Irregularities in nuclear radii at magic numbers

TL;DR

This study uses advanced nuclear models to predict and analyze irregularities in nuclear radii at magic numbers, revealing new anti-kinks at $ extit{ ext{l}s}$-closed magic numbers and confirming known kinks at $jj$-closed magic numbers.

Contribution

It introduces a novel prediction of anti-kinks at $ extit{ ext{l}s}$-closed magic numbers using density-dependent potentials from chiral effective field theory.

Findings

Kinks at $jj$-closed magic numbers in charge and matter radii.

Anti-kinks at $ extit{ ext{l}s}$-closed magic numbers.

Consistency with experimental data in Ca, Sn, and Pb isotopes.

Abstract

Influence of magic numbers on nuclear radii is investigated via the Hartree-Fock-Bogolyubov calculations and available experimental data. With the $\ell s$ potential including additional density-dependence suggested from the chiral effective field theory, kinks are universally predicted at the $jj$-closed magic numbers and anti-kinks (\textit{i.e.} inverted kinks) are newly predicted at the $\ell s$-closed magic numbers, both in the charge radii and in the matter radii along the isotopic and isotonic chains where nuclei stay spherical. These results seem consistent with the kinks of the charge radii observed in Ca, Sn and Pb and the anti-kink in Ca. The kinks and the anti-kinks could be a peculiar indicator for magic numbers, discriminating $jj$-closure and $\ell s$-closure.

Equations

$$v_{ij}^{(\mathrm{LS})}=\sum_{n}\big{(}t_{n}^{(\mathrm{LSE})}P_{\mathrm{TE}}+t_{n}^{(\mathrm{LSO})}P_{\mathrm{TO}}\big{)}f_{n}^{(\mathrm{LS})}(r_{ij})\,\mathbf{L}_{ij}\cdot(\mathbf{s}_{i}+\mathbf{s}_{j})\,,$$

$$\begin{array}{rl}{v}_{ij}^{(\mathrm{LS}\rho )}=& 2iD[\rho ({\mathbf{R}}_{ij})]{\mathbf{p}}_{ij}\times \delta ({\mathbf{r}}_{ij}){\mathbf{p}}_{ij}\cdot ({\mathbf{s}}_i+{\mathbf{s}}_j);\\ & D[\rho (\mathbf{r})]=w_1\frac{\rho (\mathbf{r})}{1+d_1\rho (\mathbf{r})}.\end{array}$$

Full text

I Introduction

As finite quantum many-body systems, atomic nuclei show notable irregularities in their properties. The most typical and significant example is magic numbers, which have been identified in irregularities of masses and excitation energies ref:Hey04 . The magic numbers are fundamental to nuclear structure physics. Furthermore, they are relevant to the origin and abundance of elements, forming waiting points in several processes of nucleosynthesis in the universe ref:syn97 . Proton magicity in large neutron excess plays a key role in the structure of the neutron-star crust as well ref:NV73 ; ref:ODC08 . Developments of the radioactive nuclear beams in recent decades have disclosed that magic numbers are not so rigorous as once expected. While some of the known magic numbers disappear, new magic numbers come out far off the stability ref:SP08 . It is highly desired to comprehend magic numbers all over the nuclear chart.

Though interesting and significant, quantum irregularities as arising at the magic numbers are often an obstacle to constructing an accurate and practical theoretical model. One of the successful methods to handle quantum many-body problems is the density functional theory, as well-developed for bound electronic systems in attractive external fields ref:ED11 . However, whereas the Hohenberg-Kohn theorem guarantees the existence of an energy density functional (EDF) that gives exact ground-state energies ref:HK64 , it is crucial to remove irregularities properly, as done by the Kohn-Sham method for the electronic systems ref:KS65 . Although there have been many attempts to construct nuclear EDFs in terms of the nucleonic densities and quasi-local currents (e.g. Ref. ref:UNEDF2 ), none have been as successful as in the electronic systems. If a nuclear EDF could be developed that attains accuracy at a comparable level to the electronic EDF, it would provide us with a unified theoretical framework for many-fermion systems. Full understanding of irregularities such as magic numbers is crucial also in this respect.

