Simple methods for calculating activity of a parent-progeny system

Abstract

Based on the original work of Rutherford (Radio-activity, 1905) and Bateman (Proc Camb Philos Soc 15:423–427, 1910), the authors designed two schemes consisting of explicit equations as simple methods for accurately obtaining activity of ⁹⁰Sr and ⁹⁰Y before they reach secular equilibrium. Application of the methods to the ⁹⁰Sr/⁹⁰Y system where ⁹⁰Sr and ⁹⁰Y are not in equilibrium will substantially reduce the time needed for determining activity of ⁹⁰Sr because neither sequential measurements of ⁹⁰Sr or ⁹⁰Y (up to about 2 weeks) nor waiting for ⁹⁰Sr and ⁹⁰Y to reach equilibrium (more than 3 weeks) will be needed. We also implemented the explicit equations for decay/ingrowth correction of progeny’s activity and applied them to the ⁹⁵Zr/⁹⁵Nb system. Using the equations, the authors corrected activity concentrations of ⁹⁵Nb to a designated reference time from the activity concentrations measured from samples at different times, for instance, 2, 8, 15, and 29 days after a reference time. During those measurement times, ⁹⁵Nb and ⁹⁵Zr were not in equilibrium. The corrected ⁹⁵Nb activity concentrations were within an accuracy of − 10%.

Introduction

For emergency response to a nuclear incident, the ⁹⁰Sr/⁹⁰Y system is especially important because ⁹⁰Sr is one of the radionuclides of most concern due to its hazardous nature. The capability of providing accurate and rapid ⁹⁰Sr analytical result will have a great impact on public health during a nuclear emergency response. Rutherford in 1905 presented differential equations to describe the decay of a system consisting of a parent and two successive progenies and deduced a solution to show how the three radioisotopes change with time [1]. In 1910, Bateman expanded the system for more successive progenies and provided a solution with broader applications [2]. There are also some other efforts in providing more general solutions to Bateman equations [3, 4] and algebraic expressions of the equations [5]. We designed schemes and rearranged Rutherford’s equations into explicit formulas which can be directly applied to calculate the ⁹⁰Y ingrowth as part of a ⁹⁰Sr/⁹⁰Y parent-progeny system before it reaches secular equilibrium.

It was also found that correcting ⁹⁵Nb activity from a measurement time to a reference time is not a straightforward practice [6] and equations for determining ⁹⁵Nb activity were provided in the referenced article. The provided equations considered ^95mNb in the system and involved more in-depth considerations of start and end of acquisition times while obtaining the integrated form of the Bateman equation to determine the ⁹⁵Nb activity. Later, even more in-depth formulas were communicated for ⁹⁵Zr and ⁹⁵Nb chronometry of a nuclear event [7]. During participation in a recent inter-laboratory comparison organized by the US Federal Radiological Monitoring and Assessment Center (FRMAC), it occurred to us that it would benefit the radioanalytical community if explicit equations could be presented for accurately calculating activity considering decay/ingrowth for systems that are not in equilibrium. We implemented two explicit equations and applied the equations to the ⁹⁵Zr/⁹⁵Nb system: (1) using the measurement data of both ⁹⁵Zr and ⁹⁵Nb to demonstrate the method’s effectiveness and degree of accuracy; (2) assuming the initial progeny’s activity was zero and calculating the progeny’s activity using measured parent’s activities only. Comparison of results obtained from (1) and (2) can yield information for nuclear forensics.

