1. Introduction

Both fast neutron radiography (FNR) and thermal neutron radiography (TNR) are important non-destructive testing techniques.^[1-3] These 2 methods have similar mechanisms and complementary effects; however, compared to TNR, the neutron in the FNR method has higher penetrability, and can better detect light materials in thick heavy metals, which extends the application of neutron radiography. Presently, there exists a widely recognized international non-destructive testing standard, ASTME545-86, for TNR; however, no quantitative analytical standard has been established for the detection ability of FNR due to its scattering effect and low detection efficiency.^[4-6] Generally speaking, the FNR technique is not as mature or developed as the TNR technique, which makes it crucial to develop new analysis approaches and criterion for FNR testing.

In recent years, international research has mainly focused on the construction and optimization of large-scale imaging systems,^[7-9] the development of conversion screens, ^[10-12] parameter simulations,^[13-15] and preliminary applications.^[16-18] In terms of spatial resolution, FNR uses the KLASENS method to determine the modulation transfer function (MTF) of the system. In terms of contrast sensitivity, FNR uses tiered samples (made from the same materials, but with differing levels of thickness) for preliminary quantitative analysis.^[19-21] Researchers have not yet solved an inherent problem in FNR's contrast sensitivity, i.e., the longitudinal detection limit of FNR when a fixed-thickness sample made from a certain material is irradiated by neutrons with the same energy and fixed intensity. As such, in this study, we conducted an experiment on the contrast sensitivity of 14 MeV FNR generated by the 600kV Cockcroft-Walton accelerator in the Key Laboratory of Nuclear Data of the China Institute of Atomic Energy.

2. Modeling and Analysis

The contrast sensitivity of an FNR system is defined as the minimum discernible variance in the thickness of a fixed-thickness sample along the incidence direction of the detection beam. Contrast sensitivity serves as an index for judging a detection system’s capacity for identifying the longitudinal thickness of a sample. If the thickness variance of a sample is less than the contrast sensitivity of the system, the sample is deemed undetectable. If the variance is equivalent to the contrast sensitivity, it is necessary to consider the comprehensive influence of irradiation time and scattering on the statistical fluctuation of the image contrast. Contrast sensitivity is primarily a TNR technical index, where its expression can be modified according to the characteristics of FNR applications.

Figure 1 shows the schematic of FNR data conversion. In Figure 1, I₀ represents the intensity of incident collimated neutron beams, x is the thickness of the sample, I is the intensity of the neutron beam transmitted through the sample, the yellow rectangle represents a thin scintillator as a convertor, Q is the number intensity of fluorescence photons converted from the neutron beam by the convertor, and P stands for the pixel value of the image on the camera. When passing through the sample, the intensity of the neutron beam I has 2 parts: directly-attenuated neutrons and scattered neutrons, where the latter depends on the geometry and material function of the sample. Therefore, I can be expressed as:

Figure 1:Full-screen View

Schematic of the data conversion of fast neutron radiography (FNR).

where µ is the macroscopic cross-section, µ_s is the material cross-section, and S(x, y, z, µ_s) is the geometry and material function of the sample.

The fast neutrons continue to react with the converter. In the experiment, we used a thin BC400 plastic scintillator plate (consisting of carbon and hydrogen) as the convertor. The reactions between fast neutrons and carbon (or hydrogen nuclei) generate recoil nucleons and protons.

According to energy and momentum conservation principles, the equation of recoil nucleus energy E_A can be expressed as:

where E represents the incident neutron energy, θ isthe angle between the recoil nucleon and the incident neutron, A is the mass number of the recoil nucleon, and $K_{\text{b}}=\frac{4A}{（1+A{）}^2}$. E_A represents the energy of a recoil carbon nucleon if A=12, and E_Ais a recoil proton if A=1. The ratio of K_b between a recoil carbon nucleon and a recoil proton is 1:3.57. Therefore, the energy of a recoil proton is much larger than that of a recoil carbon nucleon under the same initial conditions. Even for the same energy, a recoil proton generates more photon yield than a recoil carbon nucleon because of different reaction cross-sections and ionization energy. Accordingly, the reaction of fast neutrons with a scintillator BC400 can be simplified as an interaction between neutrons and protons.

During the reaction process, the recoil protons impact the BC400 scintillator and generate fluorescence photons, whose intensity Q is:

Where ε(E) is the energy deposition function of the recoil proton energy (E) in the BC400 scintillator, σ_BC is the reaction cross-section of the fast neutrons with hydrogen in the BC400 scintillator, m is the inherent nucleon density of the BC400 scintillator, n is the coefficient used to compensate for the recoil effect of carbon nucleons on photon intensity Q.

