A new measurement of the 122Sb half-life

Following significant discrepancies observed when decay-correcting 122Sb γ-peak count rates to a reference time, we looked at the literature supporting the presently recommended 2.7238(2) d (1σ) 122Sb half-life value as the source of these discrepancies. Investigation revealed that the value was derived from an inconsistent dataset and was published without reporting details of the experiment nor the uncertainty budget. In this work we performed a new measurement of the 122Sb half-life by measuring the 122Sb decay of neutron-activated antimony samples using state-of-the-art γ-detection systems characterized in terms of efficiency drift and random pulse pile-up. The measurement was carried out in two different laboratories with the same method. The resulting 2.69454(39) d and 2.69388(30) d (1σ) 122Sb half-life values are in agreement at the evaluated 10–4 relative combined standard uncertainty level but are significantly lower (1.07% and 1.10% lower, respectively) than the preexisting recommended value.


Introduction
Antimony has two stable isotopes ( 121 Sb and 123 Sb) that can be detected by neutron activation via nuclear reactions 121 Sb(n,γ) 122 Sb and 123 Sb(n,γ) 124 Sb and quantified by counting γ-photons emitted by 122 Sb and 124 Sb at the interference-free energies 564.2 keV and 602.7 keV, respectively. Preliminary measurements carried out to test the possibility of determining the 121 Sb/ 123 Sb ratio at 0.1% relative uncertainty level unexpectedly showed a linear negative drift of 2% over about 200 h for the 122 Sb 564.2 keV γ-peak count rate after correction for decay. Notably, the stability of the decay corrected 124 Sb 602.7 keV γ-peak count rate obtained with the same data did not show the same drift of the detection system. This experimental evidence suggested a potential bias affecting the presently recommended 2.7238(2) d 122 Sb half-life, t 1/2 [1], adopted for the decay correction.
Here and hereafter uncertainties in parentheses are standard uncertainties (k = 1). A literature review revealed that the recommended t 1/2 ( 122 Sb) value is based on the most recent result of an inconsistent dataset published in 1990 [2] without reporting details on the experiment nor the uncertainty budget which are compulsory to support the claimed 10 -4 relative standard uncertainty level [3].
In this work we aimed to refine the knowledge of the t 1/2 ( 122 Sb). We repeated the measurement of the 122 Sb halflife by taking advantage of state-of-the-art digital signal processing γ-spectrometers coupled to high-purity germanium detectors characterized in terms of efficiency drift and random pulse pile-up. This paper reports the adopted measurement method and the experiments carried out to measure the t 1/2 ( 122 Sb). The uncertainty budget is included and the result is compared with the published values.

Measurement method
The approach used to measure the t 1/2 ( 122 Sb) is based on a method previously applied for the determination of the t 1/2 ( 31 Si) [4] and consists of repeated observations of the exponential decay of the activity performed via γ-spectrometry measurements. A single observation is a sequence of repeated count rate measurements performed by recording consecutive γ-spectra of neutron-activated samples. The (full-peak) 122 Sb γ-photon count rate at the beginning of the ith count of the jth sequence,C ij , starting at t dij is computed using the formula (1) found in [4] adjusted to include the effect of a linear drift of the γ-photon detection efficiency: where = ln (2)∕t 1∕2 is the 122 Sb decay constant, n ij is the full-peak net count, t dij is the decay time, t cij is the (real) counting time, is the (constant) relative variation of efficiency per unit time, ij and f ij are the dead time and pile-up correction factors, respectively. Specifically, ij = t cij ∕ t cij − t dead ij and f ij = e (tdead ij ∕t cij ) , where t dead ij is the dead time and is the pile-up constant.
If the activated sample is fixed at steady distance from the detector end-cap during the recording of a sequence, the γ-photon count rate is proportional to the activity via the γ-emission yield multiplied by the γ-photon efficiency of the detection system. Accordingly, the nonlinear equation modeling C ij versus t dij is where C j t d1j is the expected value of the count rate at the starting time of the first count of the jth sequence, t d1j , and ij is the error term.
