# High-resolution neutron spectroscopy using backscattering and neutron spin-echo spectrometers in soft and hard condensed matter

## Abstract

The instruments best suited to performing high-energy-resolution neutron spectroscopy are spin-echo spectrometers and backscattering spectrometers. The development of these experimental techniques dates back almost half a century, and most major neutron scattering facilities operate mature spectrometers of one or both classes. Recent advances in instrumentation and neutron sources are enhancing their performance and expanding their capabilities, with the objective of enabling researchers to tackle new and more complex problems. In this Technical Review, we assess the current state of the art in high-energy-resolution neutron spectrometers, showcasing their role in the study of nanoscale dynamics in soft and biological materials, as well as disordered magnets.

## Key points

• Neutron scattering is a uniquely powerful experimental method to study the structure and dynamics in materials.

• Neutron spin-echo and backscattering spectrometers probe the dynamics in materials on the picosecond and nanosecond timescales and provide spatial information on the motions over ångström to nanometre length scales.

• High-resolution scattering studies have contributed to a wide range of scientific fields, from nanosized data storage and carbon sequestration to the efficiency of modern batteries and drug delivery systems.

• The state of the art of these instruments is constantly evolving, in particular in improving the resolution and accelerating the rate at which data can be acquired.

The dynamic processes that occur on the nanoscale in many materials, ranging from proteins and quantum magnets to catalysts, are crucial to their functionality and are often essential to understanding their bulk properties. A wide range of characterization tools can be used to probe the complex set of dynamic modes that are found in materials, including dielectric spectroscopy, muon spin spectroscopy, NMR or Mössbauer spectroscopy. However, the scattering of quantum particles or radiation, such as visible light, X-rays or neutrons, provides information on both the timescales and length scales of motion. Among these probes, thermal and cold neutrons have wavelengths and energies on the order of interatomic distances and excitation energies in matter, respectively; thus, neutron spectroscopy is uniquely positioned to study the behaviour in materials on the nanometre length scale and nanosecond timescale1,2. In this Technical Review, we focus on neutron spectroscopy, specifically, high-energy-resolution methods, which are used to probe dynamic processes that occur on timescales from picoseconds to hundreds of nanoseconds. High energy resolution is an arbitrary term used by the neutron scattering community, with no exact quantitative definition. In general, this term refers to spectrometers that have an energy resolution in the μeV range, specifically neutron spin-echo (NSE) and backscattering spectrometers. Figure 1 shows the temporal and spatial scales that can be accessed using these neutron spectrometers compared with those of more commonly used experimental techniques.

## Key aspects of neutron scattering

As neutrons have no charge, they are insensitive to the charge in the electron cloud of atoms and are scattered directly by the nuclei. The interaction potential between a neutron and the nucleus, quantified as a scattering cross section, is isotope dependent3 (Supplementary Table 1); for example, there is a notable difference in the scattering cross sections of hydrogen and its less-common isotope, deuterium (the incoherent cross sections are 80.26 × 10–28 m2 and 2.05 × 10–28 m2 for hydrogen and deuterium, respectively). Thus, neutron scattering is an indispensable tool in the study of time and space correlations in soft matter and biological systems, which are rich in hydrogen atoms. Subsequently, hydrogen–deuterium substitution is commonly used to control the source of the neutron scattering signal to, for example, make certain atoms, molecular structures or system components invisible without (to a first approximation) affecting the sample chemistry. In addition, a neutron also has a spin, which couples to the magnetic moments of a material through a dipolar interaction. This spin–moment coupling makes neutron scattering an irreplaceable probe for magnetism.

The interaction between a neutron and a nucleus is described by the Fermi pseudopotential through a single parameter, the scattering length b, which determines the scattering probability. The scattering potential of an atom depends on its nuclear spin, which is usually disordered. Therefore, there is often a random variation of the scattering lengths of a specific atom, owing to the random population of different isotopes and the randomness of the nuclear spin direction. This variation enables nuclear neutron scattering to be separated into a so-called coherent component (related to the average scattering length of an element), which provides information on the relative position and motion of different atoms, and an incoherent component (related to the random deviation from the average), which yields information on the single particle dynamics of the atoms. Thus, the coherent and incoherent scattering lengths of an atom are defined by the average of b over all isotopes and spin states and by its root mean square deviation, respectively. The coherent and incoherent scattering cross sections are defined by 4π times the square of the coherent and incoherent lengths, respectively.

Owing to the neutron spin, neutron beams can also be spin polarized, and the polarization can be manipulated by magnetic fields. This capability is used in the spin-echo technique, which is discussed below. Coherent scattering does not alter the spin of the neutron, whereas incoherent scattering (when it originates from a distribution of nuclear magnetic moments rather than the presence of various isotopes) has a 2/3 probability of flipping the neutron spin. As the scattering of a neutron by a magnetic moment depends on the relative orientations of the neutron spin and the wavevector transfer Q, controlling the beam polarization in an experiment offers another degree of freedom that is relevant for magnetic neutron scattering.

The interaction of neutrons with matter is relatively weak; thus, neutrons can penetrate through most materials and rarely cause damage or changes to the sample under study (beam heating is rarely an issue at millikelvin temperatures or above). Furthermore, complex and bulky sample environment equipment can surround the sample to enable measurements under various conditions, including tension or shear, very low temperature (millikelvin) or extreme heat (<2,000 °C), high pressure (on the order of gigapascals), high magnetic fields (<26 T) or a combination of these. Unfortunately, neutron scattering methods are limited by the low flux of available neutron beams, and the beams are not easily focused. Therefore, experiments generally require large samples (with a mass in the range of hundreds of milligrams to grams). In addition, a few isotopes are strong neutron absorbers, which poses a problem if one of these isotopes is present in a sample.

