Targets

The ^natCu target (3.15 g) was placed in polyethylene target holders and irradiated by LCS γ beams. The alignments of the target and FED with the LCS γ-ray beam were adjusted using a MiniPIX X-ray pixel detector for collimation. The detailed specifications are provided in Table 1.

The target holder has a 10-mm in diameter window. Considering that the size of the LCS γ-ray beams was approximately 4 mm in diameter at the target position, a 10 mm diameter window was sufficient for the target to be measured, avoiding the influence of neutrons from polythene.

Measurements

The SLEGS facility features a new FED with 26 proportional counters arranged in three concentric radii within a polyethylene moderator shielded by a 2 mm Cd sheet [23]. The counters, with an effective length of 500 mm and filled with ³He gas at a pressure of 2 atm, were read out through the Mesytec MDPP-16 digitizers and MVME DAQ. Figure 3 shows the efficiency curves of each ring and the total efficiency curve simulated by GEANT4 using a real detector configuration. For the neutron evaporation spectrum, the total detector efficiency increases from 35.64% at 50 keV to 42.32% at 1.65 MeV, and then decreases slowly to 39.05% at 4 MeV [28]. The efficiency calibrated using the ²⁵²Cf source is 42.10 ± 1.25%, corresponding to an average neutron energy of 2.13 MeV. In our experiment, we used the ring-ratio technique to obtain the average energy of neutrons produced by the (γ, n) reaction and then estimated the detector efficiency using its calibration curve [29, 30].

Fig. 3Full-screen View

(Color online) Total detector efficiency and efficiencies of individual rings. Detector efficiency curves were simulated using neutron evaporation spectra and monochromatic neutrons. The red dots are given by the neutron spectrum described by the Maxwell-Boltzmann distribution, at the average neutron energy (T = 1.42 MeV) of ²⁵²Cf [28]

Data Analysis method

In the monochromatic approximation, the photoneutron cross section can be expressed by the integral equation [31]: ${\displaystyle {\int }_{S_{\text{n}}}^{E_{\text{max}}}}{n}_{\gamma }\left(E\right)\sigma (E)\text{d}E=\frac{N_{\text{n}}}{N_{\gamma }{N}_{\text{t}}\xi {ϵ}_{\text{n}}}\mathrm{,}$ (1) where $n_{\gamma }\left(E\right)$ is the energy distribution of the LCS γ-ray beam normalized in the integration region. σ(E) represents the photoneutron cross section, N_n the number of detected neutrons, N_t the number of target nuclei per unit area; and $N_{\gamma }$ the number of incident γ-ray with energies above the neutron threshold. The self-attenuation coefficient ξ is given by [32] $\xi =\frac{\mu t}{1-e^{-\mu t}}\mathrm{,}$ (2) where μ is the linear attenuation coefficient of the sample and t is the thickness of the sample. The photoneutron cross section in the monochromatic approximation is calculated by ${\sigma }_{\left(\gamma \mathrm{,}\text{n}\right)}^{E_{\text{max}}}=\frac{N_{\text{n}}}{N_{\gamma }{N}_{\text{t}}\xi {ϵ}_{\text{n}}}\mathrm{.}$ (3) In the experiment, the laser pulse cycle was 1000 μs (50 μs on + 950 μs off). Owing to the energy dispersion of LCS γ-ray beams, the monochromatic approximation is insufficient for determining photoneutron cross sections. When neutrons were counted with the FED array, the flat-efficiency regions for each detector ring were determined, considering the neutron energy and detector parameters. The median method was used to establish the optimal efficiency points and improve neutron count statistics.

To solve this unfolding problem, the integral in Eq. (1) is approximated as the summation of each $\gamma$ beam profile, resulting in a system of linear equations ${\sigma }_{\text{f}}=\text{D}\sigma$ [33, 34]. The folding iteration method was used to solve this underdetermined system. Starting with a constant trial function ${\sigma }_0$, the folded vector ${\sigma }_{\text{f}}^0=\text{D}{\sigma }^0$ was calculated [35-38]. The next trial input function ${\sigma }_1$ was obtained by adding the difference between the experimental spectrum ${\sigma }_{\text{exp}}$ and the folded spectrum ${\sigma }_{\text{f}}^0$ to $\text{D}{\sigma }^0$ after spline interpolation to match the vector dimensions. The iteration proceeds with [39-41] ${\sigma }^{i+1}={\sigma }^i+\left({\sigma }_{\text{exp}}-{\sigma }_{\text{f}}^i\right)\mathrm{.}$ (4) The iteration continues until convergence, when ${\sigma }_{\text{f}}^{i+1}$ approximates σ_exp within statistical errors. Convergence was assessed by calculating the reduced χ² between ${\sigma }_{\text{f}}^{i+1}$ and σ_exp, with typical convergence achieved in approximately three iterations, resulting in a reduced χ² value of approximately 1.

