1 Polychromatic neutron phase contrast imaging of weakly absorbing samples enabled by phase retrieval

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Polychromatic neutron phase contrast imaging of weakly absorbing

samples enabled by phase retrieval

Maja Østergaarda1, Estrid Buhl Naverb1, Anders Kaestnerc, Peter K. Willendrupde, Annemarie

Brüelf, Henning Osholm Sørensendg, Jesper Skovhus Thomsenf, Søren Schmidte, Henning Friis

Poulsend, Luise Theil Kuhnb* and Henrik Birkedala*

aDepartment of Chemistry and iNANO, Aarhus University, Gustav Wieds Vej 14, Aarhus, Denmark

bDepartment of Energy Conversion and Storage, Technical University of Denmark, Fysikvej 310,

Kongens Lyngby, Denmark

cLaboratory for Neutron Scattering and Imaging, Paul Scherrer Institute, Villigen, Switzerland

dDepartment of Physics, Technical University of Denmark, Fysikvej 307, Kongens Lyngby, Denmark

eEuropean Spallation Source ERIC, P.O. Box 176, Lund, Sweden

fDepartment of Biomedicine, Aarhus University, Wilhelm Meyers Allé 3, Aarhus, Denmark

gXnovo Technology ApS, Galoche Allé 15, Køge Denmark

Correspondence email: luku@dtu.dk; hbirkedal@chem.au.dk

1These authors contributed equally

Synopsis Neutron imaging enhanced by retrieval of propagation-based phase contrast is described.

Abstract We demonstrate the use of a phase retrieval technique for propagation-based phase contrast

neutron imaging with a polychromatic beam. This enables imaging samples with low absorption contrast

and/or improving the signal-to-noise ratio to facilitate e.g. time resolved measurements. A metal sample,

designed to be close to a pure phase object, and a bone sample with canals partially filled with D2O were

used for demonstrating the technique. These samples were imaged with a polychromatic neutron beam

followed by phase retrieval. For both samples the signal-to-noise ratio were significantly improved and in

case of the bone sample, the phase retrieval allowed for separation of bone and D2O, which is important for

example for in situ flow experiments. The use of deuteration-contrast avoids the use of chemical contrast

enhancement and makes neutron imaging an interesting complementary method to X-ray imaging of bone.

Keywords: Phase contrast imaging, neutron imaging, bone, tomography, phase retrieval

1. Introduction

There is a continued need for improved 3D imaging methods to analyse complex multi-length-scale

structures of a plethora of technologically or biologically important materials. In this connection, neutron

imaging is of interest due to the good penetration capabilities, the non-continuous dependence of

interaction cross sections with atomic number and the possibilities for adjusting material contrast by

controlling isotope compositions. The primary principle of image formation with neutron imaging is

attenuation contrast from absorption. This is not always the best way to provide contrast, especially for

samples consisting of materials with very similar absorption cross sections or very low absorption. For

this case, it is useful to apply the principle of phase contrast that harnesses the real part of the refractive

index. Phase contrast imaging can be performed in several ways, using various kinds of interferometry

(Pushin et al., 2017; Strobl et al., 2019) or propagation-based phase contrast (Fiori et al., 2006; Paganin et

al., 2019). The benefit of the latter is that no gratings or other specialised elements are needed to measure

the phase shift. Therefore, propagation-based phase contrast imaging has evolved to a very powerful tool

in X-ray imaging (Alloo et al., 2022; Bidola et al., 2017; Wieland et al., 2021; Yu et al., 2021) but has

only been investigated for neutrons in a very few cases (Allman et al., 2000; Jacobson et al., 2004;

Lehmann et al., 2005; McMahon et al., 2003).

When neutrons pass through a sample, they are refracted due to local variations in the refractive index.

Features in the sample with different refractive indices will act as neutron lenses that either focus or

diverge the neutrons from that point in the sample. These changes of the neutron directions are small and

thus difficult to observe at short distances between sample and detector but they do become progressively

easier to detect when increasing the distance between sample and detector. The build-up of phase contrast

requires a coherent beam, meaning that the beam divergence must be smaller than the changes caused by

refractive features. The propagation-based phase contrast imaging only gives relative information about

the sample, which means that a phase-retrieval technique is required in order to obtain the projected

density of the sample (Paganin et al., 2019).

Here we explore propagation-based phase contrast imaging first in a model sample with very low

absorption and secondly in bone, which is a hierarchically structured material.

Bone is replete with blood vessels and cells situated in lacunae interconnected by canaliculi only a few

hundred nm in diameter (Wittig et al., 2022). Together, these form a vast fluid-containing network.