If the detector is in normal working conditions ≪ 1 , and the experiments are carried out to assure t cij < t 1∕2 and rij = ij C ij t dij ≪ 1 , i.e. the error term normalized to C ij t dij , Eq. (1) simplifies to and Eq. (2) can be adjusted to get the linear equation is the natural logarithm of the count rate C ij t dij normalized to C 1j t d1j and corrected for efficiency drift, and m j = ln C j t d1j C 1j t d1j is the natural logarithm of the count rate C j t d1j normalized to C 1j t d1j .
The t 1/2 ( 122 Sb) and C j t d1j values are obtained from the (common) slope, , and intercepts, m j , respectively, of the j straight lines fitted to Y ij versus t dij − t d1j data. Since C ij t dij and C 1j t d1j depends on via (3), the final value is obtained iteratively until convergence. It is worth noting that rij corresponds to the relative error of n ij , which in turn depends on counting statistics. As a result, the standard (1) deviation of rij can be kept constant (and small) by properly increasing t cij during the activity decay. This removes the need to assign varying weights to the fitted Y ij data when γ-spectra are recorded at constant t cij [5] and allows performing an unweighted fit.
While n ij , t cij , t dead ij , t dij and t d1j values adopted in Eqs. (3) and (4) are determined during the observation of the 122 Sb decay, and are parameters previously measured during the characterization of the detection system carried out by repeated count rate measurements of a long-lived γ-photon source.

Experimental
We carried out two separated experiments, the first at the Istituto Nazionale di Ricerca Metrologica (INRIM) and the latter at the National Institute of Standards and Technology (NIST), using detection systems consisting of a coaxial germanium detector connected to a digital multichannel analyzer.
The ORTEC GEM 50P4-83 1 (relative efficiency 50%, resolution 1.90 keV FWHM at 1332 keV) detector was used at INRIM and the ORTEC GEM 40P4-S (relative efficiency 51%, resolution 1.72 keV FWHM at 1332 keV) detector was used at NIST. In both cases, the ORTEC DSPEC 502 was adopted as the digital multichannel analyzer.
The end-cap of the GEM 50P4-83 was placed inside a low-background graded lead shield located in a room of an underground laboratory with temperature controlled at 23 °C whereas the GEM 40P4-S was placed inside a low-background graded lead shield located in a room of a shielded laboratory with temperature controlled at 20 °C. In these shielding conditions, the (total) input pulse rate due to background was limited to below 15 pulses per second for both detectors. The rise time and flat top digital filter shapingtime constants of the multichannel analyzers were set to 12 μs and 1 μs, respectively, at both INRIM and NIST. The acquisitions were performed in extended live-time correction mode according to the Gedcke-Hale method to compensate for the loss of counts due processing (dead) time, with pulse pile-up rejection in automatic set threshold.
Prior to performing the experiments, we tested the longterm temporal stability of the γ-photon efficiency of the detection systems by recording a single (uninterrupted) sequence of successive counts of the 661.8 keV γ-photons emitted by a 137 Cs source kept at a fixed position from the detector end-cap. Additionally, the pile-up constant was measured using the moving source method [6] by recording a number of sequences of successive counts of the γ-photons emitted by the 137 Cs source always kept at a fixed position. A supplementary moving 152 Eu source was located at different distances from the detector end-cap to change the rate of input pulses at the gate of the detection systems. Along with the sequence collected with the 137 Cs source alone, we collected 6 and 3 sequences with the 152 Eu source located in 6 and 3 different positions for the GEM 50P4-83 and GEM 40P4-S, respectively.
It is worth noting that the 661.8 keV 137 Cs γ-emission used to measure the pile-up constant is close to the 564.2 keV 122 Sb γ-emission adopted to measure the t 1/2 ( 122 Sb). This makes the second order effects due to the γ-energy dependence of the pile-up constant negligible [7]. In addition, the use of 152 Eu as a moving source is a common choice because the shape of its spectrum looks like the neutron-activated materials used in this study. Specifically, the 152 Eu spectrum shows eight main and fifteen minor γ-emissions within the range 122-1458 keV, which is to some extent similar to the seven main and ten minor γ-emissions within the range 564-2090 keV of a neutron-activated antimony sample.
