-
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 andN= 112, 136/142, are predicted to have comparable or larger deformed gap sizes than the strongest spherical gaps atZ=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 [11−13] and halo structures [14−18], and enhancement of the pseudospin symmetry in the nucleons [19]. Additionally, hypernuclei have been linked to the extension of the nuclear drip line [16, 20−22], 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 numbersΩπ[Nn3ml]=1/2+[000] ) reduces the quadrupole deformation parameterβ2 , effectively shrinking the nucleus [25−28]. Conversely, a Λ hyperon occupying the1/2−[110] Nilsson orbit derived from the p shell (Λp ) makes the nucleus more prolate, while those occupying the nearly degenerate3/2−[101] and1/2−[101] orbits make it more oblate [11, 26, 29−31]. In triaxial calculations, a Λ hyperon can soften the potential energy surface (PES) along the γ-direction (Hill-Wheeler coordinate) [32−34]. This alters the2+ excitation energy andE2 transition probability of the core nucleus [34−37]. 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 additionalΛs(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
81Λ Zr, this paper provides a theoretical understanding of the hyperon impurity effect withY32 correlation. We employ the multidimensionally-constrained covariant density functional theories (MDC-CDFTs), where the shape degrees of freedom (represented byβλμ with even μ values) are self-consistently included under the intrinsicV4 symmetry group [39−42]. This framework has been successfully applied to study tetrahedral shapes in neutron-rich Zr isotopes [42] andY32 correlations inN=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 and81Λ Zr 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. -
The CDFTs are microscopic nuclear models that have been very successful in describing properties of nuclear matter and finite nuclei [41, 45−53]. In CDFTs, baryons interact through mesons, and the Lagrangian for a Λ hypernucleus is written as
L=∑BˉψB(iγμ∂μ−MB−gσBσ−gωBγμωμ−gρBγμ→τ⋅→ρμ−eγμ1−τ32Aμ)ψB+ψΛfωΛΛ4MΛσμνΩμνψΛ+12∂μσ∂μσ−12m2σσ2−14ΩμνΩμν+12m2ωωμωμ−14→Rμν→Rμν+12m2ρ→ρμ→ρμ−14FμνFμν,
(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 , andFμν are field tensors of the vector mesonsωμ and→ρμ and photonsAμ .mσ (gσB ),mω (gωB ) , andmρ (gρB ) are the masses (coupling constants) for meson fields. Note thatσ∗ and ϕ mesons introduced for multihypernuclei [44, 54, 55] and nuclear matter [56−58] are omitted in this work because we only focus on single-Λ hypernuclei.The nonlinear coupling terms for mesons [59−61] and density dependence of the coupling constants [62−64] 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, andfmB 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 HamiltonianhB=α⋅p+VB+TB+ΣR+β(MB+SB).
(4) The Klein-Gordon equations for mesons and the Proca equation for photons are
(−Δ+m2σ)σ=−gσNρsN−gσΛρsΛ,(−Δ+m2ω)ω0=gωNρυN+gωΛρυΛ−fωΛΛ2MΛρTΛ,(−Δ+m2ρ)ρ0=gρN(ρυn−ρυp),−ΔA0=eρυp.
(5) Eqs. (3) and (5) are coupled via the scalar, vector, and tensor densities
ρsB=∑iˉψiBψiB,ρυB=∑iˉψiBγ0ψiB,ρTΛ=i∂(∑iψ†iΛγψiΛ),
(6) and various potentials
VB=gωBω0+gρBτ3ρ0+e1−τ32A0,SB=gσBσ,TΛ=−fωΛΛ2MΛβ(α⋅p)ω0,ΣR=∂gσN∂ρυρsNσ+∂gωN∂ρυρυNω0+∂gρN∂ρυ(ρυn−ρυp)ρ0+12MΛ∂fωΛΛ∂ρυρTΛω0.
