Deformed mean-field plus standard pairing model

The Hamiltonian of the deformed mean-field plus standard pairing model for either the proton or the neutron sector is given by $\widehat{H}={\displaystyle \sum _{i=1}^n}{\epsilon }_i{\widehat{n}}_i-G{\displaystyle \sum _{i{i}^{\prime }}}{S}_i^{+}{S}_{i^{\prime }}^{-}\mathrm{,}$ (1) where the sums run over all given i-double degeneracy levels of total number n, G>0 is the overall pairing interaction strength, $\left\{{\epsilon }_i\right\}$ are the single-particle energies obtained from mean-field, such as Hartree-Fock, Woods-Saxon potential, Yukawa-Folded single-particle potential, or Nilsson model. $n_i=a_{i\uparrow }^{†}{a}_{i\uparrow }+a_{i\downarrow }^{†}{a}_{i\downarrow }$ is the fermion number operator for the i-th double degeneracy level, and $S_i^{+}=a_{i\uparrow }^{†}{a}_{i\downarrow }^{†}\left[S_i^{-}={\left(S_i^{+}\right)}^{†}=a_{i\downarrow }{a}_{i\uparrow }\right]$ is the pair creation (annihilation) operator, The up and down arrows in these expressions refer to time-reversed states.

According to the Richardson-Gaudin method [36-43], the exact k-pair eigenstates of (1) with ${\nu }_{i^{\prime }}=0$ for even systems or ${\nu }_{i^{\prime }}=1$ for odd systems, in which i' is the label of the double degeneracy level that is occupied by an unpaired single particle can be written as $|k;\xi ;{\nu }_{i^{\prime }}\rangle =S^{+}\left(x_1^{(\xi )}\right)S^{+}\left(x_2^{(\xi )}\right)\cdots S^{+}\left(x_k^{(\xi )}\right)|{\nu }_{i^{\prime }}\rangle \mathrm{,}$ (2) where $|{\nu }_{i^{\prime }}\rangle$ is the pairing vacuum state with the seniority ${\nu }_{i^{\prime }}$ that satisfies $S_i^{-}|{\nu }_{i^{\prime }}\rangle =0$, and ${\widehat{n}}_i|{\nu }_{i^{\prime }}\rangle ={\delta }_{i{i}^{\prime }}{\nu }_i|{\nu }_{i^{\prime }}\rangle$ for all i. Here, ξ is an additional quantum number for distinguishing different eigenvectors with the same quantum number k and $S^{+}\left(x_{\mu }^{\left(\xi \right)}\right)={\displaystyle \sum _{i=1}^n}\frac{1}{x_{\mu }^{\left(\xi \right)}-2{\epsilon }_i}{S}_i^{+}\mathrm{,}$ (3) in which the spectral parameters $x_{\mu }^{\left(\xi \right)}\left(\mu =\mathrm{1,}\mathrm{2,}\dots \mathrm{,}k\right)$ satisfy the following set of Bethe ansatz equations (BAEs): $1+G{\displaystyle \sum _i\frac{{\text{Ω}}_i}{x_{\mu }^{\left(\xi \right)}-2{\epsilon }_i}}-2G{\displaystyle \sum _{{\mu }^{\prime }=1(\ne \mu )}^k}\frac{1}{x_{\mu }^{\left(\xi \right)}-x_{{\mu }^{\prime }}^{\left(\xi \right)}}=\mathrm{0,}$ (4) where the first sum runs over all i levels and ${\text{Ω}}_i=1-{\delta }_{i{i}^{\prime }}{\nu }_{i^{\prime }}$. For each solution, the corresponding eigenenergy is given by $E_k^{\left(\xi \right)}={\displaystyle \sum _{\mu =1}^k}{x}_{\mu }^{\left(\xi \right)}+{\nu }_{i^{\prime }}{\epsilon }_{i^{\prime }}\mathrm{.}$ (5) In general, according to the polynomial approach in Refs. [45-48], one can find solutions of Eq. (4) by solving the second-order Fuchsian equation [44] as $A(x)P^{″}(x)+B(x)P^{\prime }(x)-V(x)P(x)=\mathrm{0,}$ (6) where $A(x)={\displaystyle {\prod }_{i=1}^n}\left(x_{\mu }^{\left(\xi \right)}-2{\epsilon }_i\right)$ is an n-degree polynomial, $B(x)/A(x)=-{\displaystyle \sum _{i=1}^n\frac{{\text{Ω}}_i}{x_{\mu }^{\left(\xi \right)}-2{\epsilon }_i}}-\frac{1}{G}\mathrm{,}$ (7) V(x) are called Van Vleck polynomials [44] of degree n-1, which are determined according to Eq. (6). They are defined as $V(x)={\displaystyle \sum _{i=0}^{n-1}}{b}_i{x}^i\mathrm{.}$ (8) The polynomials P(x) with zeros corresponding to the solutions of Eq. (4) is defined as $P(x)={\displaystyle \prod _{i=1}^k}\left(x-x_i^{\left(\xi \right)}\right)={\displaystyle \sum _{i=0}^k}{a}_i{x}^i\mathrm{,}$ (9) where k is the number of pairs. bi and ai are the expansion coefficients to be determined instead of the Richardson variables xi. Furthermore, if we set ak=1 in P(x), the coefficient $a_{k-1}$ then equals the negative sum of the P(x) zeros, $a_{k-1}=-{\displaystyle {\sum }_{i=1}^k{x}_i^{\left(\xi \right)}}=-E_k^{\left(\xi \right)}$.

