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Our previous study has revealed the crucial role of pairing correlation in fission barriers of Th, U, and Pu isotopes[17]. The exactly solvable pairing model has been shown to be effective in investigating the properties of fissioning systems. In this paper, we use the efficient new iterative algorithm to systematically analyze the fission barriers of Th, Pa, U, Np, Pu, Am, Cm, Bk, and Cf isotope chains using the exactly solvable pairing model. The potential energy is calculated in the macroscopic-microscopic frame, using the Cassini ovaloids shape parametrization combined with the liquid drop model + Woods-Saxon potential[24].
Geometrically, the Cassini ovaloids which is proposed by Pashkevich[24–30] are obtained by rotating the curve around the
$ z $ axiswhere
$ z $ and$ \rho $ being cylindrical coordinates,$ R_{0} $ is the radius of the spherical nucleus. The constant$ a $ is determined by volume conservation, which means that the family of shapes in Eq. (11) is only related to the elongation deformation parameter$ \epsilon $ .The deviation of the nuclear surface from Cassini ovaloids is defined by expansion of
$ R(x) $ in series in Legendre polynomials$ P_{m}(x) $ [24, 26],In the mac-mic framework, the total energy of a nucleus can be written as the sum of macroscopic and microscopic terms as follows,
where the macroscopic term
$ E_{\rm{LD}}(N, Z) $ is approximated by the standard liquid drop model with neutron number$ N $ and proton number$ Z $ . When calculating the potential energy surface, we only consider the energy$ E_{\rm{B}}(N, Z, \epsilon, \alpha_{m}) $ with the shape parameter$ \{\epsilon, \alpha_{m}\} $ .Here, the deformation correction energy is defined as
$ E_{\rm{def}}(N, Z, \epsilon, \alpha_{m}) = [B_{s}-1]E_{s}^{0}+[B_{c}-1]E_{c}^{0} $ , where$ B_{s} $ and$ B_{c} $ are functions of the$ \{\epsilon, \alpha_{m}\} $ [24]. The spherical surface energy is written as$ E_{s}^{0} = \alpha_{s}A^{\frac{2}{3}} $ with$ \alpha_{s} = 16 $ MeV.$ E_{c}^{0} = \alpha_{c}\frac{Z^{2}}{A} $ with$ \alpha_{c} = 0.71 $ MeV, in which$ A $ is the mass number, is the spherical Coulomb energy. The microscopic terms consist of two terms, the shell correction energy$ E_{\rm{shell}}^{\nu(\pi)}(N, Z, \epsilon, \alpha_{m}) $ proposed by Strutinsky[31-32]. Here, we consider$ 200 $ single-particle levels for the shell correction energy calculations. The pairing correction energy$ E_{\rm{pair}}^{\nu(\pi)}(N, Z, \epsilon, \alpha_{m}) $ is obtained from Eq. (1) with doubly-degenerate levels,$ \nu (\pi) $ denotes the neutron (proton) pairs. For the pairing correction energy, we perform$ 29 $ single-particle levels around the neutron Fermi level and$ 16 $ single-particle levels around the proton Fermi level. The multi-dimensional potential energy surface is minimized in the elongation parameter$ \epsilon $ and the multipole deformation parameters ($ \alpha_{m}, m = 3, 4, 5, 6 $ ) simultaneously.Generally, the pairing interaction strength is determined by the empirical formula or by fitting the odd-even mass differences[33-34]. The recent study[17] reveals that pairing correlation plays an essential role in the fission barrier height. Therefore, the odd-even mass differences (ground-state property) and the height of barriers (excited states property) should be used as experimentally observable quantities to determine the values for pairing interaction strength in the fission process. In this work, realistic values of pairing interaction strengths for isotope chains considered are obtained by fitting the experimental odd-even mass differences and the heights of the inner and outer barriers. The odd-even mass difference is defined as follows:
Figure 1 shows that our model calculations accurately reproduce the experimental odd-even mass differences for 235-245Pu and 241-250Cm isotopes with pairing strengths of