During the conversion process, we regarded each spot on the convertor that generated photons as a point of light source that illuminated at a full 4π solid angle. Blocked by the converter and influenced by the transmission path, the camera only recorded photons entering the camera lens at a very small solid angle. Ultimately, the fluorescence photons passed through the air, were reflected by the mirror, and transmitted through the lens before being recorded by the CCD, which inevitably led to some attenuation. Considering all of these factors, we chose a parameter L to represent the equivalent distance from the light source to the camera, and coefficients K₁, K₂,and K₃to take into account airborne transmission efficiency, the reflection efficiency of the mirror, and the transmittance capacity of the lens, respectively. The unit-time pixel value P/T detected by the CDD camera can be expressed as:

where $\vartheta \left(e\right)$represents a positive correlation function between the pixel value P and the photon energy e, h stands for the Planck constant, v stands for the photon frequency, and T is the irradiation time.

If the imaging system is free of stray light and the camera works stably under electrical refrigeration, the background image B(w,v) with no neutron beam depends solely on the pixel coordinates w and v, and is constant with respect to time. By combining Equations (2) and (4), the pixel value P is:

where $\lambda =\frac{\epsilon \left(E\right)\vartheta \left(e\right)hv{\sigma }_{BC}mn}{4\pi L^2}{K}_1{K}_2{K}_3$is a parameter related to the entire testing and imaging system.

In our experiment, we only considered variances in thickness Δx along the direction of the neutron beam. By differentiating both sides of Equation (5), we obtained

By setting ΔP as the minimum signal in the pixel value that can be detected by the computer system after digital statistics (which is related to the fluctuation of the system’s counting statistics) we were able to calculate the minimum variance in thickness Δx as:

If the scattering term $\dot{S}\left(x,y,z,{\mu }_s\right)$ had a minor effect on the pixel value, then we were able to simplify Equation (7) as:

As shown in Equation (8), when the influence of scattering could be ignored (i.e., if the scattering was too insignificant to be considered), the contrast sensitivity of the system, (the minimum discernible thickness (Δx)) was inversely proportional to the irradiation time T, the macroscopic cross-section μ of the sample, and the intensity of the neutron source I₀. Moreover, it had a positive correlation to the minimum discernible signal of the pixel value ΔP.

3. Experimental Study

Figure 2(a) depicts the schematic diagram of the relative positions of the sample and experimental facility. We generated a 14.1 MeV fast neutron source with a flux of ~1.5×10¹⁰ n/s and an energy of 14.1 MeV using a 600kV high voltage multiplier. The neutron source emitted into the collimator from the large inlet at a 4πsolid angle, and incidents perpendicularly on the sample in a plane wave after passing through the collimator. We used a BC400 plastic scintillator plate with a size of 200 mm×200 mm×10 mm, a density of 1.032 g/cm3, and an H:C~1.103:1as the convertor. The collimator we used in the experiment was a composite structure consisting of lead, polyethylene, stainless steel, and red copper. It had a length of L=147 cm, with a maximum diameter at the outlet of D=8 cm, and a minimum diameter inside of d=5.14 cm. Based on the results of the experiment and the MC simulation, the flux ratios between the neutrons and the γ-ray near the collimator inlet (on the neutron source side) and the outlet were ~1:2.81*10^-3 and ~1:5.68*10^-4, respectively. The ratio between the neutrons and the γ flux at the collimator outlet was approximately 25:1. Based on the experimental and simulation data, we determined that the X-ray and the γ-ray contribution to the dose rate on the BC400 scintillator was relatively low when compared to that of the neutrons. Futhermore, based on the simulation, the average conversion efficiency of the X-ray and the γ-ray with a 1 cm thickness on the BC400 scintillator was only 0.2%; whereas the neutron efficiency was approximately 2.2%, which was much higher than the gamma efficiency. The peak center of the BC400 scintillation light spectrum was located in the UV range (maximum wavelength 423 nm), which could barely be recorded by the CCD camera. In short, the experimental images’ contribution to the pixel values from the X-ray and the γ-ray were relatively small compared to that of the neutrons.

Figure 2:Full-screen View

Schematic diagram of the FNR experiment (a) a photo of the lead sample; (b) a photo of the polyethylene sample (c)

The camera was a 1024×1024-pixel high-sensitivity scientific CCD. Each pixel corresponded to a 0.084 mm×0.084 mm area on the converter. The sample was located 2 cm from the outlet of the collimator. The system collimation ratio was 294, and the geometric dullness of the system Ug was ~0.0029 cm. We processed all of the images in the experiment after dark field deduction and flat field analysis had occurred. The testing times of the dark field and flat field were equivalent to the exposure time of the experiment images.

