1. Introduction

Inelastic neutron scattering is a technique commonly used in condensed matter research to study atomic and molecular motion, magnetic and crystal field excitations. In this type of scattering, the exchange of energy and momentum between the incident neutron and the sample causes both the direction and the magnitude of the neutron’s wave vector to change.^[1]Neutrons interact via short-range interactions. They have zero charge and nonzero spin, hence they can interact with materials via nuclear and magnetic interactions. Since probabilities of both interactions (i.e. cross sections) are small, neutrons can penetrate well into the bulk of the sample. Neutrons primarily interact with the nucleus itself and potentially with magnetic fields from unpaired electrons. When neutron is scattered by a nucleus with nonzero spin, the interaction strength is determined by the relative orientations of neutron and nuclear spins. For condensed matter studies, nuclear interaction and magnetic-dipole interaction are the most important.

Inelastic neutron scattering is now used by researchers in virtually any field that involves the condensed state. The applications are widely common in materials science, engineering, physics and particularly in catalysis, polymers, magnetism, hydrogen-in-metals, hydrogen bonding, and quantum fluids. It is a common technique for example for investigating vibrational spectra of glasses or for validating the results of ab initio calculations. It is also a valuable tool for the investigation of atomic motions in solids and liquids or for the determination of phonon dispersion curves of metals, which is critical in understanding their thermodynamics properties.^[1]

2. Theory

2.1.  Inelastic and elastic scattering

When neutron collides with a nucleus, it can be either scattered, transmitted or absorbed through a nuclear reaction. Each neutron can be described in terms of its wave vector k, which points along the neutrons trajectory and has a magnitude related to the wavelength of the neutron as,^[2]

\( k = \frac{2\pi}{\lambda}. \)

Neutron energy (E_n), wavelength and the wave vector are related by

\( E_n = k_BT = \frac{m_nv_n^2}{2} = \frac{\hbar^2k_n^2}{2m_n}, \)

where k_B is the Boltzmann constant, m_n and v_n are the mass and velocity of the neutron, respectively.

Neutron scattering is characterized in terms of momentum and energy conservation between the incident and scattered neutron and the scattering object. Momentum transfer (Q) can be expressed in terms of the wave vectors of the initial and scattered neutrons^[3]as

\( Q = k_i - k_f, \)

where indices i and f refer to the initial and final states of the scatterer, respectively. Relation between Q and the wave vectors is illustrated in Figure 1. In elastic scattering, magnitude of the incident and final wave vectors are equal (k_i = k_f), and the momentum transfer is given by^[3]

\( Q = \frac{4\pi\sin(\theta )}{\lambda}, \)

where θ is the scattering angle, as presented in figure 1. Energy of the incident neutron is not changed during the scattering process. However, in inelastic scattering, the wave vectors are not equal (k_i ≠ k_f) and the neutrons either gain or lose energy as illustrated in figure 1.

Figure 1. Scattering triangles for elastic neutron scattering and inelastic neutron scattering showing the relationship between momentum transfer (Q) and initial/final wave vectors. Figure by Pinja Kangas. License: CC BY-SA 4.0.

During an inelastic collision, neutron is absorbed, activated to an excited short lived energy state and then re-emitted back down to the ground state. Momentum is conserved, but kinetic energy of the system is changed as energy is transferred between the sample and the neutron. The energy transfer (E_n) can be obtained as the difference between incident and final neutron energies as^[4]

\( E_n = \hbar\omega = \frac{\hbar^2k_i}{2m_n} - \frac{\hbar^2k_f}{2m_n} = E_i - E_f \)

2.2.  Cross sections

The probability of interaction between an incident neutron and target nucleus, i.e. scattering probability is described in terms of cross section (σ), which ultimately gives information about the material’s atomic structure and dynamics. Scattering is also depends on the type of target nucleus and the neutron speed. Scattering cross sections are defined by the neutrons scattered to all directions as a ratio of the total scattered neutron count rate to the incident flux,^[3]

\( \sigma_s = \frac{(\text{total scattered neutron count rate})}{\Phi} \)

Where Φ is the incident flux. In an idealized case, for a single fixed nucleus, the cross-section can be obtained as

\( \sigma = 4\pi b^2 \)

where b is the scattering length of the nucleus, which essentially measures the strength of the interaction between the neutron and the nucleus. Inelastic scattering cross section is measured in barns (1barn = 10^-28 m²) and is relatively small for light nuclei.^[2]There are no clear trends for the scattering lengths within the periodic table, only a mild increase in scattering length as one moves across the periodic table.^[3]

3. Techniques and facilities

3.1.  Triple axis spectrometer

Triple axis spectrometer (TAS) allows for the determination of the energy transfer by analysis of the wavelength as it utilizes Bragg diffraction in selecting the neutron wavelength.^[3] A typical TAS measurement geometry is schematically presented in Figure 2, where monochromatization, sample interaction and analyzing make up the three axes.