Substitution Measurement method

In this experiment, the natural Cu target (^natCu) has an isotopic abundance of 69.15% for ⁶³Cu and 30.85% for ⁶⁵Cu. The one-neutron (S_n) and two-neutrons (S_2n) separation energies for ⁶³Cu are 10.86 and 19.74 MeV, respectively. And for ⁶⁵Cu, S_n and S_2n become 9.91 MeV and 17.83 MeV, respectively [42-44]. Within the energy range where the one-neutron separation energy thresholds of ⁶³Cu and ⁶⁵Cu overlap, the FED detector measures neutrons from both the ⁶³Cu(γ, n) and ⁶⁵Cu(γ, n) reactions as $N_{{}^{\text{nat}}\text{Cu}}=N_{{}^{63}\text{Cu}}+N_{{}^{65}\text{Cu}}\mathrm{,}$ (5) where $N_{{}^{\text{nat}}\text{Cu}}$ represents the result obtained from direct measurement, whereas $N_{{}^{63}\text{Cu}}$ needs to be calculated by combining the monoenergy cross section ${\sigma }_{\left(\gamma \mathrm{,}\text{n}\right)}^{{}^{63}\text{Cu}}$ with the current energy spectrum $n_{\gamma }\left(E_{\gamma }\right)$; the detailed calculation process is shown in Eq. (6), where d is the thickness of target. $\begin{array}{c}{N}_{{}^{63}\text{Cu}}=N_{\text{t}}g{ϵ}_{\text{n}}\frac{{\displaystyle {\int }_{S_{\text{n}}}^{E_{\max }}}{n}_{\gamma }\text{d}{E}_{\gamma }\left(1-e^{-{\mu }_{{}^{\text{nat}}\text{Cu}}{\rho }_{{}^{\text{nat}}\text{Cu}}d}\right)}{{\mu }_{{}^{\text{nat}}\text{Cu}}{\rho }_{{}^{\text{nat}}\text{Cu}}d}\\ \times {\displaystyle {\int }_{S_{\text{n}}}^{E_{\max }}}{n}_{\gamma }\left(E_{\gamma }\right){\sigma }_{\left(\gamma \mathrm{,}\text{n}\right)}^{{}^{63}\text{Cu}}\left(E_{\gamma }\right)\text{d}{E}_{\gamma }\mathrm{.}\end{array}$ (6) By substituting the neutron count of $N_{{}^{63}\text{Cu}}$ into Eq. (7), the quasi-monoenergetic photoneutron cross section data of ⁶⁵Cu can be obtained as ${\sigma }_{\left(\gamma \mathrm{,}\text{n}\right)}^{{}^{65}\text{Cu}}={\displaystyle {\int }_{S_{\text{n}}}^{E_{\text{max}}}}{n}_{\gamma }\left(E\right)\sigma (E)\text{d}E=\frac{N_{{}^{\text{nat}}\text{Cu}}-N_{{}^{63}\text{Cu}}}{N_{\gamma }{N}_{\text{t}}\xi {ϵ}_{\text{n}}}\mathrm{.}$ (7) The monochromatic cross section of the ⁶⁵Cu(γ, n)⁶⁴Cu reaction was derived using a deconvolution iteration method. Figure 4 compares the quasi-monoenergetic and monochromatic cross sections of ⁶⁵Cu.

Fig. 4Full-screen View

(Color online) Cross sections of ⁶⁵Cu(γ, n) measured at SLEGS. The dots are the folded cross section and the line with shaded area is the unfolded (monochromatic) cross section

According to Eq. (7), statistical uncertainty is mainly caused by N_n. Methodological uncertainty arises from the extraction algorithm N_n and deconvolution method incorporating the simulated BGO response matrix. Systematic uncertainty, which is the main source of the total error, includes the γ-flux uncertainty from the copper attenuator, the uncertainty of the target thickness, and the uncertainty of the FED efficiency. Table 2 summarizes the systematic and methodological uncertainties.