If the value of x approaches twice the single-particle energy of a given level δ, i.e., $x=2{\epsilon }_{\delta }$, one can rewrite Eq. (6) in doubly degenerate systems with Ω_i=1 as [45, 46, 48] ${\left(\frac{P^{\prime }\left(2{\epsilon }_{\delta }\right)}{P\left(2{\epsilon }_{\delta }\right)}\right)}^2-\frac{1}{G}\left(\frac{P^{\prime }\left(2{\epsilon }_{\delta }\right)}{P\left(2{\epsilon }_{\delta }\right)}\right)={\displaystyle \sum _{i\ne \delta }\frac{\left[\left(\frac{P^{\prime }\left(2{\epsilon }_{\delta }\right)}{P\left(2{\epsilon }_{\delta }\right)}\right)-\left(\frac{P^{\prime }\left(2{\epsilon }_i\right)}{P\left(2{\epsilon }_i\right)}\right)\right]}{2{\epsilon }_{\delta }-2{\epsilon }_i}}\mathrm{.}$ (10) In Ref. [50], a new iterative algorithm is established for the exact solution of the standard pairing problem within the Richardson-Gaudin method using the polynomial approach in Eq. (10). It provides efficient and robust solutions for both spherical and deformed systems at a large scale. The key to its success is determining the initial guesses for the large-set nonlinear equations involved in a controllable and physically motivated manner. Moreover, one reduces the large-dimensional problem to a one-dimensional Monte Carlo sampling procedure, which improves the algorithm's efficiency and avoids the nonsolutions and numerical instabilities that persist in most existing approaches. Based on the new iterative algorithm, we applied the model to study the actinide nuclei isotopes, where an excellent agreement with experimental data was obtained [51-53].

The Fourier shape parametrization

Recent studies demonstrated that the developed Fourier parametrization of deformed nuclear shapes was highly effective in capturing the essential features of nuclear shapes, particularly up to the scission configuration [54, 22]. Current research indicated that combining this innovative Fourier shape parametrization with the LSD + Yukawa-Folded macro-microscopic potential-energy framework was exceptionally efficient [52, 53, 55, 56]. This work primarily adopted the macro-microscopic framework outlined in Refs. [52, 53], where the single-particle energies $\left\{{ϵ}_i\right\}$ in the model Hamiltonian (1) were derived from the Yukawa-Folded potential.

