Introduction

Radioactive gases ²²²Rn and ²²⁰Rn are colorless, tasteless, and odorless. They are universally present in the walls, floors, and ceilings of buildings, as well as in the surrounding soil, before permeating into the cracks of a building and entering indoor spaces [1, 2]. Their decay products typically accumulate in closed spaces or poorly ventilated areas such as basements, mines, and warehouses [3, 4]. The generated progeny atoms recombine with air ions and other air impurity clusters to form clusters of size 0.5 nm – 5 nm, which are referred to as radioactive unattached particles [5-7]. After this process, the clusters attach to submicron-sized aerosol particles in the air within 1-100 s, forming radioactive aerosols or attached particles with sizes ranging from 100 nm to 500 nm [8, 9]. During breathing, radioactive progeny aerosols are inhaled into the respiratory organs. Some aerosols remain adsorbed in the respiratory system, where they continue to undergo radioactive decay [10]. During this process, the progeny of ²²²Rn and ²²⁰Rn can cause continuous tissue damage, leading to lung cancer. The inhalation doses are contributed by their progeny, but not by the gas itself [11]. Therefore, numerous researchers conducted studies on detectors for measuring ²²²Rn and ²²⁰Rn progeny concentrations.

Various detectors have been developed for measuring alpha particles emitted from ²²²Rn and ²²⁰Rn progeny in air, including ZnS(Ag)-coated scintillation cells [12, 13], the imaging plate (IP) [14], solid-state nuclear track detectors (SSNTD) [15, 16], silicon semiconductor detectors [17, 18], and others. An alpha spectrometer, equipped with ULTRA ion-implanted silicon semiconductor detectors, features a high-energy resolution, low background, and appropriate detection efficiency. Therefore, it is frequently employed for the identification and quantitative analysis of mixtures of radionuclides based on the alpha particles emitted during decay [19, 20].

Previously, methods based on air filtration followed by measurements using an alpha spectrometer have been widely explored for the determination of mixed ²²²Rn and ²²⁰Rn progenies in environmental air [21-25]. Harley et al. proposed a method that includes a 60-min sampling period, with counts considered at five intervals within 2440 min, to estimate the individual concentrations of ²²²Rn and ²²⁰Rn progeny. Hence, the lung dose due to ²²⁰Rn progeny [21] is assessed. A comparable method employed in high-background areas in Yangjiang County, China, involved 30-min sampling, followed by five measurement intervals within 560 min, and ensuring on-site measurement work completion within ten hours [22]. For more accurate measurement of the concentrations of mixed ²²²Rn and ²²⁰Rn progeny, a set of optimum time intervals was obtained by comparing the uncertainties in the total alpha spectroscopy data across the three sets of measurement intervals. Each set of time intervals lasted 560 min and was calculated using the weighted least-squares method [23]. To assess the reliability of this air-filtration method, the sensitivity of the measurements of ²²²Rn and ²²⁰Rn concentrations in air to variations in alpha counting at three and five intervals (560 min) was examined [24]. They indicated that the ²²²Rn progeny reached saturation activities on the filter after 3 h of filtering, whereas the saturation activities of ²²⁰Rn were achieved after three days owing to the different half-lives in the ²²⁰Rn chain. The influences of ²²²Rn and ²²⁰Rn concentrations, filtering duration, and choice of measuring interval (560 min) on the relative standard deviations were further analyzed [25]. Their study indicated that satisfactory results using the total count method should be expected only in the case of high radon and thoron progeny concentrations in air.

In contrast to the total alpha-counting method, which requires five counting intervals, the alpha spectroscopy method enables the direct differentiation of alpha particles with varying energies emitted by ²¹⁸Po, ²¹⁴Po, ²¹²Bi, and ²¹²Po. This is widely preferred, particularly in the context of modern requirements for high measurement accuracy [26-29]. In 1978, Kerr introduced an α spectrometry technique to measure the activity concentration of short-lived radon and thoron progenies. This method, termed as Kerr–Tn, utilizes an inverse matrix and obtains five alpha counts within the region of interest (ROI). Three counting intervals were established, each with a sampling time of 10 min: 2–12 min, 15–30 min, and 200–220 min. The intervals were initiated after the completion of sample collection [30]. Compared to the Kerr-Tn method, both the PKU-Tn and Wicke–Tn methods exhibit the advantage of synchronizing the first counting interval with the sampling time. With an extended 30-min sampling time and a total measurement duration of 180 min, this results in an increase in methodological sensitivity by nearly 8.97 times and 3.56 times, respectively [31]. However, adopting a method of simultaneous sampling and measurement may lead to broadening of the measured alpha energy spectrum because there is a need for a distance between the detector and filter membrane to allow for gas circulation. This in turn affects the accuracy of the measurements [32]. Furthermore, previous fitting methods demonstrated significant advantages in alpha energy spectrum measurements. For the deconvolution of the alpha-particle spectra, improved peak shape formulae were implemented as fit functions in a spreadsheet application, and the optimum fit parameters were searched using built-in optimization routines [33]. Considering the entire spectrum background, background noise, and peaks with low statistical counts, a model was developed by fitting the typical shape of the alpha peak [34].

