Loading [MathJax]/jax/output/HTML-CSS/jax.js

Tetrahedral shape and Lambda impurity effect in 80Zr with a multidimensionally constrained relativistic Hartree-Bogoliubov model

Figures(6) / Tables(2)

Get Citation
Dan Yang and Yu-Ting Rong. Tetrahedral shape and Lambda impurity effect in 80Zr with a multidimensionally constrained relativistic Hartree-Bogoliubov model[J]. Chinese Physics C. doi: 10.1088/1674-1137/ad8e40
Dan Yang and Yu-Ting Rong. Tetrahedral shape and Lambda impurity effect in 80Zr with a multidimensionally constrained relativistic Hartree-Bogoliubov model[J]. Chinese Physics C.  doi: 10.1088/1674-1137/ad8e40 shu
Milestone
Received: 2024-07-31
Article Metric

Article Views(984)
PDF Downloads(23)
Cited by(0)
Policy on re-use
To reuse of subscription content published by CPC, the users need to request permission from CPC, unless the content was published under an Open Access license which automatically permits that type of reuse.
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Email This Article

Title:
Email:

Tetrahedral shape and Lambda impurity effect in 80Zr with a multidimensionally constrained relativistic Hartree-Bogoliubov model

    Corresponding author: Yu-Ting Rong, rongyuting@gxnu.edu.cn
  • 1. Department of Physics, Guangxi Normal University, Guilin 541004, China
  • 2. Guangxi Key Laboratory of Nuclear Physics and Technology, Guangxi Normal University, Guilin 541004, China

Abstract: This study investigated the tetrahedral structure in 80Zr and Lambda (Λ) impurity effect in 81ΛZr using the multidimensionally constrained relativistic Hartree-Bogoliubov model. The ground states of both 80Zr and 81ΛZr exhibit a tetrahedral configuration, accompanied by prolate and axial-octupole shaped isomers. Our calculations reveal that there are changes in the deformation parameters β20, β30, and β32 upon Λ binding to 80Zr, except for β32 when Λ occupies p-orbits. Compared to the two shape isomers, the Λ particle exhibits weaker binding energy in the tetrahedral state when occupying the 1/2+[000](Λs) or 1/2[110] single-particle state. In contrast, the strongest binding occurs for the Λ particle in the 1/2[101] state with tetrahedral shape. Besides, a large Λ separation energy may not necessarily correlate with a significant overlap between the density distributions of the Λ particle and nuclear core, particularly for tetrahedral hypernuclei.

    HTML

    I.   INTRODUCTION
    • Tetrahedral symmetry, a prevalent arrangement in molecules and metallic clusters, can also manifest in nuclei. This specific symmetry arises from nonaxial octupole deformations β32. Similar to the well-established concept of magic numbers for spherical and prolate/oblate nuclei, specific configurations of protons and neutrons that fill full shells are predicted to be associated with enhanced stability for nuclei adopting a tetrahedral shape. These configurations effectively close certain energy shells, leading to tighter binding energy for the nucleus. Consequently, nuclei with specific neutron (proton) numbers, N(Z)= 16, 20, 32, 40, 56−58, 70, 90−94 and N= 112, 136/142, are predicted to have comparable or larger deformed gap sizes than the strongest spherical gaps at Z=20, 28, 40, or 50 [1].

      Among nuclei predicted to exhibit a tetrahedral shape, 80Zr serves as a prime example. Previous theoretical calculations have predicted a low-energy tetrahedral configuration for 80Zr alongside its known prolate ground state [2, 3]. Later, the Hartree-Fock-Bogoliubov calculations with Gogny interations found that the ground state of 80Zr is tetrahedral carried by pure β32 deformation [4, 5]. However, experimental observations based on decay properties indicate a highly deformed, non-tetrahedral shape for this nucleus [6, 7]. Recent mass measurement found that it is more strongly bound than predicted and might be classified as a deformed doubly magic system [8]. However, this observation does not resolve the discrepancy regarding its shape. Tagami et al. [4] proposed that the observed "superdeformed" band in 80Zr might be an excited state, while the tetrahedral ground state remains elusive.

