Improvement of Neutron Diffraction at Compact Accelerator-driven Neutron Source RANS Using Peak Profile Deconvolution and Delayed Neutron Reduction for Stress Measurements

Abstract

Neutron diffraction is a powerful non-destructive method for evaluating the microscopic structure and internal stress of metal plates as a bulk average. Precise neutron diffraction measurements with a high-intensity neutron beam have already been carried out at large-scale neutron facilities. However, it is not easy to provide users with enough experimental opportunities. We are working on upgrading the neutron diffractometer with techniques of time-of-flight to enable stress measurements at RIKEN accelerator-driven compact neutron source (RANS). To improve neutron diffraction resolution, delayed neutrons, which expand neutron beam pulse width, should be suppressed. However, it is difficult to separate the delayed neutrons experimentally. In this study, a new analysis method has been proposed to deconvolute the diffraction peak from the delayed neutron component. Moreover, a new collimator system, called decoupled collimator system, has been developed to reduce the number of delayed neutrons. The diffraction patterns from a powder sample of pure body-centered cubic iron were measured with the decoupled collimator and the diffraction peak of {211} reflection was analyzed by the new analysis method using a model function of a single exponential decay function convoluted with a Gaussian function. By this method, the decoupled collimator system has been confirmed to achieve a smaller measurement limit of lattice strain Δε than a small-aperture polyethylene collimator system and a non-collimator system. The currently available Δε was 6.7×10⁻⁴, this means that the internal stress up to 130 MPa can be well evaluated for steel materials with a Young’s modulus of 200 GPa at RANS.

1. Introduction

High-strength steels are widely used to achieve weight reduction while ensuring collision safety in the manufacturing of transport equipment such as automobiles, aircraft, and ships. On the other hand, residual stress acting on high-strength steels may lead to the delayed fracture and fatigue fracture due to hydrogen embrittlement and the reduction of fatigue strength by plastic working. To avoid such fracture and improve the structure reliability during long service, the generation process analysis and the quantitative optimization control of residual stresses are being widely investigated.

Crystallographic diffraction measurement via quantum particles such as neutrons, X-rays, and electrons is a useful method to analyze the microscopic structure of metals non-destructively and quantitatively. For example, the internal stresses of metals can be evaluated as the relative change of lattice plane spacing by measuring the wavelength and diffraction angle of quantum particles from samples. These diffraction methods can extract not only macroscopic stress but also intergranular and intragranular stresses. Therefore, it is possible to analyze the generation mechanism of macroscopic stresses from the change of the microscopic structure.^1,2) The volume fraction of constituent phases can also be determined from the relative intensity of their diffraction peaks. High-precision measurements have been performed by changing the type of quantum particle and measuring the crystallographic texture to precisely determine the volume fraction of the retained austenite phase in high-strength steels, which must well be controlled for the development of modern advanced steels.^3,4,5) Using quantum beam diffraction, we can also obtain the texture of metallic materials simply by rotating the sample or detectors, or by enclosing the sample with detectors while fixing the optical system. In situ texture measurements while increasing the temperature and deforming the sample to obtain the orientation distribution function in a short time are being developed.^6,7,8) The standard deviation of the wavelength and the diffraction angle from samples provide information of lattice defects in metal crystals. The dislocation density and crystallite size can be quantitatively evaluated by line-profile analysis, in which several factors affect the diffraction peak width.^9,10)

X-ray diffraction and electron backscattering diffraction are suitable for evaluating the surface of metal materials with μm-order depth because the transmittance of quantum beams through the metals is low. Neutron diffraction can provide bulk-averaged information on the microstructure of such metal plates with mm-order thickness, which are used for manufacturing automobile bodies, because of the high penetrability. It is important that the neutron diffraction method enable broad applications for complementary research by taking advantage of the characteristics of each quantum particle.

Large experimental facilities of neutron beam irradiation are widely used worldwide.¹¹⁾ In Japan, studies on material sciences are carried out^12,13,14,15) using various neutron diffractometers with a pulsed neutron beam at the high-intensity proton accelerator facility J-PARC¹⁶⁾ and a continuous neutron beam at the research reactor JRR-3.¹⁷⁾ Experiments on precise neutron diffraction can be performed with a high-intensity neutron beam in these large facilities. However, it is not easy to provide users with enough experimental opportunities. If neutron diffractometers with a compact neutron source become widespread, on-site neutron diffraction experiments can be performed at laboratories and factories frequently and even daily as with X-ray and electron diffraction measurement. This accelerates development of new materials and new processes in laboratories, and improvements of production methods in factories.

A compact accelerator-driven neutron source (CANS) is promising for on-site measurement with a neutron beam. Several CANSs have been developed in Japan and are utilized for research on, for example, material engineering, medical irradiation, and infrastructure maintenance.¹⁸⁾ We have also developed RIKEN compact accelerator-driven neutron source (RANS) and its utilization technology to enable its easy on-site use. RANS can generate a pulsed neutron beam with nm-order wavelength by irradiating neutron production target with a pulsed proton beam and moderating neutrons within the polyethylene moderator. The neutron beam intensity of CANSs is lower than that of the large facilities. Therefore, resolution of diffraction peaks is lower, and required measurement time is longer. These problems must be overcome to achieve precise neutron diffraction measurement. However, there is a trade-off between the resolution and the intensity of the diffraction peaks; therefore, it is difficult to improve the resolution while maintaining the intensity and performing measurement in a realistic measurement time.