Nuclear radii are basic physical quantities, directly linked to the density distributions that are essential ingredients of an EDF. The measured matter radii of stable nuclei are proportional to the one-third of the mass number $A$ in the first approximation, which manifests the saturation of the nucleon-number densities ref:Hey04 . Deviation from this simple rule carries interesting information concerning nuclear structure. For instance, some of the nuclei in the vicinity of the neutron drip line have significantly large root-mean-square (rms) matter radii, indicating neutron halos ref:halo . Although the nuclear Hamiltonian should keep the rotational invariance, a number of nuclei are deformed rather than spherical in their intrinsic states. The nuclear deformation has been verified by the distinctly large radii compared to those of the nearby spherical nuclei, together with other observables ref:BM1 .

Since nuclei with magic proton ($Z$) or neutron numbers ($N$) are usually spherical and those without magicity often depart from the sphericity, it is not surprising that the radii become relatively small at magic $Z$ or $N$. In practice, kinks have been observed at magic $N$ in the charge radii in many isotopes ref:Ang13 . However, kinks have been found at magic numbers even when nuclei stay spherical. A well-known example is a kink at $N=126$ in the isotope shifts of Pb ref:AHS87 . As discussed below, the relevance of the magicity to the nuclear radii has not been elucidated sufficiently. It is significant to perceive the presence and mechanism of irregularities in the radii at magicity.

In this article, I shall discuss the relationship between the radii of spherical nuclei and the magicity, emphasizing possible roles of the three-nucleon ($3N$) interaction. Originating from the nucleonic interaction, nuclear EDFs are often associated with effective interactions. A valuable guide for nuclear EDFs will be obtained by appreciating how the nucleonic interaction affects nuclear structure. Based on the predictions in which the $3N$-force effects are taken account of, irregularities in radii are proposed as an experimental tool that is useful for recognizing characters of individual magicity, as well as for identifying magic numbers. Note that magic numbers can well be identified by no single observable, and the consistency among relevant physical quantities should be checked carefully. Whereas the magic numbers are usually identified via energies, irregularities in radii could be important as well, both experimentally and theoretically.

II Effects of spin-orbit potential on nuclear radii

The spin-orbit ($\ell s$) splitting of the nucleon orbitals is essential to the magic numbers. Although it must be traced back to the nucleonic interaction, the origin of the $\ell s$ splitting has not been understood sufficiently ref:AB81 . While many-body correlations induced by the two-nucleon ($2N$) interaction were suggested to enhance the $\ell s$ splitting ref:LS-UMOA , there was an argument from the quark-meson coupling model that the $\rho$-meson exchange could contribute ref:GT04 . Via an ab initio calculation applying the quantum Monte Carlo method, the importance of the $3N$ interaction in the $\ell s$ splitting was indicated ref:LS-MC . It was argued recently, based on the chiral effective-field theory ($\chi$EFT), that the $3N$ interaction may account for the missing part of the $\ell s$ splitting ref:Koh13 . The $\ell s$ splitting is linked to the $\ell s$ potential at the mean-field (MF) level.

It has been recognized from the experimental data that kinks often come out in the neutron-number ($N$) dependence of the nuclear charge radii at $N=28,50,82$ ref:Ang13 and likely at $N=126$ ref:BFI18 , while not so at $N=20$. It should be noticed that the numbers $28,50,82$ and $126$ are the $jj$-closed magic numbers, in which a $j=\ell+1/2$ orbit is fully occupied while its $\ell s$ partner is empty. One of the well-known examples of the kinks is found at $N=126$ in the Pb isotopes [see Fig. 1(d) below]. Since deformation is unlikely around 208Pb, the kink implies that the proton wave-functions are influenced by neutrons which occupy the single-particle (s.p.) levels above the $N=126$ shell gap. However, not all the neutron s.p. orbitals above the shell gap distribute so broadly as to account for the observed kink. It has been pointed out that neutron occupancy on the $0i_{11/2}$ orbit is relevant to the kink, whereas the s.p. function of $n1g_{9/2}$, the lowest s.p. level above $N=126$ ref:TI , is not broad enough in the self-consistent MF calculations ref:TBP93 ; ref:GSR13 . The spatial distributions of the s.p. orbits are affected by the $\ell s$ potential to a certain extent. The $\ell s$ potential is repulsive (attractive) for a nucleon occupying a $j=\ell-1/2$ ($j=\ell+1/2$) orbital, tending to shift the wave function outward (inward). This effect is the stronger for the higher $\ell$. Thereby occupation of a $j=\ell-1/2$ orbit (e.g. $0i_{11/2}$) yields a larger radius than the occupation of surrounding orbitals. Sizable occupation on $n0i_{11/2}$ in $N>126$ broadens the neutron wave-function and may induce a rapid rise of the charge radii through the attraction between protons and neutrons.

