absorption due to nuclear resonance processes which lead to nuclear reactions releasing charged particles, for example in ³He, ⁶Li, and ¹⁰B. Therefore, these reactions form the basis of neutron detectors. Strong resonance absorption with emission of gamma rays occurs, for example, in the elements Cd and Gd, which are often used in neutron shielding materials. However, care must be taken because such resonances are strongly energy dependent. For example, while Cd has an absorption cross section of about 20 kbarn at a wavelength of 0.064 nm (0.64 Å), this drops to only 8 barn at 0.02 nm (0.2 Å). In practice, absorption may even spoil a neutron diffraction experiment, which can be circumvented by combined use of annular samples, optimized incident neutron wavelength and high-intensity diffractometers or, alternatively, by using less absorbing isotopes, ⁸ or flat-plate sample holders. ⁹

2.1 Constant-wavelength (CW) neutron powder diffraction

To understand how reactor-based neutron diffraction (NPD) was born from a convergence of quantum mechanics and X-ray crystallography and became a practical tool during the Manhattan project, it is important to place the method development in the historical context. By the time neutrons were discovered by Chadwick in 1932, ¹² both the De Broglie hypothesis of wave-particle duality had been confirmed and X-ray crystallography was well established already. In fact, the number of crystal structures reported by then based on X-ray diffraction (XRD) was in the hundreds. It was thus only a question of time before neutron diffraction by means of crystalline materials was proposed and demonstrated. Already in 1936, Elsasser considered neutron diffraction theoretically ¹³ and Preiswerk and von Halban, ¹⁴ and Mitchell and Powers ¹⁵ experimentally demonstrated that for polycrystalline iron and single-crystal MgO, respectively, neutron diffraction indeed occurred according to the Laue–Bragg conditions. However, the low intensities of the Ra–Be sources available at that time precluded any quantitative crystal structure analysis. The situation radically changed after nuclear reactors were built as a part of the Manhattan project. The first nuclear reactors CP-1 and X-10 went critical in 1942 and 1943, at what later became the Argonne and Oak Ridge National Laboratories, respectively. Both reactors produced high neutron fluxes such that neutron diffraction experiments began almost immediately, within months after the criticality was achieved. ^{16 , 17}

Nuclear reactors generate a white beam with an approximately Maxwellian distribution of neutrons in energy or wavelength (Figure 2). To perform constant-wavelength (CW) neutron diffraction similarly to constant-wavelength X-ray diffraction, the beam should be coherent and collimated, and it must be monochromatized first. This can be achieved either with mechanical velocity selectors or with single crystal monochromators (Figure 5). In both cases, neutrons with a particular wavelength or energy are selected from the white beam as illustrated in Figure 2, by the shaded area. With velocity selectors the band position is controlled by the speed of rotation of neutron-absorbing blades. With monochromators, on the other side, it is controlled by the single-crystal angle in the incident beam via Bragg’s law (λ = 2d sin θ) for a fixed lattice plane with lattice spacing d _hkl . As can be seen from Figure 2, only a small fraction of the neutrons produced by the reactor is utilized in CW neutron-diffraction experiments. While velocity selectors are still being used in the design of some instruments, e.g., for small-angle neutron scattering, single-crystal monochromators proved to be more suitable for neutron powder diffractometers, e.g., with respect to the desired instrumental resolution and the possibility to realize focusing optics gaining intensity. Conceptually, the two-axes design of CW neutron powder diffractometers has not changed since the instruments built at the first reactors. ¹⁸ The “two-axes” refer to one axis of rotation of single-crystal monochromator and another one as regards the detector around the sample, usually dubbed transmission Debye–Scherrer geometry. Nowadays position-sensitive detectors instead of point detectors are being used, making data acquisition far more efficient. A typical design using a “banana-like” detector is schematically illustrated in Figure 5.

Figure 5:

Schematic sketch of a two-axis neutron diffractometer (left); α ₁, α ₂, and α ₃ indicate beam divergence (typically defined by Soller collimators), β is the mosaic spread of the single-crystal monochromator. Scattering geometry (right) relating propagation vectors of the incoming and outgoing (diffracted) beams (k and k ₀, respectively) and scattering angle of Bragg’s law (θ). See also Section 2.1.

The monochromatic beam is sent onto the sample, and the distribution of the intensity of the beam scattered by the sample is measured as a function of the angle 2θ. Both positions and intensities of the characteristic diffraction peaks observed at particular 2θ (Bragg) angles contain information about the crystal structure and magnetic ordering of the sample material. The sharpness of the peaks in neutron powder diffraction (NPD) data is defined by the divergence of the beam, and the trade-off between resolution and intensity is controlled by the instrument optics. How the divergences α ₁, α ₂, α ₃, the mosaic spread of the monochromator β, and the take-off angle 2θ _M (Figure 5) affect the full widths at half maximum (FWHM) of the diffraction peaks and overall beam intensity (luminosity), was worked out by Caglioti and coworkers, ¹⁹ which thereby created a solid basis for the optimal design of CW neutron powder diffractometers. ²⁰

Analysis of NPD data was originally performed by extracting structure factors from integrated intensities of the diffraction peaks, essentially the same procedure as for single crystals up to the present day. In fact, the very first crystal structures of NaH and NaD quantitatively studied by NPD were analysed that way. ²¹ Nonetheless, this very approach works only for simple data sets with well-resolved diffraction peaks. For complex structures and/or low-resolution diffraction data, peak overlap hinders extracting individual structure factors. To overcome this obstacle, the approach to powder diffraction data analysis was reversed and, instead, the diffraction profile is calculated based on the input structural model and then compared to the experimental data; for further improvement, the model gets iteratively adjusted to minimize the difference between model and experiment. This approach was developed in the late 1960s by Loopstra, van Laar, and Rietveld and is now commonly referred to as the Rietveld method. ^{22 , 23 , 24 , 25} Historically speaking, the conditions were favorable because of the nearly exactly Gaussian peak shapes of those diffractometers, e.g., in use at the neutron High Flux Reactor in Petten, The Netherlands, allowing for computing the so-called least-squares refinements by means of the upcoming computational power of that time. The method made tremendous impact and led to wide adoption of neutron powder diffraction, as well illustrated by the very rapid rise in the number of crystal structures reported based on neutron powder diffraction data from early 1970s (Figure 6). The Rietveld method has been implemented and further developed in dozens of software codes, and as of today the most widely used programs, e.g., FullProf, GSAS-II, TOPAS, Jana, Maud, can be used not only to refine crystal and magnetic structures but also extract a lot of other information as regards phase fraction, stress/strain, particle size, texture, extended defects such as stacking faults, and a lot more. ²⁶

Figure 6:

Number of the annual entries in the Inorganic Crystal Structure Database based on neutron powder diffraction data.

Neutron powder diffraction combined with the Rietveld method proved to be so successful and in high demand that today most neutron scattering facilities have at least two neutron powder diffractometers: one optimized for high resolution aiming at clarifying complex crystal and magnetic structures and another one optimized for high intensity aiming at in situ or operando experiments on structurally known matter and small (down to a few mg) samples. Some specialized neutron diffractometers are also installed on “hot” and “cold” neutron sources with neutron beam spectra shifted to shorter and longer wavelengths (Figure 2), thereby focusing at short-range and long-period crystal and magnetic structures, respectively.

