SLEGS γ beam Spectrum

In parallel with the ⁶³Cu irradiation, the γ-ray spectrum after copper attenuation was measured online using a BGO detector. Figure 3 shows an exemplary γ-ray spectrum detected at the slant scattering angles (θ_L) of 91°, 113°, and 140°. To obtain the SLEGS γ-ray spectrum in front of the irradiation target, a direct unfolding method was employed in combination with a known response function of the BGO detector, which was obtained by a GEANT4 simulation [26]. The resulting (unfolded) γ-ray spectrum is shown in Fig. 3. The folded-back spectrum is consistent with the γ-ray spectrum measured by the detector, suggesting reliable reproduction of the γ-ray spectrum before the irradiation target. The γ-ray spectrum was integrated and corrected for the Cu attenuation factor to obtain the γ beam flux at each slant scattering angle.

Fig. 3Full-screen View

(Color online) Typical γ-ray spectra measured by the BGO detector (black line), folded-back γ-ray spectra (red line), and corresponding γ-ray spectra incident on the target (blue line) at θ_L = 91°, 113°, and 140°

⁶³Cu Target

The diameter, thickness, and purity of the ⁶³Cu target were a 10 mm, 1.5 mm, and 99.8%, respectively. For the ⁶³Cu isotope sample, the purity of the ⁶³Cu target was determined by inductively coupled plasma-mass spectrometry. The uncertainty of the thickness was estimated to be 0.01 mm.

Neutron Detection

The number of (γ, n) reactions was determined by detecting the reaction neutrons using the calibrated FED, which comprised 26 sets of ³He proportional counters embedded in a polyethylene moderator. The proportional counters were arranged in three concentric rings positioned 65 mm, 110 mm, and 175 mm from the beam axis. All the sensitive volumes of the ³He proportional counters were cylindrical in shape with the same length of 500 mm and inflated with ³He gas at 2 atm. While the counters in Ring-1 (inner ring) were 1 inch in diameter, those in Ring-2 (middle ring) and Ring-3 (outer ring) were 2 inches in diameter. The bodies of ³He proportional counters were made of stainless steel for a lower α emission rate. The inner polyethylene moderator was 450 mm × 450 mm × 550 mm (along the beam direction) and was surrounded by additional polyethylene plates with cadmium to suppress the background neutrons [27]. These background neutrons were subtracted from the duty cycle of the laser pulse. In our experiments, the duty cycle is set to 50 μs per laser period of 1000 μs. The polyethylene moderation effect significantly broadened the time distribution of the neutrons detected by the ³He proportional counters. However, a flat interval of the time distribution that was only contributed by background neutrons was identifiable. Then, the number of neutrons (N_n) was directly extracted by subtracting the time-normalized background. Further details are available in [28].

The average energy of the reaction neutrons was obtained using the “ring-ratio technique” originally developed by Berman and Fultz [29] and used to determine the detection efficiency. Fig. 4(a) shows the simulated efficiency curve by GEANT4 with realistic detector configuration. For the neutron-evaporation spectra, the total detector efficiency increases from 35.64% at 50 keV to 42.32% at 1.65 MeV and then falls slowly to 39.05% at 4 MeV. The efficiency calibrated using a ²⁵²Cf source was 42.10 ± 1.25%, corresponding to an average neutron energy of 2.13 MeV. In our experiments, we used the ring-ratio technique to obtain the average energy of neutrons produced by (γ, n) reactions and then estimated the detector efficiency using its calibrated curve of the detector efficiency. The curve for the efficiency ratio of Ring-3 to Ring-1 is illustrated in Fig. 4(b) [28].

Fig. 4Full-screen View

(Color online) (a) Total detector efficiency and the efficiencies of individual rings. The detector efficiency curves were simulated by neutron-evaporation spectra. The red dots are given by the neutron spectrum described by the Maxwell-Boltzmann distribution, $P(E)\propto E^{1/2}\cdot \text{exp}(-E/T)$, at the average neutron energy (T = 1.42 MeV) of ²⁵²Cf. (b) Ring-ratio curve of the FED