Based on the relationship between various parameters in the sample system and the pixel value of the image as stated in Equation (5), we used 2 types of samples consisting of different materials and varied thickness. We then used different irradiation times for the 2 types of samples in the experiments. We chose lead and polyethylene as the materials for the 2 samples, because the former (lead) has a weak scattering effect of fast neutrons, while the scattering effect of the latter (polyethylene) is strong. Both samples were arranged in the shape of a triangular prism with a 5 cm×10 cm×5√5 cm right triangle as the base, and a thickness of 5 cm, as shown in Figures 2(b) and 2(c), which also illustrate the direction of the transmitted neutrons.

Our analysis consisted of 3 steps: (1) First, we compared the influence of the different sample materials on the image pixel values; (2) Second, we derived the relationship between the irradiation time and the image pixel values, and (3) Third, we determined how the variations of sample thickness affected the pixel values for a certain material. Once we determined all 3 relationships, we were able to establish the corresponding discriminant of the system contrast sensitivity.

3.1 Experimental Study on Influential Factors in Pixel Values

Figure 3(a)-(c) shows the fluorescence images of the lead sample’s irradiation times at 1200 s and 1800 s, and the irradiation time of the polyethylene sample at 1800 s, respectively. The change of thickness in the images is presented in a horizontal direction. We subtracted the background of all of the fluorescence images to ensure equal irradiation times. We also filtered the medians before saving the data in ASCII format. Figure 3(d)-(f) shows the experimental data and fitting curves of the pixel values versus the sample thickness for the corresponding configurations. By visually comparing the images in Figure 3(a)-(c), one can see that Figure 3(c) has the lightest coloring and superior gray scale layering, while Figure 3(a) has the darkest coloring and poorest gray scale layering.

Figure 3:Full-screen View

Fluorescence images of the lead sample after 1200 s of irradiation (a) and 1800 s of irradiation (b), and a fluorescence image of the polyethylene sample after 1800 s of irradiation (c). The experimental data and simulation curves of the lead sample after 1200 s of irradiation (d) and 1800 s irradiation (e), and the experimental data and simulation curves of the polyethylene sample after 1800 s of irradiation (f).

When analyzing the pixel values of the images, we chose a specific area marked as a rectangle with red edges, as shown in Figure 3(a)-(c). By averaging the pixel values along the width of the rectangle, we obtained the experimental data of the pixel values with respect to changes in thickness (along the length of the rectangular area) as shown in Figure 3(d)-(f).

In order to choose an appropriate function to fit the experiment data of the pixel value referred to in Equation (5), we eliminated the term B(w,v) T because we had already subtracted the background images from the data. We then ignored the scattering term S(x, y, z, µ_s) because it did not have an explicit expression. Next, we fitted the function using Y=KAe^-µx, and evaluated the reliability of the fitted curve based on the goodness-of-fit R². In Y=KAe^-µx, Y, K, A, µ, x, and x are related to the pixel value P,system parameter λ, neutron accumulation TI₀, material of the sample, and the thickness of the sample, respectively. Table 1 shows the fitting functions of the experimental data in the rectangular areas in Figure 3(a)-(c) and the goodness-of-fit of each curve.

Table 1Full-screen View

Fitting Functions and Goodness-of-Fit for the Experimental Data in the Rectangular Areas Depicted in Figure 3(a)-(c)

Name	Sample	Fitting Function	Goodness-of-Fit
a	1200 s Lead Sample	Y = 840.3130e-0.0131x	R² = 0.9925
b	1800 s Lead Sample	Y=1211.3447e-0.0131x	R² = 0.9943
c	1800 s Polyethylene Sample	Y=1330.3464e-0.0087x	R² = 0.9902

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Judging by the goodness-of-fit data, the adoption of function Y=KAe^-µx provided consistently high degrees of fitting, which indicates that choosing the exponential function Y as the fitting function was appropriate. However, compared to the lead sample of 1200 s of irradiation time (Figure 3(a)) and 1800 s of irradiation time (Figure 3(b)), the goodness-of-fit of the 1800-s-irradiation-time polyethylene sample (Figure 3(c)) was the lowest. This may be the result of polyethylene having a stronger scattering effect on fast neutrons (compared to lead samples), where the scattering term S(x, y, z, µ_s) may have had a relatively larger contribution to the pixel values of the images, and should be treated carefully rather than being ignored.

