Bragg-edge imaging, which is also known as neutron radiography, has recently emerged as a novel crystalline characterization technique. Modelling of this novel technique by incorporating various features of the underlying microstructure (including the crystallographic texture, the morphological texture, and the grain size) of the material remains a subject of considerable research and development. In this paper, Inconel 718 samples made by additive manufacturing were investigated by neutron diffraction and neutron radiography techniques. The specimen features strong morphological and crystallographic textures and a highly heterogeneous microstructure. A 3D statistical full-field model is introduced by taking details of the microstructure into account to understand the experimental neutron radiography results. The Bragg-edge imaging and the total cross section were calculated based on the neutron transmission physics. A good match was obtained between the model predictions and experimental results at different incident beam angles with respect to the sample build direction. The current theoretical approach has the ability to incorporate 3D spatially resolved microstructural heterogeneity information and shows promise in understanding the 2D neutron radiography of bulk samples. With further development to incorporate the heterogeneity in lattice strain in the model, it can be used as a powerful tool in the future to better understand the neutron radiography data.

## I. INTRODUCTION

Neutrons have the advantage of being electronically charge-free and achieving deep penetration (higher than that of X-rays), and therefore, they are often used in a non-destructive measurement. Neutrons are considered to form an ideal “probe” to explore the structure of bulk materials. By keeping records of the trace and the quantity of either the neutrons reflected by (neutron diffraction) or the neutrons transmitted through (neutron transmission) certain lattice planes of the material, statistical information regarding the microstructure of the material can be obtained. Specifically, wavelength-dependent neutron transmission imaging, also referred to as neutron radiography or Bragg-edge imaging, exhibits a sudden drop in the transmitted intensity (a sharp increase in the total cross section) when the neutron wavelength, λ_{hkl}, is equal to 2d_{hkl}, where d_{hkl} is the inter-planar distance for the {hkl} planes in the crystalline structure,^{1,2} as enunciated by Bragg's law. This is known as the Bragg-edge phenomenon. The wavelength dependent neutron spectrum has been studied extensively over the past several decades.^{3–16} It was only in recent years that wavelength resolved transmission imaging was applied to spatially resolved studies of crystalline features and has received growing attention due to its promising capability for providing quantitative spatial information regarding the internal structure of bulk samples.^{17,18} Thanks to the development of boron-doped micro-channel plate (MCP) neutron detectors with sufficient time resolution and detection efficiency,^{19,20} wavelength-resolved neutron radiography has been used in the investigation of the spatially resolved lattice strain,^{21,22} the crystallographic orientation distribution,^{23–27} and phase distribution^{17,28} in materials. An attractive capability of the Bragg-edge imaging lies in the potential to visualize the crystallographic texture and the inhomogeneities through the sample thickness.^{12} Theoretical and mathematical approaches to fit and interpret the transmission spectra have been proposed and applied to several engineering materials.^{29–33} However, many of the current approaches do not take into account all the microstructural features and the associated heterogeneity due to the high level of complexity for both measurements and analyses.^{12} The objective of the modelling developed in this paper is to extend currently used models by including the grain morphology in addition to the crystallographic texture in a statistical full-field model, so that it can potentially be used as a complimentary tool in the future to better understand the measured neutron radiography data.

Similar to the neutron diffraction single peak fitting, the narrow Bragg edge can be fitted using an analytical function based on a normalized Gaussian integral convoluted with a cut-off decaying exponential.^{2,34,35} The fitting method allows quantitative determination of the position (with a resolution of ∼100 *μ*ε), width, and height of the Bragg edge.^{29,36,37} Single-pixel neutron radiography data give the distribution of the crystallographic texture and the average d-spacing of lattice planes for grains with the studied lattice plane perpendicular to the beam direction through the thickness of the material. However, the heterogeneity of the microstructure through the thickness is quite often unknown, and the influence of the identified microstructure feature on the neutron radiography data is complex to analyze. The validity of the fitting procedure for Bragg edge analysis, especially for materials after plastic deformation,^{38} needs to be investigated based on the detailed microstructure in the material. The neutron transmission calculation based on the orientation distribution function (ODF),^{39} i.e., calculation based on the crystallographic texture, is well established.^{23,40} This method is called the statistical method in our paper, and it has been used for the verification of the incomplete pole figure measurements^{40} and for crystallographic texture analysis.^{23} In this paper, a novel three-dimensional (3D) statistical full-field model is introduced to represent not only the grain orientation distribution (or the ODF) in the studied sample but also the morphological texture (grain shape). Such a statistical full-field model is 3D spatially resolved, and it is widely used in simulations of the deformation of polycrystalline materials.^{41} Here, we have employed it for the first time for the Bragg-edge imaging and the total cross section calculations. In comparison to the statistical model,^{23,40} the 3D statistical full-field model has the added benefit of being able to potentially capture the grain size and variation of the d-spacing (and therefore the lattice strain) due to deformation/residual strain in the material. Making use of the well-developed neutron transmission physics, the neutron radiography of samples with a complex microstructure can be understood. As an initial demonstration of the 3D statistical full-field modelling approach, we present the results of our predictive microstructure model and compare them with experimental measurements using additively manufactured (AM) Inconel 718 samples.^{42}