Methods

Based on the original work of Rutherford [1] and Bateman [2], for a successive radioisotope decay system, a progeny’s activity can be expressed as follows:

$$A_2(t)=A_2\left(t_{\text{ref}}\right)e^{-{\lambda }_2\left(t-t_{\text{ref}}\right)}+\frac{{\lambda }_2{p}_1{A}_1\left(t_{\text{ref}}\right)}{{\lambda }_2-{\lambda }_1}\left[e^{-{\lambda }_1\left(t-t_{\text{ref}}\right)}-e^{-{\lambda }_2\left(t-t_{\text{ref}}\right)}\right]$$ (1)

where A₁(t_ref) is the activity of the parent at a reference time t_ref, A₂(t) is the activity of the progeny at time t, λ₁ is the decay constant of the parent, λ₂ is the decay constant of progeny, and p₁ is the branching ratio of the parent decaying to the progeny. In Rutherford’s solution, A₂(t_ref) = 0 was assumed. If we establish a very clear connection between the data provided by an instrument’s analysis report and the inputs needed for calculating a progeny’s activity using the equations below based on Eq. (1), it will be easier for a radioanalytical analyst to accurately perform decay and ingrowth corrections for systems that have not reached secular equilibrium or in which the equilibrium does not exist. For this purpose, we define

$$t_{m0}\equiv \text{ Start of data acquisition, }\Delta t\equiv t_{m0}-t_{\text{ref}}.$$

The activity of the progeny at a reference time t_ref can be calculated by

$$A_2\left(t_{\text{ref}}\right)=\left\{A_2\left(t_{m0}\right)-\left[\frac{{\lambda }_2{p}_1{A}_1\left(t_{\text{ref}}\right)}{{\lambda }_2-{\lambda }_1}\left(e^{-{\lambda }_1\Delta t}-e^{-{\lambda }_2\Delta t}\right)\right]\right\}{e}^{{\lambda }_2\Delta t}$$ (2)

where

$$A_2\left(t_{m0}\right)=\frac{{\lambda }_2}{1-e^{-{\lambda }_2\Delta t_{mf0}}}\{\Delta t_{mf0}{A}_2-\frac{{\lambda }_2{A}_1\left(t_{m0}\right)}{{\lambda }_2-{\lambda }_1}\times \left[\frac{\left(1-e^{-{\lambda }_1\Delta t_{mf0}}\right)}{{\lambda }_1}-\frac{\left(1-e^{-{\lambda }_2\Delta t_{mf0}}\right)}{{\lambda }_2}\right]\},$$ (3)

A₂ is the uncorrected progeny’s activity at the beginning of data acquisition, not corrected for ingrowth during the measurement, and t_mf is the end time of data acquisition, Δt_mf0 ≡ t_mf − t_m0. Specifically, A₂ is the net counts divided by the acquisition time interval and the detector efficiency. For a pair of gamma-ray emitters, the net counts also need to be divided by the branching ratio of the peak energy. The above equation for calculating A₂(t_m0) is used to correct for the decay and ingrowth effect of a progeny during the data acquisition period. Some instrument software has implemented a built-in correction factor in A₂ by performing $A_2/\left[\left(1-e^{-{\lambda }_2\left(t_{mf}-t_{m0}\right)}\right)/{\lambda }_2\Delta t_{mf0}\right]$ for the data acquisition period and A₂/e^−λΔt for the time interval between the start of data acqusition and reference time. Those built-in correction factors consider the progeny’s decay only, disregarding the feeding from parent. Therefore, when using Eq. (2), for the software that has the above built-in decay correction factors, the correction needs to be removed by multiplying A₂ by $\left(1-e^{-{\lambda }_2\left(t_{mf}-t_{m0}\right)}\right)/{\lambda }_2\left(t_{mf}-t_{m0}\right)$ and e^−λΔt to accurately account for decay/ingrowth. If a progeny’s activity change during the data acquisition is negligible (⁹⁵Nb’s case), Eq. (2) with A₂(t_m0) ≈ A₂ can be directly used for calculating the progeny’s activity, which is a very simple form for correcting decay and ingrowth effects. For instance, the error introduced to $A_{{95}_{\text{Nb}}}\left(t_{\text{ref}}\right)$ because of not accounting decay/ingrowth during acquisition is only about 1% $\left(A_{{95}_{\text{Nb}}}{e}^{-{\lambda }_{{95}_{\text{Nb}}}\Delta t}=A_{{95}_{\text{Nb}}}{e}^{-0.02\left(d^{-1}\right)\times 0.5(d)}\approx 0.99A_{{95}_{\text{Nb}}}\right)$ for a 12 h acquisition time and about 0.1% for a 2 h acquisition time. However, Eq. (3) must be used for obtaining A₂(t_m0) if a progeny’s activity change during the data acquisition is not negligible. Equation (2) clearly shows that to accurately obtain the activity of a progeny at a reference time, the only needed inputs are: (1) the detected activity of progeny A₂; (2) the detected activity of parent A₁(t_ref) at the reference time t_ref; (3) the time interval Δt = t_m0 − t_ref; (4) Δt_mf0 = t_mf − t_m0; (5) the branching ratio p₁; and (6) the decay constants of both isotopes, λ₁ and λ₂. The equation for calculating the A₂(t_ref)’s uncertainty can be found in the supplementary materials.