The nuclear surface is expanded in terms of a Fourier series of dimensionless coordinates as follows: $\begin{array}{c}\frac{{\rho }_{\text{s}}^2\left(z\right)}{R_0^2}={\displaystyle \sum _{n=1}^{\infty }}[a_{2n}\cos \left(\frac{(2n-1)\pi }{2}\frac{z-z_{\text{sh}}}{z_0}\right)\\ +a_{2n+1}\sin \left(\frac{2n\pi }{2}\frac{z-z_{\text{sh}}}{z_0}\right)]\mathrm{,}\end{array}$ (11) where ρ_s(z) is the distance from a surface point to the symmetry z-axis, and R₀ = 1.2A^1/3 fm is the radius of a corresponding spherical shape with the same volume. The shape's extension along the symmetry axis is 2z₀, with the left and right ends located at z_min = z_sh - z₀ and z_max = z_sh + z₀, respectively. The parameter z₀ represents half the shape's extension along the symmetry axis and is determined by volume conservation, while z_sh is set such that the center of mass of the nuclear shape is at the origin of the coordinate system. Based on the convergence properties discussed in Ref. [22], the first five terms a₂, …, a₆ are retained as a starting point, and the parameters an are transformed into deformation parameters qn as follows: $\begin{array}{l}{q}_2=a_2^{\left(0\right)}/a_2-a_2/a_2^{\left(0\right)}\mathrm{,}\\ q_3=a_3\mathrm{,}\\ q_4=a_4+\sqrt{{\left(q_2/9\right)}^2+{\left(a_4^{\left(0\right)}\right)}^2}\mathrm{,}\\ q_5=a_5-\left(q_2-2\right)a_3/\mathrm{10,}\\ q_6=a_6-\sqrt{{\left(q_2/100\right)}^2+{\left(a_6^{\left(0\right)}\right)}^2}\mathrm{,}\end{array}$ (12) where $a_n^{\left(0\right)}$ are the Fourier coefficients for the spherical shape. Higher-order coordinates q₅ and q₆ are generally set to zero within the accuracy of the current approach. The set of qi parameters has explicit physical significance in describing the shape of the fissioning nucleus: q₂ denotes the elongation, q₄ represents the neck parameter, and q₃ indicates the left-right asymmetry.

Additionally, the non-axial deformation of nuclear shapes is described as follows, assuming that the surface cross-section at a given z-coordinate is elliptical with semi-axes a(z) and b(z): ${\varrho }_{\text{s}}^2\left(z\mathrm{,}\phi \right)={\rho }_{\text{s}}^2\left(z\right)\frac{1-{\eta }^2}{1+{\eta }^2+2\eta \cos (2\phi )}\mathrm{,}$ (13) where $\eta =\frac{b-a}{b+a}$ characterizes the non-axial deformation. Volume conservation requires that ${\rho }_{\text{s}}^2\left(z\right)=a(z)+b(z)$, with the condition $ab={\rho }_{\text{s}}^2\left(z\right)$ ensuring volume conservation for non-axial deformations. The semi-axes are then given by: $\begin{array}{l}a(z)={\rho }_{\text{s}}\left(z\right)\sqrt{\frac{1-\eta }{1+\eta }}\mathrm{,}b(z)={\rho }_{\text{s}}\left(z\right)\sqrt{\frac{1+\eta }{1-\eta }}\mathrm{.}\hfill \end{array}$ (14) This description of non-axial shapes using the parameters q₂ and η is more general than the commonly used Bohr parametrization (β, γ). For spheroidal shapes, both descriptions are equivalent. However, as shown in Fig. 1, where the two parametrizations are compared, the periodicity of nuclear shapes by a 60° rotation angle is similar in both (q₂, η) and (β, γ) planes. It is important to note that this regularity is disrupted when higher multipolarity deformations qn (n>2) are considered, making the (η, q₂, q₃, q₄, q₆) shape parametrization substantially more general than the 3-dimensional $\left({ϵ}_2\mathrm{,}{ϵ}_4\left(\gamma \right)\mathrm{,}\gamma \right)$ parametrization used in Ref. [59, 60]. The two parametrizations coincide only in the special case of spheroidal shapes.