In this paper, we have employed neutron diffraction to characterize the crystallographic texture distribution in the sample, which was then used to build a 3D statistical full-field model to calculate the corresponding neutron radiography. The modelling results are compared with the measured total cross section and the Bragg-edge imaging. The microstructure heterogeneity of the sample was investigated, and its effect on the radiography is discussed.

## II. MATERIALS AND METHODS

In this work, Inconel 718 samples made by additive manufacturing using rotary atomized powder of the Ni base superalloy were studied. The bulk blocks were built using electron beam melting (EBM) at the Oak Ridge National Laboratory (ORNL) Manufacturing Demonstration Facility (MDF). The microstructures of the AM Inconel blocks were controlled by assigning specific EBM parameters to obtain preferred crystallographic orientations.^{43} Additive manufacturing is popular due to its unprecedented potential in controlling the crystallographic microstructure. However, AM samples often lack in-depth understanding of the link between the process to make them and the resulting microstructure. Neutron diffraction and transmission are rigorous nondestructive characterization methods that can be used for studying the microstructure of the AM samples.

A sample of 5.5 × 5.5 × 12 mm^{3} was studied with the longitudinal direction perpendicular to the build direction as shown in Fig. 1(a). The chemical composition of the material is 18.8% Fe, 19.4% Cr, 3.05% Mo, 5.03% Nb, 0.11% Co, 0.005% Mn, 0.033% Cu, 0.49% Al, 0.95% Ti, 0.031% Si, 0.005% C, 0.001% S, 0.0055% P, and 0.0016% B, Bal. Ni (weight percentage). The samples feature both strong crystallographic and morphological textures, which is common for the AM materials.^{44,45} The grain size is ∼50–100 *μ*m in the direction perpendicular to the build direction, while along the build direction, it is ∼1 mm or even larger. The microstructure in the sample is heterogeneous. The crystallographic texture of the sample was measured on the VULCAN engineering diffractometer located at the Spallation Neutron Source (SNS) at ORNL. A sketch of the experimental setup for the texture measurement can be found in Fig. 1(b). The angular coverage of the two detectors (bank 1 and bank 2) is ±15° in the vertical direction and ±11.5° in the horizontal direction. The two detectors are used to collect the diffracted beams from polycrystalline grains.^{46} For the texture measurement, the sample was rotated horizontally and tilted with respect to the incident beam to obtain a full reconstruction of the pole figure (for the pole figure measurement shown in Fig. 1(b), Ω=0°–45° with 5° intervals and for each Ω, θ = 0°–330° with 30° intervals). The detector coverage area in the pole figure for Ω= 45° and θ=0° is shown in Fig. 1(b). A 5 × 5 × 5 mm^{3} gauge volume is determined by the incident slits and the receiving collimators, and the location of the gauge volume varies for different Ω. The measured pole figures were represented in the sample frame x-y-z as given in Fig. 1(c). Strong crystallographic texture with a dominant cube texture component, with ⟨200⟩ parallel with the three sample axes, is seen. Note that the neutron diffraction is based on the interaction between neutrons and the crystal structure in a defined gauge volume shown in Fig. 1(b). During the rotation for the crystallographic texture measurement, the position of the involved gauge section, i.e., gauge volume in the sample, varies slightly. Only one part of the sample belongs to the gauge volume.

