Investigating the minimum detectable activity concentration and contributing factors in airborne gamma-ray spectrometry

Investigating the minimum detectable activity concentration and contributing factors in airborne gamma-ray spectrometry 1 2 first-author 1 2 corresp sunkun@stu.cdut.edu.cn sunkun@stu.cdut.edu.cn sunkun@stu.cdut.edu.cn 1 2 https://orcid.org/0000-0002-4273-6062 1 1 2 https://orcid.org/0000-0001-8840-7793 1 2 https://orcid.org/0000-0002-8726-2198 1 2 3 3 The College of Nuclear Technology and Automation Engineering, Chengdu University of Technology, Chengdu 610059, China The College of Nuclear Technology and Automation Engineering, Chengdu University of Technology, Chengdu 610059, China Key Laboratory of Applied Nuclear Techniques in Geosciences of Sichuan Province, Chengdu University of Technology, Chengdu 610059, China Key Laboratory of Applied Nuclear Techniques in Geosciences of Sichuan Province, Chengdu University of Technology, Chengdu 610059, China China Coal Geology Group Co., Ltd., Beijing 100040, China China Coal Geology Group Co., Ltd., Beijing 100040, China In this study, the theory of minimum detectable activity concentration (MDAC) for airborne gamma-ray spectrometry (AGS) was derived, and the relationship between the MDAC and the intrinsic efficiency of a scintillation counter, volume and energy resolution of scintillation crystals, and flight altitude of an aircraft was investigated. To verify this theory, experimental devices based on NaI and CeBr 3 scintillation counters were prepared, and the potassium, uranium, and thorium contents in calibration pads obtained via the stripping ratio method and theory were compared. The MDACs of AGS under different conditions were calculated and analyzed using the proposed theory and the Monte Carlo method. The relative errors found via a comparison of the experimental and theoretical results were less than 4%. The theory of MDAC can guide the work of AGS in probing areas with low radioactivity. Airborne gamma-ray spectrometry (AGS) Minimum detectable activity concentration (MDAC) Sensitivity 1.　Introduction 1 The minimum detectable activity concentration (MDAC) is the lowest activity concentration (Bq/g) that can be reliably detected under certain measurement conditions [ 1 1 ] and is an important technical specification of airborne gamma-ray spectrometry (AGS) [ 2 2 ] . Both the IAEA guidance and Chinese industry standards specify that before using AGS, parameters such as the detection sensitivity to the ground and the MDAC of AGS should be determined to ensure the accuracy of measurement results [ 3 3 , 4 4 ] . Numerous studies have been conducted to determine the factors that influence the MDAC in AGS. Roy Pöllänen et al. studied the relationship between the minimum detectable activity and flight altitude. The minimum detectable activity of a detector for 137 Cs and 60 Co sources was obtained from data measured at different flight altitudes [ 5 5 ] . Casanovas et al. researched the MDAC of NaI and LaBr 3 for 131 I and 137 Cs sources [ 6 6 ] . Tang et al. calculated the MDAC of high-purity germanium (HPGe) and LaBr 3 for a variety of artificial radionuclides using the Monte Carlo (MC) method and investigated the effects of source term size, flight altitude, and gamma-ray energy on MDAC [ 7 7 , 8 8 ] . Ni et al. researched the effects of flight altitude, gamma-ray energy, background level, and other factors on MDAC using extensive measured data [ 9 9 ] . The above results showed that MDAC positively correlates with flight altitude, gamma-ray energy, and background level. Under the same measurement conditions, MDACs of different detectors also varied. The above studies used experimental or simulated data to analyze the relationships between MDAC and the aforementioned factors. In addition, the theory for MDAC indicates that MDAC is influenced by the absolute detection efficiency, background level, and emission probability of gamma photons [ 1 1 ] . However, no theory indicates the relationship between MDAC and intrinsic efficiency, gamma-ray energy, energy resolution, total volume, and flight altitude. The authors began with the basic principle of AGS, then derived a theory for MDAC considering detection sensitivity to ground, and finally analyzed the relationship between MDAC and the above factors based on this theory. Using the Geant4 software, the intrinsic efficiencies of NaI and CeBr 3 scintillation counters with different volumes were simulated for different energies. These intrinsic efficiencies were calculated using MDAC theory. Experimental devices with a NaI scintillation counter (one 10 cm × 10 cm × 40 cm) and a CeBr 3 scintillation counter (six φ4.5 cm × 5 cm) were established. Based on the elemental content of the calibration pads obtained using the experimental device, the accuracy of the theory was evaluated. Finally, the MDAC of AGS was calculated at different intrinsic efficiencies, energy resolution levels, total volumes, and flight altitudes, and the experimental devices were used for field flight experiments. Accurate evaluation of the MDAC of AGS and in-depth research on the factors influencing MDAC can guide the work of AGS in measuring low radioactivity areas. 