The lead samples with 1200-s-irradiation-times and 1800-s-irradiation-times had the same fitting value of µ (0.0131), while the 1800-s-irradiation-time polyethylene sample had a fitting value of µ (0.0087). This demonstrates that the parameter µ depends on the materials of the sample, and is free of the constraints of time. When considering the value of parameter KA, we depended on the system parameter λ, the irradiation time T, and neutron intensity I_0. We determined that if all of the tests were achieved in one system and had stable neutron sources and equivalent irradiation times, the KA values were the same for all samples, regardless of material type. However, the fitted data of the 1800-s-irradiation-time lead sample and the 1800-s-irradiation-time polyethylene sample had KA values of 1211.34 and 1330.34, respectively. The disparity in KA values between the lead and polyethylene samples with equal irradiation times implies that the scattering term S(x, y, z, µ_s) exerts influence on the pixel values, or that the drift of the neutron beam intensity I₀ for a relatively long time (thousands of seconds) has an impact on KA values. Overall, for Figure 3(a)-(c), the pixel value undergoes a negative exponential function with respect to approximate thickness x. Our analysis showed that Y=KAe^-µx can satisfy the fitting of pixel value curves under certain limitations, such as the scattering effect. Moreover, the results of our experiment agree well with the theoretical derivation of the pixel values.

3.3 Experimental Verification of the Reliability of Contrast Sensitivity Discriminants

In order to verify the reliability of the contrast sensitivity discriminants, we conducted an analysis based on the fitted curve Y=1211.3447 e^-0.0131x with high goodness-of-fit for the lead sample with an 1800 s irradiation time. When the sample’s thickness X₀ was 40 mm, the corresponding pixel value Y₀ was 717.3 under the conditions of the experiment. Based on Equation (10), pixel values fall into the range of 717.3±26.8, and were considered undetectable. The minimum detectable concave depth X_c and convex height X_v were 2.8 mm and 2.9 mm, respectively, using Equations (11) and (12). Considering systematic errors, we aimed at the minimum discernible thickness of 3 mm for a 40mm thick lead sample in the FNR system.

For the experiment, we designed 2 cuboid lead samples with geometrical proportions of 60 mm×40 mm×20 mm. One of the samples had 5 grooves with different depths on the 60 mm×40 mm surface (as shown in Figure 4(a)). The depth and width for each slot were the same: 1 mm, 2 mm, 3 mm, 4 mm, and 5 mm for the 5 grooves, respectively. During the experiment, the 2 samples were stacked to form a block measuring 60 mm×40 mm×40 mm. The experimental conditions included the neutron beam intensity and irradiation times being consistent with those in Equations (11) and (12). Figure 4(b) shows the fluorescence image of the lead sample in which the grooves (easily distinguishable by eye) have depths of 3 mm, 4 mm, and 5 mm, respectively.

Figure 4:Full-screen View

A picture of the lead sample for verification of results (a); an image of the experimental verification results (b); and a statistical diagram of the pixel value data in the rectangular area (c).

We vertically added the pixel values in the red rectangle in Figure 4(b) using the software Andor Solis for Imaging. We then plotted a statistical diagram of the pixel values in the horizontal axis, from which one can see 3 peaks which correspond to the groove depths of 3 mm, 4 mm, and 5 mm, respectively. Since the pixel values associated with the 1 mm and 2 mm deep grooves were equivalent to the statistical fluctuations, they were undetectable. We only analyzed data from the central area of the images because they were more creditable than the areas near the boundary.

In conclusion, the minimum detectable thickness of the 40mm thick lead sample was ~3 mm, which demonstrated the reliability of Equations (11) and (12). Moreover, the contrast sensitivity discriminant theory stated above was applicable for evaluating the minimum detectable thickness of a fix-thickness low-scattering sample in an FNR testing system. When applying such a method, attention must be paid to the applicable range and conditions of the criterion.

4. Conclusion

We conducted a study on the contrast sensitivity of a 14 MeV FNR system using the 600kV Cockcroft-Walton accelerator in the China Institute of Atomic Energy. We derived an equation for pixel values with respect to system sample parameters and irradiation time, and determined the theoretical expression of contrast sensitivity. We selected lead and polyethylene samples to derive the relationship between pixel values and irradiation times, and sample materials and thickness.; We then established the discriminants of contrast sensitivity and the reliability of the fitted curves. Afterwards, we verified the discriminants using a 40 mm thick lead sample.