$ G^{\nu} = 0.12 $ MeV and$ G^{\pi} = 0.18 $ MeV, respectively. However, for the 231-239U isotopes, theoretical results show a deviation from the experimental data. To investigate this further, we analyzed the odd-even mass differences and the height of barriers for the 231-239U isotopes using pairing interaction strengths of$ G^{\nu} = 0.12(G^{\pi} = 0.18) $ MeV and$ G^{\nu} = 0.15(G^{\pi} = 0.18) $ MeV, as shown in Fig. 2. We found that the odd-even mass differences for the 224-238U isotopes were reproduced remarkably well by the model calculations with the pairing interaction strength$ G^{\nu} = 0.15 (G^{\pi} = 0.18) $ MeV, as shown in Fig. 2(a). On the other hand, as shown in Fig. 2(b) and (c), the results of the inner and outer fission barriers with$ G^{\nu} = 0.12(G^{\pi} = 0.18) $ MeV calculated in the current model are very close to the corresponding experimental values. In this work, taking into account the calculation results of the odd-even mass differences and the height of barriers in Fig. 2, we determined that the realistic values of pairing interaction strengths for the U isotope chains considered are$G^{\nu} = 0.12 (G^{\pi} = 0.18)$ MeV.Figure 1. The odd-even mass differences (in MeV) for 231-239U235-245Pu and 241-250Cm. Experimental values are denoted as "Expt.", which are taken from Ref. [35], and theoretical values calculated in the exactly solvable pairing model are denoted as "Theor.". (color online)
Figure 2. The odd-even mass differences (in MeV), inner and outer fission barriers heights for U isotopes with given pairing interaction strengths. Experimental values are denoted as "Expt." and the theoretical values calculated in the present model are denoted as "Theor.". Experimental data are taken from Refs. [35, 36] (in MeV). A typical uncertainty in the experimental values, as suggested by the differences among various compilations, is of the order of ±0.5 MeV[36]. (color online)
In this paper, we calculate the root-mean-square deviation
$ \sigma $ (in MeV) of the odd-even mass differences to measure the overall variation in a data set. By analyzing the variation of$ \sigma $ caused by increasing pairing interaction strengths, we demonstrate that the odd-even mass difference can be used as an experimentally observable quantity to determine the realistic values of pairing interaction strength. The root-mean-square deviation$ \sigma $ is defined asin which
$ { E}_\mu^{\rm{Theor.}} $ are the theoretical values,$ {E}_\mu^{\rm{Expt.}} $ are the corresponding experimental estimates.$ {\cal{N}} $ is the number of the experimental data involved. Table 1 displays that the root-mean-square deviation decreasing from$\sigma_{\rm{U}} \sim 0.432$ MeV to$\sigma_{\rm{U}} \sim 0.329$ MeV by increasing the pairing strengths$ G^{\pi} $ from$ 0.10 $ to$ 0.12 $ MeV and$ G^{\nu} $ from$ 0.15 $ to$ 0.18 $ MeV. For Pu isotopes, when$ G^{\nu} = 0.15 $ MeV and$ G^{\pi} = 0.10 $ MeV, the theoretical value of the odd-even mass difference deviates from the experimental results with the root-mean-square deviation$ \sigma_{\rm{Pu}} = 0.223 $ MeV. By increasing the strengths to$ G^{\nu} = 0.12 $ MeV and$ G^{\pi} = 0.18 $ MeV,$ \sigma_{\rm{Pu}} $ decreases to$ 0.060 $ MeV. For Cm isotope chain, the value of$ \sigma_{\rm{Cm}} $ changes in a small region.$ G^{\nu} = 0.10 $ MeV and$ G^{\pi} = 0.18 $ MeV seem to yield a better result. The results in Table 1 observed that the odd-even mass difference can serve as one of the effective quantities to determine the neutron and proton pairing strengths.$ G^{\nu} = 0.10 $ $ G^{\nu} = 0.10 $ $ G^{\nu} = 0.12 $ $ G^{\pi} = 0.15 $ $ G^{\pi} = 0.18 $ $ G^{\pi} = 0.18 $ $ \sigma_{\rm U} $ 0.432 0.430 0.329 $ \sigma_{\rm Pu} $ 0.223 0.046 0.060 $ \sigma_{\rm Cm} $ 0.027 0.019 0.033 Table 1. Root-mean-square deviations
$ \sigma $ (in MeV) of the odd-even mass differences in given pairing strengths (in MeV) for the 231-239U, 235-245Pu and 241-250Cm isotopes. -
In our study, published in Ref. [17], we conducted a systematic analysis of fission barriers and static fission paths in Th, U, and Pu isotopes using the deformed mean-field plus standard pairing model. Our results indicate that the pairing interaction plays different roles in different stages of the fission processes. Specifically, for 226Th, we found that the neutron pairing significantly affects the height of the inner barrier, while proton pairing has a greater effect on the outer barrier height. Building on this analysis, we extended our study to include a systematic investigation of fission barriers in nuclei spanning 227-232Th, 230-234Pa, 231-240U, 233-238Np, 235-245Pu, 239-245Am, 244-249Bk, 241-250Cm, and 250-253Cf isotopes. By varying the pairing strength under our current model, we aimed to further elucidate the role of pairing interactions in the height of barrier. The results in Table 2 show that the height of the inner barrier is significantly affected by neutron pairing, while proton pairing affects the height of the outer barrier in Th, Pa, and U isotopes. For example, the root-mean-square deviation of the inner barrier changes by 30.9% in Th isotopes when the neutron pairing strengths
