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Symmetry
Volume 16
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10.3390/sym16050631
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Review

Pseudo-Spin Symmetry and the Hints for Unstable and Superheavy Nuclei

1
School of Nuclear Science and Technology, Lanzhou University, Lanzhou 730000, China
2
Frontier Science Center for Rare Isotope, Lanzhou University, Lanzhou 730000, China
3
Joint Department for Nuclear Physics, Lanzhou University and Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China
4
School of Physical Science and Technology, Southwest University, Chongqing 400715, China
*
Author to whom correspondence should be addressed.
Symmetry2024,16(5), 631;https://doi.org/10.3390/sym16050631
Submission received: 18 April 2024 /Revised: 4 May 2024 /Accepted: 6 May 2024 /Published: 20 May 2024
(This article belongs to the Special IssueRestoration of Broken Symmetries in the Nuclear Many-Body Problem)

Abstract

:
The pseudo-spin symmetry (PSS) provides an important angle to understand nuclear microscopic structure and the novel phenomena found in unstable nuclei. The relativistic Hartree–Fock (RHF) theory, that takes the important degrees of freedom associated with theπ-meson andρ-tensor (ρ-T) couplings into account, provides an appropriate description of the PSS restoration in realistic nuclei, particularly for the pseudo-spin (PS) doublets with high angular momenta (l˜). The investigations of the PSS within the RHF theory are recalled in this paper by focusing on the effects of the Fock terms. Aiming at common artificial shell closures appearing in previous relativistic mean-field calculations, the mechanism responsible for the PSS restoration of high-l˜ orbits is stressed, revealing the manifestation of nuclear in-medium effects on the PSS, and thus, providing qualitative guidance on modeling the in-medium balance between nuclear attractions and repulsions. Moreover, the essential role played by theρ-T coupling, that contributes mainly via the Fock terms, is introduced as combined with the relations between the PSS and various nuclear phenomena, including the shell structure and the evolution, novel halo and bubble-like phenomena, and the superheavy magicity. As the consequences of the nuclear force in complicated nuclear many-body systems, the PSS itself and the mechanism therein can not only deepen our understanding of nuclear microscopic structure and relevant phenomena, but also provide special insight into the nature of the nuclear force, which can further enrich our knowledge of nuclear physics.

