-
Nuclear halo is characterized by weak binding and large spatial extension due to the considerable occupation of low-l (s- or p-wave) orbitals by valence nucleon(s) close to the threshold of the particle emission[6, 10, 17, 36−37, 69−70]. The various shapes of atomic nuclei originate from different quantum correlations. For example, the quadrupole deformation (ellipsoidal shape) is related to the $ Y_{20} $ correlation and the $ Y_{30} $ correlation for octupole deformation (pear-shaped). For a weakly bound nucleus in the light mass region, if the configuration of valence nucleons contains the mixing of $ sd $ or $ pf $ orbtials, a deformed halo might be developed. In neutron-rich B, C, Ne, Na, and Mg isotopes, such conditions can be satisfied such that deformed halos are formed. The DRHBc theory has been used to study the ground-state properties, including deformations, matter radii, density profiles, and single-particle levels, of deformed halo nuclei in these isotopic chains. In this section, first, we give a typical example to show how the halo forms in a deformed weakly bound nucleus and then we summarise the application of the DRHBc theory on deformed halos and the comparison with the available experimental information.
The first application of the DRHBc theory is to study the deformed halo structures in the magnesium isotopic chain. With the density functional NL3[71], the DRHBc calculations predicted that 46Mg is the neutron drip line nucleus[22] and 44Mg is a deformed halo nucleus. The two-neutron separation energy $S_{ 2{\mathrm{n}}}$ of 44Mg is very small, only 0.44 MeV, and the quadrupole deformation parameter $ \beta_2 $ is 0.32, meaning that the ground state has a prolate shape. Fig. 2(a) shows the proton and neutron two-dimensional density distribution of 44Mg. Since 44Mg is an extremely weakly bound neutron-rich nucleus, the neutron density is more spatially extended than the proton density. In the MF approach, the configuration and formation mechanism of the halo nucleus can be obtained by analyzing the structure of the valence orbitals. In Fig. 3, the neutron SPLs of 44Mg in the canonical basis near the neutron Fermi level ($ \lambda_n $) is shown. Near the neutron Fermi level, the valence neutrons occupy weakly bound $ \frac{3}{2}^- $, $ \frac{1}{2}^- $ levels, and some orbitals in the continuum. There are p-wave components in the levels $n = 3,\; 4,\; 6$, and the occupation number of the p-wave orbitals is about 2, which leads to the halo structure of this nucleus.
Figure 2. Density distributions of 44Mg with the z-axis as the symmetry axis: (a) The proton ($ x < 0 $) and the neutron ($ x > 0 $) densities, (b) the density of the neutron core, and (c) the density of the neutron halo. In each plot, a dotted circle is drawn to guide the eye[22]. (color online)
Figure 3. Single neutron levels with the quantum numbers $ \varOmega^\pi $ around the chemical potential (dotted line) in the canonical basis for 44Mg versus the occupation probability $ v^2 $. The order n, $ \varOmega^\pi $, and the main DWS components for orbitals close to the threshold are also given. The dashed line corresponds to the BCS formula with an average pairing gap[22]. (color online)
In the DRHBc theory description of magnesium isotopes, 42Mg is a deformed two-neutron halo nucleus[40], and its halo structure is caused by the p-wave dominated valence neutrons. However, the heaviest observed magnesium isotope is 40Mg up to now[72]. Therefore the existence of deformed halos in 42, 44Mg still requires further experimental confirmation. Whether 40Mg is a deformed halo nucleus remains an open question. The DRHBc theory predicts that 40Mg is not a deformed halo nucleus. It should be noted that the triaxial deformation plays a role in the description of 40Mg[73] and the influence of triaxial deformation on single particle structure should be investigated in future. In Mg isotopes, a single-neutron halo has already been observed experimentally in 37Mg[24]. Considering the blocking effect, the DRHBc calculations with different density functionals have shown that 37Mg is a deformed one-neutron halo nucleus[46] with p-wave.
For the B isotopic chain, the DRHBc theory has predicted the existence of deformed neutron halos in 17, 19B[43−44] with the valence neutrons occupying the s-wave orbital. The s-wave component of the two valence neutrons in 17B obtained from quasi-free neutron knockout reactions is only 9(2)%[43]. The deformation of 17B has been studied in 2005[74] and the quadrupole deformation parameter of neutrons is about 0.6[75]. The DRHBc theory predicts that 17B is well deformed with the s-wave component of 14% and the calculated neutron separation energy and radius are consistent with experimental data. The measurements of the $ B(E1) $ excitation in 19B by using Coulomb breakup reactions confirmed the presence of a neutron halo[76]. Assuming the two-neutron separation energy being 0.5 MeV, a three-body model suggests that 19B is a two-neutron halo nucleus caused by the s-wave with the percentage of 35%. The antisymmetrized molecular dynamics and relativistic mean-field calculations have predicted that 19B is deformed[77−79]. The DRHBc calculations show that the halo in 19B is crucially related to the deformation effects and the valence neutrons are mainly dominated by s- and d-waves with the s-wave percentage of 36%. Furthermore, the two-neutron halos in 17, 19B exhibit shape decoupling effects, prolate cores and oblate halos. The neutron drip line of B isotopic chain is located at 19B and experimentalists have observed 20B and 21B as resonance states[80], which might be dominated by the channels 19B+n and 19B+2n, respectively. Therefore it will be very interesting to study the structure of the resonant states in 20, 21B based on a proper description of halo structure in 19B.