For the cases when only the parent existed at an established time, e.g., the effective time of a nuclear event, t₀, the activity of the immediate progeny can also be obtained using Eq. (2) with the inputs of only the parent’s activity A₁(t₀) and A₁(t_ref) respectively. In this case, A₂(t₀) = 0 and

$$A_2\left(t_{m0}\right)=\frac{{\lambda }_2{p}_1{A}_1\left(t_0\right)}{{\lambda }_2-{\lambda }_1}\left[e^{-{\lambda }_1\Delta t_T}-e^{-{\lambda }_2\Delta t_T}\right]$$ (4)

where Δt_T ≡ t_m0 − t₀. Therefore, Eq. (2) becomes

$$A_2\left(t_{\text{ref}}\right)=\frac{{\lambda }_2{p}_1}{{\lambda }_2-{\lambda }_1}{e}^{{\lambda }_2\Delta t}\{A_1\left(t_0\right)\left[e^{-{\lambda }_1\Delta t_T}-e^{-{\lambda }_2\Delta t_T}\right]-A_1\left(t_{\text{ref }}\right)\left[e^{-{\lambda }_1\Delta t}-e^{-{\lambda }_2\Delta t}\right]\}$$ (5)

Both A₁(t₀) and A₁(t_ref) can be calculated from the parent activity A₂(t_m0) using $A_1\left(t_0\right)=A_1\left(t_{m0}\right)e^{-{\lambda }_1\Delta t_T}$ and $A_1\left(t_{ref}\right)=A_1\left(t_{m0}\right)e^{-{\lambda }_1\Delta t}$. Evaluation of A₂(t_ref) obtained using Eqs. (2) and (5) may yield valuable nuclear forensics information. For instance, if A₂(t_ref)_Eq.(2) = A₂(t_ref)_Eq.(5), then the progeny did not exist at the beginning of the incident; if A₂(t_ref)_Eq.(2) ≠A₂(t_ref)_Eq.(5), then the progeny existed at the beginning of the incident and the ratio A₂(t_ref)_Eq.(2)/A₂(t_ref)_Eq.(5), can be evaluated.

While activity was used as the measurand (unit = Bq) in Eqs. (1), (2), (3), (4), and (5), the equations and the following derived equations also hold true for activity concentration and can be directly used for calculating activity concentration (e.g., Bq/kg) should the unit be required for reporting purposes.

⁹⁰Sr/⁹⁰Y system

There has been substantial research conducted for ⁹⁰Sr analysis [8–20]. One of the representative methods for obtaining ⁹⁰Sr activity requires separating other elements, including yttrium, from strontium, then taking the difference of activity obtained using Cerenkov counting ⁸⁹Sr/⁹⁰Y and liquid scintillation counting (LSC) ⁸⁹Sr/⁹⁰Sr/⁹⁰Y, and then sequential measurements for ⁹⁰Sr determination considering the ingrowth of ⁹⁰Y or measuring either ⁹⁰Y or ⁹⁰Sr when they reached secular equilibrium [21–27]. Sequential measurements or waiting for equilibrium to be reached takes weeks and requires significantly more laboratory resources for obtaining reliable data, which will result in the delayed analytical results needed urgently during a nuclear emergency response. Below, we present two schemes with explixit equations that can be used for rapidly and accurately obtaining the ⁹⁰Sr activity. First, we will evaluate the ⁹⁰Y ingrowth effect on a ⁹⁰Sr measurement.

