Experimental set-up

In this study, a large-area windowless multiwire gas-flow proportional counter [15] developed by the China Institute of Atomic Energy (CIAE) was used as the detector. The working principle of this detector is based on the primary ionization of α and β particles in the sensitive volume of the proportional counter, which produces positive ions and electrons. These particles are then accelerated by an electric field to gain sufficient energy to trigger secondary ionization with gas molecules. The number of electron–ion pairs produced by this process is proportional to the number of electron–ion pairs produced by the primary ionization, resulting in a linear amplification of the primary ionization pulse signals from the α and β particles.

For the single-channel counting measurement requirements of this experiment, the electronic equipment included a high-voltage DC power supply, current-type pre-amplifier, pulse-shaping circuit, timing single-channel analyzer, monostable control unit, and timer/counter. A diagram of the instrument is shown in Fig. 2. P-10 gas (a mixture of 90% Ar and 10% CH₄) was used as the operating gas at a flow rate of 100 mL/min. The working high voltage for the α-plane source was 2100 V, whereas that for the β-plane source was 2800 V. Measurements were performed on four plane sources (²⁴¹Am, ²³⁸Pu, ⁹⁰Sr, and ²⁰⁴Tl). According to oscilloscope observations, the pulse width of the pulse signal output from the main amplifier was generally less than 2 μs. The output pulse width of the single-channel analyzer was approximately 650 ns, whereas the pulse width of the monostable control unit had an adjustable range of 0.4–4 μs.

Fig. 2Full-screen View

Actual size of a well-prepared line drawing

Dead time measurement

The surface emission rate of plane sources generally does not exceed 5000 s^-1, resulting in a dead time that produces leakage counts that may have a small effect on the count rate measurements [16-22]. However, because the windowless proportional counter is an absolute measurement device, it is necessary to measure the dead time of the system to correct for leakage counts. The dead time provided by the monostable control unit within the measurement system is a non-extended dead time; however, the system dead time is also affected by various components. Therefore, it is necessary to measure the system dead time experimentally. ICRU Report 52 [23] recommends three methods for measuring the dead time. In this experiment, a two-source two-pulse method was used to measure the dead time of the system. In this method, pulse signals are generated from two different radioactive sources and input into the detector separately to obtain the measurement count rates of the two sources, n₁ and n₂. The two sources are then measured simultaneously using a detector to obtain the mixed measurement count rate n₁₂. Ideally, the real count rate of three sets of pulse count rates ${n^{\prime }}_1$, ${n^{\prime }}_2$, and ${n^{\prime }}_{12}$ should satisfy ${n^{\prime }}_1+{n^{\prime }}_2={n^{\prime }}_{12}$, and according to the formula for the real count rate of a non-extended dead time system, $n^{\text{'}}=n/(1-\tau n)$, the dead time formula for the two-source method is finally obtained: $\tau =\frac{n_1+n_2+n_{12}}{2n_1{n}_2}\mathrm{.}$ (2)

Threshold correction factor measurements

To avoid the influence of noise on count measurement, a discriminating voltage threshold must be set. Because the energy of α particles is high and the discriminating threshold is set low, small changes in the α-particle count rate measurement can be ignored. However, the energy spectrum of β sources is continuously distributed, and it is difficult to distinguish their low-energy signal from noise signals using the pulse amplitude. Consequently, setting the discriminating voltage threshold may result in a loss of the β-particle count rate, and a threshold correction for β sources must be performed [5, 24-26].

Some National Metrology Laboratories [27, 28] that use multichannel analyzers to construct measurement systems typically calibrate the site corresponding to the lower threshold by using 10% of the site of the ⁵⁵Fe full-energy peak. This is primarily designed to correct for count-rate variations owing to fluctuations in the gain of the working gas. Although a measurement system based on a single-channel analyzer cannot set a lower threshold by determining the absolute energy, it is possible to set different lower discriminating voltage thresholds to obtain the count rate at different discriminating voltage thresholds, and then extrapolate to obtain the count rate when the threshold is zero.

