Shape coexistence in atomic nuclei

Shape coexistence in nuclei appears to be unique in the realm of finite many-body quantum systems. It differs from the various geometrical arrangements that sometimes occur in a molecule in that in a molecule the various arrangements are of the widely separated atomic nuclei. In nuclei the various “arrangements” of nucleons involve (sets of) energy eigenstates with different electric quadrupole properties such as moments and transition rates, and different distributions of proton pairs and neutron pairs with respect to their Fermi energies. Sometimes two such structures will “invert” as a function of the nucleon number, resulting in a sudden and dramatic change in ground-state properties in neighboring isotopes and isotones. In the first part of this review the theoretical status of coexistence in nuclei is summarized. Two approaches, namely, microscopic shell-model descriptions and mean-field descriptions, are emphasized. The second part of this review presents systematic data, for both even- and odd-mass nuclei, selected to illustrate the various ways in which coexistence is observed in nuclei. The last part of this review looks to future developments and the issue of the universality of coexistence in nuclei. Surprises continue to be discovered. With the major advances in reaching to extremes of proton-neutron number, and the anticipated new “rare isotope beam” facilities, guidelines for search and discovery are discussed.

Figure 1

Energies for the different $n\mathrm{p}\mathrm{\text{−}}n\mathrm{h}$ configurations in the Ca isotopes. Open squares (dashed lines) correspond to the lowest $0^{+}$ unperturbed $n\mathrm{p}\mathrm{\text{−}}n\mathrm{h}$ configuration corrected for the monopole interaction energy (see text), the open circles (full lines) correspond to the lowest $0^{+}$ state from diagonalizing in the $n\mathrm{p}\mathrm{\text{−}}n\mathrm{h}$ subspaces separately, and the diamonds show the three lowest-lying $0^{+}$ states from the fully mixed calculation. From 95.

Figure 2

Schematic view of the $2\mathrm{p}\mathrm{\text{−}}2\mathrm{h}$ neutron excitations from the $sd$ shell into the $fp$ shell (upper panel). In the lower part, the energy gap which equals the difference of the monopole energy for the normal and $2\mathrm{p}\mathrm{\text{−}}2\mathrm{h}$ intruder $0^{+}$ configuration (left panel) and the correlation energy in the normal and $2\mathrm{p}\mathrm{\text{−}}2\mathrm{h}$ intruder $0^{+}$ configuration [as derived from Eq. (7)] (right panel) are given for the Mg ($Z=12$) nuclei. From 99.

Figure 3

Relative position of the lowest normal and lowest neutron $2\mathrm{p}\mathrm{\text{−}}2\mathrm{h}$ intruder $0^{+}$ states, resulting from diagonalizing in the separate subspaces. From 99, and 97.

Figure 4

The different energy terms, contributing to the energy of the lowest proton $2\mathrm{p}\mathrm{\text{−}}2\mathrm{h}$ $0^{+}$ intruder state for heavy nuclei. On the right-hand side, a schematic view of the excitation is given. On the left-hand side, the unperturbed energy, the pairing energy, the monopole energy shift, and the quadrupole energy gain are presented, albeit in a schematic way.

Figure 5

Illustration of the energy surface $E(\beta )$ for a single collective parameter (quadrupole deformation $\beta$) which follows from self-consistent Hartree-Fock-(Bogoliubov) calculations.

Figure 6

Particle-number projected mean-field (small dotted curve and marked as “Mean-field”) as well as the particle- and angular-momentum projected energy curves (full, long-dashed, etc. lines), up to spin $J=10$ for ${}^{186}\mathrm{Pb}$, as a function of the quadrupole deformation variable ${\beta }_2$. From 151.

Figure 7

Energy spectrum for the lowest-lying positive-parity bands in ${}^{186}\mathrm{Pb}$ with even angular momentum $J$ and $K=0$ as a function of the quadrupole deformation parameter ${\beta }_2$. The particle- and angular-momentum projected energy curves are given as a reference. The excitation energy is derived from measuring the energy relative to the energy of the $0_1^{+}$ ground state. From 151.