$ G^{\nu} $ is varied by approximately 20%. In contrast, a variation of the proton pairing strengths$ G^{\pi} $ by the same amount results in a 10.3% change in the root-mean-square deviation of the inner barrier. For the outer barrier height, neutron pairing has a 5.9% effect on the root-mean-square deviation, while proton pairing has a 40.0% effect when varied by approximately 20%.Nuclei $ \sigma_{\rm{inner}} $ $ \sigma_{\rm{inner}} $ $ \sigma_{\rm{outer}} $ $ \sigma_{\rm{outer}} $ $G^{\nu} \approx 20\%$ $G^{\pi} \approx 20\%$ $ G^{\nu}\approx 20\% $ $G^{\pi} \approx 20\%$ 227-234Th 30.9% 10.3% 5.9% 40.0% 230-234Pa 35.5% 16.1% 19.8% 39.3% 231-240U 125.6% 41.4% 12.8% 57.1% 235-245Pu 24.6% 22.0% 6.52% 10.9% 241-250Cm 28.7% 65.5% 22.3% 4.9 % 250-253Cf 67.7% 105% 77.5% 37.6% Table 2. Percentage change of root-mean-square deviation
$ \sigma $ (in MeV) of the theoretical barrier heights as compared to the experimental value for the 227-232Th, 230-234Pa, 231-240U, 235-245Pu, 241-250Cm and 250-253Cf isotopes when the neutron pairing strength ($ G^{\nu} $ ) and proton pairing strength ($ G^{\pi} $ ) are varied by approximately 20%.In contrast, for the Pu isotopes, we found that the height of the inner and outer barriers is affected equally by neutron and proton pairing. Specifically, a variation of approximately 20% in the pairing strengths
$ G^{\nu} $ and$ G^{\pi} $ results in a 24.6% and 22.0% change in the height of the inner barrier, respectively, and a 6.52% and 10.9% change in the height of the outer barrier.Interestingly, our analysis of the heavier isotopes of Cm and Cf shows that the role of neutron and proton pairing in fission barrier heights is reversed compared to the Th, Pa, U, and Pu isotopes. Specifically, our results indicate that neutron pairing has a greater influence on the height of the outer barrier, while proton pairing has a greater influence on the height of the inner barrier. For example, when the neutron pairing strength
$ G^{\nu} $ is varied by approximately 20%, the root-mean-square deviation of the inner barrier height changes by 67.7% for Cm isotopes, while a variation of the proton pairing strength$ G^{\pi} $ by the same amount results in a 105.0% change in the height of the inner barrier for Cm isotopes.These findings indicate that the role of neutron and proton pairing in fission barrier heights is not universal across all isotopes. Instead, the effect of pairing interactions can vary significantly depending on the specific isotopes being studied.
Following the previous analysis[17], we advance a systematic study on fission barriers of nuclei 227-232Th, 230-234Pa, 231-240U, 233-238Np, 235-245Pu, 239-245Am, 244-249Bk, 241-250Cm and 250-253Cf isotopes with the pairing strength parameters
$ G^{\nu} = 0.12 $ MeV and$ G^{\pi} = 0.18 $ MeV under the current model. The root-mean-square deviation$ \sigma $ of the fission barrier height is used to measure the overall variation in theoretical values. As shown in Table 3, the deviations of the calculated inner barrier heights from the experimental estimates systematically less than 0.5 MeV$ \sigma_{\rm{inner}}<0.5 $ MeV, except for Th and Pa isotopes. While the calculated outer barrier heights yield an excellent agreement of the experimental data in those isotope chains, namely,$ \sigma_{\rm{outer}}<0.201 $ MeV.Nuclei $ \sigma_{\rm{inner}} $ $ \sigma_{\rm{outer}} $ $ \sigma_{\rm total} $ 227-232Th 2.269 0.462 1.366 230-234Pa 1.449 0.146 0.797 231-240U 0.422 0.290 0.356 233-238Np 0.068 0.211 0.139 235-245Pu 0.362 0.463 0.412 239-245Am 0.171 0.722 0.447 241-250Cm 0.277 0.657 0.467 244-249Bk 0.063 0.367 0.215 250-253Cf 0.214 0.108 0.161 Table 3. Root-mean-square deviation
$ \sigma $ (in MeV) of the theoretical barrier heights as compared to the experimental value for the 227-232Th, 230-234Pa, 231-240U, 233-238Np, 235-245Pu, 239-245Am, 244-249Bk, 241-250Cm and 250-253Cf isotopes.Figures 3~5 shows that the theoretical barrier heights calculated by our model for the actinide isotope chains considered are close to the experimental estimates. The detailed results presented in Table 3 reveal that the deviations of the calculated inner barrier heights from the experimental estimates are systematically less than