To improve the resolution of the diffraction peaks, pulse width and angular dispersion of the neutron beam should be suppressed. There are three causes to expand the pulse width of the neutron beam: the proton pulse width irradiating the production target, the moderation time of the neutrons inside the moderator, and the neutron scattering from flight tube placed between the moderator to the sample. The angular dispersion of the neutron beam depends on emission position and angle of the neutrons on the moderator surface. In addition, the angular dispersion of the diffraction neutrons due to spread of scattering point inside the sample also cause to decrease the resolution. So far, we have employed a setup that introduce polyethylene collimators with a small aperture into the neutron flight path to suppress the angular dispersion of the neutrons. The reason for using the polyethylene collimators is to shield short wavelength neutrons which cause background noise not derived from diffraction and coming from all direction in the experimental hall. With the setup, the resolution of the diffraction peak has been evaluated by changing the proton pulse width and the sample width.¹⁹⁾ We have also successfully measured the volume fraction of the austenite phase and the texture via neutron diffraction with RANS for the sample size of 10 mm³ and using the proton pulse width of 30 μs. These previous studies have shown that results with precision comparable to the measurements at J-PARC can be achieved with RANS, although the required measurement time was longer.^20,21)

As next cause to decrease the resolution, devices such as reflector surrounding the moderator and the polyethylene collimator are focused on. Delayed neutrons due to scattering from these devices expand the width of the neutron beam irradiating the sample, accordingly, the resolution of the diffraction peaks becomes low. Not only the moderator and the reflector which are also discussed in the large facilities, but also the flight tubes and the collimators placed downstream of the moderator should not be ignored in the CANSs because the distance from the moderator to the sample is short. Furthermore, delayed time of the delayed neutrons is about several tens microseconds and the delayed time is comparable with the proton pulse width in RANS, therefore, it is difficult to separate the delayed neutrons with detectors experimentally.

In this study, we have developed two methods to improve the resolution by reducing the delayed neutron component for the purpose to increase measurement accuracy of the diffraction peak center. First, an analysis method has been proposed to deconvolute original diffraction peak from the delayed neutron component by defining a model function describe the delayed neutron shape. Second, a new collimator system to reduce the number of delayed neutrons has been developed. Finally, the measurement accuracy of the lattice strain has been evaluated by employing the analysis method and the new collimator system.

2. Principle

2.1. Stress Measurement by Neutron Diffraction

The lattice strain used to determine the internal stress is obtained by measuring the relative change of lattice plane spacing in crystalline materials via neutron diffraction and using from Bragg’s law,   

$$\lambda =2d_{hkl}sin{\theta }_{hkl},$$(1)

where λ is the wavelength of the neutrons, d_hkl is the lattice spacing of the atomic planes characterized by Miller indices {hkl}, and 2θ_hkl is the diffraction angle of the neutrons. When the wavelength of the neutron beam is fixed, the diffraction angle of the neutrons gives the lattice spacing; this method is called angular dispersion measurement. When the diffraction angle is fixed, the lattice spacing is extracted from the wavelength of the measured neutrons, and this method is called energy dispersion measurement. The lattice strain is directly related to the relative change of lattice plane spacing, which is calculated from the wavelength and the diffraction angle as follows:   

$$\epsilon =\frac{d-d_0}{d_0}=\frac{\lambda -{\lambda }_0}{{\lambda }_0}=-cot{\theta }_0\left(\theta -{\theta }_0\right),$$(2)

where ε is the strain, and d₀, λ₀, and θ₀ are the reference value of the lattice spacing, the wavelength, and the diffraction angle in the absence of the internal stresses, respectively. Finally, the internal stresses are calculated from the measured lattice strain by Hooke’s law.

2.2. Time-of-flight (TOF) Method to Measure Neutron Wavelength with Pulsed Neutron Beam

The nm-order wavelength suitable for the microstructure analysis of steel materials by the diffraction is given by the following de Broglie wave function:   

$$\lambda =\frac{h}{p}=\frac{ht}{mL},$$(3)

where h is Planck’s constant, p is the momentum of the neutrons, m is the neutron mass, t is the time of flight (TOF) of the neutrons from generation point to detection point, and L is the flight path length of the neutrons. The neutron wavelength is determined by measuring the time of flight t for the flight path length L. The time of flight of the neutrons can be measured by defining the start time as the generated timing of the pulsed neutron beam. Accelerator-driven neutron sources can generate pulsed neutron beams by the irradiation of neutron production targets with accelerated pulsed ion beams, in the case of RANS, the beryllium target is irradiated with protons.

2.3. Dispersion of Time of Flight of Diffracted Neutrons

The measured time of flight of neutrons with a particular wavelength has uncertainty due to several factors. For example, the pulse width of the neutron beam slightly changes the start time of the time-of-flight measurement, resulting in dispersion of the time of flight. The neutron pulse width mainly depends on the pulse width of the ion beam to generate the neutron beam and moderation time inside moderator in which the neutrons slow down to the nm-order wavelength. Variation in diffraction points inside the finite volume of the sample also affects the dispersion of the time of flight because the variation varies the actual flight path length from the nominal one.