      A hypernucleus is composed of nucleons and hyperons. Due to the additional strangeness degree of freedom, hyperons can penetrate deeply into the nucleus, acting as sensitive probes of various nuclear properties. These include effects like shrinkage [9, 10], modification of cluster [1113] and halo structures [1418], and enhancement of the pseudospin symmetry in the nucleons [19]. Additionally, hypernuclei have been linked to the extension of the nuclear drip line [16, 2022], increase of fission barrier heights [23, 24], and modification of shapes. These hyperon impurity effects are associated with nuclear shapes and the hyperon single-particle levels. In axial and reflection-symmetric nuclei, a Λ hyperon occupying the lowest s-state (represented by Λs or Nilsson quantum numbers Ωπ[Nn3ml]=1/2+[000]) reduces the quadrupole deformation parameter β2, effectively shrinking the nucleus [2528]. Conversely, a Λ hyperon occupying the 1/2[110] Nilsson orbit derived from the p shell (Λp) makes the nucleus more prolate, while those occupying the nearly degenerate 3/2[101] and 1/2[101] orbits make it more oblate [11, 26, 2931]. In triaxial calculations, a Λ hyperon can soften the potential energy surface (PES) along the γ-direction (Hill-Wheeler coordinate) [3234]. This alters the 2+ excitation energy and E2 transition probability of the core nucleus [3437]. Additionally, Λ hyperons can split the rotational bands of the corresponding nucleus when they occupy p-orbits [35]. In calculations assuming axial symmetry but reflection asymmetry, with increasing octupole deformation, the additional Λs(Λp) hyperons become more concentrated around the bottom (top) of the pear-shaped nucleus [38]. However, to date, no further studies have investigated exotic tetrahedral hypernuclei.

      Therefore, this work investigated the shape of 80Zr and the Λ hyperon impurity with tetrahedral shapes. Although it is difficult to synthesis the hypernucleus 81ΛZr, this paper provides a theoretical understanding of the hyperon impurity effect with Y32 correlation. We employ the multidimensionally-constrained covariant density functional theories (MDC-CDFTs), where the shape degrees of freedom (represented by βλμ with even μ values) are self-consistently included under the intrinsic V4 symmetry group [3942]. This framework has been successfully applied to study tetrahedral shapes in neutron-rich Zr isotopes [42] and Y32 correlations in N=150 isotones [43] and has been extended to hypernuclear properties [27, 28, 30, 32, 44]. The remaidner of this paper is organized as follows. In Sec. II, we introduce the MDC-CDFTs for hypernuclei. In Sec. III, we use MDC-CDFTs to calculate the potential energy surfaces (PESs) of 80Zr and 81ΛZr and discuss their shapes. The impurity effect in a tetrahedral nucleus is analyzed based on Λ separation energies, charge radii, and density distributions. Finally, a summary is given in Sec. IV.

    II.   THEORETICAL FRAMEWORK
    • The CDFTs are microscopic nuclear models that have been very successful in describing properties of nuclear matter and finite nuclei [41, 4553]. In CDFTs, baryons interact through mesons, and the Lagrangian for a Λ hypernucleus is written as

      L=BˉψB(iγμμMBgσBσgωBγμωμgρBγμτρμeγμ1τ32Aμ)ψB+ψΛfωΛΛ4MΛσμνΩμνψΛ+12μσμσ12m2σσ214ΩμνΩμν+12m2ωωμωμ14RμνRμν+12m2ρρμρμ14FμνFμν,

      (1)

      where B represents baryon (neutron, proton or Λ), and MB is the corresponding mass. σ, ωμ, and ρμ are scalar-isoscalar, vector-isoscalar, and vector-isovector meson fields coupled to baryons, respectively. Aμ is the photon field. Ωμν, Rμν, and Fμν are field tensors of the vector mesons ωμ and ρμ and photons Aμ. mσ (gσB), mω (gωB) , and mρ (gρB) are the masses (coupling constants) for meson fields. Note that σ and ϕ mesons introduced for multihypernuclei [44, 54, 55] and nuclear matter [5658] are omitted in this work because we only focus on single-Λ hypernuclei.

