# Deriving Quantitative Crystallographic Information from the Wavelength-Resolved Neutron Transmission Analysis Performed in Imaging Mode

## Abstract

**:**

## 1. Introduction

## 2. Bragg-Edge/Dip Profile Analysis for Quantitative Evaluation of Crystalline Microstructural Information

#### 2.1. Information Included in Bragg-Edge/Dip Neutron Transmission Spectrum

_{hkl}is crystal lattice plane spacing (d-spacing) of the crystal lattice plane {hkl} and θ

_{hkl}is Bragg angle for the crystal lattice plane {hkl}. Wavelength-resolved neutron transmission imaging experiments use polychromatic neutrons, while the wavelengths are analysed by the TOF method etc. Therefore, neutrons in a polycrystalline material can be diffracted at various wavelengths from λ = 2d

_{hkl}sin0° to λ = 2d

_{hkl}sin90° and there are no diffracted neutrons at λ > 2d

_{hkl}sin90°. As a result, Bragg-edge appears at λ = 2d

_{hkl}. Thus, d

_{hkl}can be deduced from the Bragg-edge wavelength λ = 2d

_{hkl}. In addition, the elastic macro-strain ε in crystal lattice can be evaluated by:

_{0}is d-spacing without strain/stress. At the same time, broadening of the d

_{hkl}distribution relating to micro-strain (relating to dislocation density etc.) can be deduced from broadening of Bragg-edge.

_{hkl}, the wavelength-dependent intensity reflects the angle-dependent intensity of diffracted neutrons, according to Equation (1) which indicates the relation between λ and θ

_{hkl}. In other words, crystallographic anisotropy (orientation distribution) and preferred orientation due to texture can be evaluated from shape of wavelength-dependent transmission spectrum. The intensity increase caused by multiple diffraction due to the primary extinction effect inside a crystallite reflects the crystallite size. Of course, the whole Bragg-edge pattern reflects the number of crystalline phases and the crystal structure of each phase.

_{hkl}sinθ

_{hkl}(θ

_{hkl}can be changed depending on the crystal orientation of a grain). Thus, the number of grains and the crystal orientation of each grain can be deduced from the Bragg-dip pattern.

#### 2.2. Profile Calculation Model for Bragg-Edge Transmission Spectrum Analysis: Algorithm of the RITS Code

- Automatic calculation function for multiple elements, multiple crystalline phases and 230 crystal structure space groups [5].

#### 2.2.1. Neutron Transmission and Total Cross-Section

_{0}(λ) as follows:

_{0}(λ) do not contain the background noise. On the other hand, Tr(λ) is theoretically expressed by:

_{tot}is total cross-section, ρ is atomic number density and t is thickness, respectively. Note that the projected atomic number density ρt can be evaluated in a transmission spectroscopy. By using the refined projected atomic number density ρt, the volume fractions of each crystalline phase can be deduced and mapped [9].

#### 2.2.2. Coherent Elastic Scattering (Nuclear Bragg Scattering) and the Crystal Structure Factor

_{hkl}(λ) [18,21,22] for strain analysis, the March-Dollase preferred orientation function P

_{hkl}(λ) [18,22,23] for texture analysis and Sabine’s primary extinction function E

_{hkl}(λ) [22,24,25] for crystallite size analysis. Here, V

_{0}is unit cell volume of crystal structure and F

_{hkl}is the crystal structure factor that is defined by the same definition of crystallography [7]:

_{n}is site occupancy of nth atom, b

_{n}is scattering length of nth atom, (x

_{n}, y

_{n}, z

_{n}) is fractional coordinate of nth atom, the exponential is the Debye-Waller factor and B

_{iso}is isotropic atomic displacement parameter, respectively.