Total dispersion σ of the diffraction peaks as a function of the time of flight is estimated by the following equation:   

$$\sigma =\sqrt{{\sigma }_{\mathrm{p}}^2+{\sigma }_{mod}^2+{\sigma }_{\mathrm{z}}^2+\dots },$$(4)

where, σ_p is the dispersion due to the pulse width of the ion beam, which is the proton beam in RANS, σ_mod is the dispersion caused by the moderation time of the neutrons, and σ_z is the dispersion from the variation in the diffraction position within the finite volume of the sample. In a previous study,²¹⁾ the resolution of the diffraction peak originating from the σ_p and the σ_z were evaluated, and the moderation time distributions were also calculated by changing thickness of the moderator for the evaluation of the σ_mod. Main causes expanding the σ_mod are the moderator and reflector around the moderator; therefore, various decoupled moderators are developed in the large facilities. However, another causes due to devices placed downstream the moderator is also not negligible in the CANSs facilities.

2.4. Delayed Neutrons Expanding Dispersion σ_mod

The time dispersion of the neutron beam incident to the sample is also caused by delay from the reflector around the moderator, and flight tubes and collimators placed downstream the moderator. The reflector plays a role of increasing the intensity of the neutron beam irradiating the sample by returning neutrons emitted in all directions around the moderator to the beam line. However, the returned neutrons from the reflector reach the sample with a delay than the neutrons directly from the moderator to the sample. Neutrons with a large emission angle from the moderators and the reflectors also arrive to sample with a delay because the actual flight path become longer than nominal one due to scattering from inner wall of the flight tubes. Polyethylene collimators with a small aperture have been installed into the flight tubes to suppress the angular dispersion of the neutrons at the RANS beam line. The reason for using the polyethylene collimators is to shield short wavelength neutrons which cause background noise not derived from diffraction and coming from all direction in the experimental hall. However, the polyethylene collimators also may cause neutrons reaching the sample late by decelerating the neutrons inside the polyethylene collimator. The neutrons that arrive to the sample with a delay than the original arrival time due to such as reason are defined as delayed neutrons.

Delayed time of such delayed neutrons in the CANS, that the distance from the moderator to the sample is short, is about several tens microseconds and the delayed time is comparable with the pulse width of the ion beam. Therefore, effect of the delayed neutrons cannot be ignored, furthermore, it is difficult to separate the delayed neutrons with detectors experimentally. In this study, we propose two methods to solve this problem; a new collimator system to reduce the number of the delayed neutrons, and a new fitting analysis method to deconvolute the Gaussian distributions including the diffraction peak information from exponential decay component, which describe the delayed neutron component.

3. Experiment

3.1. Accelerator-driven Compact Neutron Source RANS

RANS consists of three parts, a proton linear accelerator, a target station, and a measurement space. Figure 1 shows the entire system. Protons are accelerated with the accelerator to an energy of 7 MeV. Maximum intensity of the proton beam is 100 μA. Because the pulse width and frequency of the beam can be varied in the ranges from 8 μs to 200 μs and from 20 Hz to 200 Hz, respectively, these parameters can be optimized for each experiment. The proton beam irradiates a neutron production target, which is a beryllium foil with a thickness of 0.3 mm in the center of the target station. A cross-sectional view of the target station around the neutron production target is shown in Fig. 2. Neutrons with a maximum energy of 5 MeV are produced by a nuclear reaction between the beryllium and the protons. The fast neutrons are moderated in a polyethylene block moderator with a thickness of 40 mm, and thermal neutrons with wavelengths from 0.05 nm to 0.4 nm, which are suitable for the microstructure analysis of steel materials, are introduced into the measurement space. Figure 3 shows the neutron spectrum from the moderator at the position 5 m from the moderator simulated by the PHITS code.²²⁾

Fig. 1.

Photo of entire RANS system. (Online version in color.)

Fig. 2.

Cross-sectional view of target station of RANS. (Online version in color.)

Fig. 3.

Neutron energy spectrum at position 5 m from production target simulated by PHITS code [22]. (Online version in color.)

3.2. Beam Line and Detector Setup of Time-of-flight Diffraction Experiment in RANS

The beam line setup from the moderator to the detector is shown in Fig. 4. The thermal neutrons pass through the internal tunnel in the target station with an aperture size of 160×160 mm² and a length of 1.465 m. Borated-polyethylene flight tubes with an aperture size of 250×250 mm² and a total length of 3 m were placed downstream of the exit of the target station. The neutrons pass through the flight tubes and arrive to the sample. The beam spot size was formed into a rectangle of 20 mm length in the horizontal direction and 45 mm length in the vertical direction at the sample position by a slit made of B₄C ceramic installed between the flight tubes and the sample. The diffracted neutrons were detected by a position-sensitive detector, called PSD for short, mounted at a distance of 305 mm from the sample at an angle of 90° from the beam axis. The PSD had a structure of eight proportional counters arranged in the vertical direction. The size of one proportional counter tube was 600 mm length in the horizontal direction and 12.7 mm diameter. The maximum acceptance for diffraction angle 2θ with the PSD was from 45° to 135°. The proportional counter tubes were filled with ³He gas at 10 standard atmospheres. The position and time resolution of the detectors were about 10 mm and μs-order, respectively. The total flight path length was 5644 mm.

