# Small Angle Scattering in Neutron Imaging—A Review

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## Abstract

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## 1. Introduction

#### 1.1. Basic Concept of Imaging

_{0}, which is described by the Beer–Lambert law

_{t}= N(σ

_{a}+ σ

_{s}) in Equation (1) is the total macroscopic cross section, referred to as the linear attenuation coefficient, where N is the number density of atoms and σ

_{t,}σ

_{a}and σ

_{s}are the microscopic total, absorption and scattering cross sections respectively. Correspondingly this description assumes that scattered intensities are scattered to large angles and hence do not reach the transmission detector. This can lead to significant deviations of this simplified description from transmission imaging results, where a relatively large detector is placed close to the sample. While small angle scattering has hardly any impact on the transmission image due to involved angles and investigated length scales, incoherent scattering contributions and diffraction from crystalline structures close to the forward direction can be significant [9].

#### 1.2. Basic Concept of SANS

_{2}, comparable to the collimation distance l

_{1}, downstream of the sample. This way only intensities scattered at small angles θ (app. 0.3 to 5 deg) out of the direct beam are detected. The modulus of the scattering vector

**q**, which is the difference of the incoming wavevector

**k**

_{0}and the scattered wavevector

**k**is defined as

_{s}= R

_{D}/l

_{2}is measured through the distance l

_{2}of the detector from the sample and the distance R

_{D}from the direct beam at which the scattered intensity is recorded. The scattering angle and scattering vector are related to the length scale D

_{s}of the scattering structure through the Bragg equation and the scattering vector definition. Correspondingly the structure sizes probed by SANS D

_{s}= 4π/q range from a few nm to a few 100 nm corresponding to a q-range between about 1 and 0.01 nm

^{−1}. The scattering function S(

**q**) can be expressed as a Fourier transform of the scattering structure described by the even function of its scattering length density-density correlation function γ(

**r**) [1]

**q**)d

**q**= 1 by n

_{0}. Hence, with the total small angle scattering probability for a two phase system

_{i}being the bound coherent scattering lengths of the constituting elements of a homogeneous phase with a phase fraction φ, and with the sample thickness t the intensity distribution scattered at small angles out of the direct beam can be written as

**q**) = S(−

**q**) and the scattering function is invariant with respect to an exchange of the two phases (Babinet principle) [1].

#### 1.3. Initial Approaches of SANS in Imaging

## 2. Beam Modulation Techniques for SANS and Imaging

#### 2.1. Neutron Grating Interferometry (NGI)

_{2}referred to as Talbot distances. For the conventionally used gratings inducing a phase shift of π, the period of the self-image, i.e., the modulation of the interference pattern is half the period of the grating. In order to provide sufficient beam coherence for diffraction at a micrometer sized structure a source grating, which is an absorption grating (G

_{0}), is installed in the divergent neutron beam close to the pinhole, which defines the real space spatial resolution of the imaging instrument. The period and distances are chosen such that the interference patterns related to individual beams from G

_{0}are superposed constructively at the chosen Talbot distance, i.e., p

_{0}= pL

_{1}/L

_{2}, where p

_{0}is the period of the source grating and L

_{1}is the distance from source grating to phase grating. Because the period of the interference pattern in the micrometer range cannot be detected directly with conventional state-of-the-art imaging detectors with pixel sizes >10 μm, another absorption grating (G

_{2}) with a period p is installed at the chosen Talbot distance L

_{2}and the transversal beam modulation is translated into an intensity variation in each pixel of the imaging detector by stepwise scanning of one grating over about one period. The distances between source and phase gratings are typically of the order of meters and the Talbot distance of the order of centimeters. The sample can be placed before or after the phase grating, but for optimum spatial resolution performance should remain close to the detector. The Talbot–Lau interferometer is optimized for a certain wavelength with regards to the phase shift in the phase grating and with respect to the Talbot distance. However, the requirements for monochromatisation, i.e., wavelength resolution (dλ/λ ≈ 10%) and beam divergence are very modest, which enables to work with significantly brighter beams than in any other technique sensitive in the respective angular range. This hence enables efficient imaging measurements with high spatial resolution.

_{max}− I

_{min})/(I

_{max}+ I

_{min}), where I

_{max}and I

_{min}are the maximum and the minimum intensity of a modulation period across the beam. Visibility is lost through redistribution of intensity from the bright field (modulation maxima) to the dark field (modulation minima), which is the case in particular when the beam is scattered symmetrically like in small angle scattering or when magnetic features affect spin-up and spin-down components of the neutron conversely through refraction. However, also conventional refraction can due to limited direct spatial resolution lead to dark-field effects.

