Introduction
Considering the generation of a radioactive plume or the deposition of radioactive materials on the surrounding soil may be necessary in the case of an accident involving the leakage of radioactive material at a nuclear power plant or a radiation-related facility. Environmental radiation monitoring systems have been established to detect these events, but identifying the cause of increases in the dose rate only by measuring the rate is difficult. The energy spectrum of γ rays needs to be measured to evaluate the exposure dose and to determine the leaked radioisotopes at the accident site and the surrounding environment.
The exposure dose from γ rays at a site is obtained by multiplying the γ-ray energy spectrum by the dose-per-fluence conversion coefficient [1], but a method for calculating the exposure dose using a weighting factor called the G(E) function [2] is utilized since measuring the energy spectrum is generally difficult. The G(E) function is considered a weighting factor that compensates for the difference between the response characteristics of the detector to γ rays and those to the dose. The dose rate can be accurately obtained with a simple calculation using this method. However, the radioisotope that contributes to increasing the dose cannot be determined because the fluence information at each γ-ray energy is lost in the calculation. One of the methods for assessing the air dose rate for each radioisotope is in situ measurement using a portable high-purity germanium (HPGe) semiconductor detector with a high energy resolution [3, 4]. With this method, the dose rate due to artificial radioisotopes leaked during an accident or natural radioactivity is evaluated by multiplying the fluence of each γ-ray energy calculated by the HPGe semiconductor detector using a weighting factor that considers the concentration distribution of radionuclides in soil. However, HPGe semiconductor detectors have high installation and operation costs due to the need for a cooling mechanism using liquid nitrogen or an electric cooler, which causes difficulties in emergency monitoring.
NaI(Tl) scintillators, despite having difficulties with radioisotope identification due to their poor energy resolution, demonstrate the advantages of increased sensitivity and decreased measurement time compared with HPGe semiconductor detectors due to the ease of manufacturing their large crystals at a low cost. Additionally, they require no cooling mechanism, making them suitable for emergency environmental radiation monitoring with a short activation time and easy operation. Therefore, a low cost, easy-to-handle γ-ray spectrometer can be fabricated by improving the energy resolution of the NaI(Tl) scintillator to separate and quantify γ-rays within an energy range of several tens of keV. Furthermore, obtaining the energy spectrum of the γ-ray fluence rate and the dose rate simultaneously is possible by incorporating these spectrometers into environmental radiation monitoring systems. This capability allowed leaked radioisotopes to be determined and the event’s cause to be estimated, contributing to reducing exposure doses in emergencies.
One method for enhancing the energy resolution of radiation detectors is known as the unfolding method, which estimates the source energy spectrum from a measured pulse-height distribution. Several unfolding methods have been investigated using response functions, such as inverse matrix methods, spectrum stripping methods [5], and iterative methods [6, 7]. Each method requires appropriate initial guesses and accurate response functions to perform unfolding correctly.
The γ rays come from environmental radioactivity and unknown artificial radioactive materials in radiation monitoring. Estimating the source energy spectrum is difficult in many cases. Hence, algorithms that do not depend on initial guesses are preferable. The iterative approximation method [5, 8], predominantly used in environmental radiation monitoring, is suitable for this study because a measured pulse-height distribution is utilized as an initial guess.
Conversely, the γ-ray energy available for evaluating the response function using a radiation source is limited; thus, obtaining this function by some calculation is necessary. Therefore, the function to improve the unfolding performance needs to be precisely calculated. The energy bin width of response functions used in conventional environmental γ-ray monitoring is ≥50 keV, which is insufficient to separate and quantify γ rays with energies as close as several tens of keV. Therefore, response functions with smaller energy bin widths are warranted. Monte Carlo calculations cannot completely reproduce the detector output pulse; thus, calculating the deposit energy distribution of the detector is predominant due to γ rays and folding with measured energy resolutions. However, the response function for scintillators is not accurate using the method because it cannot reproduce photoelectric peak shape deviation from a Gaussian distribution due to the effect of the energy dependence of scintillation efficiency.
Therefore, this study improved the performance of γ-ray spectroscopy using an unfolding method with a response function that includes the energy dependence of scintillation efficiency for the NaI(Tl) scintillator.