Fig. 4.

Beam line setup from moderator to detectors. (Online version in color.)

3.3. Collimator Setup to Reduce Delayed Neutrons

To maximize the beam intensity, the apertures of the tunnel and the flight tubes should be as large as the moderator surface. On the other hand, the large aperture increases the number of the delayed neutrons scattering from the inner wall of the tunnel and the flight tubes. So far in the RANS beam line, the small aperture polyethylene collimators were installed to the flight tube and neutron diffraction measurement were performed with using the neutron beam whose actual flight path does not change significantly from nominal value. Because the polyethylene is used for the collimator to shield short wavelength neutrons from the production target and to avoid background noise in the experimental hall, the neutrons with the short wavelengths being moderated in the polyethylene might arrive the sample as the delayed neutrons.

A new collimator system, called decoupled collimator system, has been developed to suppress the delayed neutrons. Figure 5(a) shows a schematic view of the decoupled collimator system. Two components played a role to reduce the delayed neutrons. One is a neutron shield of a B₄C rubber sheet on the inner wall of the tunnel of the target station. As the inner shield, two collimators, with a 3.5 mm thick B₄C rubber sheet on the inner wall, were installed in the tunnel of the target station. One collimator with an aperture of 140 mm diameter and 500 mm length was installed on the upstream side, and the other collimator with an aperture of 120 mm diameter and 500 mm length was installed on the downstream side. The inner shield of the B₄C rubber sheet suppressed the number of delayed neutrons from the inner wall of the tunnel. However, the delayed neutrons from the inner wall of the flight tubes still reached the sample. Thus, a slit was installed at the end of the flight tubes to reduce the number of remaining delayed neutrons from the inner wall of the flight tubes. The slit was a 15-mm-thickness B₄C lubber sheet with an aperture of 50 mm×50 mm. Finally, the decoupled collimator system can select the neutrons emitted from the moderator surface and suppress the angular spread and the time dispersion of the neutron beam incident to the sample. The angular spread was about 1°.

Fig. 5.

Schematic view of collimator setup. (a) Decoupled collimator. (b) Without collimator. (c) Polyethylene collimator. (Online version in color.)

To evaluate the decoupled collimator system, diffraction measurements with two other collimator systems were also performed. Figures 5(b) and 5(c) show the collimator system. Figure 5(b) shows the system with no collimator between the moderator and the B₄C ceramic slit. The number of the delayed neutrons from the inner wall increase although the total number of neutrons irradiating the sample becomes larger because of the effect of the reflector. Figure 5(c) shows the case of the polyethylene collimators with a small aperture installed in the flight tubes used in RANS so far. Although the intensity of the neutrons irradiated to the sample decreases, the delayed neutrons, which have long actual flight path due to scattering from the inner wall of the tunnel and the flight tubes, is suppressed even more than that by the decoupled collimator. However, several delayed neutrons moderated by the polyethylene collimators may reach the sample. To suppress the number of the delayed neutrons, cadmium sheets with a length of 725 mm were installed on the inner wall of the polyethylene collimator on the most downstream side. One polyethylene collimator with an aperture of 50 mm×50 mm and a length of 1550 mm was installed on the upstream side, and the other polyethylene collimator with an aperture of 35 mm × 35 mm and a length of 1450 mm was installed on the downstream side.

The summary of the aperture size of each collimator system is shown in Table 1. The slit size of the slit made of B₄C ceramic installed between the flight tubes and the sample to form the beam spot size was fixed to the size of 10 mm length in the horizontal direction and 32 mm length in the vertical direction. The role of the ceramic slit is to change the beam spot size according to the sample size. Therefore, the ceramic slit was not used to cut the delayed neutrons in this study.

Table 1. Aperture size at exit for each part with each collimator system.

	Aperture size of exit
	Tunnel of target station	Flight tube 1	Flight tube 2
Decouple collimator	120 mm diameter	250 × 250 mm2	50 × 50 mm2
Without collimator	160 × 160 mm2	250 × 250 mm2	250 × 250 mm2
Polyethylene collimator	160 × 160 mm2	50 × 50 mm2	35 × 35 mm2

3.4. Samples and Measurement Time

Diffraction from a powder sample of pure body-centered cubic iron (b.c.c. iron) was measured to evaluate the decoupled collimator. The powder sample was filled in a sample cell made of vanadium film. The cell was cylindrical with a diameter of 8 mm and a height of 45 mm. Background noise, which is not derived from diffraction and come around the sample and the detector from all direction in the experimental hall, should be subtracted from the powder sample diffraction data. The background noise was obtained by measuring neutrons with a setting placing a blank cell at the sample position. The measurement times of the b.c.c. iron sample and the blank cell were both 30 min. The diffraction intensity of the b.c.c. iron was extracted by normalizing the time-of-flight distribution of the diffraction neutrons with a time-of-flight distribution of the neutron beam. The distribution of the neutron beam was obtained by measuring a time-of-flight distribution of the incoherent scattering neutrons from a vanadium metal. The vanadium metal sample was a cylinder with a diameter of 15 mm and a height of 35 mm. The measurement times of the vanadium metal sample and a background measurement for the vanadium sample data with a setting not placing any samples were both 60 min.

