Shell and isospin effects in nuclear charge radii
The shell effect and isospin effect in nuclear charge radii are systematically investigated and a four-parameter formula is proposed for the description of the root-mean-square (rms) charge radii by combining the shell corrections and deformations of nuclei obtained from the Weizsäcker-Skyrme mass model. The rms deviation with respect to the 885 measured charge radii falls to 0.022 fm. The proposed formula is also applied for the study of the charge radii of super-heavy nuclei and nuclear symmetry energy. The linear relationship between the slope parameter of the nuclear symmetry energy and the rms charge radius difference of S - Si mirror pair is clearly observed. The estimated slope parameter is about MeV from the coefficient of the isospin term in the proposed charge radius formula.
pacs:21.10.Ft, 21.65.Ef, 21.10.Dr, 27.90.+b
As one of the basic nuclear properties, the root-mean-square (rms) charge radii of nuclei are of great importance for the study of nuclear structures (1); (2) and nucleus-nucleus interaction potentials (3); (4). On one hand, the rms charge radii of nuclei can be self-consistently calculated by using microscopic nuclear mass models such as the Skyrme Hartree-Fock-Bogoliubov (HFB) model (5); (6) and the relativistic mean-field (RMF) model (7); (8). The HFB21 model (6) can reproduce the 782 measured charge radii (9) with an rms deviation of 0.027 fm. On the other hand, the rms charge radii of nuclei are also frequently described by using mass- and isospin-dependent (or charge-dependent) phenomenological formulas (10); (11); (12); (13); (14); (15). Although these microscopic and phenomenological models can successfully describe the nuclear charge radii of most nuclei, the parabolic charge radii trend in the Ca isotope chain due to the shell closure of and cannot be reasonably well reproduced (2). The shell effect directly influences the deformations of nuclei and thus affects the nuclear rms charge radii. To consider the shell effect, an empirical shell correction term which is a function of the numbers of valence nucleons was introduced in the phenomenological charge radius formulas (9); (15), assuming the proton magic numbers and neutron magic numbers . Obviously, the fine structure of nuclear charge radii for nuclei with semi-magic numbers such as and can not be well described by the parameterized formulas. Microscopic shell corrections and the influence of nuclear deformations should be considered in the formula.
In addition to the shell effect, the isospin effect also plays a role for the nuclear charge radii. The nuclear symmetry energy, in particular its density dependence, has received considerable attention in recent years (16); (17); (18); (19); (20); (21); (22); (23). The nuclear symmetry energy probes the isospin part of nuclear force and intimately relates to the structure character of neutron-rich and neutron-deficient nuclei. The density dependence of nuclear symmetry energy has been extensively investigated by using various models and experimental observables, such as the microscopic dynamics models (16); (17), the nuclear mass models (24); (25); (26); (6); (15), the pygmy dipole resonance (27); (28), the neutron star observations (21); (29); (30), and so on. In particular, the neutron skin thickness of Pb is found to be a sensitive observable to constrain the slope parameter of nuclear symmetry energy at the saturation density, since the linear relationship between the slope parameter and the neutron skin thickness of Pb was clearly observed (20); (31). However, it is difficult to precisely measure the neutron radius of Pb in experiments, which results in large uncertainty of the extracted slope parameter. Comparing with the neutron radii of nuclei, the rms charge radii of nuclei can be measured with relatively high accuracy. It would be helpful if the slope parameter can be determined from the charge radii of nuclei. It is known that in the absence of Coulomb interactions between the protons, a perfectly charge-symmetric and charge-independent nuclear force would result in the binding energies of mirror nuclei (i.e. nuclei with the same atomic number but with the proton number and neutron number interchanged) being identical (32); (33). At this case, the neutron skin thickness of a neutron-rich nucleus approximately equals to the proton radius difference (in absolute value) of this nucleus and its mirror partner. It is therefore interesting to investigate the correlation between the charge radius difference of mirror nuclei and the slope parameter of the nuclear symmetry energy.
In this work, we attempt to propose a phenomenological formula for the global description of the nuclear rms charge radii by combining the deformations and shell corrections of nuclei obtained from the Weizsäcker-Skyrme mass model (24); (25) which is based on the macroscopic-microscopic method together with the Skyrme energy-density functional and mirror constraint from the isospin symmetry.