⁶⁵Cu photoneutron reaction cross section

The ${}^{65}\text{Cu}{\left(\gamma \mathrm{,}\text{n}\right)}^{64}\text{Cu}$ photoneutron cross section data measured at SLEGS were compared with the existing experimental and evaluated data in Fig. 5. Although the overall trends were consistent, significant discrepancies were observed in the absolute values. The experiment by Fultz et al. [13] at Lawrence Livermore National Laboratory (LLNL) is closest to the results in this work. They used BF₃ neutron detectors and measured in the energy range of 9.34 to 27.78 MeV. Data from Katz et al. [11] using BR source at the BETAT accelerator in Canada with a 22 MeV endpoint energy, are notably higher than the LLNL data. Antonov’s [12] measurements using the BR source at JINR are significantly higher than those of other datasets in terms of both neutron threshold and cross-sectional values. The cross section for ⁶⁵Cu(γ, n)⁶⁴Cu reaction measured by the monoenergetic LCS-γ source at SLEGS is listed in Table 3. Data errors include statistical, methodological, and systematic uncertainties. The total uncertainties at different Eγ are also listed in Table 3.

Fig. 5Full-screen View

(Color online) Cross section of ⁶⁵Cu(γ, n)⁶⁴Cu reaction. The solid circles denote the measured results with the natural abundance (30.85%) of ⁶⁵Cu. The results measured using BR γ rays by Katz [11] and Antonov [12] are plotted as solid green diamonds and purple squares, respectively. The results measured using PAIF γ rays by Fultz [13] are plotted as solid blue triangles. The TENDL-2021 evaluation is indicated by a sky-blue solid line. The calculated results by the increased (+5.00%) and decreased (-5.00%) abundance of ⁶⁵Cu are indicated by the purple line segment and gray dotted line. The inset figure highlights the differences in cross sections at the peak positions for these isotopic abundances

Isotopic abundance variations influence cross sections and cause discrepancies in the results. The photoneutron cross sections changed regularly with alterations in ⁶⁵Cu abundance. Based on data from the 2025 SLEGS experiment, comparing the cross section of ⁶⁵Cu at 35.85%, natural abundance (30.85%), and 25.85%, the photoneutron cross sections decreased as the abundance of ⁶⁵Cu increased (e.g., from 25.85% to 35.85%) and increased as the abundance decreased (see the inset in Fig. 5). This occurs across all energy ranges and is most noticeable near the threshold and peak positions in the cross section distribution. It is suggested that the isotopic abundances in the target material should be determined prior to analysis of the substitution data. The sensitive change in isotopic abundance also suggests that it is capable of adopting an enhanced isotopic target to determine the cross section (γ, n) in addition to the pure isotopic target.

As discussed in Ref. [45], the ratios of the integral cross sections provide a clear indication of the systematic differences among the various data compilations. The integral cross sections in S_n and S_max regions are as follows: ${\sigma }^{\text{int}}={\displaystyle {\int }_{S_{\text{n}}}^{S_{\text{max}}}}\sigma (E)\text{d}E\mathrm{.}$ (8) Based on these experimental data, the integral ratios of the photoneutron reaction cross section were calculated for energy ranges from S_n to 15 MeV, 15 MeV to S_2n, and S_n to S_2n, and the results are presented in Table 4. In the energy range of S_n to 15 MeV, the measured results show a difference of only 0.8% from the Fultz data and a discrepancy of less than 0.4% from theoretical calculations of TENDL-2021, while discrepancies with other datasets exceed 40%. In the 15 MeV to S_2n range, the minimum difference between this study and the Katz [11] data is 0.4%, and the differences with other datasets exceed 20%. In general, the neutron threshold and the peak position of the cross section demonstrate good consistency with the results measured by Fultz [13] and the evaluation in TENDL-2021 [46]. The neutron threshold exhibits favorable agreement with Katz data [11]. However, there are notable differences in both the neutron threshold and peak position compared with the data measured by Antonov [12].