4. Results

Figure 6 shows the time-of-flight diffraction spectra from the powder sample of the b.c.c. iron in the diffraction angle range 2θ from 64° to 116°, which corresponds to a distance of ± 150 mm from the center of the detector in the horizontal direction. The spectra have already been after subtracting the background noises. The flight path length from the sample to the detector and the diffraction angle change depending on the detected position of the diffracted neutrons. These differences were corrected by a time focusing method.²³⁾ Neutron yields, obtained by integrating the diffraction peak from the {211} reflection between 2.7 ms and 3.3 ms, were compared for each collimator system. The yield of neutrons with the decoupled collimator system was 1.6 times larger than that for the polyethylene collimator system but 0.57 times that in the no-collimator system.

Fig. 6.

Diffraction spectra of powder sample of b.c.c. iron as a function of the time of flight obtained in the diffraction angle range 2θ from 64° to 116°. (Online version in color.)

Figure 7(a) shows the time-of-flight distributions of the diffraction intensity extracted by normalizing with the time-of-flight distributions of the neutron beam obtained by measuring the incoherent scattering from the vanadium metal. In the time-of-flight distributions of the {211} reflection peak shown in Fig. 7(b), the diffraction peaks have an asymmetric shape characterized in the time-of-flight measurement.²⁴⁾ The reason for this is that the delayed neutrons make a tail structure at side of slower in time. In the case of the no collimator, the delayed neutron component increased, and the height of the diffraction peak was lower than another collimator setup.

Fig. 7.

(a) Diffraction spectra of powder sample of b.c.c. iron normalized by TOF distributions of neutron beam intensity obtained by measuring incoherent scattering from vanadium metal. (b) Diffraction peak of {211} reflection. (Online version in color.)

5. Discussion

First, a new analysis method to deconvolute original diffraction peak by defining a model function describe the delayed neutron component is discussed. To take into account the asymmetric shape due to the delayed neutrons, a model function obtained from exponential functions convoluted with a symmetric peak function such as a Gaussian function is generally used for peak fitting analysis.^24,25,26) In this study, a model function F(t) obtained from a single exponential decay function convoluted with a Gaussian function is used.   

$$\begin{array}{l}F\left(t\right)={\int }_{-\infty }^{+\infty }f\left(t^{\prime }\right)g\left(t-t^{\prime }\right)d{t}^{\prime }\\ \hspace{1em}\hspace{1em}=Aexp\left(-\beta t\right)\mathrm{e}\mathrm{r}\mathrm{f}\mathrm{c}\left(\frac{-\left(t-\left(t_c+{\sigma }^2\beta \right)\right)}{\sqrt{2}\sigma }\right)+C \end{array}$$(5.1)

  

$$f\left(t\right)=\{\begin{array}{c}0:t<0\\ \beta exp\left(-\beta t\right):t\ge 0\end{array}$$(5.2)

  

$$g\left(t\right)=\frac{I}{\sqrt{2\pi }\sigma }exp\left(\frac{-{\left(t-t_c\right)}^2}{2{\sigma }^2}\right)$$(5.3)

  

$$A=\frac{I\beta }{2}\text{exp}\left(\frac{{\sigma }^2{\beta }^2}{2}+\beta t_c\right)$$(5.4)

Where, β is the time constant of the exponential decay function, σ is the standard deviation of the Gaussian function, t_c is the peak centroid of the Gaussian function, I is the normalization factor and C is the offset constant to describe the background noise distribution. F(t) can be separated into the exponential part and the complementary error function part. The exponential part describes the delayed neutron component by using the parameter β. The complementary error function part, which is the cumulative distribution function of the Gaussian function, includes information of the diffraction peak. Therefor the diffraction peak information can be separated from the delayed neutrons component by the fitting analysis using the function F(t). In this study, the function h(t), which is a differential function of the complementary error function, was used to determine the lattice spacing t_c by extracting component forming the diffraction peak from β describing the delayed neutrons component.   

$$h\left(t\right)=\frac{I}{\sqrt{2\pi }\sigma }exp\left(\frac{-{\left(t-{t^{\prime }}_c\right)}^2}{2{\sigma }^2}\right)$$(6.1)

  

$${t^{\prime }}_c= t_c+{\sigma }^2\beta$$(6.2)

Figure 8 shows result of the fitting to the diffraction peak of the {211} reflection measured with the decoupled collimator. The red line is the result of fitting the measured data with F(t) and the blue dotted line is the distribution of h(t). The parameters and the estimation errors determined by the fitting are given in Table 2.

Fig. 8.

Results of fitting to diffraction peak of {211} reflection measured with decoupled collimator. The red line is the fitting to the measured data with F(t) and the blue dotted line is the distribution of h(t)+C. (Online version in color.)