For the C isotopic chain, the DRHBc theory predicts that 15C and 22C are deformed halo nuclei[25, 42]. The ground state of 15C has a prolate shape and the valence neutron is partially occupied with s-wave, leading to the appearance of a halo. The neutron core is nearly spherical while the neutron halo has an elongated shape, demonstrating the shape decoupling effect. Recent calculations using time-dependent density functional theory have shown that the deformed halo structure of 15C enhances fusion cross-sections below the Coulomb barrier in the fusion reaction with 232Th[81]. For 22C, the halo configuration of this nucleus is not completely determined up to now experimentally. The measurement in 2010[82] shown that the matter radius extracted from the cross section is 5.4 fm, indicating a huge halo. The two-neutron removal reaction in 2012[83] supported that the valence neutrons are dominated by s-wave. But the following experimental work[84] shows a matter radius of ($ 3.44 \pm 0.08 $) fm. The theoretical descriptions of this nucleus vary from model to model[60, 85−97]. The DRHBc theory calculations show that the ground state of 22C has an oblate shape and predict a “shrunk halo” due to the amplitude of s-wave in valence neutrons being 25%, which is smaller than the other theoretical predictions[25]. In addition, this work[25] also demonstrates that various exotic nuclear phenomena, including deformed halos, changes of shell-closure due to the inversion of SPLs, and shape decoupling effects coexist in this single nucleus 22C. As for 19C, the DRHBc calculations show an oblate shape for its ground state without a neutron halo. The calculated ground state spin-parity, $ {\frac{3}{2}}^+ $, does not agree with the experiment $ {\frac{1}{2}}^+ $. However, in the DRHBc calculations, an excited state of 19C with prolate shape (referred as 19C*) has the spin-parity of $ {\frac{1}{2}}^+ $ and exhibits a deformed one-neutron halo.
We summarize the DRHBc predictions for s-wave halo in Table 1 and some available experimental information is also given for comparison. Besides the studies mentioned above, the DRHBc theory is also applied to the newly observed Na isotopes 39Na and has predicted that 39,41Na have deformed halos[45]. The self-consistent description of deformed halos using the DRHBc theory is also combined with reaction models to examine the reaction observables, such as interaction cross section and parallel momentum distributions in breakup reaction[101]. With the Fock terms considered, the deformed relativistic Hartree-Fock-Bogoliubov theory has been developed recently[61] and it has been applied to the halo in 11Be[102]. It should be noted that the predictions of deformed halos can also be made by using non-relativistic MF models[103−106] and an “egg-like” halo has been predicted[103]. Other methods, such as Green's function method[107−108], complex scaling method[109], and the analytical continuation of the coupling constant[110] based on self-consistent MF calculation are also used to investigate the halo structures in deformed nuclei.
Nuclei $ \beta_2 $ $S_{ 2{\mathrm{n} } }$ ($S_{ \mathrm{n} }$)/MeV $ R_{{\mathrm{m}}} $/fm Percentage of s-wave Shape decoupling DRHBc DRHBc Exp. DRHBc Exp. DRHBc 17B 0.52 1.78 1.38(21) 3.03 3.00(6)[99] 14% prolate core with oblate halo 19B 0.36 0.22 0.089(56) 3.27 3.11(13) [99] 36% prolate core with oblate halo 15C 0.26 1.22 1.2181(8) 2.64 2.54(4) [100] 36% spherical core with prolate halo 19C* 0.37 0.04 0.563 3(915) 3.05 3.16(7)[100] 66% prolate core with oblate halo 22C −0.26 0.40 0.035(20) 3.25 3.44(8)[84] 25% oblate core with prolate halo Table 1. The quadrupole deformation parameter $ \beta_2 $, two- or one- neutron separation energy $S_{ 2\rm{n}}(S_{ {\rm n}})$, root-mean-square matter radius $R_{\rm{m}}$, and the amplitude of s-wave components of the halo nuclei in born and carbon isotopes from the DRHBc calculations. Available experimental values are included for comparison. The experimental data of neutron-separation energies are taken from AME2020[98].
Shape Decoupling Effects and Rotation of Deformed Halo Nuclei
doi: 10.11804/NuclPhysRev.41.2023CNPC56
- Received Date: 2023-08-14
- Rev Recd Date: 2024-03-01
- Available Online: 2024-03-15
- Publish Date: 2024-03-20
-
Key words:
- exotic nuclei /
- deformed halo /
- shape decoupling effect /
- nuclear mass /
- rotational excitation /
- density functional theory
Abstract: With the development of radioactive-ion-beam facilities, many exotic phenomena have been discovered or predicted in the nuclei far from the $\beta$ stability line, including cluster structure, shell structure, deformed halo, and shape decoupling effects. The study of exotic nuclear phenomena is at the frontier of nuclear physics nowadays. The covariant density functional theory (CDFT) is one of the most successful microscopic models in describing the structure of nuclei in almost the whole nuclear chart. Within the framework of CDFT, toward a proper treatment of deformation and weak binding, the deformed relativistic Hartree-Bogoliubov theory in continuum (DRHBc) has been developed. In this contribution, we review the applications and extensions of the DRHBc theory to the study of exotic nuclei. The DRHBc theory has been used to investigate the deformed halos in B, C, Ne, Na, and Mg isotopes and the theoretical descriptions are reasonably consistent with available data. A DRHBc Mass Table Collaboration has been founded, aiming at a high precision nuclear mass table with deformation and continuum effects included, which is underway. By implementing the angular momentum projection based on the DRHBc theory, the rotational excitations of deformed halos have been investigated and it is shown that the deformed halos and shape decoupling effects also exist in the low-lying rotational excitation states of deformed halo nuclei.
Citation: | Xiangxiang SUN, Shangui ZHOU. Shape Decoupling Effects and Rotation of Deformed Halo Nuclei[J]. Nuclear Physics Review, 2024, 41(1): 75-85. doi: 10.11804/NuclPhysRev.41.2023CNPC56 |