2.1 Detector Response Function

Figure 1 shows the process of alpha particle emission from a nuclide to the signal generated in a multi-channel analyzer (MCA). In general, this process can be considered a signaling system. For a thin source, self-absorption can be ignored, so the emissivity of alpha particles with energy E (in keV or MeV) generated by nuclide decay can be represented by an original energy delta function δ(E) [13] [see Eq. (1)]. Alpha particles then lose energy due to the absorption of air, entrance window, and dead layer of the detector; in addition, incomplete charge collection results from the voltage bias and doping level of P and N regions, which is commonly assumed to be an exponential function [5, 13] [see Eq. (2)]. In the detector, electron-hole–pair statistical fluctuation occurs for ionization, leading to the pulse height, which is in direct proportion to the electron-hole pairs, also exhibiting statistical fluctuation. Moreover, the electron noise of the preamplifier increases the fluctuation. These fluctuations can commonly be considered a Gaussian distribution [14] [see Eq. (3)].

Fig. 1.Full-screen View

Process of alpha spectrum formation.

We thus write

where E_k is the mean energy of alpha particles (in keV or MeV), σ is the standard deviation of the Gaussian, and τ is the parameter of the exponential function expressed as the rate of the exponential component.

As a signaling system, alpha particles δ(E) go through two system processes of f_e(E) and f_g(E). Thus, the final nuclear signal from the MCA is a convolution of these three functions:

$f\left(E\right)=\delta \left(E-E_k\right)\text{*}{f}_e\left(E\right)\text{*}{f}_g\left(E\right) = f_e\left(E\right)*f_g\left(E-E_k\right) = \underset{-\infty }{\overset{+\infty }{{\displaystyle \int }}}{f}_e\left(E-x\right)f_g\left(x-E_k\right)dx = \underset{-\infty }{\overset{+\infty }{{\displaystyle \int }}}\frac{1}{\tau }\text{exp}\left(\frac{E-x}{\tau }\right)\times \frac{1}{\sigma \sqrt{2\pi }}\text{exp}\left(\frac{-{\left(x-E_k\right)}^2}{2{\sigma }^2}\right)\text{dx}\text{.}$

The integrals may be solved by introducing the complementary error function erfc, which is not representable with elementary functions:

$\text{erfc}\left(x\right)=\frac{2}{\sqrt{\pi }}\underset{x}{\overset{\infty }{{\displaystyle \int }}}{e}^{-t^2}dt,$

where

$x = \frac{E-E_k}{\sqrt{2}\sigma }+\frac{\sigma }{\sqrt{2}\tau }.$

The solution can be written as

$f\left(E\right) = \frac{\text{1}}{\text{2τ}}\text{exp}\left(\frac{E-E_k}{\tau }+\frac{{\sigma }^2}{2{\tau }^2}\right)\text{erfc}\left[\frac{1}{\sqrt{2}}\left(\frac{E-E_k}{\sigma }+\frac{\sigma }{\tau }\right)\right].$

f(E) is the so-called detector response function (DRF), which actually is a probability density function whose integral in the full energy region is 1, so for a certain intensity A of alpha particles, the detector response function can be described as

where the DRF is determined by four parameters: mean energy E_k, Gaussian standard deviation σ, exponential parameter τ, and the area A of the peak. Figure 2 shows the DRF lines with A=1 and different E_k, σ, and τ values. Note that the mean energy E_k mainly determines the position of the peak. The tailing of the low energy and width of the peak are determined by τ and σ, respectively. The greater the τ value, the greater the tailing, and the larger the σ value, the wider the peak.

Fig. 2.Full-screen View

(Color online) Variation of detector response function curves with parameters shown in legend.

However, n characteristic alpha rays are emitted from a nuclide with a branching ratio. Therefore, the branching ratio can be used in the DRF as a fixed parameter [9]. On the other hand, a single exponential function could not adequately deal with a complex alpha spectrum [6]. Thus, Eq. (4) was expanded as the following expression with two exponentials:

$f\left(E\right) = A{\displaystyle \sum }_{k = 1}^n{I}_k\left\{\frac{\eta }{2{\tau }_1}\cdot \text{exp}\left(\frac{E-E_k}{{\tau }_1}+\frac{{\sigma }^2}{2{\tau }_1{}^2}\right)\cdot \text{erfc}\left[\frac{1}{\sqrt{2}}\left(\frac{E-E_k}{\sigma }+\frac{\sigma }{{\tau }_1}\right)\right]+\frac{1-\eta }{2{\tau }_1}\cdot \text{exp}\left(\frac{E-E_k}{{\tau }_2}+\frac{{\sigma }^2}{2{\tau }_2{}^2}\right)\cdot \text{erfc}\left[\frac{1}{\sqrt{2}}\left(\frac{E-E_k}{\sigma }+\frac{\sigma }{{\tau }_2}\right)\right]\right\},$

where I_k is the branching ratio of the kth alpha energy emitted from the nuclide, which can be quoted from the appropriate nuclear data sheet; τ₁ and τ₂ are the parameters of the exponentials; η is the proportion of the exponential; and all curves of the same nuclide are assumed to have the same parameters of σ, τ₁, τ₂, and η [9].

2.2 Fitting Method

The substance of spectrum unfolding is obtaining the DRF parameters, and the normal method of obtaining them is nonlinear least-squares curve fitting [8, 9]. In order to remove the heteroscedasticity of the spectral data caused by the nonconformity of each channel’s count rate, the WLS method can be applied to fit the alpha spectrum data.

