Aging Process

The half-life of ²²²Rn is relatively short, so the waiting time needed for 7–10 half-lives to elapse is not very long. Based on exponential decay, ²²²Rn levels can be reduced to background levels over this time period. Therefore, a common method to eliminate ²²²Rn contamination is the 30-day aging process, in which the sample is stored in a freezer. This method is based on the principle that approximately 99.6% of ²²²Rn will decay over that time.

Gas Proportional Counters Methods

Historically, the first methods of ²²²Rn removal were dedicated to removing radon from CO₂, such as by slow fractional distillation at controlled temperatures (de Vries, 1957). There was also the application of gas proportional counters to monitor ²²²Rn concentration in the Rn–CO₂ mixture (Pazdur, 1979).

Liquid Scintillation Counters Methods

There are several methods of purifying benzene, involving vacuum pumping at low temperatures. Hood et al. (1989) reported that ²²²Rn can be removed during dynamic vacuum recovery of benzene at −78°C. Tudyka et al. (2021) described a method of removing ²²²Rn using four freeze–pump–thaw cycles at either −20°C (with a H₂O ice and NaCl mixture) or −72°C (with a dry ice and ethanol mixture) to bring ²²²Rn contamination to the background level.

Another method involves calculating the correction based on additional measurements. This method uses the very short half-life of ²¹⁴Po, and if two consecutive counts are registered within a certain time window, they can be interpreted as ²¹⁴Bi and ²¹⁴Po decays, respectively, referred to as a ²¹⁴Bi/²¹⁴Po pair decay event (Theodórsson, 2005b; Tudyka et al., 2015). A liquid scintillation counter with the ability to measure pulse height, pulse shape, and time between subsequent pulses, such as the ICELS system (Theodórsson, 2005a) equipped with a multichannel analyser that records amplitude of pulse and time between pulses (Tudyka and Bluszcz, 2011), or the Multicell spectrometer (Tudyka et al., 2017), is required for this method.

Mathematical and Computational Methods

The literature lacks mathematical or computational methods that do not require additional measurements or chemical treatment, except for the linear regression fitting method described by Gupta and Polach (1985). By using the radioactive decay equation, one can obtain:

After taking natural logarithm, Eq. (1.1) can be written as

However, this method should be interpreted with caution and can only be used as a means of verifying the presence of ²²²Rn in the sample. It is not suitable for modern-day processing standards. Firstly, the transformation of data from exponential form to linear form through linearization alters the probability density distribution (Box and Cox, 1964). Secondly, introducing the variable decay constant λ for ²²²Rn adds an additional degree of freedom to the calculations, which should be avoided.

Therefore, there is a lack of mathematical methods for estimating overcounting induced by daughter isotopes of ²²²Rn. In this paper, direct exponential fitting is proposed, taking advantage of efficient modern-day numerical calculations.

Uranium Series Radionuclides

The uranium series, named after its parent isotope ²³⁸U, is one of three naturally occurring decay chains. A detailed table of the uranium series, starting from ²²⁶Ra (a parent radionuclide of ²²²Rn) and ending with ²¹⁰Pb, is presented in Table 1, along with the ¹⁴C radionuclide for comparison. Uncertainties are provided in parentheses, and they refer to the last digit or the last two digits of the presented value.

Within this chain, two isotopes of interest are ²²⁶Ra and ²²²Rn, which can be found in underground water, as well as in drinking water (Cothern and Rebers, 1990) and construction materials of buildings (Nero and Nazaroff, 1984). As a result, these isotopes can be present in the atmosphere of a laboratory and in water used for chemical pretreatment, which can lead to sample contamination.

The first of the aforementioned radionuclides, ²²⁶Ra, has a half-life comparable to that of ¹⁴C. As such, if it contaminates a sample, it will remain there indefinitely from a radiocarbon dating standpoint.

It is worth noting that the daughter radionuclides of ²²²Rn, namely ²¹⁸Po, ²¹⁴Pb, ²¹⁴Bi and ²¹⁴Po, all have very short half-lives of less than an hour. As a result, one decay of ²²²Rn is followed by four additional decays (Table 1).

The next isotope in the uranium chain, ²¹⁰Pb, has a relatively long half-life and typically its contribution within the ¹⁴C counting window is considered negligible.

