In situ measurement of environmental γ radiation dose rates of key nuclides for large radioactive surface sources

NUCLEAR ELECTRONICS AND INSTRUMENTATION

In situ measurement of environmental γ radiation dose rates of key nuclides for large radioactive surface sources

Ze-Qian Wu，

Jian Sun，

Wei-Qi Huang，

Bai-Rong Wang，

Jin-Xing Cheng，

Jiang-Feng Wu，

Yong-Hong Wang，

Biao Yuan，

Sheng Qi，

Kun Shan

Nuclear Science and Techniques

Vol.36, No.1

Article number 3

Published in print Jan 2025

Available online 16 Dec 2024

DOI：10.1007/s41365-024-01549-4

281014

To monitor nuclear and radiation emergencies, it is crucial to obtain accurate in-situ measurements of the environmental γ radiation dose rate from key radionuclides, particularly for large radioactive surface sources. The methods currently used for measuring dose rates are inadequate for obtaining the dose rates of key radionuclides and have large angular response errors when monitoring surface sources. To address this practical problem, this study proposes three methods for measuring the dose rate: the weighted peak total ratio, mean value regression, and numerical integration methods. These methods are based on energy-spectrum measurement data, and they were theoretically derived and numerically evaluated. Finally, a 1-m-long hexagonal radioactive surface source was integrated into a larger surface source. In-situ measurement experiments were conducted on a large radioactive surface source using a dose-rate meter and a portable HPGe γ spectrometer to analyze the errors of the three aforementioned methods and verify their validity.

Environmental γ radiation dose rateHPGe γ spectrometerKey radionuclidesRadioactive surface source

Introduction

For radiation monitoring and protection, it is crucial to measure the absorbed dose rate of γ radioactive airborne particles. This helps to evaluate the extent of radiation exposure and ensure the safety of individuals and the surrounding environment. After the occurrence of a nuclear accident, it is necessary to monitor the dose rate of the environment around the nuclear power plant [1]. Through the use of data assimilation and large-model predictions, it is possible to support the drawing of radiation field posture maps [2-8] and evaluate the consequences of the accident [9, 10], providing a further basis for decision-making in the delineation of emergency operation areas [11, 12]. Typical air-absorbed gamma dose rate detectors include ionization chambers [13], GM counters [14], and proportional counters [15, 16], as well as silicon p-i-n photodiodes (Si-PINs) [17], field effect transistors (FETs) [18], scintillator detectors [19-22], and semiconductor detectors [23-25]. The "G(E) function" method is commonly employed for in-situ dose rate measurements using portable gamma spectrometers in emergency monitoring situations for nuclear and radiation incidents. This method was proposed in 1966 by Moriuch et al. [26, 27] of the Japan Atomic Energy Research Institute (JAERI) and is described in ICRP Report No. 53. [28] The G(E) function method relies on gamma energy spectral data. This method involves converting the count rate spectrum into a dose-rate spectrum using experimental scaling of the G(E) function curve. Numerous researchers have dedicated efforts to studying, calculating, refining, and validating the precision of dose-rate measurement techniques, including the G(E) function [29-36]. In addition, Kulhar et al. [37] have conducted research to validate the accuracy of radiation-sensitive field (RSF) measurements, which are commonly used for total ionizing dose measurements [38]. Research is currently being conducted on radiation-sensitive field-effect transistors (RadFETs), which are commonly used to measure total ionization doses. This study aims to develop a technique for extracting real-time dose rate data. Fricano et al. [39] studied N-doped optical fibers, while Kwon et al. [40] focused on a 64-channel scintillation fiber system that was used to measure the dose rates in intense radiation environments. Considering cerium-doped air-clad optical fibers, Bahou et al. [41] explored their application for X-ray dose-rate measurements up to 100 keV. This type of fiber offers several advantages including a compact size, flexibility, and resistance to electromagnetic interference. Pavelić et al. [42] conducted a study on a SiPM-based detector for measuring dose rates in pulsed radiation fields. Čerba et al. [43] investigated the same topic, whereas Ji et al. [44] focused on studying an unmanned radiation monitoring system that included dose rate measurement and radiation map creation on a UAV. Their research addressed the challenges of emergency radiation monitoring in the event of a nuclear power plant accident.

