The experimental-theoretical method

An experimental-theoretical method independent of neutron multiplicity sorting was proposed in Ref. [28, 29] to obtain partial photoneutron cross sections free of systematic uncertainties. The method is based on using neutron yield reaction cross section σ(γ, xn) data as the initial experimental information,(1)and for the total photoneutron reaction cross section σ(γ, sn),(2)T. Kawano et al. [23] pointed out when the emission of charged particles is negligible in experimental photonuclear reaction, the measured one-neutron emission cross section is identical to that for the production of the (Z, A-1) nucleus. In this case, the measured one-neutron emission cross section σ(γ, 1nX),(3)where X stands for anything except i-neutrons.

Figure 1 compares the theoretical and experimental data of σ(γ, 1nX) and σ(γ, 2nX) for ¹⁴⁴Nd. The experimental data were obtained using quasimonoenergetic annihilation photons and a Gd liquid scintillation detector at the Saclay laboratory, while the theoretical data were calculated using the modern Hauser-Feshbach nuclear reaction code TALYS (version 1.96) [30]. The discrepancy between the theoretical and experimental data may originate from substantial systematic uncertainties associated with the neutron multiplicity sorting method employed at Saclay.

Fig. 1Full-screen View

(Color online) (a) (γ, 1nX) reaction, (b) (γ, 2nX) reaction. Comparison of the experimental cross sections for ¹⁴⁴Nd measured at the Saclay laboratory with TALYS calculations using default parameters

Due to these systematic uncertainties, Varlamov et al. [29] introduced the transition multiplicity function Fi, defined as a reliability criterion for partial photoneutron cross sections in the form of a ratio,(4)to facilitate the evaluation of the experimental partial photoneutron cross sections.

According to Eq. (4), values such as F₁>1.0 or F₂>0.50 cannot be considered reliable. Fi values larger than the mentioned top limits indicate that the experimental sorting of neutrons between partial reactions has been carried out with large systematic uncertainties; therefore, the obtained reaction cross sections are not reliable. It should also be emphasized that, because Fi is defined purely as a ratio of cross sections, its values must always be positive.

The evaluated partial photoneutron cross sections σ^eval(γ, inX) are obtained by multiplying the experimental photoneutron yield cross section σ^exp(γ, xn) given in Eq. (1) by the theoretical functions computed with the theoretical code,(5)where the σ^th(γ, xn) is the theoretical photoneutron yield cross section and the σ^th(γ, inX) is the theoretical partial photoneutron reaction cross section.

The differences between the experimental and the evaluated cross sections were determined separately for reactions (γ, 1nX) and (γ, 2nX),(6)(7)In Refs. [31-36], the experimental partial photoneutron cross sections obtained by quasimonoenergetic annihilation photons of many atomic nuclei (^90,91,92,94Zr, ¹¹⁵In, ^112-124Sn, ¹⁵⁹Tb, ^{186,188,189,190,192}Os, ²⁰⁸Pb, etc.) were analyzed using an experimental-theoretical method, and obtained new evaluation data.

γ strength function models

The key to obtaining the theoretical Fi values is to determine the photoneutron yield cross sections of ^{142-146,148,150}Nd. These cross sections were calculated with common classical γ strength function models in TALYS, and the results were compared by means of χ² analysis,(8)where N is the total number of experimental points, σ_th, σ_exp and σ_err are the theoretical value of the neutron yield cross sections, the experimental measurement value of the neutron yield cross sections, and the uncertainty of the experimental measurement value, respectively.

The theoretical photoneutron yield cross sections σ(γ, xn) of ¹⁴⁴Nd, calculated using Eq. (1), are shown in Fig. 2, along with the experimental data [37-47]. γ-ray strength functions play a crucial role in describing transitions involving γ rays in nuclear reactions [48, 49]. Figure 2 demonstrates that the choice of γ-ray strength function has a considerable impact on the calculated photoneutron yield cross sections.