Neutron beams are currently produced by either fission reactors or spallation sources. Once a free particle, a neutron passes a moderator volume, which reduces or moderates its energy to that required by a specific instrument. Moderators generally contain light atoms or molecules, such as water or liquid hydrogen. The mean energy of a neutron beam emanating from a room-temperature water moderator is ~25 meV (with a wavelength (λ) of 1.8 Å); hence, neutrons having approximately this energy are termed thermal neutrons. To probe slow dynamics, such as those described in the following, typically requires cold neutrons of less than ~5 meV (with λ 4 Å) obtained, for example, from a liquid hydrogen moderator. In general, the wavelength dependence of the neutron flux can be described by a Boltzmann distribution that peaks in the 2–4 Å range and decreases at long wavelengths as λ–5.

### Neutron spectroscopy

In a traditional neutron scattering experiment4, the quantity measured is the double differential scattering cross section, where σ is the total scattering cross section (equation 1)

$$\frac{{{\rm{d}}}^{2}\sigma }{{\rm{d}}\varOmega {\rm{d}}E}\propto \frac{| {{\bf{k}}}_{{\rm{s}}}| }{| {{\bf{k}}}_{{\rm{i}}}| }S({\bf{Q}},E)$$
(1)

which is the probability that the neutron is scattered in the solid angle, dΩ, and exchanges an energy, E = Es – Ei = (ħks)2/2mn –  (ħki)2/2mn, and a momentum ħQ = ħks – ħki, with the sample. Here, ħ is the reduced Planck’s constant; mn is the neutron mass; ks and ki are the wavevectors of the scattered and incoming neutron, respectively; and Es and Ei are the energies of the scattered and incoming neutron, respectively. In the energy range of usual high-energy-resolution neutron spectrometers, |ki| = |ks| = 2π/λ and |Q| = Q = (4π/λ)sinθ, which relates the magnitude of the wavevector transfer, Q, to the neutron wavelength, λ, and the scattering angle, 2θ.

According to the well-established van Hove picture5, the dynamic structure factor S(Q,E) is the time and space Fourier transform of the two-point correlation function between the atomic positions in the sample (equation 2)

$$S({\bf{Q}},E)={\int }_{-\infty }^{\infty }\sum _{{\rm{i}},{\rm{j}}}\langle {b}_{{\rm{i}}}{b}_{{\rm{j}}}\exp \{i{\bf{Q}}\,[{{\bf{r}}}_{{\rm{i}}}(t)-{{\bf{r}}}_{{\rm{j}}}(0)]\}\rangle \exp (-i\frac{Et}{\hbar }){\rm{d}}t$$
(2)

where bi and bj, and ri and rj are the neutron scattering lengths and positions of atoms i and j, respectively. A similar equation can be written for the magnetic scattering from unpaired electrons2. Equation 2 demonstrates the inverse relationship between Q and distance (or length scale) and between E and time (t), implying that better energy resolution enables the study of slower dynamic processes.

If only spatial correlations are of interest, measurements are taken using diffractometers, with which no energy analysis of the scattered neutrons is performed (essentially, S(Q,E) is integrated over E). However, to probe dynamic modes in a sample, it is necessary to use a neutron spectrometer that measures the energy transfer E.

In materials, motions can be broadly characterized as either excitations that take place at a defined energy transfer E (observed as a distinct peak intensity in S(Q,E)), for which there is an inherent restoring force, such as periodic natural modes (phonons or magnons) or vibrational modes; or as a relaxation, for which there is no restoring force and the inherent dynamics are diffusive in nature. In the latter case, the observed scattering signal appears as a broadening of the so-called elastic line, which is referred to as quasi-elastic neutron scattering (QENS). QENS is the main measurement associated with the instruments discussed herein. The elastic line represents zero energy transfer, where ‘zero’ is the smallest energy transfer that can be measured by a given spectrometer and is also referred to as the energy resolution of the instrument. The elastic line contains information on motions that are too slow to be measured. In general, any spectrometer can be used to probe vibrational or relaxational dynamics; however, because of the typical energy scales involved, vibrations are most commonly studied on instruments with energy resolutions of the order of meVs, such as three-axis, direct-geometry or vibrational neutron spectrometers. The diffusional and transport processes that occur in biological and liquid-based systems, relaxations in polymers and colloidal systems, interfacial fluctuations in complex fluids, and relaxation phenomena in both structural and spin glasses as well as frustrated magnets all occur at slower timescales (tens of picoseconds to hundreds of nanoseconds). To probe such relaxational dynamics, spectrometers with a μeV energy resolution (that is, a high energy resolution), such as backscattering and NSE are needed.

In the following, we discuss, in turn, the fundamental principles of NSE and backscattering instruments, the current status and recent advances in technology, and future directions for their development. We then showcase the applicability of these techniques to study relaxations at the nanoscale using examples from the fields of soft and hard condensed matter, specifically polymer science, biophysics and magnetism. However, the scientific questions that can be addressed using this technique go well beyond these systems and include, among others, diffusion in liquids, conduction in energy-relevant and functional materials, and diffusion in solids, such as catalysts. A broader overview of the scientific opportunities offered by these techniques can be found elsewhere6,7,8,9.

## Neutron spin-echo spectrometers

NSE spectrometers provide the highest energy resolution of all neutron spectroscopy techniques. Of the currently operational instruments (Table 1), IN15 at the Institut Laue–Langevin (ILL; France)10 offers the highest resolution of ~1 neV. This means that IN15 can probe dynamic processes with characteristic times of up to almost a microsecond, thus making it the ideal technique to observe slow relaxations in polymers (such as reptation), dynamics such as bending in amphiphilic membranes and fluctuations in soft colloidal systems or normal modes in proteins in solution. In hard matter, it is used to probe spin glasses, critical magnetic fluctuations or emergent excitations in quantum magnetism.