    1. Introduction

    The global development of new-generation radioactive-ion-beam (RIB) facilities and advanced nuclear detectors has significantly expanded the field of research in nuclear physics, extending it from a few stable nuclei to several thousand unstable ones. This has greatly enriched nuclear science [1,2,3,4,5,6]. In contrast to the stable nuclei, numerous nuclear phenomena have been observed in the unstable nuclei, including the dilute matter distributions, the halo structure [7,8,9,10]; the emergence of new magicity and the disappearance of traditional magic shells [11,12,13,14,15,16,17,18]; and the occurrence of the central density depression, the bubble-like structure [19,20,21,22,23,24,25,26,27,28,29,30,31,32]; etc. These observations challenge our conventional understanding of nuclear physics. Moreover, a significant number of unstable nuclei are involved in the rapid neutron capture process (r-process) in the synthesis of heavy elements. This indicates that the unstable nuclei play a crucial role in understanding the origin of the elements in the universe [33,34].
    One of the most challenging issues in nuclear physics is the exploration of the existence limit of very heavy nuclei, which are known as the superheavy elements (SHEs), withZ⩾104, and the so-called stability island of superheavy nuclei (SHN). Experimentally, the discoveries of new elements up toZ=118 have been reported in Refs. [35,36]. It is evident that an enhanced shell effect is observed with increasing atomic numberZ, which may indicate the emergence of a proton magic shell atZ⩾120. This is evidenced by the increasing survival probabilities observed in SHEs fromZ=114 to 118 [37]. However, the current experimental conditions limit the ability to synthesize and precisely measure the expected SHN. Consequently, there is a need for pioneering theoretical studies to advance our understanding on both unstable nuclei and superheavy ones.
    The pseudo-spin symmetry (PSS) [38,39] represents a significant phenomenon in nuclear structure [40,41,42,43]. It is characterized by the quasi-degeneracy of two single-particle (s.p.) orbits with the quantum numbers (n,l,j =l + 1/2) and (n−1,l + 2,j =l + 3/2). The pseudo-spin (PS) doublet is designated by a set of quantum numbers (n˜ =n−1,l˜ =l + 1,j˜ =j =l˜± 1/2) [38,39]. Significant research has been conducted to understand the origin of the PSS, which is recognized as a relativistic symmetry [40,42,44], and the pseudo-orbitl˜ is simply the orbital angular momentum of the lower component of the Dirac spinor [44]. According to the effective field theory, the nuclear force contains both strong attractive and repulsive components, which are described, respectively, by the exchanges of the scalar and vector mesons [45]. The exact PSS condition is then generalized asS(r)+V(r)=0 [44] ordS(r)+V(r)/dr=0 [46]. It is important to note that both conditions indicate a balance between strong attractive and repulsive nuclear interactions, which are propagated mainly by the scalar and vector mesons, respectively.
    On the other hand, the delicate in-medium balance between nuclear attractions and repulsions is crucial to comprehend how a number of nucleons form a bound nucleus, a fundamental challenge in nuclear physics. The mean-field potential for nucleons in a nucleus is then approximated as a counteraction between the scalar and vector potentials, namely,S(r)+V(r). Consequently, any variation in this balance would directly influence the microscopic nuclear structure. In light of its origin, the PSS can be a valuable tool for elucidating the nature of the nuclear force and various nuclear phenomena. For instance, the pseudo-spin orbit (PSO) splitting, which measures the breaking of the PSS, can be helpful in comprehending the shell structure and the evolution [47,48,49], the halo structure [50], the superheavy nuclei [51,52], the nuclear superdeformed configurations [53,54], etc.
    At the level of the mean-field approach [55], the meson-exchange diagram of the nuclear force [56] can be divided into the Hartree and Fock terms. The significantπ-meson contributes only via the Fock terms. The relativistic mean-field (RMF) theory [55,57], which contains only the Hartree terms, has the advantage of providing a self-consistent treatment of strong spin–orbit coupling in nuclei, in contrast to the non-relativistic Skyrme or Gogny Hartree–Fock models. The incorporation of the Bogoliubov transformation with the RMF theory, namely, the relativistic Hartree–Bogoliubov (RHB) theory [58,59,60,61,62,63], unifies the descriptions of both mean-field and pairing correlations. Furthermore, the continuum effects are taken into account automatically, which promises extensive reliability in the description of novel phenomena in unstable nuclei, such as the novel halo phenomena observed in both spherical and deformed nuclei [64,65,66,67,68]. The relativistic scheme has facilitated numerous investigations into the origin of the PSS [44,46,69], the relationship between PSS and spin-orbit effects [70], the PSS’s dynamical properties [71,72,73] and non-perturbative nature [74,75,76], the PSS in deformed nuclei [77,78,79,80,81,82,83], the PSS in hypernuclei [84,85], and the resonant states [83,86,87,88,89,90,91,92,93,94,95,96], etc.