Monochromatic Cross Section

The experimental formula for the photoneutron cross section is given by [30, 31] ${\displaystyle {\int }_{S_{\text{n}}}^{E_{\text{max}}}}{n}_{\gamma }\left(E_{\gamma }\right)\sigma \left(E_{\gamma }\right)\text{d}{E}_{\gamma }=\frac{N_{\text{n}}}{N_{\gamma }{N}_{\text{t}}\xi {ϵ}_{\text{n}}g}\mathrm{,}$ (1) where nγ(Eγ) is the spectral distribution of the normalized LCS γ beam; σ(Eγ) is the photoneutron cross section; Nn is the number of neutrons detected; Nt is the number of target nuclei per unit area; Nγ is the number of γ-rays incident on the target; ϵ_n is the neutron detection efficiency; and ξ=(1-eμd)/μd is a correction factor for a thick-target measurement. Here, μ is the linear attenuation coefficient of γ photons in a target of thickness d. The factor g represents the fraction of γ flux above the neutron threshold S_n: $g=\frac{{\displaystyle {\int }_{S_{\text{n}}}^{E_{\text{max}}}}{n}_{\gamma }\left(E_{\gamma }\right)\text{d}{E}_{\gamma }}{{\displaystyle {\int }_0^{E_{\text{max}}}}{n}_{\gamma }\left(E_{\gamma }\right)\text{d}{E}_{\gamma }}\mathrm{.}$ (2) The incident γ energy distribution was used to determine the cross section σ(Eγ), which is a function of the γ energy Eγ. Specifically, the incoming γ beam spectra were used to determine nγ(Eγ). The γ-energy distribution was normalized to unity: ${\displaystyle {\int }_{S_{\text{n}}}^{E_{\text{max}}}}{n}_{\gamma }\left(E_{\gamma }\right)\text{d}{E}_{\gamma }=1$. The measured ${\sigma }_{\text{exp}}^{E_{\text{max}}}$ for an incoming γ beam with maximum energy E_max is given by the convoluted cross-section: ${\sigma }_{\text{exp}}^{E_{\text{max}}}={\displaystyle {\int }_{S_{\text{n}}}^{E_{\text{max}}}}{n}_{\gamma }\left(E_{\gamma }\right)\sigma \left(E_{\gamma }\right)\text{d}{E}_{\gamma }=\frac{N_{\text{n}}}{N_{\gamma }{N}_{\text{t}}\xi {ϵ}_{\text{n}}g}\mathrm{.}$ (3) Therefore, we refer to the quantity on the right side of Eq. (3) as the monochromatic cross section. However, owing to the energy spread of the LCS γ beam (Fig. 3), the monochromatic approximation cannot be used to describe the real photoneutron cross section.

Unfolded Cross Section

The deconvoluted Eγ-dependent photoneutron cross-section, σ(Eγ), must be extracted from the integral of Eq. (3). Each measurement characterized by E_max corresponds to the folding of σ(Eγ) with the measured beam profile $n_{\gamma }\left(E_{\gamma }\right)$. Following Ref. [32], we unfold σ(Eγ) according to Eq. (3): ${\sigma }_{\text{f}}={\text{D}}_{\sigma }\mathrm{,}$ (4) where ${\sigma }_{\text{f}}$ represents the folded cross section with beam profile D. The indices i and j of matrix element D_ij correspond to E_max and Eγ, respectively. The set of equations is given by: ${\left[\begin{array}{c}{\sigma }_1\\ {\sigma }_2\\ ⋮\\ {\sigma }_N\end{array}\right]}_{\text{f}}=\left[\begin{array}{ccccc}{D}_{11}& D_{12}& \cdots & \cdots & D_{1M}\\ D_{21}& D_{22}& \cdots & \cdots & D_{2M}\\ ⋮& ⋮& ⋮& ⋮& ⋮\\ D_{N1}& D_{N2}& \cdots & \cdots & D_{NM}\end{array}\right]\left[\begin{array}{c}{\sigma }_1\\ {\sigma }_2\\ ⋮\\ ⋮\\ {\sigma }_M\end{array}\right].$ (5) Each row of D corresponds to the γ beam profile corresponding to E_max. The σ vector ${[{\sigma }_i]}_{\text{f}}$ (i=1, 2, 3..., N) on the left-hand side of Eq. (5) is the folded cross-section, referred to as the experimental monochromatic cross-section, whereas the vector $[{\sigma }_j]$ (j=1,2,3..., M) on the right-hand side is the unfolded cross-section to be determined. In the present experiment, the number of monochromatic cross sections was N=44. The energy profile of the γ beam was simulated in M=200 energy bins. The number of unfolded cross-sections was equal to M. Figure 3 presents a visual representation of the response matrix D for the case of ⁶³Cu. There are N=44 γ-beam spectra, and only three are shown as blue lines in Fig. 3 as examples. As the system of linear equations in Eq. (5) is under-determined, the σj vector cannot be obtained by matrix inversion. We determined σj using an iterative folding method, which can be summarized as follows:

(1) As our starting point, we choose a constant trial function σ⁰ for the zeroth iteration. This initial vector is multiplied by D to obtain the zeroth folded vector ${\sigma }_{\text{f}}^0=\text{D}{\sigma }^0$,

(2) The next trial input function, σ¹, is established by adding the difference between the experimentally measured spectrum σ_exp and folded spectrum ${\sigma }_{\text{f}}^0$ to σ⁰. To add the folded and input vectors, we first perform a spline interpolation on the folded vector, and then interpolate to ensure that the two vectors have equal dimensions. The new input vector is ${\sigma }^1={\sigma }^0+\left({\sigma }_{\text{exp}}-{\sigma }_{\text{f}}^0\right)\mathrm{.}$ (6) (3) The above steps are iterated i times, yielding ${\sigma }_{\text{f}}^i=\text{D}{\sigma }^i\mathrm{,}$ (7) and ${\sigma }^{i+1}={\sigma }^i+\left({\sigma }_{\text{exp}}-{\sigma }_f^i\right)\mathrm{,}$ (8) until convergence is achieved. Thus, ${\sigma }_{\text{f}}^{i+1}\approx {\sigma }_{\text{exp}}$ is within the statistical uncertainties. To check the convergence quantitatively, we calculated the reduced χ² of ${\sigma }_{\text{f}}^{i+1}$ and σ_exp after each iteration. The experiment was terminated when the reduced χ² value approached unity.

Figure 5 shows the monochromatic ${\sigma }_{\text{exp}}^{E_{\text{max}}}$ and unfolded σ(Eγ) for ⁶³Cu(γ, n) reaction. Table 1 lists the σ(Eγ) values at E_max and their uncertainties respectively. According to Eq. (3), the statistical uncertainty is primarily induced by N_n. Because the incident γ-ray count was sufficiently high, its statistical uncertainty was negligible. The methodological uncertainty was approximately 1.8%, which was induced by the extraction algorithm N_n (1.5%) and unfolding methodology incorporating the simulated BGO response matrix (~1%). The systematic uncertainty was estimated to be 3.15%. This was due to the neutron detector efficiency (3.02%), γ flux attenuation and incident γ spectrum unfolding (0.90%), and target areal density (0.10%). In our study, the total uncertainty included statistical, systematic, and methodological uncertainties. The unfolded σ(Eγ) had a total uncertainty of approximately 4%, except for the γ-energy region with σ(Eγ) less than 7.5 mb (i.e., S_n < Eγ < 11.5 MeV).

Fig. 5Full-screen View

(Color online) ⁶³Cu(γ, n) reaction cross section as a function of the incident γ energy, Eγ. The dots indicate the monochromatic cross section, while the line with the shadow area indicates the unfolded cross section

⁶³Cu(γ, n) Reaction Cross Section

Here, we compare our measurements with available experimental [4, 6-9] and evaluated [33] data. The results are presented in Fig. 6. The uncertainty of our unfolded cross-sections is comparable to those of Fultz et al. [7] and Sund et al. [9] with monochromatic photons obtained from positron annihilation in flight. Moreover, it was significantly better than those of Owen et al. [8], Berman et al. [6] and Plaisir et al. [4] obtained with bremsstrahlung radiation. Our measurements are consistent with the data of Berman et al. [6], although the latter have only a few data points. Moreover, our data agree well with those of other groups when S_n < Eγ < 15 MeV. However, our data were visibly higher than those of other studies at Eγ > 15 MeV.

