During irradiation, the γ-ray spectrum after attenuation was continuously monitored using a BGO detector positioned downstream of the target. A representative spectrum measured at a laser incident angle of is shown in Fig. 3(a). After obtaining the BGO measured spectrum (black solid line), the SLEGS γ-ray spectrum (red dashed line) incident on the target was calculated by considering the thickness and attenuation coefficients of both the copper attenuator and target material. The spectral distribution was unfolded using a direct unfolding method based on the detector response matrix simulated with GEANT4 [32]. The blue dashed line in Fig. 3(a) represents the reconstructed spectrum, which is obtained by folding the incident gamma spectrum with the detector response matrix, showing good agreement with the measured BGO response spectrum. The total number of γ-rays contributing to the reaction, , was obtained by integrating the incident gamma spectrum above , as shown by the red shaded region of Fig. 3(a). The unfolded energy profiles for the γ-ray beams used in the measurements are shown in Fig. 3(b). The intensity difference between the measured and unfolded spectra is mainly caused by a significant attenuation from both the copper attenuator ( mm thickness) and target during experiment (5 h).

The physical parameters of the target are listed in Table 1.The target purity was determined by an inductively coupled plasma-mass spectrometer. The thickness uncertainty was estimated to be 0.01 %.

Neutron counts ( ) were determined with the FED. It consists of 26 ³He proportional counters embedded in a polyethylene moderator with dimensions of 450 mm × 450 mm× 550 mm. This polyethylene moderator is covered with a 2 mm thick Cd sheet to absorb environmental neutrons and then further wrapped with a 50 mm thick layer of polyethylene. The proportional counters were arranged in three concentric rings to determine the average neutron energy via the ring-ratio (RR) method [33]. The inner ring, positioned at 65 mm from the center, contains 6 1-inch counters. The middle ring (110 mm from center) has 8 2-inch counters, and the outer ring (175 mm from center) holds 12 2-inch counters. All the proportional counters were filled with 2 atm of ³He gas. The total FED efficiency in the plateau region was approximately 40%. The FED was calibrated using a neutron source, and its efficiency uncertainty was determined to be 3.02% [34].

The DAQ system of the FED enables recording of the pulse height and timestamp of signals from detectors. For determination, accurate thresholds of the analog-to-digital converter spectra for each ³He proportional counter were assigned to distinguish the γ background from the neutrons. Then, the constant bremsstrahlung-induced background was distinguished from the LCS γ beam-induced neutrons by its time distribution [34]. In this work, the CO₂ laser operated at an average power of 5 W, with a pulse period of 1000 µs and pulse width of 50 µs. can be extracted by directly subtracting the time normalized background. This method, as presented in our previous work [35], provides a practical approach to estimating the average neutron energy, which can then be used to determine the FED efficiency for detecting the neutrons emitted from (γ, n) reactions.

The total experimental uncertainties included statistical, methodological, and systematic uncertainties. The statistical uncertainty was primarily induced by and , which encompass the corresponding statistical fluctuations. Because the incident had a sufficiently high count rate in our study, it contributed negligibly to the statistical uncertainty. The methodological uncertainty consists of two main components: the extraction algorithm and unfolding procedure using the simulated BGO response matrix. The systematic uncertainties were primarily due to the external γ-ray induced by the copper attenuator, target thickness, and FED detection efficiency. The methodological uncertainty and systematic uncertainties are summarized in Table 2.

The experimental photoneutron cross section measured with quasi-monochromatic γ-ray beams is given by the following expression [36, 37]:

1

where represents the normalized energy profile of the incident γ-ray beams. The term denotes the monochromatic cross section, which is the quantity to be determined. corresponds to the number of detected neutrons, while is the number of γ-rays incident on the target above . The average FED efficiency, , is derived using the RR technique [35]. represents the number of target nuclei per unit area. The correction factor in the target self-attenuation term is given by , where μ is the attenuation coefficient for incident γ-ray, and t is the target thickness. Finally, g is the fraction of the γ-ray flux above (see Fig. 3(a)). Its expression is given by

2

Note that the measured quantities described above are commonly referred to as monochromatic approximations. In reality, they correspond to the convolution of the true energy-dependent cross sections with the spectral distribution of the incident photon beam. Specifically, the measured cross section represents the folding of the excitation function with the γ-ray energy spectrum, which is the monochromatic approximation cross section in the experiment.

