A new low-uncertainty measurement of the ³¹Si half-life

The stability and pile-up factor of the detection system was checked and measured, respectively, by processing the γ-spectra recorded during the sequences of counts of the ¹³⁷Cs source.

The count rate of the ¹³⁷Cs 661.8 keV γ-photons measured at the ith count of the jth sequence, C_ij(t_dij), and corrected for decay at the starting time of the first count of the first sequence, t_d11, is

$$\begin{eqnarray}&&{{C}_{ij}}\left({{t}_{\text{d11}}}\right)=\frac{{{C}_{ij}}\left({{t}_{\text{d}ij}}\right)}{{{\text{e}}^{-\lambda ~ \left({{t}_{\text{d}ij}}-{{t}_{\text{d}11}}\right)}}},\end{eqnarray} \tag{ 4 }$$

where C_ij(t_dij) is obtained from (1) using the ¹³⁷Cs decay constant, λ(t_1/2  =  30.08y), and the net count of the 661.8 keV γ-peak, n_ij. Since the γ-photons counting was performed with a constant detection efficiency, C_ij(t_d11) is constant as well.

In addition, the following model applies:

$$\begin{eqnarray}&&{{C}_{\text{pu}ij}}\left({{t}_{\text{d11}}}\right)={{\hat{C}}_{ij}}\left({{t}_{\text{d11}}}\right) ~ {{\text{e}}^{-\mu \left({{t}_{\text{dead}ij}}/{{t}_{\text{c}ij}}\right)}}+{{\varepsilon}_{ij}},\end{eqnarray} \tag{ 5 }$$

where ${{\hat{C}}_{ij}}\left({{t}_{\text{d} ~\text{11}}}\right)$ is the expected value of C_ij(t_d11) and ${{C}_{\text{pu}ij}}\left({{t}_{\text{d11}}}\right)=\frac{\lambda {{n}_{\,ij}}}{\left(\text{1}-{{\text{e}}^{-\lambda \,{{t}_{\text{c}ij}}}}\right)\,{{\text{e}}^{-\lambda ~ \left({{t}_{\text{d}ij}}-{{t}_{\text{d}11}}\right)}}}{{\delta}_{ij}}$ is the γ-photons count rate measured at the ith count of the jth sequence uncorrected for pile-up and corrected for decay at t_d11.

The single sequence of 750 counts collected for the stability test was performed at a fixed relative dead time/input pulse rate, 0.077 (i.e. 7.7% of lost counts)/2000 s⁻¹. Therefore, j  =  1 (single sequence) and i  =  1, 2, ..., 750 (counts); the δ_i1 and f_i1 values are constant. In the case of a stable detection system, the ratio of the C_i1 (t_d11) value, obtained from equation (4), to the average of the C_i1 (t_d11) values, C_ave1 (t_d11), must vary due to counting statistics around the expected unity value without a deterministic drift. This was the case of the ratios obtained in this study and plotted in figure 3.

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The average relative uncertainty due to counting statistics of the 661.8 keV γ-peak net count and evaluated by the WAN32 analysis algorithm is 1.048  ×  10^–3 whereas the experimental standard deviation of the ratios is 1.064  ×  10^–3. The experimental standard deviation is 1.5% higher than the expected standard deviation according to Poisson statistics. This (small) difference might be due to medium frequency instability of the detection system.

The fourteen sequences of 34 counts collected for the measurement of the pile-up factor were performed with the detection system working at seven different relative dead times/input pulse rates, i.e. 0.077/2000 s⁻¹ (1st and 8th sequence), 0.116/3000 s⁻¹ (2nd and 9th sequence), 0.153/4000 s⁻¹ (3rd and 10th sequence), 0.189/5000 s⁻¹ (4th and 11th sequence), 0.220/6000 s⁻¹ (5th and 12th sequence), 0.253/7000 s⁻¹ (6th and 13th sequence), and 0.288/8000 s⁻¹ (7th and 14th sequence). Therefore, j  =  1, 2, ..., 14 (sequences) and i  =  1, 2, ..., 34 (counts); the δ_ij and f_ij values are changing according to equation (2).

Linear least squares regression is applied to estimate the pile-up factor, µ, by fitting a straight-line to the natural logarithm of equation (5) averaged with respect to the 34 counts of the jth sequence, ${{C}_{\text{pu}\,\text{ave}j}}\left({{t}_{\text{d11}}}\right)$, and normalized to the ${{C}_{\text{pu}11}}\left({{t}_{\text{d11}}}\right)$ value. Expressly, the measured variable, y_ij, is $\ln \left[{{C}_{\text{pu} ~\text{ave}j}}\left({{t}_{\text{d11}}}\right)\text{/}{{C}_{\text{pu}11}}\left({{t}_{\text{d11}}}\right)\right]$, the independent variable x_ij is t_{dead avej}/t_{c avej} and the estimated parameters are the intercept $\ln \left[{{\hat{C}}_{ij}}\left({{t}_{\text{d11}}}\right)\text{/}{{C}_{p\text{u}11}}\left({{t}_{\text{d11}}}\right)\right]$ and the slope  −µ. Since the main contribution to the uncertainty of ${{C}_{\text{pu}ij}}\left({{t}_{\text{d11}}}\right)$ is the fixed 0.1% relative uncertainty of the 661.8 keV γ-peak net count, the standard deviation of the error term of y_ij, ${{\varepsilon}_{ij}}/{{C}_{\text{pu} ~\text{ave}j}}\left({{t}_{\text{d11}}}\right)$, is constant and close to $0.001/\sqrt{34}$.

