Islands of shape coexistence from single-particle spectra in covariant density functional theory

Using covariant density functional theory with the DDME2 functional and labeling single-particle energy orbitals by Nilsson quantum numbers, a search for particle-hole (p-h) excitations connected to the appearance of shape coexistence is performed for $Z=38$ to 84. Islands of shape coexistence are found near the magic numbers $Z=82$ and $Z=50$, restricted in regions around the relevant neutron midshells $N=104$ and $N=66$, respectively, in accordance with the well-accepted p-h interpretation of shape coexistence in these regions, which we call neutron-induced shape coexistence, since the neutrons act as elevators creating holes in the proton orbitals. Similar but smaller islands of shape coexistence are found near $N=90$ and $N=60$, restricted in regions around the relevant proton midshells $Z=66$ and $Z=39$, respectively, related to p-h excitations across the three-dimensional isotropic harmonic-oscillator magic numbers $N=112$ and $N=70$, which correspond to the beginning of the participation of the opposite parity orbitals $1i_{13/2}$ and $1h_{11/2}$, respectively, to the onset of deformation. We call this case proton-induced shape coexistence, since the protons act as elevators creating holes in the neutron orbitals, thus offering a possible microscopic mechanism for the appearance of shape coexistence in these regions. In the region around $N=40$, $Z=40$, an island is located on which both neutron p-h excitations and proton p-h excitations are present.

Figure 1

Energies (in MeV) of $1h_{11/2}$ single-particle proton orbitals relative to the Fermi energy obtained by CDFT for $Z=78–84$ isotopes. For $N\approx 98–110$ the orbital 11/2[505] (normally lying below $Z=82$) is vacant, thus creating two holes. Panel (a) is adapted from Ref. [37]. In this figure, as well as in all subsequent ones, the spherical shell-model quantum labels $n{l}_j$ [25] are shown in order to indicate the levels from which the reported deformed Nilsson orbitals $K\left[N{n}_z\mathrm{\Lambda }\right]$ [26] come from. See Sec. 3 for further discussion.

Figure 2

Energies (in MeV) of $3s_{1/2}$ and $2d_{3/2}$ single-particle proton orbitals relative to the Fermi energy obtained by CDFT for $Z=78–84$ isotopes. For $N\approx 98–110$ the orbitals 1/2[400] and 3/2[402] (normally lying below $Z=82$) are vacant, thus creating four holes. Panel (a) adapted from Ref. [37]. See Sec. 3 for further discussion.

Figure 3

Energies (in MeV) of $1h_{9/2}$ single-particle proton orbitals relative to the Fermi energy obtained by CDFT for $Z=78–84$ isotopes. For $N\approx 98–110$ the orbitals 1/2[541] and 3/2[532] (normally lying above $Z=82$) are occupied, thus creating four particle excitations. Panel (a) is adapted from Ref. [37]. See Sec. 3 for further discussion.

Figure 4

Energies (in MeV) of $2d_{5/2}$ single-particle proton orbitals relative to the Fermi energy obtained by CDFT for $Z=78–84$ isotopes. All orbitals (normally lying below $Z=82$) are occupied, except 5/2[402], which gets empty for the Pt isotopes with $N=100–108$. See Sec. 3 for further discussion.

Figure 5

Energies (in MeV) of $1g_{7/2}$ single-particle proton orbitals relative to the Fermi energy obtained by CDFT for $Z=78–84$ isotopes. All orbitals (normally lying below $Z=82$) are occupied. See Sec. 3 for further discussion.

Figure 6

Potential-energy surface (PES) for the nucleus ${}_{80}{}^{182}\mathrm{Hg}_{102}$, obtained by CDFT using the DDME2 functional used throughout the present work. All energies are normalized with respect to the binding energy of the corresponding ground state. The PES looks very similar to Fig. 18(a) of Ref. [83], produced for the same nucleus within the IBM-CM framework. See Sec. 3 for further discussion.

Figure 7

Energies (in MeV) of single-particle proton orbitals relative to the Fermi energy obtained by CDFT for Te ($Z=52$) isotopes. 2p-2h proton excitations are seen for $N=64–68$. (a) The orbital 9/2[404] of $1g_{9/2}$ (normally lying below $Z=50$) is vacant, while (b) the orbital 3/2[422] of $1g_{7/2}$ (normally lying above $Z=50$) is occupied. Adapted from Ref. [37]. See Sec. 3 for further discussion.

Figure 8

Energies (in MeV) of single-particle proton orbitals relative to the Fermi energy obtained by CDFT for Zr ($Z=40$) isotopes. 6p-6h proton excitations are seen for $N=38–40$. (a) The orbitals 1/2[301] of $2p_{1/2}$, as well as 3/2[301], 5/2[303] of $1f_{5/2}$ (normally lying below $Z=40$) are vacant, while (b) the orbitals 1/2[440], 3/2[431], 5/2[422] of $1g_{9/2}$ (normally lying above $Z=40$) are occupied. Adapted from Ref. [37]. See Sec. 3 for further discussion.