      The nonlinear coupling terms for mesons [5961] and density dependence of the coupling constants [6264] introduced two different approaches to give proper saturation properties of the nuclear matter, extending the investigation to hypernuclei. For nonlinear coupling effective interactions, nonlinear meson fields are added to the Lagrangian in Eq. (1) [55, 65, 66]. For density-dependent effective interactions, the coupling constants are dependent on the total baryonic density ρυ as follows:

      gmB(ρυ)=gmB(ρsat)fmB(x),x=ρυ/ρsat,

      (2)

      where m labels mesons, ρsat is the saturation density of nuclear matter, and fmB describes the density dependence behaviour [28, 67]. Because nonlinear coupling methods are widely used, in the following, we only introduce the detailed framework adopting density-dependent coupling constants.

      Starting from the Lagrangian (1) with density-dependent coupling constants, the equations of motion can be derived via the variational principle. The Dirac equation for baryons reads

      hBψiB=εiψiB,

      (3)

      where εi is the single-particle energy, ψiB is the single-particle wave function, and the single-particle Hamiltonian

      hB=αp+VB+TB+ΣR+β(MB+SB).

      (4)

      The Klein-Gordon equations for mesons and the Proca equation for photons are

      (Δ+m2σ)σ=gσNρsNgσΛρsΛ,(Δ+m2ω)ω0=gωNρυN+gωΛρυΛfωΛΛ2MΛρTΛ,(Δ+m2ρ)ρ0=gρN(ρυnρυp),ΔA0=eρυp.

      (5)

      Eqs. (3) and (5) are coupled via the scalar, vector, and tensor densities

      ρsB=iˉψiBψiB,ρυB=iˉψiBγ0ψiB,ρTΛ=i(iψiΛγψiΛ),

      (6)

      and various potentials

      VB=gωBω0+gρBτ3ρ0+e1τ32A0,SB=gσBσ,TΛ=fωΛΛ2MΛβ(αp)ω0,ΣR=gσNρυρsNσ+gωNρυρυNω0+gρNρυ(ρυnρυp)ρ0+12MΛfωΛΛρυρTΛω0.

      (7)

      The rearrangement term ΣR is present in the density-dependent CDFTs to ensure energy-momentum conservation and thermodynamic consistency [68].

      In this work, the Bogoliubov transformation is implemented to describe the pairing correlation between nucleons. A separable pairing force of finite range in the spin-singlet channel [69, 70], i.e.,

      V=Gδ(RR)P(r)P(r)1Pσ2,

      (8)

      is adopted, where G is the pairing strength, R and r are the center of mass and relative coordinates, respectively, and P(r) is the Gaussian function. The equal filling approximation [71] is adopted for the single Λ hyperon. Then, one can calculate the pairing field Δ and construct the relativistic Hartree Bogoliubov (RHB) equation as

      d3r(hBλΔΔhB+λ)(UkVk)=Ek(UkVk),

      (9)

      where λ is the Fermi energy, and Ek and (Uk,Vk)T are the quasi-particle energy and wave function, respectively. In MDC-CDFTs, instead of treating the pairing with BCS approximation after solving Eq. (3), Eq. (9) is directly solved by the multidimensionally-constrained relativistic Hartree Bogoliubov (MDC-RHB) model.

      The (hyper)nuclear shapes are charaterized by the deformation parameters βλμ as

      βλμ=4π3ARλQλμ,

      (10)

      where R=1.2A1/3 fm with mass number A. Qλμ is the multipole moment of the intrinsic densities and is calculated as follows:

      Qλμ=d3rρυ(r)rλYλμ(Ω),

      (11)

      where Yλμ(Ω) represents spherical harmonics with Euler angle Ω=(ϕ,θ,ψ).

      In the MDC-RHB model, thanks to the usage of an axial symmetric harmonic oscillator basis in solving the RHB equation (9), one can keep or break the axial and reflection symmetries easily. Therefore, four kinds of symmetries can be imposed: (a) axial-reflection symmetry with only β20; (b) non-axial but reflection symmetry with β20 and β22; (c) axial symmetry but reflection asymmetry with β20 and β30; and (d) non-axial and reflection asymmetry with β20, β22, β30 and β32, which are labeled as Kπ, Kπ, Kπ, and Kπ, respectively.

      Note that higher-order deformations are included naturely in MDC-CDFTs [72, 73] with the V4 symmetry, but we only discuss the deformations up to λ=3 in this work.