⁹⁰Y ingrowth effect on Sr-90 measurement

After strontium is separated from other elements, if ⁹⁰Sr could be decoupled from ⁸⁹Sr, then the remaining activity is actually the combined activity of ⁹⁰Sr and ⁹⁰Y. The combined activity can be obtained using Eq. (6):

$$A{\left(t_{m0}\right)}_{Sr-90/Y-90}=A_{Sr-90}\left(t_{\text{sep}}\right)e^{-{\lambda }_1\Delta t}+\frac{{\lambda }_2{A}_{Sr-90}\left(t_{\text{sep}}\right)}{{\lambda }_2-{\lambda }_1}\left[e^{-{\lambda }_1\Delta t}-e^{-{\lambda }_2\Delta t}\right]$$ (6)

where Δt = t_sep – t_m0 is the time interval between the start of the measurement time, t_m0, and the effective moment of ⁹⁰Sr separation, t_sep. Using Eq. (6), we obtain

$$\begin{gathered} \frac{A{\left(t_{m0}\right)}_{Sr-90/Y-90}-A_{Sr-90}\left(t_{m0}\right)}{A_{Sr-90}\left(t_{m0}\right)}\cdot 100 \\ =\left\{\frac{{\lambda }_2}{{\lambda }_2-{\lambda }_1}\left[1-e^{-\left({\lambda }_2-{\lambda }_1\right)\Delta t}\right]\right\}\cdot 100 \end{gathered}$$ (7)

Equation (7) can be used to evaluate the ⁹⁰Y ingrowth with time. For instance, one hour after strontium separation from other elements, there will be about 1% ⁹⁰Y due to ingrowth, and after 12 h, there will be about 12% ⁹⁰Y due to ingrowth. For accurately obtaining ⁹⁰Sr activity before ⁹⁰Sr/⁹⁰Y secular equilibrium is reached, we designed the following two analytical schemes:

Scheme 1 Isolating Yttrium

To obtain the ⁹⁰Sr activity in a sample, the sample needs to be homogenized and divided into two subsamples. The steps for isolation and measurement of the two subsamples are shown in Fig. 1 below:

Fig. 1.

Steps for obtaining ⁹⁰Sr activity using Isolating Yttrium scheme

Step 1 Separate yttrium from other elements in subsample 1 at t_sep1 and count ⁹⁰Y at t_m1 with either a gas proportional counter or liquid scintillation counter.

Step 2 Separate yttrium from other elements in subsample 2 at t_sep2 and count ⁹⁰Y at t_m2 with either a gas proportional counter or liquid scintillation counter.

At t_sep2, the ⁹⁰Y activity in subsample 2 came from two sources:

Source 1: The existing ⁹⁰Y activity at t_sep1:

$$A_{{90}_Y}\left(t_{\text{sep}1}\right)\to A_{{90}_{\text{Y}}}\left(t_{\text{sep}1}\right)e^{-{\lambda }_2\Delta t_2}$$ (8)

Source 2: The ⁹⁰Y activity produced from ingrwoth:

$$A_{{90}_{\text{Sr}}}\left(t_{\text{sep}1}\right)\to \frac{{\lambda }_2{A}_{{90}_{\text{Sr}}}\left(t_{\text{sep}1}\right)}{{\lambda }_2-{\lambda }_1}\left[e^{-{\lambda }_1\Delta t_2}-e^{-{\lambda }_2\Delta t_2}\right]$$ (9)

