Neutron backscattering

Neutron backscattering spectroscopy (NBS) is a neutron scattering technique used to study atomic and molecular motion. It is distinguished by its ultra-high energy resolution, typically in the micro-electronvolt (μeV) range, which allows for the observation of processes on a nanosecond timescale.^[1] The technique achieves this high resolution by using Bragg diffraction from crystals at a scattering angle very close to 180° (backscattering), a geometry where the wavelength spread of the diffracted neutrons is minimized.^[2]

Neutron backscattering spectroscopy is an inelastic neutron scattering technique that improves the energy resolution of neutron spectrometers by approximately two orders of magnitude compared to conventional methods.^[1] The method exploits the reflection of neutrons from a near-perfect crystal back toward their source direction to achieve very high energy resolution for studying slow dynamical processes in condensed matter.^[3] This technique is particularly valuable for investigating phenomena such as slow motions in complex liquids, jump diffusion, and quantum rotational tunneling.^[2]

The accessible energy transfer range typically extends from -15 to +15 μeV, which corresponds to timescales from approximately 0.1 to 10 nanoseconds.^[4] This makes backscattering spectroscopy uniquely suited for probing slow relaxation processes that are too fast for neutron spin echo but too slow for a conventional triple-axis spectrometer.^[5]

The development of neutron backscattering spectroscopy emerged from the recognition that operating spectrometers at extreme scattering angles could significantly improve energy resolution. The theoretical foundation was established by Heinz Maier-Leibnitz and his colleagues in the 1960s. They demonstrated that the wavelength spread (Δλ) of a Bragg-diffracted neutron beam decreases as the scattering angle (2θ) approaches 180°.^[6]

The first operational backscattering spectrometer was constructed at the Jülich Research Centre in 1966 by B. Alefeld and colleagues.^[7] This prototype instrument demonstrated the feasibility of achieving μeV energy resolution but suffered from low neutron flux. Subsequent developments at high-flux reactor facilities such as the Institut Laue-Langevin (ILL) led to the construction of IN10 in 1972 and IN13 in 1978, which established backscattering as a routine technique for the scientific community.^[8]

Neutrons used in backscattering spectroscopy typically have wavelengths of 6–7 Å, corresponding to energies of approximately 2 meV.^[2] The neutron's properties make it an ideal probe for this technique:

• Zero electric charge, allowing deep penetration into materials without interacting with electron shells.
• Magnetic moment, enabling studies of magnetic dynamics.
• Mass comparable to atomic nuclei, facilitating efficient energy transfer during scattering events.
• Wavelength comparable to interatomic distances, making it sensitive to atomic-scale structure and motion.^[9]

The de Broglie wavelength (λ) of a neutron is related to its energy (E) by the equation:

${\displaystyle \lambda ={\frac {h}{\sqrt {2m_{n}E}}}}$

where h is the Planck constant and m_n is the neutron mass.^[10]

The principle of backscattering relies on Bragg's law: λ = 2d sin θ, where d is the lattice spacing of the monochromator or analyzer crystal and θ is half the scattering angle.^[11]

For inelastic scattering, the energy transfer (ℏω) and momentum transfer (Q) are related to the incident (k_i) and final (k_f) wavevectors:

${\displaystyle \hbar \omega ={\frac {\hbar ^{2}}{2m_{n}}}(k_{i}^{2}-k_{f}^{2})}$

${\displaystyle {\vec {Q}}={\vec {k_{i}}}-{\vec {k_{f}}}}$

In a backscattering geometry, the momentum transfer is maximized for elastic scattering, where Q = 2k.^[10]

The dynamic structure factor S(Q,ω) measured in backscattering experiments typically contains three components:^[9]

• Elastic scattering: A delta function, δ(ω), arising from the static (time-averaged) structure of the sample.
• Quasielastic scattering: A broadened component centered at ω = 0 that results from slow, diffusive, or random motions.
• Inelastic scattering: Discrete peaks at finite energy transfer (ω ≠ 0) that correspond to vibrational excitations or de-excitations (phonons, magnons, etc.).