#### 2.2. Spin-Echo Modulation (SEM)

_{1}and B

_{2}and the distance of the fields L

_{1}and L

_{2}to the detector, the spin polarization is a sinusoidal function of the transversal position on the detector only [16,31,32,33]. Hence, the installation of a spin filter, i.e., polarization analyser after the second precession field yields a spatially intensity modulated beam on the detector. The key condition is B

_{1}L

_{1}= B

_{2}L

_{2}and provides a beam modulation with a period.

## 3. Dark-Field Imaging (DFI)

#### 3.1. Modulated Beam Dark-Field Contrast Theory

_{S}and V

_{0}are the visibilities measured with and without the sample, respectively, is

_{max}− I

_{min})/(I

_{max}+ I

_{min}) is the visibility of the modulation for an open beam (V

_{0}) and a sample measurement (V

_{s}) in each pixel, respectively, and Δω is the specific phase shift of the modulation with respect to a specific scattering angle, respective scattering vector, and can be expressed by the set-up parameter ξ and the scattering vector q. ξ = λL

_{s}/p has been formulated for the first time in this context [42], being the autocorrelation length, i.e., the specific correlation length probed by a particular set-up. This parameter is characterized by the utilized wavelength λ, the modulation period p and the effective sample to detector distance L

_{s}. Taking into account transmitted neutrons, the path integral through an extended sample and multiple scattering effects the final expression can be written as

_{s}/p, which can be realized through scans of the wavelength, like in a ToF approach, a scan of the sample distance, which however also affects the real space spatial resolution, and the period of the interference pattern, which cannot be achieved with a fixed grating set-up.

#### 3.2. Quantitative Dark-Field Contrast Imaging

## 4. Discussion

## 5. Outlook

#### 5.1. Progress in Grating-Based DFI

#### 5.2. Progress in Spin-Echo Modulated DFI

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 2.**Double crystal diffractometer for imaging (

**Left**) and results (

**Right**) of differential phase (matrix,

**Top**) and dark-field contrast (

**Bottom**) tomographies of a reference sample filled with different sizes and concentrations of beta-carotine particles in solution as visualized in the tomography (color code); Results Reproduced from [10], with the permission of AIP Publishing.

**Figure 3.**Modulated beam imaging set-ups schematic: Talbot–Lau grating interferometer (

**Top**) and spin-echo modulation set-up (

**Middle**) as well as a novel development referred to as far-field grating interferometer (

**Bottom**).

**Figure 5.**Quantitative neutron grating interferometry dark-field imaging (NGI-DFI): (

**a**) spherical polystyrene particles diluted in H2O/D2O; loss of concentration observed over time (sedimentation) of 4 μm particles (

**a1**); results corresponding to form factor of spherical particles with dimeter of 2 μm measured (

**a2**); (

**a3**,

**a4**) corresponding dark-field images at 3.3 Å and 5 Å, respectively [44]; (

**b**) DFI of polystyrene spheres sedimenting in an aqueous dilution where a diluted phase a concentrated phase and a quasi-crystalline phase can be identified and in particular a depletion zone (C) between high concentration and crystallization regions is found [46]; (a1) to (a4) are reproduced with permission of the International Union of Crystallography from [44].

**Figure 6.**Time-of-flight dark-field imaging of steel welds: (

**a**) real space correlation functions of three distinct areas of a 1.5 mm thick steel weld; (

**b**) photograph of the sample(s) with the color coded regions of interest analysed in (

**a**) and (

**c**); (

**c**) simultaneously recorded wavelength dependent transmission data of two color coded regions of interest (

**b**) in the weld, displaying the Bragg edge pattern around the Fe(110) Bragg edge.

**Figure 7.**Resolution ranges and limitations correlation length and direct real space resolution for different DFI approaches; (

**a**) modelled data from wavelength dispersive NGI-DFI and time-of-flight (ToF) SEM-DFI; (

**b1**) real space image of the SEM-DFI [34] and (

**b2**) of ToF NGI-DFI; (

**c**) curves from (

**a**) on logarithmic correlation length scale and including ranges of different neutron techniques to explore the corresponding size ranges (note these apply only to the size range, i.e., x-axis).

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**MDPI and ACS Style**

Strobl, M.; Harti, R.P.; Gruenzweig, C.; Woracek, R.; Plomp, J.
Small Angle Scattering in Neutron Imaging—A Review. *J. Imaging* **2017**, *3*, 64.
https://doi.org/10.3390/jimaging3040064

**AMA Style**

Strobl M, Harti RP, Gruenzweig C, Woracek R, Plomp J.
Small Angle Scattering in Neutron Imaging—A Review. *Journal of Imaging*. 2017; 3(4):64.
https://doi.org/10.3390/jimaging3040064

**Chicago/Turabian Style**

Strobl, Markus, Ralph P. Harti, Christian Gruenzweig, Robin Woracek, and Jeroen Plomp.
2017. "Small Angle Scattering in Neutron Imaging—A Review" *Journal of Imaging* 3, no. 4: 64.
https://doi.org/10.3390/jimaging3040064