Fig. 6Full-screen View

(Color online) Unfolded cross section curve for ⁶³Cu(γ, n) together the available experimental and evaluated data [6-9, 33]

The integral ratio of two cross-section curves reflects their systematic differences [34]. The total cross-section integrated over the energy region of interest is defined as follows: ${\sigma }^{\text{int}}={\displaystyle {\int }_{E_{\text{min}}}^{E_{\text{max}}}}\sigma \left(E_{\gamma }\right)\text{d}{E}_{\gamma }\mathrm{.}$ (9)

We conducted experimental measurements on ¹⁹⁷Au(γ, n) and ¹⁵⁹Tb(γ, n) reactions at SLEGS [28]. The ¹⁹⁷Au(γ, n) reaction data were compared with those reported by Itoh et al. [30]. The resulting σ^int difference was ~0.4%, suggesting the reliability of SLEGS in the measurement procedure and data analysis [28]. We calculated σ^int of ⁶³Cu(γ, n) reactions for different laboratories within S_1n < Eγ < 15 MeV and 15 MeV < Eγ < S_2n. The results are shown in Table 2. For S_1n < Eγ < 15 MeV, the relative difference between the σ^int value and those of the others is 4–13%, except for the data of Owen, for which the difference is 28%. In contrast, for 15 MeV < Eγ < S_2n, our data are larger than the others by a factor of 0.13–0.35. Table 2 shows that the data of Owen are evidently lower than the others within the two aforementioned energy regions. Consequently, it is not a priority. Overall, our measurements are expected to clarify the inconsistency between the available experimental data for the ⁶³Cu(γ, n) reaction.

Radiative ⁶²Cu(n, γ) Cross Section

γSF [35, 36] is a statistical quantity employed in the Hauser-Feshbach model of the compound nuclear reaction. The γSF in the de-excitation mode aids in determining the radiative (n, γ) cross-sections that are directly relevant to the s-process nucleosynthesis of elements heavier than iron. The downward γSF for dipole radiation at a given energy Eγ is defined as [37] $\overrightarrow{f_{X1}}\left(E_{\gamma }\right)=E_{\gamma }^{-3}\frac{\langle {\Gamma }_{X1}\left(E_{\gamma }\right)\rangle }{D_{\mathcal{l}}}\mathrm{.}$ (10) Here, X is either electric (E) or magnetic (M); $\langle {\Gamma }_{X1}\left(E_{\gamma }\right)\rangle$ is the average radiation width; and $D_{\mathcal{l}}$ is the average level spacing for s-wave ($\mathcal{l}=0$) or p-wave ($\mathcal{l}=1$) neutron resonances.

In contrast, γSF in the excitation mode for dipole radiation [37] is defined by the average cross section for E1/M1 photoabsorption ${\sigma }_{X1}\left(E_{\gamma }\right)$ to the final states with all possible spins and parities [36]: $\overleftarrow{f_{X1}}\left(E_{\gamma }\right)=\frac{E_{\gamma }^{-1}}{g_J{\left(\pi \hslash c\right)}^2}\langle {\sigma }_{X1}\left(E_{\gamma }\right)\rangle \mathrm{.}$ (11) Here, the spin factor gJ = (2J + 1)/(2J₀ + 1), where J = 1 and J₀ = 0 (ground state).

Above the neutron separation energy, except at energies near the neutron threshold, the total upward γSF can be determined by substituting ${\sigma }_{X1}\left(E_{\gamma }\right)$ with the experimental (γ, n) cross-sections that dominate the photoabsorption cross-sections. According to the principle of detailed balance [38] and the generalized Brink hypothesis, the equality of the upward and downward γSF, $f_{X1}\left(E_{\gamma }\right)=\overrightarrow{f_{X1}}\left(E_{\gamma }\right)=\overleftarrow{f_{X1}}\left(E_{\gamma }\right)$, connect the (upward) (γ, n) cross section ${\sigma }_{\gamma \text{n}}$ to the (downward) γSF by [37] $f_{X1}\left(E_{\gamma }\right)=\frac{1}{g_J{\pi }^2{\hslash }^2{c}^2}\frac{{\sigma }_{\gamma \text{n}}\left(E_{\gamma }\right)}{E_{\gamma }},$ (12) where $1/g_J{\pi }^2{\hslash }^2{c}^2=8.674\times {10}^{-8}{\text{mb}}^{-1}{\text{MeV}}^{-2}$. This relation yields the experimentally constrained γSF from the measured ⁶³Cu(γ, n) reaction data, as indicated by the red dots in Fig. 7.

