A comparison of two calibration methods of the ²²⁹Th spike for uranium age dating by ²³⁰Th/²³⁴U radiochronometer

Abstract

²³⁰Th/²³⁴U radiochronometer was commonly used for uranium age determination which could help to reveal the origin of the nuclear material. ²²⁹Th was a relatively appropriate spike for measuring ²³⁰Th by isotope dilution mass spectrometry and its concentration needs to be calibrated before using as spike. A prepared ²²⁹Th solution which was used as Th isotope spike for age dating of CRM U850 sample were calibrated separately by ²³²Th and ²³⁰Th standard materials. The calibrated results were compared and evaluated.

Introduction

Uranium age could be defined as the time since the uranium material was last subjected to a separation process capable of separating uranium from its radioactive-decay daughters [1, 2]. It is a dynamic fingerprint of nuclear material and could help to reveal the origin of the nuclear material and trace back the possible route of the sample to its source, which could be verified against the declared origin in nuclear forensics [3, 4].

Several isotope ratios such as ²³⁰Th/²³⁴U, ²³¹Pa/²³⁵U and ²¹⁴Bi/²³⁴U have been successfully applied for the determination of the age of uranium [5,6,7]. However, in case of ²³¹Pa/²³⁵U lack of suitable isotopic spike inhibits its application, while for ²¹⁴Bi/²³⁴U ratio the method is restricted only to highly enriched uranium (HEU) samples or old nuclear materials due to the slow ingrowth of this daughter nuclide [8]. ²³⁰Th/²³⁴U isotope ratio was commonly used for uranium age dating. For most samples of uranium, ages determined with this ²³⁰Th-²³⁴U chronometer are considered as “model ages” by R. W. Williams [9], because they are based on the assumptions of (a) some initial amount of ²³⁰Th in the sample, and (b) closed-system behavior of the sample since production. Under the condition of meeting these assumptions the accuracy of dates determined with the ²³⁰Th-²³⁴U chronometer only relies upon accurate ²³⁰Th and ²³⁴U measurements.

In the developed isotopic dilution mass spectrometric methods for determining ratio of ²³⁰Th/²³⁴U, ²²⁹Th solution was used as spike for ²³⁰Th. It was usually prepared by separating the ²²⁹Th from old ²³³U solution [7,8,9]. The ²²⁹Th solution should be calibrated by reverse isotope dilution measurement with another isotope standard material before use. Usually, it is a long and complicated process to determine the age of uranium material.

Both the ²³⁰Th and ²³²Th isotope standard reference material could be used for ²²⁹Th calibration [10]. The aim of this study was to develop a more exact ²²⁹Th calibration method for the age determination of uranium materials using ²²⁹Th solution as spike by isotope dilution multi-collector inductively coupled plasma mass spectrometry.

Experimental

Instrumentation and materials

The mass spectrometric analysis was carried out using an inductively coupled plasma mass spectrometer equipped with 7 Ion counters, 1 Daly detector and 9 Faraday cups (Micromass Isoprobe-T, GV). The 1.5 mg uranium oxide reference material (CRM 850) was dissolved in 10 mL HNO₃ for age analysis. The ²³³U isotopic standard (IRMM051) was used to spike the samples for ²³⁴U measurements. ²²⁹Th spike for ²³⁰Th analysis were calibrated by natural Th-solution (Relative standard uncertaintyof certified value is 0.3% (k = 1), SPEXertificate) and ²³⁰Th standard material provided by LLNL (prepared from NIST 4342A, Relative standard uncertainty of certified value is 0.19% (k = 1)). Standard solutions (prepared from ERM-199, CRM U030, CRM U200 and working standard UTB900) were used to correct for instrumental mass discrimination and relative detector gain factors. The TEVA (100–150 μm) extraction chromatographic resin was used for chemical separation. All sample preparation procedures were carried out under clean room conditions (Class 100,000).

