1 Introduction

Shape coexistence in nuclei is an old, but still very exciting, research subject in nuclear physics [1]. Since the discovery of a large jump in the mean-squared charge radius, which is associated with a dramatic change in shape, considerable effort has been devoted to investigate nuclei exhibiting different shapes, especially at low excited energy [2]. Over the years, significant progress has been made toward a unified view of shape coexistence within the same atomic nuclei both theoretically and experimentally, though to fully understand such phenomena is still one of the greatest challenges faced by theories of nuclear structure [3].

Experimentally, shape coexistence in nuclei has been observed by using different techniques, ranging from optical [2] and laser spectroscopy [4, 5] to in-beam spectroscopy [6, 7] and decay spectroscopy [8, 9]. Further, more and more new experimental have data become available to date through lifetime measurements [10, 11] and even by using new techniques, such as Coulomb excitation of postaccelerated radioactive beams. Until now, it has evolved from the early interpretation of Morinaga [12] into a phenomenon that appears all through the nuclear landscape, including in light nuclei as well as in heavy nuclei [1, 3]. On the theoretical side, the most fundamental picture to describe an atomic nucleus is to regard it as a system of many nucleons interacting through an in-medium effective nuclear force. The nuclear properties stem from the interplay of the low multipoles of such interaction which generate nuclear mean field, characterized by a nuclear shell structure, and the high multipoles which scatter the nucleons out of their independent orbitals, i.e. the nuclear pairing and proton-neutron correlations. Indeed, the origin of shape coexistence is attributed to the balance between two opposing nuclear force components, e.g., the stabilizing effect of closed shells and subshells, causing the nucleus to retain a spherical shape, and the residual interactions between protons and neutrons (especially the proton-neutron interaction), driving the nucleus into a deformed shape. Based on this, up to date, several kinds of ways are developed to understand the shape-coexisting structure in nuclei, including the calculations within large-scale shell model approach [13, 14], mean-field approaches based on energy density functional [15-19], and algebraic approaches, e.g., the interacting boson model (IBM) [20-23].

In addition, due to the semi-magic characteristic of the N=104 (between 82 and 126 closed shells) isotonic chain, these isotones have been paid great attention to for a long time. There is a general understanding that nuclear collectivity is highest when the number of nucleons outside closed shells is largest, e.g., around midshell. A formal rationale for this rule is provided by the simple seniority scheme [24]. As the doubly midshell nucleus, ¹⁷⁰Dy has the largest ground-state deformation, consisting with the above rule [25]. However, the situation becomes more complicated when the evolution of the proton number is considered. The shape-coexisting phenomenon at low spin in N=104 isotones ¹⁸⁶Pb [26], ¹⁸⁴Hg [27], and ¹⁸²Pt [28] has been unambiguously confirmed in experiments. Naturally, extending the study and revealing the evolution law of the nuclear properties in this isotonic chain are desirable and of interest to some extent.

At present, 22 members from Z=63 to 84 have been discovered in this N=104 isotonic chain, including 3 stable, 12 proton-rich, and 7 neutron-rich isotones [29]. In this investigation, we have performed the systematic total Routhian surface (TRS) calculations for nine even-even N=104 isotopes ranging from ¹⁷⁰Dy to ¹⁸⁶Pb in which at least three yrast levels are identified experimentally, focusing on the possible shape coexistence with rotation. Prior to this work, we also performed the systematic studies in tungsten [30], osmium [31], and superheavy [32] isotopes by using similar calculations, but paying attention to different physics.

The article is organized as follows. In Sec. 2 we present a brief description of the theoretical formalism used to obtain the main ingredient of the present study, i.e., the potential energy surface (PES) calculations. Section 3 is devoted to the numerical calculated results and discussions for the present work. Finally, a brief summary is given in Section 4.

2 Theoretical descriptions

The present TRS approach is based on the macroscopic-microscopic model and cranking shell model [33, 34]. Such an approach has several standard components, each one individually familiar from the literature. Therefore, we would like to briefly outline the unified procedure and provide some necessary references for the readers.

The total Routhian, which is called "Routhian" rather than "energy" in a rotating frame of reference, is the sum of the energy of the non-rotating state and the contribution due to cranking. That is, the total Routhian can generally be written as

where the energy of the non-rotating state, E^ω=0, consists of a macroscopic part and a fluctuating microscopic one. The macroscopic energy is obtained from the standard liquid-drop model (LDM) [35] with the parameters used by Myers and Swiatecki. The microscopic correction part, which arises because of the nonuniform distribution of single-particle levels, is calculated by means of the well-known Strutinsky method [36]. Note that the Strutinsky shell-correction method has been considered as a major leap forward in the nuclear many-body problem, which can optimize the liquid-drop energy and give relatively high descriptive power.