The straight line fitting the data and the residuals are shown in figures 4(a) and (b), respectively. The error bars confirm that the logarithm of uncertainties due to counting statistics are constant. Thus, the straight-line fitting is recommended instead of the exponential function fitting required in the case of non-adapted uncertainties.

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Since the residuals of three out of fourteen data are somewhat larger than the expanded uncertainty due to counting statistics, there must be an additional influence factor affecting the estimated 4.45(8)  ×  10^–3 pile-up factor, µ. Here and hereafter the uncertainties (k  =  1) in parentheses apply to the last respective digits. Furthermore, the µ value obtained using the ¹⁵²Eu as interfering source might be biased due to a possible energy dependence of the correction [10]. In fact, the ¹⁵²Eu γ-spectrum includes 23 γ-peaks with energies between 120 keV and 1500 keV; the resulting effect is different with respect to the pile-up occurring in the case of single 1266.1 keV ³¹Si γ-peak counting.

It may be observed that the residuals confirm the application of the pile-up correction defined in (2) at least up to relative dead times/input count rates close to 0.300/8300 s⁻¹. In addition, the fraction of the 38 µs dead time per pulse equivalent to µ is 0.17 µs and represents the ultimate limit of the pile-up rejection system.

The decay constant of ³¹Si was determined by processing the γ-spectra recorded during the twelve sequences of counts of the neutron activated ³¹Si source. The duration of a single sequence and the number of performed counts, N, depended on the ³¹Si activity at the start of the observation. Specifically, the N values were 26 (1st sequence), 29 (2nd sequence), 25 (3rd sequence), 28 (4th sequence), 26 (5th sequence), 32 (6th sequence), 25 (7th sequence), 14 (8th sequence), 27 (9th sequence), 31 (10th sequence), 25 (11th sequence) and 22 (12th sequence).

The relative dead time/input pulse rate varied, on average, from 0.30/8300 s⁻¹ (t_c  =  460 s) to 0.06/1400 s⁻¹ (t_c  =  33 min) whereas the maximum and the minimum values, 0.31/8600 s⁻¹, and 0.03/550 s⁻¹, were reached at the first and last count of the 6th sequence which lasted 9.2 h. Furthermore, the ratio of the background count, b, to the net count, n, of the ³¹Si 1266.1 keV γ-photons, α, was in the range between 0.010 and 0.035. Since the collected net count values were close to 10⁴, the background counts beneath γ-peak ranged from 100 to 350 (170 in the examples of the 1266.1 keV γ-peaks in figure 2), corresponding to γ-peak background rates from 9  ×  10^–4 s⁻¹ to 0.8 s⁻¹, respectively.

The count rate of the 1266.1 keV γ-photons at the beginning of the ith count of the jth sequence, Cij(t_dij), is computed according to (1), where j  =  1, 2, ..., 12 and i  =  1, 2, ..., N. The δ_ij and f_ij values are obtained from (2).

Linear least squares regression is applied to estimate λ, and the corresponding t_1/2, by fitting twelve straight lines to the twelve series of data obtained by taking the natural logarithm of (3) normalized to the ${{C}_{1j}}\left({{t}_{\text{d}1j}}\right)$ value. Specifically, the measured variable, y_ij, is $\ln \left[{{C}_{ij}}\left({{t}_{\text{d}ij}}\right)\text{/}{{C}_{1j}}\left({{t}_{\text{d1}j}}\right)\right]$, the independent variable x_ij is t_dij and the estimated parameters are the twelve intercepts, $\ln \left[{{\hat{C}}_{j}}\left({{t}_{\text{d}1j}}\right)\text{/}{{C}_{1j}}\left({{t}_{\text{d1}j}}\right)\right]$, and the shared slope, −λ. Since y_ij depends on λ, the solution is obtained iteratively until convergence.

The main contribution to the uncertainty of C_ij(t_dij) is the 1% relative uncertainty of the 1266.1 keV γ-peak net count. Therefore, the standard deviation of the error term of y_ij, ε_ij/C_ij(t_dij), is constant and equal to 0.01.