²²⁹Th spike calibration by ²³²Th

The primary ²²⁹Th solution which is as ²²⁹Th spike used for the ²³⁰Th determination was prepared by separating the Th from an old ²³³U solution. The ²²⁹Th concentration in the primary solution obtained was determined by reverse isotope dilution ICP-MS measurement applying a 0.870(7) µg g^{−1 232}Th standard solution which was obtained by successively diluting the certified natural Th-solution with 2% HNO₃ solution. Three aliquots of 0.5 mL ²²⁹Th solution were added in 0, 5 and 2.5 mL 0.870(7) µg g^{−1 232}Th standard solution respectively. After being isotopic equilibrium, ²²⁹Th/²³²Th atom ratios in three samples were measured by MC-ICP-MS. Both the ²²⁹ Th and ²³²Th ionic beams were collected by faraday cups (FC).

²²⁹Th spike calibration by ²³⁰Th

The primary ²²⁹Th solution was successively diluted with 2% HNO₃ solution to get around 100 pg g^{−1 229}Th solution (named as solution2). Each dilution was approximately 100-fold and 30-fold respectively. Three aliquots of 0.8 mL solution2 were added in around 0, 80 and 80 pg ²³⁰Th standard reference solution which was from gravimetrically diluting ²³⁰Th standard reference material (NIST 4342A) respectively. After being isotopic equilibrium, ²²⁹Th/²³⁰Th ratios in three samples were measured by MC-ICP-MS. Both the ²²⁹Th and ²³⁰Th ionic beams were collected by ionic counters (IC).

Sample preparation for ²³⁴U determination

Approximately 15 μg aliquot of the dissolved uranium oxide sample (CRM 850) was used for the analysis. The sample was diluted twice successively with 2% ultrapure nitric acid solution. Each dilution was approximately tenfold and 100-fold respectively. The ²³⁴U concentration following the second dilution was determined by isotope dilution ICP-MS measurement using ²³³U standard as spike.

Sample preparation for ²³⁰Th determination

The separation method of Th from U in Ref. [2] was adopted. Approximately 300 μg aliquot of the dissolved uranium oxide sample (CRM 850) was placed in a Teflon cup and spiked with approximately 2 pg of ²²⁹Th spike gravimetrically, respectively. The Th-content of the sample was separated from uranium by 1.6 mL TEVA resin. In order to further purify the sample, the extraction chromatographic separation was repeated.

Data evaluation

All raw data were corrected taking into account instrumental mass bias using linear correction. The overall uncertainty was calculated taking into account the uncertainty of the weight measurements, spike concentrations, isotope ratios measurement and half-lives according to “A Guide to Expression of Uncertainty in Measurement” (ISO/1995).

Results and discussion

The ²²⁹Th concentration in the working solution obtained was determined twice by reverse isotope dilution ICP-MS measurement applying certified natural Th-solution with different additive quantum. CRM U200 was used to correct instrumental mass bias. The results were showed in Table 1. The twice determined isotope concentrations were identical within the range of number of significant digit.

Table 1 Isotope concentrations of ²²⁹Th spike calibrated by ²³²Th

The ²²⁹Th concentration in the working solution were obtained according to the gravimetrical dilution and the calibrated Th isotope concentration in dilution2 which was repetitively determined twice by reverse isotope dilution ICP-MS measurement applying ²³⁰Th standard reference solution. CRM U005 and CRM U200 were used to correct instrumental mass bias and relative detector gain factors. The results were showed in Table 2. The relative deviation between two determined isotope concentrations was in 2%.