During the process of microscopic calculations, the single-particle energies and wave functions are obtained by solving the Schrödinger equation of the stationary states for an average nuclear potential of Woods-Saxon (WS) type including a central field, a spin-orbit interaction, and the coulomb potential for the protons [37]. The deformed WS potential is generated numerically at each (β₂,γ,β₄) deformation lattice by using the cranked parameters [38, 39]. The Hamiltonian matrix is built by means of the axially deformed harmonic oscillator basis with the principal quantum number N ≤ 12 and 14 for protons and neutrons, respectively. Then its eigenvalues and eigenfunctions are calculated in terms of a standard diagonalization procedure.

The pairing correlation is treated using the Lipkin-Nogami (LN) approach [40, 41] in which the particle number is conserved approximately and thus the spurious pairing phase transition encountered in the usual Bardeen-Cooper-Schrieffer (BCS) calculation can be avoided. Not only monopole but also doubly stretched quadrupole pairings are considered. The monopole pairing strength, G, is determined by the average gap method [42] and quadrupole pairing strengths are obtained by restoring the Galilean invariance broken by the seniority pairing force [34, 43].

To calculate the cranking contribution, the rotation Hamiltonian, Hω ($\equiv H-\omega J_x$), will be adopted instead of the static Hamiltonian, H. The resulting cranked-Lipkin-Nogami (CLN) equation will take the form of the well-known Hartree-Fock-Bogolyubov-like (HFB) equation and can be solved by using the HFB Cranking (HFBC) method [44].

Finally, one can obtain the TRS after the numerical calculated Routhians at fixed ω are interpolated using a cubic spline function between the lattice points. Then, the nuclear properties, including the equilibrium deformation and shape coexistence, can be deduced from it.

3 Results and discussion

During the process of the actual calculations, the Bohr shape deformation parameters and Cartesian quadrupole coordinates, X=β₂cos(γ+30°) and Y=β₂sin(γ+30°), were used [45], where the parameter β₂ specifies the magnitude of the quadrupole deformation, while γ specifies the asymmetry of the shape. The three sectors [-120°, -60°], [-60°, 0°], and [0°, 60°] obviously represent the same triaxial shapes, but they denote respectively rotation about the long, medium, and short axes at non-zero cranking frequencies, according to the Lund convention [46]. In Fig. 1 we give an example of the present calculation corresponding to the nucleus ${}_{80}^{184}$ Hg₁₀₄. One can see in this figure that there are three minima in the energy surface. The deepest one, i.e., the ground state, carries deformation parameters β₂ = 0.14 and γ =-60°, indicating that in this state the nucleus is an oblate spheroid. Such calculation agrees with the experimental data [5, 47]. The second minimum with an excited energy E_exc 0.52MeV corresponds to a prolate shape with deformation β₂ = 0.26. The difference of the relative energy of this prolate and oblate states is qualitatively supported by a two-band-mixing model estimation 0.33MeV [48, 49]. Thus, it can be said that the present calculation reproduces the shape coexistence in this nucleus. Furthermore, it seems that the superdeformed prolate minimum below 2 MeV is developed at β₂=0.45. Indeed, with increasing rotational frequency, such superdeformed minimum, as discussed below, evolves toward the yrast state.

Figure 1:Full-screen View

(Color online) Calculated ground-state potential energy surface for the nucleus ${}_{80}^{184}$Hg₁₀₄. The energy difference between neighboring contours is 200 keV. At each given (β₂, γ) point, the PES has been minimized with respect to the deformation parameter, β₄. The solid circle, square, and triangle denote the first, second, and third minima, respectively.