2.　Theory Derivation 1 2.1　Detection Sensitivity of AGS to Ground 2 The detection sensitivity to the ground ( Sen , cps/(μg/g)) describes the degree of change in the net counts of full-energy peaks n p (cps) in the gamma-ray spectrum caused by a change in radioactive elements C (μg/g) in a geological body, and it is given by [ 10 10 ] : <math display="block" id="M1"><mrow><mi>S</mi><mi>e</mi><mi>n</mi><mo>=</mo><mfrac><mrow><mtext>d</mtext><msub><mi>n</mi><mtext>p</mtext></msub></mrow><mrow><mtext>d</mtext><mi>C</mi></mrow></mfrac><mo>,</mo></mrow></math> where d C is the change in the radioactive element content C in a geological body, and d n p is the change in the net count of the full-energy peaks n p for this radioactive element. It is assumed that the radioactive substance in a geological body is uniformly distributed. The density of soil is given by ρ (g/cm 3 ), V is volume (cm 3 ), m is mass (g), the content of certain radioactive elements is given by C (g/g); A is specific activity (Bq/g), T 1/2 is half-life (s), A γ is atomic weight (g/mol), λ is decay constant (s -1 ); N A is Avogadro constant, number of gamma rays with energy E released by each decay of a radioactive element is given as p γ (pcs), linear attenuation coefficient of gamma rays with energy E is given by μ 0 (m -1 ) for soil, and linear attenuation coefficient of gamma rays with energy E is μ 1 (m -1 ) for air. Additionally, the number of gamma rays with energy E in a volume of soil is represented by N V (cps/cm 3 ). In the spherical coordinate system, the solid angle of a gamma-ray spectrometer is given by 2 θ 0 , S p is cross-sectional area of the scintillation crystal (m 2 ), H represents the flight altitude of AGS (m), R is detection radius for the ground (m) ( R = H ×tan θ 0 ), and the intrinsic efficiency of the gamma-ray spectrometer is given by ε inp . Therefore, the net count rate of the full-energy peak ( n p ) for gamma rays with energy E in a gamma-ray spectrometer is expressed as [ 10 10 , 11 11 ] : <math display="block" id="M2"><mrow><msub><mi>n</mi><mtext>p</mtext></msub><mo>=</mo><msub><mi>ε</mi><mrow><mtext>inp</mtext></mrow></msub><msub><mi>S</mi><mtext>P</mtext></msub><mstyle displaystyle="true"><mrow><munderover><mo>∫</mo><mn>0</mn><mrow><mn>2</mn><mi>π</mi></mrow></munderover><mrow><mstyle displaystyle="true"><mrow><munderover><mo>∫</mo><mn>0</mn><mrow><msub><mi>θ</mi><mn>0</mn></msub></mrow></munderover><mrow><mstyle displaystyle="true"><mrow><munderover><mo>∫</mo><mrow><msub><mi>r</mi><mn>0</mn></msub></mrow><mrow><mi>r</mi><mn>1</mn></mrow></munderover><mrow><mfrac><mrow><msub><mi>N</mi><mi>V</mi></msub></mrow><mrow><mn>4</mn><mi>π</mi><msup><mi>r</mi><mn>2</mn></msup></mrow></mfrac><msup><mi>e</mi><mrow><mo>−</mo><msub><mi>μ</mi><mn>0</mn></msub><mrow><mo>(</mo><mrow><mi>r</mi><mo>−</mo><msub><mi>r</mi><mn>0</mn></msub></mrow><mo>)</mo></mrow><mo>−</mo><msub><mi>μ</mi><mn>1</mn></msub><msub><mi>r</mi><mn>0</mn></msub></mrow></msup></mrow></mrow></mstyle></mrow></mrow></mstyle><mtext>d</mtext><mi>V</mi></mrow></mrow></mstyle><mo>=</mo><mfrac><mrow><msub><mi>ε</mi><mrow><mtext>inp</mtext></mrow></msub><mo>×</mo><msub><mi>S</mi><mtext>P</mtext></msub><mo>×</mo><mi>ρ</mi><mo>×</mo><mi>C</mi><mo>×</mo><msub><mi>p</mi><mi>γ</mi></msub><mo>×</mo><mi>Ψ</mi><mrow><mo>(</mo><mrow><msub><mi>μ</mi><mn>1</mn></msub><mi>H</mi><mo>,</mo><msub><mi>θ</mi><mn>0</mn></msub><mo>,</mo><msub><mi>μ</mi><mn>0</mn></msub><mi>L</mi></mrow><mo>)</mo></mrow></mrow><mrow><mtext>2</mtext><mtext>.40</mtext><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mtext>-</mtext><mn>24</mn></mrow></msup><mo>×</mo><msub><mi>A</mi><mi>γ</mi></msub><mo>×</mo><msub><mi>T</mi><mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></mrow></msub><mo>×</mo><mn>2</mn><msub><mi>μ</mi><mn>0</mn></msub></mrow></mfrac><mo>,</mo></mrow></math> where 2.40×10 -24 = 1/4.17×10 23 = 1 / ( N A ×ln2), and <math display="block" id="M3"><mtable columnalign="left"><mtr><mtd><mi>Ψ</mi><mrow><mo>(</mo><mrow><msub><mi>μ</mi><mn>1</mn></msub><mi>H</mi><mo>,</mo><msub><mi>θ</mi><mn>0</mn></msub><mo>,</mo><msub><mi>μ</mi><mn>0</mn></msub><mi>L</mi></mrow><mo>)</mo></mrow><mo>=</mo><mi>Φ</mi><mrow><mo>(</mo><mrow><msub><mi>μ</mi><mn>1</mn></msub><mi>H</mi></mrow><mo>)</mo></mrow><mo>−</mo><mi>cos</mi><msub><mi>θ</mi><mn>0</mn></msub><mi>Φ</mi><mrow><mo>(</mo><mrow><msub><mi>μ</mi><mn>1</mn></msub><mi>H</mi><mi>sec</mi><msub><mi>θ</mi><mn>0</mn></msub></mrow><mo>)</mo></mrow><mo>−</mo></mtd></mtr><mtr><mtd><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mi>Φ</mi><mrow><mo>(</mo><mrow><msub><mi>μ</mi><mn>0</mn></msub><mi>L</mi><mo>+</mo><msub><mi>μ</mi><mn>1</mn></msub><mi>H</mi></mrow><mo>)</mo></mrow><mo>+</mo><mi>cos</mi><msub><mi>θ</mi><mn>0</mn></msub><mi>Φ</mi><mrow><mo>[</mo><mrow><mrow><mo>(</mo><mrow><msub><mi>μ</mi><mn>0</mn></msub><mi>L</mi><mo>+</mo><msub><mi>μ</mi><mn>1</mn></msub><mi>H</mi></mrow><mo>)</mo></mrow><mi>sec</mi><msub><mi>θ</mi><mn>0</mn></msub></mrow><mo>]</mo></mrow><mo>,</mo></mtd></mtr></mtable></math> where <math id="M4"><mrow><mi>Φ</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mn>0</mn><mrow><mfrac><mi>π</mi><mn>2</mn></mfrac></mrow></msubsup><mrow><msup><mi>e</mi><mrow><mo>−</mo><mi>x</mi><mi>sec</mi><mi>θ</mi></mrow></msup><mi>sin</mi><mi>θ</mi><mtext>d</mtext><mi>θ</mi></mrow></mrow></mstyle></mrow></math> is the Ginger function; and, when x = 0, Φ(0) = 1; when x = ∞, Φ (∞) = 0. Φ ( x ) can be obtained from the Ginger function table [ 11 11 ] . Combining Eqs. (1) and (2), we obtain: <math display="block" id="M5"><mrow><mi>S</mi><mi>e</mi><mi>n</mi><mo>=</mo><mfrac><mrow><msub><mi>S</mi><mtext>P</mtext></msub><mo>×</mo><mi>ρ</mi><mo>×</mo><msub><mi>p</mi><mi>γ</mi></msub><mo>×</mo><msub><mi>ε</mi><mrow><mtext>inp</mtext></mrow></msub></mrow><mrow><mtext>2</mtext><mtext>.40</mtext><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mtext>-</mtext><mn>24</mn></mrow></msup><mo>×</mo><msub><mi>A</mi><mi>γ</mi></msub><mo>×</mo><msub><mi>T</mi><mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></mrow></msub><mo>×</mo><mn>2</mn><msub><mi>μ</mi><mn>0</mn></msub></mrow></mfrac><mo>×</mo><mi>Ψ</mi><mrow><mo>(</mo><mrow><msub><mi>μ</mi><mn>1</mn></msub><mi>H</mi><mo>,</mo><msub><mi>θ</mi><mn>0</mn></msub><mo>,</mo><msub><mi>μ</mi><mn>0</mn></msub><mi>L</mi></mrow><mo>)</mo></mrow><mtext> (cps/</mtext><mi>μ</mi><mtext>g/g))</mtext><mtext>.</mtext></mrow></math> 2.2　MDAC of AGS 2 The activity concentration of a radionuclide that releases gamma rays with energy E is given by A E (Bq/g). In the instrumental spectrum, A E is derived from the net count rate of the full-energy peak n INP and the detection sensitivity to the ground, as: <math display="block" id="M6"><mrow><msub><mi>A</mi><mtext>E</mtext></msub><mo>=</mo><mfrac><mrow><mi>C</mi><mo>×</mo><mn>4.17</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>23</mn></mrow></msup></mrow><mrow><msub><mi>A</mi><mi>γ</mi></msub><mo>×</mo><msub><mi>T</mi><mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></mrow></msub></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>4.17</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>23</mn></mrow></msup></mrow><mrow><msub><mi>A</mi><mi>γ</mi></msub><mo>×</mo><msub><mi>T</mi><mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></mrow></msub></mrow></mfrac><mo>×</mo><mfrac><mrow><msub><mi>n</mi><mrow><mtext>INP</mtext></mrow></msub></mrow><mrow><mi>S</mi><mi>e</mi><mi>n</mi></mrow></mfrac><mo>.</mo></mrow></math> In the instrumental spectra, the net peak count is equal to the full-energy gross count peak minus the background count. Figure 1 Figure 1 shows the peak patterns of the full-energy peaks in the instrumental spectrum with different energy resolutions. It is assumed that the shape of the full-energy peak is described by a Gaussian function (the standard deviation of the Gaussian function is σ) [ 1 1 ] . Meanwhile, the background of the instrumental spectrum is described by a linear function. According to the instrumental spectrum, the left and right channels of the range of interest are expressed as i L and i R , while the counts of the left and right channel are given by f L and f R , respectively, the peak channel is i E , and the count of the peak channel is f E . The background count n B can be written as: <math display="block" id="M7"><mrow><msub><mi>n</mi><mtext>B</mtext></msub><mtext>=</mtext><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>×</mo><mrow><mo>(</mo><mrow><msub><mi>f</mi><mtext>L</mtext></msub><mo>+</mo><msub><mi>f</mi><mtext>R</mtext></msub></mrow><mo>)</mo></mrow><mo>×</mo><mrow><mo>(</mo><mrow><msub><mi>i</mi><mtext>R</mtext></msub><mo>−</mo><msub><mi>i</mi><mtext>L</mtext></msub></mrow><mo>)</mo></mrow><mo>.</mo></mrow></math> It is assumed that both the left and right boundaries of the full-energy peak are on the channels that correspond to 0.1 times the peak position of the full-energy peak <math id="M8"><mrow><msub><mi>i</mi><mtext>R</mtext></msub><mo>−</mo><msub><mi>i</mi><mtext>L</mtext></msub><mo>=</mo><mn>2</mn><mi>σ</mi><msqrt><mrow><mn>2</mn><mi>ln</mi><mn>10</mn></mrow></msqrt></mrow></math> . Combining Eq. (6) and the definition of the energy resolution η [ 10 10 , 11 11 ] , we obtain: <math display="block" id="M9"><mrow><msub><mi>n</mi><mtext>B</mtext></msub><mtext>=</mtext><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>×</mo><mrow><mo>(</mo><mrow><msub><mi>f</mi><mtext>L</mtext></msub><mo>+</mo><msub><mi>f</mi><mtext>R</mtext></msub></mrow><mo>)</mo></mrow><mo>×</mo><mrow><mo>(</mo><mrow><mi>η</mi><mo>×</mo><msub><mi>i</mi><mtext>E</mtext></msub><mo>×</mo><mn>0.549</mn></mrow><mo>)</mo></mrow><mo>.