The threshold extrapolation curves are influenced by low-energy noise interference and the characteristics of the beta smooth continuum spectrum and can only be assumed to be a straight line over a small threshold interval. As a rule of thumb, the normalized value (predicted value/fitted intercept) of the fitted curve should not exceed 90%. The threshold extrapolation curve fitted using the least-squares method [29-33] is a straight line, where x represents the voltage threshold and y represents the measured count rate. The equation for the fitted curve is as follows: $y=px+q\mathrm{,}$ (3) where p and q represent the slope and intercept of the line, respectively. This equation is solved as follows: $p=\frac{j{\displaystyle \sum }{x}_i{y}_i-\left({\displaystyle \sum }{x}_i\right)\left({\displaystyle \sum }{y}_i\right)}{n{\displaystyle \sum }{x}_i^2-{\left({\displaystyle \sum }{x}_i\right)}^2}\mathrm{,}$ (4) $q=\frac{\left({\displaystyle \sum }{x}_i^2\right)\left({\displaystyle \sum }{y}_i\right)-\left({\displaystyle \sum }{x}_i{y}_i\right)\left({\displaystyle \sum }{y}_i\right)}{j{\displaystyle \sum }{x}_i^2-{\left({\displaystyle \sum }{x}_i\right)}^2}\mathrm{,}$ (5) where i=1,2, ..., j. To fit a straight line, a minimum of j=5 measurements must be substituted into the equation to complete the fit with degrees of freedom ν=j-2.

Evaluation by the GUM method

The evaluation process of the GUM method is divided into two types of evaluations: A and B. This study focuses on two uncertainty components: system dead time and voltage threshold correction factor, and the process is a Type B evaluation. The uncertainty obtained from the Type B evaluation of uncertainty can be obtained using Eq. (6). $u_{\text{B}}=a/k=\max (\text{Δ}a)/k\mathrm{,}$ (6) where a represents the half-width of the interval of the possible measurement values, or the maximum fluctuation $\max (\text{Δ}a)$ if the coverage interval distribution is asymmetric [6, 34], and k is the coverage factor. Therefore, obtaining the coverage interval and probability distribution of the components is crucial.

a. Evaluation of system dead-time uncertainty

In the GUM method, the monostable control unit produces a square wave as the output signal. The pulse width of this signal is typically measured using an oscilloscope, which requires debugging the oscilloscope’s resolving power and sampling time interval. After debugging, the system dead-time interval half-width a is determined based on the time interval of the index value, as shown on the oscilloscope. Because the probability distribution function of the system’s dead time is unknown, it is generally assumed that the error follows a rectangular distribution, in which case $k=\sqrt{3}$.

b. Evaluation of voltage threshold correction factors uncertainty

In this study, the threshold correction curves were fitted using the least-squares method. The expected corrections obtained from the calibration curves and their standard uncertainties were calculated using their intercepts, slopes, estimated variances, and covariances. Multiple sets of thresholds and the corresponding measurements were selected and substituted into Eqs. 4 and 5 to obtain the p and q values, which were substituted into the following equation: $s^2=\frac{{\displaystyle \sum }{\left(y_i-p{x}_i-q\right)}^2}{j-2}$ (7) $s^2\left(p\right)=\frac{j{s}^2}{j{\displaystyle \sum }{x}_i^2-{\left({\displaystyle \sum }{x}_i\right)}^2}$ (8) $s^2\left(q\right)=\frac{s^2{\displaystyle \sum }{x}^2}{j{\displaystyle \sum }{x}_i^2-{\left({\displaystyle \sum }{x}_i\right)}^2}$ (9) $r(p\mathrm{,}q)=\frac{-{\displaystyle \sum }{x}_i}{\sqrt{j{\displaystyle \sum }{x}_i^2}}$ (10) where $s(p)$ is the standard deviation of the slope, $s(q)$ is the standard deviation of the intercept, s² is the estimated variance of the fitted curve, and $r(p\mathrm{,}q)$ is the correlation coefficient between the p and q estimates. This provided the combined standard uncertainty [10] of the measured count rate (y) corresponding to the specified prediction threshold (x) on the fitted curve. $u_{\text{c}}^2\left(y\right)=u^2\left(q\right)+x\times u^2\left(p\right)+2x\times u\left(p\right)\times u\left(q\right)\times r\left(p\mathrm{,}q\right)$ (11) The coverage interval [35, 36] for each measured count rate on the fitted curve was obtained as $\left[p{x}_i+q+u_{\text{c}}\left(y_i\right)\mathrm{,}p{x}_i+q+u_{\text{c}}\left(y_i\right)\right]$. In addition, because yi is an experimental measurement, and the result of $\left(p{x}_i+q\right)$ is the predicted value of the fitted curve, the voltage threshold correction factor f_d is expressed as follows: $f_d=\frac{q}{px+q}.$ (12) This yields an asymmetric coverage interval for f_d: $\left[\frac{q}{px+q+u_{\text{c}}\left(y\right)}\mathrm{,}\frac{q}{px+q-u_{\text{c}}\left(y\right)}\right]$. According to the GUM, because the probability distribution function of f_d is unknown, its coverage interval distribution is assumed to be rectangular by default. According to Eq. (7), the uncertainty components of the voltage threshold correction factors are obtained as follows: $u\left(f_{\text{d}}\right)=\frac{\max \left(\frac{q}{px+q-u_{\text{c}}\left(y\right)}-\frac{q}{px+q}\mathrm{,}\frac{q}{px+q}-\frac{q}{px+q+u_{\text{c}}\left(y\right)}\right)}{\sqrt{3}}.$ (13) c. Evaluation of the combined standard uncertainty