#### 2.2.3. Edge Profile Function R_{hkl}(λ) for Macro/Micro-Strain Correction/Analysis

_{hkl}(λ), is the Jorgensen-type edge profile function [18,21,22]. This function can describe the profile change near Bragg-edge due to instrumental wavelength resolution and micro-strain (relating to dislocation density etc.). The expression [4] is:

_{hkl}is a broadening parameter representing the Bragg-edge profile. In the RITS code, this parameter is separated to the instrumental wavelength resolution part σ

_{0}(the standard deviation of (asymmetric) Jorgensen-type edge profile function) and the micro-strain part σ

_{1}’ (the standard deviation of (symmetric) complementary error function), respectively [6]:

_{1}’

_{,hkl}represents the standard deviation and w

_{hkl}represents FWHM (full width at half maximum) of the d-spacing distribution expressed by the Gaussian profile. Moreover, α

_{hkl}is the rising parameter of Bragg-edge and β

_{hkl}is the decaying parameter of Bragg-edge. By determining the instrumental wavelength resolution parameters σ

_{0,hkl}, α

_{hkl}and β

_{hkl}before material characterization, the micro-strain parameter σ

_{1}’

_{,hkl}(w

_{hkl}) of a specimen can be evaluated from the Bragg-edge profile [6].

_{hkl}can be evaluated, as well as pioneer works [26,27] using the Kropff-type edge profile function which is simpler than the Jorgensen-type, without the distortion due to micro-strain effect [4,6]. It is also a feature of RITS that w

_{hkl}can be also evaluated at the same time. In the single Bragg-edge analysis mode of the RITS code, three-stage fitting method [26] is adopted for easy and accurate profile fitting. Note that the three-stage fitting method is a step-by-step fitting algorithm for spectrum of three wavelength regions before/around/after a Bragg-edge. The data-analysis examples are described in Section 5 and Section 6.

#### 2.2.4. March-Dollase Type Preferred Orientation Function P_{hkl}(λ) for Crystallographic Texture Correction/Analysis

_{hkl}(λ), is the March-Dollase function for all θ

_{hkl}[18,22]. This function can describe the shape change over whole transmission spectrum due to preferred orientation and degree of crystallographic anisotropy. The expression [1] is:

_{hkl}(α,β) corresponds to so-called the pole figure of {hkl} with the latitude angle (= Bragg angle) α and the longitude angle β. The function inside the integral of Equation (16) is the March function. In other words, in the March-Dollase model, the pole figure is described by the March model. In addition, Equation (16) represents that α (Bragg angle θ

_{hkl}) dependent orientation distribution is observable but the distribution is integrated and averaged over all β; β-dependent orientation distribution is not observable in the Bragg-edge transmission spectroscopy. A of Equation (18) represents angle between the plane-normal vector of considering diffraction <hkl> and the preferred orientation vector <HKL>.

_{hkl}(λ) is 1 that represents no texture (isotropic orientation distribution). If R is close to 0 or ∞, strong texture exists in a specimen (anisotropic orientation distribution). The other is the preferred orientation vector <HKL>. When R is less than 1, <HKL> represents the preferred orientation parallel to the neutron incident/transmission direction (α = θ

_{hkl}= 90° direction). When R is greater than 1, <HKL> represents the preferred orientation perpendicular to the neutron incident/transmission direction (α = θ

_{hkl}= 0° direction). The data-analysis examples are described in Section 3 and Section 4.

#### 2.2.5. Sabine’s Primary Extinction Correction Function E_{hkl}(λ) for Crystallite Size Analysis

_{hkl}(λ), is Sabine’s function [22,24,25]. This function can describe the intensity increase over whole transmission spectrum (decrease of diffraction intensity) due to multiple diffraction of neutrons inside a crystallite. Thus, this effect can reflect the size of crystallite. The expression [1] is:

_{hkl}(λ) is 1 (no extinction). If S becomes larger, E

_{hkl}(λ) becomes smaller (extinction becomes stronger, total cross-section becomes smaller and transmission intensity becomes larger). The data-analysis examples are described in Section 3 and Section 4.

#### 2.3. Bragg-Dip Pattern Analysis Method

#### 2.3.1. Database Matching Method for Fast Determination of the Number of Crystalline Grains and Their Crystal Orientations

- Fast determination of crystal orientation without any initial estimation.
- Data obtained from multiple grains can be analysed. In this case, the number of grains and their crystal orientations are individually determined. (Of course, there is a limit of acceptable number).