Adding the two sources together yields the ⁹⁰Y activity at t_sep2:

$$A_{{90}_{\text{Y}}}\left(t_{\text{sep}2}\right)=A_{{90}_{\text{Y}}}\left(t_{\text{sep}1}\right)e^{-{\lambda }_2\Delta t_2}+\frac{{\lambda }_2{A}_{\text{9}{0}_{\text{Sr}}}\left(t_{\text{sep}1}\right)}{{\lambda }_2-{\lambda }_1}\left[e^{-{\lambda }_1\Delta t_2}-e^{-{\lambda }_2\Delta t_2}\right]$$ (10)

Rearranging Eq. (10), one can derive the following equation for calculating the ⁹⁰Sr activity:

$$A_{{90}_{\text{Sr}}}\left(t_{\text{sep}1}\right)=\frac{{\lambda }_2-{\lambda }_1}{{\lambda }_2\left[e^{-{\lambda }_1\Delta t_2}-e^{-{\lambda }_2\Delta t_2}\right]}\left\{A_{{90}_{\text{Y}}}\left(t_{\text{sep}2}\right)-A_{{90}_{\text{Y}}}\left(t_{\text{sep}1}\right)e^{-{\lambda }_2\Delta t_2}\right\}$$ (11)

$$A_{{90}_{\text{Sr}}}\left(t_{\text{sep}1}\right)=\frac{{\lambda }_2-{\lambda }_1}{{\lambda }_2\left[e^{-{\lambda }_1\Delta t_2}-e^{-{\lambda }_2\Delta t_2}\right]}\times \left\{A_{{90}_{\text{Y}}}\left(t_{m2}\right)e^{{\lambda }_2\Delta t_3}-A_{{90}_{\text{Y}}}\left(t_{m1}\right)e^{-{\lambda }_2\left(\Delta t_2-\Delta t_1\right)}\right\}$$ (12)

where $A_{{90}_{\text{Y}}}\left(t_{\text{sep}1}\right)=A_{{90}_{\text{Y}}}\left(t_{m1}\right)e^{{\lambda }_2\Delta t_1}$, $A_{{90}_{\text{Y}}}\left(t_{\text{sep}2}\right)=A_{{90}_{\text{Y}}}\left(t_{m2}\right)e^{{\lambda }_2\Delta t_3}$, t_sep1 ≡ Yttrium separation time of subsample 1, t_m1 ≡ Start of ⁹⁰Y data acquisition for subsample 1, t_sep2 ≡ Yttrium separation time of subsample 2, t_m2 ≡ Start of ⁹⁰Ydata acquisition for subsample 2, Δt₁ ≡ t_m1 − t_sep1, Δt₂ ≡ t_sep2 − t_sep1, Δt₃ ≡ t_m2 − t_sep2, $A_{{90}_{\text{Y}}}\left(t_{m1}\right)$ is measured ⁹⁰Y activity of subsample 1 at t_m1, $A_{{90}_{\text{Y}}}\left(t_{m2}\right)$ is the measured ⁹⁰Y activity of subsample 2 at t_m2.

Scheme 2 (1) Isolating Strontium, (2) Isolating Yttrium

As shown in Fig. 2, the following steps are used for obtaining ⁹⁰Sr activity from ingrowth of ⁹⁰Y activity.

Fig. 2.

Steps for obtaining ⁹⁰Sr activity using Scheme 2

Step 1, separate strontium and retain the strontium portion.

Step 2, wait a certain time, e.g., a half day, separate yttrium from the retained strontium portion.

Step 3, count extracted yttrium with either gas proportional counter or liquid scintillation counter.