In the NSE technique, the velocity changes experienced by the neutrons as they interact with matter are determined by measuring the change in the Larmor precession of their spins in a magnetic field (Box 1). NSE spectrometers are commonly, and most efficiently, employed to measure the motions of nanosized structures, such as polymers, proteins and self-assembled complexes; however, measurements on molecular liquids and solid materials are also performed to investigate structural relaxations, the single particle dynamics of hydrogen atoms or spin dynamics.

Traditionally, NSE spectrometers have been located at neutron research reactors (Table 1) and most are of the traditional Mezei type (Mezei NSE). The first time-of-flight NSE spectrometer, and the only Mezei-type instrument operating at a pulsed source, was built at the Spallation Neutron Source (SNS) at the Oak Ridge National Laboratory (ORNL; United States)11. This spectrometer was also the first to employ superconducting precession coils. Currently, two new NSE spectrometers are being developed at the pulsed source in the Material and Life Science (MLF) Experimental Facility at the Japan Proton Accelerator Research Complex (J-PARC)12. One of these is a modulation of intensity by zero effort (MIEZE) instrument, which enables measurements on samples that cause a depolarization of the neutron beam, such as ferromagnets and superconductors, which cannot easily be accomplished using classical Mezei-type instruments. A similar MIEZE instrument is available at the Resonance Spin Echo for Diverse Applications (RESEDA) at Forschungs-Neutronenquelle Heinz Maier-Leibnitz (FRM II; Garching, Germany), which also offers a longitudinal resonance spin-echo option13.

The most important area of technical development in NSE spectrometers concerns the design of the precession coils and correction elements to improve the homogeneity of the field integral, which determines the upper limit of the Fourier times that can be probed. With modern precession coils, it is possible to come close to the theoretical limit of ~200 ppm (ref.14), and the inhomogeneity can be further reduced by a factor of ~300 through the use of state-of-the-art correction elements (known as Pythagoras coils)14. Recent upgrades of IN15 (ref.15) have produced a maximum field integral of 1.2 Tm, which is four times larger than what was previously available on that spectrometer. In combination with the high flux and long wavelength available to IN15, Fourier times of ~600 ns (ref.10) can routinely be achieved, and it is even possible, under favourable experimental conditions, to reach 1 μs. A similar performance has been achieved with the superconducting precession coils at the Jülich-neutron spin-echo (J-NSE)-Phoenix instrument at FRM II16. This instrument can reach a maximum field integral of 1.5 Tm, and preliminary results indicate a successful reduction of the field inhomogeneities by a factor 2.5 compared with the previous design. With this field integral, the J-NSE-Phoenix spectrometer can reach ~500 ns with 17 Å neutrons16.

In addition to the improvements in energy resolution, considerable effort has been made to widen the detection area of NSE instruments. Typical Mezei-type instruments have a detector with an area of ~1,000 cm2, which covers a scattering angle range of ~5° (in comparison, in a typical backscattering spectrometer, >1 steradian is covered with an analyser crystal). This area is appropriate when the signal of interest is concentrated in a small region of the Q space, such as a small-angle neutron scattering pattern, but not when the neutron signal is spread uniformly over Q, such as is the case for incoherent dynamics or in some cases for magnetic spin dynamics (paramagnetic scattering). This desire to perform simultaneous measurements in a large detector has led to the development of a 30° wide detection option for the IN11 instrument17, which has been successful despite the consequent limitation on the attainable Fourier time. In a further development to reach detection coverage as high as 180° in the Q plane, anti-Helmholtz coils were first employed on SPAN at the Hahn-Meitner Institute (Berlin, Germany)18, and eventually led to the construction of the WASP (wide angle spin-echo spectrometer) at the ILL19, which is currently under commissioning. On WASP, the detector coverage is 150° (in the scattering plane) × 2.5° (in the vertical direction) and the maximum field integral is 0.27 Tm, which provides a maximum Fourier time of 10 ns at 6 Å. Covering more than three orders of magnitude in time, WASP is an extremely promising tool for the investigation of slow dynamics over a wide Q range.

## Backscattering spectrometers

Backscattering spectrometers9 (Box 2) have a slightly coarser energy resolution than those of NSE spectrometers, achieving a resolution on the order of μeVs, and in contrast to NSE instruments, use unpolarized neutron beams. The information on the dynamic processes accessed by backscattering spectrometers is complementary to that probed using NSE instruments, but with an upper timescale limit of ~10 ns. Thus, backscattering instruments are suitable for probing relaxations in glass-forming systems (such as alpha and beta relaxations), dynamic signatures of phase transitions, molecular reorientations and rotational tunnelling, diffusion in bulk and confined liquids, as well as diffusion in proteins. In general, these instruments are most commonly used to measure the incoherent scattering signal from hydrogenated materials to determine the single particle dynamics of the hydrogen atoms in the sample.

Neutron backscattering spectrometers (Box 2) use crystals in a perfect (or near-perfect) backscattering geometry (that is, a Bragg angle of 90°) to analyse the scattered neutron energy. Table 2 highlights current backscattering instruments and their key characteristics.