Fig. 1Full-screen View

(Color online) Relationsheep between the elongation parameter q₂ and the nonaxiality parameter η [22, 54], and the traditional Bohr deformation parameters β and γ is taken from [57, 58]

It is essential to stress that different points in the (β, γ), and (q₂, η) planes can correspond to identical shapes when higher qn (n>2) degrees of freedom are neglected, differing only in the interchange of coordinate system axes. For example, the point $(\beta =\mathrm{0.4,}\gamma =0)$ corresponds to (q₂ = 0.42, η = 0) in the new parametrization, representing the same shape as $(\beta =\mathrm{0.4,}\gamma {=120}^{\circ })$, which corresponds to (q₂ = -0.21, η = 0.16) in the new parametrization.

When analyzing potential energy landscapes that include triaxial degrees of freedom, it is crucial to avoid treating as distinct configurations points in the (q₂, η) deformation plane that are merely rotational images of each other at $\gamma {=60}^{\circ }$.

In this study, the dynamic process of nuclear fission will be described in the three-dimensional deformation space $\left(\eta \mathrm{,}{q}_2\mathrm{,}{q}_4\right)$ using the Fourier shape parametrization.

The potential energy

This study calculated the PESs for the isotopes ¹⁷⁰Pt, ¹⁷²Hg, and ¹⁷⁴Pb in a three-dimensional deformation space $\left(\eta \mathrm{,}{q}_2\mathrm{,}{q}_4\right)$ and analyzed the impact of pairing interactions on the shape coexistence of these isotopes. The results were obtained over the following grid points in the deformation parameter space: $\begin{array}{cc}\eta \in [\mathrm{0.00,}0.20]& \text{Δ}\eta =0.02\\ q_2\in [-\mathrm{0.60,}0.85]& \text{Δ}{q}_2=0.05\\ q_4\in [-\mathrm{0.30,}0.30]& \text{Δ}{q}_4=\mathrm{0.03.}\end{array}$ (15) As indicated in the literature [22], the q₃ degree of freedom has no significant impact on the description of shape coexistence for the isotopes discussed in this paper. Therefore, in this study, q₃ was set to 0, and for each point on the PES, q₄ was minimized to find the energy extremum. The potential energy of the system was calculated within the macro-microscopic approach in this work. The total energy $E_{\text{total}}\left(N\mathrm{,}Z\mathrm{,}{q}_n\right)$ of a nucleus with a given deformation is calculated as $E_{\text{total}}\left(N\mathrm{,}Z\mathrm{,}{q}_n\right)=E_{\text{LD}}\left(N\mathrm{,}Z\mathrm{,}{q}_n\right)+E_{\text{B}}\left(N\mathrm{,}Z\mathrm{,}{q}_n\right)\mathrm{,}$ (16) where $E_{\text{LD}}\left(N\mathrm{,}Z\mathrm{,}{q}_n\right)$ was the macroscopic term obtained by the LSD model with proton number Z and neutron number N [61]. In the current calculation for the potential-energy surface, we just considered the energy $E_{\text{B}}\left(N\mathrm{,}Z\mathrm{,}{q}_n\right)$ related to the shape parameter $\left\{q_2\mathrm{,}{q}_4\right\}$. $E_{\text{B}}\left(N\mathrm{,}Z\mathrm{,}{q}_n\right)=E_{\text{shell}}\left(N\mathrm{,}Z\mathrm{,}{q}_n\right)+E_{\text{pair}}\left(N\mathrm{,}Z\mathrm{,}{q}_n\right)\mathrm{.}$ (17) The microscopic term consisted of the shell correction energy $E_{\text{s}hell}^{\nu \left(\pi \right)}\left(N\mathrm{,}Z\mathrm{,}\left\{{\epsilon }_i\right\}\mathrm{,}{q}_2\mathrm{,}{q}_4\right)$ proposed by Strutinsky [62, 63], and the pairing interaction energy $E_{\text{pair}}^{\nu \left(\pi \right)}\left(N\mathrm{,}Z\mathrm{,}\left\{{\epsilon }_i\right\}\mathrm{,}{q}_2\mathrm{,}{q}_4\right)$ calculated from Eq. (1). Here, ν (π) was the label of the neutron (proton) sector. In the current study, we considered 18 deformed harmonic-oscillator shells in Yukawa-Folded single-particle potential to obtain single-particle levels for the microscopic calculations. For the pairing interaction energy, we performed 29 single-particle levels around the neutron Fermi level and 22 single-particle levels around the proton Fermi level.