#### 2.3.2. Validity of Evaluated Crystal Orientation

## 3. Texture and Crystallite-Size Imaging by Rietveld-Type Bragg-Edge Analysis

#### 3.1. Experimental

- Two rolled plates (Samples A and B in Figure 4). Relation between neutron transmission direction and rolling direction are perpendicular.
- Welded plate (Sample D in Figure 4). Relation between neutron transmission direction and rolling direction are perpendicular. (Relation between neutron transmission direction and normal direction (ND) are parallel.)
- Welded plate (Sample C in Figure 4). Relation between neutron transmission direction and rolling direction (RD) are parallel.

_{4}moderator installed at the Hokkaido University Neutron Source (HUNS) [35,36] in Japan. HUNS is a pulsed neutron experimental facility based on 1.2 kW electron LINAC, which is one of compact accelerator driven neutron sources [37]. The distance from the neutron source to the sample/detector position was about 6 m. The maximum neutron flux is less than 10

^{4}n/cm

^{2}/s and the cold-neutron wavelength resolution of the TOF method is about 3% due to the coupled moderator and the short neutron flight distance. The collimator ratio (L/D) is 60 (beam angular divergence of 16.6 mrad).

^{10}B-type GEM (gas electron multiplier) detector developed by Prof. Uno of High Energy Accelerator Research Organization (KEK) [38]. The GEM detector has the pixel size of 800 μm and the detection area is 10.24 cm × 10.24 cm. By using this detector, samples shown in Figure 4 were simultaneously measured. The measurement time were 5.0 hours for sample beam and 3.3 hours for open beam, respectively.

#### 3.2. Spectrum Fitting Analysis Results

_{110}= 90°. In other words, <110> faces to the neutron incident/transmission direction, namely, the preferred orientation parallel to the beam direction at the RD rolled area is <110>. This is consistent with the well-known preferred orientation of rolling textures [34]. On the other hand, at the ND rolled area, {110} Bragg-edge transmission intensity has a valley near λ = 0.35 nm (θ

_{110}= 60°). This means <110> faces to the direction near θ

_{110}= 60° and the other preferred orientation exists along the beam direction. According to the RITS analysis, this preferred orientation was estimated as <111> [1]. This is also consistent with the well-known preferred orientation of rolling textures [34]. Since the transmission shape of welded zones is close to ideal Bragg-edge transmission spectrum shape, it is estimated that there are weak texture at welded zones.

#### 3.3. Imaging Results

#### 3.4. Check by Optical Microscope and Neutron Diffraction

## 4. Crystalline Phase Imaging with Texture/Extinction Corrections

#### 4.1. Experimental

#### 4.2. Spectrum Fitting Analysis Result

#### 4.3. Imaging Results

#### 4.4. Check by Neutron Diffraction

## 5. Imaging of Crystal Lattice Plane Spacing (Macro-Strain) and Its Distribution’s Broadening (Micro-Strain)

#### 5.1. Experimental

_{2}moderator. The accelerator power of the proton synchrotron was 120 kW during this experiment. L/D was 337 (the beam angular divergence was 3 mrad) due to collimator setup in the beam-line and the estimated cold neutron flux was about 10

^{6}n/cm

^{2}/s. The cold-neutron wavelength resolution was about 0.35%. The neutron flight path length from the moderator to the detector was about 14 m. The used neutron TOF-imaging detector was

^{10}B-MCP (micro channel plate) type detector developed by Dr. Anton S. Tremsin of University of California at Berkeley in USA [46]. This detector has the pixel size of 55 μm and the detection area was 1.4 cm × 1.4 cm. The measurement time were 3.0 hours for open beam of the 3 mm quenched rod experiment, 4.0 hours for open beam of the 5 and 7 mm quenched rod experiments, 9.0 hours for 3 mm quenched rod, 8.0 hours for 5 mm quenched rod and 5.0 hours for 7 mm quenched rod, respectively.

#### 5.2. Spectrum Fitting Analysis Results

_{0,110}, α

_{110}and β

_{110}(see Section 2.2.3) were determined. Then, d

_{110}(crystal lattice plane spacing, relating to macro-strain) and w

_{110}(broadening of d

_{110}’s distribution, relating to micro-strain) were evaluated and visualized.