$ 0.5 $ MeV ($ \sigma_{\rm{inner}}<0.5 $ MeV), except for Th and Pa isotopes. Moverover, the calculated outer barrier heights show excellent agreement with the experimental data in those isotope chains, with$ \sigma_{\rm{outer}}<0.201 $ MeV.Figure 3. Calculated inner and outer fission barrier heights for 227-232Th, 230-234Pa and 231-240U. Theoretical values are compared to the experimental data in Ref. [36] (in MeV). (color online)
Figure 4. Calculated inner and outer fission barrier heights for 233-238Np, 235-245Pu and 239-245Am. Theoretical values are compared to the experimental data in Ref. [36] (in MeV). (color online)
Figure 5. Calculated inner and outer fission barrier heights for 241-250Cm, 244-249Bk and 250-253Cf. Theoretical values are compared to the experimental data in Ref. [36] (in MeV). (color online)
In many other theoretical studies, the calculated inner barrier heights of light Th and Pa isotopes are also systematically lower than the experimental estimates[36–40]. Based on the analysis above, we conclude that neutron pairing significantly impacts the inner barrier height for Th and Pa. The significant deviation in the inner barrier height of Th and Pa in Fig. 3 may be due to the strong neutron pairing interaction strength adopted. This observation may provide new insights and understanding of the mentioned anomaly for light actinides.
However, we acknowledge that there are conceptual difficulties in comparing calculated and experimental estimates due to the multi-dimensionality problem, i.e., the theoretical description of the fission process relates to a large number of deformation parameters. In addition, our study needs to provide more information on the effect of the proton and neutron correlation on the fission barrier height. Therefore, further verification of this hypothesis is necessary.
In particular, as shown in Fig. 4 for the Pu isotopes and Fig. 5 for the Cm isotopes, the theoretically calculated inner barrier heights exhibit odd-even staggering. Furthermore, as demonstrated in Fig. 6 (red points), the odd-even staggering vanishes when the pairing correction energy is neglected. This result confirms that the odd-even staggering of the inner barrier heights for Pu and Cm isotopes is indeed caused by the pairing interaction.
Figure 6. Calculated inner fission barrier heights for 235-245Pu and 241-250Cm. Theoretical values and theoretical values without the pairing correction energy compared to the experimental data in Ref. [36] (in MeV). (color online)
Based on the results presented in this paper, we recommend using the odd-even mass differences (a ground-state property) and the height of barriers (an excited-state property) as experimentally observable quantities to determine the realistic values for pairing interaction strengths in the fission process. Numerical analysis of these observables can provide further insights into the impact of proton and neutron correlations on the fission barrier height.
Fission Barriers of Actinide Isotopes in the Exactly Solvable Pairing Model
doi: 10.11804/NuclPhysRev.40.2023013
- Received Date: 2023-02-09
- Rev Recd Date: 2023-03-19
- Available Online: 2024-02-04
- Publish Date: 2023-12-20
Abstract: In this paper, we investigate the impact of pairing correlations on the fission barriers of Th, Pa, U, Np, Pu, Am, Cm, Bk, and Cf isotopes using an exactly solvable pairing model. Our results show that the pairing correlation plays a crucial role in determining the fission barrier height. Specifically, we find that the role of neutron and proton pairing in fission barrier heights is not universal across all isotopes, and the exact nature of the interaction depends on the specific isotopes being studied. Our calculated barrier heights are consistent with experimental data, and we propose using the odd-even mass difference(ground-state properties) and barrier height(excited-state properties) as experimentally observable quantities to determine the pairing interaction strengths in the fission process.
Citation: | Xin GUAN, Wanqiu JIANG, Tiancong WANG, Jinhuan ZHENG, Meiyan ZHENG, Feng PAN. Fission Barriers of Actinide Isotopes in the Exactly Solvable Pairing Model[J]. Nuclear Physics Review, 2023, 40(4): 502-510. doi: 10.11804/NuclPhysRev.40.2023013 |