The quasielastic component often follows a Lorentzian profile:

${\displaystyle S_{QE}(Q,\omega )={\frac {1}{\pi }}{\frac {\Gamma (Q)}{\omega ^{2}+\Gamma (Q)^{2}}}}$

where Γ(Q) is the half-width at half-maximum (HWHM), which is related to the characteristic relaxation time (τ) of the motion by Γ = ℏ/τ.^[9]

The energy resolution in backscattering is determined by three main contributions:^[12]

1. **Primary extinction**: Intrinsic width of the Bragg reflection from perfect crystals, with δE/E ≈ 10^-6. 2. **Mosaic spread**: Spread in crystal plane orientations, where δE/E ≈ η·cot θ, with η being the mosaic width. 3. **Beam divergence**: Angular spread of the neutron beam, where δE/E ≈ α²/8 for a divergence angle α.

The total resolution is the quadratic sum of these contributions.^[12]

Continuous neutron beams from research reactors provide a stable flux for backscattering measurements. Notable reactor-based instruments include:^[13]

• IN16B at ILL (France) - flux: 1.3×10⁴ n/cm²/s, resolution: 0.75 μeV
• HFBS at NIST (USA) - flux: 1.0×10⁴ n/cm²/s, resolution: 0.8 μeV
• SPHERES at FRM II (Germany) - flux: 8×10³ n/cm²/s, resolution: 0.65 μeV

Pulsed neutron beams from spallation sources enable time-of-flight methods for energy discrimination. Key instruments include:^[4]

• BASIS at SNS (USA) - flux: 2×10⁴ n/cm²/s, resolution: 3.5 μeV
• IRIS at ISIS (UK) - flux: 5×10³ n/cm²/s, resolution: 1 μeV
• DNA at J-PARC (Japan) - flux: 1×10⁴ n/cm²/s, resolution: 1.8 μeV

Traditional backscattering spectrometers employ fixed monochromator and analyzer crystals arranged in an exact backscattering geometry.^[13] The IN16B spectrometer at ILL exemplifies this design with:

• Si(111) monochromator cooled to 200 K
• A large, spherically arranged Si(111) analyzer array covering 2 steradians
• A phase space transformation chopper for flux enhancement^[14]

Modern instruments at spallation sources combine backscattering analyzers with time-of-flight techniques.^[15] The BASIS spectrometer employs:

• 84 Si(111) analyzer crystals at a scattering angle of 2θ = 176° (near-backscattering)
• Bandwidth choppers for elastic line discrimination
• A dynamic range of ±100 μeV with 3.5 μeV resolution^[4]

Silicon crystals are predominantly used in backscattering applications due to their crystalline perfection and suitable lattice parameters.^[6] Common crystal reflections include:

• Si(111): d = 3.135 Å, λ = 6.27 Å, E = 2.08 meV
• Si(311): d = 1.637 Å, λ = 3.27 Å, E = 7.64 meV
• Ge(111): d = 3.266 Å, λ = 6.53 Å, E = 1.92 meV

The neutron energy can be varied by scanning the temperature of the monochromator, which changes the lattice spacing d through thermal expansion:^[13]

${\displaystyle d(T)=d_{0}[1+\alpha (T-T_{0})]}$

where α ≈ 2.6×10^-6 K^-1 for silicon.

Modern backscattering spectrometers can accommodate diverse sample environments to study materials under various conditions:^[16]

• Cryostats: Temperature range of 0.05–800 K
• Pressure cells: Up to 10 kbar for studies of liquids and soft matter
• Humidity chambers: 0–100% relative humidity control
• Magnetic fields: Up to 14 T for quantum phenomena
• Sample changers: Automated multi-position systems for high-throughput experiments

Backscattering experiments typically employ three measurement protocols:^[2]

1. **Fixed window scans**: Energy-integrated intensity is measured as a function of temperature. 2. **Elastic scans**: Elastic intensity is measured as a function of Q at a fixed temperature. 3. **Inelastic scans**: The full dynamic structure factor, S(Q,ω), is measured at selected temperatures.