Current technology frequently provides an overall dose rate in real-world scenarios. Unfortunately, there is a lack of extensive research on calculation methods for determining the dose rate of a particular nuclide, particularly in real-world situations such as radiation monitoring during nuclear accidents and emergency responses. A monitoring object commonly takes the form of a sizable surface source, and the conventional approach is susceptible to the influence of angular response factors, resulting in significant discrepancies in the outcomes [45]. Because various radionuclides have varying impacts on the biological effects of radiation, radiation protection strategies, diagnosis of radiation accidents, and emergency responses, it is important to consider these factors. Additionally, the dose-to-Curie (DTC) method allows the direct conversion of measured dose rates into activity. This is particularly useful when dealing with radioactive surface sources because the DTC method enables the direct conversion of dose rates into activity [33, 46]. Furthermore, the DTC method can directly convert the measured dose rate into the activity. This is important for precisely measuring the dose rate of crucial nuclides when the monitored object is a radioactive surface source.

In this study, we address a practical issue with the traditional dose-rate measurement method when dealing with surface sources. We focus on the inability to obtain the key nuclide dose rate and the large angular response error. Our study aims to develop a methodology for dose rate measurement and proposes three methods: weighted peak-to-total ratio, mean regression, and numerical integration. To validate our findings, we conduct experiments using a dose-rate meter. The method is also tested experimentally using a dose-rate meter and portable HPGe γ spectrometer for radioactive surface sources.

Method

2.1

G(E) function

The G(E) function was determined by establishing the relationship between the γ-ray absorbed dose–rate equation. Eq. (1) represents the average energy received by the detector when a photon with energy E₀ enters it. $\bar{I_{d}} (E_{0}) = \int_{0}^{E_{0}} f (E, E_{0}) d E,$ (1) where f(E, E₀) is the response function of the monoenergetic photon and E is the integral variable of the integral. When the position of the detector is changed to the location where the air medium is placed, the average energy granted to the air by the γ photon is $\bar{I_{a}} (E_{0}) = k {(μ_{en} (E_{0}) / ρ)}_{a} E_{0},$ (2) where k is a constant. The variation caused by the medium is not a fixed multiple, but rather a result of energy alteration. The equation describing this system can be derived using Eq. (3). $\dot{D} = k_{1} \bar{I_{a}} (E_{0}) = k_{2} \int_{0}^{E_{0}} f (E, E_{0}) G (E) d E,$ (3) where $\dot{D}$ is the absorbed dose of air and k₁ and k₂ are constants. The value $\dot{D}$ was calculated theoretically and the curve of the G(E) function was obtained through experiments or simulations of the response function of monoenergetic photons.

To determine the theoretical dose rate at a specific location, Eq. (4) can be calculated for the known activity of a standard radioactive point source. $\dot{D} = k_{tran} φ E {(μ_{en} / ρ)}_{a},$ (4) where $\dot{D}$ is the absorbed dose rate in air, nGy/h; k_tran is the conversion factor, which converts keV/s to nGy/h, with a value of 1.6×10^-7×3600; E is the characteristic γ-ray energy, keV/s; and ${(μ_{en} / ρ)}_{a}$ is the mass-energy absorption coefficient of γ-rays in air for this energy, m²/kg. ϕ is the γ-ray fluence rate corresponding to energy m^-2 s^-1 without considering air scattering, which can be calculated using Eq. (5): $φ = \frac{A \cdot P_{r}}{4 π r^{2}},$ (5) where A is the activity of the standard radioactive point source, Bq; P is the emissivity of the characteristic γ-rays of the nuclide with energy E; and r is the distance between the point source and the detector, m.

To calibrate the G(E) function curves, a series of Monte Carlo (MC) simulations were conducted using 12 radionuclides and 15 characteristic γ-rays. Each simulation involved 12 measurements of 10,000 s to generate the energy spectra. A DETECTIVE-EX-100T HPGe γ spectrometer manufactured by ORTEC was used as the simulated detector. The dead layer and cold-finger sizes of the crystals were characterized using a multipoint source experiment [47]. The goal was to determine the response function and calculate the theoretical dose-rate produced by each γ ray at a distance of 30 cm. The calculation results are presented in Table 1.