Traditional backscattering spectrometers are based on ‘strained’ Si(111) crystals (for which the Bragg condition is satisfied at λ= 6.27 Å), offering an energy resolution of the order of ~1 μeV. Currently, the highest-resolution backscattering spectrometer in operation is IN16B at the ILL, which has an energy resolution of ~0.3 μeV, achieved using polished Si(111) crystals20. However, as IN16B with polished Si(111) suffers from an order of magnitude loss in intensity, the most commonly used setup uses strained crystals to give an energy resolution of ~0.75 μeV. IN16B also has the option of using strained Si(311) crystals that effectively double the width of the resolution and the dynamic range with only a slight reduction in intensity and also increase the maximum accessible Q from 1.8 Å–1 to 3.8 Å–1. This increase is of importance when fitting the data to obtain information on the geometry of the motions being probed. An even larger Q range is obtainable by using CaF2(422) crystals to analyse neutrons with a wavelength of 2.23 Å, as on the IN13 instrument21. IN13 therefore provides an unparalleled extended Q range of 0.2–4.9 Å–1, but with a coarser energy resolution of ~8 μeV than that acheiveable with Si(111).

The newest additions to the worldwide suite of Si(111)-based backscattering spectrometers are found at spallation sources rather than reactors: the backscattering silicon spectrometer (BASIS at SNS-ORNL)22 and the biomolecular dynamics spectrometer (DNA at MLF, J-PARC)12. Previously, backscattering instruments at pulsed sources were based on pyrolytic graphite, namely IRIS23 and OSIRIS24 at the ISIS Neutron and Muon Facility (United Kingdom), which due to a combination of primary and secondary spectrometer design characteristics are limited to a coarser resolution of ~20 μeV (but offer a wider dynamic range and higher intensity). IRIS introduced mica as an analyser to improve the energy resolution but at the high expense of intensity. Si(111) offers a better resolution (for example, achieving an energy resolution of 3.5 μeV) but requires an increased primary flight path length (for example, 84 m for BASIS compared with 36 m for IRIS). Further gains in resolution have been achieved by researchers at DNA by using a pulse-shaping chopper.

More recently, a so-called backscattering and time-of-flight spectrometer (BATS) option has been realized on IN16B25; this setup keeps the Si(111) analyser but adds a time-of-flight option to extend the energy window to ±340 μeV with only a slight coarsening of the resolution to 1.2 μeV. This was a highly innovative upgrade, requiring technological progress in chopper disk construction, which has increased the flexibility of the operation of the instrument, enabling new scientific challenges to be addressed, such as the study of more complex systems with dynamics that span many decades.

## Dynamics in soft and biological matter

High-energy-resolution neutron spectrometers provide unique information relevant to understanding the dynamics in soft matter and biophysics research, for example, in addressing structure–dynamics–function relationships in proteins, transport processes in membranes (biological or synthetic) and in the design of new polymeric materials with controlled, tunable and ‘smart’ properties. As most biological and soft matter materials are rich in hydrogen, isotopic substitution (primarily between hydrogen and deuterium) is a powerful approach to study the dynamics of a specific part of a molecule or a component in a multicomponent system5,10,26,27. Furthermore, separation of the neutron–sample interaction into coherent and incoherent scattering provides additional information about structurally relevant dynamics and cooperative motions. Measurements on backscattering and NSE spectrometers are ideally suited to probe many of the dynamic modes and relaxation processes that occur in soft materials and biological molecules, from the very local and fast rotations or vibrations that are essential precursors to the slower and larger-scale chain dynamics or diffusional processes that determine macroscopic behaviour.

### Polymers

Polymer relaxations have been studied with high-resolution neutron spectroscopy for at least 50 years5,7,28,29, resulting in well-established relaxation maps for, at least, linear homopolymers. Since the 2000s, interest in more complex polymer-based systems, such as polymer electrolytes30, polymers of complex architectures (for example, dendrimers)31, mixtures of two or more polymers (blends)32 or mixtures with hard33,34,35 or soft36,37 components (nanocomposites), and associating polymers38, has been driven primarily by the interest in possible applications. In these systems, the underlying molecular dynamics and the relationship with the macroscopic, that is, rheological behaviour, is still not fully understood. These more complex systems exhibit fundamental features of interest, such as surface, confinement and nanostructuring effects39. In polymer nanocomposites, for example, high-energy-resolution neutron spectroscopy measurements have shown that the dynamics of polymer chains at the surface of nanoparticles with attractive interactions with the polymer are not frozen. However, the translational dynamics of the chains is suppressed and the Rouse dynamics altered40, in agreement with results from confined polymer chains39,40. In addition, measurements have also highlighted the existence of two competing effects on the reptation tubes of polymers in polymer nanocomposites33: tube dilation due to chain disentanglement and a reduction in the reptation tube owing to geometrical confinement, the latter contribution being dominant for nanoparticle volume fractions of more than ~35% (ref.41). As revealed by NSE measurements, the dilation of the reptation tube can explain the anomalous softening of some polymer nanocomposites. The dilation can originate from, for example, the disentangling of polymer chains in nanocomposites with dynamically asymmetric polymers35,37; here, dynamical asymmetry refers to the interplay between a polymer with a high glass transition temperature (Tg), such as poly(methylmethacrylate), and a polymer with a low Tg, such as poly(ethylene oxide). A similar dilation of the reptation tube has also been observed in athermal polymer nanocomposites (systems in which the net enthalpic interactions between the surface of the nanoparticles and the polymer chains are negligible, that is, neither attractive nor repulsive) when the size of the nanoparticles is smaller than the entanglement mesh size42. This study provides a straightforward example of the microscopic insight neutron scattering techniques can provide in the study of polymer composite materials and therefore the main neutron data are reported in Fig. 2 to demonstrate typical experimental results.