To validate our results and further explored the efficacy of the exactly solvable pairing model, we also calculated the PESs for the isotopes considered under the BCS approximation. The pairing interaction was determined as the difference between the BCS energy [24] and the single-particle energy sum and the average pairing energy [64]. $E_{\text{pair}}=E_{\text{BCS}}-{\displaystyle \sum _{i=1}^k}{\epsilon }_i-{\tilde{E}}_{\text{pair}}\mathrm{.}$ (18) In the BCS approximation the ground-state energy of a system with an even number of particles and a monopole pairing force was given by $E_{\text{BCS}}={\displaystyle \sum _{i=1}^k}2{\epsilon }_i{v}_k^2-G{\left({\displaystyle \sum _{i=1}^k}{u}_i{v}_i\right)}^2-G{\displaystyle \sum _{i=1}^k}{v}_i^4\mathrm{,}$ (19) where the sums run over the pairs of single-particle states contained in the pairing window defined below. The coefficients vi and $u_i=\sqrt{1-v_i^2}$ were the BCS occupation amplitudes.

The average projected pairing energy, for a pairing window of width 2Ω, which is symmetric in energy with respect to the Fermi energy, is equal to $\begin{array}{c}{\tilde{E}}_{\text{pair}}=-\frac{1}{2}\tilde{g}{\widetilde{\text{Δ}}}^2+\frac{1}{2}\tilde{g}G\widetilde{\text{Δ}}\arctan \left(\frac{\text{Ω}}{\widetilde{\text{Δ}}}\right)-\log \left(\frac{\text{Ω}}{\widetilde{\text{Δ}}}\right)\widetilde{\text{Δ}}\\ +\frac{3}{4}G\frac{\text{Ω}/\widetilde{\text{Δ}}}{1+{\left(\text{Ω}/\widetilde{\text{Δ}}\right)}^2}/\arctan \left(\frac{\text{Ω}}{\widetilde{\text{Δ}}}\right)-\frac{1}{4}G.\end{array}$ (20) Here $\tilde{g}$ was the average single-particle level density and $\widetilde{\text{Δ}}$ the average paring gap corresponding to a pairing strength G $\widetilde{\text{Δ}}=2\text{Ω}\exp \left(-\frac{1}{G\tilde{g}}\right)\mathrm{.}$ (21)

Influence of pairing interactions on the shape coexistence of ¹⁷⁰Pt, ¹⁷²Hg and ¹⁷⁴Pb isotopes

Figure 2 shows the PESs of ¹⁷⁰Pt projected onto the (q₂, η) plane for different pairing interaction strengths Gν (MeV), while the proton pairing interaction strength is fixed at Gπ = 0.100 MeV. Gν and Gπ represent the neutron and proton pairing interaction strengths (MeV), respectively. The energy is minimized in the q₄ direction and q₃ is set to 0 and normalized to zero energy at the ground-state value. The choice of Gν varying from 0.03 to 0.145 MeV, and Gπ = 0.100 MeV, were based on the fact that our calculations in the next section, when employing Gν = 0.145 MeV, and Gπ = 0.100 MeV, closely matched the experimental odd-even mass differences for the ¹⁷¹Pt to ¹⁸⁰Pt isotopes. Therefore, this range was selected to study the effects of pairing strength variations on the shape coexistence. The red lines represent the corresponding (β,γ) coordinates, with γ coordinates distributed within 0≤γ≤180°. The β coordinate values are taken as 0.1, 0.2, 0.3 ..., etc.