Wavelength-dependent neutron radiography was conducted at the SNAP beamline, located also at SNS at ORNL. An MCP detector with a field of 28 × 28 mm^{2} and a pixel size of 55 × 55 *μ*m^{2} was utilized for the transmission radiography experiments. The pixel size is close to the minimum dimension of the grain size in the studied material. The time resolution of the MCP detector was 10 *μ*s, corresponding to a Δλ of 0.002 Å. The source-to-detector distance was calibrated as follows: we conducted Bragg-edge imaging on an Inconel 718 powder with known lattice planar spacing, which was determined by neutron diffraction. Then, the Bragg-edge spectra were analyzed by a well-known phenomenological function^{34–36} to determine the position of the Bragg-edge for the {hkl} planes as a function of the time of flight (TOF). With the planar spacings determined by neutron diffraction and Bragg-edge imaging, we can derive the distance by the following relation:

where *D* is the distance between the source and the detector, *T* represents the TOF of neutrons, *d* indicates the delay, and *λ* is the wavelength. Figure 2 shows the wavelength *vs*. TOF plot with the linear fitting for several different {hkl} planes. The source-to-detector distance was identified to be 1593 ± 1 cm. The pinhole aperture was 10 × 10 mm^{2} and was positioned upstream in the SNAP cave. Samples sat as close as possible to the MCP detector. Further details of the experimental radiography setup are presented elsewhere.^{2,47}

The Bragg-edge theory developed by Vogel,^{48} Boin,^{49,50} and Santisteban *et al.*^{23,40} was used to calculate the total neutron cross section as a function of the wavelength. The neutron relative transmission, *Tr*, is the ratio of the detected neutron intensity, *I*, over the incident neutron intensity, *I _{0}*, and is given by

where *n* is the number of cells per unit volume, σ_{total} is the total neutron cross section, and *z* is the sample thickness. The total cross section σ_{total} is composed of the absorption cross section (σ_{absorption}) and the scattering cross section (σ_{scattering}) which is the sum of the coherent and incoherent scattering cross sections, σ_{coh} and σ_{inc}, respectively. Since the incoherent part of the inelastic scattering cross section is a good approximation for the coherent part in the thermal neutron range,^{51} σ_{total} is approximated by using Eq. (3). Here, $\sigma \xafcoh$ and $\sigma \xafinc$ are the average cross section terms and S describes the effects of the neutron energy and the spatial nuclei arrangement

An approximation for the first term on the right-hand side of Eq. (3) was given by Boin^{49} as shown in Eq. (4). σ_{absorption} is calculated following Ref. 49:

The elastic coherent scattering cross section can be calculated using

where M_{hkl} is the multiplicity factor of the {hkl} plane and F_{hkl} is the corresponding structure factor. In comparison to Ref. 52, the powder extinction effect and the Bragg-edge broadening due to the neutron pulse shape are neglected in our calculation. The R factor depends on the neutron wavelength and the crystallographic texture in the material^{23} and is given by

P in Eq. (6) is the orientation density at each position $\pi 2\u2212arcsin\lambda 2dhkl,\u2009\psi $ in the {hkl} pole figure. The first term in the parenthesis is the angular deviation from the center of the pole figure (polar angle), and $\psi $ is the angle in the circumferential direction (azimuth angle), which ranges from 0 to 2π in radians. If P is equal to one, it corresponds to random orientation density. If it is equal to two, then it is twice the random orientation density. In Eq. (5), a total of 13 crystallographic planes are calculated: {111}, {200}, {220}, {311}, {222}, {400}, {331}, {420}, {422}, {511}, {333}, {440}, and {531}.

The above equations are used to calculate the total cross section for a given crystallographic texture. To calculate the 2D Bragg-edge imaging, a 3D statistical full-field model was introduced to model the actual microstructure as shown in Fig. 3(a). A total of 91 125 (45 × 45 × 45) elements were used to represent the microstructure. The elongated grain shape with an aspect ratio equal to 10 is captured in the model. The pole figures shown in Fig. 1(c) were used to calculate the ODF,^{53} which was used to generate 550 grain orientations in the model. In Fig. 3(b), the pole figures generated by the 91 125 elements are shown, which are similar to the measured pole figures shown in Fig. 1(c). The maximum intensity in the {111} and {220} pole figures is lower than the measured one, and the possible reasons will be given in the discussion section. Each element is treated as a coherent zone featuring a single grain orientation, and there are totally 45 coherent zones through the thickness for each pixel in the calculated Bragg-edge imaging. Several thousands of elements that represent the sample are commonly used in the crystal plasticity models for statistical representation of the volume element (SRVE) in order to quantitatively represent the sample.^{41,54}

Figure 3(a) indicates how the neutron radiograph was measured at different angles to the beam direction. By successively rotating the sample around its z-axis in 30° intervals (β ranges from 0° to 90°), a total of 4 neutron transmission spectra corresponding to different β values were calculated. For the total cross section calculation, the crystallographic texture needs be represented in a plane normal to the incident beam. Three Euler angles in Bunge's convention^{39} were employed to rotate the sample, as given in Table I. The Euler angles for each β are used to rotate the pole figure represented in the x-y plane to a plane perpendicular to the neutron beam.