### Proteins

Proteins are multiscale entities and their dynamics occur over several decades, from the atomic to the macroscopic scale. High-resolution neutron spectroscopy provides key insight into the dynamics of these systems at the atomic and molecular level. Over the ångström length scale at timescales shorter than a nanosecond, and following the seminal work of Doster and co-workers43, much work has been devoted to investigating the so-called dynamical transition (Td) in protein powders, its relationship with the dynamics of the solvent and its connection to the onset of protein function44. Studies have mainly focused on probing the onset of anharmonic motions and the relationship between these motions and the onset of biological activity using simple globular and model proteins. The effect of external parameters, such as pressure, temperature and salt concentration, provides insight into biological adaptation to extreme environments45. For example, the adaptive properties of extremophiles remain mysterious, but using neutron spectroscopy, three behaviours have been observed: reduced internal dynamics at high pressure, especially above Td, resulting from a roughening of the energy landscape46; stabilizing effects of high salt concentrations on the hydrogen-bonding network in halophiles without penetration into the biomass47; and an entropic argument to explain the increased flexibility in the folded state of thermophiles compared with that of their mesophilic counterparts48. Although much work continues to be done on simple or model proteins, researchers are successfully extending their investigations using backscattering spectrometers, in particular, to more biologically relevant systems by studying complex structures such as cancer cells49 or neural tissues50.

At longer length scales, on the order of several nanometres, the motions of large domains are fundamental to the function of proteins as biological machines, enabling molecular groups to reach the required proximity or co-enzymes to access the catalytic sites. Knowledge of these dynamics could be crucial for the design of new drugs. To this end, unique experimental information can be obtained using NSE spectrometers owing to their nanoscale space–time sensitivity and the possibility of probing, through the coherent neutron signal, the conformational dynamics of proteins (that is, the motion of one domain relative to another)17. Such studies have highlighted how large domain motions are overdamped (relaxational) movements and can often be described in terms of normal modes. In addition, it is possible to extract information on the friction experienced by the domains and/or the elastic constant51 of the flexible linker. NSE measurements can also be performed on intrinsically disordered52 or denaturated proteins53, revealing whether polymer-like models apply. The methods and recent results on internal protein dynamics have been reviewed elsewhere54,55. Despite the relevance of these studies, access to protein domain motions with the NSE technique is limited by several practical difficulties: the size of the protein and its domain need to be such to provide a sufficient signal; a relatively large amount of protein is required; and the analysis of the data to disentangle the conformational dynamics from other processes, such as diffusion and rotation as well as inter-particle interactions, is often complex and might benefit from the use of molecular dynamics simulations56. Combining NSE data with that from backscattering experiments can offer some help if, for example, the data is for perdeuterated protein powders; without solvent, the global motion of the protein molecules is suppressed, thereby enabling a clearer characterization of the collective internal dynamics. Observed collective motions on the 100 ps timescale are pathways for the larger-scale domain motions that are accessible with NSE57 in protein solutions as indicated by results from molecular dynamics simulations, which are in good agreement with the experimental data. Another example of the study of the collective motions of proteins58 in the subnanosecond timescale involved the careful analysis of QENS data from human deoxy-haemoglobin and CO-bound haemoglobin in solution, revealing that although the deoxy-haemoglobin performs simple translational and rotational rigid-body movements, the CO-bound form exhibits additional large-scale motions that result from the ligation state of the haem group.

Over the interparticle length scale, the mass transport of the proteins themselves is the most relevant process. The diffusion of proteins in solution is most commonly measured with dynamic light scattering59, but high-resolution neutron spectroscopy provides a measure60 of the short-time diffusion coefficient of the protein inside the first neighbour shell, even in systems of high concentrations, that is, ‘crowded’ systems61. This measurement enables a description of protein diffusion based on theories and methods borrowed from the colloid community, which have led to the discovery of dynamic clusters of proteins60,62,63. However, such structuring of a protein is not long lived, as small fluctuations can break this organization. Besides the fundamental value of the research, this line of work has important implications for the pharmaceutical industry because, although short lived, the clusters affect the solution viscosity64, which is an important parameter for the administration of therapeutic proteins, such as monoclonal antibodies65.

### Lipids

Another area of relevance for NSE and backscattering techniques is the investigation of the dynamics of lipid systems. Although much of the neutron scattering work on these samples is structural, high-energy-resolution neutron spectroscopy has played an important part in characterizing both simple model membranes and ones with biologically relevant entities. Understanding the microscopic dynamics in the complex mixture of lipids that constitute biological membranes could be the key to understanding many of the cellular processes that are regulated by lipid–lipid, lipid–protein or protein–protein interactions. At an individual level, the localized motions of lipid chains and self-diffusion of lipid molecules throughout the hydrated environment are accessible with backscattering spectroscopy, enabling a detailed atomic-level description of different lipids and revealing fundamental effects that contribute to the phase behaviour. In a simple case, a comparison of saturated and unsaturated lipids shows that double bonds restrict the mobility of the chains and may explain the lower bending and compressibility moduli in unsaturated membranes66. Increasing in complexity, lipid membranes can undergo various, and sometimes sizeable and physiologically important, changes when small molecules or peptides are present. Examples include antimicrobial peptides67, neurotoxic amyloid-β,36 melittin68, the anaesthetic ethanol69 or even cholesterol under pressure70. Furthermore, new insight into the role of rafts was obtained through NSE measurements of the lateral diffusion coefficients in a phase-separated lipid mixture. In agreement with structural and dynamical data from molecular dynamics simulations, the measurements indicate that rafts function as buffers of membrane properties and that a change in the complex lipid compositions in response to a variation in external parameters (such as temperature) may provide an advantageous stabilizing effect71.