Theoretical work on optimal design, ²⁰ decades of experience in experimentation, and in-depth knowledge exchange between neutron scattering facilities led to some common features in instruments. Most modern CW neutron diffractometers employ focusing Ge or graphite monochromators, providing wavelengths in the range of about 1.5–2.5 Å and 2.5–5 Å, respectively, ³He gas detectors, either multiple tubes or continuous multiwire tanks, and also interchangeable slits or Soller collimators ²⁷ needed to adjust the resolution/intensity balance for different samples. To increase throughput, many diffractometers are equipped with automatic sample changers. As a result, data collection typically takes less than a few hours per data set and many CW NPD experiments are so streamlined that most neutron scattering facilities offer Mail-in (also known as Express or Rapid) access for which users post their samples to the facility and receive NPD data, collected at room and/or several low-temperature points, within a few days/weeks only. Also, high-temperature ovens and different kinds of high-pressure cells are usually routinely available. Given that there are about 40 constant-wavelength powder diffractometers around the world and data analysis with the Rietveld method is similar to the analysis of ubiquitous X-ray powder diffraction data, CW NPD certainly has the lowest entry barrier among all the neutron scattering techniques and should be routinely used by any research group working with crystalline materials, in particular when X-ray diffraction reaches its limits. Further information on the CW NPD can be found in ref . More particular resources for the instruments are found in Table 2.

2.2 Time-of-flight method and high-resolution neutron powder diffraction

To begin with, the striking advantage of the time-of-flight (TOF) method in neutron diffraction experiments is to utilize a wide range of the entire energy spectrum whereas the constant-wavelength angle-dispersive method uses only a part of the neutron spectrum by cutting out the rest, say, by using a monochromator (see Section 2.1). This was pointed out by Egelstaff (UK) in the 1950s already, and the intensity gain over the angle-dispersive method was discussed shortly after. ⁵² The TOF diffraction experiments were carried out for the first time using a so-called Fermi chopper at Swierk, Poland, ^{53 , 54 , 55} then using the pulsed reactor IBR-1 in collaboration with Dubna researchers. ⁵⁶ Later on, the TOF method was continuously developed in electron accelerator facilities in which accelerated electron beams targeted to heavy metals generated neutrons through the (γ, n) reaction (see Section 1.1). Three facilities, Rensselaer linac (USA), Tohoku University linac (Japan), and Harwell linac (UK), independently carried out the TOF neutron diffraction and started fundamental research on the generation and scattering of pulsed neutrons. ^{57 , 58 , 59 , 60} In addition, a TOF Debye–Scherrer diffractometer with multiple detector banks was constructed by Kimura and coworkers ^{58 , 59} to conduct crystal and magnetic diffraction experiments on metals and oxides.

While many pulsed-neutron source projects were proposed but not funded, spallation neutron projects using proton accelerators opened the next era. Four spallation facilities first opened user programs since the 1980s: IPNS (1981–2008, Argonne National Laboratory, USA), KENS (1980–2006, High Energy Accelerator Research Organization, Japan), LANSCE (from 1983 until now, Los Alamos National Laboratory, USA), and ISIS (from 1984 until now, Rutherford Appleton Laboratory, UK).

The recent important advance in TOF diffraction comes from the worldwide construction of high-intensity proton accelerators: ISIS (UK), SNS (from 2007 until now, Oak Ridge National Laboratory, USA), J-PARC MLF (from 2008 until now, Japan Atomic Energy Agency and High Energy Accelerator Research Organization, Japan), China Spallation Neutron Source (from 2018 until now, Institute of High Energy Physics, China) and the coming European Spallation Source (ESS, Sweden). The many scientific publications of work based on measurements at these large-scale research facilities provide a fine testimony to the success of the TOF neutron-diffraction technique.

In spallation, pulsed protons colliding with a heavy metal target effectively produce high-energy pulsed neutrons (see Section 1.1), which are then moderated to energies in the range of 0.1 meV–100 eV using a moderator. This energy range corresponds to the wavelength range of 0.03–30 Å by the de Broglie relationship, λ = h m v = h 2 m E , as previously defined in Section 1.1. Although neutrons with various energies (wavelengths, velocities) are generated at the same time, the time when the neutrons are scattered by the sample and finally reach the detectors is different, thereby reflecting the difference in neutron velocity.

The TOF method measures the elapsed time for a neutron from a defined state with time t ₀ = 0, e.g., after the spallation impact and slowing down at the moderator or by passing a chopper window creating a pulsed beam, until the time where the neutron gets scattered by the sample, and finally reaches the detector. We may therefore say that each neutron event (release, propagation, scattering, and detection) can be distinguished and recorded in a neutron scattering experiment. Typically, 10⁶–10⁸ neutron events are accumulated for a crystal structure analysis and a time channel on the order of 10⁴ μs. The neutron-event data are usually converted to histograms with linearly or logarithmically binned time channels of a few μs intervals for further data analysis such as Rietveld refinement.

Since the neutron velocity, v, does not change for the elastic Bragg scattering, v can be calculated by v = L t in TOF diffraction (L: total flight path, t: elapsed time). Combining this equation with the de Broglie relationship, we get λ = h t m L . By using the Bragg condition, λ = 2dsinθ (d: lattice spacing, 2θ: scattering angle), we obtain

which shows that the time of arrival of the neutron at the detector is proportional to the d spacing (see Figure 7), an exceptionally simple and beautiful relationship.

As given by Figure 7, it appears that slower and longer-wavelength neutrons belonging to one neutron pulse arrive at the detectors after the newly released neutrons (from a later neutron pulse but with faster and shorter wavelengths) have also arrived. This unwanted phenomenon is called frame overlap, so the measurable wavelength range is limited by the frame overlap. There are several methods to extend the limit of the measurable d range due to frame overlap. As can be seen from equation (8), a Bragg reflection with a larger d can be observed using lower-angle detector banks. The measurable d range is often extended by lowering frequencies using mechanical choppers; multiple disk choppers in lower frequencies remove faster neutrons and a t ₀ chopper removes burst neutrons with very high energies which otherwise contribute to the background.

Figure 7:

A conceptual diagram in TOF diffractometry with a pulsed source and 25 Hz frequency. The Bragg reflection positions, t ₁, t ₂, …, are a function of d, flight path, L1 + L2, and a scattering angle, 2θ.

As many neutron-diffraction practitioners do, it is convenient to remember the following formula:

to achieve the best diffraction strategy.

Although the advent of the Rietveld method allows extraction of structural information from overlapping Bragg reflections, a too severe overlap makes the analysis results unreliable, especially for very small lattice spacings. In contrast, the use of high-resolution diffractometers with a narrow Bragg peak full width at half maximum (FWHM), ∆d, may lead to new discoveries because the diffraction patterns contain a plethora of information about the structure.

In the early days of TOF diffraction, the TOF technique was not thought to be suited to high-resolution powder diffraction, ²⁰ interestingly enough. In fact, the 150 m flight-path high-resolution diffractometer with the resolution of 2 × 10⁻³ needed a large amount of sample and measurement time. ⁶¹ However, due to technological developments in neutron transportation, detection, diffractometer design, sources, etc., the TOF method is very well suited to high-resolution powder diffraction, even though the shape of the Bragg peaks appears as being a little more complex than in the constant-wavelength scenario.

