Evolution of shell structure in exotic nuclei

The atomic nucleus is a quantum many-body system whose constituent nucleons (protons and neutrons) are subject to complex nucleon-nucleon interactions that include spin- and isospin-dependent components. For stable nuclei, several decades ago, emerging seemingly regular patterns in some observables could already be described successfully within a shell-model picture that results in particularly stable nuclei at certain magic fillings of the shells with protons and/or neutrons: $N$, $Z=8$, 20, 28, 50, 82, 126. However, in short-lived, so-called exotic nuclei or rare isotopes, characterized by a large $N/Z$ asymmetry and located far from the valley of $\beta$ stability on the nuclear chart, these magic numbers, viewed through observables, were shown to change. These changes in the regime of exotic nuclei offer an unprecedented view at the roles of the various components of the nuclear force when theoretical descriptions are confronted with experimental data on exotic nuclei where certain effects are enhanced. This article reviews the driving forces behind shell evolution from a theoretical point of view and connects this to experimental signatures.

Figure 1

The nuclear chart as a function of neutron and proton number $N$ and $Z$. Each nucleus is represented by a box specified by $Z$ and $N$. Dark-blue squares indicate stable nuclei. Exotic nuclei experimentally observed as of 2012 are shown by light-blue squares, while light-green squares denote those predicted by a theoretical model ([172]). The ${}^{11}\mathrm{Li}$ nucleus is highlighted in purple. A possible path of the $r$ process is indicated schematically by the green arrows. (Inset) Number of bound neutron-rich exotic nuclei as a function of $Z$ based on [172]. The light- and dark-green parts count nuclei with two-neutron separation energy $S_{2n}>$ and $<2\mathrm{MeV}$, respectively. Adapted from [226], and [222].

Figure 2

(a) Nucleon density distribution $\rho (r)$ and (b) mean potential $U(r)$ are shown as a function of the distance from the center of the nucleus r. (c) Single-particle energies for a harmonic oscillator (HO) potential well, with an added ${\ell }^2$ term and a spin-orbit (SO) interaction $\overrightarrow{\ell }·\overrightarrow{s}$. Shell-gap categories are shown by HO and SO. The $N$ label refers here to the oscillator shell $N=2(n-1)+\ell$, with $n-1$ the number of nodes of the radial wave function and $\ell$ the orbital angular momentum. Adapted from [245].

Figure 3

Schematic illustrations of (a) closed-shell and single-particle states in a Hartree-Fock picture and (b) single-particle states with additional neutrons in a valence orbit (small blue circles). The large yellow circle indicates a subshell gap. The green wavy line denotes the monopole interaction.

Figure 4

Systematics of the first $2^{+}$ levels, for (a) stable and long-lived nuclei, and (b) all nuclei included measured up to 2016 ([243]).

Figure 5

Schematic illustration of shell evolution from Ni to Ca for neutron orbits. Light-blue circles represent protons. The wavy line implies the interaction between the proton $1f_{7/2}$ and the neutron $1f_{5/2}$ orbit. The numbers in circles indicate (semi)magic numbers. From [231].

Figure 6

Contributions of the monopole, pairing, and all other terms of the $NN$ interaction to the ground-state energy of Mg isotopes with even $N$. The contribution from the single-particle energy (SPE) is included also. The other terms are dominated by a quadrupole interaction. The upper panel shows the cumulative contributions, while the lower one shows the variations of pairing and the other terms. For $N=20$, the full calculation, including the excitations to the $pf$ shell, is shown in comparison to the calculation without such excitations. For the latter, the small arrow indicates the ground-state energy because the “all other terms” calculation gives a repulsive contribution (a positive value). See the text for details.

Figure 7

A schematic visualization of monopole matrix elements for the two-body interaction $v$.

Figure 8

Implication of ${\widehat{\tau }}_j^{+}{\widehat{\tau }}_{j^{\prime }}^{-}$ terms. (a) and (c) are for the $\{{\widehat{\tau }}_j^{+}{\widehat{\tau }}_{j^{\prime }}^{-}+{\widehat{\tau }}_j^{-}{\widehat{\tau }}_{j^{\prime }}^{+}\}$ and $:{\widehat{\tau }}_j^{+}{\widehat{\tau }}_j^{-}:$ cases in the isospin scheme, respectively. (b) and (d) are similar to (a) and (c), respectively, in the proton-neutron scheme. The magnetic substates are indicated by $m$ and $m^{\prime }$.