The regression analysis was performed with the Levenberg Marquardt algorithm of the OriginLab^® software using the line model in global multi-data fit mode [11]. The twelve straight lines fitting the data and the residuals plotted with respect to t_dij  −  t_d1j are shown in figures 5(a) and (b), respectively. The error bars confirm that the logarithm of uncertainties due to counting statistics are constant.

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The resulting ³¹Si decay constant, λ, is 4.4103  ×  10^–3 min⁻¹ with an associated (fitting) uncertainty of 0.0046  ×  10^–3 min⁻¹. It should be noted that the multi-data fitting of twelve repeated observations of ³¹Si lasting about 3 half-lives allowed to average the effect of counting statistics and to reach a 0.1% (fitting) uncertainty.

In the case of an ideal linear relationship between the activity of the source and the counting rate of the detection system, the residuals are expected to be pure stochastic and due to the Poisson nature of radioactive decay. Thus, the only contribution to the uncertainty of λ would be the fitting. However, medium- (cyclic) and low-frequency (drift) deviations must carefully be considered because they can significantly affect the uncertainty also when they are hidden in the residuals. The most important sources of these deviations are (i) instability of the detection efficiency, (ii) pile-up and dead time, (iii) geometrical changes in source positioning, (iv) spectral interference and (v) background signals [2].

The effect of counting statistics can be anticipated using equations (34) and (37) reported in [2] and based on the number, N, of activity measurements, A, and associated relative uncertainties, u(A)/A, repeated at regular time during an observation lasting T. Although in this study the activity (count rate) measurements, were not equally spaced in time, the resulting relative uncertainty, u(λ)/λ, with T  =  400 min, N  =  26 and u(A)/A  =  0.01 is about 0.003 for a single sequence. The corresponding value for the mean of twelve of these sequences, 0.0009, is close to the evaluated (fitting) relative uncertainty, 0.0010.

The results of the stability test didn't show significant additional variability from any cyclic variations or drift of the detection efficiency system other than the expected statistical 0.001 relative standard deviation. Thus, 30% of this standard deviation is conservatively assigned as potential maximum drift of the system during a single sequence and the corresponding 0.0003 is given as a reasonable upper limit for the contribution to u(A)/A.

The pile-up correction, f, obtained from (2) with a pile-up factor, µ, of 4.45  ×  10^–3 and a relative dead time, t_dead/t_c, of 0.30 (at the start count of a sequence) and 0.06 (at the end count of a sequence) is 1.0013 and 1.0003, respectively. Due to the energy dependence of µ, a conservative 30% error is assigned to the (corrected) 0.001 relative variation of f between the start and end counts of a sequence and leads to an average relative uncertainty, $\langle u(A)/A\rangle$, i.e. at t  =  0 and t  =  400 min, of 0.000 15 [2]. Since the counting time, t_c, and the dead time, t_dead, are known with a negligible uncertainty, the contribution to u(A)/A of the dead time correction, δ, obtained from (2) is negligible as well.

During a single observation, the position of the ³¹Si source was fixed with respect to the detector, therefore the effect of geometrical changes in source positioning is absent.

Given the high-purity of the silicon sample used as a source, the 1266.1 keV γ-peak was free from spectral interference. The Compton continuum underlying the γ-peak was approximately flat and due to natural background (see figure 2). To achieve the 0.01 relative uncertainty due to counting statistics, the net count of the γ-peak, n, reached a constant value (close to 10⁴) in every count. Hence, the effect of a possible and undisclosed departure of the γ-peak from the Gaussian shape is considered constant during the sequence. In addition, the upper limit for the background contamination fraction of the fitted γ-peak area is fixed at 10% of the maximum background to the net count ratio, α, occurred during the experiments, i.e. 0.0035. The 30% variation of α usually observed between the start and end of a sequence leads to an average relative uncertainty, $\langle u(A)/A\rangle$, i.e. at t  =  0 and t  =  400 min, of about 0.000 53.

In agreement with [2], a propagation factor 2/λT is assumed for the u(A)/A due to (i) variations or drift of the detection efficiency system, (ii) pile-up correction and (iii) background contamination. The corresponding relative uncertainties, u(λ)/λ, are 0.000 34, 0.000 18 and 0.000 60, respectively; these values and the corresponding contributions to the combined uncertainty are listed in table 1.

In summary, an overall relative uncertainty of 0.0013 is evaluated for λ. The main contributors are counting statistics, 68%, and background contamination, 22%.

The presently adopted ³¹Si half-life value, 157.36(26) min [5], is the weighted average of ten selected values reported in high quality peer-reviewed journals: 163.0(58) min [12], 157.5(5) min [6], 157.2(6) min [13], 158.4(12) min [14], 159(1) min [15], 157.3(5) min [16], 157.1(7) min [17], 155.5(10) min [18], 157.3(13) min [19] and 170(10) min [20]. These values are shown in figure 6 and compared with the result obtained in this study, i.e. 157.16(20) min.

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