Figure 8

The main regions of nuclear shape coexistence discussed in Sec. 3 are shown in relationship to closed shells. Regions A, F: see Sec. 3b1; regions B, C, D, and E: see Sec. 3b2; region G: see Sec. 3a8; region H: see Sec. 3a5; region I: see Sec. 3a3; region J: see Sec. 3a2; region K: see Sec. 3a4; and region L: see Sec. 3a1.

Figure 9

Isotope shift systematics for the Os-Pb isotopes. The large increases and “staggering” in the $N\sim 104$ region are the result of states in these nuclei with different quadrupole deformations. Note particularly ${}^{185}\mathrm{Hg}$, which possesses the largest known isomer shift. The trend in $\delta ⟨r^2⟩$ with $N$ (and therefore $A$) for a droplet model is shown. From 320.

Figure 10

Systematics of excited states in the even-Hg isotopes. Note the “parabolic intrusion” of the closely spaced bands of states (marked with solid lines) with $J=0,2,4,\dots$ centered on ${}^{182}\mathrm{Hg}$ ($N=102$). The data are taken from Nuclear Data Sheets. Some recent lifetime data can be found in 222.

Figure 11

Schematic view of characteristic states, in particular, emphasizing high-spin isomeric states, as indicators of shape coexistence (“high-$M$” should read “high-$J$”). From 144.

Figure 12

Systematics of excited states in the even-Pb isotopes. Note the parabolic intrusion of states centered on ${}^{186}\mathrm{Pb}$ ($N=104$). Heavy downward arrows indicate states connected by $E0$ transitions. Upward pointing arrows indicate excitations above which states have been omitted. Lifetimes of the $J=12$ seniority isomers are given: Note that this state coexists with intruder states in ${}^{188}\mathrm{Pb}$, particularly with a $K$ isomer shown in Fig. 23 (discussed further in the text). The data are from 301, 145, 146, 434, 460, and Nuclear Data Sheets. There are recent lifetime data for ${}^{188}\mathrm{Pb}$ (134; 220, 221) and ${}^{186}\mathrm{Pb}$ (220, 221).

Figure 13

Systematics of excited states in the odd-Pb isotopes. Heavy vertical downward arrows indicate states connected by $E0$ transitions. Diagonal arrows indicate states populated in $\alpha$ decay, with hindrance factors shown in the boxes. Vertical upward arrows indicate energies above which states with positive parity have been omitted. Dashed lines connect states with probable similar structures. The data are from 116, 18 (${}^{187}\mathrm{Pb}$ $\alpha$-decay hindrance factors), 33, 147, 596 (${}^{191,193}\mathrm{Pb}$ $\alpha$-decay hindrance factors), and Nuclear Data Sheets.

Figure 14

Systematics of $\alpha$-decay hindrances to excited $0^{+}$ states in the even-Rn, even-Po, even-Pb, even-Hg, and even-Pt isotopes: The ground-state-to-ground-state hindrance factors (numbers given in the boxes) are all normalized to unity. The ${}^{200}\mathrm{Rn}(0_1^{+})\to {}^{196}\mathrm{Po}(0_2^{+})$ $\alpha$-hindrance factor of $\sim 92$ is the highest $0^{+}\to 0^{+}$ value known. Note the ${}^{188}\mathrm{Po}(0_1^{+})\to {}^{184}\mathrm{Pb}(0_2^{+})$ $\alpha$-hindrance factor of 0.08: This implies that the normalization is to a strongly hindered (ground-state-to-ground-state) transition. The data are from Nuclear Data Sheets [except for ${}^{188}\mathrm{Pb}(\alpha ){}^{184}\mathrm{Hg}$, the hindrance factor is from 626, but see comment in Nuclear Data Sheets].

Figure 15

Systematics of $\alpha$-decay hindrances between the odd-At, odd-Bi, and odd-Tl isotopes: The hindrance factors (numbers given in the boxes) are normalized to the neighboring even-even ground-state-to-ground-state $\alpha$-decay strengths. Note the manner in which the low hindrance factors pick out the intruder states on either side of the $Z=82$ shell closure ($3s_{1/2}$ intruder states in the Bi and At isotopes and $1h_{9/2}$ intruder states in the Tl isotopes). The data are from 311 and Nuclear Data Sheets [except for ${}^{183,189}\mathrm{Tl}$, the energies of the $9/2^{-}$ states are from 16 and from 117, respectively].