Fig. 2Full-screen View

(Color online) Potential energy surface of ¹⁷⁰Pt projected onto the (q₂, η) plane under different pairing interaction strengths Gν (MeV), while the proton pairing interaction strength is fixed at Gπ = 0.100 MeV. The energy is minimized in the q₄ direction and q₃ is set to 0 and normalized to zero energy at the ground-state value. The ground-state deformation is represented by a red dot

In Figs. 2 (a)-(d), the PESs of ¹⁷⁰Pt are shown for different values of the neutron pairing interaction strength Gν, while the proton pairing interaction strength is fixed at Gπ = 0.100 MeV. The values of Gν are: 0.030, 0.070, 0.105, and 0.145 MeV. It can be seen that the ground-state of the ¹⁷⁰Pt isotope is located at (q₂ ≈ 0.150,η = 0), indicating a prolate shape for different pairing strengths. The other minimum at (q₂ ≈ -0.150,η = 0.04, γ=120°) illustrated in Fig. 2 is simply a reflection of the ground-state minimum.

It is noteworthy to highlight the existence of two distinct shape isomers in ¹⁷⁰Pt with different pairing strengths. The first is an oblate shape isomer located at (q₂ = -0.400, η = 0), with an energy approximately 3.900 MeV above the ground-state. The second is a triaxial shape isomer at (q₂ ≈ 0.600, η ≈ 0.060 (γ ≈ 10°)), positioned around 4.0 MeV above the ground-state. These isomers represent the local minima on the potential energy surface that are separated from the ground-state by energy barriers, highlighting the complex deformation characteristics of the nucleus. With an increase in pairing strength, both shape isomers become shallower. When the pairing strength Gν reaches 0.145 MeV, the oblate isomer disappears (see Fig. 2(d)).

As shown in Figs. 3(a)-(d), the PESs for different pairing interaction strengths demonstrates the evolution of the triaxial minimum at (q₂ = 0.150, η = 0.020) to the oblate minimum at (q₂ = 0.100, η = 0.040) as the pairing interaction strength increases. The nucleus of ¹⁷²Hg is nearly γ-unstable, with the energy difference between different points in the ground-state valley not exceeding approximately 0.4 MeV. Additionally, three shape isomers are visible in the (a)-(d) maps: a prolate isomer at (q₂ ≈ 0.600, η = 0), E ≈ 5.0 MeV; a triaxial isomer at (q₂ ≈ 0.400, η = 0.100), E ≈ 4.0 MeV, and oblate one at (q₂ ≈ -0.45, η = 0), E ≈ 4.0 MeV. These local minima are separated by energy barriers of approximately 1 MeV in height. As the pairing strength increases, all shape isomers gradually become shallower. By Gν = 0.145 MeV and Gπ = 0.100 MeV (Fig. 3(d)), the triaxial isomer at (q₂ ≈ 0.400, η = 0.100) disappeared.

Fig. 3Full-screen View

(Color online) Same as Fig. 2, but for ¹⁷²Hg

The PESs of ¹⁷⁴Pb, as presented in Figs. 4(a)-(d), reveal that a prolate ground-state (q₂ ≈ 0.150, η = 0) (in Fig. 4(a)) tend to become spherical (in Fig. 4(d)) as the pairing interaction strength increases. The shape isomers observed here are particularly interesting: a prolate shape at (q₂ = 0.600, η = 0, E ≈ 5.0 MeV and a slightly triaxial oblate shape at (q₂ = 0.450, η = 0.020, E ≈ 3.9 MeV in Fig. 4(a), and (b), respectively. As the pairing strength increased, both shape isomers gradually became shallower. When Gν = 0.145, MeV, and Gπ = 0.100 MeV (Fig. 4(d)), they almost disappeard. Overall, regardless of pairing strength, there was no indication of robust shape coexistence in this nucleus.

Fig. 4Full-screen View

(Color online) Same as Figs. 2 and 3, but for ¹⁷⁴Pb

Figures 5 illustrate the PESs projections of ¹⁷⁰Pt, ¹⁷²Hg, and ¹⁷⁴Pb under realistic pairing interaction strengths, with Gν = 0.145 MeV, and Gπ = 0.100 MeV under both Exact and BCS pairing schemes.