Figure 9

Experimental energy levels of $N=9$ isotones, some of which are regarded as neutron $2s_{1/2}$ and $1d_{5/2}$ single-particle energies. See the text. Adapted from [303].

Figure 10

(a) Schematic picture of shell structure on top of the ${}^{14}\mathrm{C}$ core. (b) Two more protons (red solid circles) are added into the $1p_{1/2}$ orbit. Experimental levels are identified as single-particle states: green level for $1d_{5/2}$ and pink level for $2s_{1/2}$. Numbers near the levels are energies relative to $1d_{5/2}$. The wavy lines imply proton-neutron interactions. Solid arrows indicate the changes of SPEs. The dash-dotted line denotes the neutron threshold, and downward dashed arrows indicate one-neutron separation energy ($S_{\mathit{n}}$).

Figure 11

Monopole matrix elements of central Gaussian interactions of Eq. (47) for all $(S,T)$ channels with an equal strength parameter (i.e., $+166\mathrm{MeV}$; see the text). One of the orbits is $1g_{9/2}$, and the other is shown.

Figure 12

Monopole matrix elements of delta interactions for $(S,T)$ channels. See the caption of Fig. 11.

Figure 13

Monopole matrix elements of central Gaussian and $\delta$ interactions for the ($S=1$, $T=0$) channel. The opposite sign should be taken for actual values. The orbit labeling is abbreviated as g9 for $1g_{9/2}$, etc. The orbits are from the valence shell for (a) $A=100$ and (b) $A=70$.

Figure 14

Neutron ESPEs for (a) ${}^{30}\mathrm{Si}$ and (b) ${}^{24}\mathrm{O}$, relative to the $2s_{1/2}$ (shown as $1s_{1/2}$) orbit. The dotted line connecting (a) and (b) is drawn to indicate the change of the $1d_{3/2}$ (shown as $0d_{3/2}$) level. (c) The major interaction producing the basic change between (a) and (b). (d) The process relevant to the interaction in (c). From [224].

Figure 15

Triplet-even potential due to the tensor force for various interaction models. Adapted from [228].

Figure 16

Intuitive picture of the tensor force acting on two nucleons.

Figure 17

(a) Schematic picture of the monopole interaction produced by the tensor force between a proton in $j_{>,<}=l\pm 1/2$ and a neutron in $j_{>,<}^{\prime }=l^{\prime }\pm 1/2$. (b) Exchange processes contributing to the monopole interaction of the tensor force. From [228].

Figure 18

Intuitive picture of the tensor force acting on two nucleons on orbits $j$ and $j^{\prime }$. From [228].

Figure 19

Intuitive picture of the tensor force acting between two nucleons in a one-dimensional model. The relative-motion wave function is shown for (a) head-on collision and (b) parallel motion cases. In (a) [(b)], the change is shown as the relative momentum $k$ becomes larger (smaller). See the text for explanation. Adapted from [223].

Figure 20

Monopole matrix elements of the tensor force in the $T=0$ channel. The orbit labeling is abbreviated as g9 for $1g_{9/2}$, etc. The orbits are from the valence shell for $A=100$.

Figure 21

Monopole matrix elements of the tensor force in the $T=0$ channel. The orbit labeling is abbreviated as f7 for $1f_{7/2}$, etc. The orbits are from the valence shell for $A=70$.

Figure 22

Monopole matrix elements of various forces for (a)–(d) pf and (e)–(h) sd shells. In (b), (d), (f), and (h), the tensor-force effect is subtracted from the others, and results from a Gaussian central force are shown. Adapted from [230].

Figure 23

Diagrams for the $V_{\mathrm{MU}}$ interaction. From [230].

Figure 24

(Left panel) Proton ESPEs of Cu isotopes predicted by the $V_{\mathrm{M}U}$ interaction (solid lines). Dashed lines are obtained only with the central-force part. The neutron number in the $1g_{9/2}$ orbit is equal to $N-40$, as the filling scheme is assumed. From [230]. (Right panel) Same quantities by the A3DA-m Hamiltonian used by [260]. From [260].

Figure 25

Energy of the lowest levels from experiment and shell-model calculations. From [98].