Figure 2. Schematic illustration of a typical TAS-spectrometer geometry. (Figure: Pinja Kangas)

During a TAS experiment, a monochromatic crystal (first axis) is used to set the incident neutron wave vector and the monochromatic beam is then scattered from the crystalline sample (second axis). The intensity of the scattered beam is reflected by the analyzer crystal (third axis).^[4] Neutrons are characterized before and after hitting the sample, as the first single crystal monochromator with plane spacing d_c1 will refract only neutrons with a particular wavelength to the sample. By changing the incident angle of the first monochromator θ_B1, the selected wavelength may be varied. Neutrons are scattered from the sample by an angle 2θ, and they may lose or gain energy in respect to the incident energy. Measurement results can be expressed as a spectra in terms of a scattering function, which is related to the momentum transfer and the energy transfer between neutrons and the sample. The experimental spectra yields information about the dynamics of the sample, such as lattice vibrations.

3.2.  Time-of-flight spectrometer

Time-of-flight (TOF) spectrometer utilizes pulsed neutron sources. A typical arrangement of a TOF spectrometer with a linear geometry is shown in Figure 3.

Figure 3. Schematic illustration of a linear TOF spectrometer geometry. (Figure: Pinja Kangas)

The source emits a pulse of neutrons at t₁, while simultaneously the chopper is open and the neutrons can propagate to the sample. The neutrons will reach the detector at t₂. The initial position and velocity of a neutron pulse is fixed, and the time that it takes for the neutrons to be detected (t₂) after the pulse is measured.^[3] Once the signal for neutrons hitting the sample is defined, the energy exchange to sample can be analyzed by recording the arrival of the neutrons at the detector from a given distance from the sample. In contrast to three-axis spectroscopy, the energy can then be determined simultaneously for all scattering angles.^[5]

3.3.  Backscattering spectrometer

Backscattering spectrometer includes a monochromator and an analyzer, which are both operated at backscattering conditions, which enables a high resolution within µeV range.^[6]Due to the high resolution, backscattering spectrometers can be used to slow observe nuclear motion, which is relatively slower than for example lattice vibrations typically seen with TAS. However, in back-scattering spectroscopy, it is not possible to vary the diffraction angles of the monochromator or analyzer in order to select wavelengths as it would impair the resolution. Backscattering experiments can be generally used to study atomic or molecular motion in a nanosecond timescale. The term backscattering refers to the scattering from the monochromator and analyzer crystals and not from the sample.^[5]

3.4. Spin-echo spectrometer

Neutron spin echo is a type of time-of-flight technique. It uses the Larmor precession (the precession of the magnetic moment of an object about an external magnetic field) of the neutron spin to measure the change in the energy of the neutron upon scattering from some dynamical process in condensed matter. The method makes polarized neutrons precess in uniform opposite magnetic fields before and after the sample so that those having different wavelengths end up with the same spin orientation at the analyzer position.^[5]

4.  References

1.	1 2 Steward F. Parker (1999), Inelastic neutron scattering, applications, Encyclopedia of Spectroscopy and Spectrometry. https://doi.org/10.1006/rwsp.2000.0381
2.	1 2 Pynn R. (2009) Neutron Scattering—A Non-destructive Microscope for Seeing Inside Matter. In: Liang L., Rinaldi R., Schober H. (eds) Neutron Applications in Earth, Energy and Environmental Sciences. Neutron Scattering Applications and Techniques. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-09416-8_2
3.	1 2 3 4 5 6 Fernandez-Alonso, F., & Price, D. L. (2013) Neutron scattering – fundamentals San Diego, CA : Academic Press
4.	1 2 Voigt, J., (2012) Inelastic scattering: Lattice, magnetic and electronic excitations, Manuel Angst, Thomas Brückel, Dieter Richter, Reiner Zorn (Eds.): Scattering Methods for Condensed Matter Research: Towards Novel Applications at Future Sources. ISBN: 978-3-89336-759-7
5.	1 2 3 Neumann, D. A., & Hammouda, B. (1993). Ultra-High Resolution Inelastic Neutron Scattering. Journal of research of the National Institute of Standards and Technology, 98(1), 89-108. https://doi.org/10.6028/jres.098.007
6.	1 Meyer, A., Dimeo, R. M., Gehring, P. M. & Neumann, D. A. (2003) The high-flux backscattering spectrometer at the NIST center for neutron research. Review of Scientific Instruments 74 2759-2777 https://doi.org/10.1063/1.1568557