Values of the air absorbed dose rate for different energies of γ-rays at 30 cm from the detector

Nuclide	Theoretical activity (Bq)	Radiant energy (keV)	Fluence rate (m^-2 s^-1)	Dose rate (nGy/h)
²⁴¹Am	7.28×10⁴	60	2.31×10⁴	2.38
¹⁵⁵Eu*	1.72×10³	87	4.70×10²	5.50×10^-2
¹⁵⁵Eu*	1.72×10³	105	3.10×10³	4.36×10^-2
⁵⁷Co	2.34×10⁴	122	1.77×10⁴	2.98
¹³²Te	2.27×10⁴	228	1.77×10⁴	6.27
¹³¹I	2.46×10⁴	364	1.77×10⁴	1.08×10¹
¹³³I	2.32×10⁴	530	1.77×10⁴	1.62×10¹
¹³⁷Cs	2.30×10⁴	662	1.73×10⁴	1.92×10¹
⁵⁴Mn	1.60×10³	835	1.42×10³	1.92
²²Na	1.56×10⁴	511	2.47×10⁴	1.71×10¹
²²Na	1.56×10⁴	1274	1.38×10⁴	2.68×10¹
⁶⁰Co	1.73×10⁴	1173	1.53×10⁴	2.72×10¹
⁶⁰Co	1.73×10⁴	1332	1.53×10⁴	3.01×10¹
¹²⁴Sb	4.08×10⁴	1691	1.77×10⁴	4.31×10¹
²⁴Na	2.00×10⁴	2754	1.77×10⁴	5.89×10¹

To achieve a more precise understanding of the relationship between the G(E) function and energy, the G(E) function was expanded into a higher-order polynomial of the logarithm of energy. To prevent overfitting or underfitting of the G(E) function, a polynomial order of reasonable size should be determined, and after theoretical analyses and computational attempts, it was determined that the order of 10 is fitted with higher accuracy and no overfitting phenomenon, as described in Eq. (6). $G (E) = \sum_{p = 1}^{10} A_{p} {(\log_{10} E)}^{p - 1},$ (6) where Ap is a polynomial coefficient and E is the energy of the γ-rays represented by the corresponding channel site, keV.

Combining Eq.(3) and Eq.(6), Ap can be directly utilized because k₂ in Eq.(3) and Ap in Eq.(6) are both constants. By expanding the integral sign, Eq. (7) can be obtained. To reduce the difficulty of matrix inversion in subsequent calculations, the channel sites in the energy spectrum data were combined into 200 channels with an energy interval of 15 keV per channel. ${\dot{D}}_{j} = \frac{1}{T} \sum_{p = 1}^{10} A_{p} [\sum_{i = 1}^{200} N_{j} (E_{i}) {(\log_{10} E_{i})}^{k - 1}],$ (7) where ${\dot{D}}_{j}$ is the air absorbed dose rate of the characteristic γ-rays in the j^th energy spectrum, nGy/h; T is the measurement time, s, and here the measurement time is equivalent to 10000 s; Ap is a polynomial coefficient; Ei is the energy magnitude corresponding to the i^th channel, keV; and Nj is the counts in the i^th channel in the j^th energy spectrum. Eq.(7) may be transformed and solved using the least squares method. By introducing the intermediate variable Bj,p we obtain $B_{j, p} = \sum_{i = 1}^{200} N_{j} (E_{i}) \cdot {(\log_{10} E_{i})}^{k - 1} .$ (8) Bringing Eq.(8) into Eq.(7), ${\dot{D}}_{j} = \frac{1}{T} \sum_{p = 1}^{10} A_{p} B_{j, p} .$ (9) The relative deviation between the derived dose rate and the theoretical true value was calculated using the principle of least squares. The partial derivative of the squared value was set to zero. $\frac{\partial \sum_{j = 1}^{12} {(\frac{\frac{1}{T} \sum_{p = 1}^{10} A_{p} B_{j, p}}{{\dot{D}}_{j_true}} - 1)}^{2}}{\partial A_{p}} = 0$ (10) where ${\dot{D}}_{j_true}$ is the theoretical true dose rate. By subsequent derivation, matrixization, and solving the matrix equations, the values of the polynomial coefficients A₁, A₂, ldots, and A₁₀ of the G(E) function can be determined, as shown in Table 2.

Total weighted peak ratios for different energies of γ-rays

Polynomial coefficients of the G(E) function	A₁	A₂	A₃	A₄	A₅
Coefficient calculation results	1.76	1.18	0.19	-0.83	-1.07
Polynomial coefficients of the G(E) function	A₆	A₇	A₈	A₉	A₁₀
Coefficient calculation results	0.03	1.17	-0.72	0.16	-0.01

The coefficient values in this table were used in the equation of the G(E) function to generate a fitted curve, as depicted in Fig. 1.