    However, limited by the Hartree approach, significant degrees of freedom associated with theπ-meson andρ-tensor (ρ-T) couplings are missing in the RMF models. Moreover, an important ingredient of the nuclear force—the tensor force—cannot be considered efficiently either. Once the Fock terms have been implemented, the relativistic Hartree–Fock (RHF) theory [97,98,99] can introduce theπ-pseudo-vector (π-PV) andρ-T couplings in a natural manner. In recent decades, RHF theory [48,100], as well as the extensive relativistic Hartree–Fock–Bogoliubov (RHFB) theory [50], has achieved comparable accuracy to the popular RMF models in describing various nuclear phenomena. Furthermore, the incorporation of Fock terms has led to significant improvements in the self-consistent description of nuclear shell evolution [49,101,102], the PSS restoration [48,100,103,104], symmetry energy [105], etc. In particular, the tensor force components can be taken into account naturally by the Fock terms [106,107,108], which is essential for extensive reliability from the stable to unstable nuclei. The RHFB theory [50], which benefits from the Bogoliubov scheme, provides a reliable description of various novel phenomena, including the bubble-like structures [31,32], halo structures [109], and new magicity [51,110,111], etc. Given these achievements, it is important to investigate systematically the relationship between the PSS and various nuclear phenomena within the RHF framework. This could provide valuable insight into the nature of the nuclear force and the properties of unstable nuclei.
    For both the RMF and RHF models mentioned above, the modeling of the in-medium effects of the nuclear force is significant to ensure the accuracy in describing various nuclear phenomena, for instance, an appropriate incompressibility of nuclear matter [112]. In the RMF theory, the in-medium effects are evaluated by either collaborating with the nonlinear self-couplings of mesons [113,114,115] or by considering the density dependencies in the meson–nucleon coupling strengths [116,117,118,119]. For the RHF theory [99], theoretical accuracy can be improved by incorporating the self-couplings of theσ-meson or scalar fields(ψ¯ψ) [120,121]. Significantly, assuming the meson–nucleon coupling strengths to be density-dependent [48,50,122], the density-dependent relativistic Hartree–Fock (DDRHF) theory has been demonstrated to achieve comparable accuracy to the RMF theory in describing nuclear structures. In this review, the density dependencies of the coupling strengths will be briefly recalled inSection 2.2. In comparison to the modeling of in-medium effects via self-coupling schemes [120,121], the density dependencies in the meson–nucleon coupling strengths, as guided by ab initio calculations [116,117,119,123], are of critical importance for the theoretical consistency of the RHF approach [122].
    Actually, the relativistic Brueckner–Hartree–Fock (RBHF) theory [43], which is based on a realistic nuclear force, can help us to understand the PSS. The relativistic effects, such as nucleon–antinucleon excitations, can be expressed in a non-relativistic framework in terms of a three-body force. The results of RBHF using the Bonn interaction are similar to those of non-relativistic Brueckner–Hartree–Fock containing the two- and three-body forces [124,125]. For relativistic ab initio calculations, the spin symmetry in Dirac sea is supported by the bare nucleon–nucleon interaction without three-body forces [126], and the evolution of the spin-orbit and PSO splittings in neutron drops can be explained by the tensor force [127,128]. Therefore, the RBHF calculations will provide crucial information, such as the splitting of PS partners, which is helpful when developing density functional theory. It is also interesting to study the microscopic nuclear structure by using RBHF theory with covariant chiral interactions [129].
    Despite their successes, both RMF calculations and RHF ones with PKOi (i=1,2,3) can lead to the emergence of artificial shell closuresN/Z = 58 and 92 [48,109], accompanied by largely overestimated binding energies aroundCe140 (Z=58) andU218 (Z=92) [130]. In fact, these artificial shell closures correspond to large PSO splitting of the high-l˜ PS doublets near the Fermi surfaces. Fortunately, the RHF Lagrangian PKA1 [48] has been developed to address this issue. It benefits from an improved in-medium balance between nuclear attractions and repulsions due to the strongρ-T coupling [104]. Currently, due to the uncertainty in the decomposition of nucleon self-energy in the ab initio calculations [43] it remains challenging to constrain the density dependencies of the coupling strengths for specific channels in an efficient manner. Alternatively, suitable physical measurements, which can guide the modeling of nuclear in-medium effects, are still welcomed for both RMF and RHF models, for instance, qualitatively by the PSS restoration.
    On the other hand, an increasing abundance of novelties in unstable nuclei can be taken as a new standard for testing the reliability of theoretical models which are established from the stable nuclei. It should be noted that some novel phenomena, such as the halo and bubble-like structures, are closely related to the restoration of the PSS [31,32,109]. Aiming at the role of the Fock terms in the PSS restoration, as well as the relationship between the PSS and various novel nuclear phenomena, the progress of the RHF descriptions of the PSS are reviewed. The manuscript is organized as follows. InSection 2, the RHF formalism is briefly introduced, and inSection 3 the influence of Fock terms on the PSS restoration is recalled. Then, inSection 4, the elimination of artificial shell closures and the physical mechanism responsible for the PSS restoration of the high-l˜ PS doublets are introduced. Focusing on the role of the Fock terms and their relation to the PSS,Section 5 reviews the RHF descriptions of various nuclear novelties, including the halo structure for Ce isotopes, the bubble-like structure, and the superheavy magicity. Finally, a summary is given inSection 6.