It is reasonable to define the resolution, ∆ d d , of the TOF powder diffractometer as composed of three variances: ∆ t t , ∆ L L and cotθΔθ. To achieve high resolution, these three variances should be small, that is, a large L, which additionally translates into a large t, and a large 2θ.

High-resolution TOF powder diffractometers with the best resolution of ca. ∆ d d ≈ 5 × 10⁻⁴ such as HRPD at ISIS, SuperHRPD at J-PARC, and HRPD at CSNS have long flight paths, about L = 80–100 m (both ∆ L L and ∆ t t are small) and backward detector banks (cotθΔθ is close to zero).

It should be emphasized that the latest high-resolution hydrogen moderator cooled to about 20 K are designed to have sharp neutron pulses with small ∆t over a wide energy range from 1 meV to 300 meV (0.5 Å to 10 Å).

Finally, Table 3 gives a list of general-purpose high-intensity and high-resolution TOF neutron diffractometers as well as instruments for special science cases at a world-wide selection of neutron sources. Future TOF instruments are currently built at the ESS in Sweden (e.g., DREAM, HEIMDAL) and at the FRM-II/MLZ in Germany (POWTEX).

2.3 Multidimensional Rietveld

We recall that constant-wavelength (CW) and time-of-flight (TOF) diffraction both probe the lattice spacing d according to Bragg’s law. While monochromatic CW diffractometers screen the scattering angle 2θ with their detector, a very simple TOF diffractometer concept would place one detector at a fixed scattering angle while the wavelength λ varies within a neutron pulse. In modern neutron TOF diffractometers such as POWGEN, SNS (USA), the detector system covers almost the entire range of scattering angles, i.e., both 2θ and λ vary. Every detector pixel at fixed 2θ measures an individual TOF pattern, and each d reflection is “moving” over the detectors with λ and within the selected wavelength band. For sticking to a one-dimensional, conventional data-treatment, the detectors are grouped into detector banks with narrower 2θ ranges in which the data are integrated along d to yield (several) TOF patterns with similar resolution Δd/d.

The sophisticated detector arrangement of POWGEN ⁶² allows to integrate the whole detector (one bank) into one d-dependent pattern which is then converted back to a TOF pattern using Bragg’s law for 2θ = 90°. This procedure is called diffraction focusing (see Figure 8, left). As a clear advantage, these reduced pattern(s) – one per wavelength-band – can be processed with all common Rietveld packages such as FullProf, ⁶⁵ GSAS-II, ⁶⁶ etc., either as single patterns each or simultaneously in a multi-pattern refinement of several wavelength-bands. However, for every instrumental setting (combinations of detector bank, wavelength range, chopper settings, etc.) an individual, averaged instrument parametrization is needed. This is because the resolution, i.e., the reflection-width Δd, is averaged during diffraction focusing. Similarly, the reflection shape, being convolutions of Gaussian, Lorentzian and other functions with higher complexity (see Section 2), and its parametrization is changing with (and within) each setting. Both effects may compromise the maximum instrument performance, so a loss of information is likely.

Figure 8:

Left: Diffraction focusing from an angular- and wavelength dispersive pattern to a conventional TOF pattern based on a CuNCN sample ^{63 , 64} measured at POWGEN. Right: Schematic drawing showing one arbitrarily chosen d _⊥ (red curve) in (2θ, λ) space being always normal to all d-reflections and the comparison to CW (blue, horizontal cut) and TOF (green, vertical cut). Adapted from ref 10 by CC BY.

For the multi-dimensional Rietveld method, the angular- and wavelength-dispersive neutron events are binned without diffraction focusing either in a (2θ, λ) coordinate system (see Figure 8, right) or in the recently introduced (d, d _⊥) coordinate system with d as d-spacing from (2θ, λ) and

being the solution to the differential equation dλ/dθ = –(2d cos θ)⁻¹. The differential equation requests d _⊥ to be perpendicular to all d-reflections. ¹⁰ Accordingly, the sinusoidal d-reflection in (2θ, λ) becomes a perfectly vertical line. By definition, the d _⊥ axis will always be perpendicular to the d-reflection in both, (2θ, λ) and (d, d _⊥), coordinate systems (see Figure 8, right). Furthermore, it will intersect the d-reflection with the smallest possible reflection-width Δd, being the instrumental resolution for an ideal sample. Consequently, in the multi-dimensional description, the instrumental resolution function depends on 2θ and λ. Such a function, based on the fundamental instrument characteristics (divergence of neutron beam, sample size, pixel size, pulse-width, etc.) of POWGEN, has been fully described. ⁶⁷

Instruments such as GEM, POLARIS, at ISIS and SuperHRPD at JPARC, as well as the future instruments POWTEX, FRM II (Germany) and DREAM, ESS (Sweden) have a very large solid-angle detector coverage. The detector geometry is very simple as the detector-surface is cylinder-like sharing its axis with the neutron beam axis, ⁶⁸ different from the heart-shaped layout at POWGEN. Hereby, the instrumental resolution function is changing even more drastically and makes the multi-dimensional description more favorable. To test the tailor-made POWTEX detectors and develop the multi-dimensional Rietveld method, one mounting-unit (Δ2θ = 90°; Δφ ≲ 9°) plus electronics was operated at POWGEN with diffraction conditions very similar to POWTEX’s secondary instrument. ⁶⁹ The detector was calibrated and the measured event data were treated using the PowderReduceP2D ⁷⁰ algorithm as implemented in Mantid ⁷¹ to yield the binned diffraction patterns in (2θ, λ) and (d, d _⊥) coordinates, given the fundamental instrument resolution function as defined similar to ref 72, and all intended to test the multi-dimensional Rietveld method implemented in a modified, yet unpublished GSAS-II version.

Powdered diamond with very sharp reflections serves as an excellent example for illustration. By refining the parameters of the structural model (unit cell, isotropic thermal displacement, background, instrumental parameters) to make the calculated pattern eventually fit the experimental neutron diffraction pattern, the multidimensional fit (see Figure 9) is already convincing by visual inspection. A real-world sample of BaZn(NCN)₂, however, not only evidences similar standard deviations compared to the conventional approach but arrives at internal structural parameters which are slightly different and make more chemical sense. For example, two formerly very similar lattice parameters (a and b) of the orthorhombic structure are better resolved (and differ more), and this also relates to even finer structural details as regards C=N double-bond lengths and N–C–N bond angles. ⁶⁹ So, the quality of structural information, i.e., the related scientific question, clearly profits from the multidimensional approach.

Figure 9:

Multi-dimensional Rietveld refinement of a powdered diamond sample. Top, from left to right: observed, calculated and difference pattern. Bottom: 1D plot of multi-dimensional refinement (left) and conventional, one-dimensional refinement using an unmodified GSAS-II (right). Blue lines indicate peak positions. Adapted from ref 69 by CC BY.