⁶⁴Cu radiative neutron capture cross section

The gamma strength function (γSF) [47] is used to describe the average probabilities of gamma decay and absorption in nuclear reactions, and is an important parameter for characterizing the nuclear reaction process. When research involves reactions with gamma rays, such as reactions (n, γ) and (γ, n), the precision of γSF is particularly crucial. According to the principle of detailed balance [48] and generalized Brink assumption [49-51], it is believed that the upward $\overleftarrow{f_{X\mathcal{l}}}\left(E_{\gamma }\right)$ is approximately equal to the downward $\overrightarrow{f_{X\mathcal{l}}}\left(E_{\gamma }\right)$. Therefore, it can be considered that $f_{X\mathcal{l}}\left(E_{\gamma }\right)\approx \overrightarrow{f_{X\mathcal{l}}}\left(E_{\gamma }\right)\approx \overleftarrow{f_{X\mathcal{l}}}\left(E_{\gamma }\right)$. The (upward) σ_γn photoneutron cross section is connected to the (downward) γSF [52] by $f_{X1}\left(E_{\gamma }\right)=\frac{1}{3{\pi }^2{\hslash }^2{c}^2}\frac{{\sigma }_{\gamma \text{n}}\left(E_{\gamma }\right)}{E_{\gamma }}\mathrm{,}$ (9) in which the constant is $1/3{\pi }^2{\hslash }^2{c}^2=8.674\times {10}^{-8}$ mb^-1MeV^-2. Using this relationship, γSF can be obtained from the cross section measured in the photoneutron reaction. Then, the γSF model in TALYS was compared with the experimentally constrained γSF. The model prediction closest to the experimentally obtained γSF is selected. Furthermore, the γSF model is constrained using the normalization parameter G_norm in the TALYS 2.0 toolkit, and the optimization of the normalization parameter G_norm is achieved by minimizing the χ² value, thus making the theoretical calculation results of the constrained γSF model more consistent with the experimentally obtained γSF values. The γSF of ⁶⁵Cu(γ, n) constrained by measured (γ, n) data is shown in Fig. 6, with χ² determined by ${\chi }^2=\frac{1}{N}{\displaystyle \sum _{i=1}^N}{\left(\frac{{\sigma }_{\text{th}\mathrm{,}i}-{\sigma }_{\text{exp}\mathrm{,}i}}{{\sigma }_{\text{err}\mathrm{,}i}}\right)}^2\mathrm{,}$ (10) where N denotes the total number of experimental data points. σ_{th, i}, σ_{exp, i} and σ_{err, i} denote the theoretical value, experimentally measured value, and experimental error of γSF at the i-th data point, respectively. The neutron capture cross section of ⁶⁴Cu, after adjustment of the optimal G_norm value, is shown in Fig. 7. Specifically, G_norm is set to 1.2 in the Kopecky-Uhl generalized Lorentzian model [53]. However, it is 1.4 in the Goriely hybrid model. In particular, when constrained by G_norm, the χ² value of the hybrid mode [54] was the smallest among all the investigated models, showing the best agreement with this set of experimental data. Similarly, Utsunomiya et al. [39] and Li et al. [44] previously conducted related research and measured the (n, γ) radiative reaction cross sections for the ^136,137Ba and ⁶²Cu isotopes. In this study, owing to the lack of experimental data on the low-lying excited states and neutron resonance spacings of ⁶⁵Cu, the constraints on the nuclear level density (NLD) model are limited, resulting in substantial theoretical uncertainties. The study of the ${}^{64}\text{Cu}(\text{n}\mathrm{,}\gamma )$ cross section is of great value for improving nuclear data and optimizing the preparation of medical isotopes. This is closely related to the measurement of the ${}^{65}\text{Cu}{\left(\gamma \mathrm{,}\text{n}\right)}^{64}\text{Cu}$ cross section through the principle of detailed balance, and the former can verify the reliability of the latter, collectively highlighting the overall significance of the research.

Fig. 6Full-screen View

(Color online) The γSF values of ⁶⁵Cu calculated using the Kopecky-Uhl generalized Lorentzian model (red solid line) and Goriely’s hybrid model (blue dash-dot line) with default parameters are compared with those obtained from the optimized Kopecky-Uhl generalized Lorentzian model (red dashed line) and Goriely’s hybrid model (blue dotted line) using the G_norm method, along with the γSF values extracted from the SLEGS experimental data (red plot points)

Fig. 7Full-screen View

(Color online) The cross sections for ⁶⁴Cu(n, γ)⁶⁵Cu are calculated using the optimized Opecky-Uhl generalized Lorentzian model (red shaded area)and Goriely’s hybrid model (purple shaded area). The shaded area represents the results considering six different nuclear-level density models within the TALYS 2.0 [51]