## III. RESULTS AND DISCUSSION

The calculated total cross sections in comparison to the experimental counterparts are shown in Fig. 4(a). The prediction is good in the sense that (1) the maximum and minimum total cross sections at different β values are nicely captured and (2) variations of the total cross section profile at different β values are captured as well. There are no {111} Bragg edges at β=0°, 30°, and 90° in the measurements. The model gives no {111} Bragg edges for β=0° but shows small {111} Bragg edges for β=30° and 90°. This may be due to the fact that the measured texture used in computing the total cross sections corresponds to a small volume of the sample; however, the experimental total cross sections are from the whole sample covered by the beam, and this is especially a problem for samples with a heterogeneous microstructure, as will be shown in Fig. 5. Thus, certain crystallographic texture features can be missing in the crystallographic texture measurement due to the heterogeneous microstructure in the AM sample. The total cross section is very sensitive to the crystallographic texture in the sample. The R factor distribution for different {hkl} planes [c.f. Equation (6)] as a function of the wavelength at β=0° is shown in Fig. 4(b). One can see that the calculated profile of the total cross section is highly dependent on the R factor distribution (each peak in the R-factor profile contributes to a peak in the total cross section profile) which is sensitive to the detailed features of the pole figures, i.e., the crystallographic texture. The R factor at each given value r in Fig. 4(b) for a given {hkl} pole figure in Eq. (6) corresponds to the average orientation density in the whole circle with radius r. At the same time, there is one to one relation between the value r in Fig. 4(b) and the corresponding wavelength as shown in the table in Fig. 4(b).

To further understand the quality of the total cross section prediction for the currently studied AM sample, Fig. 5 plots the total cross sections of the sample measured using smaller gauge volume at different positions for β = 0° and 90°, respectively. Indeed, the microstructure heterogeneity can be obviously seen in Fig. 5. As mentioned earlier, the AM sample features a heterogeneous microstructure, and therefore the total cross section is quite different at different gauge locations. Then, the differences between the measured and the calculated total cross sections in Fig. 4(a) can be due to the following reasons: (1) The microstructure in the AM sample is highly heterogeneous, and inconsistences in the measured pole figures during the texture measurement are expected due to the variation of the gauge volume during rotation of the sample. This will reduce the accuracy of the ODF calculation. (2) The measured texture corresponds to a small part of the sample, while the measured total cross section is from the whole sample, and this is especially a problem for a sample with a heterogeneous microstructure. Interestingly, there is a {111} Bragg edge at β = 90° for gauge position 1 shown in Fig. 5(b), and it is closer to the calculated total cross section than those in other gauge positions.

Figure 6(a) shows the relative transmission spectra as a function of the neutron wavelength for two orientations (β = 0° and 90°) of the AM sample, which was obtained by averaging the data in a volume denoted by the red dotted rectangle in Fig. 6(b). For comparison, a relative transmission spectrum based on the Inconel 718 structure with random crystallographic texture is shown in Fig. 6(a). The experimental spectra are quite different from those with random crystallographic texture, which confirms the preferential distribution of grain orientations, i.e., high density of grains with ⟨200⟩ parallel with the beam direction. The absence of the {111} grains leads to almost featureless contrast distribution in the energy-dependent transmission radiograph [Fig. 6(b)] at λ = 4.110 Å. Relative transmission radiographs obtained at other wavelengths (2.088 and 3.599 Å) reveal apparently distinct features. The selected three wavelengths are close to Bragg edges of {111}, {200}, and {311}. The radiographs at β = 0° display certain features of the microstructure.^{55} The radiograph at β = 90° exhibits a significantly different contrast variation with a stripe-like feature. The neutron radiograph was calculated to understand the experimental results. Both the stripe-like and the spotted features from an assembly of the columnar grains averaged through the sample thickness are nicely reproduced in the model. The obtained neutron radiograph at different wavelengths and at different angles β is therefore influenced by both the strong morphological and crystallographic texture. The relative transmission in Fig. 6(a) at λ = 4.110 Å is the highest and that at λ = 2.088 Å is lower than that at λ = 4.110 Å, while at λ = 3.599 Å, it corresponds to a large range of intensity. This trend is captured in the calculated radiograph. At β=0°, there is almost no contrast in the calculated relative transmission at a wavelength equal to 4.110 Å. The spots in the calculated image are due to the fact that there are only 45 grain orientations through the thickness, and then, the pole figure generated by the 45 grains can give certain intensity with ⟨111⟩ parallel with the beam for certain pixels. For a wavelength equal to 4.110 Å at β = 90°, the calculated radiograph still gives certain banded patterns. This is due to the fact that the calculated transmission in Fig. 4(a) at β = 90° gives a {111} Bragg edge although there is no {111} Bragg edge in the experiment for the whole sample. In the model, the {111} Bragg edge corresponds to a {111} peak intensity of 0.5 times the random case. Possible reasons for the additional {111} Bragg edge in the model were explained earlier. Interestingly, in Fig. 5(b), there is a small {111} Bragg edge at gauge position 1, and indeed, there are minor dark features in that part of the sample shown in Fig. 6(b) indicated by the arrow. Note that the dark patterns correspond to low relative transmission which corresponds to a high total cross section. The dark pattern in Fig. 6(b) indicated by the arrow should correspond to the position with the {111} Bragg edge.

