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We used a square shaped thin grade-1 copper metal sheet with a dimension of 1.0×1.0 cm2 and thickness of 0.0125 cm wrapped in an aluminum foil for irradiation purposes. This aluminum foil was used as a relative sample to normalize the neutron flux with the IRDFF-1.05 library's known cross section value for the
$ ^{27} {\rm{Al}}$ (n,α)$ ^{24} {\rm{Na}}$ reaction. A pure$ {\rm{K}}_{2} {\rm{SO}}_{4} $ powder was taken as a sample for potassium irradiation; this circular potassium sample has a diameter of 1 cm and thickness of 0.2 cm. Further details on the samples acquired for this investigation are summarized in Table 1.Isotope Isotope abundance (%) Isotope weight in the sample/mg Thickness/cm density/(g/cm3) Number of target atoms/( $10^{-4}$ atoms/b)$^{41}{\rm{K}}$ 6.7302 ± 0.0044 370.5 ± 0.1 0.2 2.66 1.612 $^{65}{\rm{Cu}}$ 30.85 ± 0.15 116.5 ± 0.1 0.0125 8.96 3.330 $^{27}{\rm{Al}}$ 100 22.5 ± 0.1 0.0025 2.70 5.019 Table 1. Sample details for this investigation.
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Neutron irradiation in this experiment was performed at the Bhabha Atomic Research Center (BARC)'s PURNIMA neutron generator facility in Mumbai [18]. The t(d,n)α fusion process was used to create neutrons during which d+ ions with an energy of 140 ± 5 keV and a current of 60 µA were accelerated onto a Ti-t (Titanium-tritide) target to create neutrons in the advancing direction with a flux value of 9.42×107 n/cm2/s on the aluminum sample. The position order of the samples during the experiment was Al-Cu-K. The samples of aluminum and potassium were positioned at zero degrees with respect to the deuteron beam at distances of 1.1 cm and 1.2 cm from the neutron target, respectively. The aluminum used was square shaped, while the potassium was in the shape of a circular pellet. A GEANT4 simulation was performed to estimate the variation in neutron flux for both samples owing to the difference in their shape and distance with respect to neutron target [19]. The simulation was performed incorporating the deuteron beam energy, deuteron beam profile and emittance, tritium profile of the Ti-t target, and the solid angle between the neutron target and the sample. The neutron flux observed on the potassium sample was 82.13% of the neutron flux observed on the aluminum sample, as calculated by GEANT4. The energy of the neutrons and the associated uncertainty were calculated using two-body kinematics described in [20].
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The irradiated samples were removed from the irradiation room and placed in the counting room for further cooling. Then, the samples were pasted on a perplex plate and taken to the gamma counting room to measure for induced activity. The irradiation time (
$t_{\rm ir}$ ) was set in such a way that it covered the time until the induced activity reached the saturation state as different reactions in this study have different half-lives. Similarly, the irradiated samples were counted by providing the appropriate cooling time ($t_{\rm co}$ ) before measuring the counts; this ensures that the short half life γ-rays disintegrate and do not contribute to the primary characteristic γ-rays used for the cross section estimation. The timing factor parameters ($t_{\rm ir}$ ,$t_{\rm co}$ ) of this experiment with the measured counting time ($t_{\rm ms}$ ) factor are listed in Table 2. A lead-shielded 185-cc HPGe detector system was used to detect the induced activity counts.Reaction $t_{\rm ir}/{\rm s}$ $t_{\rm co}/{\rm s}$ $t_{\rm ms}/{\rm s}$ $^{65}{\rm{Cu}}$ (n,α)$^{62m}{\rm{Co}}$ 8525 1468 250 $^{41}{\rm{K}}$ (n,α)$^{38}{\rm{Cl}}$ 8525 2208 356 $^{65}{\rm{Cu}}$ (n,2n)$^{64}{\rm{Cu}}$ 8525 6620 693 $^{27}{\rm{Al}}$ (n,α)$^{24}{\rm{Na}}$ 8525 11268 2030 Table 2. Timing factor parameters of this experiment.