As mentioned above, the co-existing shapes correspond to the different minima in the calculated nuclear energy (or Routhian) surface. Once the energy difference between the different minima is small enough, the populating probability of such states will be comparable and no one will be strongly favored. They could be observed simultaneously in experiment which means shape coexistence. Therefore, to understand such a co-existing phenomenon, it is necessary to reveal how the minima are generated in the TRS calculations. It has been well understood that the calculated minina (ground or shape-coexisting states) and maxima (or saddle points) can be attributable to the nonuniform distribution of the single-particle levels in the vicinity of the Fermi surface, namely, the shell effect [36]. In general, the minimum corresponds to a region of low level density, i.e., a region with shell gap, whereas saddle points will usually are developed in the vicinity of level crossings, the region of high level density. If several regions with low level density appear in the calculated energy surface, the corresponding minima could be produced. Figures 2 and 3 show the calculated neutron single-particle levels along β₂ and γ deformation degrees of freedom, respectively, for the selected ${}_{80}^{184}$ Hg₁₀₄ nucleus. One can see from Fig. 2 that the large gaps near the Fermi level, corresponding to low single-particle level density, actually appear at the β₂ positions of three minimum, as seen in Fig. 1. Also, it can be seen that there is a large energy gap at γ =-60°. Of course, under rotation the single-particle levels will be rearranged due to the Coriolis effect and the energy minima and the relative difference between them can be affected correspondingly.

Figure 2:Full-screen View

(Color online) An example of the calculated ground-state WS single-neutron levels in function of the quadrupole deformation β₂ near the Fermi surface for ${}_{80}^{184}$Hg₁₀₄. The positive (negative) parity levels are denoted by the solid (dashed) lines. Part curves near the Fermi level are labeled with the Nilsson labels Ω[NnzΛ]. Note that in the left panel the γ value equals -60°, which is equivalent to the negative β₂ values. Large gaps near the Fermi-surface positions labeled "104" at oblate (obl.), prolate (pro.), and super-deformed prolate (sd-pro.) shapes deserve noticing.

Figure 3:Full-screen View

(Color online) Similar to Fig. 2, but versus γ deformation degree of freedom. The ground-state equilibrium deformations (0.144,-0.005) are adopted for (β₂,β₄) parameters. The three domains [-120°, -60°], [-60°, 0°], and [0°, 60°] are symmetric. For further comments see the text.

In order to further check the validity of the phenomenological WS mean field generated the single-particle states, as shown in Fig. 4, we have investigated the proton and neutron Fermi energy levels, with the energy of the last occupied single-particle level, for the N=104 isotones synthesized in experiment. These levels are to some extent related to the difficulty degree of nucleon separation and nuclear stability. In general, if the energies of both proton and neutron Fermi energy levels below zero are close to the average one-nucleon separation energy ( 8) MeV, the nucleus usually has the largest stability. Of course, the Coulomb repulsive force between positively charged protons together with the Coulomb barrier above zero potential energy may slightly modify the general law. Indeed, it is clearly seen in Fig. 4 that the trend of the calculated proton and neutron Fermi energy levels determined by the universal potential parameters is rather consistent with that of the proton and neutron separation energies. As expected, the nuclei are stable at the positions where the proton and neutron Fermi energy levels are close to each other, e.g. in ¹⁷⁴Yb and ¹⁷⁶Hf. The nuclear stabilities, i.e., denoted by half-lives, are also in good agreement with the distances of the calculated single-particle levels from the zero-energy point. That is, as the proton or neutron Fermi level evolves toward the top of the potential well, the half-life decreases exponentially. From this figure, if the symmetric trends can arbitrarily adopted, one can deduce that in the proton-deficient side of this isotopic chain there may exist several relatively long-life nuclei which can be measured though it is difficult to synthesize due to the scarce reaction mechanism. However, in the proton-rich side, it seems that the proton drip line is accessed though the Coulomb barrier can extend the existence of the proton-rich nuclei to some extent. At this moment, the proton Fermi energy level has begun to appear positive, e.g., in ¹⁸⁸Po, indicating the occurrence of a quasibound or unbound structure. Then, the standard way of extracting the shell correction may break down for such a weakly bound nucleus since the contribution from the particle continuum will be of importance [50]. In the present work, we consider the positive-energy spectrum as quasibound states to perform the calculations of the pairing and shell corrections. Under this situation, even the pairing window covers the positive energy states, and it is assumed that particles do not scatter into the continuum by the residual pairing force. In other words, the present single-particle picture does not give the true nuclear ground or excited states and it only serves as the set of basis functions for the shell and pairing calculations. Certainly, such a procedure should not be considered as satisfactory. The Wigner-Kirkwood expansion and Green function method, beyond the scope of our work, have been suggested to deal with the continuum states [50, 51], which deserve to be considered in our future work.