For the occupation on $n0i_{11/2}$ above $N=126$, the s.p. energy difference relative to $n1g_{9/2}$ is crucial. In the calculations with the conventional Skyrme EDFs, $n0i_{11/2}$ lies significantly higher than $n1g_{9/2}$, preventing the visible kink from emerging. Via comparison to the relativistic mean-field (RMF) calculations which yield a kink at 208Pb ref:SLR94 , it was found that the $n0i_{11/2}$ occupation is related to the isospin partitions of the $\ell s$ potential ref:SLKR95 ; ref:RF95 , which should originate from a certain channel of the nucleonic interaction. Still, it has been difficult to reproduce the kink unless $n1g_{9/2}$ and $n0i_{11/2}$ are nearly degenerate or even inverted ref:GSR13 , incompatible with the observed energy levels ref:TI . On the contrary, if there is a significant contribution of the $3N$ interaction to the $\ell s$ potential as suggested by the $\chi$EFT, it makes the $\ell s$ potential stronger in the nuclear interior than in the exterior. Then the difference in the radial distribution between the $\ell s$ partners is grown further, because the wave-functions of the $j=\ell-1/2$ ($j=\ell+1/2$) orbitals slide outward (inward) so as to be influenced by the $\ell s$ potential to a lesser (greater) extent. This effect on the s.p. functions has been confirmed in Fig. 1 of Ref. ref:NI15 . The enhanced difference of the wave-functions improves $N$-dependence of the charge radii in Pb with little influence on the s.p. energies ref:NI15 . Similarly, the kink of the charge radii in Ca at $N=28$ is pronounced and a kink is predicted in Sn at $N=82$ ref:Nak15 [see also Fig. 1(a) and (c) below]. Both kinks have been observed in recent experiments ref:shift-Ca52 ; ref:shift-Sn134 . The kink at 48Ca has been obtained by ab initio methods with the $\chi$EFT interactions as well ref:shift-Ca52 .

There are two types of nuclear magic numbers: the $\ell s$-closed magic numbers and the $jj$-closed ones. While magicity is normally indicated by irregularities in energies that do not discern between the $\ell s$-closed and the $jj$-closed magic numbers, the irregularities in the nuclear radii may work as a peculiar indicator. The $jj$-closed magic numbers occur after a high-$j$ orbit with $j=\ell+1/2$ is fully occupied, and its $\ell s$ partner with $j=\ell-1/2$ starts occupied above the magic numbers. Even though the $j=\ell-1/2$ orbit does not always lie lowest above the magic number, its occupancy is sizable owing to the pair correlation. Therefore the nuclear radii increase relatively slowly below the magicity and more rapidly above it, producing a kink. On the other hand, the $\ell s$-closed magic numbers occur after a $j=\ell-1/2$ orbit is filled, and a $j=\ell+1/2$ orbit with higher $\ell$ starts being occupied above it. It is then expected that the nuclear radii increase rapidly below the $\ell s$-closed magic numbers, and increase more slowly or even decrease above it. Thus an inverted kink emerges at the $\ell s$-closed magic numbers, which will be called ‘anti-kink’ in contrast to the kink at the $jj$-closed magic numbers. As well as the magicity itself, its character, i.e. whether it is $jj$- or $\ell s$-closed, may be examined by qualitative behavior of the nuclear radii. The kinks and the anti-kinks are expected to be visible or pronounced as an effect of the $3N$ interaction, as shown in Sec. III.

While accurate data can be obtained for the charge radii, experimental data on the matter radii have been reported for some isotopic chains (e.g. Ref. ref:OST01 ). More abundant data including unstable nuclei are expected in future experiments using hadronic probes. Nuclear matter radii are an average reflecting the radial distributions of all the constituent nucleons. It is also intriguing whether and how the neutron magicity influences isotopic variation of the nuclear matter radii, which directly reflect the radial distributions of neutrons. The same holds for the proton magicity under the isotonic variation.

It should be kept in mind that deformation can be another source of irregularities in the nuclear radii. As the deformation is suppressed at the magic numbers, it tends to produce a kink, not an anti-kink. For the $\ell s$-closed magicity, the effects of the s.p. functions and the deformation may act competitively, possibly obscuring the anti-kinks. Halos, which could emerge in vicinity of the drip lines, also give rise to irregularity in nuclear radii. However, it will be feasible to investigate the magicity via the radii, by choosing a series of spherical nuclei not too close to the drip lines.