Time-of-flight discrimination is used at spallation sources to separate elastically scattered neutrons from the background.^[4] The arrival time is:

${\displaystyle t_{elastic}={\frac {L_{1}+L_{2}}{v_{0}}}}$

where L₁ and L₂ are the primary and secondary flight paths, and v₀ is the incident neutron velocity.

Standard data reduction procedures include:^[17]

1. **Empty cell subtraction**: To remove scattering from the sample container. 2. **Vanadium normalization**: To correct for detector efficiency, using a sample of vanadium which scatters neutrons almost perfectly incoherently. 3. **Multiple scattering correction**: To account for neutrons that scatter more than once within the sample. 4. **Absorption correction**: To compensate for the attenuation of the neutron beam by the sample.

The corrected intensity is given by:

${\displaystyle I_{corr}={\frac {I_{sample}-I_{empty}}{I_{vanadium}}}\cdot A_{abs}\cdot M_{mult}}$

The instrumental resolution function, R(Q,ω), is typically measured using a vanadium standard, which scatters purely elastically at room temperature.^[17] The measured spectrum is a convolution of the true scattering function and the resolution function:

${\displaystyle I_{meas}(Q,\omega )=S(Q,\omega )\otimes R(Q,\omega )}$

Deconvolution methods include Fourier deconvolution, maximum entropy methods, and Bayesian inference.^[17]

The dynamic structure factor S(Q,ω) is analyzed by fitting it with model functions:^[9]

• **Lorentzian model** for simple diffusion:

${\displaystyle S(Q,\omega )=A_{0}(Q)\delta (\omega )+{\frac {1}{\pi }}A_{1}(Q){\frac {\Gamma (Q)}{\omega ^{2}+\Gamma (Q)^{2}}}}$

• **Stretched exponential** for complex relaxation, typically fitted in the time domain:

${\displaystyle I(Q,t)=A\exp[-(t/\tau )^{\beta }]}$

where β < 1 indicates a non-exponential relaxation process.^[18]

Common models used to analyze the Q-dependence of quasielastic scattering include:^[9]

• **Jump diffusion (Chudley-Elliott model)**:

${\displaystyle \Gamma (Q)={\frac {D_{jump}Q^{2}}{1+D_{jump}Q^{2}\tau _{0}}}}$

where D_jump is the jump diffusion coefficient and τ₀ is the residence time between jumps.

• **Continuous diffusion (Fick's law)**:

${\displaystyle \Gamma (Q)=DQ^{2}}$

where D is the translational diffusion coefficient.

• **Rotational diffusion**:

${\displaystyle S_{rot}(Q,\omega )=j_{0}^{2}(Qa)\delta (\omega )+\sum _{l=1}^{\infty }(2l+1)j_{l}^{2}(Qa)L(\omega ,l(l+1)D_{rot})}$

where j_l are spherical Bessel functions and D_rot is the rotational diffusion coefficient.^[9]

Standard analysis packages for backscattering data include:^[17]

• DAVE (NIST): A comprehensive suite for data reduction and analysis.
• LAMP (ILL): Large Array Manipulation Program.
• Mantid (ISIS/SNS): A framework for neutron and muon data analysis.
• QENS (JCNS): A specialized library for quasielastic neutron scattering analysis.

Molecular dynamics simulations provide complementary insight by calculating intermediate scattering functions that can be directly compared with experimental data:^[19]

${\displaystyle I(Q,t)=\langle \sum _{i,j}\exp[i{\vec {Q}}\cdot ({\vec {r}}_{i}(0)-{\vec {r}}_{j}(t))]\rangle }$

Backscattering spectroscopy has revealed quantum tunneling in numerous systems:^[20]

• Methyl group rotation in molecular crystals (tunnel splitting: 0.1–10 μeV)
• NH₄⁺ rotation in ammonium salts (splitting: 1–100 μeV)
• Hydrogen tunneling in metals (activation energy: 10–100 meV)