The CAMAC based multi-parameter data acquisition and data analysis programme LAMPS [21] was used for data acquisition, with a low detector dead time. Table 3 contains information on the decay data (retrieved from the ENSDF library) required for data analysis [22]. The characteristic γ-ray produced from the residues of the reactions with a high gamma-ray intensity was used to calculate the neutron activation cross section. The obtained γ-ray spectra from the HPGe detector for all the given reactions are presented in Figs. 1-2. The γ peak counts for
$ ^{41} {\rm{K}}$ (n,α),$ ^{65} {\rm{Cu}}$ (n,α), and$ ^{65} {\rm{Cu}}$ (n,2n) are 130 ± 11.401 (1642.68 keV), 53 ± 7.280 (1163.50 keV), and 98 ± 9.899 (1345.77 keV), respectively. The statistics associated with the cross section of the$ ^{65} {\rm{Cu}}$ (n,2n) reaction may be improved by increasing the counting period. Detailed information related to the calibration and efficiency calculation of the HPGe detector, including its uncertainty quantification and coincidence summimg-effect, is explained in our previous study [23]. The parameters and their correlation coefficients, as given in Section II.(C) of [23], were used for the detector efficiency calculations in the current study. Table 4 summarizes the obtained efficiency value, its uncertainty, and its correlation matrix. This correlation matrix will be used to calculate the total uncertainty in the observed cross section and the correlation matrix between various reaction cross sections.Reaction Residue product Half-life ( $t_{1/2}$ )$E_{\gamma}$ /keV$I_{\gamma}$ (%)Reference $^{65}{\rm{Cu}}$ (n,α)$^{62m}{\rm{Co}}$ 13.86 ± 0.09 min 1163.50 70.5 ± 1.4 [24] $^{41}{\rm{K}}$ (n,α)$^{38}{\rm{Cl}}$ 37.230 ± 0.014 min 1642.68 32.9 ± 0.5 [25] $^{65}{\rm{Cu}}$ (n,2n)$^{64}{\rm{Cu}}$ 12.701 ± 0.002 hr 1345.77 0.475 ± 0.011 [26] $^{27}{\rm{Al}}$ (n,α)$^{24}{\rm{Na}}$ 14.997 ± 0.012 hr 1368.62 99.9936 ± 0.0015 [27] Table 3. Decay data with associated uncertainties used for the samples and reference reactions.
Figure 1. Gamma-ray spectra from the HPGe detector with a cooling period of (a) 1468 s & (b) 2208 s.
Reaction $E_{\gamma}$ /keVEfficiency Correlation matrix $^{65}{\rm{Cu}}$ (n,α)$^{62m}{\rm{Co}}$ 1163.50 0.00897 ± 0.00023 1.0000 $^{41}{\rm{K}}$ (n,α)$^{38}{\rm{Cl}}$ 1642.68 0.00810 ± 0.00032 0.9125 1.0000 $^{65}{\rm{Cu}}$ (n,2n)$^{64}{\rm{Cu}}$ 1345.77 0.00846 ± 0.00027 0.9679 0.9860 1.0000 $^{27}{\rm{Al}}$ (n,α)$^{24}{\rm{Na}}$ 1368.62 0.00842 ± 0.00028 0.9626 0.9892 0.9997 1.0000 Table 4. Interpolated efficiency of the HPGe detector for the corresponding γ-ray energy of the samples, the reference reactions, and their correlation matrix.
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Using the following neutron activation formula, the neutron induced cross section (
$ \sigma_{s} $ ) for the$ ^{65} {\rm{Cu}}$ (n,α), (n,2n) and$ ^{41} {\rm{K}}$ (n,α) reactions were measured with reference to the$ ^{27} {\rm{Al}}$ (n,α) reaction cross section ($ \sigma_{Al} $ ):$ \small\sigma_{s} = \sigma_{\rm Al} \times \left[\frac{A_{s} \varepsilon_{\rm Al}I_{\rm Al}}{A_{\rm Al} \varepsilon_{s}I_{s}}\right] \times \left[\frac{\lambda_{s}f_{\rm Al}}{\lambda_{\rm Al}f_{s}}\right] \times \left[\frac{a_{\rm Al}N_{\rm Al}}{a_{s}N_{s}}\right] \times \frac{C_{{\rm self}(s)}}{C_{\rm self(Al)}}, $
(1) where
$\sigma_{\rm Al}$ is the reference cross section, ($A_{s,\rm Al}$ ,$I_{s,\rm Al}$ &$\varepsilon_{s,\rm Al}$ ) are the experimentally generated parameters defined as the photo-peak counts of the residues' characteristics γ-rays, γ-ray intensity, and measured HPGe detector efficiency value for the sample and reference reactions, respectively. ($\lambda_{s,\rm Al}$ ,$f_{s,\rm Al}$ ) are the timing factor parameters, where the timing factor ($f_{s,\rm Al}$ ) for the sample and reference reactions was calculated using the following equation:$ \small f_{s,\rm Al} = (1-{\rm e}^{-{\lambda}t_{\rm ir}}) \times ({{\rm e}^{-{\lambda}t_{\rm co}}}) \times (1-{\rm e}^{-{\lambda}t_{\rm ms}}), $
(2) where
$\lambda_{s,\rm Al}$ is the decay constant of the sample and reference reactions, respectively, and symbols$t_{\rm ir}$ ,$t_{\rm co}$ , and$t_{\rm ms}$ denote the timing parameters of this experiment, as described in subsection II.C and listed in Table 2. The samples parameters ($a _{s,\rm Al}$ ,$N _{s,\rm Al}$ ) define the isotopic abundance and number of atoms of a specific isotope in a specific sample, and${{C}}_{\rm self}$ denotes the self-attenuation factor. The IRDFF library's interpolated neutron reference cross section value for the$ ^{27} {\rm{Al}}$ (n,α) reaction at 14.92 ± 0.02 MeV is 0.1092 ± 0.000398 barns. The estimated neutron flux from the given IRDFF reference cross section and γ-ray activity of the$ ^{24} {\rm{Na}}$ residue product is 9.42×107 n/cm2/s, which is corrected through the γ-ray self attenuation process. The subsequent section provides further information on the correction factor for the self attenuation of γ-rays. -