</mo></mrow></math> When n INP = 2.71+4.65 <math id="M10"><mrow><msqrt><mrow><msub><mi>n</mi><mtext>B</mtext></msub></mrow></msqrt></mrow></math> (with a confidence interval of 95%), it can be considered that n INP is the count of the full-energy peak in the instrumental spectrum, and this count is contributed by the lowest activity concentration of the radioactive element that can be identified by the detector [ 10 10 ] . Therefore, the MDAC of the measurement system should be: <math display="block" id="M11"><mrow><mi>M</mi><mi>D</mi><mi>A</mi><mi>C</mi><mo>=</mo><mfrac><mrow><mn>4.17</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>23</mn></mrow></msup></mrow><mrow><msub><mi>A</mi><mi>γ</mi></msub><mo>×</mo><msub><mi>T</mi><mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></mrow></msub></mrow></mfrac><mo>×</mo><mfrac><mrow><msub><mi>n</mi><mrow><mtext>INP</mtext></mrow></msub></mrow><mrow><mi>S</mi><mi>e</mi><mi>n</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mrow><mo>(</mo><mrow><mtext>2</mtext><mtext>.71+4</mtext><mtext>.65</mtext><mo>×</mo><msqrt><mrow><mrow><mo>(</mo><mrow><msub><mi>f</mi><mtext>L</mtext></msub><mo>+</mo><msub><mi>f</mi><mtext>R</mtext></msub></mrow><mo>)</mo></mrow><mo>×</mo><mi>η</mi><mo>×</mo><msub><mi>i</mi><mtext>E</mtext></msub></mrow></msqrt></mrow><mo>)</mo></mrow><mo>×</mo><mn>2</mn><msub><mi>μ</mi><mn>0</mn></msub></mrow><mrow><msub><mi>S</mi><mtext>P</mtext></msub><mo>×</mo><mi>ρ</mi><mo>×</mo><msub><mi>p</mi><mi>γ</mi></msub><mo>×</mo><msub><mi>ε</mi><mrow><mtext>inp</mtext></mrow></msub><mo>×</mo><mi>Ψ</mi><mrow><mo>(</mo><mrow><msub><mi>μ</mi><mn>1</mn></msub><mi>H</mi><mo>,</mo><msub><mi>θ</mi><mn>0</mn></msub><mo>,</mo><msub><mi>μ</mi><mn>0</mn></msub><mi>L</mi></mrow><mo>)</mo></mrow></mrow></mfrac><mo>.</mo></mrow></math> To analyze MDAC more easily, let 2 μ 0 /( ρ × p γ ) = k , where k is a constant. Eq. (9) can then be simplified as: <math display="block" id="M12"><mrow><mi>M</mi><mi>D</mi><mi>A</mi><mi>C</mi><mo>≈</mo><mi>k</mi><mo>×</mo><mfrac><mrow><mn>2.71</mn><mtext>+</mtext><mn>4.65</mn><msqrt><mrow><mrow><mo>(</mo><mrow><msub><mi>f</mi><mtext>L</mtext></msub><mo>+</mo><msub><mi>f</mi><mtext>R</mtext></msub></mrow><mo>)</mo></mrow><mo>×</mo><mi>η</mi><mo>×</mo><msub><mi>i</mi><mtext>E</mtext></msub></mrow></msqrt></mrow><mrow><msub><mi>S</mi><mtext>P</mtext></msub><mo>×</mo><msub><mi>ε</mi><mrow><mtext>inp</mtext></mrow></msub><mo>×</mo><mi>Ψ</mi><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow></mrow></mfrac><mo>.</mo></mrow></math> From Eq. (9), with increases in the intrinsic efficiency and cross-sectional area of scintillation crystals, the MDAC value decreases, and with increases in the flight altitude and energy resolution values of the scintillation counter, the MDAC value increases. The MDAC of AGS can be optimized by reducing the flight altitude and the value of the energy resolution and increasing the intrinsic efficiency and the cross-sectional area of the scintillation crystals. 3.　Simulation Analysis 1 3.1　Model of an AGS Detector 2 The scintillation counter is the core unit of AGS. It mainly includes scintillation crystals, photomultiplier tubes, and associated electronics modules. The types of scintillation crystals that can be used in AGS are NaI, CeBr 3 , and LaBr 3 [ 5 5 - 8 8 5-8 , 12 12 - 14 14 12-14 ] . 138 La in LaBr 3 scintillation crystals is radioactive, and its characteristic peak at 1.44 MeV overlaps with the full-energy peak of 40 K at 1.46 MeV. In the radioactivity exploration work using AGS, 40 K was the object isotope. Therefore, we constructed two experimental devices: a scintillation counter based on CeBr 3 scintillation crystals (referred to as the CeBr 3 scintillation counter) and a scintillation counter based on NaI (Tl) scintillation crystals (referred to as the NaI scintillation counter). All scintillation counters used the automatic spectrum stabilization method of a 241 Am source [ 15 15 ] . The automatic spectrum stabilization method of the 241 Am source is a dynamic peak correction technique that uses the full-energy peak for the 59.5 keV gamma rays emitted from the Am source as a reference peak and employs the algorithm built into the energy spectrometer to correct the peak position. Sketches of the scintillation counters are shown in Fig. 2 Fig. 2 . The CeBr 3 scintillation counter was equipped with six φ4.5 cm × 5 cm CeBr 3 scintillation crystals, and the output spectrum is the synthetic spectrum of the six scintillation crystals (the synthesis is linearly cumulative). CeBr 3 scintillation crystals were produced by the Beijing Glass Research Institute, and the energy resolution of the crystals was 4% (at gamma-ray energy of 0.662 MeV). The NaI scintillation counter was equipped with one 10 cm × 10 cm × 40 cm scintillation crystal. NaI scintillation crystals were produced by Saint-Gobain Crystals, and the energy resolution of the crystals was 8.7% (at gamma-ray energy of 0.662 MeV). The box of the scintillation counter was made of a carbon fiber sheet (thickness of 0.15 cm). The spaces between the scintillation crystals were filled with foam (density of 0.95 g/cm 3 ). Moreover, the scintillation counter was hung below an unmanned aerial vehicle. 