    2. Relativistic Hartree–Fock Model

    In order to provide a comprehensive overview of the theoretical framework, the general formalism of the RHF theory, including the effective Hamiltonian and energy functional, are recalled briefly in the following. In this review, the modeling of nuclear in-medium effects represents a significant topic. In order to address this issue, the density dependencies of the meson–nucleon coupling strengths will be introduced. These will be further detailed by analyzing the density-dependent feature of the popular RMF and RHF Lagrangians.

    2.1. Hamiltonian and Energy Functional

    In the context of the meson-exchange diagram [56], the nuclear force is posited to be propagated by massive virtual mesons. In the RMF and RHF approaches, two isoscalar mesons, namely,σ andωμ, and two isovector ones, namely,ρ→μ andπ→, are considered. These mesons, possessing the following quantum numbers (IP,τ), are considered to be the effective fields which propagate the nucleon–nucleon interactions:
    σ(0+,0),ωμ(1−,0),ρ→μ(1−,1),π→(0−,1).
    The spin, parity and isospin of selected mesons are represented byI,P, andτ, respectively. Specifically, theπ-meson, with a low mass of about 138 MeV, is responsible for the long-range part of the nucleon–nucleon (NN) interaction. It is also thought to be the main origin of the tensor force in the relativistic framework [99]. The scalar mesonσ, with a mass about 500 MeV, provides the intermediate-range attraction of theNN interaction. The repulsive inner part of theNN potential is thought to arise mainly from the exchange of a massive vector meson, theω-meson. The repulsion can be understood by analogy with the Coulomb interaction, which is repulsive between electrons of the same charge [131]. The attraction and repulsion contributed by theσ- andω-mesons, respectively, primarily determine the binding of a nuclear many-body system. The isovector–vectorρ→-meson is introduced to describe the isovector nature of the nucleon–nucleon interaction. The photon fieldAμ accounts for the electromagnetic interactions between protons. In this review, arrows are employed to represent isovectors and bold type is used to denote the space vectors.
    When restricting to the mean-field approach, the meson-exchange diagram of the nuclear force can be divided into the Hartree and Fock terms, as illustrated inFigure 1. Despite the fact that the two-body interactions propagated by mesons are not instantaneous, the retardation effects are ignored in the Hartree–Fock description of nuclear ground states, i.e., neglecting the time component of the four–momentum carried by the meson [99]. In fact, the masses of theσ-,ω-, andρ-mesons, which have several hundred MeV, are much larger than the energy transfers. Such an approximation is valid for heavy mesons, and also to a lesser extent for theπ-meson. Following the standard procedure as described in Ref. [99], the Hamiltonian of nuclear systems can be expressed in the following form:
    H=∫drψ¯(r)−iγ·∇+Mψ(r)+12∑ϕ∫drdr′ψ¯(r)ψ¯(r′)ΓϕDϕ(r−r′)ψ(r)ψ(r′),
    where the symbolϕ represents various two-body interaction channels, including the Lorentz scalar (σ-S), vector (ω-V,ρ-V,A-V), tensor (ρ-T), vector–tensor (ρ-VT), and pseudo–vector (π-PV) couplings. The corresponding vertexΓϕ(r,r′) can be expressed in the following form:
    Γσ-S≡−gσ(r)gσ(r′),
    Γω-V≡gωγμrgωγμr′,
    Γρ-V≡gργμτ→r·gργμτ→r′,
    Γρ-T≡14M2fρσνkτ→∂kr·fρσνlτ→∂lr′,
    Γρ-VT≡12Mfρσkντ→∂kr·gργντ→r′+12Mgργντ→r·fρσkντ→∂kr′,
    Γπ-PV≡−1mπ2fπτ→γ5γμ∂μr·fπτ→γ5γν∂νr′,
    ΓA-V≡e24γμ(1−τ)rγμ(1−τ)r′.
    For the meson and photon fields, the propagatorDϕ is of the following form:
    Dϕ=14πe−mϕ|r−r′||r−r′|,DA=14π1|r−r′|.