Fig. 1

Fitted curve of the G(E) function. The curve shows that the G(E) function is not monotonically increasing, but a minimum value occurs at low energy

Based on the G(E) function obtained from the calculations, the total γ dose rate can be determined from practical measurements using $\dot{D} = \sum_{i = 0}^{n} {\dot{D}}_{i} = \sum_{i = 0}^{n} n (E_{i}) \cdot G (E_{i}),$ (11) where $\dot{D}$ is the total air absorbed dose rate, nGy/h; ${\dot{D}}_{i}$ is the air absorbed dose rate corresponding to the energy in the i^th channel, nGy/h; n(Ei) is the net count rate of γ-rays corresponding to the energy in the i^th channel, cps; and G(Ei) is the value of G(E) function of the γ-rays corresponding to the energy in the i^th channel.

2.2

Methods for calculating dose rates for specific key nuclides

2.2.1

Weighted peak-to-total ratio

Considering the diverse properties of radiation and its effects on biology, it is crucial to adopt different strategies to diagnose radiation-related accidents and establish standards for radiation protection. This includes measuring the dose rate of specific key nuclides in addition to the overall dose rate of all γ-ray energies. Obtaining this directly using the traditional G(E) function is challenging. Consequently, we introduce the concept of a weighted peak-to-total ratio. To obtain the ratio of the weighted net counts to the full spectrum counts within the region of interest (ROI), we multiplied the measured energy spectrum by the G(E) function. These values remain consistent in the same measurement environment. Consequently, the weighted peak-to-total ratio was adjusted using a standard source and matched to the energy to generate a graph showing the relationship between the weighted peak-to-total ratio and energy. In practice, the dose rate of a single nuclide can be determined by calculating the net count in the ROI associated with that nuclide. It is important to consider that, when dealing with radionuclides that emit γ-rays of different energies, the absorbed dose rate of each energy should be calculated individually. The weighted peak-to-total ratio was calculated using Eq. (12): $F (E) = \frac{\sum_{i = a}^{b} n (E_{i}) \cdot G (E_{i})}{\sum_{i = 0}^{8192} n (E_{i}) \cdot G (E_{i})},$ (12) where F(E) is the weighted peak-to-total ratio of γ-rays with energy E; n(Ei) is the count rate of the γ-rays corresponding to the i^th channel, cps; G(Ei) is the G(E) function value of the γ-rays corresponding to the i^th channel; a is the channel site corresponding to the left end of the ROI; and b is the channel site corresponding to the right end of the ROI. The calculated weighted peak-to-total ratios of different energies of γ-rays are listed in Table 3.

Weighted peak-to-total ratio for different γ-ray energies

Nuclide	Radiant energy (keV)	Weighted peak-to-total ratio
¹³²Te	228	0.41
¹³¹I	364	0.60
¹³³I	530	0.60
¹³⁷Cs	662	0.54
⁵⁴Mn	835	0.46
⁶⁰Co	1173	0.37
⁶⁰Co	1332	0.33
¹²⁴Sb	1691	0.28
²⁴Na	2754	0.20

Based on the above data, after taking the logarithm of the weighted peak-to-total ratios, the weighted peak ratios were fitted by the least squares method using the following fourth-order polynomial: $\frac{1}{F (E)} = m_{1} E^{4} + m_{2} E^{3} + m_{3} E^{2} + m_{4} E + m_{5},$ (13) where m₁, m₂, and m₃ are the fitting coefficients and E is the characteristic γ-ray energy and keV. The fitting coefficients were obtained as m₁=1.152×10^-12, m₂=-7.008×10^-9, m₃= 1.435 ×10^-5, m₄= -1.005×10^-2, m₅= 3.945, and the regression coefficient was R₂ = 0.9902. The fitting results are shown in Fig. 2.

Fig. 2

Fitted relationship of the weighted peak-to-total ratio with a characteristic γ-ray energy

Based on the obtained total weighted peak ratios, the dose-rate contributions of the individual nuclides can be calculated using Eq. (14) for the measured energy spectra of mixed nuclides: ${\dot{D}}_{nuclide_tra} = \frac{1}{F (E)} \cdot \sum_{i = a}^{b} n (E_{i}) \cdot G (E_{i}),$ (14) where ${\dot{D}}_{nuclide_tra}$ is the dose rate of a single nuclide, nGy/h; a is the channel site corresponding to the left end of the ROI; b is the channel site corresponding to the right end of the ROI; F(E) is the weighted peak-to-total ratio of γ-rays of energy E; and n(Ei) is the count rate of γ-rays of the energy corresponding to the channel in the i^th channel, cps.