Based on the consideration of the nuclear saturation property, the nuclear charge radius is usually described by the law: , where is the mass number. Considering the quadrupole and hexadecapole deformations of nuclei, the rms charge radius of a nucleus can be approximately written as (11)
For a better description of the charge radii of light nuclei and nuclei far from the -stability line, the mass- and isospin-dependent radius coefficient was introduced (10), i.e. with the isospin asymmetry . In addition to the mass- and isospin-dependence of nuclear charge radii, it is found that the shell effect also plays a role for some nuclei (15). In Fig. 1, we show the rms charge radii of Ca isotopes and the corresponding shell corrections of nuclei from the Weizsäcker-Skyrme (WS*) mass model (25). One sees that the parabolic trend of the rms charge radii of Ca isotopes between and 28 seems to be consistent with that of the corresponding shell corrections, which implies that considering the shell effect could be helpful for a better description of nuclear charge radii.
Considering the influence of shell effect in nuclei, we propose a modified four-parameter formula for the description of the nuclear charge radius ,
where denotes the shell corrections of nuclei from the WS* mass model with which the 2149 known masses in AME2003 (34) can be reproduced with an rms deviation of 441 keV and the shell gaps for magic nuclei can also be well reproduced. The term in Eq.(2) which is different from the isospin term in the available phenomenological radius formulas will be discussed later. Based on the 885 measured rms charge radii for nuclei (35) with together with the deformations , and shell corrections of nuclei obtained from the WS* mass model (25), and searching for the minimal rms deviation
between the experimental data and model calculations, we obtain the optimal values for the four parameters which are listed in Table 1. Comparing with the rms deviation between the 885 measured radii and the HFB21 calculations (6) which is 0.026 fm, the corresponding result in this work falls to 0.022 fm. With the microscopic shell corrections, the rms deviation of the charge radii can be reduced by . From Fig. 1(a), one sees that the known rms charge radii of Ca isotopes can be reproduced reasonably well.
For the isospin term in the charge radius formulas, the forms , , and were proposed in Refs.(15), (10) and (9), respectively. Here, denotes the corresponding isospin asymmetry of nuclei along the -stability line. We note that the rms deviation can be further reduced by about with a new form , comparing with the result using the form . In Fig. 2, we show the isospin dependence of the relative rms charge radii. Here the relative rms charge radius of a nucleus is given by without considering the influence of nuclear deformations. Comparing with the linear form, the form gives relatively better results for the extremely neutron-rich nuclei since the decreasing trend of the charge radii gradually weakens with the increase of isospin asymmetry. We also note that the value of nuclear radius constant in Eq.(2) which relates to the saturation properties of symmetric nuclear matter and neutron matter is very close to the value (1.2257 fm) proposed in (36).
Based on the proposed four-parameter nuclear charge radius formula, the known experimental data are systematically investigated. In Fig. 3, we show the difference between the experimental data and model calculations for the 885 rms charge radii of nuclei. One sees that the trend of the differences is similar to each other from the two quite different models, which is due to that the obtained deformations of nuclei with the HFB21 mass model are comparable with the results from the WS* mass model. The calculated rms deviation with the proposed formula is only 0.022 fm which is smaller than the results of HFB21 by . Here, we also present the results of the microscopic RMF model in Fig. 4 for comparison. In Ref. (7), the rms charge radii of even-even nuclei with are systematically calculated by using the RMF model with the force NL3. For the measured rms charge radii of 343 even-even nuclei, the rms deviation with the proposed formula is 0.016 fm which is significantly smaller than the result (0.026 fm) of the RMF calculations. For nuclei with charge number and , the results of the proposed formula are better than those of the RMF calculations. It is partly due to that the shell corrections and deformations for nuclei with new magic numbers such as , 16 and for nuclei with sub-shell closure are reasonably well described by the WS* mass model.
In Fig. 5, we show the comparisons of the rms charge radii for Ca, Ni, Zr and Pb isotopes from three models. For the Ca isotopes, neither the HFB21 model nor the RMF model reproduce the trend of the experimental data. For the Ni isotopes, the experimental data are systematically over-predicted by the HFB21 model and under-predicted by the RMF model. For the doubly-magic nucleus Ni, the calculated quadrupole deformation of nucleus with the HFB21 model. For other Ni isotopes, the obtained values from the HFB21 calculations are significantly larger than the results of WS* which results in the over-predicted results for the Ni isotopes. According to the RMF calculations, we note that the binding energies of Ni are systematically under-predicted by about 4 MeV which might affect the reliable description of the rms charge radii. For the Zr isotopes, one sees that the rms charge radii for nuclei with , 56 and 58 can be remarkably well reproduced with the proposed formula since the shell corrections and deformations of these nuclei (with shell or sub-shell closure) are reasonably well described by the WS* mass model. For the neutron-deficient Pb isotopes, the rms charge radii are significantly over-predicted by the HFB21 calculations and the kink at can not be reproduced. The global trend of the rms charge radii for the nuclei in Fig. 5 especially the kinks at the magic numbers can be well described by using the proposed nuclear charge radius formula.