To complete the measurement within a relatively short period of time and adapt to simultaneous measurements at comparatively lower ²²²Rn and ²²⁰Rn progeny concentrations, a new mathematical model based on the radioactive decay law was proposed by performing a non-linear fitting on the counts of alpha particles with different energies over continuous measurement periods to deduce the concentrations of ²²²Rn and ²²⁰Rn progeny in air. Several measurements were performed using the alpha spectrometer. The results of the proposed method are then compared with those of the total count method [25]. This indicates that the newly proposed method shows more potential advantages in the fast simultaneous measurement of relatively low concentrations of ²²²Rn and ²²⁰Rn progeny in air.

Materials and methods

2.1

alpha emitter from ²²²Rn and ²²⁰Rn progeny

The decay chains of ²²²Rn and ²²⁰Rn are shown in Fig. 1 [35, 36], excluding branches with a low probability. There are some short-lived alpha emitters of radon and thoron progeny in the air, including RaA (²¹⁸Po, 6.0 MeV), RaC’ (²¹⁴Po, 7.69 MeV), ThA (²¹⁶Po, 6.78 MeV), ThC (²¹²Bi, 6.05 MeV), and ThC’ (²¹²Po, 8.78 MeV). Given the air filtration method employed in this study, progeny with extremely short half-lives, such as ²¹⁴Po and ²¹²Po, quickly reach equilibrium with their parents during the sampling process. Additionally, it is assumed that ²¹²Pb originates directly from ²²⁰Rn due to the short half-life of ²¹⁶Po. In reality, ²¹²Bi emits an alpha only 36% of the time, whereas the remaining 64% of the time it emits a beta. This is followed almost immediately by an alpha through the decay of ²¹²Po (with a half-life of 0.3 s). Furthermore, ²¹²Po decays with an alpha energy of 8.78 MeV and ²⁰⁸Tl decays via beta decay only. It is impossible to distinguish the activity on the filter because of 6.05 MeV alpha particles from ThC and 6.0 MeV alpha particles from RaA. However, the activity can be partitioned between RaA and ThC using the 8.78 MeV alpha particle activity from ThC’ [30].

Fig. 1Full-screen View

(Color online) Decay chains of ²²²Rn and ²²⁰Rn

2.2

Measurement method of the progeny of ²²²Rn and ²²⁰Rn via alpha spectroscopy

The new method of simultaneously measuring the progeny concentrations of ²²²Rn and ²²⁰Rn by using an alpha spectrometer includes three time intervals: sampling duration (T₁), time interval (T₂) between the end of sampling and beginning of the measuring the progeny of ²²²Rn and ²²⁰Rn on the filter membrane, and time duration (T₃) for alpha spectrometer measurements from the beginning to the end. Three assumptions are made: 1) the flow rate remains steady during the sampling process; 2) the concentrations of RaA, RaB, RaC, ThB, and ThC in the environment remain constant throughout the sampling process; and 3) the filter exhibits the same collection efficiency for different ²²²Rn/²²⁰Rn progenies, and its self-absorption can be considered negligible. This procedure is illustrated in Fig. 2 and can be described as follows:

Fig. 2Full-screen View

(Color online) Process of simultaneously measuring the progeny concentrations of ²²²Rn and ²²⁰Rn using the alpha spectrometer

2.2.1

Sampling

Before the sampling begins, the number of atoms in ²²²Rn and ²²⁰Rn progenies on the filter membrane is zero. The switch of the vacuum pump is turned on to draw air containing the progeny of ²²²Rn and ²²⁰Rn into the sampler through a flow meter, where ²²²Rn and ²²⁰Rn are collected on the filter. Subsequently, air is returned to the chamber with mixed concentrations of ²²²Rn and ²²⁰Rn. The sampling duration lasts for fifteen minutes. During the sampling process, the quantity of ²²²Rn and ²²⁰Rn progeny on the filter membrane continuously increases. However, this quantity also decreases owing to the decay of parent nuclides and their decay. This can be expressed using the following system of differential equations. $\begin{array}{l}\{\begin{array}{l}\text{d}{N}_{\text{RaA}}/\text{d}t=QG{C}_{\text{RaA}}/{\lambda }_{\text{RaA}}-N_{\text{RaA}}{\lambda }_{\text{RaA}}\hfill \\ \text{d}{N}_{\text{RaB}}/\text{d}t=QG{C}_{\text{RaB}}/{\lambda }_{\text{RaB}}+N_{\text{RaA}}{\lambda }_{\text{RaA}}-N_{\text{RaB}}{\lambda }_{\text{RaB}}\hfill \\ \text{d}{N}_{\text{RaC}}/\text{d}t=QG{C}_{\text{RaC}}/{\lambda }_{\text{RaC}}+N_{\text{RaB}}{\lambda }_{\text{RaB}}-N_{\text{RaC}}{\lambda }_{\text{RaC}}\hfill \\ \text{d}{N}_{\text{ThB}}/\text{d}t=QG{C}_{\text{ThB}}/{\lambda }_{\text{ThB}}-N_{\text{ThB}}{\lambda }_{\text{ThB}}\hfill \\ \text{d}{N}_{\text{ThC}}/\text{d}t=QG{C}_{\text{ThC}}/{\lambda }_{\text{ThC}}+N_{\text{ThB}}{\lambda }_{\text{ThB}}-N_{\text{ThC}}{\lambda }_{\text{ThC}}\hfill \end{array}\hfill \end{array}$ (1) where G is the filtration efficiency, Q is the flow rate, C_RaA, C_RaB, C_RaC, C_ThB, C_ThC are the activity concentrations of RaA, RaB, RaC, ThB and ThC, respectively. λ_RaA, λ_RaB, λ_RaC, λ_ThB, λ_ThC are decay constants. N_RaA, N_RaB, N_RaC, N_ThB, N_ThC are the number of atoms of RaA, RaB, RaC, ThB and ThC on the filter membrane during the sampling process.