3.2　MC Simulation 2 The MC technique is a powerful tool for simulating particle transport. In this study, the detection efficiency of AGS was calculated using the Geant4 [ 16 16 ] . Owing to limitations in the manufacturing process and the production costs of scintillation crystals, the maximum volume of a single-crystal has an upper limit. At present, the largest volume sizes of CeBr 3 and NaI scintillation crystals available on the market are φ7.62 cm × 7.62 cm and 10 cm × 10 cm × 40 cm, respectively. Numerous research results and application examples have shown that the combination of multiple scintillation crystal arrays is effective for increasing the volume of scintillation counters [ 2 2 , 17 17 , 18 18 ] . Thus, we increased the volume of scintillation crystals using arrays. Table 1 Table 1 lists information about different scintillation counters, including crystal type, single-crystal size, crystal number, and total volume. We simulated the intrinsic efficiency of scintillation counters with different volumes (as shown in Table 1 Table 1 ). 3 1 NaI 3 1 CeBr 3 1 1 Single-crystal size  1 1 Number(pcs) 1 1 Total volume (cm 3 ) 1 1 Single-crystal size 1 1 Number(pcs) 1 1 Total volume (cm 3 ) 1 1 φ7.62 cm×7.62 cm 1 1 1 1 1 347.3 1 1 φ4.5 cm×5 cm 1 1 2 1 1 159.0 1 1 1 1 2 1 1 694.6 1 1 1 1 4 1 1 317.9 1 1 1 1 4 1 1 1,389.3 1 1 1 1 6 1 1 476.9 1 1 10 cm×10 cm×20 cm 1 1 1 1 1 2,000 1 1 1 1 8 1 1 635.9 1 1 10 cm×10 cm×40 cm 1 1 1 1 1 4,000 1 1 1 1 16 1 1 1,271.7 1 1 1 1 2 1 1 8,000 1 1 1 1 32 1 1 2,543.4 1 1 1 1 4 1 1 16,000 1 1 1 1 64 1 1 5,086.8 The intrinsic detection efficiency is the ratio of the number of particles identified by the detector to the number of particles injected into the detector. When simulating the intrinsic efficiency, the physical structure of the detector presented in Sect. 3.1 was used to model the detector. For CeBr 3 scintillation crystals, an Al-Mg alloy was used as the material of the crystal shell (thickness of 0.2 cm, density of 2.7 g/cm 3 ). The scintillation crystals were placed in a rectangular pattern and equally spaced. The distance between adjacent scintillation crystals was 0.8 cm, and the distance of the outermost crystal from the carbon fiber plate was 0.87 cm. For NaI scintillation crystals, the material of the crystal shell was stainless steel (density of 7.8 g/cm 3 ). For crystals with dimensions of 10 cm × 10 cm × 40 cm, the thickness of the stainless steel was 0.05 cm. The distance between adjacent scintillation crystals was 2 cm, and the distance of the outermost crystal from the carbon fiber plate was 1 cm. For crystals with dimensions of φ7.62 cm × 7.62 cm, the thickness of the stainless steel was 0.2 cm. The pattern used for the placement of the scintillation crystals was the same as that used for the CeBr 3 crystals. The number of particles identified by the detector is the sum of the full-energy peak in the energy spectrum. The number of particles injected into the detector was the sum of primary particles injected into the bottom and sides of the detector box. The number of initial particles sampled was set to 1 × 10 7 . The shape of the radioactive source is round platform (radius of 110 cm, height of 60 cm). The energy was set to a single value (1.46, 1.76, and 2.61 MeV). Air was used as the filling medium outside the detector box. 3.3　Simulation Result 2 Using Geant4 software, the intrinsic efficiencies of NaI and CeBr 3 for gamma-ray energies of 1.46, 1.76, and 2.61 MeV were simulated for different total volumes ( Table 2 Table 2 ). 4 1 NaI 4 1 CeBr 3 1 1 Total volume (cm 3 ) 3 1 Energy (MeV) 1 1 Total volume (cm 3 ) 3 1 Energy (MeV) 1 1 1 1 1.46 1 1 1.76 1 1 2.62 1 1 1 1 1.46 1 1 1.76 1 1 2.62 1 1 347.3 1 1 4.28 1 1 3.53 1 1 2.82 1 1 159.0 1 1 3.39 1 1 3.15 1 1 1.83 1 1 694.6 1 1 5.61 1 1 4.96 1 1 3.73 1 1 317.9 1 1 4.21 1 1 3.56 1 1 2.65 1 1 1,389.3 1 1 7.60 1 1 6.67 1 1 5.13 1 1 476.9 1 1 4.97 1 1 4.45 1 1 3.14 1 1 2,000 1 1 12.25 1 1 10.82 1 1 8.19 1 1 635.9 1 1 5.52 1 1 4.63 1 1 3.52 1 1 4,000 1 1 16.75 1 1 15.33 1 1 12.42 1 1 1,271.7 1 1 6.63 1 1 5.74 1 1 4.27 1 1 8,000 1 1 20.53 1 1 18.80 1 1 15.88 1 1 2,543.4 1 1 6.99 1 1 6.20 1 1 4.72 1 1 16,000 1 1 23.09 1 1 21.56 1 1 18.54 1 1 5,086.8 1 1 8.97 1 1 7.91 1 1 6.04 As shown in Table 2 Table 2 , the intrinsic efficiency varies with the total volume. The larger the volume, the higher the intrinsic efficiency. The intrinsic efficiency is difficult to derive from theory and is independent of the measurement conditions. Therefore, the effective detection efficiency ε eff (% ∙ m 2 ) is defined as ε eff = ε inp × S p . This indicates that the