Figure 16

Systematics of excited states in the even-Po isotopes. Seniority isomers with $J=8$ are indicated by their half-lives. Evidently, the intruding deformed states become the ground states for ${}^{190,192,194}\mathrm{Po}$. The data are from 247 and Nuclear Data Sheets. There are recent lifetime data for ${}^{194}\mathrm{Po}$ (220, 221).

Figure 17

Systematics of the odd-Tl $\pi (1\mathrm{p}\mathrm{\text{−}}2\mathrm{h})$, odd-Bi $\pi (2\mathrm{p}\mathrm{\text{−}}1\mathrm{h})$, and even-Pb $\pi (2\mathrm{p}\mathrm{\text{−}}2\mathrm{h})$ intruder-state energies (the even-Pb energies are divided by 2). Note the strong correlation of the three quantities: This is discussed further in the text. For the even-Pb isotopes, the candidate $\pi (4\mathrm{p}\mathrm{\text{−}}4\mathrm{h})$ states are shown as solid squares. The data are from Nuclear Data Sheets.

Figure 18

Systematics of bands observed built on the lowest $9/2^{-}$ and $13/2^{+}$ states in the odd-Tl isotopes. Strongly coupled bands with small $B(E2)$ values and decoupled bands with large $B(E2)$ values are observed (the values are given in $[e\mathrm{b}{]}^2$ in the boxes, these can be converted to W.u. for $A\approx 188$ by $\approx \times 160$). Mixing and repulsion of two $9/2^{-}$ configurations in ${}^{183}\mathrm{Tl}$ are indicated. The parabolic energy trends are discussed in the text. The data are from 458, 103, 87, and Nuclear Data Sheets.

Figure 19

Multiple-coexisting structures in ${}^{187}\mathrm{Au}$. The upper part of the figure shows the states assigned as the coupling of the $1h_{11/2}$ proton hole to a ${}^{188}\mathrm{Hg}$ core for which there are $\pi (2\mathrm{h})$ and $\pi (2\mathrm{p}\mathrm{\text{−}}4\mathrm{h})$, $0_1^{+}$ and $0_2^{+}$ states, respectively, i.e., $\pi (3\mathrm{h})$ and $\pi (2\mathrm{p}\mathrm{\text{−}}5\mathrm{h})$ states. The lower part of the figure shows states assigned as the coupling of the $1h_{9/2}$ proton particle to a ${}^{186}\mathrm{Pt}$ core for which there are $\pi (4\mathrm{h})$ and $\pi (2\mathrm{p}\mathrm{\text{−}}6\mathrm{h})$, $0_2^{+}$ and $0_1^{+}$ states, respectively, i.e., $\pi (1\mathrm{p}\mathrm{\text{−}}4\mathrm{h})$ and $\pi (3\mathrm{p}\mathrm{\text{−}}6\mathrm{h})$ states. Details of the coupling of particles and holes to the Pt and Hg cores are given by 509, 508; a simple perspective is provided by 380, 381, and 553. Transitions with observed $E0$ components are marked by arrows. The data are from Nuclear Data Sheets.

Figure 20

Systematics of bands built on the lowest $9/2^{-}$ and $13/2^{+}$ states in the odd-Au isotopes. Decoupled bands with large $B(E2)$ values (given in $[e\mathrm{b}{]}^2$ in the boxes) are observed. The $7/2^{-}$ states result from the $2f_{7/2}$ intruder configuration. Details are discussed in the text. The data are from 299, 298, 605, and Nuclear Data Sheets.