Materials and Methods
1. Experimental Setup
The γ-ray spectrometer contains a Φ76.2 mm×76.2 mm cylindrical NaI(Tl) scintillator (12B12; OKEN), Φ76.2 mm photomultiplier tube (R1847; Hamamatsu), and a tube base (TB-5; Amptek). The photomultiplier tube, surrounded by a permalloy magnetic shield, was optically coupled to the NaI(Tl) scintillator with optical glasses. The tube base consisted of a preamplifier, digital pulse processor, multi-channel analyzer, and high voltage power supply, with control and data readout from a personal computer via a universal serial bus cable.
The operating voltage of the photomultiplier tube was set to 800 V. The radiation source was placed at the side of the NaI(Tl) scintillator to assess the accuracy of environmental γ-ray measurements. This study focused on measuring the fluence rate caused by γ-rays incident parallel to the detector from the environment. Measurements were conducted in the laboratory with radiation sources positioned at distances of 1, 2, and 3 m from the detector center (Fig. 1).
2. Unfolding Method
The pulse-height distribution of the γ rays measured using the NaI(Tl) scintillator is the convolution of the incident γ-ray energy spectrum and response function (Fig. 2). The pulse-height distribution of the NaI(Tl) scintillator, when the energy range is divided into N parts, is expressed as follows:
where Mi denotes the i-th element of the pulse-height distribution of the NaI(Tl) scintillator, Sj indicates the j-th element of the source energy spectrum, and Rij represents the response function coupling the i-th element of the pulse-height distribution with the j-th element of the source energy spectrum [5].
The technique of assuming the source energy spectrum from the measured pulse-height distribution is known as the unfolding method. The source energy spectrum can be obtained by multiplying the measured energy spectrum by the inverse of the response matrix if the response function of the NaI(Tl) scintillator is known (Fig. 2). This technique is the inverse matrix method. However, the inverse matrix method has the problems of oscillations and negative values appearing in the obtained source energy spectrum. We utilized the iterative approximation method [7, 8] to avoid these problems.
The calculation procedure of the iterative approximation method is presented as follows. The initial guess of the source energy spectrum is the measured pulse-height distribution, as shown in the following equation [8]:
where the superscript indicates the number of iterations. The initial guess of the source spectrum represented by
S i ( 1 )
The fraction of the pulse height distribution obtained by the Equation (3) to the measured ones are multiplied by the source energy spectrum to update them as shown in the following equation, as:
The source energy spectrum will converge by repeating the calculations in Equations (3) and (4) approximately 1,000–10,000 times.
3. Response Functions
The energy spectrum of a source obtained by unfolding depends on response function accuracy. This study obtained the response function by weighting the nonlinearity of the scintillation efficiency onto the energies of secondary electrons calculated using Monte Carlo simulation code EGS5 [9]. Fig. 3 illustrates a flowchart of the response function derivation method in this study.
Fig. 4 demonstrates the calculation geometries of the response function. We modeled the NaI(Tl) scintillator, magnesium oxide (MgO) used as a reflector, borosilicate glass used as an optical window, photomultiplier tube, permalloy as a magnetic shield, and aluminum housing that constitutes the detector in calculating the response function (Fig. 4A). Table 1 shows the composition and density of the detector components. We set the γ-ray irradiation conditions to parallel irradiation from the side of the cylindrical NaI(Tl) scintillator (Fig. 4B) since we detect γ rays emitted from the surroundings of the detector in environmental radiation monitoring and the axis of the cylindrical detector is usually installed vertically. The energy range of the response function was 0–3 MeV with a 5 keV mesh.
Scintillation light in NaI(Tl) scintillators is produced due to energy deposition onto the crystal by the electrons generated from the interaction with incident γ rays. The output pulse of a NaI(Tl) scintillator is generated by converting light into an electrical pulse using a photomultiplier tube. The total charge of the output pulse for a single γ-ray detection is expressed as follows:
where ΔE denotes the energy deposited through electrons produced by the interaction with incident γ rays, S(E) indicates the relative scintillation efficiency considering nonlinearity associated with electron energy (Fig. 5) [10, 11], C represents the light collection efficiency, q demonstrates the quantum efficiency at the photocathode, G denotes the multiplication factor of the photomultiplier tube, and e indicates the elementary charge. Equation (5) is a modification of Equation (1) in [12] to consider the effect of nonlinearity in the scintillation efficiency based on the energy of the secondary electrons produced by the interaction with γ rays.
Moreover, q and G are constants for the same operating conditions of the photomultiplier tube, QTot is proportional to the light output expressed as the product of ΔE and S(E). The range of electrons in the NaI(Tl) scintillator is sufficiently short enough C can be considered constant for the produced electrons. The product of ΔE and S(E), i.e., the light output shown in Equation (5), was calculated using EGS5, and these were summarized for each incident γ ray. Fig. 6 demonstrates the calculated light output distribution for several γ-ray energies. The horizontal axis is normalized to the electron equivalent energy.