With this separation sequence, the detected ⁹⁰Y activity will be only originated from ingrowth. Therefore, the activity can be expressed as follows:

$$A_{\text{Y}-90}\left(t_m\right)=A_{\text{Y}-90}\left(t_{\text{Y}-\text{sep}}\right)e^{-{\lambda }_2\Delta t_2}=\frac{{\lambda }_2{A}_{\text{Sr}-90}\left(t_{\text{Sr}-\text{sep}}\right)}{{\lambda }_2-{\lambda }_1}\left[e^{-{\lambda }_1\Delta t_1}-e^{-{\lambda }_2\Delta t_1}\right]e^{-{\lambda }_2\Delta t_2}$$

and

$$A_{\text{Sr}-90}\left(t_{\text{Sr}-\text{sep}}\right)=A_{\text{Y}-90}\left(t_m\right)\frac{{\lambda }_2-{\lambda }_1}{{\lambda }_2\left[e^{-{\lambda }_1\Delta t_1}-e^{-{\lambda }_2\Delta t_1}\right]}{e}^{{\lambda }_2\Delta t_2}$$ (13)

Once A_Sr-90(t_Sr-sep) is obtained, using the Scheme 2 as an example, the ⁹⁰Sr activity at a reference time is,:

$$\begin{gathered} A_{\text{Sr}-90}\left(t_{\text{ref}}\right)=A_{\text{Sr}-90}\left(t_{\text{Sr}-\text{sep}}\right)e^{{\lambda }_1\Delta t_3} \\ =A_{\text{Y}-90}\left(t_m\right)\frac{{\lambda }_2-{\lambda }_1}{{\lambda }_2\left[e^{-{\lambda }_1\Delta t_1}-e^{-{\lambda }_2\Delta t_1}\right]}{e}^{{\lambda }_2\Delta t_2+{\lambda }_1\Delta t_3} \end{gathered}$$ (14)

where t_Sr-sep is the time when strontium is separated, t_Y-sep is the time when yttrium is separated, t_m is the start of ⁹⁰Y data acquisition, t_ref is a designated reference time of interest for ⁹⁰Sr, Δt₁ = t_Y-sep – t_Sr-sep, Δt₂ = t_m – t_Y-sep, Δt₃ = t_Sr-sep – t_ref, A_Y-90(t_m) is the ⁹⁰Y activity measured at t_m, A_Sr-90(t_Sr-sep) is the ⁹⁰Sr activity calculated at t_Sr-sep using Eq. (13), A_Sr-90(t_ref) is the ⁹⁰Sr activity calculated at t_ref using Eq. (14).

We may use the Eqs. (13) or (14) to provide some guidance on the correlation of needed detection limit for ⁹⁰Y, ⁹⁰Y ingrowth time, and the time between yttrium separation and measurement to the ⁹⁰Sr activity level of interest. Equation (12) can also be used for the same purpose.

Using schemes 1 and 2, we may accurately and rapidly obtain the activity of both ⁹⁰Sr and ⁹⁰Y. Therefore, if an additional portion of a sample is used for measuring the total activity of ⁸⁹Sr/⁹⁰Sr/⁹⁰Y, A_total(t_m), at the same time as that for ⁹⁰Sr and ⁹⁰Y, then we may use the following equation A_Sr−89(t_m) = A_total(t_m) – A_Sr−90(t_m) – A_Y−90(t_m) to obtain the activity of ⁸⁹Sr.

⁹⁵Zr/⁹⁵Nb system

The decay scheme of ⁹⁵Zr to ⁹⁵Nb includes two decay routes:

1. ${}^{95}\text{Zr}\stackrel{\text{branching ratio }:0.9892}{\to }{}^{95}\text{Nb and}$

2. ${}^{95}\text{Zr}\stackrel{\text{branching ratio }:0.0108}{\to }{}^{95\text{m}}\text{Nb and }\stackrel{\text{branching ratio }:0.975}{\to }{}^{95}\text{Nb}.$