Fig. 2Full-screen View

(Color online) The experimental photoneutron yield cross sections σ(γ, xn) of ¹⁴⁴Nd obtained at Saclay laboratory and compared with the ten γ strength function models calculations in TALYS (1, Kopecky-Uhl generalized Lorentzian [37], 2, Brink-Axel Lorentzian [38, 39], 3, Hartree-Fock-Bogoliubov tables [40], 4, Hartree-Fock BCS tables [41], 5, Goriely’s hybrid model [42], 6, Goriely T-dependent HFB [43], 7, T-dependent RMF [44], 8, Gogny DIM HFB+QRPA [45], 9, SMLO [46], 10, Skyrme HFB+QRPA [47])

As shown in Fig. 3, for , the Brink-Axel Lorentzian (BAL) model is very close to Goriely’s T-dependent HF model and Goriely’s hybrid model, while for , the BAL and the SMLO models describe the data equally well. The BAL model generally results in the lowest χ² values over all isotopes, with only a minor deviation at A = 150, where Goriely’s hybrid model performs marginally better. The average χ² value obtained using the BAL model is 11.41. To ensure consistent and systematic treatment across the entire isotopic chain, the BAL model was adopted in the present analysis. A quantitative comparison shows that the average relative deviations in neutron multiplicity functions (F₁, F₂) between other γSF models and BAL are 10.60% 12.53% 10.69% 9.21% 8.79% 13.59% 7.37% 6.86% and 10.15% for models 1 through 10 (excluding 2), indicating that moderate differences exist among the models and warrant consideration in model-based evaluations.

Fig. 3Full-screen View

(Color online) The χ² values obtained with the experimental photoneutron yield cross sections σ(γ, xn) and compared with the ten γ strength function models in TALYS

The Brink-Axel Lorentzian model is based on the Brink-Axel hypothesis [50], which states that the photon absorption cross section is independent of the excitation energy of a nuclear system and is an assumption used in nuclear structure studies and calculations. This model offers a solid theoretical basis for our analysis, balancing the reliability and precision of neutron interaction modeling.

The parameters of γ-strength function models for each nuclide are listed in Table 1, where E, Γ and σ are the energy center value, width, and strength of the formant, respectively.

Using the Brink-Axel Lorentzian model in TALYS, the theoretical neutron multiplicity functions were obtained using Eq. (4) and are shown in Fig. 4. These results were compared with the experimental neutron multiplicity functions derived from the ¹⁴⁴Nd data measured at the Saclay laboratory.

Fig. 4Full-screen View

(Color online) The experimental neutron multiplicity functions F₁ (a) and F₂ (b) for the isotope ¹⁴⁴Nd, obtained from data measured at the Saclay laboratory, are compared with TALYS calculations performed with default parameters, except that the γ strength function was chosen by the BAL model

Up to the (γ, 2nX) reaction threshold of S_2n=13.94 MeV, remains equal to 1. Once the (γ, 2nX) channel opens, diminishes in correspondence with the competition from the growing σ(γ, 2nX) and shrinking σ(γ, 1nX) cross sections, eventually approaching zero.

There is a clear discrepancy between the experimental and calculated values of F₁ and F₂. The value is consistent with value only when the γ energies are below the two-neutron separation energy S_2n. For γ energies above S_2n, the shows an overestimated trend and shows an underestimated trend. This suggests that in the Saclay (γ, xn) measurements of Nd isotopes, (γ, 2nX) events may have been erroneously classified as (γ, 1nX) events.

Evaluation of isotopes ^{144-146,148,150}Nd

For the isotope ¹⁴⁴Nd, a comparison between σ^eval(γ, inX), σ^exp(γ, inX) and the existing evaluation data is shown in Fig. 5. The Fig. 5(a) and 5(b) indicate the comparison of ¹⁴⁴Nd (γ, 1nX) reaction and ¹⁴⁴Nd (γ, 2nX) reaction, respectively. The Fig. 5(c) shows that the values of σ^exp(γ, inX) - σ^eval(γ, inX). The integrated cross section σ^int containing the energy ranges of only 1n and 1n + 2n are obtained and presented in Table 2.

