Introduction

Advances in the digital electronics and software used in gamma-spectrometry systems are providing unprecedented possibilities for data analysis and interpretation [1,2,3,4,5,6,7]. A new frontier is emerging, where high-performance multi-channel analyzers (MCAs) are becoming readily available at affordable cost to radiometrology laboratories. One such example is the Canberra Lynx MCA that operates pulse height analysis (PHA), multichannel scaling (MCS), multispectral scaling (MSS) and time-stamped list (TLIST) modes. The latter allows comprehensive logging of detector events with 100–200 ns timing resolution [3]. Such TLIST data is especially useful for capturing the maximum data from a measurement, and can be post-processed to provide PHA, MCS and MSS data. As the raw data is preserved, the processing can be applied on varying time-scales to identify the radioactive in-growth and decay of parent-daughter radionuclides according to their half-life (e.g. 214Pb/214Bi, 212Pb/212Bi, 140La/140Ba, 95Zr/95Nb). For a given radionuclide, these count rate changes are described by the Bateman first-order differential equations [8]. The equation can be solved using non-linear least-squares fitting and the Levenberg–Marquardt algorithm [9, 10]. The Levenberg–Marquardt algorithm was selected as it provides a more robust fitting algorithm than other methods such as the Gauss–Newton algorithm [11]. Its application herein provides a novel technique for simultaneous calculation of the parent-daughter atoms from the radioactive decay of only the daughter radionuclide.

The approach has been demonstrated using naturally occurring radionuclides (NOR) collected using a high volume air sampler. Amongst the NOR are the short-lived radon progeny (214Pb, 212Pb, 214Bi, 212Bi, 218Po, 216Po, 214Po, 212Po and 208Tl) with half-lives ranging from 3.0 × 10−7 s to 10.64 h [12,13,14,15]. Together with 222Rn and 220Rn, they represent 56.8% of average radiation dose received by man from natural sources [16, 17]. Their abundance in the environment and parent–daughter couplings (214Pb/214Bi, 212Pb/212Bi) makes them convenient for study, and a useful proxy for other parent–daughter radionuclides that are more difficult to obtain (e.g. fission products such as 140La/140Ba, 95Zr/95Nb). Their accurate measurement (especially for 220Rn progeny) has also often been hindered by their short half-life, highly heterogeneous distribution, low environmental concentrations and overlapping 222Rn and 220Rn distributions [18,19,20,21]. Their measurement is also important within the atmospheric sciences, as it is a major source of atmospheric ions near the earth’s surface. These are important for a range of processes including nucleation of water drops necessary for rain and formation of thunderstorms [18], tracers of atmospheric transport processes [22,23,24,25,26,27,28], initiation of atmospheric electrical phenomena [29,30,31] and diffusion of solid matter [32].

Methodology

Experimental setup

Measurements were performed using a Canberra Broad Energy Germanium (BEGe) gamma-spectrometer (model BE6530) at the Atomic Weapons Establishment (Reading, UK). The detector was controlled by a Canberra Lynx MCA with high voltage set at +4500 V, a rise time of 5.6 us and flat top of 0.8 us. The instrument was situated within a low-background shield of aged lead (<25 Bq/kg 210Pb). Ambient radon concentrations were minimized using high laboratory air flow with HEPA filtration. Acquisition of the TLIST data was performed using the Canberra Lynx Software Development Kit and custom (C++) acquisition software written using the Microsoft Visual Studio.NET application [33]. This allows all events interacting with the BEGe to be logged to a comma-separated text file for data analysis after acquisition. The measurement sample was prepared by the collection of radon progeny (and other NOR) using a Senya Snow White air sampler with Macherey–Nagel MN85/90 filter. This was operated over a 14 day period to sample 242286.3 m3 of air (with an approximate flow rate of 730 m3 h−1). After collection, the sample was promptly folded into a calibrated geometry and measured immediately for 2 days.

Data analysis

The TLIST data was processed using custom (Visual Basic) software written using the Microsoft Visual Studio.NET application. This converted the TLIST data into a series of Canberra CAM (.cnf) format files with 4 and 30 min acquisition times. Count durations were selected for measuring the half-life of 214Pb/214Bi and 212Pb/212Bi respectively. The time divisions were based on the TLIST event time which is automatically corrected for dead time. The spectra were analyzed using the Canberra Genie 2000 Gamma Acquisition and Analysis software (version 3.4) to provide the net peak counts for the gamma-energies of interest (Table 1). Igor Pro (version 7.01) was then used for data analysis, including least squares fitting using the Levenberg–Marquardt algorithm. The calculated values were corrected for gamma-emission abundance, detector efficiency and true coincidence summing (TCS). The detector efficiency and TCS corrections were calculated using the Randomised Iterative Monte-Carlo Model for ENSDF Records (RIMMER) described elsewhere [34,35,36].