The previous two subsections provide the system dead-time and voltage threshold correction factors and their uncertainty components, which can be used to obtain the relative uncertainty of each component. According to the GUM method, the relative combined standard uncertainty u(E) of the surface emission rate can be derived from Eq. 1 using the law of propagation of uncertainty proposed by Stanga [37]. $u(E)=\frac{{\left(E+B\right)}^2}{RE}\sqrt{u^2\left(R\right)+\frac{R^2}{{\left(E+B\right)}^2}{u}^2{f}_{\text{d}}+{\tau }^2{R}^2{u}^2\left(\tau \right)+u^2\left(B\right)\frac{R^2{B}^2}{{\left(E+B\right)}^4}}$ (14) Here, R and B belong to the Type A evaluation of uncertainty, both of which were obtained from experimental measurements. u(R), u(B), u(fd), and u(τ)) are the relative standard deviations of R, B, fd, and τ, respectively.

Evaluation by the MCM

The MCM provides a general numerical approach for evaluating the measurement uncertainty for models with any number of input quantities that can be characterized by a PDF and a single output quantity. In this study, MATLAB was used as the Monte Carlo simulation platform to model the measurement between the output quantities Y and input quantities X₁, …, X_N. According to Equations (1), (2), and (12), a new measurement function for the surface emission rate E can be obtained as follows: $E=\frac{R}{1-\frac{n_1+n_2-n_{12}}{2n_1{n}_2}R}\frac{q}{px+q}-B.$ (15) The probability distributions, best estimates, standard uncertainties, and coverage intervals of the propagated input quantities Xi for each measurement model were analyzed. The probability distributions for each input quantity are listed in Table 1.

Finally, the estimated and standard uncertainty of Y were calculated from the PDF of Y and the coverage interval of Y at a 95% coverage probability (k=2).

GUM method

a. System dead time uncertainty The system dead-time measurements were performed using two α-plane sources with a count rate of 5000 s^-1. Ten measurements were taken per set of experiments, and the statistical standard deviations obtained as the measurement uncertainties are listed in Table 2.

The measured values in Table 2 were substituted into Eq. (2) to obtain the system dead-time values. According to the method described in Sect. 2.4.1, the half-widths of the interval of the system dead time, a= 0.2 μs, and k = $\sqrt{3}$, are used to obtain the uncertainty of the measurement of the system dead time, as presented in Table 3.

b. Voltage threshold correction factor uncertainty The threshold correction curves of the ⁹⁰Sr and ²⁰⁴Tl plane sources were measured by increasing the threshold voltage from 0.040 V. The measured count rate at each set of voltages was recorded, and the obtained data are shown in Figs. 3 and 4.

Fig. 3Full-screen View

The coordinate plots of β-source (⁹⁰Sr) show eight coordinate points, and five coordinate points from 0.10 to 0.18 V were used for curve fitting, with the fitted coordinate points and fitted straight line marked in red

Fig. 4Full-screen View

The coordinate plots of β-source (²⁰⁴Tl) show eight coordinate points, and five coordinate points from 0.10 to 0.18 V were used for curve fitting, with the fitted coordinate points and fitted straight line marked in red