2.4 Texture

An inherent feature of most polycrystalline materials and aggregates, be they natural (like rocks) or synthetically produced (like metals and ceramics), is the preferred orientation of the constituting crystallites. This preferred orientation distribution of the crystallites is called the crystallographic texture. ⁷² Because of the anisotropic properties of most single crystals, texture influences mechanical and physical properties such as strength, conductivity, magnetic susceptibility, light refraction, or wave propagation in materials. Deformation, casting, welding, and heat input may change the texture and, thus, the knowledge of the texture and its evolution is a key issue in property optimization in materials science. ⁷³ In addition, texture provides a wealth of information on geological processes and the manufacturing routes of archaeological artefacts. ⁷⁴

The ultimate aim of texture analysis is to quantitatively obtain the three-dimensional (3D) orientation distribution of the crystallite grains in a polycrystalline material with respect to a sample coordinate system. Besides the shape this is often defined in terms of directions of the forming work acting on the sample, like the rolling direction in metals after cold processing or foliation in rocks. In general, the resulting orientation distribution function (ODF) can be given as the volume distribution of grains dV in an orientation increment dg defined by the Euler angles.

In diffraction the integral intensity of the scattered beam is proportional to the volume fraction of crystallites within the sample volume fulfilling the Bragg condition. By rotating the sample through all possible orientations, the intensity distribution across a polar sphere can be measured. In a diffractometric texture measurement this is usually mapped to a stereographic projection and yields the so-called pole figure. Manipulation of the sample orientation is carried out using Eulerian cradles or, more recently, robots. ^{75 , 76} As the information on the rotation of crystallites around the scattering vector is lost in such a measurement, a pole figure only represents a 2D projection of the true ODF. Therefore, several pole figures are needed to calculate the ODF using pole figure inversion. ^{72 , 77}

Since texture analysis using neutron diffraction was first used by Brockhouse in 1953 in the attempt to determine the magnetic structure of nickel, ⁷⁷ it has seen a wide range of applications ranging from the earth sciences ⁷⁸ to applied industrial processing technologies. ⁷³ Texture analysis can be carried out on neutron powder and single-crystal diffraction data; hence it is applicable at all neutron sources, that is, reactor, spallation and accelerator-based. In contrast to other methods, neutron diffraction usually measures complete pole figures. ⁷⁹ Due to the weak interaction of neutrons with matter, effects of absorption are almost negligible and large sample volumes in the order of several cm³ can be probed (see Section 1.2). This ensures an accuracy better by almost one order of magnitude than by using X-rays, simply due to the large number of sampled grains, but it also allows for coarse grained materials to be investigated. Non-destructive bulk texture analysis needs no special sample preparation such that neutron diffraction has shown its enormous potential as an efficient tool for measuring multi-phased materials. Using TOF diffractometers with multi-detector systems adds the benefit that many Bragg reflections can be recorded simultaneously, and fewer sample rotations are necessary for fast texture measurements. ⁸⁰ Especially for crystallites of low symmetry with many closely spaced or even overlapping reflections (a very typical scenario found for many rocks or similar geological samples), TOF neutron diffraction in combination with advanced data analysis methods like Rietveld texture analysis is essential. ⁸¹ Finally, neutron diffraction uniquely offers the possibility to study the magnetic texture of polycrystalline materials.

2.5 Single-crystal neutron diffraction

As mentioned before, powder diffraction is one of the most successful and fastest methods to obtain structural information on the atomic scale. The applications are numerous and cover all the important fields of research from solid-state physics, chemistry, materials science and mineralogy, as well as biology and macromolecular sciences. Nevertheless, diffraction on single crystals has the ability to contribute additional details about the spatial distribution of the various intensities from differently orientated lattice planes. A highly schematic picture of both powder and single-crystal diffraction in presented in Figure 10. The latter method allows to record the specific intensities of different lattice planes separately, even if their d values and respective |Q| values are the same or so similar that they overlap in powder diffraction experiments. Examples where the spatially resolved assignment of intensities to individual lattice planes becomes particularly advantageous are given below.

Figure 10:

Constant wavelength diffraction patterns for powder (left) showing Debye–Scherrer rings per d lattice-spacings and single crystal (right) showing distinct spots; in the single crystal case the sample needs to be rotated in order to fulfill Bragg’s law for one reflection after each other; in the ideal powder sample a random distribution of lattice-planes is naturally present.

Many applications benefit from combining the advantage of spatial information preservation through single crystal diffraction with the excellent discrimination between Q-dependent and Q-independent neutron interactions. One example are the mean-square displacements (MSD) – also going under the name atomic displacement parameters (ADP) in the Debye–Waller part of the structure factors – whose temperature dependence can reveal information about thermal oscillations and static positional disorder. Access to large Q with neutrons allows for very accurate studies that might help to carefully validate density-functional based phonon simulation studies. ⁸²

Another example is the Q-dependent magnetic scattering length respective form factor (see Section 1.2). The direction-dependent observation of the magnetic and non-magnetic intensity components of individual lattice planes allows for very precise studies on the structural character of macroscopic order phenomena (ferro-, ferri- or antiferromagnetism) and the orientation of the magnetic moments, in particular as regards potentially important materials for information technology. ⁸³ By spin-polarizing the magnetic moment of neutrons in the beam, this allows for even more detailed recording of the vectoral components of magnetic elements (e.g., 3D polarimetry, spin-flip-ratio measurements), which cannot be discussed in greater detail within the scope of this primer, ^{84 , 85} but please see Section 3.4.

Resolving twinned structures is another domain of single-crystal diffraction needed to clarify effects of crystallographic symmetry lowering. The latter is a consequence of crystallographic transformation by group-subgroups relationships, ^{86 , 87} accompanied by domain formations often caused by small displacements of the light elements; on the atomic scale, the higher symmetry has gone but macroscopically it is (almost) retained. Typically, a reduction in symmetry is accompanied by additional reflections (so-called superstructure reflections). If they are caused by small displacements of light elements they can easily be missed upon using X-ray diffraction. Furthermore, twinning causes the creation of domains in a single crystal whose spatial orientations to each other follow specific so-called twin laws. ⁸⁸ The overall intensity distribution of a twinned single crystal is thus the result of a complicated – often only partial – overlap of the intensity patterns of the different domains as well as their volumetric fractions and may also include reflection splitting (for pseudo-merohedral twins). In powder diffraction, intensities are plotted against d-spacing and |Q| respectively. Additional spatial information like the angular relationships between equivalent lattice planes of different domains can be accurately recorded with single crystal diffraction. This makes it much easier to correctly assign the measured intensities (including magnetic ones) to the intensity patterns of the different domains and thereby to identify their true space group. ⁸³