Table 1 Gamma-energies and corrections for short-lived radon progeny
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Mathematical theory

The (N1) atoms of 214Pb and 212Pb were determined from the radioactive decay of each isotope from the time sequence spectra. It was assumed that the parent 222Rn and 220Rn (and 218Po and 216Po) were removed (or decayed) during air sampling and that the lead isotopes were unsupported. As radioactive decay follows the Bateman equation:

$$N_{1}^{t} = N_{1}^{0} \left[ {e^{{ - \lambda_{1} t}} } \right],$$
(1)

where N represents the number of atoms at time t, and λ 1 is the decay constant; the gradient of a plot of N t1 versus e −λt will equal the number of initial atoms (a linear line equation of the form y = mx). To calculate the daughter (N 2) atoms of 214Bi and 212Bi, it is necessary to consider daughter growth (and decay) from the N1 parent, and the decay of the initial N 2 atoms. Thus, the N 2 atoms can be expressed as:

$$N_{2}^{t} = \frac{{\lambda_{1} }}{{\lambda_{2} - \lambda_{1} }}N_{1}^{0} (e^{{ - \lambda_{1} t}} - e^{{ - \lambda_{2} t}} ) + N_{2}^{0} e^{{ - \lambda_{2} t}} .$$
(2)

This equation can be manipulated to give a non-linear plane equation of the form z = mx + ny:

$$N_{2}^{t} = N_{1}^{0} \left[ {\frac{{\lambda _{1} (e^{{ - \lambda _{1} t}} - e^{{ - \lambda _{2} t}} )}}{{\lambda _{2} - \lambda _{1} }}} \right] + N_{2}^{0} [ {e^{{ - \lambda _{2} t}} } ].$$
(3)

As the equation is non-linear due to N 2 growth and decay from the N 1 parent it is not suitable for solving using multiple linear regression techniques [15, 37]. Instead non-linear least-squares data fitting using the Levenberg–Marquardt algorithm was applied to fit the coefficients m and n. The algorithm is an iterative procedure that minimizes the value of Chi square (χ 2) from initial m and n guess values (β):

$$f\left( {x^{t} , \beta + \delta } \right) \approx f\left( {x^{t} ,\beta } \right) + J^{t} \delta ,$$
(4)

where x t is the dependent value at time t (x or y values for parent–daughter calculations). For each iteration, the parameter β is replaced by a new estimate β + δ. J t δ is the Jacobian matrix:

$$J_{i} = \frac{{\delta f(x^{t} ,\beta )}}{\delta \beta }.$$
(5)

Measurement uncertainty was incorporated into the Chi square calculation to improve fitting and provide accurate error estimates for the fit coefficients:

$$\chi^{2} = \mathop \sum \limits_{i} \left( {\frac{{N^{\text{fit}} - N^{t} }}{{N_{\text{err}}^{t} }}} \right)^{2} ,$$
(6)

where N fit and N t are fitted and original measurements and N terr is the measurement uncertainty of the measurements.

Results and discussion

Parent calculations

Although solvable using linear regression, non-linear least squares fitting using the Levenberg–Marquardt algorithm was applied for (N 1) parent calculations from the time sequence data (Fig. 1.). This was used to validate the approach and showed excellent agreement with linear N 01 calculations (Table 2). For 214Pb, there were 4.75 × 108 ± 0.6% (351.9 keV) and 4.95 × 108 ± 0.7% (295.2 keV) atoms calculated, and 4.75 × 108 ± 0.6% (238.6 keV) atoms for 212Pb using the Levenberg–Marquardt algorithm. Both techniques calculated the same result within 3 significant figures, and all values were within 0.07% difference. Only the uncertainty was higher using linear regression for 214Pb (295.2 keV) at 0.8% compared to 0.7% using the Levenberg–Marquardt algorithm. Decay correction of the first time sequence measurement (as for conventional analysis) was also within 1.8% (351.9 keV), 2.3% (295.2 keV) and 0.6% (238.6 keV) of the Levenberg–Marquardt calculated values. Notably, the time sequence approach significantly reduced N 01 measurement uncertainty from 1.8 to 0.6%, 2.3 to 0.7% and 0.7 to 0.2% respectively.