#### 5.3. Imaging Results

_{110}(crystal lattice plane spacing, relating to macro-strain) and w

_{110}(broadening of d

_{110}’s distribution, relating to micro-strain) of each quenched depth rod. Note that two same-type rods were simultaneously measured. d

_{110}and w

_{110}become larger at the rim (martensite) zone due to dislocation-density’s increase and/or fine microstructures caused by solid solution of carbon atoms into the crystal lattice. In addition, the depth of large d

_{110}and w

_{110}zone from the outer rim surface seems to correspond to the aimed quenched depth (3 mm, 5 mm and 7 mm). Actually, according to the Vickers hardness measurements, positions of the boundary between high d

_{110}/w

_{110}zone and low d

_{110}/w

_{110}zone correspond to positions indicating Hv 450 (hardness boundary between martensite and ferrite) [6]. In other words, this type of imaging also indicates crystalline phase imaging of martensite in a ferritic steel.

_{110}at the same positions of each specimen. This result indicates the Vickers hardness is proportional to w

_{110}. Since the Vickers hardness is also proportional to ferrite/martensite ratio, imaging of w

_{110}obtained by this method can quantitatively visualize distributions of the martensite phase and the hardness in a ferritic steel.

#### 5.4. Current Status at Compact Accelerator Driven Pulsed Neutron Sources

_{hkl}and w

_{hkl}by using Bragg-edge transmission imaging were successfully demonstrated [47]. The d

_{hkl}and w

_{hkl}values visualized at HUNS corresponded to those visualized at J-PARC although the total measurement time required 73 hours [47].

#### 5.5. Check by Neutron Diffraction

_{110}[3] and Figure 15b shows comparison results of w

_{110}relating to micro-strain [8], between Bragg-edge transmission experiment and neutron diffraction experiment. The simultaneous experiment of transmission and diffraction was performed at J-PARC MLF BL19 “TAKUMI” [49]. The sample was an α-Fe plate of 5 mm thickness under tensile testing. In-situ measurements depending on the tensile load were carried out. Since the observed macro-strain direction is perpendicular to the tensile direction, negative macro-strain was observed. Incidentally, the Bragg-edge transmission data were measured by

^{6}Li-glass scintillator pixel detector developed by Hokkaido University and KEK [50].

_{hkl}and w

_{hkl}obtained by Bragg-edge transmission imaging are reliable. In particular, for the latter (micro-strain), this means that Bragg-edge broadening analysis method can be based on the same method used for neutron diffraction. Therefore, the dislocation density analysis [51] is expected for also Bragg-edge neutron transmission method in the future.

## 6. Development of Tensor CT Algorithm for Macro-Strain Tomography

#### 6.1. Algorithm Based on the Maximum Likelihood - Expectation Maximization

_{φψ}observed along a direction φ and ψ tilted from a certain axes set, can be described by:

_{11}, ε

_{22}and ε

_{33}are normal strains along the axes 1, 2 and 3, respectively. ε

_{12}, ε

_{23}and ε

_{31}are shear strains from axis 1 to axis 2, from axis 2 to axis 3 and from axis 3 to axis 1, respectively. In this paper, the macro-strain “scalar” components means these parameters. These angle-unchangeable components have to be reconstructed individually from angle-changeable ε

_{φψ}information (projection data). Since these six components are connected by the sine/cosine angle-dependent coefficients as a function of the angles φ and ψ, ε

_{φψ}is angle-changeable. Incidentally, in case of axial-symmetric distribution, ε

_{φψ}can be described by using the circular coordinates (r,θ) for the position in the tomographic cross-section:

_{θθ}is macro-strain scalar component along the hoop direction and ε

_{rr}is macro-strain scalar component along the radial direction.