The reader may recall the main advantages of diffraction with neutrons covered in Section 1.2, that is, the comparatively strong scattering lengths of many light elements (H, Li, O, N, etc.) and the fact that they are independent from the scattering vector Q = k−k ₀, the difference between outgoing and incoming wave vectors k and k ₀, respectively, also dubbed k _f and k _i in the literature (see Figure 5, right). The length of the vector Q relates to the lattice-spacing by d = 2 π | Q | . Hence, single-crystal neutron diffraction, in particular, may localize those atoms precisely, for instance in molecules or in ion conductors for energy applications ⁸⁹ or minerals for environmentally friendly applications. ^{90 , 91} As also covered before, a keener look reveals that NPD suffers from strong incoherent scattering in the case of hydrogen which leads to an enhanced background and to a massive deterioration in the signal-to-noise ratio. While this problem may be solved by exchanging hydrogen by deuterium since deuterium often behaves chemically very similarly to protium (see Section 1.2), such approach is not always applicable, for instance, when studying natural minerals. Furthermore, exchanging isotopes may significantly impact the chemical behavior, e.g., for hydrogen-bridging bonds, or affect the crystal structure dynamics due to the higher mass of D compared to H such that protium-deuterium replacement may shift the temperatures of phase transitions or even generate new phase transitions. ⁹² Single-crystal diffraction, however, suffers much less from this issue of incoherent hydrogen scattering, as the diffracted Bragg intensities concentrate on well located small areas in reciprocal space, so they are not distributed over the full Debye–Scherrer rings. In many cases, single-crystal neutron diffraction preserves a good signal-to-noise ratio even for non-deuterated samples and allows to solve the structures of small non-deuterated molecules. ^{82 , 93}

Furthermore, the nuclear scattering lengths of neighboring atoms often differ considerably (e.g., Mn vs. Fe, see Section 1.2 and Figure 3). This is an interesting feature for studies on mixed crystals, where X-ray diffraction often fails to discriminate the site distribution (and occupancies) of elements with similar electron configurations. ^{94 , 95} Instead, using the different scattering lengths of various isotopes of the same element (like in the H/D exchange) can improve contrast variation. At the same time, the absorption cross sections of most elements in the periodic table are very small and allow neglecting absorption correction even in large single crystals of few millimeter size, very much different from single-crystal X-ray diffraction. There is another effect dubbed extinction affecting intensities and going back to microstructural features which also requires different approaches for correction in X-ray and neutron scattering.

Of course, neutron single-crystal diffraction does not only offer advantages; there is also a price to be paid. To begin with, neutron sources deliver significantly lower neutron fluxes compared to the high photon fluxes of synchrotron radiation sources. At the same time, the interaction probability of neutrons is much lower (nuclear instead of electron scattering, see Section 1.2). In order to preserve reasonable counting statistics for neutron experiments, they are performed with a worse resolution (Δd/d ratio) from the source/monochromator and much larger sample volumes (millimeter sized instead of micrometer sized) compared to the X-ray counterparts. Thus, growing suitably sized single crystals for neutron single-crystal diffraction can be very challenging. Even then, measurements at neutron facilities require significantly more time than at synchrotron facilities.

The larger sample volumes also affect single-crystal-typical effects such as multiple scattering at the same (dubbed extinction effect, see above) or different (dubbed Renninger effect) sets of lattice planes. Both effects do also exist for X-ray diffraction, but special attention must be paid to extinction in order to avoid misinterpretations, especially for very good and large crystals, while the Renninger effect happens more often due to the poorer energy resolution.

We close this section with a few remarks concerning instrumentation: most neutron single-crystal diffraction experiments at reactor (= nuclear fission) sources are carried out in angular dispersive mode which requires a monochromatic beam. By rotating the sample, the intensities of one set of lattice planes are collected one after each other, putting the detector on the corresponding Bragg angle (or using an area detector). Nonetheless, the energy-dependent velocities of neutrons also allow for the so-called time-of-flight method (see Section 2.2), an energy dispersive technique that prospers from the evolution of spallation sources which generate high peak intensities from a pulsed neutron flux. The fixed single crystal sample is then surrounded by a large area detector. Combining the positions and times of arrival of the diffracted neutrons on the detector reveal the corresponding lattice planes in a comparatively short period of time. This makes the technique most suitable for experiments with limited movability of the sample, for instance experiments at temperatures in the milli-Kelvin range and within strong magnets.

Recently, considerable effort has been made to investigate diffuse scattering with neutrons. Further information can be found in Section 3.1 or, e.g., in ref 96.

3.1 Diffuse scattering

Generally speaking, all diffraction data include both the information from the average structure and from any deviations from the average lattice or spin structure. The latter part is called diffuse scattering – see ref 97 for a readable historical overview – and is due to spatial fluctuations in occupation and atomic positions and the respective magnetic moments. The isotropic background of Laue scattering ⁹⁸ from a binary solid solution provides the simplest example. The truly ideal case is found in the incoherent scattering of isotopic mixtures or randomness of nuclei with spin. For real solid solutions, like ZrTi, a modulation of the Laue intensity is observed indicating a deviation from simple randomness, such as preferential near-neighbor pairing. Diffuse scattering often shows very characteristic features, serving as fingerprints of the underlying defect structure and short-range correlations that impact material properties, e.g., the short-range order in alloys, which appears as precursor states near ordering phase transitions. The diffuse scattering from single crystals between the Bragg peaks is typically very weak in intensity, when compared to Bragg scattering, unless integrated over the full solid angle. Accordingly, modern neutron instrumentation tends to use large area detectors, which is obviously most efficient for measuring diffuse scattering.

There are several approaches to reveal the microscopic origin of diffuse scattering, either by Fourier analysis (of short-range order) in terms of pair correlation functions, ⁹⁹ or by modeling with a certain assumed motif, ¹⁰⁰ or by trying to establish a real-space model by the reverse Monte Carlo method ^{101 , 102} and, similarly, in the case of spin structures by the Spinvert software. ¹⁰³ This approach is limited to simple cases, however, because the intensity is a sum of possibly indistinguishable terms from many pair correlations in multi-component systems. An alternate approach, PDF analysis (see Section 3.2), provides model-free insights into real space as given by the superposition of the pair correlation function.

Some neutron instruments are dedicated to diffuse scattering studies, e.g., the SXD – the single crystal diffractometer at the ISIS spallation source. ¹⁰⁴ A large volume in reciprocal space is explored by neutron time-of-flight Laue diffraction in a single setting with an area detector using a pulsed, polychromatic beam. The instrument CORELLI at the spallation neutron source SNS in Oak Ridge uses a pseudo-statistical chopper for an efficient separation of elastic diffuse scattering from structural disorder, while discriminating thermal inelastic background. ¹⁰⁵

3.2 Pair distribution function analysis

Whenever structures deviate from perfect crystallinity such as in nanostructured materials, disorder evolves and gives rise to diffuse scattering (see section before). Common powder diffraction experiments relying on the interpretation of only the Bragg peaks therefore face limitations in the structural characterization of such disordered materials, liquids, and glasses. To then achieve more insight, one needs to introduce the pair distribution function (PDF) which is the probability of finding two atoms in the sample separated by distance r, weighted by their scattering lengths b _i . The PDF is accessed by Fourier transformation of diffraction data collected to high momentum transfer Q. ¹⁰⁶ Hereby, we use not only scattering from Bragg peaks, but additionally any diffuse scattering in-between and underneath the Bragg peaks, resulting in the terminology total scattering. It is important to note that the mathematical definitions of radial and pair distribution functions vary slightly between different publications, very unfortunately, and that G(r) is commonly used for the X-ray PDF while for neutron PDF (nPDF) further definitions co-exist employing D(r) or T(r). Please see ref 102 for a comparison of nomenclatures.