From Equation (5) the uncertainty of QTot can be described:
where σ denotes the standard deviation and L indicates the light output expressed as the product of ΔE and S(E). This equation was obtained from the uncertainty propagation law with each term in Equation (5) as independent variables. Here, the sum of the products of ΔE and S(E) is L; thus, we summarized it as L in the equation.
Fig. 7 shows the σTot obtained by Gaussian fitting of the measured photoelectric peak for several radioisotopes. Similarly, σL was obtained by Gaussian fitting of the peak areas corresponding to the photoelectric absorption peaks for the light output distribution (Fig. 6). The energy dependence of σTot (E) was obtained by fitting with the following equation [13]:
where p0 represents fluctuations due to the circuit noise being constant and independent of the light output, p1 denotes fluctuations in the quantum efficiency and electron amplification in the photomultiplier tube being proportional to the square root of the light outputs, and p2 indicates fluctuations in the light collection efficiency being proportional to the light outputs [13]. The fitting results indicated that p0=−4.497×10−6, p1=5.733×10−4, and p2=2.148×10−4. Similarly, the energy dependencies of σL (E) were obtained by fitting using Equation (7), resulting in p0=−1.980×10−7, p1=1.159×10−5, and p2=1.271×10−6. Fig. 7 illustrates the energy dependencies of σTot and σL with fitting results.
The uncertainty of QTot is expressed as in Equation (6); thus, the response function is obtained by folding with σTot, when the deposit energy distribution is calculated using Monte Carlo simulations. This study calculated the light output distribution, including the uncertainty of scintillation efficiency by EGS5. Therefore, the response function was obtained by folding the light output distribution using the standard deviation calculated from the difference between σTot and σL as shown in the following equation, as:
The energy dependencies of σFolding(E) were obtained by using Equation (8), resulting in p0=−4.299×10−6, p1=5.617× 10−4, and p2=2.135×10−4.
Fig. 8 shows a comparison of the measured pulse-height distribution of 137Cs, 133Ba, 88Y, and 60Co, and the response function, including the energy dependence of scintillation efficiency, and the response function without the energy dependence of scintillation efficiency. Both response functions were confirmed to reproduce the measurement results for the peak region. Conversely, the response function, including the energy dependence of scintillation efficiency, reproduces the measurement results well for the Compton continuum region, whereas the response function that does not include the energy dependence of scintillation efficiency is an overestimation. No significant difference was observed between the two response functions for the 0.081 MeV peak of 133Ba. This is because the energy dependence of scintillation efficiency is relatively small due to the narrow energy range of generated secondary electrons. Therefore, considering the energy dependence of scintillation efficiency makes improving the reproducibility of the response function possible.
We obtained the response matrix shown in Fig. 9 by applying the above calculations up to 3 MeV with a 5 keV mesh.
Results and Discussion
We evaluated the results of unfolding the response function to determine the optimal number of iterations in unfolding. Fig. 10 shows the results of unfolding the response function for γ-ray energies of 0.660–0.665 MeV. We observed that the peak converged toward the source energy bin because this is the unfolding of the response function, and the fluence rate asymptotically approached 1 cm−2 s−1 with an increase in the number of iterations. The peak of the unfolding spectrum should ideally converge to a single bin of 0.660–0.665 MeV, but it was slightly spread out to nearby energy bins (Fig. 10). However, the fluence rate was reduced by a factor of 10−5 at energies of >3 bins away from the source energy bin, considering the case of 1,000 iterations as an example, and the effect on the measurement was almost negligible. Therefore, we defined the region of interest (ROI) to be within two bins from the source energy bin (five bins in total, including the source energy bin). Fig. 11 illustrates the association between the number of iterations and the integrated fluence rate within the ROI obtained by unfolding the response function. A larger number of iterations are required for convergence at higher source energies, but the unfolding result was confirmed to converge at 10,000 iterations within the energy range of up to 3 MeV, which is the measurement target. Based on this, we set the number of unfolding iterations to 10,000 in this study.
Fig. 12 illustrates a comparison of the measured pulse-height distribution, calculated response functions, and γ-ray fluence rate obtained using the unfolding method. The points indicate the measured pulse-height distribution, the dashed line represents response functions, and the solid line indicates unfolding results. The measurement time for each source was 3,000 seconds.