effective detection efficiency is the product of the intrinsic efficiency and cross-sectional area of the scintillation crystal (m 2 ). For example, the effective detection efficiency of six φ 4.5 cm × 4.5 cm CeBr 3 scintillation crystals for 1.46-MeV gamma rays is 0.047 % ∙ m 2 . Using the data presented in Tables 1 Tables 1 and 2 2 , the effective detection efficiencies of the two scintillation counters were calculated for different energies at different volumes. In the exploration of radioactivity via AGS, the elemental contents of K, U, and Th in the ground are mainly obtained using the total energy peak count rates of 40 K (1.46 MeV), 214 Bi (1.76 MeV), and 208 Tl (2.61 MeV). The volume of six φ 4.5 cm × 5 cm CeBr 3 scintillation crystals used to count K is taken as an example. It is assumed that the atmospheric pressure is one standard atmosphere, air temperature is 20°C, density of air is 1.205 kg/m 3 [ 2 2 ] , density of the geological body is 2.3 kg/m 3 , line reduction factor of the geological body for a 1.46-MeV gamma-ray is 12.3 /m, line reduction factor of the air for a 1.46-MeV gamma-ray is 0.0129 /m, number of gamma rays produced in each decay of 40 K is 0.11, half-life of 40 K is 4.01×10 16 s, abundance of 40 K in the geological body is 0.014 %, and background count rate is 4.47 cps. Calculated from Eq. (10), at the ground surface, the MDAC of the CeBr 3 scintillation counter at 40 K is 52.69. Based on the above parameters, the MDACs of CeBr 3 at 40 K were calculated for changes in the effective detection efficiency, energy resolution, and flight altitude ( Table 3 Table 3 ). 1 1 Effective detection efficiency 1 1 MDAC 1 1 Energy resolution 1 1 MDAC 1 1 Flight altitude 1 1 MDAC 1 1 0.02 1 1 131.5 1 1 3 1 1 41.4 1 1 0 1 1 52.9 1 1 0.12 1 1 21.9 1 1 4 1 1 47.8 1 1 20 1 1 104.0 1 1 0.22 1 1 12.0 1 1 5 1 1 53.4 1 1 40 1 1 166.5 1 1 0.32 1 1 8.2 1 1 6 1 1 58.5 1 1 60 1 1 252.7 1 1 0.42 1 1 6.3 1 1 7 1 1 63.2 1 1 80 1 1 373.4 1 1 0.52 1 1 5.1 1 1 8 1 1 67.6 1 1 100 1 1 539.5 1 1 0.62 1 1 4.2 1 1 9 1 1 71.7 1 1 120 1 1 766.2 1 1 0.72 1 1 3.7 1 1 10 1 1 75.5 1 1 140 1 1 1,099.2 According to the results in Tab. 3 Tab. 3 for 40 K, if the effective detection efficiency is doubled or the flight altitude is reduced by approximately half, the MDAC value of AGS will be reduced by approximately half. Improving the energy resolution can also improve the MDAC of AGS. 4.　Experiment and Result Analyses 1 4.1　Experimental Verification 2 The physical experimental validation was mainly based on actual measurements of the calibration pads using the experimental device. The theory and the stripping ratio method were employed to calculate the elemental content in the calibration pads, and the calculated results were compared with the actual contents to determine relative errors and test the theory. The calibration pads are located in Luojiang County, Sichuan Province, and belong to the nuclear industry radiation testing and protection equipment measurement and certification station. Stripping ratios are recommended in the IAEA guidelines and Chinese industry standards [ 3 3 , 4 4 ] . The calibration pads include the background pad, K pad, U pad, Th pad, and mixture pad. Figure 3 Figure 3 shows the spectra measured using the CeBr 3 scintillation counter for the mixture pad. The measured spectra are in the range of 0.41–2.81 MeV. The difference between Eqs. (5) and (10) involves taking the value of n INP ; when n INP was 2.71+4.65 <math id="M13"><mrow><msqrt><mrow><msub><mi>n</mi><mtext>B</mtext></msub></mrow></msqrt></mrow></math> , the value derived from Eq. (10) was the MDAC; when n INP was taken as the net peak count of the full-energy peak, the value derived from Eq. (5) represents the activity concentration. Eq. (5) was used to calculate the contents of K, U, and Th in the mixture pad, which were also calculated from the stripping ratios. The relative errors between the actual contents of K, U, and Th and those listed above obtained from the two different methods were calculated ( Tab. 4 Tab. 4 ). The net peak count was calculated using the energy window method recommended by the IAEA guidelines and Chinese industry standards (i.e., the sum of the energy spectrum within the range of the energy window minus the background counts within that range) [ 3 3 , 4 4 ] . The ranges of energy windows for K, U, and Th are 1.370–1.570, 1.66–1.86, and 2.41–2.81 MeV, respectively. 1 1 Type of scintillation crystal 3 1 CeBr 3 3 1 NaI 1 1 1 1 K 1 1 U 1 1 Th 1 1 K 1 1 U 1 1 Th 1 1 Actual content 1 1 2.49% 1 1 61.69 μg/g 1 1 183.93 μg/g 1 1 2.49 % 1 1 61.69 μg/g 1 1 183.93 μg/g 1 1 Content obtained from stripping ratios 1 1 2.41% 1 1 63.67 μg/g 1 1 190.99 μg/g 1 1 2.61 % 1 1 59.23 μg/g 1 1 180.18 μg/g 1 1 Relative error vs. actual content (%) 1 1 -3.21 1 1 3.21 1 1 3.84 1 1 -4.82 1 1 -4.00 1 1 -2.04 1 1 Content obtained by theory 1 1 2.39% 1 1 60.72 μg/g 1 1 190.40 μg/g 1 1 2.41 % 1 1 64.32μg/g 1 1 190.10 μg/g 1 1 Relative error vs. actual content (%) 1 1 -4.00 1 1 -0.16 1 1 3.52 1 1 -3.27 1 1 4.26 1 1 3.35 Table 4 Table 4 shows that the relative errors between the actual contents and the contents obtained using the stripping ratios were less than 5%, and the relative errors between the actual contents and the contents obtained using Eq. (5) were less than 4%. The comparison results demonstrate that calculations using Eq. (5) are reliable. 