Originally, the short-range order and hydrogen bonding networks in bulk liquids ¹⁰⁷ and around ions in solution ¹⁰⁸ have been commonly investigated, as well as the structure of glasses. ¹⁰⁹ The general popularity of PDF analysis exploded with the need for structural characterization of nanomaterials ¹¹⁰ often lacking sufficient crystallinity. For clusters of a few atoms, quantum dots, nanocrystals, but also bulk materials with nanoscale domains, the PDF approach will excel in extracting the interatomic distances, coordination numbers, local vibrational motion, local structural motifs and chemical short-range order in the sample, as well as nanoparticle size. Because of the constant, Q-independent neutron scattering lengths, in contrast to the decaying X-ray atomic form factor, nPDF studies can measure to high momentum transfer Q _max, readily up to 50 Å⁻¹. Hence, smaller changes in interatomic distances are accessible simply because of better resolution in the PDF, as defined by Δr = 2π/Q _max.

The collection of PDF data is based on a powder diffraction experiment up to high Q of at least ≈ 20 Å⁻¹. To achieve the latter, short neutron wavelengths are employed, often combined with extensive detector coverage to collect the entire angular range simultaneously. State-of-the art nPDF instruments are found at spallation sources enabling high flux at high neutron energies, i.e., at CSNS, ISIS, SNS, and JSNS. At present, only the hot source of the ILL features a reactor-based PDF instrument. A very recent review on nPDF recapitulates facilities, data treatment, and analysis with examples. ¹¹¹

One recent nanomaterials example showcasing the power of nPDF analysis was the revelation of a much higher concentration of surface oxygen defects in CeO₂ rod-shaped nanocrystals compared to ceria nanocubes. The refinement of the nPDF data distinguished surface from bulk oxygen defects and required two different phases, one of bulk CeO₂ with Frenkel defects and one of surface Ce₃O_5+x species, see Figure 11. Further, a smaller number of surface defects has been observed upon exposure of the nanorods to SO₂ being possibly the reason for sulfur poisoning of these catalytic materials. ¹¹² The high sensitivity of neutrons towards light elements such as H, C, O and Li is exploited in nPDF studies of the local structure of fuel cells, heterogeneous catalysts, ferroelectrics, and battery materials. For instance, nPDF helped to clarify that an increase in Co concentration in proton-conducting oxides of the type La_0.9Sr_0.1Sc_1−x Co _x O_3−δ impacts the local structural disorder and proton location. ¹¹³ This field of energy materials comprises the need for in situ and operando measurements (see Section 3.5).

Figure 11:

Left: Neutron PDF refinement of ceria nanorods with two phases: ceria with Frenkel-type defects by applying a numerical nanorod shape correction γ ₀(r) and surface defects modelled as Ce₃O_5+x. The overall fit (red) to the experimental data (black) results in the difference curve (blue, in offset), showing the shape envelope function γ ₀(r) (purple), as well as the PDF of the contribution of surface defects (green, in offset). Right: Scheme of the surface species Ce₃O_5+x as spherical nanoparticles on the surface of the ceria nanorods together with their crystal structures. Adapted from ref 112 by kind permission of the American Chemical Society.

The pair-correlation function, i.e., the Fourier transform of the scattering data from reciprocal to real space also provides a valuable approach to the analysis of magnetic diffuse scattering. Magnetic short-range order and correlations can efficiently be determined by acquiring diffraction data and utilizing the PDF formalism. ¹⁰⁶ For powder diffraction, this magnetic PDF (mPDF) analysis ¹¹⁴ proceeds by modelling the structural pair correlation function which is then subtracted from the experimental pair correlation function. The difference then serves as a first model for the magnetic pair correlation such that subsequent refinement techniques can be used to iteratively model both the structural and magnetic contributions. The mPDF approach was utilized recently to model the short-range order in MnO ¹¹⁵ and frustrated magnets. ¹¹⁶ Likewise, for single crystals, the 3D-mPDF method can be utilized to determine the three-dimensional magnetic correlations. The analysis proceeds by removing both the structural and magnetic Bragg peaks, for example through the punch-and-fill or KAREN method, and then employs a 3D-Fourier transform of the remaining diffuse magnetic scattering to obtain the magnetic correlations in 3D. This method was recently showcased on MnTe single-crystal data to determine the real-space correlation in the short-range ordered state of this antiferromagnet. ¹¹⁷ We expect that PDF and mPDF analysis may see further advances and progress when combined with use of polarized neutrons, and when appropriate, dedicated instrumentation will become available for users.

3.3 Polarized neutrons

For revealing magnetic diffuse scattering, it is often helpful to use polarized neutrons as probes and possibly polarization analysis on the scattered neutrons. Polarized neutron diffraction has been established in the 1960s already. ^{118 , 119} Clearly, knowing about the direction and change of the neutron’s magnetic moment upon scattering will provide more detailed structural information. Furthermore, the higher quality data may justify using the lower flux of polarized experiments. This enables clean separation of magnetic diffuse scattering and elimination of any background, e.g., through the so-called XYZ analysis, applicable within the orientational average for powder diffraction using multi-detectors. ¹²⁰ With polarized neutrons and polarization analysis it is possible to distinguish nuclear scattering from magnetic scattering, to probe sensitively the orientation of magnetic moments, and further to reveal what remains hidden to unpolarized studies, that is, the interference of nuclear and magnetic amplitudes, and the handedness (also dubbed chirality) in magnetic configurations. Single crystals and polarized neutrons are necessary requisites to directly reveal chiral properties by magnetic diffuse scattering from disordered spin structures. ^{84 , 121}

Polarized neutrons are valuable to enhance the intensity signal |N ± P∙M _⊥ | ² for very small moments by interference of nuclear and magnetic amplitudes, N and M _⊥ , as obtained from the intensity difference under reversal of the neutron beam polarization P. Note that the ⊥ index in M _⊥ refers to only the components of M perpendicular to Q, which are visible to neutrons. ¹²² The magnetic structure can be established in a two-step process by first determining the nuclear, non-magnetic crystal structure. Moreover, the linear response of M _⊥ to an applied magnetic field H is proportional to the susceptibility and can be obtained at the atomic scale by crystallographic analysis of the Bragg intensities. This analysis includes preferred orientations, anisotropies, and spin- and magnetization densities, which can be extracted efficiently even from powder data using 2D detectors ^{123 , 124} and from magnetic nanoparticles. ¹²⁵ One can readily extend this approach to ferrimagnetic samples and determine weak magnetic amplitudes from small magnetic moments, even with high precision. Similar to the electron density distribution obtained from X-ray Bragg intensities, magnetic form factors and spin densities can be determined from magnetic Bragg intensities in neutron diffraction.

It is noteworthy that using polarized neutrons and polarization analysis, one can also distinguish spin-incoherent from coherent scattering, a possibility that may be of interest and relevant particularly for hydrogenous materials. However, the need for more specialized instrumentation and lower flux still limits applications.