The measurement of the t 1/2 ( 122 Sb) was carried out by recording counts of the 122 Sb 564.2 keV γ-photons emitted from two different high-purity antimony samples after neutron activation. At INRIM, two 1 mg samples were prepared by pipetting and drying in five subsequent steps 1 mL of a 1000 µg mL −1 Sb solution (99.9999% purity) on filter papers placed into polyethylene vials. The two samples were irradiated and the activity decay measured. Each neutron irradiation lasted 1 h and was carried out at a thermal neutron flux of about 6 × 10 12 cm −2 s −1 in the central thimble of the 250 kW TRIGA Mark II reactor operated by the University of Pavia. After activation, each sample was fixed and counted at about 20 cm from the end cap of the detector.
At NIST, two 50 mg samples of high-purity (99.999%) Sb in metal shot form deposited as-is into polyethylene vials were used. Each neutron irradiation lasted 2 min and was carried out at a thermal neutron flux of about 2 × 10 13 cm −2 s −1 in the RT-2 irradiation position at the NIST Center for Neutron Research (NCNR). After activation, each sample was fixed and counted at approximately 20 cm from the end cap of the detector.
Both at INRIM and NIST, the counting time window was adjusted on line by the acquisition software to achieve a 0.1% relative standard uncertainty due to counting statistics of the 137 Cs 661.8 keV γ-peak net count in the stability and pile-up measurements and of the 122 Sb 564.2 keV γ-peak net count in the t 1/2 ( 122 Sb) measurements. In details, the software performs repeated fits of the γ-peak to evaluate the uncertainty of the net count and stops the acquisition when the target 0.1% relative uncertainty is reached.

Results and discussion
The Hyperlab software [8] was used to process the collected spectra and obtain the full-peak net count of the 137 Cs 661.8 keV and 122 Sb 564.2 keV γ-emission. The fitting algorithm evaluates also the net count uncertainty based on Poisson statistics, P n ij .

Stability and pile-up of the detection systems
The stability and pile-up constant of the detection systems was checked and measured, respectively, by analysing the γ-spectra collected during the sequences of counts of the 137 Cs source. The sequences performed to check the stability of the GEM 50P4-83 and GEM 40P4-S consisted of 74 and 93 counts per sequence, N c , respectively, collected in an observation time of 480 h and 470 h. Both the detection systems worked with a 1.4% relative dead time.
The equation adopted to determine the (constant) relative variation of efficiency per unit time, , is obtained by adjusting Eq. (2) to where Y i1 = C i1 t di1 C 11 t d11 1 e − (tdi1−td11) is the count rate C i1 t di1 normalized to C 11 t d11 and corrected for decay, m 1 = C 1 t d11 C 11 t d11 is the count rate C 1 t d11 normalized to C 11 t d11 , ̃= m 1 and ri1 = i1 C i1 t di1 is the error term normalized to C i1 t di1 . In this case, is the 137 Cs decay constant, j = 1 and i = 1, 2,…, N c (number of counts per sequence). The C i1 t di1 and C 11 t d11 values are calculated using Eq. (3) by substituting for convenience f i1 = f 11 = 1, i.e. = 0.
The Y i1 versus t di1 − t d11 values for the GEM 50P4-83 and GEM 40P4-S are plotted in Fig. 1a and b, respectively. The expected 0.1% relative standard deviation of Y i1 , due to counting statistics, is evaluated by P n i1 ∕n i1 .
The value is calculated from the slope, ̃ , and intercept, Since the rate of input pulses at the gate of the detection systems is constant during the stability measurement, outcomes are independent of the value.