Figure 4:Full-screen View

(Color online) (a) Calculated neutron and proton Fermi energy levels for even-even N=104 isotones with the calculated equilibrium shapes as input quantities. (b) Available proton and neutron separation energies for the N=104 isotones. (c) Available half-lives of the N=104 isotones, indicating the nuclear instabilities. Note that the half-life for ¹⁷⁴Yb and ¹⁷⁶Hf are greater than 10¹⁵s.

Table 1 presents our calculated deformation parameters for even-even N=104 isotones, which have at least three observed yrast levels, together with partial experimental data and/or other acceptable theoretical results for comparison. The phenomenological or empirical energy ratio, R_4/2 [57], and P-factor [58] are also given to evaluate the present calculations and nuclear properties. The energy ratio, $R_{4/2}=E_{4_1^{+}}/E_{2_1^{+}}$, is 3.3 for a well-deformed axially symmetric rotor, 2.5 for an γ-unstable vibrator, and 2.0 for a spherical vibrator, which are corresponding to SU(3), O(6), and U(5) dynamic symmetries in the algebraic view of the IBM [23, 57, 59], respectively. Moreover, it is pointed out that 1.82 is the so-called Mallmann critical point which is the separatrix between single-particle and collective characteristics [60] and 3.0 is the shape/phase transition point to quadrupole deformed nuclei [61, 62]. As another more sensitive phenomenological quantity related to nuclear collectivity and deformation, the P-factor is given by Casten et al. [63], which is defined by $P\equiv N_{\text{p}}{N}_{\text{n}}/(N_{\text{p}}+N_{\text{n}})$, where N_p and N_n are the numbers of valence protons and neutrons, respectively; the product N_pN_n indicates the number of p–n interactions and the summation N_p+N_n denotes the number of pairing interactions. It was pointed out that the transition to deformation generally occurs when P ≈ 4-5, that is, each valence nucleon interacts with about 4-5 nucleons of the other type. Indeed, as expected, the collectivity increases rapidly as the proton number moves away from the Z=82 closed shell due to the large number of valence neutrons, N_n, in this midshell isotopes. One can see that just after three nuclei from the proton closed-shell nucleus, ${}_{82}^{186}$ Pb₁₀₄, the R_4/2 ratios and P factors are already more than the shape transition points 3.0 and 4, respectively, indicating the onset of large collectivity and deformation [57]. Note that the small R_4/2 values of ¹⁸⁴Hg and ¹⁸⁶Pb, which are less than the critical value 1.82 in the Mallmann plot, show a possible single-particle excitation. From Table 1, one can notice that our calculated equilibrium deformation parameters are basically in agreement with other theoretical results and available data. It should be pointed out that other theoretical results are respectively obtained based on the fold-Yukawa (FY) single-particle potential, the finite-range droplet model (FRDM) [52, 53], the Hartree-Fock-BCS (HFBCS) [54], and the extended Thomas-Fermi plus Strutinsky integral (ETFSI) methods [55], and the experimental values β₂ are deduced from the reduced transition probabilities, B(E2) [56]. Especially, all the theories reproduce the oblate and spherical ground state for ¹⁸⁴Hg and ¹⁸⁶Pb, respectively, except for the ETFSI calculations where the superdeformed prolate shapes are responsible for the ground states in these two nuclei. However, our calculation, as shown in Fig. 1, actually presents a superdeformed prolate minimum in ¹⁸⁴Hg though its excited energy is somewhat high. In addition, it can be seen that the calculated β₄ parameters have the similar evolution trends, especially with the same signs. It is worth mentioning that our calculation and the FY+FRDM calculation are based on the framework of the macroscopic-microscopic models, the HFBCS is based on the self-consistent Hartree-Fock method, and the ETFSI calculation intends to combine the advantages of the macroscopic-microscopic and self-consistent methods. Despite this, no one theory can fully reproduce the data, or even be close to them always, as seen in Table 1. Nevertheless, the macroscopic-microscopic methods actually show a relatively high descriptive power to a large extent.