To obtain the energy-dependent photoneutron cross section , it is essential to extract it from the integral form given in Eq. (1). Each measured cross section represents a convolution of the true cross section with the normalized incident beam profile . In this study, we adopted the unfolding algorithm proposed in Ref. [38] to extract from Eq. (1):

3

where the quantity represents a monochromatic approximated cross section array with each element corresponding to the monochromatic approximated cross section measured at discrete beam energies ( ). The array σ contains the monochromatic cross sections. The matrix D is constructed from the normalized spectral distribution of the incident γ-ray beams with energy spanning from to at each independent measurement. Eq. (4) is the expanded form of Eq. (3). The number of rows (N) in D corresponds to the number of discrete beam energies, while the number of columns (M) represents the number of bins within the incident γ-ray spectrum. For the case, we choose N = 25 and M = 1000.

4

An unfolding iteration method is employed to extract the monochromatic cross section σ:

(1) As our starting point, we choose a constant trial function for the zeroth iteration. This initial vector is multiplied with D , and we get the zeroth folded vector = .

(2) The next trial input function, , is established by adding the difference of the experimentally measured spectrum, , and the folded spectrum, , to :

5

It is worth noting that the dimension of (M) is much larger than those of and (N). To solve Eq. (5), it is necessary to expand the dimensions of and to M by interpolation.

(3) The above steps are iterated i times, giving

6

and

7

The value between and is recorded in each iteration. The iterative process is stopped once convergence is achieved. This means that within statistical uncertainties.

Figure 4 presents the monochromatic cross sections and unfolded energy-dependent cross sections for the (γ, n) reaction. The values at each , together with their uncertainties, are summarized in Table 3. These uncertainties account for statistical, systematic, and methodological contributions. In our case, the total uncertainty of the unfolded is approximately 4%, except in the low-energy region of MeV, where drops below 12.10 mb.

We compare the currently measured data for the (γ, n) reaction with available experimental data from the EXFOR database [12−15] and TENDL-2023 evaluations. The results are shown in Fig. 5(a). Because earlier measurements employed different types of γ-ray sources together with different experimental techniques, their results are analyzed and discussed separately. To quantitatively analyze these differences, we further calculate the ratios of the data obtained in the current study to the experimental data shown in Fig. 5(b).

Katz et al. [12] reported photoneutron cross sections at the SANS laboratory in Canada in 1951, using bremsstrahlung photons with endpoint energies between 11.00 and 22.00 MeV. Antonov et al. [13] performed a similar measurement at JINR in Russia with bremsstrahlung radiation endpoint energies ranging from 14.68 to 24.55 MeV. Compared with the results in the present work, both datasets show substantial deviations at lower energies and within the GDR peak region. For example, a significant difference exists in their GDR peak positions, magnitudes, and widths.

Fultz et al. [14] measured the (γ, n) data at the Livermore Laboratory in 1964 using quasi-monoenergetic annihilation photons and a BF₃-filled FED neutron detector, covering the energy range of 9.34−27.78 MeV. This measurement exhibits a good agreement with our data below 15 MeV, although the data of Fultz et al. remain systematically lower than our data by a factor of approximately 0.4 in the GDR region.

Recently, Jiao et al. [15] performed the photoneutron cross section measurement with a natural Cu target using the quasi-monoenergetic γ-ray beams at the SLEGS. Combining the measured natural Cu data and previously measured data [39], the (γ, n) cross sections within the energy range of 11.28−17.58 MeV were successfully extracted using the subtraction method [15]. Their result shows good agreement with the present measurement using an isotopic target, as shown in Fig. 5.

We should note that early measurements with bremsstrahlung γ-ray sources [12, 13] suffered from limited accuracy due to the broad photon spectra and additional uncertainties introduced in the unfolding procedures. Although the Livermore data were obtained using quasi-monoenergetic annihilation photons, systematic discrepancies still appear when compared with the Saclay measurements that employed the same type of γ-ray beams. The SLEGS experiment [15] did not measure the cross sections near of , which led to a slight difference between the monochromatic cross sections obtained through the subtraction method and those from the present experiment.

γSF [40] is a nuclear statistical quantity used to describe the nuclear electromagnetic response. In the de-excitation mode, γSF quantifies the average probability of a nucleus emitting a γ-ray of a given energy during de-excitation, which we refer to as downward γSF and can be defined by [41]

8

Here, the symbol X is either electric (E) or magnetic (M), is the average radiation width, and is the average level spacing for s-wave ( = 0) or p-wave ( = 1) neutron resonances.