    whereϕ=σ,ω,ρ,π. In the above expressions,M andmϕ are the masses of the nucleon and meson respectively, and the symbolsgϕ (ϕ=σ,ω,ρ) andfϕ′ (ϕ′=ρ,π) represent the meson–nucleon coupling strengths.
    Under the mean-field approach, the no-sea approximation is considered as usual, which amounts to neglecting the contributions from the Dirac sea [112]. Consequently, in terms of the particle creation and annihilation operators, namely,cα† andcα, defined by the positive-energy solutions of the Dirac equation, the Dirac fieldψ that describes nucleons can be quantized as
    ψ(x)=∑αψα(x)e−iεαtcα.
    whereψα represents the Dirac spinor of positive-energy stateα. According to the no-sea approximations, the Hartree–Fock ground state of nuclear systems can be defined as
    HF=∏lAcl†−,
    where the symbol− represents the vacuum state, andA denotes the nuclear mass number.
    Eventually, starting from the meson-exchange diagram of the nuclear force, combined with the quantization of the nucleon field (11), the expectation of the Hamiltonian (2) with respect to the Hartree–Fock ground stateHF gives the energy functionalE of nuclear systems [99,132]:
    E=HFHHF=Ek+∑ϕEϕD+EϕE,
    including the kinetic energyEk, and the Hartree and Fock terms of the potential energies from various coupling channelsϕ, named asEϕD andEϕE, respectively. In accordance with the variational principle, the single-particle Dirac equations of nucleons can be derived from the energy functional (13), which allows for the derivation of two types of self-energies, the local Hartree and non-local Fock ones [133,134]. It should be noted that in the RMF models, only the Hartree termsEϕD are considered. Consequently, the degrees of freedom associated with theπ-PV andρ-T couplings, which contribute mainly via the Fock diagrams, cannot be taken into account explicitly.
    In open-shell nuclei, the pairing correlations play a significant role in determining the properties of the ground state, particularly for the nuclei situated close to the drip line [58,61,64,65,66]. In general, the pairing correlations are treated using the BCS [135] or Bogoliubov [136,137,138] methods. In particular, for unstable nuclei, the one- and/or two-nucleon separation energies may be less than 1 MeV. For such weak-bound systems it is crucial to use an appropriate treatment of pairing correlations and continuum effects. The Bogoliubov scheme offers a unified treatment of both mean fields and pairing correlations, which is a distinct advantage over the traditional BCS method. Starting from the quantization (11) and combining the Bogoliubov transformation [139], one can also quantize the Dirac spinor in the Bogoliubov quasi-particle space as follows:
    ψ(x)=∑kψkU(x)e−iEktβk+ψk˜V(x)eiEktβk†,
    where the indexesk andk˜ represent the time-reversal conjugated states, theU andV components of the Bogoliubov quasi-particle spinors are denoted byψU andψV, andEk is the quasi-particle energy. By considering the expectation of the Hamiltonian with respect to the Bogoliubov ground state, namely, the quasi-particle vacuumβkHFB=0, one can obtain a unified energy functional that contains the kinetic, potential, and pairing energies:
    E=HFBHHFB=Ek+∑ϕEϕD+EϕE+Eϕpp,
    whereEϕpp represents the pairing energy contributed by the coupling channelϕ. From the full energy functional presented above, one can derive the RHFB equations by applying the variational principle.
    In practice, the phenomenological pairing force was adopted, including the zero-rangeδ-force [58], and finite-range Gogny force [50,58] and separable one [140]. The details of the RHFB formalism are provided in Refs. [50,134,141]. In the following, we will focus on the RHB and RHFB calculations, with the aim of discussing novel nuclear phenomena.