Combining polarization analysis with Bragg and diffuse scattering will be possible at the new ESS diffractometers, ¹²⁶ the polarized instrument MAGIC and in further future the instrument DREAM. ¹²⁷ These will enable new types of experiments combining atomic and magnetic PDFs, for which just recently proof-of-principle experiments were carried out on MnTe at HYSPEC, SNS. ¹²⁸

3.4 Magnetism

The first pioneering studies of determining magnetic structures ¹²⁹ were carried out by analyzing powder-diffraction data using unpolarized neutrons, similar in style to regular structural refinements covered in Section 2.1, and this most frequent approach persists until today. ^{22 , 65} The magnetic structure information is found by the magnetic cross-section which imposes that only magnetization perpendicular to the scattering vector Q is observed. Moments parallel to the scattering vector are not observed, in principle. For layered materials in particular, magnetic intensity for Bragg peaks perpendicular to the layers vanishes for purely axial moments, see Figure 12, left. For antiferromagnets, in many cases the magnetic Bragg peaks do not overlap with structural peaks (see below, however), witnessing a new periodicity popping up from the diffractogram. For ferromagnets, however, magnetic Bragg peaks are superimposed on top of some nuclear peaks. The simplest approach to identify the magnetic peaks is thus to measure the diffraction pattern above and below the ordering temperature. Alternatively, by applying a saturating external magnetic field, ferromagnetic peaks can be made to appear or vanish, since magnetic scattering amplitudes vanish if moments align along Q, to cleanly separate atomic and magnetic structure information.

Figure 12:

Left: Structures and neutron powder diffraction data of hexagonal MnTe and 5%-Li:MnTe, with basal and axial spins, respectively. The [0001] magnetic peak is present only when moments are perpendicular to the c-axis. Adapted from ref 130 by kind permission of the American Physical Society. Right: Diffuse scattering from Ho₂Ti₂O₇ spin ice measured with polarization analysis in the spin-flip channel (A), non-spin-flip channel (B), and their sum (C) as would be observed without polarization analysis. Adapted from ref 131 by kind permission of AAAS.

Having no new periodicity in a magnetic structure does not necessarily imply ferromagnetism. So-called k = 0 structures, for which the unit cells of lattice and spin structure coincide, can also be antiferromagnetic. A most useful convention is to describe magnetic structures by their propagation vectors k that capture the magnetic periodicity. The definition of a multi-k structure requires more than a single k vector. For high-symmetry lattices it can be difficult to distinguish a multi-k structure from equal domain populations of a single-k structure. These structures have gathered much attention because they yield real-space spin textures, i.e., skyrmions. Unambiguous identification of a multi-k magnetic structure requires additional information either from local-probe methods ¹³² or from the spin-dynamics. ¹³³ A rational or irrational ratio of k to the reciprocal lattice parameters corresponds to a commensurate and incommensurate magnetic structure, respectively. For structural analysis, these can be treated similarly and are typically related to spiral structures. Extending our experimental repertoire to add polarization analysis (see Section 3.3) may facilitate the separation of magnetic from nuclear scattering and help to distinguish anisotropies in spin correlations. In view of the more complex structural patterns of spin spiral structures, neutron polarization analysis of diffraction from single crystals clearly opens new possibilities, revealing the vector chirality, M _⊥(−k) × M _⊥(k), i.e., the turning of helices and cycloids that remains hidden to unpolarized neutrons. The most precise tool is spherical polarimetry, see the recently developed software Mag2Pol. ¹³⁴

Deviations away from the average magnetic structure yield diffuse magnetic scattering. An exemplary origin of such diffuse scattering is the magnetic disorder arising from geometric frustration of magnetic interactions. Antiferromagnetically coupled moments on an atomic triangle are an important example because triangles are a most common structural motif and building block in nature. Local spin anisotropies can impose strong frustration also on ferromagnetic, dipolar interactions. A most prominent example is the diffuse magnetic scattering from a single crystal of the ternary oxide Ho₂Ti₂O₇, ¹³¹ also going under the name “spin-ice”. This magnetic analogue of water ice shows the same topology of disorder and ground state entropy. ¹³⁵ In this spin structure, violations of the so-called ice rules ¹³⁶ create remarkable defects, magnetic monopoles, at least in a topological sense. The distinct features are pinch points arising from the effective long-range Coulomb-like interactions, appearing as ridges of magnetic diffuse scattering through the (002) reciprocal lattice points. These are revealed using polarization analysis, with polarization perpendicular to the scattering plane to measure in-plane moments in spin-flip mode, see Figure 12, right.

3.5 In situ and operando studies, sample environment

Many characterization techniques yield a rather static picture of matter at an atomic level. While this does correspond to highly valuable information, structures will nonetheless change over time, i.e., dynamics, phase transitions and chemical reactions are not covered. In situ measurements collecting data while external parameters (e.g., T, p, gas flow, electric or magnetic field) change can shed light on such processes. If the function or wear of devices under realistic operating conditions are investigated in a time-resolved manner, we also speak of operando studies. They give a realistic picture of materials and their changes at an atomistic level over time in devices under “real-world” conditions. This relates to, for instance, mapping hydrogen uptake in hydrogen storage materials, lithium intercalation in battery materials, oxygen defects in oxygen ion conductors for fuel cells, changes in microstructure in alloys, or phase formation in chemical synthesis, just to name a few examples. ^{137 , 138} Such deep insights into time-resolved processes through operando studies contribute significantly to the optimization of whatever devices.

Obviously, in situ and operando investigations rely on controlling the external parameters to be varied. This is provided by sample environment, which may obstruct the primary or diffracted beam. Since neutrons interact weakly with most materials and have a high penetration depth, they allow for bulky sample environment and are thus ideally suited for in situ and operando studies. In addition, neutron scattering is non-destructive but highly sensitive for light elements, especially hydrogen, lithium, nitrogen and oxygen, and so neutrons may probe many interesting materials properties such as crystal structure, magnetism, atomic diffusion or vibrational properties. As light atoms such as H or Li are highly relevant for “energy materials”, neutrons are important in this respect, too.

Eventually, the high transparency of most sample environments for neutrons also facilitates the use of cryostats down to the milli-Kelvin range, not achievable by synchrotron radiation.

The sample environment for controlling the external parameters may be of a general nature and available at many neutron beamlines, e.g., furnaces (up to 2,000 K or more), cryostats (T down to mK ¹³⁹ ), magnets (up to 26 T), pressure cells (up to 100 GPa ^{140 , 141} ), or highly specialized equipment such as tensile stress testing machines, electrochemical cells, ¹⁴² or gas-pressure cells. ¹⁴³ The combination of two or more parameters to be varied is also quite common, such as cryomagnet, pressure cell in a cryostat, electric fields in a furnace or heating in gas flow. To give even deeper insight, in situ neutron diffraction may be combined with simultaneous characterization, for example by spectroscopy (Raman, IR, mass spectrometry).