To determine the self-absorption correction factor of the γ-ray interaction within a sample with a thickness (r), the following equation, proposed by [28], was used:
$ C_{\rm self} = \frac{\mu_{m}\rho r}{1-\exp(-\mu_{m}\rho r)}, $
(3) where
$ \mu_{m} $ and ρ are the mass attenuation coefficient and sample density, respectively, retrieved from the XMuDat ver. 1.01 [29] programme. The thickness and density of the sample are given in Table 1. Table 5 provides details regarding the γ-ray self attenuation factor for each given sample.Sample ${\rm{E}}_{\gamma}$ /keV${{C} }_{\rm self}$ Cu 1163.50 1.0030 ± 0.00015 1345.77 1.0028 ± 0.00014 ${\rm{K}}_{2}{\rm{SO}}_{4}$ 1642.68 1.013 ± 0.0014 Al 1368.62 1.0002 ± 0.00001 Table 5. Self attenuation correction factor for γ-ray interactions within each sample.
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The essential factors involved in the determination of cross sections are the γ-ray photo-peak counts (
$A _{s,\rm Al}$ ), γ-ray intensity ($I _{s,\rm Al}$ ), efficiency ($\varepsilon_{s,\rm Al}$ ), timing factor ($f _{s,\rm Al}$ ), isotopic abundance ($a _{s,\rm Al}$ ) (the fractional uncertainty in the isotopic abundance ($a _{\rm Al}$ ) of$ ^{27} {\rm{Al}}$ was not considered because this element is mono-isotopic with 100% isotopic abundance), number of target atoms ($N _{s,\rm Al}$ ), and reference cross section ($\sigma_{\rm Al}$ ). The fractional uncertainty (%) from all of these variables was used to transmit the overall uncertainty in the measured reaction cross section. The timing factor ($f _{s,\rm Al}$ ) fractional uncertainty was calculated using the method described in Sec. 4.1.3 of Ref. [30]. In the calculation of the timing factor, we considered the uncertainty in the residues' decay constants because the uncertainties in the$t_{\rm ir}$ ,$t_{\rm co}$ , and$t_{\rm ms}$ timings were negligible. Table 6 highlights the fractional uncertainties in various variables contributing to the measured reaction cross section, which will be used to propagate the covariance matrix between the distinct reaction cross sections. All the irradiated samples were measured with the same detection system and used the same relative reaction cross section. Hence, the detector efficiency and reference cross section accuracy is the same for all three reactions, which shows that the reaction cross sections are correlated with each other. Therefore, once the fractional uncertainties were calculated, the next part of the covariance analysis was to generate the correlation coefficients between each of the attributes associated with the different reactions. The value of the coefficients lies between$ - 1 \leqslant {\rm{Cor}}( \bigtriangleup{x} , \bigtriangleup{x} ) \leqslant + 1$ . The correlation coefficients between the attributes linked to the various reactions are summarized in Table 7, where i, j, and k indicate$ ^{65} {\rm{Cu}}$ (n,α)$ ^{62m} {\rm{Co}}$ ,$ ^{41} {\rm{K}}$ (n,α)$ ^{38} {\rm{Cl}}$ , and$ ^{65} {\rm{Cu}}$ (n,2n)$ ^{64} {\rm{Cu}}$ , respectively. We propagated the total uncertainty and the covariance matrix between two reactions, e.g., ($ \sigma_{s_i},\sigma_{s_j} $ ), from the values given in Tables 6−7 by summing the matrices of 12 subsets (attributes) using the following equation:attributes
(x)fractional uncertainties (%) $^{65}{\rm{Cu}}$ (n,α)$^{62m}{\rm{Co}}$
($\bigtriangleup{x_i}$ )$^{41}{\rm{K}}$ (n,α)$^{38}{\rm{Cl}}$
($\bigtriangleup{x_j}$ )$^{65}{\rm{Cu}}$ (n,2n)$^{64}{\rm{Cu}}$
($\bigtriangleup{x_k}$ )$A_{s}$ 13.7361 8.7706 10.1015 $A_{\rm Al}$ 1.1311 1.1311 1.1311 $I_{s}$ 1.9858 1.5197 2.3157 $I_{\rm Al}$ 0.0015 0.0015 0.0015 $N_{s}$ 0.0858 0.0270 0.0858 $N_{\rm Al}$ 0.4444 0.4444 0.4444 $a_{s}$ 0.4862 0.0653 0.4862 $\varepsilon_{s}$ 2.5641 3.9506 3.1914 $\varepsilon_{\rm Al}$ 3.3254 3.3254 3.3254 $f_{s}$ 0.2526 0.0134 0.0004 $f_{\rm Al}$ 0.0054 0.0054 0.0054 $\sigma_{\rm Al}$ 0.3644 0.3644 0.3644 $C_{\rm self}$ 0.0152 0.0142 0.1390 Table 6. The fractional uncertainties (%) of the different attributes related to the reaction cross sections measured in this study.