Fig. 7Full-screen View

(Color online) Comparison of the γSF of ⁶³Cu calculated using the Brink-Axel Lorentzian model (blue dashed line) and SMLO model (black dashed line) for the E1 strength in TALYS with the γSF extracted from our data (red dots). γSF values (red solid line) optimized using G_norm. The spin-flip and scissor model for the M1 strength is indicated by the pink dashed line

In TALYS (version 1.96) [39, 40], various phenomenological and microscopic models have been established to describe the γSF. The Brink-Axel Lorentzian model and simple modified Lorentzian (SMLO) model [41] for the E1 strength closely approximate the experimental values. The blue and black lines in Fig. 7 represent the γSFs calculated using the aforementioned two models. The pink lines in Fig. 7 shows the spin-flip and scissor model of the M1 strength [42]. Although these two models have contributed to advancements in calculating the γSF, discrepancies between model predictions and experimental observations remain. To improve the predictive accuracy of these models, we refined the Brink-Axel Lorentzian model by incorporating the normalization factor G_norm for γSF available in TALYS. G_norm was optimized by minimizing χ² and aligning the theoretical calculations of the γSF more closely with the experimental data. The expression for χ² is given by ${\chi }^2=\frac{1}{N}{\displaystyle \sum _{i=1}^N{\left(\frac{{\sigma }_{\text{th}\mathrm{,}i}-{\sigma }_{\text{exp}\mathrm{,}i}}{{\sigma }_{\text{err}\mathrm{,}i}}\right)}^2}\mathrm{.}$ (13) where N represents the total number of experimental data points, and σ_th,i, σ_exp,i, and σ_err,i denote the theoretical value, experimental data, and experimental error of the γSF for the i-th data point, respectively. By adjusting G_norm, we find that the χ² value reaches a minimum of 1.46 when G_norm = 1.2. The red solid line in Fig. 7 represents the optimized γSF values, demonstrating closer agreement with the experimental data.

The radiative (n, γ) cross-section strongly depends on the γSF and is sensitive to the nuclear level density (NLD) model employed. We extracted the experimentally constrained γSF from our newly measured ⁶³Cu(γ, n) reaction data and then optimized the Brink-Axel Lorentzian model for E1 strength in TALYS using G_norm. Finally, the radiative (n, γ) cross section for ⁶²Cu was calculated based on the Brink-Axel Lorentzian model with G_norm optimization. The results are presented in Fig. 8 as the red band. The spin-flip and scissor model of the M1 strength was considered in the TALYS calculations. The theoretical uncertainty corresponds to the use of six NLD models [40]. A similar study was performed by Utsunomiya et al. [43], in which radiative (n, γ) cross sections of ^136,137Ba isotopes were obtained. In our study, owing to the lack of experimental data on 63Cu in terms of low-lying excited levels and neutron resonance spacings, we could not effectively constrain the NLD model. Consequently, a relatively large theoretical uncertainty was obtained. To reduce the theoretical uncertainty of the (n, γ) cross sections both the γSF and NLD models should be effectively constrained. A good example can be found in Renstrom et al. [44], in which charged-particle-induced reaction data were used to constrain the NLD model.

Fig. 8Full-screen View

(Color online) ⁶²Cu(n, γ) cross section calculated with TALYS code based on the Brink-Axel Lorentzian model with G_norm optimization. The theoretical uncertainty corresponds to the use of different NLD models [39, 40]. Additional TALYS calculations based on the Brink-Axel Lorentzian model (blue line) and the SMLO model (black line) without G_norm optimization as well as TENDL-2023 evaluations [45] (pink line) are also shown for comparison

To further investigate the ⁶²Cu(n, γ) cross section, additional TALYS calculations based on the Brink-Axel Lorentzian and SMLO models were performed without G_norm optimization. The calculated results and available TENDL-2023 evaluations [45] are also presented in Fig. 8 for comparison, which shows a good agreement between each other. To the best of our knowledge, this is the first time that the experimentally constrained ⁶²Cu(n, γ) cross section has been obtained. This supports the principle of detailed balance and generalized Brink hypothesis for ⁶³Cu isotope.