4.2　Relationship between MDAC and different factors 2 Using this theory, the values of MDAC were calculated for different values of the effective detection efficiency, energy resolution, and flight altitude, and the relationship between the values of MDAC and those of these three variables was analyzed. If we assume that the CeBr 3 scintillation counter exhibited an energy resolution of 4.9%, the volume was six φ4.5 cm × 5 cm, and intrinsic efficiency for 1.46-MeV gamma rays was 4.97%, then the calculated value for the MDAC is called MDAC 0 . To compare the calculations more easily, the results were normalized. The normalization method was used to divide the MDACs obtained in different cases by MDAC 0 . Using the measured data and detection sensitivity to ground from the stripping ratios, the values were determined using <math id="M14"><mrow><mi>M</mi><mi>D</mi><mi>A</mi><mi>C</mi><mo>=</mo><mfrac><mrow><mn>4.17</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>23</mn></mrow></msup></mrow><mrow><msub><mi>A</mi><mi>γ</mi></msub><mo>×</mo><msub><mi>T</mi><mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></mrow></msub></mrow></mfrac><mo>×</mo><mfrac><mrow><msub><mi>n</mi><mrow><mtext>INP</mtext></mrow></msub></mrow><mrow><mi>S</mi><mi>e</mi><mi>n</mi></mrow></mfrac></mrow></math> in Eq. (10). The theoretical values were calculated using <math id="M15"><mrow><mi>M</mi><mi>D</mi><mi>A</mi><mi>C</mi><mo>=</mo><mfrac><mrow><mrow><mo>(</mo><mrow><mtext>2</mtext><mtext>.71+4</mtext><mtext>.65</mtext><mo>×</mo><msqrt><mrow><mrow><mo>(</mo><mrow><msub><mi>f</mi><mtext>L</mtext></msub><mo>+</mo><msub><mi>f</mi><mtext>R</mtext></msub></mrow><mo>)</mo></mrow><mo>×</mo><mi>η</mi><mo>×</mo><msub><mi>i</mi><mtext>E</mtext></msub></mrow></msqrt></mrow><mo>)</mo></mrow><mo>×</mo><mn>2</mn><msub><mi>μ</mi><mn>0</mn></msub></mrow><mrow><msub><mi>S</mi><mtext>P</mtext></msub><mo>×</mo><mi>ρ</mi><mo>×</mo><msub><mi>p</mi><mi>γ</mi></msub><mo>×</mo><msub><mi>ε</mi><mrow><mtext>inp</mtext></mrow></msub><mo>×</mo><mi>Ψ</mi><mrow><mo>(</mo><mrow><msub><mi>μ</mi><mn>1</mn></msub><mi>H</mi><mo>,</mo><msub><mi>θ</mi><mn>0</mn></msub><mo>,</mo><msub><mi>μ</mi><mn>0</mn></msub><mi>L</mi></mrow><mo>)</mo></mrow></mrow></mfrac></mrow></math> in Eq. (10). The relative MDACs were calculated for different effective detection efficiencies, energy resolutions, and flight altitudes under the same conditions, and the differences between the theoretical curves and the measured data were compared ( Fig. 4 Fig. 4 ). The MDACs were calculated for different effective detection efficiencies under the same conditions and were normalized. As shown by the black line in Fig. 4 Fig. 4 , the calculated values were in accordance with the theoretical curve, and the relative errors between the measured and calculated values were 3.8% (CeBr 3 ) and 4% (NaI). Because the volume and energy resolution of NaI in the experiment were different from those of CeBr 3 , the values of MDAC measured with NaI were normalized to equate the volume and energy resolution of NaI to those of CeBr 3 . The effective detection efficiency was calculated as the product of the intrinsic efficiency and the cross-sectional area of the scintillation crystal. Intrinsic efficiency is an intrinsic parameter of the detector itself. Therefore, when designing a detector, it is important to select scintillation crystals with a large total volume and high intrinsic efficiency. The MDACs were calculated for the same conditions but with different energy resolutions, and the calculated values were normalized. As shown by the red line in Fig. 4 Fig. 4 , the calculated values were in accordance with the theoretical curve, and the relative errors between the measured and calculated values were 3.8% (CeBr 3 ) and 6.8% (NaI). Similarly, the MDAC of NaI was normalized. The MDAC value can be optimized by increasing the energy resolution (decreasing the η value), which is an intrinsic parameter of the detector itself. The energy resolution is influenced by the type of scintillation crystal and the design of the circuit module. Therefore, when designing the detector, it is important to choose a scintillation crystal with a low energy resolution and optimize the circuit module. In research on the effect of flight altitude on MDAC, the MDAC value at 100 m was used as a baseline value and to normalize the other MDAC values. As shown by the blue line in Fig. 4 Fig. 4 , the calculated values are in accordance with the theoretical curve. The