Let us look at a few examples to demonstrate the power of in situ and operando neutron diffraction. Rechargeable batteries are often based on lithium (de)intercalation into solid electrode materials, for which neutrons are an excellent probe. In situ and operando investigations were crucial in explaining failure mechanisms in lithium manganite-based rechargeable lithium-ion batteries. An unfavorable capacity fading observed for LiMn₂O₄, but not for lithium-rich Li_1+x Mn_2–x O₄, was tracked down to different reaction pathways. Samples with x = 0.0 suffer large volume changes upon formation of three distinct phases, while samples with x = 0.1 show a continuous lithium (de)intercalation within the same phase (Figure 13, left) resulting in a lower mechanical stress during cycling and, therefore, much less pronounced capacity fading for the non-stoichiometric Li_1+x Mn_2–x O₄. ¹⁴⁴ Due to the large penetration depth, such operando investigations may be done on commercial battery packs without any disassembly or further preparation, thus providing real-world conditions, ¹⁴² as said before. In hydrogen storage materials, additives are often used in order to improve reaction rates of hydrogen uptake and release. There are numerous examples where in situ neutron diffraction could explain these findings by identifying reaction intermediates, e.g., AlB₂ for Al additives to LiBH₄, or mixed imides Li₂Mg(ND)₂ and Li₂Mg₂(ND)₃ in the Li–N–H system. Hydrogen incorporation into magnetic materials is another example, where in situ neutron experiments as a function of temperature and hydrogen gas pressure can reveal the complex interplay between hydrogen content, atomic and magnetic order (Figure 13, right). ¹⁴⁵ Nitrides are a natural target for neutron diffraction due to the large scattering contribution of nitrogen atoms for thermal neutrons. For example, following the ammonolysis of iron powder by time-resolved in situ neutron diffraction unraveled the formation mechanism of the giant magnetic moment material α’’-Fe₁₆N₂. In combination with thermal analysis, forming the iron-rich nitride was found to be kinetically controlled. This helped to optimize the synthesis and increase the yield of this metastable solid to above 90 %. ¹³⁷

Figure 13:

Left: in situ neutron powder diffraction of Li_1+x Mn_2–x O₄ cathode materials upon lithium deintercalation in a lithium-ion battery revealing three distinct phases for x = 0 and the continuous lithium deintercalation for x = 0.1, thus explaining lower mechanical stress and less capacity fading in the latter. Adapted from ref 144 by kind permission of the American Chemical Society. Right: in situ neutron powder diffraction data (λ = 1.8680(2) Å) of the reaction of FePd₃ with deuterium gas as a function of temperature and deuterium gas pressure yield deuterium content, atomic and magnetic order by Rietveld analysis, revealing their complex interplay in this ferromagnet. Adapted from ref 145 by CC BY.

Many more examples could be described in detail, for example exotic behavior of matter under extreme conditions, mimicking conditions found deep in the Earth’s mantle in geochemistry, studying magnetic ordering at low temperature or the kinetics of chemical reactions.

The time resolution of the method used has to match that of the reaction to be followed. For in situ and operando neutron diffraction the resolution typically ranges from milliseconds to hours and is largely determined by the neutron flux on the sample and the detector efficiency. This obviously limits studies with high time resolution to specialized high-flux instruments. For fast reactions (within seconds) usually only qualitative phase analysis and lattice parameter determination may be performed, while for slower ones (within minutes) the full Rietveld analysis yielding complete crystal structures is feasible given modern neutron diffraction instruments. ¹³⁸ Future developments in neutron instrumentation will allow even better time resolution. Due to the lower brilliance of the source, present-day neutron diffraction cannot compete with synchrotron diffraction in terms of time resolution in general, but often provides extremely valuable information in the accessible parameter space.

Hence, in situ and operando neutron diffraction are powerful tools for investigating the structural properties and behavior of materials. These techniques yield detailed information on phase transitions and chemical processes including phase formation, reaction intermediates and their crystal and micro-structures in real time, providing in-depth insights into their physical and chemical properties. This not only applies to basic research but also to more applied questions such as industrial processes or the function and wear of functional materials and devices.

3.6 High pressure

High-pressure experiments are an attractive tool in a rather broad research area, targeting all kinds of physical probes. Unlike chemical pressure – created by a too small atom sitting on a too large site, hence volume contraction –, high pressure is a clean and controllable external parameter. Squeezing the interatomic distance can provide a means to control and understand various physical properties, such as superconducting transitions, and even start pressure-dependent chemical reactions. ¹⁴⁶ It is also an indispensable tool in the search for novel materials and planetary science. Basically, the high-pressure experiments are incompatible with neutron experiments requiring larger sample volumes, since the pressure generated is determined by the force applied divided by the area, which inevitably reduces the volume of the sample. Developing unique high-pressure cells, however, has facilitated its application to neutron experiments.

In neutron diffraction under high pressure, the following three points are essential. First, the sample volume should be kept as large as possible to counteract the weak interaction of neutrons with matter. Second, detector coverage should be secured to acquire the signal from the small sample. Finally, background countermeasures are necessary to reduce parasitic scattering from cells, anvils, and pressure-transmitting media around the sample.

Let us take the example of a powder diffraction experiment using a Paris–Edinburgh (PE) cell, ^{140 , 147} which has expanded the possibilities of neutron experiments under high pressure (Figure 14). First, using the anvils with an indentation at the tip, a sufficient sample volume of 0.088 mL is ensured for neutron experiments up to 10 GPa. Next, the detector coverage is secured by a frame design with limited shadow area and the anvils that ensure a ±7° vertical window even under high loads. Finally, measures to record clean patterns are employed according to the source used and the purpose of the experiment. In a reactor source, the angle-dispersive method is applied with a “through the gasket” geometry. The use of cubic boron nitride, BN, an anvil material that does not show strong Bragg peaks, contributes to a clean diffraction pattern. On the other hand, the TOF diffraction at a pulsed neutron source can access a wide energy range even with a limited diffraction angle. In this case, the “through the anvil” geometry is generally applied. The prospective area contributing to the signal is confined by the incident collimator and the anvil gap, effectively reducing the parasitic signal from surrounding materials. The PE cells have been installed in neutron facilities worldwide and allow us to conduct Rietveld refinement as well as PDF analysis under extreme conditions.

Figure 14:

Paris-Edinburgh cell and scattering geometries with the inset showing a magnified view of the sample and anvils.

In addition to the PE cell, various high-pressure devices have been developed. For example, a large single crystal diamond anvil cell (DAC) was developed for SNAP at the SNS, and the pressure generated is exceeding 100 GPa. ^{141 , 148} The DAC enables the gas to be sealed inside the sample chamber, so it is expected to play a significant role in the future structural elucidation of superconducting hydrides in which hydrogen plays an important role. ¹⁴⁹ At PLANET at J-PARC, a large volume multi-anvil press for high-temperature and high-pressure experiments has been installed, ¹⁵⁰ and research on iron hydrides is conducted to estimate the amount of hydrogen contained in the Earth’s core. ¹⁵¹ A very chemical example is given by the high-pressure (2 GPa) synthesis and structural elucidation of the ubiquitous “non-existing” compound carbonic acid, H₂CO₃, for which a deuterated sample was used for diffraction in a specially built 0.4 mL hybrid clamped cell, followed by density functional-based structure solution. ¹⁵² While most high-pressure studies are on powder samples, some examples of single-crystal structural studies should be mentioned: a single-crystal DAC that can be used complementary for both X-rays and neutrons, ¹⁵³ and a DAC with polycrystalline diamonds as an anvil to overcome the problem of cell absorption correction. ¹⁵⁴ In addition, neutron polarimetry experiments at ILL succeeded in determining a complex spiral magnetic structure in a multiferroic material at 4 GPa using a nonmagnetic high-pressure cell. ¹⁵⁵ With the development of the unique cell, high-pressure neutron diffraction experiments are expected to continue to expand their range of applications.