Correlation coefficient ( $\bigtriangleup{x}$ ,$\bigtriangleup{x}$ )$A_{s}$ $A_{\rm Al}$ $I_{s}$ $I_{\rm Al}$ $N_{s}$ $N_{\rm Al}$ $a_{s}$ $\varepsilon_{s}$ $\varepsilon_{\rm Al}$ $f_{s}$ $f_{\rm Al}$ $\sigma_{\rm Al}$ $C_{\rm self}$ Cor( $\bigtriangleup{x_i}$ ,$\bigtriangleup{x_i}$ )1 1 1 1 1 1 1 1 1 1 1 1 1 Cor( $\bigtriangleup{x_i}$ ,$\bigtriangleup{x_j}$ )0 1 0 1 0 1 0 0.9125 1 0 1 1 0 Cor( $\bigtriangleup{x_i}$ ,$\bigtriangleup{x_k}$ )0 1 0 1 1 1 1 0.9679 1 0 1 1 0 Cor( $\bigtriangleup{x_j}$ ,$\bigtriangleup{x_j}$ )1 1 1 1 1 1 1 1 1 1 1 1 1 Cor( $\bigtriangleup{x_j}$ ,$\bigtriangleup{x_k}$ )0 1 0 1 0 1 0 0.9860 1 0 1 1 0 Cor( $\bigtriangleup{x_k}$ ,$\bigtriangleup{x_k}$ )1 1 1 1 1 1 1 1 1 1 1 1 1 Table 7. Correlation coefficient between attributes related to the different reactions cross sections determined at a neutron energy of 14.92 ± 0.02 MeV.
$ {\rm Cov}(\sigma_{s_i},\sigma_{s_j}) = \mathop \sum \limits_i \mathop \sum \limits_j \bigtriangleup{x_i}\times {\rm Cor}(\bigtriangleup{x_i},\bigtriangleup{x_j}) \times \bigtriangleup{x_j} . $
(4) From the above equation, we generated a [3×3] covariance matrix. Then, the total uncertainty in the measured cross section was obtained using the following formula:
$ (\bigtriangleup\sigma_{s_i})^{2} = {\rm Cov}(\sigma_{s_i},\sigma_{s_i}) . $
(5) We used the equation below to propagate the correlation matrix [3×3] between the reaction cross sections from the total uncertainty and covariance matrix.
$ {\rm Cor}(\sigma_{s_i},\sigma_{s_j}) = \frac{{\rm Cov}(\sigma_{s_i},\sigma_{s_j})}{(\bigtriangleup\sigma_{s_i})^{2} \times (\bigtriangleup\sigma_{s_j})^{2}} . $
(6) A similar procedure was used in literature [23, 30-32]. Table 8 lists the measured reaction cross sections, their total uncertainties, and their correlation matrix.
Reaction Present data [ $\sigma_{s}$ ]$\Delta\sigma_{s}$ (%)Correlation matrix $^{65}{\rm{Cu}}$ (n,α)$^{62m}{\rm{Cu}}$ 0.00404 ± 0.00059 14.57 1.0000 $^{41}{\rm{K}}$ (n,α)$^{38}{\rm{Cl}}$ 0.02509 ± 0.00260 10.37 0.1451 1.0000 $^{65}{\rm{Cu}}$ (n,2n)$^{64}{\rm{Cu}}$ 1.03082 ± 0.11776 11.42 0.1237 0.2119 1.0000 Table 8. Experimentally determined reaction cross sections (in barns) with their total uncertainty value and correlation matrix at a neutron energy of 14.92 ± 0.02 MeV.
Measurement of (n,α) and (n,2n) reaction cross sections at a neutron energy of 14.92 ± 0.02 MeV for potassium and copper with uncertainty propagation
- Received Date: 2021-06-10
- Available Online: 2022-01-15
Abstract: Experimentally measured neutron activation cross sections are presented for the