The solution to Eq. 1 shows the relationship between the accumulated number of atoms of RaA, RaB, RaC, ThB and ThC on the filter membrane at the end of the sampling time T₁ and the activity concentrations of RaA, RaB, RaC, ThB, and ThC in air.

2.2.2

Filter membrane processing

After completing the sampling, the filter membrane was removed and placed in the sample cell of the alpha spectrometer for measurement with a one-minute time interval. In this process, the ²²⁰Rn gas did not remain on the filter membrane, and its progeny ²¹⁶Po on the filter membrane completely decayed. Therefore, there was no spectral interference from the alpha particles emitted by the decay of ²²²Rn and ²¹⁶Po during the measurement. At the end of sampling, the quantity of progeny of ²²²Rn and ²²⁰Rn on the filter membrane began to decrease because of decay. Meanwhile, the quantity of progeny nuclides also increases owing to the decay of the parent nuclides. This can be expressed by the following differential equation: $\{\begin{array}{l}\text{d}{N}_{\text{RaA1}}/\text{d}t=-N_{\text{RaA1}}{\lambda }_{\text{RaA}}\hfill \\ \text{d}{N}_{\text{RaB1}}/\text{d}t=N_{\text{RaA1}}{\lambda }_{\text{RaA}}-N_{\text{RaB1}}{\lambda }_{\text{RaB}}\hfill \\ \text{d}{N}_{\text{RaC1}}/\text{d}t=N_{\text{RaB1}}{\lambda }_{\text{RaB}}-N_{\text{RaC1}}{\lambda }_{\text{RaC}}\hfill \\ \text{d}{N}_{\text{ThB1}}/\text{d}t=-N_{\text{ThB1}}{\lambda }_{\text{ThB}}\hfill \\ \text{d}{N}_{\text{ThC1}}/\text{d}t=N_{\text{ThB1}}{\lambda }_{\text{ThB}}-N_{\text{ThC1}}{\lambda }_{\text{ThC}}\hfill \end{array}$ (2) where N_RaA1, N_RaB1, N_RaC1, N_ThB1, and N_ThC1 denote the number of atoms of RaA, RaB, RaC, ThB, and ThC, respectively, on the filter membrane during time interval T₂.

The solution to Eq. 2 establishes a relationship between the number of atoms of RaA, RaB, RaC, ThB, and ThC on the filter membrane at time T₂ and the number of atoms of RaA, RaB, RaC, ThB, and ThC at the end of the sampling period.