_{id}is a geometrical detection probability of the position i by the detector d. A

_{ijd}is the most important parameter for the tensor CT algorithm, a detection probability of jth component of the position i by the detector d. A

_{ijd}has a function to express the angular dependence of the observed quantity; in other words, this parameter corresponds to the angle-dependent (sine/cosine) coefficients in Equations (27) and (28). n indicates the weight of A

_{ijd}for the back-projection procedure in the ML-EM procedure. p

_{d}is the so-called projection data that is the line-integrated value of a physical quantity (crystal lattice plane spacing, in this study) for the neutron transmission path covered by the detector d. Equation (29) is almost same as the general scalar CT algorithm using ML-EM, except for A

_{ijd}. This means this method is a versatile method, based on simple algorithm with small limitations. By using this quite simple method, macro-strain tomography was demonstrated [54].

#### 6.2. Experimental

_{θθ}and ε

_{rr}were reconstructed by the ML-EM based tensor CT method and Equation (2). Incidentally, the neutron beam was transmitted along the x direction (see Figure 16) since this is a tomography experiment.

^{6}n/cm

^{2}/s. The cold-neutron wavelength resolution was about 0.35%. The collimator ratio L/D was 600 (the beam angular divergence was 1.6 mrad). The used neutron TOF-imaging detector was GEM [38] which has the pixel size of 800 μm and the detection area of 10.24 cm × 10.24 cm. The measurement time for sample beam was 16 hours and the measurement time for open beam was 16 hours. Only one direction measurement was performed because the sample was axial symmetric.

#### 6.3. Tomographic Imaging Results

_{θθ}, (b) radial macro-strain component ε

_{rr}and (c) x-direction macro-strain component made from (a) and (b), respectively. The theoretical values of (d) hoop macro-strain, (e) radial macro-strain and (f) x-direction macro-strain made from (d) and (e) are also shown in Figure 18.

_{ijd}in Equation (29) were assumed as sine/cosine coefficients of Equation (28). The weight n of Equation (29) was set in 16. Macro-strain values were calculated by using Equation (2) after the reconstruction of tomographic image of crystal lattice plane spacing d

_{111}of Al {111}.

## 7. Grain Orientation Imaging by Bragg-Dip Pattern Analysis

#### 7.1. Experimental

^{4}n/cm

^{2}/s. The cold-neutron wavelength resolution was about 0.35%. The collimator ratio L/D was 2400 (the beam angular divergence was 0.4 mrad) because the size of B

_{4}C slit at the 7 m position was 3 mm × 3 mm. The used neutron TOF-imaging detector was the nGEM detector used in J-PARC MLF BL22 “RADEN” [57] which is the first pulsed neutron imaging instrument in the world. The detector had the pixel size of 800 μm and the detection area of 10.24 cm × 10.24 cm. The measurement time for sample beam was 14.5 hours and the measurement time for open beam was 7.2 hours.

#### 7.2. Imaging Results

## 8. Conclusions

## Acknowledgements

## Conflicts of Interest

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**Figure 1.**(

**a**) Bragg-edge transmission spectrum and the included crystallographic information. The specimen is a polycrystalline α-Fe of 5 mm thickness. (

**b**) Bragg-dip transmission spectrum and the included crystallographic information. The specimen is a single-crystal α-Fe of 5 mm thickness. (Note that all the data were measured at J-PARC MLF.) Such transmission spectra are measured at each pixel of a neutron TOF-imaging detector. Therefore, through the transmission spectrum analyses at each pixel, various crystallographic information can be quantitatively mapped over the whole body of a measured specimen.

**Figure 2.**Scheme of the database matching method. (Note that exampled neutron transmission spectra were measured by the experiment presented in Reference [28].) Even if a few grains are stacked along the neutron transmission path, the unique crystal-lattice direction [UVW] of each grain, parallel to the neutron transmission direction, are simultaneously identified by this method. This figure schematically indicates that [UVW] of one of stacked grains (Grain 1) is [100 70 53] and [UVW] of the other of stacked grains (Grain 2) is [100 75 27].

**Figure 3.**Bragg-dip transmission spectra of (

**a**) Grain Orientation 1 and (

**b**) Grain Orientation 2 with the pattern matching results (cross × marks) and the indexing results. It is identified by the database matching method that [UVW] of Grain Orientation 1 is [100 30 8] and [UVW] of Grain Orientation 2 is [100 38 0]. These figures are reproduced with permission of the International Union of Crystallography from Reference [28].