Figure 9

Energies (in MeV) of single-particle proton orbitals relative to the Fermi energy obtained by CDFT for Sr ($Z=38$) isotopes. 4p-4h proton excitations are seen for $N=38–40$. (a) The orbitals 3/2[301], 5/2[303] of $1f_{5/2}$ (normally lying below $Z=40$) are vacant, while (b) the orbitals 1/2[440], 3/2[431] of $1g_{9/2}$ (normally lying above $Z=40$) are occupied. See Sec. 3 for further discussion.

Figure 10

Energies (in MeV) of single-particle neutron orbitals relative to the Fermi energy obtained by CDFT for $N=90$ isotones. 2p-2h neutron excitations are seen for $Z=60–64$. The orbital 1/2[660] of $1i_{13/2}$ (normally lying above $N=112$) shown in this figure is occupied, while the orbital 5/2[523] of $2f_{7/2}$ (normally lying below $N=112$) shown in the next figure is vacant. Panels (a) and (b) adapted from Ref. [37]. See Sec. 4 for further discussion.

Figure 11

Energies (in MeV) of single-particle neutron orbitals relative to the Fermi energy obtained by CDFT for $N=90$ isotones. 2p-2h neutron excitations are seen for $Z=60–64$. The orbital 5/2[523] of 2f7/2 (normally lying below $N=112$) shown in this figure is vacant, while the orbital 1/2[660] of $1i_{13/2}$ (normally lying above $N=112$) shown in the previous figure is occupied. Panels (a) and (b) adapted from Ref. [37]. See Sec. 4 for further discussion.

Figure 12

Energies (in MeV) of single-particle neutron orbitals relative to the Fermi energy obtained by CDFT for $N=90$ isotones. The orbital 3/2[521] of $1h_{9/2}$ is gradually falling with increasing $N$. For $N=90$ it is empty everywhere above $Z=58$, for $N=92$, 94 it is still empty at $Z=60–70$, while at $N=94$ it has already sunk below the Fermi energy. See Sec. 4 for further discussion.

Figure 13

Energies (in MeV) of single-particle neutron orbitals relative to the Fermi energy obtained by CDFT for$N=60$ isotones. 4p-4h proton excitations are seen for $Z=40–42$. (a) The orbitals 1/2[411] of $3s_{1/2}$ and 5/2[413] of $2d_{5/2}$ (normally lying below $N=70$) are vacant for $Z=40$, 42, while (b) the orbitals 1/2[550], 3/2[541] of $1h_{11/2}$ (normally lying above $N=70$) are occupied. Adapted from Ref. [37]. See Sec. 4 for further discussion.

Figure 14

Energies (in MeV) of single-particle neutron orbitals relative to the Fermi energy obtained by CDFT for $N=58$ isotones. 4p-4h proton excitations are seen for $Z=40–42$. (a) The orbitals 1/2[411] of $3s_{1/2}$ and 5/2[413] of $2d_{5/2}$ (normally lying below $N=70$) are vacant for $Z=40$, while (b) the orbital 1/2[550] of $1h_{11/2}$ (normally lying above $N=70$) is occupied. See Sec. 4 for further discussion.

Figure 15

Energies (in MeV) of single-particle neutron orbitals relative to the Fermi energy obtained by CDFT for $N=40$ isotones. 6p-6h proton excitations are seen for $Z=40$. (a) The orbitals 5/2[303], 3/2[301] of $1f_{5/2}$ and 1/2[301] of $2p_{1/2}$ (normally lying below $N=40$) are vacant for $Z=40$, while (b) the orbitals 1/2[440], 3/2[431], 5/2[422] of $1g_{9/2}$ (normally lying above $N=40$) are occupied. See Sec. 4 for further discussion.

Figure 16

Energies (in MeV) of single-particle neutron orbitals relative to the Fermi energy obtained by CDFT for $N=38$ isotones. 4p-4h proton excitations are seen for $Z=40$. (a) The orbitals 5/2[303], 3/2[301] of $1f_{5/2}$ (normally lying below $N=40$) are vacant for $Z=40$, while (b) the orbitals 1/2[440], 3/2[431] of $1g_{9/2}$ (normally lying above $N=40$) are occupied. Adapted from Ref. [37]. See Sec. 4 for further discussion.

Figure 17

Islands of shape coexistence (SC) found in the present study. Islands corresponding to neutron-induced SC are shown in yellow, islands due to proton-induced SC are exhibited in cyan, while islands due to both mechanisms are shown in green. See Secs. 3 and 4 for further discussion.