2.2.3

Measurement process

After processing, the filter membrane was sent to the internal cell of an alpha spectrometer. The alpha spectrometer was connected to a computer via a data cable, and the measured counts of different alpha particles under a vacuum pressure below 1000 mTorr were displayed in real time using MAESTRO software on the computer. At the beginning of alpha spectrometer measurement on the filter membrane, the number of atoms of RaA, RaB, RaC, ThB, and ThC on the filter membrane are represented as N_RaA2, N_RaB2, N_RaC2, N_ThB2, and N_ThC2, respectively. According to Eqs. (1 and 2), the activity concentrations of RaA, RaB, RaC, ThB, and ThC in air can be deduced. After the alpha spectrometer measurement was initiated, the progeny of ²²²Rn and ²²⁰Rn on the filter membrane continues to decay according to Eq. 2. During the measurement process, the number of RaA, RaB, RaC, ThB, and ThC atoms on the filter membrane at any given time t are represented as N_RaA3, N_RaB3, N_RaC3, N_ThB3, and N_ThC3, respectively, and they can be determined as follows: $\begin{array}{l}{N}_{\text{RaA3}}\left(t\right)=N_{\text{RaA2}}{e}^{-{\lambda }_{\text{RaA}}t}\hfill \end{array}$ (3) $\begin{array}{c}{N}_{\text{RaB3}}\left(t\right)=\frac{{\lambda }_{\text{RaA}}{N}_{\text{RaA2}}{e}^{-{\lambda }_{\text{RaA}}t}}{{\lambda }_{\text{RaB}}-{\lambda }_{\text{RaA}}}\\ -\left(\frac{{\lambda }_{\text{RaA}}{N}_{\text{RaA2}}}{{\lambda }_{\text{RaB}}-{\lambda }_{\text{RaA}}}-N_{\text{RaB2}}\right)e^{-{\lambda }_{\text{RaB}}t}\end{array}$ (4) $\begin{array}{c}{N}_{\text{RaC3}}\left(t\right)=\frac{-\left(\frac{{\lambda }_{\text{RaA}}{N}_{\text{RaA2}}}{{\lambda }_{\text{RaB}}-{\lambda }_{\text{RaA}}}-N_{\text{RaB2}}\right){\lambda }_{\text{RaB}}{e}^{-{\lambda }_{\text{RaB}}t}}{{\lambda }_{\text{RaC}}-{\lambda }_{\text{RaB}}}\\ +\frac{\frac{N_{\text{RaA2}}}{{\lambda }_{\text{RaB}}-{\lambda }_{\text{RaA}}}{\lambda }_{\text{RaA}}{\lambda }_{\text{RaB}}{e}^{-{\lambda }_{\text{RaA}}t}}{{\lambda }_{RaC}-{\lambda }_{RaA}}\\ +\left[N_{\text{RaC2}}+\frac{\left(\frac{{\lambda }_{\text{RaA}}{N}_{\text{RaA2}}}{{\lambda }_{\text{RaB}}-{\lambda }_{\text{RaA}}}-N_{\text{RaB2}}\right){\lambda }_{\text{RaB}}}{{\lambda }_{\text{RaC}}-{\lambda }_{\text{RaB}}}-\frac{\frac{{\lambda }_{\text{RaA}}{N}_{\text{RaA2}}}{{\lambda }_{\text{RaB}}-{\lambda }_{\text{RaA}}}{\lambda }_{\text{RaB}}}{{\lambda }_{\text{RaC}}-{\lambda }_{\text{RaA}}}\right]e^{-{\lambda }_{\text{RaC}}t}\end{array}$ (5) $\begin{array}{l}{N}_{\text{ThB3}}\left(t\right)=N_{\text{ThB2}}{e}^{-{\lambda }_{\text{ThB}}t}\hfill \end{array}$ (6) $\begin{array}{c}{N}_{\text{ThC3}}\left(t\right)=\frac{{\lambda }_{\text{ThB}}{N}_{\text{ThB2}}{e}^{-{\lambda }_{\text{ThB}}t}}{{\lambda }_{\text{ThC}}-{\lambda }_{\text{ThB}}}\\ -\left(\frac{N_{\text{ThB2}}{\lambda }_{\text{ThB}}}{{\lambda }_{\text{ThC}}-{\lambda }_{\text{ThB}}}-N_{\text{ThC2}}\right)e^{-{\lambda }_{\text{ThC}}t}\end{array}$ (7) The number of atoms of RaA, RaB, RaC, ThB, and ThC on the filter membrane in Eqs. (8, 9, 10, 11, 12) into their respective activity forms: $\begin{array}{l}{A}_{\text{RaA}}\left(t\right)={\lambda }_{\text{RaA}}{N}_{\text{RaA2}}{e}^{-{\lambda }_{\text{RaA}}t}\hfill \end{array}$ (8) $\begin{array}{l}{A}_{\text{RaB}}\left(t\right)={\lambda }_{\text{RaB}}\left[\frac{{\lambda }_{\text{RaA}}{N}_{\text{RaA}2}{e}^{-{\lambda }_{\text{RaA}}t}}{{\lambda }_{\text{RaB}}-{\lambda }_{\text{RaA}}}-\left(\frac{{\lambda }_{\text{RaA}}{N}_{\text{RaA}2}}{{\lambda }_{\text{RaB}}-{\lambda }_{\text{RaA}}}-N_{\text{RaB}2}\right)e^{-{\lambda }_{\text{RaB}}t}\right]\hfill \end{array}$ (9) $\begin{array}{c}{A}_{\text{RaC}}\left(t\right)={\lambda }_{\text{RaC}}\{\frac{-\left(\frac{{\lambda }_{\text{RaA}}{N}_{\text{RaA}2}}{{\lambda }_{\text{RaB}}-{\lambda }_{\text{RaA}}}-N_{\text{RaB}2}\right){\lambda }_{\text{RaB}}{e}^{-{\lambda }_{\text{RaB}}t}}{{\lambda }_{\text{RaC}}-{\lambda }_{\text{RaB}}}\\ +\frac{\frac{N_{\text{RaA2}}}{{\lambda }_{\text{RaB}}-{\lambda }_{\text{RaA}}}{\lambda }_{\text{RaA}}{\lambda }_{\text{RaB}}{e}^{-{\lambda }_{\text{RaA}}t}}{{\lambda }_{\text{RaC}}-{\lambda }_{\text{RaA}}}\\ +\left[N_{\text{RaC2}}+\frac{\left(\frac{{\lambda }_{\text{RaA}}{N}_{\text{RaA2}}}{{\lambda }_{\text{RaB}}-{\lambda }_{\text{RaA}}}-N_{\text{RaB2}}\right){\lambda }_{\text{RaB}}}{{\lambda }_{\text{RaC}}-{\lambda }_{\text{RaB}}}-\frac{\frac{{\lambda }_{\text{RaA}}{N}_{\text{RaA2}}}{{\lambda }_{\text{RaB}}-{\lambda }_{\text{RaA}}}{\lambda }_{\text{RaB}}}{{\lambda }_{\text{RaC}}-{\lambda }_{\text{RaA}}}\right]e^{-{\lambda }_{\text{RaC}}t}\}\end{array}$ (10) $\begin{array}{l}{A}_{\text{ThB}}\left(t\right)={\lambda }_{\text{ThB}}{N}_{\text{ThB2}}{e}^{-{\lambda }_{\text{ThB}}t}\hfill \end{array}$ (11) $\begin{array}{l}{A}_{\text{ThC}}\left(t\right)={\lambda }_{\text{ThC}}[\frac{{\lambda }_{\text{ThB}}{N}_{\text{ThB2}}{e}^{-{\lambda }_{\text{ThB}}t}}{{\lambda }_{\text{ThC}}-{\lambda }_{\text{ThB}}}\\ -\left(\frac{N_{\text{ThB2}}{\lambda }_{\text{ThB}}}{{\lambda }_{\text{ThC}}-{\lambda }_{\text{ThB}}}-N_{\text{ThC2}}\right)e^{-{\lambda }_{\text{ThC}}t}]\end{array}$ (12) During the alpha spectrometer measurements, the activities of RaA, RaB, RaC, ThB, and ThC on the filter membrane are denoted as A_RaA, A_RaB, A_RaC, A_ThB, and A_ThC, respectively. Distinguishing between the activity on the filter attributed to 6.05 MeV alpha particles from ThC and 6.0 MeV alpha particles from RaA is unfeasible. However, the activity can be apportioned between RaA and ThC by utilizing 8.78 MeV alpha particle emissions originating from ThC’. According to the radioactive decay characteristics of ²²²Rn and ²²⁰Rn, the alpha particle counts of 8.78 MeV characteristic peak originate from 0.64×A_ThC; the alpha particle counts of the 7.69 MeV characteristic peak originate from A_RaC; and the alpha particle counts of the 6.0 MeV characteristic peak originate from A_RaA+0.36× A_ThC.