    2.2. Effective Lagrangian and In-Medium Effects

    For both the RMF and RHF theories, the meson masses and the meson–nucleon coupling strengths in the effective Hamiltonian (2) define the effective interactions (also referred to as effective Lagrangians); these are determined by fitting the bulk properties of nuclear matter and observable nuclei, such asO16,Ca40,Ca48,Ni56,Sn132,Pb208, and so on. More than that, the modeling of nuclear in-medium effects is also essential for an accurate description of nuclear properties; these are evaluated by considering the nonlinear self-couplings of mesons [113,114,115,120,121] or introducing the density dependence into the meson–nucleon coupling strengths [116,117,118,119,122]. In this review, the PSS and novel nuclear phenomena will be discussed by utilizing the density-dependent effective Lagrangians.
    For better understanding, the density dependence of the meson–nucleon coupling strengths is presented as follows. As shown in Equation (3), the Hamiltonian of nuclear systems includes theσ-S,ω-V,ρ-V,ρ-T,ρ-VT, andπ-PV meson–nucleon coupling channels. The density dependencies of the coupling strengthsgσ andgω in the isoscalar channelsσ-S andω-V are adopted as the following form [119]:
    gϕ(ρb)=gϕ(ρ0)fρ(x)ϕ=σ-S,ω-V,
    wherex=ρb/ρ0, andρ0 is the saturation density. For the functionfϕ(x), it has the following explicit form:
    fρ(x)=aϕ1+bϕx+dϕ21+cϕ(x+dϕ)2,
    whereaϕ,bϕ,cϕ, anddϕ are the parameters determining the density dependencies ofgσ andgω, and the constraint conditionsfϕ(1)=1,fϕ″(0)=0, andfσ″(1)=fω″(1) are generally introduced to reduce the number of free parameters.
    For the isovector mesons, an exponential density dependence is utilized:
    gϕ(ρb)=gϕ(0)exp−aϕx.
    In the above expression,gϕ(0) corresponds to the free-coupling constantsgρ,fρ, andfπ of theρ-V,ρ-T, andπ-PV couplings, respectively, andaϕ to the relevant density-dependent parameters.
    In order to understand the difference between the density-dependent effective Lagrangians used in this review,Table 1 presents the details of the meson–nucleon coupling channel for effective Lagrangians PKA1, PKOi (i = 1, 2, 3), DD-ME2, PKDD, and DD-LZ1, respectively. One can see that the RMF effective Lagrangians DD-ME2, PKDD, and DD-LZ1 do not contain the Fock terms. Hence, the significant degrees of freedom associated with theπ-meson andρ-T couplings cannot be taken into account. For the RHF Lagrangians, PKO2 shares the same degrees of freedom as the RMF ones, whereas PKO1 and PKO3 take theπ-PV coupling into account, and PKA1 contains both theπ-PV andρ-T couplings, the latter of which plays an important role in the PSS restoration, in particular for the high-l˜ PS doublets.