    III.   RESULTS AND DISCUSSION
    • To investigate the shape of 80Zr, we begin by calculating its two-dimensional PESs. Figure 1 shows the results obtained using the PK1 [61] effective interaction. In these calculations, specific pairing strengths (Gn=728.00 MeV fm3 and Gp=815.36 MeV fm3) are assigned to the pairing forces between neutrons and protons within the nucleus, respectively. These parameters are adjusted to reproduce the available empirical pairing gaps of 102,104Zr [42].

      Figure 1.  (color online) Two-dimensional potential energy surfaces (PESs) with (a) Kπ, (b) Kπ, (c) Kπ&β30=0.0, (d) Kπ&β30=0.1, (e) Kπ&β20=0.0, (f) Kπ&β20=0.1, and (g) Kπ&β32=0.2 symmetry imposed. The PK1 [61] effective interaction is adopted for the particle-hole channel, and Gn=728.00 MeV fm3 and Gp=815.36 MeV fm3 are adopted for the pairing channel. The location of the global energy minimum on each PES is marked by a red star.

      In Fig. 1(a), the Hill-Wheeler coordinates β2=β220+2β222 and γ=arctan(2β22/β20) with γ[0,60] are constrained instead of β20 and β22 because of the six-fold symmetry with pure quadrupole deformations. The PES reveals that the nucleus prefers a prolate shape in its ground state, with a quadrupole deformation of β2=0.49. This finding aligns with several other theoretical models [7477] and experimental observations based on decay properties [6, 7]. However, it is noteworthy that some theoretical models predict a spherical ground state for 80Zr [78, 79]. Our calculations also show the presence of other possible shapes (spherical, oblate, and triaxial) at slightly higher energies on the PES, suggesting a potential coexistence of shapes. This finding is consistent with other studies [74, 78].

      Figure 1(b) explores a different scenario, where Kπ symmetry is applied. In this case, the calculated PES reveals three possible low-energy shapes for the nucleus: prolate, oblate, and a new one called 'pear-like'. Interestingly, the pear-like shape, characterized by (β20,β30)=(0.02,0.20) deformation, has a slightly higher energy than the prolate shape. This suggests that 80Zr naturally prefers a prolate shape. However, including the reflection asymmetry reduces the energy barrier between the prolate and pear-like shapes. This makes it easier for the nucleus to fluctuate between these two shapes, potentially even favoring the pear-like shape under certain conditions.

      To explore a scenario with even less symmetry, we completely remove both reflection and axial restrictions (represented by Kπ symmetry). In Fig. 1(c), we fix β30=0.0, set β22 as a free parameter with initial value zero, and calculate E(β20,β32). Similar to the previous case, the nucleus can adopt prolate and oblate shapes. However, a new, even lower-energy minimum appears, characterized by a deformation pattern of a tetrahedron. This finding aligns with calculations using other theoretical models [4, 5].

      We investigate how the interplay between various deformation parameters (β20,β30,β32) affects the PESs of 80Zr. Figure 1(d)−(g) present the results. A key finding is that the tetrahedral minimum becomes more prominent when the β30 and β32 values are close to zero (Fig. 1(c,e)). This suggests that a more spherical shape favors the tetrahedral configuration. Additionally, the flatness of the PES with respect to β30 and β32 in Fig. 1(e) indicates a high degree of shape fluctuation between the tetrahedral and pear-like shapes. As β30 increases in E(β20,β32) (Fig. 1(d)), the correlation between Y20 and Y32 weakens. With increasing prolate deformation β20 (Fig. 1(f)), a saddle point emerges between the tetrahedral and axial-octupole energy minima. When β20 reaches 0.2 (Fig. 1(g)), neither β30 nor β32 impacts the energy minimum of 80Zr. These findings align with observations in neutron-rich Zr isotopes [42] and support the notion that octupole deformations play a significant role in near-spherical nuclei [2, 42, 80]. This further strengthens the validity of using the MDC-RHB model with Kπ symmetry as a reliable approximation for studying the ground state of 80Zr.