According to the threshold correction curve plots of the ⁹⁰Sr and ²⁰⁴Tl plane sources, the following was found: at 0.04 to 0.10 V, the curve non-linearity fluctuated greatly, mainly because the count rate contained a large number of noise signals; at 0.10 to 0.18 V, the correlation coefficient of the fitted curves had the largest value of R², the fitting effect was the best, and the normalized values of the count rate (predicted value/fitted intercept) were all greater than 90%. The normalized curve obtained is shown in Fig. 5. The above five sets of measurements were then entered into Eqs. (4) and (5), and the slopes p and intercepts q of the threshold correction curves for the ⁹⁰Sr and ²⁰⁴Tl plane sources, as well as the standard deviation s(p) of the slopes, the standard deviation s(q) of the intercepts, the estimated variances s², and the correlation coefficients r(p,q), were obtained and recorded in Table 4.

Fig. 5Full-screen View

Threshold extrapolation straight line for two β sources where the counting rates have completed normalization, including error bars for each predicted value on the fitted line

By substituting the calculation results in Table 4 into Equation (11), the uncertainty was evaluated for five sets of data in the range of 0.10 to 0.18 V, and the results are listed in Table 5.

Both relative standard uncertainties of the two β sources are minimized at the threshold x = 0.10 V. By substituting the predicted values at this threshold into Eq. (12), the voltage threshold correction factors and coverage intervals for the two sources were obtained, as shown in Table 6.

The voltage threshold correction factors and uncertainties for the two β sources were then obtained from Eq. (13), as listed in Table 7.

c. Combined standard uncertainty

The measured data were collated when the α source did not require threshold correction by setting its f_d value to 1 and its u(f_d) value to 0. The results are summarized in Table 8.

By substituting various data from Table 8 into Eq. (14), the standard and relative standard uncertainties of the four plane sources were obtained, as listed in Table 9.

Monte Carlo method (MCM)

According to the information in Table 1, the input quantity Xi for the Monte Carlo simulation was defined as follows: The input quantities (n₁, n₂, n₁₂) and their normal distributions for the system dead time were obtained from Table 2. The standard deviations of p and q were obtained from Equations (8) and (9) and were assumed to be normally distributed by default. The threshold voltage reading x = 0.10 V had a rectangular distribution with a scale component value of 0.02 V, and its coverage interval was set to [0.08, 0.12]. The measurements of count rate R and background count rate B in Table 6 were used to obtain the coverage interval of the normal distribution of these two quantities using their standard deviations. In the Monte Carlo simulation, the random sample size for each input quantity was set to M = 10⁷. The uncertainties for the system dead time, β-source voltage threshold correction factors, and combined standard uncertainty were obtained, and the results are presented in Tables 10, 11, and 12.

A normalized histogram of the distribution function of the system dead time obtained by the Monte Carlo simulation is shown in Fig. 6. A normalized histogram of the distribution function of the voltage threshold correction factors for the two β sources is shown in Fig. 7. A normalized histogram of the distribution function of the surface emission rate for the four sources is shown in Fig. 8. Each histogram contains 200 bars.

Fig. 6Full-screen View

Normalized histogram of the probability distribution function (PDF) of the system dead time by the MCM

Fig. 7Full-screen View

Normalized histograms of the PDF of the voltage threshold correction factors for the two β sources (⁹⁰Sr and ²⁰⁴Tl) by the MCM

Fig. 8Full-screen View

Normalized histogram of the PDF of the surface emission rate for the four sources (²⁴¹Am, ²³⁹Pu, ⁹⁰Sr, ²⁰⁴Tl) by the MCM

Finally, the value of E_n was used to determine whether the results of the two methods were consistent within the uncertainties. The E_n value is a statistical index used to evaluate the measurement capability of a laboratory. It is used to compare the difference between the laboratory measurement results and the reference value, as well as the measurement uncertainty. The closer the value of E_n is to zero, the closer the laboratory measurement results are to the reference values. If the absolute value of E_n is less than 1, the laboratory measurement results are considered consistent with the reference value. If the absolute value of E_n is greater than or equal to 1, it is considered that the laboratory measurement results are inconsistent with the reference value. The formula for calculating the value of E_n is shown in Eq. (16), where U_G and U_M are the expanded uncertainties (k=2) of the two methods. The results are summarized in Table 13. $E_{\text{n}}=\frac{|E_{\text{G}}-E_{\text{M}}|}{\sqrt{U_{\text{G}}^2+U_{\text{M}}^2}}$ (16)