This method enhances the conventional dose rate measurement technique by incorporating the concept of a weighted peak-to-total ratio. This allows the measurement of the dose rates for crucial nuclides. Nevertheless, the traditional dose rate measurement method does not consider the error caused by the angular response factor of the detector in its derivation. During radioactivity monitoring, there are instances where the object being monitored is a sizable source of radiation on the surface. We conducted a thorough quantitative analysis and calculations to assess the extent of error that may arise when employing this method in the presence of changes in the monitoring object.

The MC method was employed to simulate an experiment involving the detection of radioactive surface sources using a spectrometer. In this experiment, a plane area with a radius of 8 m [48] surrounded by the projected point of the spectrometer on the ground was used as a uniformly distributed radioactive surface source. The dose rate at the center of the spectrometer detector resulting from the surface source was determined through the numerical integration of the dose rate of the point source. Based on Eq. (15), it is possible to calculate the theoretical air-absorbed dose rate caused by the surface source. ${\dot{D}}_{S} = \int_{0}^{2 π} d θ \int_{h}^{\sqrt{R^{2} + h^{2}}} \frac{k_{tran} \cdot A_{S} \cdot P_{r} \cdot E \cdot {(μ_{en / ρ})}_{a} \cdot r}{4 π r^{2}} d r,$ (15) where ${\dot{D}}_{S}$ is the theoretically calculated air absorbed dose rate caused by the radioactive surface source at the center of the spectrometer detector, nGy/h; A_S is the surface activity of the radioactive surface source, Bq/m²; R is the horizontal distance between the point source and the detector, m; h is the height of the detector from the ground, m, and here h = 1 m. The spectrum is generated by MC simulation such that the product of the surface activity of the surface source and the emissivity of the characteristic γ-rays can be substituted with the quotient of the number of simulated particles and the surface area: $A_{S} \cdot P_{r} = \frac{n_{simul}}{S},$ (16) where n_simul is the number of particles used for the simulation and S is the area of the simulated radioactive surface source, m². Bringing Eq.(16) into Eq.(15) provides ${\dot{D}}_{S} = \frac{1}{π R^{2}} \int_{0}^{2 π} d θ \int_{h}^{\sqrt{R^{2} + h^{2}}} \frac{k_{tran} \cdot n_{simul} \cdot E \cdot {(μ_{en / ρ})}_{a} \cdot r}{4 π r^{2}} d r .$ (17) After integration, ${\dot{D}}_{S} = \frac{k_{tran} \cdot n_{simul} \cdot E \cdot {(μ_{en / ρ})}_{a}}{4 π R^{2}} \ln (\frac{R^{2} + h^{2}}{h^{2}}) .$ (18) The dose rates obtained from the theoretical calculations were compared with those obtained using the G(E) function method, and the results are listed in Table 4.

Results of the full-spectrum dose rate using the G(E) function

Nuclide	Radiation energy (keV)	Theoretical value of dose rate (nGy/h)	Calculated dose rate G(E) function (nGy/h)	Relative deviation
¹³²Te	228	3.68×10²	4.18×10²	13.56%
¹³¹I	364	6.31×10²	6.03×10²	-4.49%
¹³³I	530	9.51×10²	9.49×10²	-0.20%
¹³⁷Cs	662	1.16×10³	1.26×10³	8.99%
⁵⁴Mn	835	1.41×10³	1.66×10³	17.54%
⁶⁰Co	1173	1.89×10³	2.22×10³	16.98%
⁶⁰Co	1332	2.09×10³	2.42×10³	15.57%
¹²⁴Sb	1691	2.53×10³	2.94×10³	16.30%
²²Na	2754	3.46×10³	3.94×10³	14.00%