      Figure 2 compares PESs calculated with various symmetry constraints in one-dimensional PESs. In Fig. 2(a), the Kπ symmetric PES exhibits three energy minima corresponding to those in Fig. 1(a), except for the triaxial minimum. However, relaxing parity symmetry (Kπ) introduces an additional energy cost for near-spherical shapes. This is because a slightly non-spherical deformation term (β30) becomes more influential in this case, favoring a pear-shaped minimum over a perfect sphere. In the Kπ calculation, we fix β30 to zero to isolate the effect of β30 and set the initial value of β22 to zero. This leads to the spherical minimum transforming into a tetrahedral minimum. Interestingly, the oblate and prolate minima remain similar to those in the case with only β20 considered. It is worth noting that the higher energies observed with the Kπ&β30=0.0 symmetry than those with the Kπ symmetry around the prolate region disappear when the β30 and β22 deformations are allowed to vary freely. We have checked this through the PESs in the supplement: a proper initial value of β22 will make the energy of the prolate saddle point with Kπ symmetry lower than that of the Kπ one. Besides, the energy of the oblate minimum is lowered by the β22 deformation, but the ground state still exhibits tetrahedral shape.

      Figure 2.  (color online) Potential energy surfaces of (a) 80Zr with PK1, (b) 81ΛZr with PK1-Y1, (c) 80Zr with DD-ME2, and (d) 81ΛZr with DD-ME2-Y2. The calculations are performed with Kπ (blue dashed-dotted line), Kπ (red dotted line), and Kπ&β30=0.0 (black solid line) symmetries imposed.

      Our calculations explore the dependence of ground state predictions on the effective interaction used. While the PK1 functional (Fig. 2(a)) predicts prolate ground state with Kπ symmetries, the DD-ME2 [64] interaction (Fig. 2(b)) favors a spherical ground state with this symmetry. However, relaxing both the axial and reflection symmetry leads to tetrahedral ground states for both effective interactions. To further solidify this result, we calculate the ground state of 80Zr using additional effective interactions (NL3 [81], TM1 [60], PC-PK1 [82], and DD-PC1 [83]) in Table 1. Interestingly, all interactions predict a tetrahedral ground state for 80Zr. This consistency reinforces the conclusion that 80Zr exhibits a tetrahedral ground state, independent of the specific effective interaction employed.

      Interactionβ20β30β32E/MeV
      NL30.0000.0000.000664.861
      0.4930.0000.000665.191
      0.0150.1850.000665.759
      0.0000.0000.178666.524
      TM10.0000.0000.000668.674
      0.5380.0000.000664.996
      0.0170.1940.000669.800
      0.0000.0000.184670.657
      PK10.0000.0000.000665.074
      0.4910.0000.000667.061
      0.0210.2010.000666.894
      0.0000.0000.192668.336
      PC-PK10.0000.0000.000666.702
      0.4990.0000.000663.911
      0.0050.1180.000666.844
      0.0000.0000.116666.971
      DD-ME20.0000.0000.000665.675
      0.5020.0000.000663.500
      0.0120.1660.000666.476
      0.0000.0000.162667.213
      DD-PC10.0000.0000.000667.551
      0.5360.0000.000663.577
      0.0090.1510.000668.032
      0.0000.0000.144668.397

      Table 1.  Minimums of 80Zr in the cases of spherical, prolate, axial-octupole, and tetrahedral symmetry with different effective interactions using MDC-RHB model. Pairing strength Gn=1.0G0, Gp=1.12G0 with G0=728.0 MeV fm3 and effective range a=0.644 fm.

      Then, the PESs of 81ΛZr for various symmetries are studied using PK1-Y1 [65] (Fig. 2(c)) and DD-ME2-Y2 [28] (Fig. 2(d)) effective interactions. As expected, the PES for 81ΛZr with PK1-Y1 closely resembles that of its core nucleus, 80Zr (Fig. 2(a)). This indicates that a single Λ hyperon has minimal influence on the overall nuclear shape dominated by the eighty core nucleons. The inclusion of the β30 (and β32) deformation lowers the energies for near-spherical shapes, favoring a transition from a spherical to pear-like (or tetrahedral) minimum. Ultimately, the calculations predict a prolate minimum with slightly higher energy compared to the tetrahedral ground state. This suggests that 81ΛZr adopts a tetrahedral structure as the ground state, coexisting with prolate and pear-like shape isomers. Similar results are observed with the DD-ME2-Y2 interaction (Fig. 2(d)). Here, the overall PES is lowered upon binding a Λ hyperon to 80Zr, but the core nucleons continue to dictate the dominant shape. The calculations again predict a tetrahedral ground state for 81ΛZr with a pear-like isomer.