Fig. 5Full-screen View

(Color online) Potential energy surfaces of ¹⁷⁰Pt, ¹⁷²Hg and ¹⁷⁴Pb projected on the (q₂, η) plane under both Exact and BCS pairing schemes, with the energy minimized in the q₄ direction, q₃ set to 0 and normalized to zero energy at the ground-state value. The realistic pairing interaction strengths Gν = 0.145 MeV, and Gπ = 0.100 MeV are adopted. The ground-state deformation is represented by a red dot, while the coexistence minimum is indicated by a red cross

As shown in Fig. 5, the ground-state of ¹⁷⁰Pt is prolate, located at (q₂ = 0.15, η = 0) under both the Exact and BCS pairing schemes. However, BCS pairing exhibited a shallower depth for the prolate minimum compared with Exact pairing, indicating a less pronounced prolate ground-state. Furthermore, a triaxial isomer appeared at (q₂ ≈ 0.600, η ≈ 0.060 (γ ≈ 10°)) under Exact pairing, whereas it was less distinguishable in the BCS case.

The ground-state of ¹⁷²Hg (Fig. 5) is found at (q₂ = 0.10, η ≈ 0.04) as an oblate minimum, with another minimum at (q₂ ≈ -0.100, η ≈ 0.02), which exhibits γ-unstable deformation. The PES of ¹⁷²Hg provides an excellent example of an almost γ-unstable nucleus. Under Exact pairing, this γ-unstable minimum is more symmetric, with clear reflections around γ=150°, γ=30°, and γ=90°. Under BCS pairing, the γ-unstable features are less prominent, and the oblate minimum becomes more dominant. Additionally, two shape isomers are visible under Exact pairing model: a prolate isomer at (q₂ ≈ 0.600, η = 0), E ≈ 4.6 MeV, and an oblate one at (q₂ ≈ -0.45, η = 0), E ≈ 4.6 MeV. However, these changes were not distinguishable in the BCS case.

As shown in Figs. 5(c), the ground-state shape of ¹⁷⁴Pb tended to be spherical. The PES under Exact pairing revealed a nearly spherical configuration with minor prolate and oblate shape isomers. In contrast, BCS pairing resulted in a more pronounced spherical minimum and diminishes the depth of shape isomers.

In summary, as the number of protons increases, the ground-state transitions from prolate for ¹⁷⁰Pt to the coexistence of γ-unstable and oblate for ¹⁷²Hg, eventually approached a nearly spherical configuration for ¹⁷⁴Pb. The comparison between Exact and BCS pairing demonstrates that BCS pairing tends to smooth out shape coexistence and reduce the depth of shape isomer, leading to less pronounced deformation features. The differences in results between Exact and BCS pairing may be attributed to the mean-field approximation in the BCS approach, which likely simplifies the treatment of pairing interactions. This approximation is thought to smooth out shape coexistence phenomena by suppressing pairing fluctuations, energy gaps, and shell effects, potentially leading to less pronounced deformation features.

Shape coexistence analysis in the Pt isotope chain

In this paper, we investigate the PESs of the even-even ^170-180Pt isotopes using the exactly solvable deformed mean-field plus pairing model. Our analysis provides a comprehensive examination of the shape coexistence phenomena across these isotopes.