In Fig. 6, we show the predicted rms charge radii with the proposed formula for some super-heavy nuclei. The solid curve denote the results of HFB21 for Hs () isotopes which are comparable with the predictions of this work (the deviations are smaller than 0.05 fm in general). For the super-heavy nuclei 114 and 116, the extracted rms charge radii from the experimental -decay data are and fm (37), respectively. The predicted results in this work for these two nuclei are 6.17 and 6.19 fm, respectively, which are comparable with the extracted results in (37).
In this work, we simultaneously investigate the correlation between the nuclear symmetry energy and the isospin term of the charge radius formula. We study the difference of the rms charge radii between mirror nuclei, such as S - Si pair. Because of the influence of new magic numbers , 16, the calculated deformations of these two nuclei are very small with some different mass models such as the Weizsäcker-Skyrme mass model, the finite range droplet model (38), and the HFB calculations (5) adopting three widely used Skyrme forces SLy4, SkP and SkM*. We systematically calculate the difference of the rms charge radii between S and Si by using the Skyrme Hartree-Fock model with 62 different Skyrme forces in which the corresponding incompressibility coefficient for symmetry nuclear matter MeV and the saturation density fm. From Fig. 7, one can clearly see the linear relationship between the slope parameter and the difference of the rms charge radii . The Pearson’s (linear) correlation coefficient of with for the 62 Skyrme forces is 0.95, which is comparable with the corresponding value (0.97) of with the neutron skin thickness of Pb for the same forces. The values of for measured mirror pairs such as S - Ar and O - Ne are also investigated with the deformed configurational Skyrme Hartree-Fock calculations (39). We note that the obtained linear correlation coefficient for these two pairs are 0.58 and 0.69, respectively, which are significantly smaller than the value for the mirror pair S - Si. It indicates that the linear relationship between and for the two pairs S - Ar and O - Ne are not as good as that for the mirror pair S - Si due to the influence of the new magic numbers , 16. Considering the fact that the experimental uncertainty for the rms charge radius measurement is much smaller than that for the rms neutron radius, precise measurements of the rms charge radii for the pair of mirror nuclei S - Si, especially the unmeasured S, could be very helpful for the extraction of the slope parameter.
Due to the perfectly charge-symmetric and charge-independent nuclear force, the deformations and shell correction of a nucleus approximately equal to those of its mirror nucleus (40); (25). Based on the proposed nuclear charge radius formula, one obtains neglecting the influence of nuclear deformations. For the S - Si mirror pair, the estimated value of is about fm and the corresponding slope parameter is about MeV which is consistent with recently extracted results from the Fermi-energy difference in nuclei (41) and from modeling X-ray bursts and quiescent low-mass X-ray binaries (42); (43). In addition, the extracted slope parameter MeV from the Skyrme Hartree-Fock calculations together with the neutron skin thickness of Sn isotopes (44) and MeV from the global nucleon optical potentials (45) are in good agreement with the estimated result in this work.
In summary, by combining the Weizsäcker-Skyrme mass model, we propose a four-parameter nuclear charge radius formula in which the microscopic shell correction and nonlinear isospin terms are introduced. The 885 measured rms charge radii of nuclei are reproduced with an rms deviation of 0.022 fm. For the measured even-even nuclei, the rms deviation is only 0.016 fm. The parabolic charge radii trend in Ca chain due to the shell effect and the trend of Ni, Zr and Pb isotopes are reasonably well described with the formula. Through a study of the difference of the rms charge radii between mirror nuclei by using the Skyrme Hartree-Fock model with 62 different Skyrme forces, the linear relationship between the slope parameter and for the mirror pair S - Si is clearly observed, which would be helpful for the extraction of the slope parameter of the nuclear symmetry energy. The estimated slope parameter from the coefficient of the isospin term in the proposed formula is about MeV.
Acknowledgements.This work was supported by the National Natural Science Foundation of China No. 11275052.
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