III Theoretical and experimental results

Let us see how the above arguments apply to the theoretical and experimental results. To illustrate kinks and anti-kinks theoretically, I shall present results of self-consistent MF calculations, the spherical Hartree-Fock-Bogolyubov (HFB) to be precise ref:Nak06 , for nuclei having magic $Z$ or $N$. Odd-$A$ nuclei are treated in the equal-filling approximation ref:EFA ; ref:EFA2 . For the nucleonic effective interaction, the M3Y-P6 and M3Y-P6a semi-realistic interactions ref:Nak13 ; ref:NI15 are mainly employed. For comparison, results with the Gogny-D1S interaction ref:D1S , which has been one of the most widely-used interactions for the HFB calculations, are also displayed. Influence of the center-of-mass motion is corrected ref:NS02 . For the charge radii, the finite-size effects of the constituent nucleons are taken into account, up to the magnetic effects ref:FN75 . Also for reference, results of the RMF calculations for even-even nuclei with the NL3 parameter are quoted from Ref. ref:NL3 , in which some of the finite-size effects on the charge radii are ignored.

In the self-consistent MF framework, the $\ell s$ splitting is obtained primarily from the LS channel of the nucleonic interaction,

[TABLE]

within the $2N$ interaction. Here the subscripts $i$ and $j$ are indices of nucleons. $P_{\mathrm{TE}}$ ($P_{\mathrm{TO}}$) denotes the projection operator on the triplet-even (triplet-odd) two-particle states, $\mathbf{r}_{ij}=\mathbf{r}_{i}-\mathbf{r}_{j}$, $r_{ij}=|\mathbf{r}_{ij}|$, $\mathbf{p}_{ij}=(\mathbf{p}_{i}-\mathbf{p}_{j})/2$, $\mathbf{L}_{ij}=\mathbf{r}_{ij}\times\mathbf{p}_{ij}$, and $\mathbf{s}_{i}$ is the spin operator. In the M3Y-type interactions, $f_{n}^{(\mathrm{LS})}(r)=e^{-\mu_{n}^{(\mathrm{LS})}r}/\mu_{n}^{(\mathrm{LS})}r$, with $1/\mu_{n}^{(\mathrm{LS})}$ representing the range parameter ref:Nak03 . In M3Y-P6, which gives a reasonable prediction of magic numbers in a wide range of the nuclear chart including unstable nuclei ref:NS14 , the strength parameters $t_{n}^{(\mathrm{LSE})}$ and $t_{n}^{(\mathrm{LSO})}$ derived from Paris $2N$ force ref:M3Y-P are multiplied by a factor $2.2$, to reproduce the level sequence around 208Pb. On the other hand, analysis based on the $\chi$EFT suggests that the $3N$ interaction enhances the LS channel so that it should become stronger as the nucleon density increases ref:Koh13 . Hinted by this result, in M3Y-P6a a density-dependent term $v^{(\mathrm{LS}\rho)}$ is added instead of enhancing $t_{n}^{(\mathrm{LSE})}$ and $t_{n}^{(\mathrm{LSO})}$, which is represented as LSrho_correction

[TABLE]

Here $\rho(\mathbf{r})$ is the isoscalar nucleon density and $\mathbf{R}_{ij}=(\mathbf{r}_{i}+\mathbf{r}_{j})/2$. The density-dependent coefficient $D[\rho]$ carries effects of the $3N$ interaction. The parameter $w_{1}$ is fitted to the $n0i_{13/2}$-$n0i_{11/2}$ splitting with M3Y-P6 at 208Pb. Then the s.p. energies, as well as the binding energies, do not change from those of M3Y-P6 significantly. The parameter $d_{1}$ does not have physical importance, and $d_{1}=1.0\,\mathrm{fm}^{3}$ is assumed ref:Nak15 . As all the channels except the LS one are identical between M3Y-P6 and M3Y-P6a, comparison of their results will clarify effects of the $3N$ LS term [i.e. $v^{(\mathrm{LS}\rho)}$] in place of the naive enhancement of the LS channel by an overall factor. While the form of Eq. (2) is consistent with the $\chi$EFT analysis ref:Koh13 by which the qualitative effects of the $3N$ interaction could be investigated, the strength is not equal to that derived in Ref. ref:Koh13 . Both M3Y-P6 and M3Y-P6a contain realistic tensor channels based on the $G$-matrix ref:M3Y-P , which also have influence on the $\ell s$ splitting in some nuclei ref:NS14 ; ref:Vtn ; ref:SC14 ; ref:Nak10 ; ref:NSM13 .