The energy of decay is another factor of importance in radiocarbon dating with the LSC technique. ¹⁴C decays through β⁻ decay, with a continuous energy spectrum and a weighted mean energy of 49.47(10) keV and a maximum or endpoint energy of 156.475(4) keV. As a result, isotopes that undergo α decay, namely ²²⁶Ra, ²²²Rn, ²¹⁸Po and ²¹⁴Po, usually will be separated from the radiocarbon energy spectra (see major decay branches in Table 1).

Isotopes that undergo β⁻ decay, such as ²¹⁴Pb and ²¹⁴Bi, have much greater weighted mean β⁻ decay energy that ¹⁴C. Therefore, they are mostly separated from the radiocarbon energy spectra. However, statistically some decays will have energy that overlap the ¹⁴C energy spectra. The over-counting of uranium series radionuclides in the ¹⁴C window is therefore the result of ²¹⁴Pb and ²¹⁴Bi decays (other rare decay modes mentioned in Table 1 are negligible).

Quantulus 1220™ Spectrometer

Quantulus 1220™ is a commonly used ultra-low background spectrometer. Ultra-low level background is obtained by using a passive shield (asymmetrical 630 kg lead shield), active shield (allowing subtraction of scintillations registered simultaneously by photomultipliers in antico-incidence shield filled with mineral oil liquid scintillator and registered by photomultipliers directed onto sample), radio frequency noise discriminator and pulse amplitude comparator (allowing subtraction of pulses whose amplitude is highly different in both photomultipliers directed onto sample; Pawlyta et al., 1997). Measured spectra on the Quantulus 1220™ are spread with logarithmic conversion into 1024 channels (Fig. 1). Hence, radionuclides from uranium series are measured in α/β spectra. The Quantulus 1220™ spectrometer can be used to many experiments such as for determining content of these radionuclides in drinking water (e.g., Schönhofer, 1989; Salonen and Hukkanen, 1997) or for determining ²¹⁰Pb and ²¹⁰Po count rates in the sample (Kim et al., 2008).

Fig 1.

Alpha and beta particle spectra of radioactive nuclides in a Quantulus 1220™ liquid scintillation counter with counting windows used at the Gliwice Laboratory (Poland). In most cases, peak from ²¹⁰Pb would not be observed in radiocarbon dating due to its low probability.

The spectra in the Quantulus 1220™ at the Gliwice Laboratory are divided into three windows, after Tudyka and Pazdur (2010): the ³H window (channels 1–290), the ¹⁴C window (channels 291–580) and the radon and daughter products window (channels 580–1024), as shown in Fig. 1.

Exponential Function Fitting Method

We propose a contemporary approach, distinct from linear regression fitting, which involves direct weighted exponential function fitting utilizing a numerical algorithm to Eq. (1.1).

In most cases, the count rate induced by radon and daughter radionuclides in the ¹⁴C window is too small to directly fit an exponential function to the ¹⁴C window. However, in the radon and daughter products window, presence of ²²²Rn and its daughter isotopes is usually noticeable if the sample is measured soon enough after benzene synthesis. Therefore, two fittings into two different windows can be performed using the following formulas:

The radon and daughter nuclides-free count rate in the ¹⁴C window is calculated as

This calculation, subtracting two independent values, will be referred to as the subtraction approach later in the paper.

Since some values in Eq. (2.1) are interdependent, a different approach is more suitable, which involves calculating the values in the radon and daughter products window and the combined ¹⁴C and radon and daughter products window as a system of equations:

This approach, treating some values as interdependent, will be referred to as the system of equations approach later in the paper. Here, E represents the correction constant, which accounts for the count rate of the radon and daughter radionuclides measured in the ¹⁴C window compared to the count rate of the radon and daughter radionuclides measured in the radon and daughter products window. Specifically, aRnRnE represents the part of initial count rate of radon and daughter radionuclides measured in the ¹⁴C window. The count rate of radon and daughter nuclides at time zero, aRn0Rn , is sample-dependent, while the value of the constant E is related to the specific measurement conditions, including settings of the liquid scintillation beta spectrometer.