At the level of the collective dynamics of lipid membranes, which determine the elastic properties and have a fundamental role in processes such as endocytosis and exocytosis as well as cell fission and fusion, measurements on large unilamellar vesicles of phospholipids represent a simple but valuable model. Modelling of the elastic properties is often based on the Helfrich expression of the bending Hamiltonian72, which has been used to derive an expression of the intermediate scattering function that can well describe the NSE experimental data73. Recent advances in the theoretical understanding of how the bending energy is dispersed through the intermonolayer friction74 has led to a better interpretation of the results from NSE studies on the bending modulus of a phospholipid bilayer, resolving previous disagreements with the results from other techniques. Moreover, it was also shown that the thickness fluctuations (peristaltic motion) of a membrane can be directly accessed by NSE75. This motion can be related to the elastic and viscous properties of the membranes76. Thus, these theoretical and experimental efforts have led to a unifying picture that enables the extraction of the main elastic and viscous parameters of the membrane through NSE measurements77. This type of neutron study has been used to investigate the effect of different components in the membrane, such as various lipids78,79,80,81, cholesterol82, drugs83, proteins84,85,86 and nanoparticles87,88.

### Complex biological systems

More realistic biological systems, which are far from the simple dilute solution limit, are often multicomponent and require the collection of multiple datasets from various techniques, including QENS. Diffusion processes in real cellular environments, which are characteristically crowded, can reveal how nature has optimized the protein content for optimum oxygen transport in red blood cells89 or simply provide information about the multiscale nature of the diffusion processes occurring in such concentrated systems90,91. The concept of ‘in-cell neutron scattering’ (that is, neutron scattering on living cells) was showcased and followed by work on living planarian flatworms92, whose metabolism relies on molecular diffusion through the body. Measurable microscopic diffusivities were surprisingly well defined in such a complex system. Of more importance to human health, backscattering data have revealed intracellular water as a secondary target in the interaction of the drug cisplatin with human nucleated cells of invasive breast carcinoma; this finding has provided a new and innovative way to optimize the pharmacodynamics of anticancer agents for breast and bone cancers49,93. Backscattering spectroscopy has also proved to be a complementary tool to overcome the experimental limit of diffusion magnetic resonance imaging, which is used to diagnose different brain diseases, by extracting information about the heterogeneity and asymmetries in brain tissue from water diffusion dynamics50. Finally, similar methods to those mentioned above for NSE on large unilamellar vesicles have been used to investigate the elastic properties in living cell membranes94.

It is important to remark here that given the complexity (multicomponent and hierarchical nature) of many biological and soft matter systems, valuable insight requires the combined use of different scattering techniques and spectrometers, often including neutrons, X-rays and light, covering different areas of the space–time map (Fig. 1) as well as other measurements and the use of computer simulations.

## Dynamics in disordered magnets

As a common magnet is cooled, one expects to observe long-range, static magnetic order (often ferromagnetic or antiferromagnetic) below a critical temperature. The Curie–Weiss temperature is a classical mean-field parameter that is indicative of the magnetic coupling within a material and provides an estimate of the freezing temperature. In these textbook magnets, investigations with traditional inelastic neutron scattering, such as three-axis spectrometers, working in energy space are making notable contributions to understanding the microscopic dynamics. Backscattering spectrometers are routinely used in the study of these magnets when low-energy excitations or small gaps in the excitation spectrum are expected, taking advantage of the very high resolution of the spectrometer95. However, for studies of more exotic magnets, such as those with quantum fluctuations or low dimensionality, high-resolution neutron spectroscopy is needed to simultaneously access the spatial and temporal correlations in the low μeV to meV range. Spin glasses, frustrated magnets and spin liquids all exhibit slow spin-relaxation phenomena and require a detailed measure of the spectral weight below 1 meV (see the example in Fig. 3a). We focus here on the impact of backscattering and NSE studies on geometrically frustrated magnets96, but it should be noted that studies of magnetism have benefited from measurements using these instruments in many other areas of research, including spin glasses97 and unconventional superconductors98.

In a geometrically frustrated magnet, the spatial connections between magnetic ions are such that not all near-neighbour exchange bonds can be satisfied simultaneously99,100,101,102,103,104,105. An intuitive example is that of an Ising antiferromagnet on a triangular or Kagome lattice (Fig. 3c). A near 2D spin arrangement like this is realized in several materials and structure types, such as delafossite102 or herbertsmithite103. Three-dimensional examples of lattices that invite geometrical frustration are the pyrochlore104 (Fig. 3b) and garnet lattices105.

One of the consequences of geometrical frustration is that these materials often have a high density of low-energy states, making them susceptible to low-energy (slow) fluctuations. In extreme cases, this macroscopic degeneracy of the ground state suppresses the expected ordering down to the lowest temperatures. Experimentally, it is observed that the material may still enter a long-range ordered state albeit at a much lower temperature106. Alternatively, a material may freeze into a spin-glass state, as observed for Y2Mo2O7 (ref.107), or the spin system may stay dynamic down to the lowest temperature that is experimentally accessible and form a ‘spin liquid’108,109. In the 2D, antiferromagnetically coupled triangular sublattice of NiGa2S4 (ref.101), more than 12 decades of relaxation time were probed (Fig. 3d) from room temperature down to 100 mK. A distinctive critical slowing down of the spin fluctuation was observed at ~8.5 K, but conventional long-range order was not found.

To date, it has been difficult to predict, from first principles alone, what happens to the spin ensemble of a frustrated magnet upon cooling. Fine details of the chemistry of the material may become important, such as site disorder or off-stoichiometry composition. This can occur even with ≤1% disorder or off-stoichiometry, which is hard to quantify by measurement. A case in point is Yb2Ti2O7, a pyrochlore magnet very close to a ferromagnetic–antiferromagnetic phase boundary. Initial results, including high-resolution neutron scattering measurements, suggested the system was dynamic to 50 mK (refs110,111,112), but it was later shown using neutron diffraction that it can also freeze, depending on minute chemical and crystallographic differences113,114.