Short measurement cycles were implemented with a measurement period T, and there are i measurement cycles. The total measurement time is T₃. In the i-th measurement cycle, the net alpha counts are obtained in their respective alpha spectrum of regions of interest, denoted by ROI-1 (8.78 MeV), ROI-2 (7.69 MeV), and ROI-3 (6.0 MeV), and they can be represented by n_ThC(i), n_RaC(i), and n_RaA+ThC(i). Owing to the short duration of each measurement cycle, the average counts for the ith measurement cycle are approximately equal to the values of A_ThC, A_RaC, and A_RaA at the midpoint of the measurement cycle, and they can be expressed as: $\begin{array}{l}\frac{n_{\text{ThC}}\left(i\right)}{T}=0.64E{A}_{\text{ThC}}\left(iT-\frac{1}{2}T\right)\\ =0.64E{\lambda }_{\text{ThC}}[\frac{{\lambda }_{\text{ThB}}{N}_{\text{ThB2}}{e}^{-{\lambda }_{\text{ThB}}\left(iT-\frac{1}{2}T\right)}}{{\lambda }_{\text{ThC}}-{\lambda }_{\text{ThB}}}\\ -\left(\frac{N_{ThB2}{\lambda }_{\text{ThB}}}{{\lambda }_{\text{ThC}}-{\lambda }_{\text{ThB}}}-N_{\text{ThC2}}\right)e^{-{\lambda }_{\text{ThC}}\left(iT-\frac{1}{2}T\right)}]\end{array}$ (13) $\begin{array}{l}\frac{n_{\text{RaC}}\left(i\right)}{T}=E{A}_{\text{RaC}}\left(iT-\frac{1}{2}T\right)\\ =E{\lambda }_{\text{RaC}}\{\frac{-\left(\frac{{\lambda }_{\text{RaA}}{N}_{\text{RaA}2}}{{\lambda }_{\text{RaB}}-{\lambda }_{\text{RaA}}}-N_{\text{RaB}2}\right){\lambda }_{\text{RaB}}{e}^{-{\lambda }_{\text{RaB}}\left(iT-\frac{1}{2}T\right)}}{{\lambda }_{\text{RaC}}-{\lambda }_{\text{RaB}}}\\ +\frac{\frac{N_{\text{RaA2}}}{{\lambda }_{\text{RaB}}-{\lambda }_{\text{RaA}}}{\lambda }_{\text{RaA}}{\lambda }_{\text{RaB}}{e}^{-{\lambda }_{\text{RaA}}\left(iT-\frac{1}{2}T\right)}}{{\lambda }_{\text{RaC}}-{\lambda }_{\text{RaA}}}\\ +[N_{\text{RaC2}}+\frac{\left(\frac{{\lambda }_{\text{RaA}}{N}_{\text{RaA2}}}{{\lambda }_{\text{RaB}}-{\lambda }_{\text{RaA}}}-N_{\text{RaB2}}\right){\lambda }_{\text{RaB}}}{{\lambda }_{\text{RaC}}-{\lambda }_{\text{RaB}}}\\ -\frac{\frac{{\lambda }_{\text{RaA}}{N}_{\text{RaA2}}}{{\lambda }_{\text{RaB}}-{\lambda }_{\text{RaA}}}{\lambda }_{\text{RaB}}}{{\lambda }_{\text{RaC}}-{\lambda }_{\text{RaA}}}]e^{-{\lambda }_{\text{RaC}}\left(iT-\frac{1}{2}T\right)}\}\end{array}$ (14) $\begin{array}{l}\frac{n_{\text{RaA}+\text{ThC}}\left(i\right)}{T}=E\left\{A_{\text{RaA}}\left(iT-\frac{1}{2}T\right)+0.36A_{\text{ThC}}\left(iT-\frac{1}{2}T\right)\right\}\\ =E\{N_{\text{RaA2}}{e}^{-{\lambda }_{\text{RaA}}\left(iT-\frac{1}{2}T\right)}\\ +0.36{\lambda }_{\text{ThC}}[\frac{{\lambda }_{\text{ThB}}{N}_{\text{ThB2}}{e}^{-{\lambda }_{\text{ThB}}\left(iT-\frac{1}{2}T\right)}}{{\lambda }_{\text{ThC}}-{\lambda }_{\text{ThB}}}\\ -\left(\frac{N_{\text{ThB2}}{\lambda }_{\text{ThB}}}{{\lambda }_{\text{ThC}}-{\lambda }_{\text{ThB}}}-N_{\text{ThC2}}\right)e^{-{\lambda }_{\text{ThC}}\left(iT-\frac{1}{2}T\right)}]\}\end{array}$ (15) where E denotes the detection efficiency of the detector for alpha particles, i denotes the counting cycle of alpha particles, and iT-1/2 T denotes the midpoint time of the i-th counting cycle.