Figure 21

Systematics of the low-lying states in the odd-Hg isotopes near $N\sim 104$. Selected $\gamma$-ray transitions are shown as vertical arrows: in ${}^{177–183}\mathrm{Hg}$, observed following $\alpha$ decay; in ${}^{185,187}\mathrm{Hg}$ observed associated with decay of isomers. $\alpha$-decay feedings (from Pb parent nuclei) are shown as diagonal arrows: solid for decays of the odd-Pb $1i_{13/2}$ isomers, open for the odd-Pb $3p_{3/2}$ ground states. $\alpha$-decay hindrance factors, normalized to the neighboring doubly even ground-state-to-ground-state transitions, are shown in boxes attached to the arrows. States are grouped as spherical and deformed: The large isomer shift observed in ${}^{185}\mathrm{Hg}$, cf. Fig. 9, is between the ground state and 104 keV isomer. The data are from 14, 15 and Nuclear Data Sheets.

Figure 22

Systematics of Nilsson states (left-hand side) and rotational energy parameters (right-hand side) in the $N=105$ isotones. Note the smooth trend of Nilsson single-particle energies from nuclei with well-deformed ground states into the spherical region. The rotational parameters indicate equal or greater deformation at $Z=80$ than at $Z=72$. The data are from 315 and Nuclear Data Sheets.

Figure 23

Systematics of $K^{\pi }=8^{-}$ isomers in the $N=106$ isotones. Rotational parameters deduced from a rigid rotor energy relationship are shown circled and reveal that this deformed structure persists to the closed-shell nucleus ${}^{188}\mathrm{Pb}$. Note also that ${}^{188}\mathrm{Pb}$ supports this deformed $K$ isomer, an oblate $K^{\pi }={11}^{-}$ isomer, and a seniority isomer, the $J=12$ isomer (cf. Figs. 11, 12): This occurrence of three completely different kinds of isomerism dramatizes the role of coexistence at $Z=82$ which cannot be described as a “collapse of the shell structure.” Half-lives of the band heads are shown. The data are from 146, 400, and Nuclear Data Sheets.

Figure 24

Systematics of the odd-In $1/2^{+}$ $\pi (1\mathrm{p}\mathrm{\text{−}}2\mathrm{h})$, odd-Sb $9/2^{-}$ $\pi (2\mathrm{p}\mathrm{\text{−}}1\mathrm{h})$, and even-Sn $\pi (2\mathrm{p}\mathrm{\text{−}}2\mathrm{h})$ intruder-state energies (the even-Sn energies are divided by 2). This systematic is similar to that shown in Fig. 17. The data are from Nuclear Data Sheets.

Figure 25

Intruder states in ${}^{114}\mathrm{Cd}$ and ${}^{118}\mathrm{Te}$ (upper part) and ${}^{116}\mathrm{Sn}$ (lower part) classified by “intruder spin,” $I$: The multiplet of states labeled, e.g., as $I=3/2$ are formed from the sequence of six active protons configurations—$\pi (6\mathrm{h})$, $\pi (2\mathrm{p}\mathrm{\text{−}}4\mathrm{h})$, $\pi (4\mathrm{p}\mathrm{\text{−}}2\mathrm{h})$, and $\pi (6\mathrm{p})$, respectively; with standard SU(2) labels and interpretation given for the various multiplets. The figure is similar to one shown in 125.

Figure 26

Isotope shifts $\delta ⟨r^2⟩$ in $\mathrm{f}{\mathrm{m}}^2$ and two-neutron separation energies $S_{2n}$ in MeV for selected isotopic chains across the $N=60$ region. From 233.

Figure 27

Shape coexistence in the $N=58$ isotones, ${}^{96}\mathrm{Sr}$ and ${}^{98}\mathrm{Zr}$. The vertical arrows indicate $E0$ transitions with their observed values for ${\rho }^2(E0)\times {10}^3$; the value for ${}^{96}\mathrm{Sr}$ is the largest known for $A>56$. The cascade of two strong $E0$ transitions in ${}^{98}\mathrm{Zr}$ is discussed in the text. The data are from 648, 538, and Nuclear Data Sheets.