Fig. 1
figure 1

214Pb and 212Pb time sequence measurements. The decay term is defined as e −λt. The gradient of each dataset is equal to the initial number of atoms and is solvable using the equation y = mx

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Table 2 Calculated 214Pb and 212Pb atoms from the N 1 dataset using non-linear least squares fitting and linear regression
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Parent–daughter calculations

Graphical representation of the equation z = mx + ny (see Eqs. 2 and 3) shows the non-linearity associated with N 2 daughter growth and decay (Fig. 2). The differences in N 1 and N 2 decay terms between 214Pb/214Bi and 212Pb/212Bi are attributable to the half-life variations, in particular the longer-lived 212Pb (t ½ = 10.6 h). Calculation of the N 01 atoms from this multidimensional N 2 dataset using non-linear least squares fitting (Table 3) was in good agreement with calculations from the N 1 dataset and measured values (compare Table 2). From the N 2 dataset, there were 4.60 × 108 ± 1.4% 214Pb atoms and 5.18 × 1010 ± 1.1% 212Pb atoms, which was within 3.2 and 0.9% of calcuations from the N 1 dataset, and 4.8 and 1.3% of measured values. The N 02 atoms were also calculated from the N 2 dataset as 6.5 × 108 ± 1.0% 214Bi atoms and 5.10 × 109 ± 15.6% 212Pb atoms. These values also compared well with measured values and were within 2.4 and 10.4% difference respectively. The higher uncertainty and measurement difference for 212Pb atoms is attributable to the increased N 2 variance. This is due to the reduced 212Bi signal caused by the lower gamma abundance (6.7% at 727.2 keV) and branching ratio (64.1%).

Fig. 2
figure 2

214Bi (left) and 212Bi (right) time sequence measurements. The N 1 decay term is defined as \(\frac{{\lambda_{1} (e^{{ - \lambda_{1} t}} - e^{{ - \lambda_{2} t}} )}}{{\lambda_{2} - \lambda_{1} }}\) and the N2 decay term as e −λt. The gradients of the curve are equal to the initial N 1 and N 2 atoms and are solvable using the equation z = mx + ny

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Table 3 Calculated 214Pb/214Bi and 212Pb/212Bi atoms from the N 2 dataset using non-linear least squares fitting
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The N 02 calculation using the Levenberg–Marquardt algorithm can be refined by utilizing the N 01 atoms calculated from the N 1 decay (i.e. from the equation y = mx). This solves the mx term of the non-linear equation z = mx + ny, such that least squares fitting is only required for n calculation. Using this approach, and the values from Table 1, there are 6.42 × 108 ± 0.8% 214Bi atoms and 4.98 × 109 ± 13.4% 212Bi atoms which is within 1.0 and 9.3% of the measured values. As with the previous calculation, the increased 212Bi difference is due to the relatively high variance in the N 2 measurements. However, as the half-life of 208Tl is relatively short (t ½ = 3.05 min) compared to the 212Bi parent, equilibrium should exist between the two isotopes during counting. Thus conversion of the 212Bi N 02 atoms to 208Tl N 03 atoms yields 2.51 × 108 ± 13.4% atoms, which is within 4.0% of the measured 208Tl atoms (583.2 keV).

Other radionuclides

Future research shall apply non-linear least squares fitting and the Levenberg–Marquardt algorithm to datasets containing other parent-daughter radionuclides, including longer-lived fission products such as 140La/140Ba and 95Zr/95Nb. As with this NOR experiment, the TLIST data shall be split into a series of time sequence components. A potential challenge will be to obtain sufficient counts for statistically significant radionuclide identification at relatively short timescales and lower activity samples. However, this problem will be largely mitigated by optimization of the time sequence count duration for the half-life of the radionuclides of interest. Measurement of longer-lived radionuclides will also enable a larger N 1 and N 2 dataset, and this may improve the N 01 and N 02 calculation.

Conclusions

A novel solution has been described that uses non-linear least-squares fitting and the Levenberg–Marquardt algorithm to determine parent-daughter atoms from the Bateman first-order differential equations. This has been demonstrated using TLIST data of short-lived radon progeny (214Pb, 214Bi, 212Pb, 212Bi) on an air filter sample. Using only 214Pb and 212Pb N 1 time sequence data, the initial N 01 atoms have been calculated within 0.6–2.3% of measured values with reduced measurement uncertainty. Simultaneous parent-daughter (214Pb/214Bi, 212Pb/212Bi) calculation from only the N 2 daughter 214Bi and 212Bi time sequence data, is within 1.3–4.8% of measured values for the N 1 parent, and 2.4–10.4% of measured values for the N 2 daughter. The best agreement was for 214Pb/214Bi calculations and also provided improvements in measurement uncertainty. This time sequence technique provides a powerful tool for radionuclide identification and quantification, with potential to redefine gamma-spectrometry analysis.