Based on Eqs. (13, 14, and 15), the number of atoms (N_RaA2, N_RaB2, N_RaC2, N_ThB2, and N_ThC2) can be determined through the nonlinear fit of the experimental data, and the calculation processes are as follows:

(1) N_ThB2 and N_ThC2 can be obtained using Eq. (13) to perform a nonlinear fitting of the experimental data for (n_TcH(i))/t.

(2) By substituting the value of N_ThC2 into Eq. (15) and employing nonlinear fitting of the experimental data for (n_RaA(i))/t using Eq. (15), N_RaA2 can be obtained as:

(3) By inserting N_RaA2 into Eq. (14) and utilizing the nonlinear fitting of the experimental data for (n_RaA+TcH)/t using Eq. (14), N_RaB2 and N_RaC2 are determined.

After obtaining the values of N_RaA2, N_RaB2, N_RaC2, N_ThB2, and N_ThC2, the activity concentrations of RaA, RaB, RaC, ThB, and ThC in air can be deduced by reversing Eqs. (1 and 2).

Results and discussion

Several experiments based on air filtration of ²²²Rn and ²²⁰Rn progeny concentrations were conducted in mixed ²²²Rn and ²²⁰Rn chamber at the University of South China. This chamber is made of stainless steel and has an inner effective volume of 3 m³. It is connected to the gas path, allowing continuous compensation for ²²²Rn and ²²⁰Rn progeny. The flow rate for each sampling experiment was calibrated to 15 ± 0.45 L min^-1 using a soap bubble flowmeter, and the sampling duration lasts for fifteen minutes. The volume of the sampled gas occupies only 8% of the entire chamber volume, and continuous progeny replenishment ensures relative stability in the measurement environment. The sampling filter membrane is a 0.8 μM Millipore AA-type microfiltration membrane with a diameter of 25 mm. The filtration efficiency of the filter membrane is 0.990 ± 0.002, and its self-absorption is considered negligible. Aerosols are generated using an aerosol generator, and an aerosol electrometer, produced by TSI Incorporated, is used to monitor the number concentration of aerosols. The aerosol concentration varied from 8×10³ cm^-3 to 10×10³ cm^-3 in the experiments.

The measurements were performed at 27 ℃ and 75% relative humidity (RH) using an alpha spectrometer. This alpha spectrometer, manufactured by ORTEC Corporation, utilizes an ion-implanted surface-passivated silicon detector with an 900 mm² probe. It realizes an energy resolution of 20 keV under a vacuum pressure below 1000 mTorr. Under these measurement conditions, the alpha particles exhibit a longer range and minimal energy loss with a measurement distance of only 1 mm and no spectral broadening occurs. Consequently, the distinct characteristic energy peaks of ²²²Rn and ²²⁰Rn progenies can be clearly distinguished using 1024 channels. The alpha spectrometer calibrates its alpha detection efficiency (E) using a ²⁴¹Am electroplated surface source. The resulting mean value for E was 33.3%, accompanied by 2.0% uncertainty.

To compare the accuracy of the results obtained from the proposed model with those of the total count method presented in Ref. [25], the measurement cycle of the alpha spectrometer was set to 1 min and lasted for more than 560 min. This method can provide alpha counts at different time intervals as required by both methods. The automatic measurement process using an alpha spectrometer was realized by writing JOB files in MAESTRO software, which can display and store real-time measurement data. A portion of the measured data is presented in Fig. 3.