The nuclear charge radii can be measured by the electromagnetic probes, e.g. the electron scattering ref:FN75 . Moreover, the mean-square differential charge radii among isotopes, denoted by $\mathit{\Delta}\langle r^{2}\rangle_{c}$, are extracted accurately from the isotope shifts ref:Ang13 . For the nuclide ${}^{A}Z$, $\mathit{\Delta}\langle r^{2}\rangle_{c}$ is defined by $\mathit{\Delta}\langle r^{2}\rangle_{c}(^{A}Z):=\langle r^{2}\rangle_{c}(^{A}Z)-\langle r^{2}\rangle_{c}(^{A_{0}}Z)$, where ${}^{A_{0}}Z$ is the reference nuclide for the fixed $Z$. In Fig. 1, $\mathit{\Delta}\langle r^{2}\rangle_{c}$ in the magic-$Z$ nuclei are plotted as a function of $N$. As reference nuclei, 40Ca, 60Ni, 120Sn and 208Pb are taken as in Refs. ref:Ang13 . Experimentally, kinks have been observed at 48Ca, 132Sn and 208Pb as already mentioned, corresponding to the neutron $jj$-closed magic numbers. In the theoretical results, interaction-dependence is found for the kinks. In Pb, the isospin-dependence of the $\ell s$ potential affects $\mathit{\Delta}\langle r^{2}\rangle_{c}$ around $N=126$ ref:SLKR95 through the s.p. energy difference $\varepsilon_{n}(0i_{11/2})-\varepsilon_{n}(1g_{9/2})$. The D1S interaction has the zero-range LS channel as the Skyrme interaction ref:Sky , yielding no apparent kink at $N=126$ in Fig. 1(d). A kink is obtained at $N=126$ with M3Y-P6, but it is weaker than the observed one. The kink becomes pronounced in the M3Y-P6a results ref:NI15 without significant change in $\varepsilon_{n}(0i_{11/2})-\varepsilon_{n}(1g_{9/2})$ from M3Y-P6, which is comparable to the observed energy difference between $(11/2)^{+}$ and $(9/2)^{+}$ at 209Pb ref:TI . Kinks universally arise at the $jj$-closed magicity with M3Y-P6a, i.e. by taking into account the $3N$-force contribution to the LS channel that affects the s.p. functions. Note that this is not the case for the RMF results of Ref. ref:NL3 . As pointed out in Ref. ref:Nak15 , a kink has been predicted at $N=82$ for the Sn chain with M3Y-P6a [Fig. 1(c)], though such a prominent kink is not seen in the other results shown here. The recent discovery of a kink at 132Sn ref:shift-Sn134 is supportive of the $3N$-force contribution to the $\ell s$ splitting. A kink is also predicted at $N=28$ for the Ni chain [Fig. 1(d)], which is generic for interactions but enhanced by introducing $v^{(\mathrm{LS}\rho)}$. Moreover, anti-kinks are grown at 40Ca and 68Ni in the M3Y-P6a results, because of the $\ell s$-closed magicity of $N=20$ and $40$. The former is indeed consistent with the recent measurement ref:shift-Ca36 as exhibited in Fig. 1(a). The anti-kinks are of particular importance in establishing effects of the magicity on the nuclear radii and roles of the $3N$ interaction in them. For the $N=40$ magicity, no obvious anti-kink is seen at 60Ca even with M3Y-P6a, since the magicity is not well kept at 60Ca ref:NS14 ; ref:Nak10 . The kink-like structure at 54Ca might be related to the $N=34$ magicity ref:Ca54_Ex2 , although it was not identified as magic in Ref. ref:NS14 .

$N$-dependence ($Z$-dependence) of the rms matter radii is depicted for the magic-$Z$ (magic-$N$) nuclei in Fig. 2 (Fig. 3). Not so many data are available for the matter radii, and it has not been easy to attain good accuracy. However, owing to the progress in experimental techniques and reaction theory, systematic measurements with good precision are promising, up to nuclei far off the $\beta$-stability. Future experiments over isotopic or isotonic chains are awaited.