Therefore, if the E constant is determined for the spectrometer, correction in the ¹⁴C window can be calculated as:

This approach is suitable because most of the time, the overcounting induced by ²²²Rn and daughter isotopes can be distinguished from the naturally occurring randomness of radioactive decay only in the radon and daughter products window. It is primarily in this window where it is possible to fit an exponential function with statistical parameters at an acceptable level, such as χ². This attempt, which involves calculating the correction constant, will be referred to as the correction constant approach later in the paper.

Subtraction Approach

Calculations were performed using a custom program written in MATLAB^® R2022b, employing direct exponential fitting. Fitting was carried out in two separate windows: the radon and daughter products window (channels 580–1024) and the combined ¹⁴C and radon and daughter products window (channels 291–1024). Regarding the GdS-4657 sample, the fitted exponential regression exhibited favourable χ² value in both windows, as depicted in Fig. 3. The count rate as+bRn was found to be 0.180(22) min⁻¹ · g⁻¹ in the radon and daughter products window and as+bC+Rn=6.45251 min−1 ⋅ g−1 in the combined ¹⁴C and radon and daughter products window. The count rate in the ¹⁴C window was calculated by subtracting the count rates obtained in the two previously mentioned windows, resulting in as+bC=6.27255 min−1 ⋅ g−1 .

Fig 3.

Exponential function fitted into (A) – radon and daughter products window, (B) – combined ¹⁴C and radon and daughter products window. Fitted functions are shown with a 68.3% confidence band.

System of Equations Approach

The second exponential fitting method involving solving of a system of equations was performed in MATLAB^® R2022b and also in two separate windows: the radon and daughter products window (channels 580–1024) and the combined ¹⁴C and radon and daughter products window (channels 291–1024), as depicted in Fig. 4. Regarding the GdS-4657 sample, the count rate in the ¹⁴C window was calculated as as+bC=6.27524 min−1 ⋅ g−1 .

Fig 4.

The results of the solved system of equations for the GdS-4657 sample. (A) – Equation corresponding to the radon and daughter products window. (B) – Equation corresponding to the combined ¹⁴C and radon and daughter products window. The fitted functions for both equations are shown with a 68.3% confidence band.

The correction constant E, obtained as 1.166(26), can potentially be utilized in the future to assess the contamination of ²²²Rn and its daughter isotopes in samples in Gliwice Laboratory. This approach, labelled as the correction constant approach, relies on the definition of the boundary between windows and, consequently, the sample quality parameter (external), abbreviated as SQP(E), which is related to a specific gamma source Compton electrons counts, and quantifies the number of channels above which 1% of the pulses from the external γ radiation source occur, and it corrects for spectral shifts caused by varying levels of quenching in synthesized benzene and instrument gain changes (McCormac, 1992). In order to obtain a comprehensive understanding of the influence of this parameter on the correction constant E, it is necessary to measure it across a larger sample series.

Dating Results

To assess the efficacy of the method, we performed supplementary measurements on the GdS-4657 sample, which was labelled as GdS-4684 after undergoing an aging process for more than a month, as described earlier. The F¹⁴C and ¹⁴C age values (Table 2) were calculated by taking into account background correction and the SQP(E) correction.

Results obtained from both correction methods are in agreement with results obtained for GdS-4684 sample within statistical significance (determined using a t-test). The uncertainty obtained using the system of equations approach is smaller.

Required Number of Measurement Cycles

To evaluate the need for extended measurements of the GdS-4657 sample, we analysed the uncertainties in the count rate, as+bC , under the assumption that the sample was measured for a specific number of cycles, starting from 10 cycles (see Fig. 5). Each data point in the graph represents a different population size, therefore the χ² value versus χcrit2 is depicted, at α = 0.05. The uncertainties reach an acceptable level only after around 35 cycles. It is worth noting that this measurement duration of 35 cycles is generally longer than the standard measurements conducted in the Gliwice Laboratory, which typically involve 24 cycles.

Fig 5.

Evolution of determined count rate, with a χ² value versus χcrit2 at α = 0.05. Each data point represents a hypothetical scenario where measurements were terminated after a specific cycle, starting from the 10th cycle for the first point, the 11th cycle for the second point, the 12th cycle for the third point, and so on. (A) – Subtraction approach. (B) – System of equations approach.