The normal least-squares method used is the unweighted least-squares method (UWLS). In the radiation spectra, different values of the ith channel have different uncertainties, which makes the least-squares curve closer to the more certain points than to the less certain points. However, the variance of each channel is approximate according to the counts of each channel, and the weight of each channel is the reciprocal of the variance [15]. Thus, the WLS method used in the present work can be derived as follows, which is a matter of making the sum of weighted residual squares the minimum [16]:

$Q = {\displaystyle \sum }_{j = 1}^n{\omega }_j{\left(N_j-f\left(x\right)\right)}^2 = {\displaystyle \sum }_{j = 1}^n\frac{1}{N_j}{\left(N_j-f\left(x\right)\right)}^2 = {\displaystyle \sum }_{j = 1}^n{\left(\frac{N_j}{\sqrt{N_j}}-\frac{f\left(x\right)}{\sqrt{N_j}}\right)}^2\to \text{min},$

where ω_j is the weight, ${\omega }_j = 1/N_j$, j=1,2,…,n, N_j is the count of the jth channel, and f(x) is the fitting function.

3.1 Experimental Spectrum Deconvolution

The alpha spectrometer is a passivated implanted planar silicon (PIPS) detector with a 600-mm² surface area (Fig. 3). The PIPS detector has thin dead layer, and therefore has high-energy resolution. The ²³⁹Pu is a thin surface source and its self-absorption can be ignored; its decay data are presented in Table 1. We considered two experimental programs to test the alpha spectrum DRF and study the influence of vacuum level and source-detector distance on parameters. First, 10 vacuum levels in the range 2000–20000 mTorr in 2000-mTorr increments (1 Torr=133.32 Pa) were used, with a 22-mm source-detector distance; second, 10 source-detector distances in the range 4–42 mm in 6-mm increments were used, at 3000 mTorr of vacuum. The testing time was 5 min, and each test was conducted five times under each condition. The voltage bias of the detector was set at 50 V with a full depletion layer; the background was less than one count per hour.

Fig. 3.Full-screen View

(Color online) Experimental alpha spectrometer with PIPS detector.

The above-mentioned WLS method was then applied to fit the experimental alpha spectrum with the DRF. In the present work, reduced chi-square (χ²) [17-19] and the correlation coefficient (R²) were used to evaluate the goodness of fit. The closer to 1 that χ² and R² are, the better the results:

${\chi }^2=\frac{1}{\nu }{\displaystyle \sum _{i=l}^r\frac{{(f_i-y_i)}^2}{y_i}}$

and

$R^2 = 1-{\displaystyle \sum }_{i = l}^r{\left(f_i-y_i\right)}^2/{\displaystyle \sum }_{i = l}^r{\left(y_i-\overline{y}\right)}^2,$

where ν is the number of degrees of freedom, ν=r–l-m, l the left-hand channel of the region of interest (ROI), r the right-hand channel of the ROI, m the number of variables in the DRF, f_i the fitting data, y_i the experimental data, and $\overline{y}$ the mean experimental data. The goodness of fit results for each spectrum are given in Table 2, showing that the χ² values were 0.4807–2.4517 and the R² values were above 0.99. For comparison, the results obtained using the UWLS method are also given in Table 2. Apparently, the results obtained by the WLS method are better than those obtained by the UWLS method. Figure 4 shows an example of one of the fitting curves compared with experimental data. The figure indicates that the results are excellent, but the residuals at the peak are slightly larger than other points.

Fig. 4.Full-screen View

(Color online) ²³⁹Pu fitting curves compared with the experimental spectrum, residual=(f_i–y_i)/$\sqrt{y_i}$.

3.2 Influence of Vacuum and Distance on Parameters

Sanchez et.al [20] studied the variations of σ and τ in terms of alpha-particle curve shape with energy and concluded that "both parameters σ and τ vary as E^m with m being a number depending on the source-detector distance." In the present work, the influence of vacuum and source-detector distance on these parameters was studied.

In the prior experiment, the full width at half maximum (FWHM) was calculated and its variations with vacuum and source-detector distance are described in Fig. 5. For vacuum varying between 2000 and 20000 mTorr, the FWHM remains approximately constant. The vacuum results for the parameters studied obtained by fitting are given in Table 3, for a source-detector distance of 22 mm. The analysis of the data shows that σ, τ₁, and τ₂ exhibit no obvious trend of variation with vacuum. The mean value of σ is 8.422±0.7894, that of τ₁ is 7.248±0.8713, and that of τ₂ is 41.653±1.9368. These results are in agreement with the FWHM being constant. Thus, one could conclude that vacuums in the range 2000–20000 mTorr have no impact on the alpha peak shape with a fixed source-detector distance. In other words, the absorption of air could be regarded as constant in this vacuum region.

Fig. 5.Full-screen View

FWHM variation (a) with vacuum and (b) with distance.

The source-detector distance parameter-fitting results at a vacuum level of 3000 mTorr are given in Table 4, and regression curve fitting is applied to σ, τ₁, and τ₂ versus source-detector distance in Fig. 6. Statistical analysis indicates that to a confidence level of 95% the τ₁ and τ₂ declined in a similar fashion with the power exponential function and σ declined linearly. Table 5 lists the fitting functions of σ, τ₁, and τ₂. It is precisely that the parameters σ, τ₁, and τ₂ vary with source-detector distance that results in the FWHM decreasing with increasing distance, as shown in Fig.5 (b); however, all three parameters do not seem to vary when the source-detector distance is larger than 22 mm.

Fig. 6.Full-screen View

Variation of fitting parameters (a) τ₁ and τ₂ and (b) σwith source-detector distance.