Table 2. Parameters and estimation errors of fitting result in Fig. 8.

tc±Δtc [ms]	2.856±0.0019
σ±Δσ [ms]	4.1±0.14×10−2
β±Δβ [ms−1]	1.28±0.055×101

Next, the decouple collimator system to reduce the delayed neutrons is evaluated by the fitting parameters with the deconvolution method while comparing to other collimator systems. To evaluate the delayed neutrons component, half-life T_1/2 calculated from the time constant with the formula T_1/2=ln(2)/β is discussed. Figure 9 shows the half-life T_1/2 and the standard deviation σ of the Gaussian function. The half-life parameter for the decoupled collimator was smaller than that for the other cases. This means that the delayed neutrons component was suppressed by the decoupled collimator. As described in Section 3.3, the delayed neutrons scattered from the inner wall of the tunnel of the target station and the flight tubes were suppressed by the B₄C rubber sheet of the decoupled collimator. The half-life parameter for the polyethylene collimator was larger than that for the decoupled collimator, although the angular spread of the polyethylene collimator should be smaller than that for the decoupled collimator. The reason for this is assumed to be that the delayed neutrons moderated by the polyethylene of the collimator have an impact on the delayed component, thus increasing the time dispersion. Note that the half-life parameter itself does not describe the decay time of the diffraction peak distribution directly. However, the half-life values roughly agree with the time dispersion of about 10 μs derived from the larger flight path length than the nominal value due to the scattering from the inner wall, and the time dispersion of about several tens of μs-order time of the moderation time in the cm-order thickness polyethylene.²¹⁾

Fig. 9.

Difference between each collimator system of half-life parameter T_1/2 calculated from time constant β of exponential decay function and standard deviation σ of Gaussian function. (Online version in color.)

The standard deviation σ of the Gaussian function did not significantly depend on the collimator system. Because the standard deviation of the Gaussian distribution, which is separate from the decay component, depends only on factors other than the collimator system, such as the pulse width of the proton beam and the sample size, the difference in the standard deviation is not expected to be large. Therefore, the result suggests that a diffraction peak component independent of the structure between the moderator and the sample was successfully extracted by being separated from the decay component obtained by using the exponential decay function.

Figure 10 shows the result of the Gaussian distributions h(t) obtained by fitting to the diffraction peak of the {211} reflection. Peak height of the Gaussian distributions without collimator was lower than the other collimator setting. The reason for this is because the intensity of the diffraction peak normalized by the time-of-flight distribution of the incoherent scattering from the vanadium metal was underestimated, as shown Fig. 7(b), due to delayed neutrons with the short wavelength, which are not derived from diffraction from the {211} lattice plane although arrive at the same timing with the neutrons diffracted from the {211} lattice plane. Figure 11 shows the schematic view of the time-of-flight distribution of the diffraction and the incoherent scattering with and without the delayed neutrons which have the short-wavelength, to describe the reason why the height of the peak without collimator was lower than the other collimator setting, as shown Figs. 7(b) and 10. In the ideal case of no delayed neutrons, the intensity of the diffraction peak is obtained by taking ratio of value D, which is integration of the diffraction peak, to value A, which is integration of the neutrons contributing the {211} reflection. However, in the actual case, value B, which is integration of the neutrons not contributing the {211} reflection, is piled up in both time-of-flight distributions, as shown in Fig. 11(b). Therefore, the intensity of the diffraction is evaluated as R′ = (B+D)/(B+A). When R > 1, that mean D > A, the intensity of the diffraction in the actual case becomes R′ < R, it means the intensity of the diffraction peak is underestimated.

Fig. 10.

Gaussian distributions h(t) obtained by fitting to diffraction peak of {211} reflection. (Online version in color.)

Fig. 11.

The reason why pulse height of the peak without collimator was lower than the other collimator setting, as shown Figs. 7(b) and 10. The schematic view of the TOF distribution of diffraction and the incoherent scattering with and without the delayed neutrons of the short-wavelength neutrons. (Online version in color.)

Finally, the measurement limit of the lattice strain is evaluated by employing the analysis method and the new collimator system. Because the lattice strain is measured as the peak shift of the diffraction peak, the measurement limit of the lattice strain Δε can be defined by the following equation using the estimation error of the center of the diffraction peak,   

$$\mathrm{\Delta }\epsilon =\frac{\mathrm{\Delta }{t}_{\text{c}}}{t_{\text{c}}}$$(7)

where Δt_c is the estimation error of the center of the diffraction peak t_c by least squares fitting analysis. Figure 12 shows a comparison of t_c, and Δt_c for each collimator system. t_c did not significantly depend on the collimator system, therefore, Δt_c should be small to make Δε smaller. Δt_c depend on the measurement accuracy of the diffraction peak. In this study, the measurement accuracy includes only statistical error, therefore, Δt_c became smaller by being the smaller statistical error. The reason why Δε without the collimator was more accurate than that with polyethylene collimator may be because the neutron yield was larger, and the statistical error was smaller. In addition, other fitting parameters, the I, the β, and the σ correlate in determining the t_c, and these parameters affect deconvolution accuracy. The higher peak intensity, the smaller peak width and the smaller delayed neurons component improve the deconvolution accuracy, then the smaller Δt_c is obtained. The decouple collimator system reduce the number of the delayed neutrons and statistical error while being the higher peak height of Gaussian distribution h(t). Thus, Δt_c for the decoupled collimator was the smallest.