Figure 26

(Left panel) Neutron ESPEs relative to the $2d_{5/2}$ orbit as a function of $Z$. Dashed and full lines have the same meaning as in Fig. 24. Adapted from [230]. (Right panel) Measured energies of the $7/2^{+}$ level relative to the $5/2_1^{+}$ states for $N=51$ isotones. The squares show states with large $1g_{7/2}$ single-neutron strength, as quoted by [93]. The circles stand for the lowest observed $7/2^{+}$ level ([86]). The present assignment for ${}^{101}\mathrm{Sn}$ is by [63]. The straight line connects the points at $Z=38$ and 50.

Figure 27

Single-particle energies of ${}^{90}\mathrm{Zr}$ and ${}^{100}\mathrm{Sn}$ calculated by (left panel) SIII and (right panel) SLy4 Skyrme interactions.

Figure 28

Change of ESPEs from ${}^{56}\mathrm{Ni}$ to ${}^{48}\mathrm{Ca}$, and to ${}^{54}\mathrm{Ca}$. The arrows indicate the change of the ESPE of each orbit. The arising magic numbers, $N=32$ and 34, are shown in black circles. The dashed line just beneath the $1f_{5/2}$ ESPE (red bar) in the right column means the $1f_{5/2}$ ESPE calculated with one neutron hole in the $2p_{1/2}$ orbit.

Figure 29

First observed $2^{+}$ levels as a function of (a) $N$ and (b) $Z$. First observed $3^{-}$ levels are shown in (a). Adapted from [290].

Figure 30

ESPEs of proton $1h_{11/2}$ and $1g_{7/2}$ orbits in Sb isotopes as functions of $N$. Assuming a Sn-isotope core, the solid lines are calculated with the tensor-force effect, whereas the dotted lines are without it. Symbols are based on experimental values from [264]. Fragmentation of single-particle strength is considered for filled circles, while bare energies are used for open symbols. See the text for relevant arguments on those values. From [222].

Figure 31

Comparison of monopole matrix elements of the zero-range tensor force of [286] to those of the $\pi +\rho$ meson-exchange tensor force. The $p$, $sd$, and $pf$ shells are covered in (a), while the valence shell relevant to $A\sim 140$ is considered in (b). The parameter ${\beta }_T=128.75\mathrm{Me}\mathrm{V}{\mathrm{fm}}^5$ is used.

Figure 32

Evolution of the proton $h_{11/2}\text{−}{g}_{7/2}$ gap in the Sb isotopes with and without the tensor term. (Upper left panel) $\pi +\rho$ meson-exchange tensor force on top of the usual Woods-Saxon potential. From [228]. (Upper right panel) A Gogny-type calculation with the tensor force (GT2) and without it (D1S). From [225]. (Lower panel) A zero-range tensor force calculation added to the SLy5 force. From [59].

Figure 33

The same as Fig. 32. (Upper panel) A zero-range tensor force calculation. (Lower panel) A relativistic mean-field calculation. From [33], and [179].

Figure 34

(a) Energy of the neutron $2s_{1/2}$ orbit relative to the $1d_{5/2}$ orbit in ${}^{15}\mathrm{C}$ and ${}^{17}\mathrm{O}$, calculated within the single-particle scheme using the $V_{\mathrm{MU}}$ interaction plus the M3Y 2b-$LS$ force (see the text). Contributions from the tensor, 2b-$LS$, and central forces are decomposed. The changes are added to the experimental $1/2_1^{+}$ level placed relative to the experimental $5/2_1^{+}$ level. The experimentally observed level of ${}^{17}\mathrm{O}$ is shown at the far right. (b) Analysis similar to (a) in terms of the shell-model calculation with the YSOX ([352]) interaction. Some expectation values obtained by the YSOX calculation corresponding to (a) are shown with respect to the shell-model eigenstates. See the caption for (a).

Figure 35

Monopole matrix elements from tensor forces in the AV8′ interaction ([244]), in low-momentum interactions obtained from the AV8′, and in the third-order $Q$-box interaction for (a) $T=0$ and (b) $T=1$. From [230].

Figure 36

(a) Neutron effective single-particle energies of the SDPF-U interaction ([216]) and their (b) $k=0$, (c) $k=1$, and (d) $k=2$ contributions with increasing proton number. From [278].