Transport of fluid in bone is very important since the fluid contains nutrients, signal molecules, and ions

essential for the entire body (Cowin & Cardoso, 2015). In addition, liquid transport is proposed to be the

main mechanism of stress sensing in bone, suggesting that the osteocytes sense changes in shear liquid

flow through the canaliculi (van Tol et al., 2020; Robling & Bonewald, 2020; Burger & Klein-Nulend,

1999). Our understanding of liquid transport through the complex porous network of large canals (mean

Harversian canal diameter 85 µm (in iliac crest) (Busse et al., 2013)), and smaller channels – all the way

down to the cellular level – remains incomplete. The potential of neutron imaging to afford insights into

liquid transport in bone is thus highly interesting, since neutron imaging enables contrast variation by

deuteration. The field of neutron imaging of bone tissue is developing (Törnquist et al., 2020; Guillaume

et al., 2021), but methods providing improved signal to noise and/or faster measurements remain in high

demand.

Neutron propagation-based phase contrast imaging has been demonstrated for near-monochromatic

neutron beams (Paganin et al., 2019) but being able to harness the higher flux accessible with pink or

even white neutron beams, see for example (McMahon et al., 2003), would increase the efficiency and

applicability of the method significantly, which is indeed the aim of the current contribution.

2. Experimental

2.1. Metal sample

To validate the use of phase retrieval of propagation-based phase contrast neutron imaging with a

polychromatic neutron beam a sample was designed to have a good neutron phase contrast signal but

weak neutron absorption contrast. The sample was made of Al and Zr sheets (coherent scattering cross

section and thermal neutron absorption cross sections of 6.44 barn and 0.185 barn, respectively, for Zr

and of 1.495 and 0.231 barn for Al) with thicknesses of 10 μm and 25 μm, respectively, such that various

thicknesses were obtained (Figure 1a and 1b). The foils were cut into three pieces each of width 4 mm

and heights ranging between 8 and 11 mm and assembled in a staircase configuration as shown in Figure

1b.

2.2. Phase retrieval

For neutron absorption imaging the projection images are formed according to the Beer-Lambert

attenuation law. In phase contrast imaging this is not sufficient to describe the full contrast mechanism.

Instead the projection  at sample-detector distance  in the Fresnel regime and as a

function of two-dimensional positional coordinates in the plane , can be described (Paganin et

al., 2019):









( 1 )

Here  is the beam intensity without sample,  is the bound coherent scattering length,  is the

wavelength of the neutron beam,  is the total neutron cross section, and  is the number density of

atoms.

Phase-retrieval from measurements using a single sample to detector distance developed for use in X-ray

propagation-based phase contrast by Paganin and co-workers (Paganin et al., 2002) is sometimes referred

to as Paganin filtering. It has recently been adapted for use in neutron phase contrast imaging (Paganin et

al., 2019).

The goal of phase retrieval is to obtain the number density  of atoms of a single-material sample

given the propagation-based phase contrast image . Paganin et al. (2019) showed that this could

be obtained by











( 2 )

Here,  and  denote the spatial Fourier and inverse Fourier transforms,  are Fourier space

spatial frequencies corresponding to , and  is







( 3 )

where  is the beam divergence. The core of this expression is the Fourier-space low-pass filter 



, which depends solely on the parameter  that determines the strength of the phase contrast

signal. The result of Eq. 2 is the number density of atoms for a given volume in the sample under the

assumption that the sample consists of a homogeneous material.

It is important that , to avoid the denominator in Eq. 2 being zero. This is achieved primarily

through collimation of the neutron beam. From Eq. 3 we get the collimation condition





( 4 )

This presents a trade-off between having a divergent beam with higher neutron intensity, and thus better

statistics, but worse phase contrast, compared to having a more coherent beam with reduced signal to

noise but a better phase contrast signal. The divergence in the experiments reported herein is described

below.

Eq. 2 is only valid for a monochromatic beam. For a polychromatic, pink, or white beam, the wavelength

dependence should in principle be treated explicitly, which would require an energy sensitive detector.

Paganin et al. (2019) derived an approximate treatment employing effective spectrally averaged quantities

as shown in Eq. 5.

 





 



( 5 )

where the spectrally-averaged quantities , , and  are defined as











( 6a )











( 6b )











( 6c )

Here,  is the energy spectrum of the emitted neutrons. We used this formulation for all phase retrieval

calculations in the present work.

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1PolychromaticneutronphasecontrastimagingofweaklyabsorbingsamplesenabledbyphaseretrievalMajaØstergaarda1,EstridBuhlNaverb1,AndersKaestnerc,PeterK.Willendrupde,AnnemarieBrüelf,HenningOsholmSørensendg,JesperSkovhusThomsenf,SørenSchmidte,HenningFriisPoulsend,LuiseTheilKuhnb*andHenrikBirkedala*aDepartme...

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