2.2.2

Mean regression method

Given the nature of a monitoring object, which is a uniformly distributed radioactive surface source with a large radius, it is important to consider the influence of angular responses. Ignoring this factor using the traditional G(E) function method can lead to significant errors. This is because γ-rays incident from different locations of the surface source are introduced and accumulate owing to the angular response, resulting in a substantial error. Additionally, to determine the dose rate of the key nuclides, a more complex and labor-intensive method, known as the weighted peak-to-total ratio, was utilized. Thus, the proposed mean regression method can be utilized to enhance the accuracy of the dose rate calculations for individual nuclides. The main concept of this method involves measuring point sources at various locations on the surface. Instead of counting the full spectrum, only net counts in the ROI were considered. This helps to establish an equational relationship with the dose rate. By calculating the value of the G(E) function at different locations and taking the mean value of these values, the error caused by the angular response was minimized. In real-world scenarios, the net count of the full energy peak multiplied by the corresponding G(E) function can be used directly as the dose rate value of a single nuclide. First, the absorbed dose rate was used to establish an equational relationship between the theoretical and calculated values of the G(E) function. $n (E_{ROI}) \cdot G (E_{ROI}) = k_{tran} E {(μ_{en} / ρ)}_{a} \cdot \frac{A \cdot P_{r}}{4 π r^{2}},$ (19) where n(E_ROI) is the net count rate within the ROI of the characteristic γ-rays, G(E_ROI) is the value of the G(E) function within the ROI of the characteristic γ-rays, and A is the activity of the standard radioactive point source, Bq. The relationship between the detection efficiency and activity can be described as $A \cdot P_{r} = \frac{n (E_{ROI})}{ε_{E} (r)},$ (20) where $ε_{E} (r)$ is the detection efficiency of the HPGe γ spectrometer for the characteristic γ-rays with energy E from a point source at distance r; n(E_ROI) is the net count rate within the ROI of the characteristic γ-rays, cps; and A is the activity of the standard radioactive point source (Bq). Bringing Eq.(20) into Eq.(19), we obtain $G (E_{ROI}) = \frac{1}{4 π ε_{E} (r) r^{2}} \cdot k_{tran} E {(μ_{en} / ρ)}_{a},$ (21) Therefore, an MC simulation model was established, and various radioactive point sources with different energies were positioned in a grid. The sources were placed vertically at intervals of SI1m starting 1 m away from the center of the detector. Horizontally, the sources were positioned between 0 and 6 m. The G(E) function was determined using Eq.(21), which relies on the detection efficiency of the detector for point sources at various locations calculated through MC simulations. The results are displayed in Table 5.

Calculation results of the G(E) function at different levels of position for different energies

Energy (keV)	0 m	1 m	2 m	3 m
228	1.83×10^-1	1.89×10^-1	2.05×10^-1	2.21×10^-1
364	4.55×10^-1	4.76×10^-1	5.07×10^-1	5.47×10^-1
530	9.22×10^-1	9.72×10^-1	1.02	1.10
662	1.38	1.44	1.53	1.64
1173	3.56	3.66	3.76	3.80
1332	4.35	4.54	4.51	4.54
1691	6.30	6.38	6.57	6.69
2754	1.36×10¹	1.38×10¹	1.40×10¹	1.39×10¹
Energy (keV)	4 m	5 m	6 m
228	2.27×10^-1	2.37×10^-1	2.49×10^-1
364	3645.48×10^-1	5.79×10^-1	5.76×10^-1
530	1.15	1.20	1.22
662	1.61	1.70	1.79
1173	4.01	4.01	4.32
1332	4.90	4.88	5.10
1691	7.23	6.40	7.28
2754	1.37×10¹	1.36×10¹	1.46×10¹

The G(E) function values obtained at different positions for γ-rays of the same energy were averaged to obtain the average G(E) function values, as shown in Table 6.

Calculation results of the G(E) function at different levels of position for different energies

Nuclide	¹³²Te	¹³¹I	¹³³I	¹³⁷Cs
Energy (keV)	228	364	530	662
Average G(E) function value	2.16×10^-1	5.27×10^-1	1.08	1.59
Nuclide	⁶⁰Co	⁶⁰Co	¹²⁴Sb	²⁴Na
Energy (keV)	1173	1332	1691	2754
Average G(E) function value	3.88	4.69	6.69	1.39×10^-1

Least-squares fitting was conducted for the average G(E) function values of the characteristic γ-rays for each energy in the energy range from 122 to 2754 keV. $G (E) = p_{1} E^{2} + p_{2} E + p_{3},$ (22) where E is the characteristic γ-ray energy and p₁, p₂, and p₃ are constants. The G(E) function curve was obtained, and the coefficients were determined as p₁= 9.946× 10^-7, p₂= 2.466× 10^-3, and p₃= -4.024× 10^-1. The regression coefficient of the fit, R² = 0.9996, indicated that the fit was good. The fitting results are presented in Fig. 3.