In Fig. 2(a), a kink is predicted at $N=14$, which is enhanced by $v^{(\mathrm{LS}\rho)}$. This corresponds to the submagic nature of $N=14$ at 22O ref:NS14 . Although this kink seems compatible with the available data ref:OST01 ; ref:Kan11 , more accurate data are desirable. For the other isotopic chains, kinks are predicted at the usual $jj$-closed magic numbers. While the kinks are weak without $v^{(\mathrm{LS}\rho)}$, they come pronounced in the M3Y-P6a results. It is mentioned that the kink at 48Ca is observed in a recent experiment TF-pv . Anti-kinks are predicted with M3Y-P6a at $N=20$ in Fig. 2(b) and at $N=40$ in Fig. 2(c), corresponding to the $\ell s$ closure. Not apparent in the other results, the anti-kinks can disclose the $3N$-force effects, although these anti-kinks are less conspicuous than the kinks at the $jj$-closed magicity.

In Fig. 3(a), an anti-kink is predicted with M3Y-P6a at $Z=20$, linked to the $\ell s$-closed magicity. No visible anti-kink is predicted at 48Ca in Fig. 3(b). This is accounted for by the inversion of the s.p. levels $p0d_{3/2}$ and $p1s_{1/2}$ ref:NSM13 . In Fig. 3(c), a kink is predicted at the $jj$-closed magic number $Z=28$. Figure 3(d) shows several irregularities. In addition to a kink at the $Z=50$ magicity, an anti-kink is predicted at $Z=58$ and a weak kink is viewed at $Z=64$. The former corresponds to the closure up to $p0g_{7/2}$ at 140Ce and the latter to the closure of $p1d_{5/2}$ at 146Gd, both of which are identified as submagic numbers in Ref. ref:NS14 , consistent with the relatively high excitation energies in measurement ref:TI . The irregularities in the radii will support their submagic nature if observed. In Fig. 3(e), a kink is predicted at $Z=82$, irrespective of the interactions. An anti-kink predicted at $Z=58$ and a weak kink at $Z=64$ are attributed to the submagic nature ref:NS14 , as in the $N=82$ case.

It is commented that the kinks in $\mathit{\Delta}\langle r^{2}\rangle_{c}$ have also been predicted with the Fayans EDF at 48Ca, 132Sn and 208Pb ref:Fay98 . The Fayans EDF and M3Y-P6a are the only two EDFs that have predicted the kink at 132Sn. The results of $\mathit{\Delta}\langle r^{2}\rangle_{c}$ have similarity to those of M3Y-P6a in qualitative respect, despite difference in the EDF forms. Whereas the relation of the kinks to the nucleonic interaction is not clear in the Fayans EDF, effects of the pairing channel have been stressed ref:shift-Sn134 . This is not necessarily contradictory to the present analysis, as the pairing plays a role in the occupation of the relevant s.p. orbits. It is of interest whether the other results with M3Y-P6a shown here are shared with those with the Fayans EDF. However, the pairing should not strengthen the anti-kinks. Future experiments around the $\ell s$-closed magicity will be significant to pin down the dominant source of the irregularities. Apart from the irregularities, the RMF results of the matter radii in Figs. 2 and 3 are considerably larger than the others in the neutron excess, whereas the charge radii are comparable. These results indicate thick neutron skins with the NL3 parameter-set in the RMF and are attributed to the strong density-dependence of the symmetry energy.

IV Summary

Influence of magic numbers on nuclear radii has been investigated via the self-consistent spherical Hartree-Fock-Bogolyubov (HFB) calculations and available experimental data. Owing to the difference in the single-particle wave-functions between $\ell s$ partners, kinks are universally expected at the $jj$-closed magic numbers both in the charge radii and the matter radii, even when nuclei stay spherical. Although the former has been recognized empirically, most of the HFB calculations do not reproduce all the kinks at $N=28$, $82$ and $126$ in the Ca, Sn and Pb isotopes. The density-dependence of the $\ell s$ potential, which can be linked to the $3N$ interaction suggested from the chiral effective field theory, yields significant contribution to the kinks which are consistent with the data. Moreover, the calculations with this density-dependence predict ‘anti-kinks’ at the $\ell s$-closed magic numbers, i.e. kinks inverted from the $jj$-closed cases. If experimentally established, the anti-kinks could be good evidence for the $3N$-force effects on the $\ell s$ splitting and may be used to investigate nuclear magic numbers, discriminating $jj$-closure and $\ell s$-closure as well as indicating magicity. Finally, it is stressed that appreciation of effects of the magic numbers on irregularities in the radii, such as the kinks and the anti-kinks, is indispensable to construct an accurate and practical theory using an energy density functional.

Acknowledgements.

Discussions with M. Fukuda, M. Tanaka, S. Shlomo and T. Inakura are gratefully acknowledged. A part of numerical calculations is performed on HITAC SR24000 at IMIT in Chiba University.