Fig. 3Full-screen View

(Color online) Alpha counts with energies of 6.0 MeV, 7.69 MeV and 8.78 MeV measured via display in Maestro software

Considering the impact of statistical fluctuations on net alpha counts and the influence of measurement intervals and cycle periods on the standard error of the calculated results, nine short counting cycles (i) were selected consecutively from the beginning of the measurement, with each counting cycle (T) set to 10 min. The background count for a measurement time of 90 min was 31 ± 3. According to Eq. (17), the critical limit (L_C) of the alpha spectrometer for the counts obtained using the new method was 13. The net counts of alpha particles emitted from ²²²Rn and ²²⁰Rn progeny were recorded based on alpha particle energies of 8.78 MeV, 7.69 MeV and 6.0 MeV, and their ROI-1 (8.78 MeV), ROI-2 (7.69 MeV), and ROI-3 (6.0 MeV) were 2–6.3 MeV, 6.3–8 MeV, and 8–10 MeV, respectively.

Origin software (OriginLab Corporation) enables users to customize functions for nonlinear fitting, ensuring the accurate modeling of complex data relationships. This capability provides enhanced precision and adaptability in nonlinear data analysis, making it a widely used tool for calculating ²²²Rn and ²²⁰Rn concentrations [41, 42]. The fitting method is based on the radioactive decay law that uses nonlinear fitting by Origin2021 and return the best-fit parameters and their statistical errors for each free parameter fitted. In this study, the fitted parameters N_RaA2, N_RaB2, N_RaC2, N_ThB2, and N_ThC2 are the primary focus. Figure 4 illustrates three experimental results for the net alpha particle counts measured by the alpha spectrometer, which varies with the measurement period. The alpha counts from RaA and ThC, both around 6 MeV, were counted, and the results of the three experiments are shown in Figs. 4b, e, and h. Given significantly shorter half-life of RaA when compared with that of ThC, their alpha counts initially decreased rapidly, followed by a slow rise. The alpha counts from ThC and RaC are presented in Figs. 4(a), (d), and (g), and (c)–(i), respectively. According to Eqs. (13, 14 and 15), the number of atoms of RaA, RaB, RaC, ThB, and ThC on the filter membrane (N_RaA2, N_RaB2, N_RaC2, N_ThB2, and N_ThC2) at the initiation of alpha spectrometer measurement was derived through non-linear fitting of alpha counts with energies of 6.0 MeV, 7.69 MeV, and 8.78 MeV using Origin2021 software. By inserting the values of N_RaA2, N_RaB2, N_RaC2, N_ThB2, and N_ThC2 into Eqs. (1 and 2) using MATLAB, the concentrations of ²²²Rn and ²²⁰Rn progenies in the air can be obtained, and the detailed data are listed in Table 1.

Fig. 4Full-screen View

(Color online) Three experimental results of net alpha particle counts measured by the alpha spectrometer (with energies of 8.78 MeV, 6.0 MeV, and 7.69 MeV) vary with the measurement period: the first experiment (a, b, c), second experiment (d, e, f), third experiment (g, h, i). i represents the i-th measurement period, with each cycle lasting 10 min

Only minor changes were observed in the number of aerosol particles and environmental parameters, and the sources of ²²²Rn and ²²⁰Rn remained unchanged during the process of conducting the three measurement experiments. Table 1 shows that R² for N_ThB2 and N_ThC2 in the first experiment, as well as for N_RaA2 in the second and third experiments, are not satisfactory. This may be due to the impact of statistical fluctuations in counts over consecutive cycles. However, based on the results of the three experiments, the differences observed in the concentrations of the same types of ²²²Rn and ²²⁰Rn progeny calculated using the same method in the three experiments may be attributed to minor variations in the quantity of aerosol particles suspended in the environment and other environmental parameters. The poor goodness of fit of one of the nuclides slightly impacts the overall experimental results.

The ²²²Rn and ²²⁰Rn progeny concentrations calculated using the new method were compared with those obtained using the total count method presented in the literature [25]. According to [23], the second set of counting intervals was selected for application to the total count method. The first and second sets of counting intervals were (1,4), (5,20), (21,40), (150,250), (360,560) and (2,5), (6,20), (21,30), (200,300), (360,560) min, respectively. The alpha counts from these two measurement intervals of the three experiments were input into the model of the total count method. Furthermore, a comparison of the results for the activity concentrations of ²²²Rn and ²²⁰Rn progeny in air, calculated via the new method and total count method, is shown in Fig. 5.