Fig. 12.

Difference from collimator system of center of the diffraction peak t_c, and estimation error Δt_c.

In conclusion, the decoupled collimator system gives the smallest measurement limit of the lattice strain as shown in Fig. 13. The measurement limit of the lattice strain with the decoupled collimator Δε was 6.7 × 10⁻⁴. This means that stress of up to 130 MPa can be measured for steel materials with a Young’s modulus of 200 GPa using RANS.

Fig. 13.

Difference from collimator system of measurement limit of the lattice strain Δε.

6. Conclusion

The new analysis method to deconvolute the diffraction peak from the delayed neutrons component and the new collimator system, called the decouple collimator system, to reduce the number of the delayed neutrons have been developed to achieve on-site precise stress measurement with the accelerator-driven compact neutron source RANS. The exponential decay function convoluted with the Gaussian function have been employed as the model function to deconvolute and determine the center of the diffraction peak. The concept of the decoupled collimator was to suppress the number of the delayed neutrons due to scattering from the inner wall of the tunnel of the target station and the flight tubes placed downstream of the target station.

The diffraction pattern from the powder sample of b.c.c. iron was decomposed by the model function and the results of the fitting parameters were compared for three collimator systems: the decoupled collimator system, the polyethylene collimator system with the small aperture, and the no-collimator system. As a result, the decoupled collimator system has achieved the smallest measurement limit of the lattice strain. The reason for this was because the number of the delayed neutrons was reduced, and the statistical error was small while being the higher peak height of the deconvoluted original diffraction peak. The measurement limit of the lattice strain with the decoupled collimator Δε was 6.7 × 10⁻⁴. This means that the internal stress up to 130 MPa can be measured for steel materials with a Young’s modulus of 200 GPa with RANS.

We are planning to measure samples with well-known stresses with RANS. To further improve the accuracy of the stress measurement, it is important more development of a beam line component to suppress the number of the delayed neutrons. Optimization of the decouple collimator including by introducing a decoupled moderator is ongoing by exploiting the advantage that the beam line component of a compact neutron source such as RANS can be change easily. In addition, a study to determine a standard model function and parameters describing the dispersion of the diffraction peak for RANS more precisely is also planned. The delayed neutrons may occur in the large facility depend on the device between the production target and the sample. Therefore the development of the precise deconvolution method will be widely applicable to improve stress measurement accuracy.