      Therefore, to investigate the effects of the Λ hyperon impurity in 81ΛZr, calculations that relax both axial and reflection symmetry constraints are necessary (denoted as Kπ). For simplicity, Fig. 3 presents results from calculations with Kπ&β30=0.0. As discussed previously, β30 influences the PES within the 0.2<β20<0.2 region but does not alter the energy minima. In this investigation, we calculate the PESs for 81ΛZr in which the Λ hyperon occupies the four lowest available orbits. These PESs are then compared to the PES of the core nucleus, 80Zr. It is important to note that while the Nilsson quantum numbers (Ωπ[Nn3ml]) are used to label the Λ single-particle states for convenience, they lose their exact meaning when both axial and reflection symmetries are broken.

      Figure 3.  (color online) Energies of 81ΛZr(80Zr) as a function of the deformation parameter β20 calculated by (a) PK1-Y1(PK1) and (b) DD-ME2-Y2(DD-ME2) effective interactions. The calculations are performed with Kπ&β30=0.0 symmetry. In each subfigure, Λ hyperon occupies the lowest s orbit and the three p orbits. The Nilsson quantum numbers Ωπ[Nn3ml] shown here are the dominant components of the corresponding Λ orbits. The inset in each subfigure shows the β32 deformation of the corresponding PES as a function of β20 deformation.

      Figure 3 reveals a strong similarity between the PESs of 81ΛZr and those of its core nucleus, 80Zr, for both PK1-Y1 and DD-ME2-Y2 interactions. The presence of a Λ hyperon occupying the 1/2+[000] orbit primarily leads to an binding energy increase of approximately 22 MeV. Interestingly, the 1/2[101] and 3/2[101] states exhibit nearly identical energies for specific shapes. This is naturally attributed to the very weak spin-orbit splitting observed in Λ hyperons, as reported in Ref. [84]. The inset of the figure depicts the β32 deformation of the PESs as a function of the β20 deformation. Notably, for all PESs, the β32 values coincide with those of 80Zr for specific β20 values.

      Table 2 provides quantitative data to explore the relationship between nuclear shape and the influence of the Λ hyperon impurity. Let us focus here on the deformation changes. When a Λ particle occupies the 1/2+[000] state, all three studied states (prolate, pear-like, and tetrahedral) exhibit a decrease in deformation compared to their counterparts in the core nucleus. For the 1/2[110](1/2[101]) state occupancy by a hyperon, the β20 deformation of the prolate state increases, while the β30 deformation of the pear-like state decreases compared to the core nucleus. Interestingly, the β32 deformation of the tetrahedral state remains almost unchanged.

      (Hyper)nucleusβ20β30β32rm/fmrc/fmE/MeVSΛ/MeVIoverlap/fm3
      PK1-Y1 [65]
      80Zr0.4914.2194.326667.019
      81ΛZr(1/2+[000])0.4834.2024.320689.76422.7450.144
      81ΛZr(1/2[110])0.4984.2184.327685.24718.2280.139
      81ΛZr(1/2[101])0.4754.2084.321681.70914.6900.124
      80Zr0.0000.0000.1924.1544.265668.332
      81ΛZr(1/2+[000])0.0000.0000.1894.1404.263690.37222.0400.142
      81ΛZr(1/2[110])0.0110.0000.1924.1514.265684.07815.7460.126
      81ΛZr(1/2[101])0.0000.0000.1924.1514.265683.95715.6250.126
      80Zr0.0210.2024.1394.252666.895
      81ΛZr(1/2+[000])0.0200.1974.1264.250688.98422.0890.143
      81ΛZr(1/2[110])0.0370.2064.1374.253682.76715.8720.127
      81ΛZr(1/2[101])0.0150.1994.1364.252682.43215.5370.125
      DD-ME2-Y2 [28]
      80Zr0.5024.2404.350663.500
      81ΛZr(1/2+[000])0.4914.2064.329686.32222.8220.149
      81ΛZr(1/2[110])0.5064.2244.337682.01118.5110.146
      81ΛZr(1/2[101])0.4834.2144.331677.81114.3110.130
      80Zr0.0000.0000.1624.1574.272667.213
      81ΛZr(1/2+[000])0.0000.0000.1514.1254.254689.97322.7600.152
      81ΛZr(1/2[110])0.0130.0000.1614.1404.261682.94815.7350.135
      81ΛZr(1/2[101])0.0000.0000.1604.1404.261682.76215.5490.136
      80Zr0.0120.1664.1474.264666.476
      81ΛZr(1/2+[000])0.0100.1544.1174.247689.35222.8760.153
      81ΛZr(1/2[110])0.0270.1774.1324.254682.40315.9270.137
      81ΛZr(1/2[101])0.0060.1614.1304.252681.94015.4640.134