    3. The PSS and the Fock Term

    Due to numerical limitations, the early RHF model does not include theρ-T coupling. Moreover, the energy functional contributed by theπ-PV coupling is relatively small compared to those contributed by theσ- andω-couplings. Consequently, despite the implementation of the Fock terms leading to significant and remarkable differences in different meson channels [122], the sums of strong attraction (σ-S) and strong repulsion (ω-V), and the balance of attraction and repulsion will be similar to those of the RMF theory after the counteraction between the Hartree and Fock terms. According to the condition of PSS conservation,S(r)+V(r)=0 [44] ordS(r)+V(r)/dr=0 [46], the PSS restoration is less influenced by the Fock term [100].
    To understand the influence of the Fock terms on the PSS, the Schrödinger-type equation deduced from the Dirac equation is quite helpful [46,100]. Under spherical symmetry, the radial Dirac equation can be derived as
    EG(r)=−ddr−κrF(r)+ΣS(r)+Σ0(r)G(r)+Y(r),
    EF(r)=+ddr+κrF(r)−2M+ΣS(r)−Σ0(r)G(r)+X(r),
    whereG(r) andF(r) are the radial wave functions of the upper and lower components, respectively, the quantityκ=±(j+1/2) withj=l∓1/2, and the scalar potentialΣS and the time component of the vector potentialΣ0 contain the contributions from the Hartree terms and the rearrangement term due to the density dependence of the meson–nucleon couplings. The non-local integralX andY terms can be transformed into the equivalent local contributionsXG,XF,YG, andYF; the details can be found in Ref. [100]. Thus, from the integral Equations (18) and (19), the Schrödinger-type equation for the lower componentF can be derived as
    d2dr2F+V1ddrF+VPCB+VPSO+V2F=−(VD−E)(ΔD−E)F,
    whereVPCB andVPSO represent the pseudo-centrifugal barrier (PCB) and pseudo-spin orbital potential (PSOP), respectively, andV1 andV2 are of more complicated forms than in the RMF case [46].
    V1≡XG−YF−1Δ−EdΔdr,
    V2≡YF1Δ−EdΔdr−XGYF−ddrYF+YGVD−E+XF(Δ−E),
    VPSO≡κr1Δ−EdΔdr−XG+YF,
    VPCB≡κ(1−κ)r2.
    In the above expressions,Δ≡ΔD+YG,V≡VD+XF, withΔD≡ΣS+Σ0 andVD≡Σ0−ΣS−2M, respectively. As indicated in Ref. [46], the PSS will be recovered under the limitation ofVPSO≪VPCB. Compared to the RMF case, the Fock terms present new contributions, namely, theXG andYF terms in theVPSO. Moreover, theV2 term is entirely due to the Fock term.
    In finite nuclei, the PSO and spin-orbit (SO) splittings show some systematics with respect to the s.p. energies [46]. Coincidentally, the RHF calculations present a similar tendency as well [100]. TakingSn132 as an example,Figure 2 shows the PSO and SO splitting as functions of the average s.p. energyE¯. Despite the minor differences in the s.p. energies, an obvious monotonous decreasing behavior ofΔEPSO with respect toE¯ is observed for both RHF and RMF models, similar to that observed forZr88 andZr120 [46]. Compared to theΔEPSO, the energy dependence ofΔESO is relatively weak. This is due to the fact that theVPSO in Equation (23) is divided by a smaller factorΔ−E thanV−E for the SO potentialVSO [46]. Specifically,Δ andE have the magnitude of a few tens of MeV, whereasV can be as large as several hundred MeV. It should be noted inFigure 2 that the PSO splitting is rescaled by dividing by(2l˜+1), namely,ΔEPSO=(El˜j=l˜−1/2−El˜j=l˜+1/2)/(2l˜+1). In the following context, the PSO splitting will not be rescaled, i.e.,ΔEPSO=El˜j=l˜−1/2−El˜j=l˜+1/2.
    InFigure 2, it is obvious that the PSS is restored well for the PS partnersν2p˜=(ν3s1/2,ν2d3/2) in both RHF and RMF calculations. Regarding the restoration conditionVPCB≫VPSO given by Ref. [46],Figure 3 presents theVPSO andVPCB of PS partnersν2p˜, both of which are scaled by the factorF2/(VD−E). For the PS partnerν3s1/2, the PSOP is much smaller than the PCB after considering cancellation at the nodal points. However, for the partnerν2d3/2, the PSOP becomes comparable with the PCB, even after considering the cancellation. This seems to be conflict with the conclusion drawn from the RMF calculations [46].
    In order to understand such inconsistency,Figure 4 shows the Fock contributions to theVPSO andV1, namely,VPSOE=κr1Δ−EdYGdr−XG+YF andV1E=XG−YF−1Δ−EdYGdr, which are scaled by the factorsF2/(VD−E) andFF′/(VD−E), respectively. It is obvious that there exist distinct cancellations for the Fock contributionsVPSOE andV1E at the nodal points, which lead to tiny contributions for the PS partnerν3s1/2. Referring to theVPSO inFigure 3, the conditionVPCB>>VPSO becomes invalid for the PS partnerν2d3/2 due to the substantial contributions from the Fock terms. Even though, such contributions are largely canceled by the Fock contributionsV1E, particularly in the interior region; seeFigure 4. Due to such counteractions among the Fock contributions, mainly in theVPSOE andV1E terms, the PSS is still properly restored for the PS partnersν2p˜, despite considerable PSOP originating from the Fock terms.
    Different from the Hartree terms, the Fock terms present non-local mean-field potentials, corresponding to the non-local termsX andY in the radial RHF Equations (18) and (19), both of which are state dependent. For explicit illustration,Figure 5 shows the radial wave functionsG andF (left panels), and theY andX terms (right panels), as functions of radial distancer. It is interesting to see that theY (X) terms and radial wave functionsG (F) have rather similar radial dependencies for both PS partnersν3s1/2 andν2d3/2. Thus, one can reach the approximate relations as
    X(r)≃X0(r)F(r),Y(r)≃Y0(r)G(r),
    where theX0 andY0 terms can be treated as the local representations of the non-local Fock terms. In fact, it is not striking to see such a result. According to the principle of density functional theory, the contributions from the exchange correlations, here the Fock terms, can be expressed universally as the functional of local density [144,145]. Consistently, following the relation (25), the non-local RHF Equations (18) and (19) can be reduced to the equivalent local ones, which are of the same form as the radial Dirac equations in the RMF calculations [46]. From this point of view, it is not difficult to understand the approximately preserved PSS given by both RHF and RMF models, despite the fact the Fock terms present substantial contributions to the pseudo-spin orbit potentials.