In this article, we focus on route (1) because about 99% of the ⁹⁵Nb activity is generated through this route. The same method can be used for ingrowth correction for ⁹⁵Nb generated from ^95mNb decay via route (2) using Eq. (15) if the activity concentration of ^95mNb at a reference time was obtained. We may also note that the amount of ⁹⁵Nb produced from route (2) is only about 1% of total ⁹⁵Nb produced from the decay of ⁹⁵Zr. Both Eq. (2) with A₂(t_m0) ≈ A₂ and Eq. (5) were used for correcting various measured activity concentrations of ⁹⁵Nb to a reference time. The following equation should be used if the ^95mNb’s contribution to the ⁹⁵Nb activity needs to be considered for a more accurate result required for metrology purpose:

$$A_{\text{Nb}-95}\left(t_{\text{ref}}\right)=\{A_{\text{Nb}-95}\left(t_{m0}\right)-\left[\frac{{\lambda }_2{p}_1{A}_{Zr-95}\left(t_{\text{ref}}\right)}{{\lambda }_2-{\lambda }_1}\left(e^{-{\lambda }_1\Delta t}-e^{-{\lambda }_2\Delta t}\right)\right]-\left[\frac{{\lambda }_2{p}_2{A}_{N\text{b}-95m}\left(t_{\text{ref}}\right)}{{\lambda }_2-{\lambda }_3}\left(e^{-{\lambda }_3\Delta t}-e^{-{\lambda }_2\Delta t}\right)\right]\}{e}^{{\lambda }_2\Delta t}$$ (15)

where

λ₁ is the decay constant of ⁹⁵Zr, λ₂ is the decay constant of ⁹⁵Nb, λ₃ is the decay constant of ^95mNb, p₁ = 0.9892 is the branching ratio of ⁹⁵Zr to ⁹⁵Nb, p₂ = 0.975 is the the branching ratio of ^95mNb to ⁹⁵Nb, t_ref is a designated reference time, t_m0 is the start of data acquisition, Δt = t_ref − t_m0, A_Nb-95(t_m0) is the measured ⁹⁵Nb activity at t_m0, and A_Nb-95m(t_ref) can be obtained using Eq. (2). The experimental descriptions, results and discussion for ⁹⁵Zr/⁹⁵Nb system are presented in the next section.

Experimental

The measurement data were obtained from a Macon soil sample (~ 800 g in a 500 mL wide mouth bottle) and an air filter sample provided by Eckert & Ziegler Analytics Inc. (EZA). “Macon soil” is the product name provided by the inter-laboratory comparison sample provider. These samples were custom-made for an inter-laboratory comparison and contained a mixture of fresh fission products with an approximate total activity of 37 kBq in the soil and 18.5 kBq in the air filter. The reference time was set at 22 days after the end of the irradiation period. Data were collected using two high purity germanium detectors (~ 67% relative efficiency) with a resolution of < 2 keV FWHM @1332 keV (⁶⁰Co). The Lynx® multi channel analyzer (MCA) (Mirion technologies (Canberra), Inc.) was used for data acquisition and Canberra Genie 2000/Apex-Gamma™ was used for data processing. The amplifier gain was adjusted to obtain approximately 0.5 keV per channel energy calibration. Detector efficiency curves were created using laboratory sourceless calibration software (LabSOCS™) (Mirion technologies (Canberra), Inc.) utilizing the custom beaker template to create models matching the sample geometries. The weighted average activity was used for calculating the activity of the identified radioisotopes. The samples were counted as received for either 12 h (day 2 and 29) or 2 h (day 8 and 15).

Results and discussion

We used Eqs. (2) and (5) to obtain the activity concentration of ⁹⁵Nb at a predetermined reference time with measured activity concentrations of ⁹⁵Zr and ⁹⁵Nb at the following days after the reference time: days 2, 8, 15, 29. The spectrometer software did not have the functionality for ingrowth correction. Therefore, measured activity concentrations of ⁹⁵Nb needed to be corrected. Table 1 gives an example of data needed from an instrument analysis report and physical constants as inputs for using Eq. (2).

Table 1.