      Table 2.  Deformation parameters (β20,β30, and β32), root mean square (r.m.s.) matter radii (rm), r.m.s. charge radii (rc), total energies (E), and Λ separation energies SΛ of 81ΛZr and density overlaps between the nuclear core and hyperon (Ioverlap) with hyperon injected into the Λs (1/2+[000]) and Λp (1/2[110] and 1/2[101]) orbits, respectively. The properties of 80Zr are also listed for comparison. The tetrahedral state (labeled with one asterisk) is calculated with Kπ&β30=0.0 symmetry, and the pear-like state (labeled with two asterisks) is calculated with Kπ symmetry.

      In addition to the deformation changes, our findings also reveal a shrinkage effect. The prolate state exhibits larger matter and charge radii compared to the tetrahedral and pear-like states. This is further illustrated in Fig. 4, which depicts the difference in core and hypernuclear charge radii (Δrc), defined as Δrc=rc(81ΛZr)rc(80Zr). Here, rc signifies the charge radius. The Δrc values range from 0.04 to 0.01 fm when the Λ particle occupies the s1/2 orbit. This shrinkage effect is smaller for p-orbit occupancy and demonstrates sensitivity to the chosen effective interaction. Calculations using the PK1-Y1 interaction predict a smaller shrinkage compared to those using DD-ME2-Y2.

      Figure 4.  (color online) Difference of the core and hypernuclear charge radii Δrc, Λ separation energy (SΛ), and density overlap of the core and hyperon Ioverlap as a function of the isomer (prolate, tetrahedral, and pear-like) states. The Λ hyperon occupies the lowest s orbit (Ωπ[Nn3ml]=1/2+[000]) and the two p orbits (1/2[110] and 1/2[101]). For convenience of comparison, panels (e) and (f) are offset downward from panel (d) by 6 and 8 MeV, respectively. The calculations are performed using PK1-Y1 and DD-ME2-Y2 effective interactions, respectively.

      The final aspect we explore is the Λ separation energy, denoted by SΛ, which represents the energy difference between the hypernucleus (81ΛZr) and core nucleus (80Zr), expressed as: SΛ=E(81ΛZr)E(80Zr). Figure 4(d) depicts this energy for the three studied states when a Λ particle occupies the s1/2 orbit. The SΛ values are around 22−23 MeV, with minor variations between the shapes. Notably, the prolate state exhibits a slightly higher separation energy compared to the other two states when calculated with the PK1-Y1 interaction. In contrast, the DD-ME2-Y2 interaction predicts the highest separation energy for the axial-octupole state. Nonetheless, the tetrahedral ground state does not exhibit the strongest binding for the Λ hyperon, regardless of the interaction. For the Λ particle occupying the 1/2[110] orbit, the SΛ value for the prolate state is approximately 2 MeV larger than those of the pear-like and tetrahedral states. Interestingly, when Λ occupies 1/2[101], the largest separation energy is consistently found in the tetrahedral shape, irrespective of the chosen effective interaction.

      Finally, the overlap integral (Ioverlap) between the core nucleon density (ρN(r)) and Λ density (ρΛ(r)) is calculated using the following equation [27, 33, 85, 86]:

      Ioverlap=ρN(r)ρΛ(r)d3r.