Studies of spin dynamics in frustrated magnets and quantum spin systems benefit from the μeV resolution:^[21]

• Spin glasses (relaxation times: 10^-9–10^-7 s)
• Low-dimensional magnets (energy gaps: 1–100 μeV)
• Molecular magnets (quantum tunneling of magnetization)

Backscattering reveals protein motions that are essential for biological function:^[22]

• The protein dynamical transition at ~200–230 K
• Methyl group rotations (τ ~ 10^-10 s)
• Side chain fluctuations (τ ~ 10^-9 s)
• Domain motions (τ ~ 10^-8 s)

The mean square displacement ⟨u²⟩, extracted from elastic scans, provides insight into protein flexibility:^[23]

${\displaystyle \langle u^{2}\rangle \approx -{\frac {3}{Q^{2}}}\ln[I_{el}(Q)/I_{el}(0)]}$

Backscattering probes polymer chain dynamics across multiple length and time scales:^[24]

• Segmental relaxation (τ ~ 10^-10–10^-8 s)
• Side group rotation (τ ~ 10^-11–10^-9 s)
• Methyl group dynamics (τ ~ 10^-12–10^-10 s)

Studies of lipid bilayers reveal:^[25]

• Lateral diffusion of lipids (D ~ 10^-7 cm²/s)
• Conformational dynamics of lipid tails (τ ~ 10^-10 s)
• Collective membrane undulations

Backscattering is uniquely suited to probe hydrogen dynamics in metal hydrides:^[26]

• Jump diffusion between interstitial sites
• Quantum tunneling at low temperatures
• Trapping at defects and interfaces

Studies of battery materials and solid electrolytes reveal:^[27]

• Li⁺ ion hopping mechanisms (τ ~ 10^-10–10^-8 s)
• Correlation effects in ion transport
• Coupling of framework dynamics to ion motion

The dynamics of molecules in nanoporous materials can be investigated, including:^[28]

• Diffusion of guest molecules in zeolites
• Dynamics of water in clays
• Hydrogen storage in metal-organic frameworks

• Ultra-high energy resolution: A full width at half maximum (FWHM) of 0.1–1 μeV enables observation of slow dynamics on nanosecond timescales.^[1]
• Bulk sensitivity: Neutrons penetrate deeply into materials, probing the entire sample volume.^[29]
• Isotope contrast: Substituting deuterium (D) for hydrogen (H) allows for selective labeling and contrast variation to highlight specific components.^[9]
• Simultaneous Q-range: Modern instruments with large detector arrays cover a wide range of momentum transfer (e.g., 0.2–2.0 Å⁻¹) simultaneously.^[4]
• Non-destructive: Low-energy neutrons typically do not cause radiation damage to the sample.^[29]
• Magnetic sensitivity: The neutron's magnetic moment directly probes spin dynamics.^[10]

• Limited neutron flux: The high resolution is achieved at the cost of flux, leading to low count rates and long acquisition times (hours to days).^[6]
• Sample size: Relatively large sample quantities (0.1–10 g) are often required, depending on the scattering cross-section.^[4]
• Fixed geometry: The backscattering condition limits flexibility in accessing the full Q-ω space compared to other techniques.^[1]
• Limited energy range: The dynamic range is typically limited to ±15–100 μeV, which restricts studies of faster dynamics.^[29]
• Background signals: Weak signals can be dominated by instrumental background, requiring careful measurement and subtraction.^[2]
• Data analysis complexity: The need to deconvolve the measured spectrum from the instrumental resolution function complicates interpretation.^[17]

Neutron spin echo (NSE) achieves even higher energy resolution (neV) but differs fundamentally:^[29]

• NSE directly measures the intermediate scattering function I(Q,t), whereas backscattering measures the dynamic structure factor S(Q,ω).
• NSE is primarily sensitive to coherent scattering, while backscattering can measure both coherent and incoherent scattering.
• NSE typically covers a lower Q-range (0.02–2.0 Å⁻¹) compared to backscattering (0.2–2.0 Å⁻¹).