Table 2 Isotope concentrations of ²²⁹Th spike calibrated by ²³⁰Th

Two calibration methods could be expressed in the Eqs. (1–2). Assumed $9_{\text{p}}$ was the ²²⁹Th concentration of primary ²²⁹Th solution and $x_{\text{p}}$ was the ²³²Th or ²³⁰Th concentration of primary ²²⁹Th solution. $\left( {{\raise0.7ex\hbox{$9$} \!\mathord{\left/ {\vphantom {9 x}}\right.\kern-0pt} \!\lower0.7ex\hbox{$x$}}} \right)_{\text{pm}}$ was the determined Th isotope ratios of primary ²²⁹Th solution. $\left( {{\raise0.7ex\hbox{$9$} \!\mathord{\left/ {\vphantom {9 x}}\right.\kern-0pt} \!\lower0.7ex\hbox{$x$}}} \right)_{\text{mm}}$ was the determined Th isotope ratios of spiked aliquot primary ²²⁹Th solution by ²³²Th or ²³⁰Th standard reference material. $m$ and $n$ were the weight of aliquot primary ²²⁹Th solution and weight of ²³²Th or ²³⁰Th standard reference material respectively. $x_{\text{s}}$ was the concentration of ²³²Th or ²³⁰Th standard reference material.

$$ \left\{ {\begin{array}{*{20}l} {\frac{{_{{9_{{\text{p}}} }} }}{{x_{{\text{p}}} }} = \left( {{\raise0.7ex\hbox{$9$} \!\mathord{\left/ {\vphantom {9 x}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$x$}}} \right)_{{{\text{pm}}}} \;\;\;\;\;\;\;\;\;\;\;(1)} \hfill \\ {\frac{{9_{{\text{p}}} \cdot m}}{{x_{{\text{p}}} \cdot m + x_{{\text{s}}} \cdot n}} = \left( {{\raise0.7ex\hbox{$9$} \!\mathord{\left/ {\vphantom {9 x}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$x$}}} \right)_{{{\text{mm}}}} \;\;(2)\;\;\;} \hfill \\ \end{array} } \right.$$

$9_{\text{p}}$ could be expressed to Eq. (3) by the solution of Eq. (1) and Eq. (2).

$$ 9_{\text{p}} = \frac{{n \cdot x_{\text{s}} \cdot \left( {{\raise0.7ex\hbox{$9$} \!\mathord{\left/ {\vphantom {9 x}}\right.\kern-0pt} \!\lower0.7ex\hbox{$x$}}} \right)_{\text{pm}} \cdot \left( {{\raise0.7ex\hbox{$9$} \!\mathord{\left/ {\vphantom {9 x}}\right.\kern-0pt} \!\lower0.7ex\hbox{$x$}}} \right)_{\text{mm}} }}{{m \cdot \left[ {\left( {{\raise0.7ex\hbox{$9$} \!\mathord{\left/ {\vphantom {9 x}}\right.\kern-0pt} \!\lower0.7ex\hbox{$x$}}} \right)_{\text{pm}} - \left( {{\raise0.7ex\hbox{$9$} \!\mathord{\left/ {\vphantom {9 x}}\right.\kern-0pt} \!\lower0.7ex\hbox{$x$}}} \right)_{\text{mm}} } \right]}} $$

(3)

Linear propagation of uncertainty on the weight of aliquot primary ²²⁹Th solution $m$, weight of ²³²Th or ²³⁰Th standard reference material $n$, the concentration of ²³²Th or ²³⁰Th standard reference material $x_{\text{s}}$, the determined Th isotope ratios of primary ²²⁹Th solution $\left( {{\raise0.7ex\hbox{$9$} \!\mathord{\left/ {\vphantom {9 x}}\right.\kern-0pt} \!\lower0.7ex\hbox{$x$}}} \right)_{\text{pm}}$, and the determined Th isotope ratios of spiked aliquot primary ²²⁹Th solution $\left( {{\raise0.7ex\hbox{$9$} \!\mathord{\left/ {\vphantom {9 x}}\right.\kern-0pt} \!\lower0.7ex\hbox{$x$}}} \right)_{\text{mm}}$ results to:

$$ u(9_{\text{p}} ) = \sqrt {\begin{array}{*{20}c} {\left( {9_{\text{p}} \cdot \frac{u(n)}{n}} \right)^{2} + \left( {9_{\text{p}} \cdot \frac{{u(x_{\text{s}} )}}{{x_{\text{s}} }}} \right)^{2} + \left( {9_{\text{p}} \cdot \frac{u(m)}{m}} \right)^{2} + } \\ {\left( {9_{\text{p}} \cdot \frac{{\left( {{\raise0.7ex\hbox{$9$} \!\mathord{\left/ {\vphantom {9 x}}\right.\kern-0pt} \!\lower0.7ex\hbox{$x$}}} \right)_{\text{mm}} }}{{\left( {{\raise0.7ex\hbox{$9$} \!\mathord{\left/ {\vphantom {9 x}}\right.\kern-0pt} \!\lower0.7ex\hbox{$x$}}} \right)_{\text{pm}} - \left( {{\raise0.7ex\hbox{$9$} \!\mathord{\left/ {\vphantom {9 x}}\right.\kern-0pt} \!\lower0.7ex\hbox{$x$}}} \right)_{\text{mm}} }} \cdot \frac{{u\left[ {\left( {{\raise0.7ex\hbox{$9$} \!\mathord{\left/ {\vphantom {9 x}}\right.\kern-0pt} \!\lower0.7ex\hbox{$x$}}} \right)_{\text{pm}} } \right]}}{{\left( {{\raise0.7ex\hbox{$9$} \!\mathord{\left/ {\vphantom {9 x}}\right.\kern-0pt} \!\lower0.7ex\hbox{$x$}}} \right)_{\text{pm}} }}} \right)^{2} + \left( {9_{\text{p}} \cdot \frac{{\left( {{\raise0.7ex\hbox{$9$} \!\mathord{\left/ {\vphantom {9 x}}\right.\kern-0pt} \!\lower0.7ex\hbox{$x$}}} \right)_{\text{pm}} }}{{\left( {{\raise0.7ex\hbox{$9$} \!\mathord{\left/ {\vphantom {9 x}}\right.\kern-0pt} \!\lower0.7ex\hbox{$x$}}} \right)_{\text{pm}} - \left( {{\raise0.7ex\hbox{$9$} \!\mathord{\left/ {\vphantom {9 x}}\right.\kern-0pt} \!\lower0.7ex\hbox{$x$}}} \right)_{\text{mm}} }} \cdot \frac{{u\left[ {\left( {{\raise0.7ex\hbox{$9$} \!\mathord{\left/ {\vphantom {9 x}}\right.\kern-0pt} \!\lower0.7ex\hbox{$x$}}} \right)_{\text{mm}} } \right]}}{{\left( {{\raise0.7ex\hbox{$9$} \!\mathord{\left/ {\vphantom {9 x}}\right.\kern-0pt} \!\lower0.7ex\hbox{$x$}}} \right)_{\text{mm}} }}} \right)^{2} } \\ \end{array} } $$

(4)

For calibration by ²³⁰Th, the uncertainty on the weight of aliquot primary ²²⁹Th solution $m$ was including of uncertainty of dilution process. Two representative uncertainty budgets for the ²²⁹Th spike calibration by ²³²Th and ²³⁰Th were given in Table 3.

Table 3 The uncertainty budget for calibrated ²²⁹Th concentration in spike

The calibrated Th isotope concentrations by two methods were close, but in calibration by ²³²Th, the twice determined isotope concentrations were identical within the range of number of significant digit and in calibration by ²³⁰Th, there was a deviation between twice measurements. The difference between two calibration methods could also be concluded by the math theory. For convenience, the computation (3) of ²²⁹Th concentration in primary solution could be simplified into Eq. (5), as assuming the amount of ²³²Th or ²³⁰Th in original primary solution could be ignored. $x$ is the added true amount of ²³²Th or ²³⁰Th standard material. $229$ is the amount of ²²⁹Th in primary ²²⁹Th solution.

$$ 229 = x \cdot \left( {{\raise0.7ex\hbox{${229}$} \!\mathord{\left/ {\vphantom {{229} x}}\right.\kern-0pt} \!\lower0.7ex\hbox{$x$}}} \right)_{\text{measure}} $$