AGS system was hung below an unmanned aerial vehicle, and the gamma-ray spectrum was measured on the ground surface to obtain background data at altitudes in the range of 50–100 m. The relative errors between the measured and calculated values of MDAC were all less than 5% (i.e., 4.65%, 4.65%, -2.41%, 4.65%, and 2.35%). The MDAC value increased as the flight altitude increased. When using AGS to measure the gamma-ray spectrum on the ground surface, the flight altitude should be minimized to optimize the MDAC of the AGS. 4.3　Experiments on-field applications 2 The field application experimental area was located in a prospective metallogenic area in Jiangxi Province, China. The experimental area is 2.2 km 2 . The experimental device was the NaI scintillation counter (see Section 3.1 for details of the experimental devices), which was installed on the bottom of an F-120 single-rotor unmanned aerial vehicle. The flight altitude and flight speed of the unmanned aerial vehicle were 80 m and 10 m/s, respectively. The AGS sampling time was set to 1 s. There are 1,960 sets of measured data, including geodetic coordinates, GPS heights, radar heights, and net peak areas of the K, U, and Th elemental full-energy peaks. The data for the net peak areas were highly corrected. Hence, only the measurement result for 1.46 MeV is shown here. According to the statistics for data measured with the NaI scintillation counter, the activity concentration of elemental K had the mean value of 110.1 Bq/g, variance of 11.88, maximum value of 153.0 Bq/g, minimum value of 73.9 Bq/g, and median value of 110.1 Bq/g. The MDAC measured with the NaI scintillation counter at 80 m for energy of 1.46 MeV was calculated as 70.1 Bq/g. All activity concentration values in the data measured by the NaI scintillation counter were greater than the MDAC values. A contour plot of the activity concentrations of elemental K distribution measured by the NaI scintillation counter in the experimental area is displayed in Fig. 5 Fig. 5 . As shown in Fig. 5 Fig. 5 , the results measured by the NaI scintillation counter show that the distribution of activity concentrations with high, medium, and low values has a certain regular character. There were no significantly high-value areas in the experimental area, and two relatively high-value areas occurred in the western and central-eastern parts of the experimental area; medium-value areas were located in the central, central-eastern, and northern parts of the experimental area, and their areas were large and connected; while low-value areas were mainly concentrated in the southern part of the experimental area and occurred to a certain extent in the eastern part of the experimental area. The experimental data from field applications demonstrate the possibility of obtaining a reliable MDAC for the detector using a theoretical formula. 5.　Conclusion 1 MDAC is an important technical specification for AGS, and clarification of the influencing factors of MDAC can provide theoretical guidance for the design of AGS systems. Although some studies have shown that MDAC is influenced by factors such as intrinsic efficiency, energy resolution, total volume, and flight altitude, there is still no unified theory that directly and concretely describes the functional relationships between MDAC and these influencing factors. Therefore, we investigated the factors influencing MDAC, derived a theory that incorporated the above parameters, and verified the accuracy of the theory. The verification results indicate that the relative errors between the results calculated from the derived theoretical formula and the actual values were less than 4%. When designing an AGS system, it is recommended that reduced flight altitude and scintillation counter high intrinsic detection efficiencies and excellent energy resolution, and large total volumes should be used to reduce the MDAC value of an AGS system. The proposed theory of MDAC can guide the work of AGS in measuring areas of low radioactivity. References L.A. 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The first draft of the manuscript was written by Yi Gu and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript. 10.1007/s41365-021-00951-6 This work was supported by the Sichuan Science and Technology Program (No. 2020JDRC0108) and the National Science Foundation of China (Nos. 41774147 and 41774190). 2021-03-24 2021-06-07 2021-06-11 2021-10-07 2021-10-01 2021-11-29 © China Science Publishing & Media Ltd. (Science Press), Shanghai Institute of Applied Physics, the Chinese Academy of Sciences, ChineseNuclear Society and Springer Nature Singapore Pte Ltd. 2021 This is a post-peer-review, pre-copyedit version of an article published in Nuclear Science and Techniques. The final authenticated version is available online at: http://dx.doi.org/10.1007/s41365-021-00951-6 http://dx.doi.org/10.1007/s41365-021-00951-6 . 2021 Nuclear Science and Techniques 10 32