(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(R−R′)P(r)P(r′)1−Pσ2,
(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′(hB−λΔ−Δ∗−hB+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 asβλμ=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 angleΩ=(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 withβ20 andβ22 ; (c) axial symmetry but reflection asymmetry withβ20 andβ30 ; and (d) non-axial and reflection asymmetry withβ20 ,β22 ,β30 andβ32 , which are labeled asKπ ,⧸Kπ ,K⧸π , and⧸K⧸π , 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 toλ=3 in this work. -
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 andGp=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 of102,104 Zr [42].Figure 1. (color online) Two-dimensional potential energy surfaces (PESs) with (a)
⧸Kπ , (b)K⧸π , (c)⧸K⧸π&β30=0.0 , (d)⧸K⧸π&β30=0.1 , (e)⧸K⧸π&β20=0.0 , (f)⧸K⧸π&β20=0.1 , and (g)⧸K⧸π&β32=0.2 symmetry imposed. The PK1 [61] effective interaction is adopted for the particle-hole channel, andGn=728.00 MeV fm3 andGp=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 andγ=arctan(√2β22/β20) withγ∈[0,60∘] are constrained instead ofβ20 andβ22 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 ofβ2=0.49 . This finding aligns with several other theoretical models [74−77] 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
K⧸π 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
⧸K⧸π symmetry). In Fig. 1(c), we fixβ30=0.0 , setβ22 as a free parameter with initial value zero, and calculateE(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 theβ30 andβ32 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 toβ30 andβ32 in Fig. 1(e) indicates a high degree of shape fluctuation between the tetrahedral and pear-like shapes. Asβ30 increases inE(β20,β32) (Fig. 1(d)), the correlation betweenY20 andY32 weakens. With increasing prolate deformationβ20 (Fig. 1(f)), a saddle point emerges between the tetrahedral and axial-octupole energy minima. Whenβ20 reaches 0.2 (Fig. 1(g)), neitherβ30 norβ32 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⧸K⧸π 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 (K⧸π ) 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⧸K⧸π calculation, we fixβ30 to zero to isolate the effect ofβ30 and set the initial value ofβ22 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 onlyβ20 considered. It is worth noting that the higher energies observed with the⧸K⧸π&β30=0.0 symmetry than those with theK⧸π symmetry around the prolate region disappear when theβ30 andβ22 deformations are allowed to vary freely. We have checked this through the PESs in the supplement: a proper initial value ofβ22 will make the energy of the prolate saddle point with⧸K⧸π symmetry lower than that of theK⧸π one. Besides, the energy of the oblate minimum is lowered by theβ22 deformation, but the ground state still exhibits tetrahedral shape.Figure 2. (color online) Potential energy surfaces of (a) 80Zr with PK1, (b)
81Λ Zr with PK1-Y1, (c) 80Zr with DD-ME2, and (d)81Λ Zr with DD-ME2-Y2. The calculations are performed withKπ (blue dashed-dotted line),K⧸π (red dotted line), and⧸K⧸π&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.Interaction β20 β30 β32 E/MeV NL3 0.000 0.000 0.000 −664.861 0.493 0.000 0.000 −665.191 0.015 0.185 0.000 −665.759 0.000 0.000 0.178 −666.524 TM1 0.000 0.000 0.000 −668.674 0.538 0.000 0.000 −664.996 0.017 0.194 0.000 −669.800 0.000 0.000 0.184 −670.657 PK1 0.000 0.000 0.000 −665.074 0.491 0.000 0.000 −667.061 0.021 0.201 0.000 −666.894 0.000 0.000 0.192 −668.336 PC-PK1 0.000 0.000 0.000 −666.702 0.499 0.000 0.000 −663.911 0.005 0.118 0.000 −666.844 0.000 0.000 0.116 −666.971 DD-ME2 0.000 0.000 0.000 −665.675 0.502 0.000 0.000 −663.500 0.012 0.166 0.000 −666.476 0.000 0.000 0.162 −667.213 DD-PC1 0.000 0.000 0.000 −667.551 0.536 0.000 0.000 −663.577 0.009 0.151 0.000 −668.032 0.000 0.000 0.144 −668.397 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 withG0=728.0 MeV fm3 and effective rangea=0.644 fm.Then, the PESs of