(5)

When the uncertainty of measurement could be ignored, we could assume that $\left( {{\raise0.7ex\hbox{${229}$} \!\mathord{\left/ {\vphantom {{229} x}}\right.\kern-0pt} \!\lower0.7ex\hbox{$x$}}} \right)_{\text{measure}}$ is the true isotope ratio value and $\Delta x$ is the distance between reference value used for calculation and true value $x$.Then ²²⁹Th concentration in primary solution could be expressed by Eq. (6).

$$ 229 = \left( {x + \Delta x} \right) \cdot \left( {{\raise0.7ex\hbox{${229}$} \!\mathord{\left/ {\vphantom {{229} x}}\right.\kern-0pt} \!\lower0.7ex\hbox{$x$}}} \right)_{\text{measure}} $$

(6)

By math operation, the Eq. (7) could be gotten from Eq. (6).

$$ 229 = 229 + 229 \cdot \frac{\Delta x}{x} $$

(7)

Similarly, when added different amount of th isotope standard expressed by $kx$. The apparent amount used for calculation is equal to the value of $k$ multiplied by sum of $x$ and $\Delta x$. The calculated expression of ²²⁹Th concentration in primary solution from Eq. (8) is the same to the Eq. (7).

$$ 229 = k\left( {x + \Delta x} \right) \cdot \left( {{\raise0.7ex\hbox{${229}$} \!\mathord{\left/ {\vphantom {{229} {kx}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${kx}$}}} \right)_{\text{measure}} $$

(8)

That means when adding different amount of Th standard material, if the uncertainty of measurement could be very little to be ignored the calibrated ²²⁹Th concentration in every determination are same. Because the $\Delta x$ is a fixed value which is independent on the measurement process. From the uncertainty budget in Table 3, we can see the contribution from the measurement to total uncertainty could be ignored relative to the standard reference material. So it could be explained the twice calibrated ²²⁹Th concentration were identical.

From the above deducing we know the deviation of reference value of standard material didn’t affect the point-to-point reproducibility of calibrated results. So it could be assumed the apparent added amount of Th isotope standard material is true value $x$ and $\Delta^{1}$ is the distance between measured isotope ratio value and true isotope ratio value. In calibrating by ²³⁰Th, when the deviation of measurement could not be ignored, the calibrated ²²⁹Th concentration in primary solution could be expressed by Eq. (9).

$$ 229 = x \cdot \left( {\left( {{\raise0.7ex\hbox{${229}$} \!\mathord{\left/ {\vphantom {{229} x}}\right.\kern-0pt} \!\lower0.7ex\hbox{$x$}}} \right)_{\text{true}} + \Delta^{1} } \right) $$

(9)

By math operation, the Eq. (10) could be gotten from Eq. (9).

$$ 229 = 229 + \Delta^{1} \cdot x $$

(10)

Similarly, when adding different amount of Th isotope standard material, the calculated Eq. (11) of ²²⁹Th concentration in primary solution is different with the Eq. (10). Deltas were different in every measurement due to fluctuant parameter. So it could be explained two calibrated results by ²³⁰Th have deviation.

$$ 229 = 229 + \Delta^{2} \cdot kx $$

(11)