Fig. 3

G(E) function curves calculated from point-source simulation experiments at different locations

The G(E) function curve is accurate only for the energy intervals between 122 and 2754 keV. Based on the value of the G(E) function, the dose rate of a single nuclide can be calculated as ${\dot{D}}_{nuclide_ave} = \sum_{i = a}^{b} n (E_{i}) \cdot G (E_{i}),$ (23) where ${\dot{D}}_{nuclide_ave}$ is the airborne absorbed dose rate of a single nuclide, nGy/h; a is the channel site corresponding to the left end of the ROI; b is the channel site corresponding to the right end of the ROI; n(Ei) is the count rate of energy γ-rays corresponding to the i^th channel, cps; and G(Ei) is the value of the G(E) function of the γ-ray energy corresponding to the i^th channel.

2.2.3

Numerical integration correction method

The measured object is considered to be a surface source that is uniformly distributed and infinitely large, as shown in Fig. 4. The absorbed dose rate can be determined through double integration, as shown in Eq. (17).

Fig. 4

(Color online) Schematic of the dose rate measurement model. The detector is set at a height of h from the ground, and the direction of the probe is downward

An equation describing the relationship between the theoretical value of the surface-source dose rate and the calculated value of the G(E) function was established. ${\dot{D}}_{S} = n (E_{ROI}) \cdot G (E_{ROI}),$ (24) where ${\dot{D}}_{S}$ is the theoretically calculated air-absorbed dose rate at the center of the detector due to the radioactive surface source, n(E_ROI) is the net count rate within the characteristic γ-ray ROI, and G(E_ROI) is the value of the G(E) function within the characteristic γ-ray ROI. When considering a monitoring object that can be considered as a uniformly distributed surface source with an almost infinite size, the limit of integration represents the distance at which the γ-ray energy effectively contributes to the spectrometer. The horizontal distance R between the point source and the detector is converted to the effective contribution distance dE and then substituted into Eq. (18). The G(E) function can be derived using Eq. (25): $G (E) = \frac{k_{tran} \cdot E \cdot {(μ_{en} / ρ)}_{a}}{4 \bar{ε_{E}}} \cdot \ln (\frac{h^{2} + d_{E}^{2}}{h^{2}}),$ (25) where $ε_{E}$ is the detection efficiency of the characteristic γ-rays with energy E from an infinite uniformly distributed surface source and dE is the effective contributing distance of γ-rays with energy E to the spectrometer [49], m. The expressions and curves of the G(E) function can be obtained by the least-squares fitting of the G(E) values obtained for different energies. The results are listed in Table 7.

Calculation results of the improved G(E) function method

Nuclide	¹³²Te	¹³¹I	¹³³I	¹³⁷Cs
Energy (keV)	228	364	530	662
Horizontal distance (m)	5.58	5.75	5.84	5.97
Straight line distance (m)	5.67	5.84	5.92	6.05
Detection efficiency	1.34E-05	9.12×10^-6	6.64×10^-6	5.48×10^-6
G(E) function	1.95×10^-1	4.89×10^-1	9.71×10^-1	1.43
Nuclide	⁶⁰Co	⁶⁰Co	¹²⁴Sb	²⁴Na
Energy (keV)	1173	1332	1691	2754
Horizontal distance (m)	6.06	6.19	6.25	6.31
Straight line distance (m)	6.14	6.27	6.33	6.39
Detection efficiency	3.47×10^-6	3.12×10^-6	2.58×10^-6	1.93×10^-6
G(E) function	3.53	4.31	6.19	1.30×10¹

The least-squares fitting of the average G(E) function values of the characteristic γ-rays for each energy level in the energy range of 122 to 2754 keV was achieved using $G (E) = p_{5} E^{2} + p_{6} E + p_{7},$ (26) where E is the characteristic γ-ray energy, keV; and p₅, p₆, and p₇ are constants where p₅ = 9.796× 10^-7, p₆ = 2.163× 10^-3, and p₇ = -3.542× 10^-1. The regression coefficient R² = 0.9996 indicates that the fit is better, as shown in Fig. 5. Based on the value of the G(E) function, Eq. (23) can be utilized to obtain the dose rate of a single nuclide.

Fig. 5

Curves of the improved G(E) function method based on the numerical integration approach

Experimental validation

It is necessary to experimentally verify the accuracy of the three dose-rate measurement methods described above. The main objective of the experiment was to measure the radioactive surface source accurately using a dose rate meter and an energy spectrometer. This process is illustrated in Fig. 6(a) and (b), where the dose rate meter provided the standard value for the dose rate measurements. The value obtained from the energy spectrum using the proposed method was assessed to determine its effectiveness. Because of the challenges in recreating the experimental environment of a large radioactive surface source without accidents, we opted to use a single hexagonal radioactive surface source with standard activity and a side length of 1 m for mobile splicing. This enabled us to simulate the measurement process of the detector on a large radioactive surface, as shown in Fig. 6(c). Each value in the diagram corresponds to a specific measurement point. The hexagonally labeled 0000 indicates the location of the detector, whereas the other numbers represent the number of circles in the first digit. Serial numbers within the circles are represented by the last three bits.