Figure 28

Shape coexistence in the $N=59$ isotones, ${}^{97}\mathrm{Sr}$ and ${}^{99}\mathrm{Zr}$. The three deformed bands with $K^{\pi }=3/2^{+},3/2^{-}$, and $9/2^{+}$ correspond to the Nilsson configurations $[411\uparrow ]$, $[541\uparrow ]$, and $[404\uparrow ]$ (with the notation $[N,n_z,\Lambda ,\Sigma ]$), respectively. The data are from 588, 589, 590, 648, and Nuclear Data Sheets.

Figure 29

Two-nucleon and multinucleon transfer strengths to $0_2^{+}$ states in the Zr isotopes, given relative to 100% for $0_1^{+}$ states. The strengths marked a) are from (${}^6\mathrm{Li}$, ${}^8\mathrm{B}$) reactions, and b) are from (${}^{14}\mathrm{C}$, ${}^{16}\mathrm{O}$) reactions. The data are from references given in Nuclear Data Sheets.

Figure 30

Two-nucleon and multinucleon transfer strengths to $0_2^{+}$ states in the Mo isotopes, given relative to 100% for $0_1^{+}$ states ($\mathrm{NS}=\mathrm{\text{not seen}}$). The data are from Nuclear Data Sheets.

Figure 31

Systematics of low-lying collective states in the $N=60$ isotones. Light vertical arrows show selected transitions with their $B(E2)$ values, and heavy vertical arrows show selected transitions with their ${\rho }^2(E0)\times {10}^3$ values. The data are from 314, 552, and Nuclear Data Sheets.

Figure 32

Electric quadrupole collectivity in ${}^{104}\mathrm{Ru}$ and ${}^{108}\mathrm{Pd}$. Arrows indicate selected transitions with their $B(E2)$ values. Boxes with a double underline give $⟨Q^2⟩$ values in units $e^2{\mathrm{b}}^2$ obtained by summing over squares of $E2$ matrix elements. Boxes with a double overline give the quadrupole moments $⟨Q⟩$ in $e\mathrm{b}$. (We refer the interested reader to the cited multi-Coulex papers for details of sign convention and determination.) A candidate $\pi (2\mathrm{p}\mathrm{\text{−}}6\mathrm{h})$ band is indicated in ${}^{108}\mathrm{Pd}$. The data are from 552, 563, and Nuclear Data Sheets.

Figure 33

Isotope shifts $\delta ⟨r^2⟩$ in ${\mathrm{fm}}^2$ and two-neutron separation energies $S_{2n}$ in MeV for selected isotopic chains across the $N=90$ region. The data are from 394 and 26.

Figure 34

Systematics of ${\rho }^2(E0)\times {10}^3$ values in the $N=90$ isotones. The large values indicate underlying coexistence of bands with different deformations that mix strongly. The level data are taken from Nuclear Data Sheets. The ${\rho }^2(E0)\times {10}^3$ values are taken from 314 for $0_2^{+}\to 0_1^{+}$, from 643 for $2_2^{+}\to 2_1^{+}$, and are calculated using lifetime data (319; 578; 384) and electron data in Nuclear Data Sheets for other transitions.

Figure 35

Low-lying states in ${}^{70–76}\mathrm{Ge}$ (upper part) and $⟨Q^2⟩$ values in $e^2{\mathrm{b}}^2$ for the $0_1^{+}$ and $0_2^{+}$ states in these Ge isotopes (lower part). The lower part is taken from 557 and the data in the upper part are from 449 and Nuclear Data Sheets. (Note that in the lower part, the value of $⟨Q^2⟩$ for the $0_2^{+}$ state in ${}^{70}\mathrm{Ge}$, cf. Table , should be 0.64 and not 0.50.)

Figure 36

Two-nucleon and multinucleon transfer strengths to $0_2^{+}$ states in the Ge isotopes, given relative to 100% for $0_1^{+}$ states ($\mathrm{NS}=\mathrm{\text{not seen}}$). The data are from 20, 64, and references given in Nuclear Data Sheets.