Figure 37

Processes involved in the discussion of 3N forces and their contributions to the monopole components of the effective interactions between two valence neutrons. The solid lines denote nucleons, the dashed lines denote $\pi$ mesons, and the thick lines denote $\mathrm{\Delta }$ excitations. Nucleon-hole lines are indicated by downward arrows. (a) The leading contribution to $NN$ forces due to $\mathrm{\Delta }$-resonance excitation, and (b) the change of SPE of the state $j$, $m$ by the process (a). The process (b) is forbidden with another nucleon with the same $j^{\prime }$, $m^{\prime }$, as shown (c), which requires the inclusion of the exchange diagram (d), which is equivalent to the FM 3N force (e). The leading $\chi \mathrm{EFT}$ 3N forces include (f) the long-range two-$\pi$-exchange parts, which take into account the excitation to a $\mathrm{\Delta }$ and other resonances, plus shorter-range (g) one--$\pi$-exchange and (h) 3N contact interactions. From [229].

Figure 38

ESPE of neutron $1d_{5/2}$, $2s_{1/2}$, and $1d_{3/2}$ orbitals measured from the energy of ${}^{16}\mathrm{O}$ as a function of $N$. The ESPEs calculated (left panel) from a $G$ matrix and (right panel) from low-momentum interactions $V_{\mathrm{low}k}$ are shown. The changes due to 3N forces based on $\mathrm{\Delta }$ excitations are highlighted by the shaded areas. Adapted from [229].

Figure 39

(Left and second from left panels) Ground-state energies of oxygen isotopes including processes shown in the second from right panel. Adapted from [229]. (Right panel) The ground-state energies calculated in several $\chi \mathrm{EFT}$ approaches ([136]). From [136].

Figure 40

$2_1^{+}$ level of Ca isotopes calculated by the CC method. From [123].

Figure 41

(Upper panel) ESPEs of neutrons calculated by [53] at subshell closures of oxygen isotopes. (Lower panel) Similarly calculated ground-state energies compared to experiment (bars). From [53].

Figure 42

Ground-state energies of Na isotopes calculated with IM-SRG (SM). The IM-SRG (SM) curves use a core reference, while the curves labeled IM-SRG (ENO) use an ensemble reference. From [296].

Figure 43

The two-neutron separation energies of Na isotopes calculated with IM-SRG. From [273].

Figure 44

ESPEs of $N=20$ isotones for neutrons obtained in the normal filling scheme. The solid (dotted) lines in (a) show the case with (without) three-nucleon forces, while the solid (dot-dashed) lines in (b) represent the case with (without) the tensor component. From [323].

Figure 45

(Left panel) Part of the Segrè chart around the IoI. The red area indicates its original picture as of [338]. Now the IoI has extended much further (see the text). (Right panel) Relative energy of the lowest normal and lowest neutron $2p\text{−}2h$ intruder $0^{+}$ states, resulting from diagonalizing in a separate subspace. Adapted from [338], and [52].

Figure 46

(Left panel) $N=20$ shell gap (green solid line) and from the SDPF-M interaction (red dashed line). Adapted from [338], and [331], respectively. (Right panel) ESPEs of $N=20$ isotones for neutrons obtained in the normal filling scheme from SDPF-M interaction. From [331].

Figure 47

ESPEs of $N=20$ isotones for neutrons obtained in the normal filling scheme (left panel) from the $V_{\mathrm{MU}}$ interaction and (right panel) from the SDPF-NR interaction. From [230], and [50], respectively.

Figure 48

Symbols indicate two-neutron separation energies for the Mg and Al isotopic chains from the 2015 TITAN experiment and the mass compilation by [4]. Shell-model calculations in the $sd\text{−}pf$ shell ([51]) are shown also by the solid and dashed lines. From [178].

Figure 49

Measured spectroscopic quadrupole moments of Al isotopes compared to shell-model calculations limited to the neutron $sd$ shell (USD) and allowing for particle-hole excitations across $N=20$ (SDPF-M). The given percentage signifies the amount of ground-state intruder configurations in the respective shell-model approaches. From [141].

Figure 50

Energy levels of ${}^{31}\mathrm{Mg}$. (a) Experimental values, (b) EEdf1 ([323]), (c) SDPF-M ([331]), (d) SDPF-U-MIX ([51]), and (e) antisymmetrized molecular dynamics $+$ generator coordinate method calculations ([162]), respectively. From [323].