Inputs needed for decay/ingrowth correction using Eq. (2), sample: air filter

Δt=tm0−tref (days)	AZr−95(tref) (Bq/kg)	ANb−95(tm) (Bq/kg)	λZr−95 (day−1)	λNb−95 (day−1)	p1
29	1.29E+03	7.68E+02	ln(2)/64.032	ln(2)/34.991	0.9892

Table 2 shows an example of the decay-and ingrowth-corrected activity concentrations of ⁹⁵Nb at the reference time from data collected at day 29 using Eqs. (2) and (5). The activity concentrations of ⁹⁵Nb at the reference time obtained from data collected at days 2, 9, and 15 are submitted in a separate supplemental data file. Without knowing the correct way of correcting the progeny’s ingrowth, one would normally either assume that the system is at secular equilibrium state and calculate the progeny’s activity concentration using the simple exponential decay law with the parent’s decay constant or assume that the progeny simply follows the exponential decay law with its own decay constant. To illustrate the errors that would have been introduced if Eq. (2) or (5) was not used, we calculated ⁹⁵Nb’s activity concentrations, also shown in Table 2, using $A_{\text{Nb}-95}\left(t_{\text{ref}}\right)=A_{\text{Nb}-95}\left(t_m\right)e^{{\lambda }_{\text{Zr}-95}\left(t_m-t_{\text{ref}}\right)}$, i.e. assuming ⁹⁵Zr and ⁹⁵Nb were at secular equilibrium. Table 2 and the supplemental data indicated that Eqs. (2) and (5) yield much more accurate results compared to the results obtained by simply using the exponential decay law especially when the measurement time is farther away from the reference time. For the soil sample, using Eqs. (2) and (5), we were able to correct the activity concentration of ⁹⁵Nb to a reference time within an accuracy of −10% compared to the ⁹⁵Nb(t_ref) value provided by EZA. For the air filter sample, we were able to correct the activity concentration of ⁹⁵Nb to a reference time within −3% accuracy comparing to the ⁹⁵Nb(t_ref) value provided by EZA.

Table 2.

Ingrowth-corrected ⁹⁵Nb activity concentration (Bq/kg) at a reference time using Eq. (2), Eq. (5), and without considering ⁹⁵Nb ingrowth, Δt = t_m0 − t_ref = 29 days. Sample: Macon soil, EZA value: 1.26E+03 Bq/kg; sample: air filter, EZA value: 5.2E+02 Bq/kg

	Equation (2)	Equation (5)	$A\left(t_m\right)e^{{\lambda }_{\text{Zr}-95}\Delta t}$
ANb−95(tref) ± 1σ (Bq/kg)	1.2E+03	1.16E+03	2.4E+03
Sample: Macon soil	± 2E+02	± 9E+01	± 3E+02
ANb−95(tref) ± 1σ (Bq/kg)	5.2E+02	5.1E+02	1.1E+03
Sample: air filter	± 9E+01	± 4E+01	± 1.3E+02

Conclusions

Based on the data obtained from the ⁹⁵Zr/⁹⁵Nb system, we may conclude that the presented methods can be used for accurately calculating the activity of a parent-progeny system. As shown in Table 2, comparing the activity concentrations of ⁹⁵Nb(t_ref) obtained using Eqs. (2) and (5) assuming ⁹⁵Nb(t = 0) not present, one can conclude that ⁹⁵Nb was not present at the irradiation day and it was produced only by the decay of ⁹⁵Zr. Therefore, comparing the activity of the progeny at a reference time using Eqs. (2) and (5) will yield information on whether the progeny existed at the beginning of a nuclear event and the ratio of the progeny’s activity obtained using Eqs. (2) and (5) will yield useful information for nuclear forensics. The schemes and analytical solutions presented for a non-equilibrated ⁹⁰Sr/⁹⁰Y system will enable a radio-analytical laboratory to provide accurate ⁹⁰Sr activity data and reduce the turnaround time from weeks to a few days for responding to a nuclear incident. The schemes 1 and 2 for rapidly obtaining ⁹⁰Sr activity by separating and measuring ⁹⁰Y are especially useful for removing ⁸⁹Sr’s interference in ⁹⁰Sr determinations.