Spin ices115 and related ferromagnetically frustrated magnets have illustrated the value of QENS for uncovering the underlying physics. The dynamic processes in these materials cover a wide dynamic range and become very slow at low temperature116,117,118,119, which makes it nearly impossible to cool a crystal of spin ice below ~0.3 K because the relaxation times diverge and heat flow becomes increasingly slow120.

The low-temperature neutron scattering signal of a classical spin ice features ‘pinch points’ in reciprocal space, for example, at the (002) and (111) positions121. These points result directly from long-range dipolar interactions and are now interpreted in terms of a ‘Coulomb phase’. The dynamic dimension of this scattering was revealed in an NSE experiment116 in which the relaxational spin dynamics above 5 K were observed to have four defining characteristics: first, the dynamics are unusually slow; second, only one timescale exists in the entire spin ensemble; third, the dynamics are independent of Q; and, last, they are thermally activated with a characteristic relaxation time of τ = τ0exp(TK/T), with τ0 ≈ 4.5 × 10–12 s and, for Ho2Ti2O7, an activation temperature (TK) of 293 K. These NSE results have been reproduced using a backscattering spectrometer118, although the analysis was more difficult than that of the NSE data owing to the signal being close to the resolution of the spectrometer. This picture survives at temperatures well above room temperature, and more than two orders of magnitude above that of the spin ice or Coulomb phase, showing that spin ice is not a typical paramagnet even at room temperature. Upon cooling below 5 K, backscattering results and measurement of the dynamic susceptibility reveal a crossover in the relaxation rates with a decrease in the activation temperature to ~10 K. This crossover was first observed in 2004 and described as a mystery117. More recently, this low-temperature region has been modelled121 as the relaxation phenomena associated with monopole creation, annihilation and propagation, which cost a few kelvin of energy.

In a related material, Nd2Zr2O7 (ref.122), high-resolution inelastic scattering was needed to identify the various components of the scattering signal, and a ‘moment fragmentation’ model was introduced123 to describe the coexistence of magnetic Bragg scattering and diffuse scattering.

Two local probes that are uniquely sensitive to slow magnetic fluctuations are muon spectroscopy and Mössbauer spectroscopy, and data from these probes are often presented alongside data from high-energy resolution neutron spectroscopy. Local probes do not offer a spatial dimension to the relaxational dynamics, but these techniques have unique merits; for example, muon spectroscopy is sensitive to very slow low-amplitude fluctuations, whereas Mössbauer spectroscopy is element specific. Reports on the application of Mössbauer spectroscopy to geometrically frustrated magnets are sparse in the literature; however, unusually slow spin dynamics were observed in some pyrochlores111,124. By contrast, a substantial body of work has been carried out on spin ices using muon spectroscopy124,125,126,127, but the interpretation of the data has often contradicted those from other techniques or has been controversial within the muon community.

Jamming and glassiness is observed in a wide variety of systems, including granular media, colloidal suspensions, molecular systems and magnets. These systems exhibit non-equilibrium transitions from a fluid-like to a solid-like state. Jamming of particles or spins, in local mimima of an exotic energy landscape, precludes further exploration of the phase space128,129, but the characteristic sudden arrest of their dynamics are readily probed using NSE and backscattering techniques130,131. Glass transition, gelation and aggregation are all closely related and only the wide coverage of Q and energy space provided by these neutron techniques can help to distinguish such dynamics132. Other probes, such as dynamic light scattering and photon correlation spectroscopy, offer complementary information to the results from neutron spectroscopy in these areas.

Finally, other studies of magnetism that have benefited from quasielastic scattering include those on vortex dynamics in type-II superconductors, dynamics of large topological structures such as skyrmions and unconventional intermetallic materials98,133,134,135. Extraordinary studies like these have been performed on magnetic systems, but at present, the experiments are not routine and models to interpret the data are limited.

## Conclusion and outlook

The backscattering and NSE techniques uniquely provide spatial and temporal information on molecular and spin dynamics occurring in a space and time range from ~1 Å up to hundreds of nanometres and from tens of picoseconds to hundreds of nanoseconds, respectively. Backscattering and NSE spectrometers have unique strengths, such as the simultaneous access to the time and space domains, the sensitivity to isotopes (in particular, the contrast between hydrogen and deuterium) and the ability to study a material at extreme conditions of temperature. We have highlighted here research in polymers, biomolecules and disordered magnets; however, the study of the dynamics within liquids, molecular motions under nanoscopic confinement or on surfaces and diffusion of hydrogen in solid-state materials are also fields in which high-energy-resolution neutron spectroscopy techniques can provide relevant information136,137. Moreover, QENS can give quantitative insight into atomic-scale proton138 and ionic conduction mechanisms in ionic liquids or solid electrolytes139,140,141. These instruments also have a role in the optimization of the thermodynamic stability, kinetics and reversibility of hydrogen storage in solid-state materials142 and the dynamics of ions and molecules in green energy materials such as polymer-based batteries30.

Although this Technical Review focuses on relaxational dynamics, high-resolution spectrometers can also probe truly inelastic features within the dynamic range of the spectrometer. These features include the tunnelling of methyl groups in soft matter143, quantum tunnelling in molecules144 and low-lying electronic spin excitations145. However, it would be remiss not to mention studies of the nuclear spin146,147,148. With energy transfers of <100 μeV and very weak signals, these studies are still time consuming and are only possible on the best backscattering spectrometers. However, future spectrometers will reduce the time required and/or an increase in the number of instruments that can probe such excitations will open up the possibility of studying these quantum effects with applications in, for example, quantum computing.