Triple-axis spectroscopy (TAS) offers greater flexibility but has lower resolution:^[30]

• Energy resolution is typically 10–100 μeV for TAS versus 0.1–1 μeV for backscattering.
• TAS allows for flexible scans along any trajectory in Q-ω space, unlike the fixed geometry of backscattering.
• TAS is better suited for studying dispersive excitations like phonons and magnons, while backscattering excels at quasielastic scattering.

Inelastic X-ray scattering (IXS) provides complementary information:^[31]

• IXS probes dynamics linked to electron density, whereas neutrons probe nuclear positions.
• IXS is limited to coherent scattering, while neutron scattering can be dominated by incoherent scattering (especially from hydrogen).
• IXS requires much smaller samples (μg) compared to neutrons (grams).

Optical spectroscopies have much higher flux but different selection rules and sensitivities:^[32]

• They are limited to optically active modes, whereas neutrons are sensitive to all modes of motion.
• Optical techniques are often more surface-sensitive, while neutrons provide bulk information.
• The Q-range accessible with light scattering is very small (near the Brillouin zone center), whereas neutrons can cover the entire zone.

Neutron backscattering experiments are conducted at large-scale facilities and require adherence to strict radiation safety protocols:^[33]

• **Biological shielding**: Instruments are enclosed in concrete and polyethylene walls to minimize radiation dose rates in experimental halls.
• **Access control**: Interlocked enclosures (hutches) prevent personnel from being exposed to the direct beam during operation.
• **Sample activation**: Samples can become radioactive after exposure to the beam. This activation is monitored, and cooling periods may be required for materials containing certain elements.
• **Dosimetry**: All personnel wear dosimeters to track their accumulated radiation dose, which is kept below regulatory limits (e.g., 20 mSv/year for occupational exposure).
• **Training**: Users must complete radiation safety training and certification before being allowed to operate the instrument.

The European Spallation Source (ESS) is expected to host advanced backscattering instruments with significantly higher performance:^[34]

• MIRACLES at ESS (under construction): This instrument aims for an energy resolution of 0.3 μeV, a flux gain of up to 30 times over existing instruments, a large dynamic range (±600 μeV), and a new analyzer geometry with 360° coverage.

Advanced concepts currently under development include polarized neutron backscattering for magnetic studies, multi-energy analyzers for simultaneous resolution options, and the use of machine learning for real-time data analysis.^[35]

A growing trend is the simultaneous or in situ measurement combining backscattering with other techniques:^[29]

• **Diffraction**: To establish direct structure-dynamics correlations.
• **Small-angle neutron scattering (SANS)**: To probe multi-scale dynamics from nanometers to micrometers.
• **Optical spectroscopy**: Combining neutron data with IR/Raman spectroscopy for a complete dynamical picture.
• **Computation**: Validating and refining molecular dynamics simulations against high-resolution neutron data.

New frontiers for backscattering applications include:^[36]

• **Quantum materials**: Probing subtle excitations in topological insulators and unconventional superconductors.
• **Energy materials**: Investigating ion transport in solid-state batteries and hydrogen dynamics in storage materials.
• **Biological systems**: Studying dynamics in live cells, drug delivery mechanisms, and intrinsically disordered proteins.
• **Extreme conditions**: Exploring dynamics at high pressures (>100 GPa) and high magnetic fields (>20 T).

• Bragg diffraction
• Dynamic light scattering
• Inelastic neutron scattering
• Mössbauer spectroscopy
• Neutron scattering
• Neutron spin echo
• Nuclear magnetic resonance
• Quasielastic neutron scattering
• Time-of-flight neutron scattering
• Triple-axis spectrometer

• NIST High Flux Backscattering Spectrometer (HFBS)
• ILL IN16B Backscattering Spectrometer
• BASIS at Oak Ridge National Laboratory
• IRIS at ISIS Neutron and Muon Source
• MIRACLES at the European Spallation Source
• DNA at J-PARC