The pile-up constant was determined using a sequence of 7 measurements on the GEM 50P4-83 and 4 measurements on the GEM 40P4-S. The GEM 50P4-83 operated at relative dead times of 8%, 12%, 15%, 19%, 22%, 25% and 29%, and the GEM 40P4-S operated at relative dead times of 1%, 11%, 20% and 30%. For the GEM 50P4-83 measurements, the number of counts per sequence was 67 for the 1st, 3rd and 4th sequence, 64 for the 2nd and 5th sequence, 60 for the 6th sequence and 61 for the 7th sequence. For the GEM 40P4-S, the number of counts per sequence was 25 for the 1st sequence, 15 for the 2nd sequence and 10 each for the 3rd and 4th sequence.
The following equation is adopted to measure the pile-up constant and is obtained by adjusting Eq. (2) after replacing the natural logarithm of the count rate C ij t dij normalized to C 11 t d11 and corrected for drift and decay, m j = ln C ij t d11 C 11 t d11 | | |ave i is the natural logarithm of the count rate C ij t d11 normalized to C 11 t d11 , t r dead j = t dead ij ∕t cij | | |ave i is the average relative dead time and rj = ij C 11 t d11 | | |ave i is the average error term normalized to C 11 t d11 . In this case, is the 137 Cs decay constant, j = 1, 2, …, 7 for the GEM 50P4-83 and j = 1, 2, …, 4 for the GEM 40P4-S and i = 1, 2, …, N c . The C ij t dij and C 11 t d11 values are calculated using Eq. (3) by substituting f i1 = f 11 = 1, i.e. = 0.
The Y j versus t rdead j values obtained in case of zero efficiency drift, i.e. = 0 h −1 , for the GEM 50P4-83 and GEM 40P4-S are plotted in the upper graph of Fig. 2a and b, respectively. The expected standard deviation of Y j , due to counting statistics, is evaluated by The is the absolute value of the slope of the straight line fitted to Y j versus t r dead j data. The fitting residuals are plotted in the lower graph of Fig. 2a and b, respectively.
The resulting values of for the GEM 50P4-83 and GEM 40P4-S are 2.51(13) × 10 -2 and 3.42(12) × 10 -2 , respectively. The uncertainty budget of the quoted uncertainties includes a contribution of approximately 70% and 95%, respectively, from counting statistics; the remaining part, 30% and 5%, are due to a possible efficiency drift within the quoted 10 -7 h −1 level uncertainties of obtained during the stability test. Specifically, the uncertainty of due to an efficiency drift is computed as the average of the absolute error of obtained by substituting in (6) = ± 80 × 10 -8 h −1 and = ± 66 × 10 -8 . It is worth noting that the residuals are in agreement with the expected 0.1% � √ N C relative standard deviation of Y j and validate the application of the pile-up correction at least up to a 30% relative dead time on the detection system. Fig. 1 The Y i1 versus t di1 − t d11 values and the fitted straight line obtained during the stability test of the GEM50P4-83 (a) and GEM 40P4-S (b). Error bars indicate a 95% confidence interval due to counting statistics

The half-life time of 122 Sb
The 122 Sb half-life time was measured by processing the γ-spectra collected during the decay of the activated antimony samples. At INRIM, the 2 sequences performed with the GEM 50P4-83 lasted 251 h and 236 h, i.e. periods corresponding to 3.9 and 3.6 times the t 1/2 ( 122 Sb), and consisted of 172 and 166 successive counts, respectively. The relative dead time varied from 20 to 4.6% and from 24 to 4.9%, respectively. At NIST the 2 sequences performed with the GEM 40P4-S lasted 274 h and 400 h, i.e. periods corresponding to 4.2 and 6.2 times the t 1/2 ( 122 Sb), and consisted of 148 and 267 successive counts, respectively. The relative dead time varied from 13 to 4.9% and from 19 to 3.4%, respectively. Equation (4) is used to measure , i.e. t 1∕2 = ln (2)∕ . In this case j = 1, 2 and i = 1, 2, …, N c . The C ij t dij and C 1j t d1j values are calculated using Eq. (3) and adopting the measured value. The t cij , t dead ij and t dij values used to calculate the dead time and pile up corrections, ij and f ij , respectively, and the fitted Y ij values via Eq. (4) have negligible uncertainties. Accordingly, uncertainty of ij is negligible and the uncertainty of f ij depends on the uncertainty of . In addition, the uncertainty of the relative variation of efficiency per unit of time, , affects the Y ij values via Eq. (4) and counting statistics affect the 122 Sb γ-photon count rate C ij t dij via Eq. (3).