Fig. 5Full-screen View

(Color online) Three experimental results of simultaneous measurements of ²²²Rn and ²²⁰Rn progeny concentrations: (a) first experiment; (b) second experiment; and (c) third experiment. (The counting interval of the new method was 90 min; the first and second set of counting intervals are (1, 4), (5, 20), (21, 40), (150, 250), (360, 560), and (2, 5), (6, 20), (21, 30), (200, 300), (360, 560) min)

Figure 5 illustrates that the activity concentrations of ²²²Rn and ²²⁰Rn progeny, calculated using the new method, are all positive, whereas the activity concentrations of RaB, RaC, and ThB, calculated using the total count method with two sets of counting intervals, exhibited negative values. The occurrence of negative concentration of ²²²Rn and ²²⁰Rn progeny is not in accordance with reality. The primary reason may be that the accuracy of calculating comparatively lower progeny concentrations in the air using the total count method is insufficient, coupled with the statistical fluctuations in the counts as well as the overlapping effect between different spectra during long-term measurements. The total count method in Ref. [25] utilizes the radioactive decay law of alpha particles within five different time intervals (560 min) to obtain the concentrations of ²²²Rn and ²²⁰Rn progeny in the air. Compared to the total count method, the advantage of the new method lies in distinguishing nuclides with overlapping spectra by utilizing the alpha particle counts of non-overlapping spectra within consecutive measurement cycles (90 min). This enables the distinction of nuclides with different energy spectra and facilitates nonlinear fitting based on the radioactive decay law to determine the concentration of each progeny. Moreover, the measurement time is reduced by 84%.

Given the close proximity of the alpha energies emitted by RaA and ThC, both around 6 MeV, their counts were not distinguished in each experiment, but were counted together. For alpha counts with energies of 6 MeV, the quantity of RaA will decay rapidly within a short period because of the much shorter half-life of RaA when compared to that of ThC. When the total count method with long time intervals is employed, the activity concentration of RaA in air may have been underestimated, leading to an overestimation of the activity concentration of ThC. However, by utilizing the continuous counts of the 8.78 MeV ThC’ spectrum within consecutive cycles and the determined decay-branching ratio, it is easier to distinguish between the spectral counts of RaA and ThC. Hence, the concentration of RaA, calculated using the total count method, is lower than that determined by the new method, whereas ThC exhibits a higher concentration when compared to the calculation by the new method, as shown in Fig. 5. Although the fitting quality of some curves in Fig. 4 is not optimal, it has minimal effect on the overall accuracy of the experiments. Detailed data are listed in Table 2.

Considering that the relative uncertainty of the detection efficiency σE/E is ± 2.0%, the relative uncertainty of the flow rate σQ/Q is ± 3.0%, and the statistical uncertainty, relative uncertainty of radon progeny concentrations provided by the new methods are all within ± 16%, and those of the thoron progeny concentrations are within ± 13%. These uncertainties are significantly lower than those calculated using the total count method. Owing to the lower concentrations of measured progeny, there are significant statistical fluctuation errors, but the statistical errors of progeny concentrations calculated by the new method are within a reasonable range using the 10-min average as the measurement value for one cycle. A low measurement uncertainty is more meaningful for accurate measurements.

Given the negative values observed in the concentration of ²²²Rn progeny calculated using the total count method in the three experiments, the equilibrium equivalent radon concentration was not considered. The ²²⁰Rn decay product concentrations were measured in terms of equivalent concentrations (EEC_Tn), as provided by the following relations [43]: $\begin{array}{l}EE{C}_{\text{Tn}}=0.913\left(C_{{}^{212}\text{Pb}}\right)+0.087\left(C_{{}^{214}\text{Bi}}\right)\hfill \end{array}$ (18) where $C_{{}^{212}\text{Pb}}$, $C_{{}^{214}\text{Bi}}$ are the activity concentrations in Bq/m³.

Table 3 shows that the EEC_Tn trends calculated by both methods are consistent, with EEC_Tn in the second experiment being slightly higher than those in the first and third experiments. The lower limit values of the standard errors calculated using the total method for the three experiments were negative, which is inconsistent with the actual situation. However, the values of EEC_Tn calculated using the new method are within the range of the standard errors of those calculated using the total method, with the relative standard deviation of EEC_Tn ranging from 5% to 8%. Given the measurement uncertainties associated with different methods and fluctuations in progeny concentrations across various sampling processes, it can be confirmed that consistency was achieved. The experiments were conducted at relatively low progeny concentrations of ²²²Rnand ²²⁰Rn. The results obtained by comparing the two methods demonstrate that the proposed method has significant advantages.

Conclusions

The concentrations of ²²²Rn and ²²⁰Rn progeny in the environment pose a risk of natural radiation exposure to the public. A novel mathematical model, utilizing a nonlinear fitting method based on the radioactive decay law, is proposed for the simultaneous measurement of ²²²Rn and ²²⁰Rn progeny concentrations in air. The experimental findings indicated that ²²²Rn and ²²⁰Rn progeny concentrations determined using the new method were all positive and exhibited a closer alignment with the actual conditions than the total count method. With their relative uncertainties within ± 16%, it can be confirmed that consistency has been achieved. Moreover, the measurement time was reduced by 84%. This novel approach exhibited improved efficacy in discriminating alpha particles with analogous energies emitted from RaA and ThC, both approximately 6 MeV, consequently yielding results with heightened accuracy. The method proposed in this study has greater potential advantages for the faster measurement of relatively low concentrations of ²²²Rn and ²²⁰Rn progeny in air.