In contrast, γSF in the excitation mode, which we refer to as upward γSF, is defined by the average cross section for / photoabsorption to the final states with all possible spins and parities [42]:

9

Here, is the planck constant, and c is the speed of light in vacuum. Above , except at energies very close to the reaction threshold, the total upward γSF can be approximated by replacing with experimental (γ, n) cross sections, which dominate the photoabsorption cross section in this energy region. This approach relies on the principle of the detailed balance [43] and on the generalized Brink hypothesis, which assumes that the average electromagnetic decay process (i.e., photo-deexcitation) can be directly related to the inverse photo-excitation and depends only on the energy of the emitted γ-ray, irrespective of the absolute excitation energy or nuclear structure properties such as spin and parity. This assumption is expressed as and provides the theoretical foundation for linking the (γ, n) cross section to the downward γSF:

10

where the constant . Using this relation, we can obtain the experimentally constrained γSF from the measured (γ, n) data, as shown in Fig. 6. Note that Eq. (10) holds only when the neutron channel in the photoneutron data dominates. In the vicinity of , the competing γ emission must be taken into account through the HF formalism.

We further performed a theoretical analysis for γSF using the TALYS software (version 2.0) [44]. Here, the strength was described by the macroscopic Brink–Axel Lorentzian (BAL) model [45, 46], and the strength was described using the default spin-flip and scissors mode [47] in TALYS. These results are also shown in Fig. 6. It is found that the theoretical γSF, by default, and experimental data have noticeable discrepancies. This suggests that additional constraints are required to further improve the theoretical description.

To improve the agreement between the theoretical calculation and experimental data, the Lorentzian parameters , , and in TALYS calculations were adjusted simultaneously by a Bayesian optimization procedure [48, 49]. The parameter spaces were chosen in a proper way such that the calculated cross sections encompass the experimental upper and lower limits. Within this parameter space, we try to identify an optimal parameter set that best reproduces the experimentally extracted γSF, which corresponds to a minimized value for chi-square ( ) between the TALYS calculations and experimental data. In our case, reads

11

where N is the number of experimental data points, and , , and represent the theoretical data, experimental data, and uncertainty of experimental data, respectively. In our case, a Gaussian process was employed as a surrogate model to describe the dependence of on model parameters in TALYS, assuming that these values follow a Gaussian distribution. At each iteration, the next candidate point was selected by maximizing the expected improvement acquisition function. Then, an optimal parameter set was determined as the one that yields a globally minimum among all data points. As the BAL model is employed, the value for reaches a minimum of 9.96 when the Lorentzian parameters = 139.31, = 16.63, and = 4.37. One can see that the optimized calculation reproduces the experimental data of γSF well (see Fig. 6).

The constrained γSF is further applied to calculate neutron capture cross sections for the inverse reaction (n, γ) . Note that, in addition to the constrained γSF, the above calculation is highly sensitive to the nuclear level density (NLD) in TALYS. As a result, we first check the level density data considering different NLD models. Fig. 7 shows six kinds of level densities for , which are predicted by available NLD models: CTM [50], BFM [51], GSM [52, 53], HF-BCS [54], Skyrme-HFB [55], and Gogny-HFB [56]. One can see that these level density curves show an overall agreement, whereas a slight variation appears when the excitation energy is above 7 MeV.

By using the adjusted BAL model ( ) and the default spin-flip and scissors mode ( ), we further calculate radiative neutron capture cross sections for . The resulting (n, γ) data are shown in Fig. 8. The shaded band in the figure reflects the data uncertainty arising from different NLD models [38, 57]. Evaluated data from the EAF-2010, ENDF/B-VIII.0, JEFF-3.3, TENDL-2023, and JENDL-5 libraries are also presented in Fig. 8 for comparison. One can see that the data obtained within the BAL model show good agreement with the EAF-2010, TENDL-2023, and JENDL-5 evaluations, whereas they are higher than the evaluated data from both ENDF/B-VIII.0 and JEFF-3.3.

According to the (n, γ) data presented above, the corresponding astrophysical reaction rates were calculated. Fig. 9(a) shows the (n, γ) reaction rate as a function of astrophysical temperature, together with the data from the JINA REACLIB database (using the recommended label ths8) [58]. To facilitate comparison, the ratios between the calculated reaction rates and those from from the JINA REACLIB database [58] are shown in Fig. 9(b). In our case, the data uncertainty is also induced by the NLD models, which is not constrained in a proper way due to the lack of experimental (n, γ) data. The comparison shows that our results agree very well with the JINA REACLIB database at astrophysical temperatures over , while they are slightly lower at lower temperatures.