The Y ij versus t dij − t d1j values obtained in case of zero efficiency drift, i.e. = 0 h −1 , for the GEM 50P4-83 and GEM 40P4-S are plotted in the upper graph of Fig. 3a and b, respectively. The relative standard deviation of Y ij , due to counting statistics, is evaluated by P n ij ∕n ij .
The ( 122 Sb) value is the (shared) slope of the two straight lines fitted to Y j versus t dij − t d1j data. The fitting residuals, plotted in the lower graph of Fig. 3a and b, are in agreement with the constant 0.1% relative standard deviation of Y j Fig. 2 The straight line fitted to the Y j versus t r dead j and the corresponding residuals obtained during the pile-up constant measurement of the GEM 50P4-83 (a) and GEM 40P4-S (b). Error bars indicate a 95% confidence interval due to counting statistics expressed in terms of error bars at 95% confidence interval.
The resulting values of t 1/2 ( 122 Sb) obtained with the GEM 50P4-83 and GEM 40P4-S are 2.69457(22) d and 2.69368(17) d , respectively; the quoted uncertainties are only due to counting statistics.

Uncertainty evaluation and net count correction due to systematic errors in γ-peak fitting
The absence of a measurement equation directly linking input parameters to the ( 122 Sb) value prevents the propagation of the uncertainties of the input quantities through a functional relationship. A propagation formula is suggested in [3] to evaluate the uncertainty. However, the proposed formula cannot be applied in practice because it assumes that count rates are measured with the same relative uncertainty at regular time intervals during the decay. As an alternative, we consider the fitting uncertainty of as the contribution due to counting statistics affecting the Y j values and we evaluate the effect of efficiency drift and pile-up correction by fitting the Y j values obtained by separately substituting in Eq. (4) and values calculated at their standard uncertainty values. Specifically, the uncertainty of due to an efficiency drift is computed as the average of the absolute error of obtained by substituting in Eq. (4) = ± 80 × 10 -8 h −1 and = ± 66 × 10 -8 for the GEM 50P4-83 and GEM 40P4-S, respectively. Similarly, the uncertainty of due to the pile-up correction is computed as the average of the absolute error of obtained by substituting in (4) = (2.51 ± 0.13) × 10 -2 and = (3.42 ± 0.12) × 10 -2 for the GEM 50P4-83 and GEM 40P4-S, respectively.
Systematic errors affecting γ-peak net counts values, n ij , and due to a non-perfect separation of the γ-peak from the underlying background are estimated by observing the peak fitting residuals obtained with the Hyperlab software. To   Fig. 3 The two straight lines fitted to the Y ij versus t dij − t d1j values and the correspondent residuals obtained during the t 1/2 ( 122 Sb) measurement with the GEM 50P4-83 (a) and GEM 40P4-S (b). Error bars indicate a 95% confidence interval due to counting statistics increase the sensitivity we overlapped the residuals of the first 20 and the last 10 γ-peaks recorded at the beginning and the end of a decay observation, respectively; data of the second sequence collected with the GEM 50P4-83 and GEM 40P4-S are considered.
Residuals of the γ-peaks at the end of the sequence, and the first and last γ-peak with the underlying backgrounds are plotted in Fig. 4; the γ-peak net count to background ratio is about 3.3% and 4.0% for GEM 50P4-83 and GEM 40P4-S, respectively. The scattering of the residuals is (largely) within ± 3σ(n), where n is the channel count, and does not reveal significant systematic trends. Therefore, we consider the net count values at the end of the sequence unbiased.