**Figure 4.**Photograph of rolled/welded α-iron specimens for demonstration of quantitative texture and crystallite-size imaging [1]. Weld exists at the center of welded plates (Sample C and Sample D).

**Figure 5.**Three types of Bragg-edge transmission spectrum with the fitting curves [1]. Due to the texture effect and the extinction effect, spectrum shape and spectrum intensity are changed. From these profile changes, degree of crystallographic anisotropy, preferred orientation and crystallite size are quantitatively deduced.

**Figure 8.**Photograph of the knife specimen composed of α-Fe and γ-Fe [44]. The left-hand side is the cutting edge.

**Figure 9.**Bragg-edge transmission spectrum of the knife specimen and the RITS fitting curves assuming single-phase and double-phase [44]. The double phase assumption is suitable for reconstruction of the experimental data. This means this specimen consists of double phase (α-Fe and γ-Fe).

**Figure 10.**Quantitative imaging of (

**a**) α-Fe phase and (

**b**) γ-Fe phase and the relating images: texture evolution of (

**c**) α-Fe and (

**d**) γ-Fe and crystallite size of (

**e**) α-Fe and (

**f**) γ-Fe [44].

**Figure 11.**Photograph of three types of quenched rod [6].

**Figure 12.**{110} Bragg-edge of the unquenched center (ferrite) zone and the quenched rim (martensite) zone, with the profile fitting curves given by the single-edge analysis mode of RITS [6].

**Figure 14.**Relation between the Vickers hardness and FWHM of d-spacing distribution, discovered by Bragg-edge neutron transmission imaging [6].

**Figure 16.**Macro-strain scalar components (hoop component ε

_{θθ}and radial component ε

_{rr}) in the axial-symmetric VAMAS sample [54].

**Figure 17.**Radial dependence of the projection data evaluated by RITS, its moving averaged (smoothed) data which were actually used for the CT image reconstruction and the theoretical values of projection data from the VAMAS cylinder [54].

**Figure 18.**Macro-strain tomography obtained by the ML-EM based tensor CT algorithm: (

**a**) hoop component, (

**b**) radial component and (

**c**) x-direction component, with (

**d**–

**f**) their theoretical values [54].

**Figure 19.**Photograph of the Si-steel plate sample for demonstration of grain orientation imaging using Bragg-dip neutron transmission method [28].

**Figure 20.**Grain orientation images obtained by Bragg-dip neutron transmission analyses, expressed by inverse pole figure (IPF). (

**a**) and (

**b**) are images of all grains. (

**c**)–(

**e**) are partial images of the images (

**a**) and (

**b**). The image (

**d**) indicates IPF map of a non-stacked grain along the neutron transmission path. The image (

**c**) indicates IPF map of one of stacked grains along the neutron transmission path and the image (

**e**) indicates IPF map of the other of stacked grains along the neutron transmission path. The image (

**a**) is a combined image of the image (

**d**) and the image (

**c**) and the image (

**b**) is a combined image of the image (

**d**) and the image (

**e**). These figures are reproduced with permission of the International Union of Crystallography from Reference [28].

© 2017 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Sato, H. Deriving Quantitative Crystallographic Information from the Wavelength-Resolved Neutron Transmission Analysis Performed in Imaging Mode. *J. Imaging* **2018**, *4*, 7.
https://doi.org/10.3390/jimaging4010007

**AMA Style**

Sato H. Deriving Quantitative Crystallographic Information from the Wavelength-Resolved Neutron Transmission Analysis Performed in Imaging Mode. *Journal of Imaging*. 2018; 4(1):7.
https://doi.org/10.3390/jimaging4010007

**Chicago/Turabian Style**

Sato, Hirotaka. 2018. "Deriving Quantitative Crystallographic Information from the Wavelength-Resolved Neutron Transmission Analysis Performed in Imaging Mode" *Journal of Imaging* 4, no. 1: 7.
https://doi.org/10.3390/jimaging4010007