81Λ Zr 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 for81Λ Zr 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 theβ30 (andβ32 ) 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 that81Λ Zr 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 for81Λ Zr with a pear-like isomer.Therefore, to investigate the effects of the Λ hyperon impurity in
81Λ Zr, calculations that relax both axial and reflection symmetry constraints are necessary (denoted as⧸K⧸π ). For simplicity, Fig. 3 presents results from calculations with⧸K⧸π&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 for81Λ Zr 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
81Λ Zr(80Zr) as a function of the deformation parameterβ20 calculated by (a) PK1-Y1(PK1) and (b) DD-ME2-Y2(DD-ME2) effective interactions. The calculations are performed with⧸K⧸π&β30=0.0 symmetry. In each subfigure, Λ hyperon occupies the lowest s orbit and the three p orbits. The Nilsson quantum numbersΩπ[Nn3ml] shown here are the dominant components of the corresponding Λ orbits. The inset in each subfigure shows theβ32 deformation of the corresponding PES as a function ofβ20 deformation.Figure 3 reveals a strong similarity between the PESs of
81Λ Zr and those of its core nucleus, 80Zr, for both PK1-Y1 and DD-ME2-Y2 interactions. The presence of a Λ hyperon occupying the1/2+[000] orbit primarily leads to an binding energy increase of approximately 22 MeV. Interestingly, the1/2−[101] and3/2−[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 theβ32 deformation of the PESs as a function of theβ20 deformation. Notably, for all PESs, theβ32 values coincide with those of 80Zr for specificβ20 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 the1/2−[110](1/2−[101]) state occupancy by a hyperon, theβ20 deformation of the prolate state increases, while theβ30 deformation of the pear-like state decreases compared to the core nucleus. Interestingly, theβ32 deformation of the tetrahedral state remains almost unchanged.(Hyper)nucleus β20 β30 β32 rm /fmrc /fmE/MeV SΛ /MeVIoverlap /fm−3 PK1-Y1 [65] 80Zr 0.491 – – 4.219 4.326 −667.019 – – 81Λ Zr(1/2+[000] )0.483 – – 4.202 4.320 −689.764 22.745 0.144 81Λ Zr(1/2−[110] )0.498 – – 4.218 4.327 −685.247 18.228 0.139 81Λ Zr(1/2−[101] )0.475 – – 4.208 4.321 −681.709 14.690 0.124 80Zr ∗ 0.000 0.000 0.192 4.154 4.265 −668.332 – – 81Λ Zr∗ (1/2+[000] )0.000 0.000 0.189 4.140 4.263 −690.372 22.040 0.142 81Λ Zr∗ (1/2−[110] )0.011 0.000 0.192 4.151 4.265 −684.078 15.746 0.126 81Λ Zr∗ (1/2−[101] )0.000 0.000 0.192 4.151 4.265 −683.957 15.625 0.126 80Zr ∗∗ 0.021 0.202 – 4.139 4.252 −666.895 – – 81Λ Zr∗∗ (1/2+[000] )0.020 0.197 – 4.126 4.250 −688.984 22.089 0.143 81Λ Zr∗∗ (1/2−[110] )0.037 0.206 – 4.137 4.253 −682.767 15.872 0.127 81Λ Zr∗∗ (1/2−[101] )0.015 0.199 – 4.136 4.252 −682.432 15.537 0.125 DD-ME2-Y2 [28] 80Zr 0.502 – – 4.240 4.350 −663.500 – – 81Λ Zr(1/2+[000] )0.491 – – 4.206 4.329 −686.322 22.822 0.149 81Λ Zr(1/2−[110] )0.506 – – 4.224 4.337 −682.011 18.511 0.146 81Λ Zr(1/2−[101] )0.483 – – 4.214 4.331 −677.811 14.311 0.130 80Zr ∗ 0.000 0.000 0.162 4.157 4.272 −667.213 – – 81Λ Zr∗ (1/2+[000] )0.000 0.000 0.151 4.125 4.254 −689.973 22.760 0.152 81Λ Zr∗ (1/2−[110] )0.013 0.000 0.161 4.140 4.261 −682.948 15.735 0.135 81Λ Zr∗ (1/2−[101] )0.000 0.000 0.160 4.140 4.261 −682.762 15.549 0.136 80Zr ∗∗ 0.012 0.166 – 4.147 4.264 −666.476 – – 81Λ Zr∗∗ (1/2+[000] )0.010 0.154 – 4.117 4.247 −689.352 22.876 0.153 81Λ Zr∗∗ (1/2−[110] )0.027 0.177 – 4.132 4.254 −682.403 15.927 0.137 81Λ Zr∗∗ (1/2−[101] )0.006 0.161 – 4.130 4.252 −681.940 15.464 0.134 Table 2. Deformation parameters (