Figure 37

The ${\alpha }_A^2$, ${\beta }_A^2$ values, from Table 3 in 595, and see text, Eq. (15), and the ${\alpha }_n^2$, ${\beta }_n^2$ values (with $n=2–6$), from Table 3 in 37, and see text, Eq. (16), respectively, describing the $0_1^{+}$ and $0_2^{+}$ states and separately describing the proton (595) and neutron (37) parts of the wave functions in ${}^{70–76}\mathrm{Ge}$. This illustrates the nucleon pairs in mixed shell-model configurations schematically using ovals around solid circles (protons) and open circles (neutrons).

Figure 38

Systematics of positive-parity states in the $N=39$ isotones, arranged to reveal the probable occurrence of coexisting prolate and oblate shapes. Energies are given relative to $9/2_1^{+}$ in ${}^{71}\mathrm{Ge}$ ($E_x=198\mathrm{keV}$), ${}^{73}\mathrm{Se}(\mathrm{gs})$, $9/2_2^{+}$ in ${}^{75}\mathrm{Kr}$ ($E_x=726\mathrm{keV}$), arbitrarily offset in ${}^{77}\mathrm{Sr}$. Band assignments are based on $\gamma$-decay branching ratios. The data are from Nuclear Data Sheets.

Figure 39

Electric quadrupole collectivity in ${}^{74}\mathrm{Kr}$ and ${}^{76}\mathrm{Kr}$. Arrows indicate selected transitions with their $B(E2)$ values. Boxes with a double underline give $⟨Q^2⟩$ values in units $e^2{\mathrm{b}}^2$ obtained by summing over squares of $E2$ matrix elements. Boxes with a double overline give the quadrupole moment $⟨Q⟩$ in $e\mathrm{b}$. The data are from 113 and Nuclear Data Sheets.

Figure 40

Two-proton transfer data which indicate a pairing structure for $0_3^{+}$ states in the even-Te and even-Xe isotopes that support shape coexistence. The excited $0^{+}$ state populations are given as a percentage of the ground-state populations. The data are from 8, 167, and Nuclear Data Sheets.

Figure 41

Shape-coexisting bands in the $N=Z$ isotopes, ${}^{40}\mathrm{Ca}$ and ${}^{56}\mathrm{Ni}$. The vertical arrows indicate excitation energies above which other excited states are observed. The state at 9735 keV in ${}^{56}\mathrm{Ni}$ decays by proton emission. The data are from 277, 293, and Nuclear Data Sheets.

Figure 42

Systematics of the $d_{3/2}$ hole intruder orbital in the $f_{7/2}$ shell. The upper part shows the collective bands built on the configuration and the lower part shows the bandhead energies relative to the ground states (odd-proton cases on the left and odd-neutron cases on the right). The numbers given on the levels for ${}^{45}\mathrm{Sc}$ and ${}^{45}\mathrm{Ti}$ in the upper part of the figure are magnitudes of $Q_0$ in $e\mathrm{b}$ (e.g., ${0.66}^6=0.66\pm 0.06$), deduced from $B(E2)$ measurements by 40. Energies are from Nuclear Data Sheets. The $3/2^{+}$ state in ${}^{45}\mathrm{Sc}$ is at 12.6 keV.

Figure 43

Selected pairs of nuclei showing a possible relationship between $1\mathrm{p}\mathrm{\text{−}}2\mathrm{h}$ intruder states (marked with solid triangles) and low-lying excited $0^{+}$ states. The data are from 116, 7, 159, 226, 532, 202, and Nuclear Data Sheets.

Figure 44

Systematics of the even-mass $N=20$ isotones. Known and possible $\nu (2\mathrm{p}\mathrm{\text{−}}2\mathrm{h})$ states are shown. In ${}^{38}\mathrm{Ar}$ this structure gives rise to a deformed band. The extrapolation of this structure (see also Fig. 45) to explain the deformed states in ${}^{32}\mathrm{Mg}$ is discussed in the text. The estimate of the $0^{+}$ $\nu (2\mathrm{p}\mathrm{\text{−}}2\mathrm{h})$ state at 2400 keV in ${}^{34}\mathrm{Si}$ is from Fig. 45 (and see the remark in the text on the $2_1^{+}$ state in ${}^{34}\mathrm{Si}$). The data are from 159, 415, 506, 635, and Nuclear Data Sheets.