Fig. 5Full-screen View

(Color online) The comparison between σ^eval(γ, inX), σ^exp(γ, inX) and the existing evaluation data; (a) the case of ¹⁴⁴Nd (γ, 1nX) reaction; (b) the case of ¹⁴⁴Nd (γ, 2nX) reaction; (c) the values of differences Δσi, i=1, 2

The average relative deviations (ARD) between the experimental/evaluated results and the cross sections from nuclear data libraries are defined as(9)(10)Here, n denotes the number of experimental points, σ_exp is the experimental cross section, σ_eval is the evaluated cross section obtained in this study, and σ_lib represents the cross section values from the evaluated nuclear data libraries. The effectiveness level of the present evaluation is defined as the difference between the two ARD values, let effectiveness level = ARD₁ − ARD₂.

As shown in Fig. 5(a) and (b), for the isotope ¹⁴⁴Nd, compared with the Saclay measurements σ^exp(γ, inX), the evaluated cross sections σ^eval(γ, inX) show improved consistency with JENDL-5, CENDL-Beta, and TENDL-2023, whereas their agreement with IAEA-2019 is comparatively poorer. These results demonstrate the effectiveness of the proposed evaluation method. For (γ, 1nX) case, the effectiveness levels for databases of JENDL-5, CENDL-Beta, TENDL-2023 and IAEA-2019 are 3.75%, -5.52%, 14.11%, and -37.21% respectively; For (γ, 2nX) case, the effectiveness levels are 22.76%, 3.27%, -27.74%, and -67.67%, respectively.

The differences between the evaluated and experimental cross sections were determined separately for the reactions (γ, 1nX) and (γ, 2nX), as shown in Fig. 5(c). At energies below the threshold S_2n of reaction (γ, 2nX), where there is little problem in neutron multiplicity sorting, the difference between the experimental and theoretical integrated cross section σ(γ, 1nX) is only 0.05% (532.70 mb and 532.40 mb, respectively). But at high energies where reactions (γ, 1nX) and (γ, 2nX) compete with each other range 13.80 to 20.21 MeV, the data on both differ markedly: mb, which is 27% smaller than (784.88 mb). mb, which is 19% larger than (which is 563.28 mb). Such large and opposite-direction divergences between the cross sections of reactions (γ, 1nX) and (γ, 2nX) convincingly demonstrate the reasons for the substantial systematic uncertainties in the experiments at the Saclay laboratory, which are due to the unreliable transmission of a large number of neutrons from channel 2n to channel 1n.

Similar to the conclusions for ¹⁴⁴Nd, the evaluated cross section data for the (γ, 1nX) and (γ, 2nX) reactions of ^{145,146,148,150}Nd were compared with the corresponding experimental results and the values from the evaluated nuclear data libraries, as shown in Fig. 6. The relative differences between integrated σ^exp(γ, 1nX) and σ^eval(γ, 1nX) are 36%, 48%, 26%, 108%, respectively. For (γ, 2nX), are 14%, 17%, 5%, 21%, respectively. The relative average differences between the evaluated cross sections and those from various nuclear data libraries (JENDL-5, TENDL-2023, IAEA-2019, and CENDL-Beta) also exhibited significant variation for ^{145,146,148,150}Nd. As shown in Table 3, consistent with the conclusions of ¹⁴⁴Nd isotope, for the isotopes ^{145,146,148,150}Nd, the values of σ^eval(γ, inX) are closer to those of JENDL-5, CENDL-Beta, and TENDL-2023, whereas the experimental data σ^exp(γ, inX) are more consistent with IAEA-2019.

Fig. 6Full-screen View

(Color online) The comparison between σ^eval(γ, inX), σ^exp(γ, inX) and the existing evaluation data; (a1) - (a4) the case of ^{145,146,148,150}Nd (γ, 1nX) reaction; (b1) - (b4) the case of ^{145,146,148,150}Nd (γ, 2nX) reaction

Evaluation of isotope ¹⁴³Nd

For the isotope ¹⁴³Nd, a comparison between σ^eval(γ, inX), σ^exp(γ, inX) and the existing evaluation data is shown in Fig. 7. The Fig. 7(a) and 7(b) indicate the comparison of ¹⁴³Nd (γ, 1nX) reaction and ¹⁴³Nd (γ, 2nX) reaction, respectively. The Fig. 7(c) shows that the values of σ^exp(γ, inX) - σ^eval(γ, inX). The integrated cross section σ^int containing the energy ranges of only 1n and 1n + 2n are obtained and presented in Table 4. Because the experimental (γ, 2nX) cross section still exists at energies below the S_2n threshold, the of the (γ, xn) reaction is larger than that of the (γ, 1nX) reaction when E^int = 9.31–15.71 MeV.