As the uncertainty budget in Table 3 shown, the largest contribution to combined uncertainty of ²²⁹Th concentration calibrated by ²³⁰Th was from sample ratio measurement. Standard reference material for Mass bias correction and concentration of ²³⁰Th standard reference material were two significant contributors to combined uncertainty of ²²⁹Th concentration. But for calibration by ²³²Th the only one significant contributor was from the reference concentration value of ²³²Th standard reference material. The difference may be due to that the precision of measurement by IC for relative low concentration sample is lower than the precision of measurement by FC for relative high concentration sample. The calibration process contribution to combined uncertainty of ²²⁹Th concentration calibrated by ²³²Th could be less with one order of magnitude than the reference concentration value of ²³²Th standard reference material. It was reflected by the twice identical determined isotope concentrations within the range of number of significant digit in calibration by ²³²Th. It seems that the calibration method by ²³²Th is superior to calibration method by ²³⁰Th. The combined uncertainty of ²²⁹Th concentration in single observation calibrated by ²³²Th is a little higher than by ²³⁰Th just because the relative uncertainty of used ²³²Th standard reference material is 8.5‰ which is much larger than 2.5‰ of ²³⁰Th standard reference material. Considering about the reproducibility of two independent measurements, the combined uncertainty of average ²²⁹Th concentration in spike calibrated by ²³⁰Th was larger than by ²³²Th.

According to the weight of spikes and determined isotope ratios, the ²³⁴U/²³⁰Th ratio could be calculated. The model age was calculated according to simplified age-dating Eq. (12) where R is the ²³⁰Th/²³⁴U atomic ratio, and λ ₁ and λ ₂ are the decay constants of ²³⁴U and ²³⁰Th, respectively. Half-lives for ²³⁰Th and ²³⁴U of 75690 ± 230 years and 245250 ± 490 years were used for these decay constants, respectively [11].

$$ t = \frac{1}{{\lambda_{1} - \lambda_{2} }}\ln \left[ {1 + \frac{{R\left( {\lambda_{1} - \lambda_{2} } \right)}}{{\lambda_{1}^{{}} }}} \right] $$

(12)

The results of age determination were shown in Table 4. The purification date was calculated: 1957-4-15 ± 209 days for CRM U850 with ²²⁹Th spike calibrated by ²³²Th and 1956-2-18 ± 301 days for CRM U850 with ²²⁹Th spike calibrated by ²³⁰Th. The U850 analyzed give the determined age older than the purification dates of record (1957-12-31). This indicates that excess ²³⁰Th in U850 standard results from incomplete purification during production.

Table 4 The determined uranium age of CRM U850

The determined model age of CRM U850 and some results in the literature were shown in the Fig. 1. The data showed good agreement with the ages determined in the earlier works by Wallenius et al. [2] and Williams and Gaffney [8] using the ratio ²³⁴U/²³⁰Th. The ²³⁰Th was determined with ²³²Th spike in the work by Wallenius et al. and the ²³⁰Th was determined with ²²⁹Th spike which was calibrated by the NIST SRM 4342A ²³⁰Th radioactivity solution in the work by Williams & Gaffney. The determined purification date of CRM U850 using the ²²⁹Th spike calibrated by ²³²Th which in this work was lying between the results in the literature was closer to the average model age of current published results than the determined purification date of CRM U850 using the ²²⁹Th spike calibrated by ²³⁰Th which was earlier than the results in the literature.

Fig. 1

Results of the age determination of CRM U850

Two representative uncertainty budgets for the age-dating analyses were given for sample CRM U850 in Table 5, 6. The largest part of the uncertainty in these analyses comes from the ²²⁹Th spike, of which the uncertainty on the Mass bias correction and ratio measurement for ²³⁰Th/²²⁹Th analysis were two significant contributors to combined uncertainty of determined U850 age. The accuracy concentration of ²²⁹Th spike has a great effect on the result of determined age of Uranium material.

Table 5 The uncertainty budget for sample U850 using the ²²⁹Th spike calibrated by ²³²Th

Table 6 The uncertainty budget for sample U850 using the ²²⁹Th spike calibrated by ²³⁰Th

Conclusions

Precisely and accurately calibrating Th isotope spike was very important to determine uranium model age by ²³⁰Th/²³⁴U radiochronometer. Using higher concentration primary Th spike solution to be directly calibrated and be used as spike for uranium age dating could get more accuracy results than diluted to be calibrated especially in the case the uncertainty of standard materials for calibration were similar. Easily available ²³²Th standard material could be used to calibrate ²²⁹Th spike to get accuracy results instead of uncommon ²³⁰Th standard material.