The response functions were confirmed to reproduce the measured pulse-height distribution, especially for the photoelectric peak, and Compton edge, especially for the response function, including the energy dependence of scintillation efficiency. The response functions underestimated the measured ones in the low-energy region of approximately 0.2 MeV, but this is considered the effect of scattered γ rays from the surroundings of the detector, which were not included in the calculation model of response functions. Furthermore, unfolding was confirmed to separate the peaks for each γ-ray energy, even for nuclides that emit multiple γ rays. The Compton continuum region converged to a peak; thus, γ rays with low emission rates became observable. The difference between the measured pulse-height distribution and calculated response functions in the low-energy region mentioned above appeared as a continuum component in the low-energy region in the unfolding results. Conversely, a split of the γ-ray peak in the unfolding spectrum is observed for 137Cs in Fig. 12A and 12B and for 60Co in Fig. 12G and 12H. Slight discrepancies in the shapes of the measured pulse-height distribution and response functions due to the reproduction accuracy of the response functions and gain fluctuations during the measurement caused this phenomenon.
We compared the γ-ray fluence rate at the detector position obtained by unfolding (ΦUnf) with that obtained from the source intensity (ΦRef) as a reference value for each γ-ray energy listed in Table 2 to evaluate the accuracy of the unfolding. The peaks in the unfolding spectrum were split due to the discrepancy between the measured pulse-height distribution and the response function, as confirmed in Fig. 12; thus, the ROIs for each γ-ray energy were set to a maximum of nine bins (0.045 MeV) to include the split peaks in fluence rate evaluation. Table 2 lists the ROIs for each γ-ray energy. The fluence rate within each ROI was integrated to obtain ΦUnf in the energy spectrum of the fluence rate obtained by unfolding. The reference value, ΦRef, was obtained from the following equation:
where L denotes the distance from the detector to the source and Iγ indicates the source γ-ray emission rate for each γ-ray energy. The relative intensity of the unfolding result to the reference value for each peak was obtained by dividing ΦUnf by ΦRef. The uncertainty of ΦUnf was identified by the standard deviation of 30 measurement repetitions. The measurement time was 100 seconds.
Fig. 13 illustrates the evaluation results for the distance from the detector to the source, which was 1 m. The horizontal axis represents the γ-ray energy and the vertical axis denotes the fraction of unfolding results to the reference value. Each point is the mean of repeated measurements and error bars are the standard deviation of ΦUnf ⁄ΦRef obtained from the repeated measurements. Discrepancies in the uncertainties were observed due to differences in the source intensity and emission rate for each energy, but the unfolding results were confirmed to be consistent with the reference value within ±10% for energies of >0.3 MeV. The unfolding results are considered to be overestimated due to the effect of scattered γ rays below 0.3 MeV. The comparison between the case with (Fig. 13A) and without (Fig. 13B) the energy dependence of scintillation efficiency in the response function demonstrated that the presence of the energy dependence reduced the variance of the unfolding results.
We measured the γ-ray fluence rate by changing the distance from the source to the detector to evaluate the dependence on source intensity. Fig. 14 shows a comparison between the results measured at 1, 2, and 3 m distances and the reference value obtained from the source intensity. The horizontal axis represents the distance from the detector to the source, and the vertical axis indicates the fluence rate. Each point indicates the fluence rate obtained by the unfolding method, and error bars are the standard deviation of ΦUnf obtained from repeated measurements. The dashed line denotes the fluence rate calculated from the source intensity and distance for each isotope using Equation (9). The uncertainty of the unfolding results was greater at measurement points with lower fluence rates, but the average values were confirmed to be in good agreement with the calculated results. The comparison between the case with (Fig. 14A) and without (Fig. 14B) the energy dependence of scintillation efficiency in the response function demonstrated that the presence of the energy dependence reduced the variance of the unfolding results even for a low fluence rate.
Conclusion
We have reached the following conclusions as a result of evaluating an unfolding method with response functions, including the energy dependencies of scintillation efficiency for a NaI(Tl) scintillator using radiation sources.
The reproducibility of the response function could be improved by including the energy dependence of scintillation efficiency in the response function calculation. The unfolding method separated peaks for each γ-ray energy, although the difference in γ-ray energies was approximately several tens keV. Moreover, the accuracy of the unfolding method was almost constant within ±10% for energies of >0.3 MeV and had linearity concerning γ-ray intensity. Reducing the variance of unfolding results by including the energy dependence of scintillation efficiency in the response function is an advantage.
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