References

• 1)   M. E.  Fitzpatrick and  A.  Lodini: Analysis of Residual Stress by Diffraction using Neutron and Synchrotron Radiation, Taylor & Francis, London, (2003), 60. https://doi.org/10.1201/9780203608999
• 2)   M. T.  Hutchings,  P. J.  Withers,  T. M.  Holden and  T.  Lorentzen: Introduction to the Characterization of Residual Stress by Neutron Diffraction, Taylor and Francis, Boca Raton, (2005), 222. https://doi.org/10.1201/9780203402818
• 3)   P. J.  Jacques,  S.  Allain,  O.  Bouaziz,  A.  De,  A. F.  Gourgues,  B. M.  Hance,  Y.  Houbaert,  J.  Huang,  A.  Iza-Mendia,  S. E.  Kruger,  M.  Radu,  L.  Samek,  J.  Speer,  L.  Zhao and  S.  van der Zwaag: Mater. Sci. Technol., 25 (2009), 567. https://doi.org/10.1179/174328408X353723
• 4)   Y.  Tomota,  N.  Sekido,  P. G.  Xu,  T.  Kawasaki,  S.  Harjo,  M.  Tanaka,  T.  Shinohara,  Y.  Su and  A.  Taniyama: Tetsu-to-Hagané, 103 (2017), 570 (in Japanese). https://doi.org/10.2355/tetsutohagane.TETSU-2017-045
• 5)   P. G.  Xu,  Y.  Tomota,  Y.  Arakaki,  S.  Harjo and  H.  Sueyoshi: Mater. Charact., 127 (2017), 104. https://doi.org/10.1016/j.matchar.2017.02.028
• 6)   T.  Tomida,  M.  Wakita,  M.  Yasuyama,  S.  Sugaya,  Y.  Tomota and  S. C.  Vogel: Acta Mater., 61 (2013), 2828. https://doi.org/10.1016/j.actamat.2013.01.015
• 7)   I.  Lischewski and  G.  Gottstein: Acta Mater., 59 (2011), 1530. https://doi.org/10.1016/j.actamat.2010.11.017
• 8)   Y.  Onuki,  A.  Hoshikawa,  S.  Sato,  P. G.  Xu,  T.  Ishigaki,  Y.  Saito,  H.  Todoroki and  M.  Hayashi: J. Appl. Crystallogr., 49 (2016), 1579. https://doi.org/10.1107/S160057671601164X
• 9)   S.  Sato,  A.  Kuroda,  K.  Satoh,  M.  Kumagai,  S.  Harjo,  Y.  Tomota,  Y.  Saito,  H.  Todoroki,  Y.  Onuki and  S.  Suzuki: Tetsu-to-Hagané, 104 (2018), 201 (in Japanese). https://doi.org/10.2355/tetsutohagane.TETSU-2017-082
• 10)   Y.  Tomota,  S.  Sato and  S.  Harjo: Tetsu-to-Hagané, 103 (2017), 73 (in Japanese). https://doi.org/10.2355/tetsutohagane.TETSU-2016-085
• 11)   D. N.  Argyriou and  A. J.  Allen: J. Appl. Crystallogr., 51 (2018), 567. https://doi.org/10.1107/S1600576718007987
• 12)   K.  Nakajima,  Y.  Kawakita,  S.  Itoh,  J.  Abe,  K.  Aizawa,  H.  Aoki,  H.  Endo,  M.  Fujita,  K.  Funakoshi,  W.  Gong,  M.  Harada,  S.  Harjo,  T.  Hattori,  M.  Hino,  T.  Honda,  A.  Hoshikawa,  K.  Ikeda,  T.  Ino, et al.: Quantum Beam Sci., 1 (2017), 9. https://doi.org/10.3390/qubs1030009
• 13)   Y.  Kubota,  J.  Kubo,  K.  Ishida,  A.  Okada,  M.  Yoshida,  T.  Saito and  H.  Suzuki: SAE Int. J. Engines, 5 (2012), 446. https://doi.org/10.4271/2012-01-0549
• 14)   P. G.  Xu,  K.  Akita,  H.  Suzuki,  N.  Metoki and  A.  Moriai: Mater. Trans., 53 (2012), 1831. https://doi.org/10.2320/matertrans.MA201203
• 15)   H.  Suzuki,  J.  Katsuyama and  Y.  Morii: J. Solid Mech. Mater. Eng., 6 (2012), 574. https://doi.org/10.1299/jmmp.6.574
• 16)   F.  Maekawa,  M.  Harada,  K.  Oikawa,  M.  Teshigawara,  T.  Kai,  S.  Meigo,  M.  Ooi,  S.  Sakamoto,  H.  Takada,  M.  Futakawa,  T.  Kato,  Y.  Ikeda,  N.  Watanabe,  T.  Kamiyama,  S.  Torii,  R.  Kajimoto and  M.  Nakamura: Nucl. Instrum. Methods Phys. Res. Sect. A, 620 (2010), 159. https://doi.org/10.1016/j.nima.2010.04.020
• 17)   H.  Suzuki,  J.  Katsuyama,  T.  Tobita and  Y.  Morii: Q. J. Jpn. Weld. Soc., 29 (2011), 294 (in Japanese). https://doi.org/10.2207/qjjws.29.294
• 18)  Japan Collaboration on Accelerator-driven Neutron Sources: Website of JCANS, http://www.jcans.net/index-j.html, (accessed 2021-08-30).
• 19)   Y.  Ikeda,  A.  Taketani,  M.  Takamura,  H.  Sunaga,  M.  Kumagai,  Y.  Oba,  Y.  Otake and  H.  Suzuki: Nucl. Instrum. Methods Phys. Res. Sect. A, 833 (2016), 61. https://doi.org/10.1016/j.nima.2016.06.127
• 20)   P. G.  Xu,  Y.  Ikeda,  T.  Hakoyama,  M.  Takamura,  Y.  Otake and  H.  Suzuki: J. Appl. Crystallogr., 53 (2020), 444. https://doi.org/10.1107/S1600576720002551
• 21)   Y.  Ikeda,  M.  Takamura,  T.  Hakoyama,  Y.  Otake,  M.  Kumagai and  H.  Suzuki: Tetsu-to-Hagané, 104 (2018), 138 (in Japanese). https://doi.org/10.2355/tetsutohagane.TETSU-2017-080
• 22)   T.  Sato,  K.  Niita,  N.  Matsuda,  S.  Hashimoto,  Y.  Iwamoto,  S.  Noda,  T.  Ogawa,  H.  Iwase,  H.  Nakashima,  T.  Fukahori,  K.  Okumura,  T.  Kai,  S.  Chiba,  T.  Furuta and  L.  Sihver: J. Nucl. Sci. Technol., 50 (2013), 913. https://doi.org/10.1080/00223131.2013.814553
• 23)   J. D.  Jorgensen,  J.  Faber, Jnr,  J. M.  Carpenter,  R. K.  Crawford,  J. R.  Haumann,  R. L.  Hitterman,  R.  Kleb,  G. E.  Ostrowski,  F. J.  Rotella and  T. G.  Worlton: J. Appl. Crystallogr., 22 (1989), 321. https://doi.org/10.1107/S002188988900289X
• 24)   M. T.  Hutchings,  P. J.  Withers,  T. M.  Holden and  T.  Lorentzen: Introduction to the Characterization of Residual Stress by Neutron Diffraction, Taylor and Francis, Boca Raton, (2005), 161. https://doi.org/10.1201/9780203402818
• 25)   R. B.  Von Dreele,  J. D.  Jorgensen and  C. G.  Windsor: J. Appl. Crystallogr., 15 (1982), 581. https://doi.org/10.1107/S0021889882012722
• 26)   J. M.  Carpenter,  R. A.  Robinson,  A. D.  Taylor and  D. J.  Picton: Nucl. Instrum. Methods Phys. Res. Sect. A, 234 (1985), 542. https://doi.org/10.1016/0168-9002(85)91005-8