Fig. 6

(Color online) (a) Direct measurement of the dose rate using a dose rate meter, (b) measurement of the energy spectrum of a radioactive surface source using a portable HPGe γ spectrometer, and (c) arrangement of the radioactive hexagonal surface source

After the splicing measurement and interpolation analysis of the radioactive hexagonal surface source, the results of the dose rate measurement using the dose rate meter and count rate measurement using the portable HPGe γ spectrometer were obtained, as shown in Table 8.

Measurement results of the dose rate meter and portable HPGe γ spectrometer

No. laps	No. points	Dose rate cumulative (nGy/h)	Proportion of dose rate (%)	Cumulative count rate (cps)	Proportion of count rate (%)
0	1	558.7	15.3	476.9	19.1
1	6	1170.5	32.0	883.1	35.4
2	12	795.2	21.8	510.6	20.5
3	18	537.6	14.7	286.5	11.5
4	24	351.2	9.6	193.8	7.8
5	30	241.4	6.6	143.1	5.7
Validation results		4568.3	100.0	2494.0	100.0

To validate the method proposed in this study and assess the advantages and disadvantages of the three different methods (weighted peak-to-total ratio, mean regression, and numerical integration methods), the overall peak count rate of γ-rays with an energy of 662 keV at ¹³⁷Cs, measured using a portable HPGe γ spectrometer, was multiplied by the weighting factors obtained from each method. This calculation yielded the estimated dose rate, which was then compared with the standard dose rate measured using a dose-rate instrument. The results of this comparison are presented in Table 9.

Comparison of the results of the weighted peak-to-total ratio method, mean regression method, and numerical integration method

Method	Count rate (cps)	G(E) function	Calculated dose rate (nGy/h)	Standard dose rate (nGy/h)	Relative error
Weighted peak-to-total ratio	2494.0	0.72	3306.4	4568.3	-27.6%
Mean-reversion	2494.0	1.59	3965.4	4568.3	-13.2%
Numerical integration	2494.0	1.43	3566.4	4568.3	-21.9%

When the measurement object satisfies an approximately infinite uniform distribution of radioactive surface sources, the most accurate results are obtained using the mean regression method with a relative error of -13.2% because this method corrects the error caused by the angular response to a certain extent. The numerical integration method provides a highly accurate result with a relative error of -21.9%. While this method theoretically addresses the error in the angular response quite effectively, it is possible that the inclusion of the effective contribution distance of the radioactive surface source introduces a significant error in the distance determination, leading to potentially large measurement inaccuracies. However, without accounting for the angular response, the use of the weighted peak-to-total ratio method resulted in a significant relative error of -27.6%. Therefore, when monitoring a radioactive surface source, it is crucial to measure the dose rates of key nuclides using the mean regression method. This method significantly enhanced the accuracy of the results.

Conclusion

In this study, we investigated two practical problems related to traditional monitoring methods. One of these problems is the limited ability to obtain the total dose rate. However, it is impossible to determine the dose rate for a particular nuclide. When dealing with a large surface source as a monitoring object, the conventional G(E) function method can be influenced by the angular response factor, resulting in significant inaccuracies. Three different methods for measuring the dose rate were proposed: the weighted peak-to-total ratio, mean regression, and numerical integration methods. The weighted peak-to-total ratio method enhances the conventional G(E) function by incorporating the concept of a weighted peak-to-total ratio. This allows the assessment of the dose rate of crucial nuclides without accounting for the impact of the angular response. The mean regression and numerical integration methods were applied to study the adjustment of key nuclide dose rates and angular responses in nuclear accident emergency scenarios involving extensive areas of uniformly distributed radioactivity. After conducting measurement experiments on radioactive surface sources using dose-rate meters and portable HPGe γ spectrometers, the analysis revealed the advantages and disadvantages of these three methods. The results indicate that the mean-value regression method is the most effective among the investigated methods. From a theoretical standpoint, the numerical integration method is considered to provide the most accurate correction of the angular response. In future research, further optimization of the error introduction in different aspects of the numerical integration method may lead to improved measurement results.

References

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