The intensity of the beams of cold neutrons (<5 meV) required for backscattering and NSE studies is increasing with the development of brighter sources and neutron transport systems. Modern instrumentation is continuously expanding the region of the space–time map covered by these spectrometers as well as increasing the data acquisition rate. These developments, together with those of the ancillary sample environment, are enabling these techniques to be used by an ever-broader user community in diverse research fields and providing complementary information on the relaxation processes (motions) to that available through other techniques, such as dielectric spectroscopy, NMR, muon spin relaxation, photon correlation spectroscopy and tracer diffusion measurements.

A notable breakthrough in backscattering instrumentation was the development of GaAs(200) analyser crystals149. At present, these crystals provide an energy resolution of 77 neV, an improvement of a factor of 10 over traditional Si(111) analyser crystals. However, the resolution is still far from the theoretical limit of 13.2 neV. Current developments aim to improve the Q resolution of backscattering spectrometers, widen the accessible energy transfer range (to enable dynamics to be probed over a wider range of timescales) and to deploy modern beam polarization techniques at these instruments. The latter will extend the capabilities of quasi-elastic neutron spectroscopy by providing a clean separation of coherent and incoherent scattering, enabling the isolation of weak magnetic signals or providing access to the directional components of the magnetization of a sample. The newest backscattering spectrometer, MIRACLES, will be built at the European Spallation Source (Sweden) and will give researchers greater flexibility in choosing between flux and resolution while also extending the dynamic range of backscattering spectrometers150.

Recent developments at IN15 and J-NSE-Phoenix have increased the uniformity and magnitude of the precession fields, improving the energy resolution to beyond 1 neV. In parallel, the development of WASP aims to mitigate in some cases the severe limitation in the data acquisition rate of the technique by covering more of the Q space simultaneously. Furthermore, the development and implementation of resonant spin-echo methods will enable application to systems currently out of reach, namely samples that depolarize the neutron beam.

High-energy-resolution neutron spectroscopy studies have a unique role in understanding the dynamics of proteins and the water of hydration, lipid membranes and self-assembled amphiphilic structures. There is considerable potential in probing both the internal dynamics of lipids and the diffusion rates, developing on recent NSE work on the elastic and viscous properties of model membranes. QENS studies of hard matter would benefit tremendously from tools to handle single-crystal samples. Progress in this area will enable studies of anisotropic interactions, exotic magnetic phases, such as skyrmions and quantum spin liquids, and low-dimensional features such as stripe phases.

This Technical Review provides a glimpse of the future potential of NSE and backscattering spectrometers. In parallel with other methods, such as X-ray photon correlation spectroscopy and molecular dynamics simulations, these high-resolution neutron spectroscopic techniques will reveal new scientific phenomena. We hope this Technical Review provides readers with understanding of the potential of quasi-elastic neutron studies in their research.

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## Acknowledgements

The authors thank E. Senses and M. Nagao for critical reading of the manuscript, QENS instrument scientists worldwide and the research community, who keep bringing new research and challenges to the facilities. A.F. acknowledges support from the Center for High Resolution Neutron Scattering, a partnership between the National Institute of Standards and Technology and the US National Science Foundation under agreement no. DMR-1508249. This research used resources at the Spallation Neutron Source, a US Department of Energy Office of Science User Facility operated by the Oak Ridge National Laboratory. The authors thank J. Hemman, Oak Ridge National Laboratory Graphics Design Group, for support with some of the figures.

## Author information

The authors contributed equally to all aspects of the article.

Correspondence to Jason S. Gardner.

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### Competing interests

The authors declare no competing interests.

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Nature Reviews Physics thanks A. Jackson and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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## Glossary

Coherent scattering

Wavevector transfer (Q)-dependent scattering that contains information about scattering structures.

Incoherent scattering

Relatively weakly wavevector transfer (Q)-dependent scattering that contains information about the movement of an individual atom in time.

Fission

A nuclear reaction in which a massive nucleus splits into smaller nuclei with the simultaneous release of energy, neutrons and other products.

Spallation

A process by which a heavy nucleus ejects smaller particles, including neutrons, after, for example, being hit with high-energy particles.

Moderator

A material used to reduce the energy of free neutrons by a large number of inelastic collisions.

Thermal neutrons

Unbound or free neutrons with an average energy of ~25 meV at room temperature.

Cold neutrons

Unbound or free neutrons with an average energy lower than ~5 meV.

Mezei NSE

A type of neutron spin-echo (NSE) spectrometer in which the velocity of each neutron is encoded in its Larmor precession in a magnetic field.

Time of flight

A technique by which the time taken for a neutron to travel a known distance is used to determine its velocity (and thus energy).

Modulation of intensity by zero effort

(MIEZE). A modified resonance neutron spin-echo technique that enables measurements to be taken under conditions that depolarize the neutron beam.

Anti-Helmholtz coils

A pair of coils in which the electrical current flows in opposite directions in the two coils, producing a high-magnetic-field gradient from the centre out.

Flipper

A device used to manipulate the neutron spin direction non-adiabatically.

Resonance NSE spectrometers

A type of neutron spin-echo (NSE) spectrometer that employs radio-frequency spin flippers to manipulate the spin of each neutron and encode its velocity.

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Gardner, J.S., Ehlers, G., Faraone, A. et al. High-resolution neutron spectroscopy using backscattering and neutron spin-echo spectrometers in soft and hard condensed matter. Nat Rev Phys 2, 103–116 (2020). https://doi.org/10.1038/s42254-019-0128-1

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