Figure 51

(Left panel) Prompt $\gamma$-ray spectrum detected with GRETINA in coincidence with the ${}^{32}\mathrm{Mg}$ projectilelike fragmentation residues identified in the S800 spectrograph. From [62]. (Right panel) Comparison to theoretical calculations with EEdf1 and SDPF-U-MIX interactions. From [323].

Figure 52

$B(E2)$ value of the $N=20$ isotones plotted as a function of $Z$. (Left panel) Shell-model calculation, where the solid line includes neutron excitations across the $N=20$ gap but the dashed line does not. From [109]. (Right panel) The measured values are confronted with the $N_p{N}_n$ scheme calculation ([48]) for $N_n=0$ ($N=20$ closed) and $N_n=12$ ($sd$ and lower $pf$ shells combined). See the text for details. Adapted from [76].

Figure 53

Spectroscopic factors from ${}^9\mathrm{Be}({}^{32,30}\mathrm{Mg},{}^{31,29}\mathrm{Mg})X$ knockout reactions to two negative-parity states of ${}^{32,30}\mathrm{Mg}$. Deduced spectroscopic factors are indicated by blue point with error bar ([310]). Single-particle occupancies obtained using the SDPF-M shell model ([331]) are represented by empty histograms. Spectroscopic factors calculated with the EEdf1 interaction are shown as red histograms ([322]).

Figure 54

Proton angular distribution for the new neutron-unbound state discovered at 1.74 MeV in ${}^{27}\mathrm{Ne}$. In comparison to reaction theory, an $\ell =3$ orbital angular momentum was assigned. From [42].

Figure 55

(a) Evidence for a reduction of the $2p_{3/2}\text{−}2p_{1/2}$ spin-orbit splitting in the $N=21$ isotonic chain at ${}^{35}\mathrm{Si}$. For comparison, the spin-orbit splitting remains unchanged between ${}^{41}\mathrm{Ca}$ and ${}^{37}\mathrm{S}$. From [44]. (b) Change of the proton density along the $N=20$ isotone line from density functional theory [relativistic mean field with the DDME2 interaction ([180])]. The vanishing proton occupation of the $s_{1/2}$ orbital leads to a central depletion in the density that has been likened to a bubble. From O. Sorlin and J. P. Ebran.

Figure 56

Schematic pictures of the doorway state for (a)–(c) $E1$ excitation and (d)–(f) a reaction induced by the removal of a proton. Dashed lines indicate the neutron threshold. Red filled circles indicate the neutron being discussed, while red open circles are neutron holes. Blue circles are protons, and crossed blue circles are absent after the initial impact of the reaction. From [321].

Figure 57

(a) ESPEs for neutron orbits in C isotopes obtained with the SFO-tls interaction. (b) Ground-state energies of C isotopes obtained with SFO-tls, SFO, and WBP ([339]) as well as experimental data. In (a), the filling scheme is taken in the order of $1p_{1/2}$, $1d_{5/2}$, $2s_{1/2}$, and $1d_{3/2}$, as this order represents rather well the configurations of actual eigenstates. The ESPEs at their closures are connected. The $N=16$ (sub)magic gap is highlighted by the yellow circle.

Figure 58

The rms radius of the halo neutron as a function of two-neutron separation energy $S_{2n}$. The blue dashed line and filled circle indicate the result obtained with the core of the closed-shell ${}^{20}\mathrm{C}$, while the red solid line and filled circle indicate the result with the core of the correlated ${}^{20}\mathrm{C}$. The result obtained from the Woods-Saxon potential ($S_n=S_{2n}/2$) without $v_{nn}$ is shown as the black dotted line. The range of $S_{2n}$ obtained from [86] is shown by green thin vertical lines. Green arrows denote the values discussed by [166]. The halo radius, ${6.79}_{-0.66}^{+0.70}\mathrm{fm}$, can be deduced from the experimental matter radius ([313]) (see the text). From [300].

Figure 59

Distribution of proton-hole strengths in ${}^{48}\mathrm{Ca}$ compared between the $(e,e^{\prime }p)$ data ([174]) and shell-model calculations with the SDPF-MU interaction. The left and right panels show the calculations with and without the cross-shell tensor force, respectively. The calculated overall spectroscopic factors are quenched by 0.7. From [329].