Fig. 7Full-screen View

(Color online) The comparison between σ^eval(γ, inX), σ^exp(γ, inX) and the existing evaluation data; (a) the case of ¹⁴³Nd (γ, 1nX) reaction; (b) the case of ¹⁴³Nd (γ, 2nX) reaction; (c) the values of differences Δσi, i=1, 2

As shown in Fig. 7(a) and (b), a similar trend is observed for ¹⁴³Nd, where σ^eval(γ, inX) shows closer agreement with JENDL-5, CENDL-Beta, and TENDL-2023 than with IAEA-2019 when compared to the experimental data σ^exp(γ, inX). This further confirms the effectiveness of the evaluation method used in this study. For (γ, 1nX) case, the effectiveness levels for databases of JENDL-5, CENDL-Beta, TENDL-2023 and IAEA-2019 are 3.43%, 2.72%, 2.93%, and -1.55% respectively; For (γ, 2nX) case, the effectiveness levels are 35.15%, 23.86%, 29.61%, and -67.91%, respectively.

The differences Δσ between the experimental and evaluated cross sections [Fig. 7(c)] obtained for partial reactions seem to be ‘mirrored’. Almost all values of (γ, 1nX) are negative, whereas those of (γ, 2nX) are positive. When the energy range 14.08 to 19.80 MeV, σ^exp-σ^eval is opposite: mb, which is 16% greater than (which is 524.94 mb). mb, which is 20% smaller than (which 178.35 mb). For the isotope ¹⁴³Nd, it is interesting to find that Δσ₁ is greater than zero and Δσ₂ is less than zero at these energies. However, the results at 18.71 MeV and 19.80 MeV exhibit opposite behaviors in Δσ₁ and Δσ₂.

Evaluation of isotope ¹⁴²Nd

For the isotope ¹⁴²Nd, a comparison between σ^eval(γ, inX), σ^exp(γ, inX) and the existing evaluation data is shown in Fig. 8. The Fig. 8(a) and (b) indicate the comparison of ¹⁴²Nd (γ, 1nX) reaction and ¹⁴²Nd (γ, 2nX) reaction, respectively. The Fig. 8(c) shows that the values of σ^exp(γ, inX) - σ^eval(γ, inX). The integrated cross sections σ^int for the energy ranges of only 1n and 1n + 2n are obtained and presented in Table 5. Similar to ¹⁴²Nd, since the experimental (γ, 2n) cross sections were still measured below the S_2n threshold, the of the (γ, xn) reaction is larger than that of the (γ, 1nX) reaction when E^int = 9.85–17.75 MeV.

Fig. 8Full-screen View

(Color online) The comparison between σ^eval(γ, inX), σ^exp(γ, inX) and the existing evaluation data; (a) the case of ¹⁴²Nd (γ, 1nX) reaction; (b) the case of ¹⁴²Nd (γ, 2nX) reaction; (c) the values of differences Δσi, i=1, 2

As shown in Fig. 8(a) and (b), for the isotope ¹⁴²Nd, the evaluated cross sections σ^eval(γ, inX) exhibit improved agreement with all four major nuclear data libraries compared with the experimental data σ^exp(γ, inX). These results provide further evidence of the applicability of the evaluation method. For (γ, 1nX) case, the effectiveness levels for databases of JENDL-5, CENDL-Beta, TENDL-2023 and IAEA-2019 are 0.64%, -2.21%, 0.95%, and 0.67% respectively; For (γ, 2nX) case, the effectiveness levels are 43.15%, 13.61%, 38.71%, and 12.90%, respectively.

As shown in Fig. 8(c) and Table 5, the values of σ^eval(γ, inX) are very close to σ^exp(γ, inX) when the γ-ray energy is lower than 17.75 MeV. When the energy is between 17.75 and 20.21 MeV, mb which is 9% smaller than (226.08 mb). mb, which is 14% larger than (46.53 mb). Notably, as shown in Fig. 8(c), the values of Δσ₁ are less than zero before 18.57 MeV and greater and becomes positive above this energy. The situation for Δσ₂ is the opposite.