β20,β30 , andβ32 ), root mean square (r.m.s.) matter radii (rm ), r.m.s. charge radii (rc ), total energies (E), and Λ separation energiesSΛ of81Λ Zr and density overlaps between the nuclear core and hyperon (Ioverlap ) with hyperon injected into theΛs (1/2+[000] ) andΛp (1/2−[110] and1/2−[101] ) orbits, respectively. The properties of 80Zr are also listed for comparison. The tetrahedral state (labeled with one asterisk) is calculated with⧸K⧸π&β30=0.0 symmetry, and the pear-like state (labeled with two asterisks) is calculated withK⧸π 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 asΔrc=rc(81ΛZr)−rc(80Zr) . Here,rc signifies the charge radius. TheΔrc values range from−0.04 to 0.01 fm when the Λ particle occupies thes1/2 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 hyperonIoverlap 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] and1/2−[101] ). For convenience of comparison, panels (e) and (f) are offset downward from panel (d) by6 and8 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 thes1/2 orbit. TheSΛ 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 the1/2−[110] orbit, theSΛ value for the prolate state is approximately 2 MeV larger than those of the pear-like and tetrahedral states. Interestingly, when Λ occupies1/2−[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 andIoverlap 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 the1/2+[000] state with the DD-ME2-Y2 interaction. Here, the prolate state exhibits a higherSΛ than the pear-like state, despite a lowerIoverlap 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 thex−z plane. However, the tetrahedral state exhibits a "clustered" feature within its nucleon density, leading to a lower density at the center of thex−z plane compared to the other shapes. This explains the smallerSΛ observed for the tetrahedral state. Conversely, the largeSΛ value for the pear-like state with DD-ME2-Y2 interaction arises from its very small radii and a more centralized nucleon density distribution. -
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 deformationsβ30 andβ32 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 surfaceE(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 the1/2+[000] and1/2−[110] states, the prolate state shows the largestSΛ with the PK1-Y1 interaction, while the DD-ME2-Y2 interaction favors the axial-octupole state. Interestingly, when the Λ occupies the1/2−[101] orbit, the tetrahedral shape consistently exhibits the strongest binding (largestSΛ ) regardless of the interaction used. Our analysis revealed a general correlation betweenSΛ 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 the1/2+[000] state with the DD-ME2-Y2 interaction. In this case, a "clustered" structure within the tetrahedral nucleon density distribution leads to a smallerSΛ despite a potentially larger overlap. -
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 ofE(β20=−0.22,β32) . The energy minima appear atβ22≈0.3 andβ30=0.0 , meaning that the triaxial shape will lower the energy minimum ofE(β20=−0.22,β32) . However, the obtained energy is still higher than the tetrahedral one. Fig. A1(b) and (c) show theE(β22,β30) from the saddle point ofE(β20=−0.14,β32) and the tetrahedral minimum, respectively. The energy minima appear atβ22=0.0 andβ30=0.0 , indicating that these two deformation parameters do not attribute to the energies at theE(β20=−0.14,β32) saddle point and tetrahedral minimum. Fig. A1(d) is theE(β22,β32) from the saddle point ofE(β20=0.2,β30) . The energy minima appear atβ22≈0.12 andβ32≈0.0 and lower than theK⧸π saddle point atβ20=0.2 . Fig. A1(e) shows the PES withβ20 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 atβ20≈0.5 is prolate in the⧸K⧸π 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 withβ20>0.2 of Fig. 2(a). The location of the global energy minimum on each PES is marked by a red star.
Tetrahedral shape and Lambda impurity effect in 80Zr with a multidimensionally constrained relativistic Hartree-Bogoliubov model
- Received Date: 2024-07-31
- Available Online: 2025-02-15
Abstract: This study investigated the tetrahedral structure in 80Zr and Lambda (Λ) impurity effect in