The pairing interaction strength, denoted as G, serves as the sole adjustable parameter within our model. It is typically determined either through empirical formulas or by fitting to experimental odd-even mass differences [65, 66]. In this study, we determined Gν by fitting the experimental odd-even mass differences for the ^171-180Pt isotope chain and Gπ by fitting the experimental odd-even mass differences for the ¹⁷⁴Pt to ¹⁷⁸Pb isotonic chain. The odd-even mass differences are computed using the following expression: $\begin{array}{c}P\left(A\right)=E_{\text{total}}\left(N+\mathrm{1,}Z\right)+E_{\text{total}}\left(N-\mathrm{1,}Z\right)\\ -2E_{\text{total}}\left(N\mathrm{,}Z\right).\end{array}$ This quantity is highly sensitive to variations in the pairing interaction strength G [67], due to the pairing interaction between nucleons. As shown in Fig. 6, by employing Gν=0.145 MeV and Gπ=0.100 MeV, our calculations closely reproduced the experimental odd-even mass differences for the ^171-180Pt isotopes, yielding a root mean square deviation of σ=0.465 MeV. Additionally, as display in Fig. 7 for the ¹⁷⁴Pt to ¹⁷⁸Pb isotonic chain, the calculations closely matched the experimental odd-even mass differences, with a root mean square deviation of σ=1.192 MeV. $\sigma =\sqrt{{\displaystyle \sum _{\mu =1}^{\mathcal{N}}}{\left(P_{\mu }^{\text{Theor}\text{.}}-P_{\mu }^{\text{Expt}\text{.}}\right)}^2/\mathcal{N}},$ (22) Here, $P_{\mu }^{\text{Theor}\text{.}}$ and $P_{\mu }^{\text{Expt}\text{.}}$ represent the theoretical and experimental values of the odd-even mass differences, respectively, and 𝒩 denotes the total number of data points.

Fig. 6Full-screen View

Odd-even mass differences (in MeV) for Pt isotopes. "Expt." represents experimental values, and "Theor." represents theoretical values. Experimental data are from [67]

Fig. 7Full-screen View

Odd-even mass differences (in MeV) for the ¹⁷⁴Pt to ¹⁷⁸Pb isotonic chain. "Expt." represents experimental values, and "Theor." represents theoretical values. Experimental data are from [67]

Next, we examine the PES of the ^170-180Pt even-even isotopes under the determined pairing interaction strengths Gν=0.145 MeV and Gπ=0.100 MeV. Figure 8 shows the PES projected onto the (q₂, η) plane. For ¹⁷⁰Pt, the ground-state exhibited a prolate deformation at (q₂=0.15, η=0). In contrast, for ¹⁷²Pt, a more deformed minimum emerged, leading to the coexistence of a triaxial shape (γ≈30°) and a nearly prolate-deformed minimum at (γ≈120°), indicative of γ-instability due to the presence of multiple low-energy configurations at different γ values. The triaxial shape is even more pronounced in ¹⁷⁴Pt, where the ground-state is triaxial with deformation parameters (q₂=0.020, η=0.10, β≈0.2, γ≈90°) and a coexisting prolate minimum at (q₂=0.15, η=0). In ¹⁷⁶Pt a γ -unstable ground-state and a prolate minimum coexisted, but by ¹⁷⁸Pt and ¹⁸⁰Pt, a well-deformed prolate minimum quickly developed, becoming the most pronounced prolate ground-state at the mid-shell.

Fig. 8Full-screen View

(Color online) Potential energy surfaces of the ^170-180Pt even-even isotopes chain, projected on the (q₂, η) plane using the exact pairing model, where the energy is minimized in the q₄ direction with q₃ set to 0, with neutron and proton pairing interaction strengths of Gν=0.145 MeV, Gπ= 0.100 MeV. The ground-state deformation is represented by a red dot, while the coexistence minimum is indicated by a red cross

The findings of this study are broadly consistent with the results of Ref. [20], which studied the ^172-194Pt isotopic chain in the framework of the interacting boson model and self-consistent HFB calculation using the Gogny-D1S interaction. Both studies identified shape coexistence in the ^172-176Pt region, with γ-unstable minima and triaxial shapes in ¹⁷⁴Pt. Additionally, both studies showed the dominance of prolate deformation in ¹⁷⁸Pt, and ¹⁸⁰Pt, with the prolate minimum becoming the most pronounced ground state at the mid-shell.

It is noteworthy that a triaxial shape isomer exists for ^170-174Pt, characterized by (q₂ ≈ 0.600, η ≈ 0.060 (γ ≈ 10°)), and positioned approximately 5.0 MeV above the ground-state. However, this triaxial shape isomer vanishes for ^176-180Pt.