      (12)

      This integral quantifies the degree of spatial overlap between these densities, as introduced in Ref. [33]. The slightly softer energy curve toward the prolate configuration in C isotopes originates from the fact that the overlap between the deformed nuclear density and spherical Λ density is maximum under the prolate configuration. Later, a correlation between SΛ and Ioverlap was found in Refs. [27, 85, 86]. In this work, as shown in Table 2 and Fig. 4(d)−(i), a larger Λ separation energy (SΛ) generally correlates with a larger overlap integral. However, an exception occurs for the case of the Λ particle occupying the 1/2+[000] state with the DD-ME2-Y2 interaction. Here, the prolate state exhibits a higher SΛ than the pear-like state, despite a lower Ioverlap value. This discrepancy can be understood by examining the nucleon and Λ density distributions, which are presented in Fig. 5 for calculations with both PK1-Y1 and DD-ME2-Y2 interactions. In all cases, the Λ hyperon resides in the center of the xz plane. However, the tetrahedral state exhibits a "clustered" feature within its nucleon density, leading to a lower density at the center of the xz plane compared to the other shapes. This explains the smaller SΛ observed for the tetrahedral state. Conversely, the large SΛ value for the pear-like state with DD-ME2-Y2 interaction arises from its very small radii and a more centralized nucleon density distribution.

      Figure 5.  (color online) Contour plot of nucleon density and density profile of the Λ hyperon in the (x,z) plane at y=0 fm for differrent shapes of 81ΛZr calculated with PK1-Y1 (a)−(c) and DD-ME2-Y2 (d)−(f) effective interactions, respectively. The Λ hyperon occupies the 1/2+[000] orbit.

    IV.   SUMMARY
    • This study investigated the Y32 correlation in 80Zr and Lambda impurity effect in tetrahedral shape using the MDC-RHB model. We found that the ground state of 80Zr exhibits a minimum energy configuration in a tetrahedral shape, along with minima in prolate and axial-octupole shapes. The octupole deformations β30 and β32 lower the PESs in near-spherical shapes. However, differentiating between the pear-like and tetrahedral shapes remains challenging due to the very flat potential energy surface E(β30,β32) in near-spherical region.

      The binding strength of the Λ particle, quantified by the Λ separation energy (SΛ), depends on the specific energy level (orbital) it occupies within the nucleus and the chosen nuclear interaction. For the Λ in the 1/2+[000] and 1/2[110] states, the prolate state shows the largest SΛ with the PK1-Y1 interaction, while the DD-ME2-Y2 interaction favors the axial-octupole state. Interestingly, when the Λ occupies the 1/2[101] orbit, the tetrahedral shape consistently exhibits the strongest binding (largest SΛ) regardless of the interaction used. Our analysis revealed a general correlation between SΛ and the overlap integral (Ioverlap) between the Lambda and nucleon densities. This suggests a stronger binding for a larger spatial overlap between these densities. However, an exception occurs for the 1/2+[000] state with the DD-ME2-Y2 interaction. In this case, a "clustered" structure within the tetrahedral nucleon density distribution leads to a smaller SΛ despite a potentially larger overlap.

    APPENDIX A
    • The 2D PESs of 80Zr at different points of Fig. 2(a) are discussed here. Fig. A1(a) shows the E(β22,β30) from the energy minimum of E(β20=0.22,β32). The energy minima appear at β220.3 and β30=0.0, meaning that the triaxial shape will lower the energy minimum of E(β20=0.22,β32). However, the obtained energy is still higher than the tetrahedral one. Fig. A1(b) and (c) show the E(β22,β30) from the saddle point of E(β20=0.14,β32) and the tetrahedral minimum, respectively. The energy minima appear at β22=0.0 and β30=0.0, indicating that these two deformation parameters do not attribute to the energies at the E(β20=0.14,β32) saddle point and tetrahedral minimum. Fig. A1(d) is the E(β22,β32) from the saddle point of E(β20=0.2,β30). The energy minima appear at β220.12 and β320.0 and lower than the Kπ saddle point at β20=0.2. Fig. A1(e) shows the PES with β20 from 0.2 to 0.5. Considering the obtained result that octupole deformations will not change the energies in this region, this figure shows us that the energy minimum at β200.5 is prolate in the Kπ calculations.

      Figure A1.  (color online) Two-dimensional potential energy surfaces (PESs) at (a) the E(β20=0.22,β32) energy minimum, (b) E(β20=0.14,β32) saddle point, (c) tetrahedral energy minimum, (d) E(β20=0.2,β30) saddle point, and (e) prolate region with β20>0.2 of Fig. 2(a). The location of the global energy minimum on each PES is marked by a red star.

Reference (86)

目录

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return