Figure 45

Systematics of multiparticle–multihole states in the $N=19$, 20, and 21 isotones. The sum of the $\nu (1\mathrm{p}\mathrm{\text{−}}2\mathrm{h})$ and $\nu (2\mathrm{p}\mathrm{\text{−}}1\mathrm{h})$ state energies strongly correlates with that of the $\nu (2\mathrm{p}\mathrm{\text{−}}2\mathrm{h})$ state energies (cf. Figs. 17, 24), supporting the extrapolation shown in ${}^{34}\mathrm{Si}$ and ${}^{32}\mathrm{Mg}$. The $N=21$ extrapolation from ${}^{35}\mathrm{Si}$ to ${}^{33}\mathrm{Mg}$ is assumed to be parallel to $N=19$. The data are from 159, 414, 624, 635, and Nuclear Data Sheets.

Figure 46

Energies of $2_1^{+}$ states for $N\ge 18$, $Z\le 24$, illustrating the $N$, $Z=20$, $N=28$ shell closures and the neutron-drip line ($B_n\sim 0$). Some features are discussed in the text. The data are from 652, 171, 357, 42, 226, 285, 139, 140, and Nuclear Data Sheets.

Figure 47

Global systematics for the quantities $B_{42}≔B(E2;4^{+}\to 2^{+})/B(E2;2^{+}\to 0^{+})$ and $E(4^{+})/E(2^{+})$ plotted against $B_{20}≔B(E2;2^{+}\to 0^{+})$ in W.u. The $E(4^{+})/E(2^{+})$ values are characteristic of nonrotational nuclei and yet the $B_{42}$ ratios are characteristic of a rigid rotor. Uncertainties in $B_{20}$ and $B_{42}$ are not shown as they would clutter the figure. The data are from Nuclear Data Sheets.

Figure 48

Illustration of the concept of “multishells.” Removal of a closed-shell “line” between two open-shell regions creates an open multishell region. For $Z=82$ this provides an explanation of the extreme deformation associated with the coexisting states observed in the Hg and Pb isotopes. This perspective may also provide an explanation of the mass regions where superdeformation is observed. The superscripts, e.g., [2, 1] attached to ${}^{186}\mathrm{Pb}^{[2,1]}$, indicate the number of proton and neutron “regular” shells forming the multishells as shown by the diagonal lines passing through the location of the isotope. The region boxed in the lower left-hand corner contains mainly $N=Z$ line cases, i.e., “symmetric” cases; the region boxed in the upper right-hand corner contains only asymmetric cases.

Figure 49

Systematics of the low-lying positive-parity states in ${}^{68–76}\mathrm{Ni}$ shown relative to the $(1g_{9/2}{)}^n$, $J=8$ states. The pattern shows that in ${}^{68}\mathrm{Ni}$ the $0_2^{+}$ state results from a strong mixing between configurations involving different pair occupancies of the $1g_{9/2}$ and $2p_{1/2}$, $2p_{3/2}$, and $1f_{5/2}$ orbitals (cf. Fig. 37). The data are from Nuclear Data Sheets.

Figure 50

A similar pattern, as the one in Fig. 49 (the Ni isotopes) for the $N=50$ isotones, also involving the same shell-model orbitals. The data are from Nuclear Data Sheets.

Figure 51

A schematic view of the intruder-state “parabolas,” shown to dramatize the way that shells and subshells suppress the emergence of low-energy collectivity in nuclei. (a) The situation where deformed structures intrude to become the ground state at the middle of a singly closed shell, e.g., ${}^{32}\mathrm{Mg}$; (b) where the ground states for a sequence of singly closed shell nuclei remain spherical, but deformed structures form excited intruder bands, e.g., the Sn and Pb isotopes; (c) where a subshell may suppress intrusion of a deformed structure from becoming the ground state or a low-lying excited band, e.g., $N=50$, 82.

Figure 52

The vertical shells of the symplectic collective model, labeled by the number of oscillator quanta and the quantum numbers of the SU(3) subgroup of the model. Some details are discussed in the text. Adapted from 494 and 88.