Residuals of the γ-peaks at the beginning of the sequence, and the first and last γ-peak with the underlying backgrounds are plotted in Fig. 5; the γ-peak net count to background ratio is about 1.4% and 0.8% for GEM 50P4-83 and GEM 40P4-S, respectively. The scattering of the residuals is (largely) within ± 3σ (n) and ± 4σ (n) for GEM 50P4-83 and GEM 40P4-S, respectively; in both cases systematic trends affecting the accuracy of the γ-peak net count are revealed, most significantly for the GEM 40P4-S.
To improve the accuracy we repeated the fit by adding one interfering γ-peak at about 561 keV for the GEM 50P4-83, and two interfering γ-peaks at about 560 keV and 568 keV for the GEM 40P4-S. The new residuals are plotted in Fig. 6 together with a representative γ-peak, the interfering γ-peaks and the underlying background. The systematic trends affecting the residuals were removed for the GEM 50P4-83 and significantly decreased for the GEM 40P4-S.
The averaged uncorrected to corrected net count ratio at the beginning of the sequence is 0.99997 and 1.00032 for the GEM 50P4-83 and GEM 40P4-S, respectively; the corresponding relative correction of the measured t 1/2 ( 122 Sb) value is − 1.2 × 10 -5 and 7.4 × 10 -5 , respectively. We conservatively assign a relative uncertainty of the t 1/2 ( 122 Sb) of 0.7 × 10 -5 and 4.3 × 10 -5 for the GEM 50P4-83 and GEM 40P4-S, respectively, corresponding to a uniform probability distribution having a half-width equal to the total γ-peak net count correction.
Uncertainties of t 1/2 ( 122 Sb) due to (1) counting statistics, (2) possible efficiency drifts, (3) pile-up and (4) γ-peak fitting are listed in Table 1 together with the combined standard uncertainty, u c (t 1/2 ), obtained as the positive square root of the sum of variances.

Conclusions
We carried out two independent measurements of the 122 Sb half-life. The uncertainties due to counting statistics, efficiency drift, pile-up and γ-peak fitting were evaluated and propagated to obtain a 10 -4 relative standard uncertainty. Although the resulting 122 Sb half-life values were in agreement, there is a 1.1% relative difference with respect to the recommended value.
The importance of adopting an accurate half-life value is fundamental in analytical chemistry measurements carried out by Neutron Activation Analysis (NAA). As an example, we reprocessed data collected during an experiment carried out to test the application of the k 0 standardisation method [18] using the new instead of the recommended 122 Sb half-life value. Specifically, the quantification of Sb in a soil reference material via the nuclear reaction 121 Sb(n,γ) 122 Sb is affected by a 2.8% relative difference when the sample is counted for 6 d after a decay of 11.7 d. In addition, attempting to calculate isotopic compositions of Sb at 0.1% uncertainty level using NAA, an incorrect 122 Sb half-life would potentially affect the results.
The paper published in 1990 [2] report the presently recommended 122 Sb half-life value together with the half-life of 41 Ar, 80m Br, 94m Nb, 101 Mo, 101 Tc, 109 Pd, 109m Pd, 122m Sb, 123m Sn, 152m Eu and 239 Np. The authors did not describe the experiments nor the elaboration of the collected data. Decay curves were plotted only in the case of 41 Ar, 122m Sb, 152m Eu and 239 Np, and without the fitting residuals, which might reveal possible uncertainty sources. In addition, Fig. 6 Fitting residuals of the first 20 γ-peaks of the second decay sequence collected with the GEM 50P4-83 (a1, a2) and GEM 40P4-S (b1, b2), and obtained with additional interfering γ-peaks. A representative γ-peak with the interfering γ-peaks is also displayed  Table 1 Uncertainties of t 1/2 ( 122 Sb) due to the influence factors The resulting combined standard uncertainty, u c (t 1/2 ), and the relative contribution of each factor, I, are also given Influence factor GEM 50P4-83 GEM 40P4-S Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/.