Based on the positive facts mentioned above, we have performed the systematic TRS calculations for the even-even N=104 isotones ranging from ¹⁷⁰Dy to ¹⁸⁶Pb. In our present calculations for even-even nuclei, the configuration is not constrained which means that the energy (or Routhian) of every point on the calculated TRS corresponds to that of the yrast state rather than a fixed configuration. For a constrained configuration, the TRS generally changes monotonically and smoothly with the increasing distance from the minimum. However, with the "real" TRS this is not the case, and it usually fluctuates and even multi-minima phenomenon occurs on it, showing the inclusion of different configurations. Our present TRS calculation can record the rotational properties of different minima, which corresponds to the yrast states at their respective deformations. Obviously, if two or more minima appear on the TRS and their energy differences are not too large, the so-called shape-coexisting phenomena may take place. Figure 5 shows the calculated total Routhian values at different minima as a function of rotational frequency, ℏω, for the selected even-even N=104 isotones. Note that at a certain rotational frequency if one configuration cannot form a minimum on the TRS, it will not be recorded, according to the present TRS calculation, without configuration tracking. As shown in Fig. 5, the bands with different configurations are arbitrarily denoted by the average deformation parameters β₂ and γ (if there is a very small shape change before and after a band crossing, the two bands are not distinguished here). One can see from this figure that multiple rotational bands are developed during the cranking process. Moreover, they may cross and the configuration of the yrast sequence may change with rotation. It seems that the weakly oblate, normal prolate, superdeformed prolate, and non-collective (γ 60° or -120°) shapes can compete and coexist especially at a certain domain of the rotational frequency.

Figure 5:Full-screen View

(Color online) The calculated total Routhians for different bands with shape-coexisting configurations in even-even N=104 isotones. The yrast and shape-coexisting configurations are denoted temporarily by average β₂ and γ values with different symbols.

Taking the calculated Routhians into account, to display the shape coexistence better, in Fig. 6 we show the TRS plots at the typical rotational frequencies, $\hslash$ ω, where the Routhians are close to each other. In principle, at such rotational frequencies, the bands (coexisting minima) can be populated experimentally with the similar probability. It is worth noting that in the present cranking calculations the rotational frequency, $\hslash$ ω, is a classical quantity, and the angular momentum quantum number is not conserved. Due to $\hslash \omega =I{\hslash }^2/J$ (J is the moment of inertia of a nucleus), the same frequency may correspond to rather different angular momenta since the different shape-coexisting bands, corresponding to different minima, usually have rather different moments of inertia. Precisely speaking, the high band density (that is, there exists multiple bands with a narrow energy domain for a given spin instead of a rotational frequency) is the favourable condition to study such shape-coexisting phenomena, especially populated by the heavy-ion fusion-evaporation reactions. However, our calculations can still provide the important references in these aspects. From this figure, it is clearly seen that 2-, 3-, and even 4-fold minima coexist in the TRS plots. Furthermore, the coexisting shapes, i.e., for ¹⁸⁴Hg₁₀₄, have been confirmed in experiments [1, 28, 64]. In ¹⁸⁶Pb, experimental evidences on energy spectra and charge radii also indicate that the ground state is spherical, but both oblate (2p-2h proton excitations) and prolate (4p-4h proton excitations) low-lying minima at almost identical excitation energies are observed [26, 65], and are in good agreement with our calculation. For several other nuclei, though the optimized shape-coexisting frequencies are somewhat high, they are still below the angular momentum limits [66], even in ¹⁷⁰Dy and ¹⁷²Er, showing possible experimental observations.

Figure 6:Full-screen View

(Color online) TRS plots in the (β₂,γ) polar coordinate system for nine even-even N=104 isotones with 66 ≤ Z ≤ 82 at typical shape-coexisting rotational frequencies. At each given (β₂, γ) point, the TRS has been minimized with respect to the deformation parameter, β₄. The solid circle, square, triangle, and diamond symbols denote the first, second, third, and fourth energy minima, respectively. For convenience, the energy is normalized with respect to the first (ground-state) minimum. The contours below 2 MeV are displayed and the difference between neighboring contours is 0.2 MeV.

4 Summary

In conclusion, the TRS calculations in (β₂, γ, β₄) deformation space have been performed for nine even-even N=104 midshell isotones ranging from ¹⁷⁰Dy to ¹⁸⁶Pb, where the phenomenological WS nuclear potential with the universal parameter set is employed. Taking the ¹⁸⁴Hg nucleus as an example, the single-particle structure and resulting energy minima are analyzed, reproducing the experimental observation. Further, our present results are evaluated by comparing them with previously published results and available data, which indicate our calculations are reliable to a large extent. Based on such a TRS approach, it is found that a systematic shape-coexisting phenomena induced by rotation may occur in this isotonic chain. The optimized rotational frequency at which the shape coexistence appears is crudely suggested according to the calculated Routhians of different bands. Though part frequencies are somewhat high (still below the predicted fission limit, e.g., in the lighter ¹⁷⁰Dy and ¹⁷²Er), it may still be deserved to study experimentally to some extent.