As discussed above, the 3D statistical full field model can account for the grain morphology to enable calculation of the relative transmission images. Another benefit of this approach is that it can incorporate the effect of heterogeneous lattice strains in the material due to deformation. When combined with constitutive models for the crystal elastic and plastic deformation of the material, the 3D statistical full field model has the capability to account for any lattice strain variations in the sample. Strain mapping studies have been reported to understand the deformation behavior of several engineering materials,^{21,29,30,38} which demonstrates the effectiveness and possibility of the energy-dependent neutron transmission measurements under various conditions, such as *in-situ* tension, crack growth, and bending tests. Those experimental 2D radiography results can be interpreted using the current 3D statistical full-field model. An example of such a calculation is given here in order to demonstrate the capability.

The result after 2% engineering compressive strain is shown in Fig. 7. Figure 7(a) shows the distribution of the true stress along the beam direction. Variations of the {200} d-spacing due to the inhomogeneous deformation are shown in Fig. 7(b). Note that in Fig. 7(b), the blue color means that there are no grains through the thickness with the grain ⟨200⟩ direction deviating from the beam direction by less than 2.5°, and the other colors indicate the average {200} d-spacing for those grains through the thickness. Then, there are no grains with the ⟨200⟩ direction deviating from the beam direction by less than 2.5° for pixel 1, while there are several such grains in pixel 2 in Fig. 7(b). Figure 7(c) gives the calculated total cross section from the two pixels and the averaged one from all pixels. It indicates that a minor shift of the {200} Bragg edge for pixels 1 and 2 corresponds to an angular accuracy of ⟨200⟩ parallel with the beam with 2.5° tolerance. The current approach coupled with the well-developed crystal-plasticity models makes it potentially possible not only to explore the dynamic behavior under diverse deformation conditions in 3D but also to help understand and interpret the 2D neutron radiography.

## IV. CONCLUDING REMARKS

A 3D statistical full-field model was introduced in the neutron transmission radiography calculation. The model can capture details of the microstructure inhomogeneity, especially for a sample after deformation. It quantitatively captures the effect of both the morphological and the crystallographic texture on the neutron transmission, with the neutron beam at different angles to the sample axis. The model gives 3D spatially resolved information of the microstructural heterogeneity and shows promise as a tool to investigate the effect of plastic deformation on the 2D neutron transmission radiography and to potentially be used as a complementary tool for validating the fitting procedure used for the Bragg edge analysis. The results presented show that for samples with a high degree of heterogeneity, such as those made using additive manufacturing, the neutron radiography measurements are sensitive to the underlying grain morphology and crystallographic texture. It is therefore important for the model to capture these microstructure features for proper interpretation of the measurements. This in turn points to the need for a greater number of elements in the model to improve the statistics associated with each pixel in the calculated Bragg-edge imaging.

## ACKNOWLEDGMENTS

This research was sponsored by the Laboratory Directed Research and Development (LDRD) Program of Oak Ridge National Laboratory, managed by UT-Battelle, LLC, for the U. S. Department of Energy. Resources at the Spallation Neutron Source, U.S. DOE Office of Science User Facilities operated by ORNL, were used in this research. Research at Manufacturing Demonstration Facility (MDF) was sponsored by the U.S. Department of Energy, Office of Energy Efficiency and Renewable Energy, Advanced Manufacturing Office, under Contract No. DE-AC05-00OR22725 with UT-Battelle, LLC. The present authors also thank Louis J. Santodonato at ORNL for final improvements on the manuscript.

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