    4. The PSS and Nuclear In-Medium Effects

    In realistic nuclei, the binding of the systems is ensured by the balance between the attraction and repulsion, which manifests itself as the PSS restoration. Therefore, any change in the balance will affect the microscopic nuclear structure. In the last section, the effects of Fock terms on the PSS restoration were discussed. It seems that the PSS restoration described by the RHF model is similar to the RMF one due to the significant cancellation between the Fock contributions, mainly theVPSOE andV1E terms, and the similarity between the radial wave functions and the non-local Fock terms [Y andX in Equations (18) and (19)]. Coincidentally, both RMF and RHF calculations with PKOi (i=1,2,3) suffer greatly from the notable PSS breaking of the high-l˜ PS doublets near the Fermi surfaces, which present artificial shell closuresN/Z=58 and 92 and lead to systematically overestimated binding energies for the nuclei aroundCe140 (Z=58) andU218 (Z=92) [130].
    With advances in numerical capability,ρ-T coupling was later introduced explicitly into the RHF model. It has been shown thatρ-T coupling contributes a rather strong attraction, which can even change the balance between the dominantσ-S andω-V channels [48]. In addition,ρ-T coupling contributes significant nuclear in-medium effects due to its strong density dependence [104]. Consequently, the RHF calculations with PKA1 show a rather different PSO splitting for the high-l˜ PS doublets close to the Fermi surfaces, accompanied by the disappearance of the artificial shell closures [49,51,109]. This is attributed to the implementation of the degree of freedom associated with theρ-T coupling. Furthermore, the calculations also benefit from this improvement, for example, in the shell structure and the evolution, in the description of novel nuclear phenomena, etc. Given these improvements, the elimination of the artificial shell closuresZ=92 andZ=58 and the physical mechanisms behind the PSS restoration of high-l˜ PS doublets will be recalled in this section by focusing on the nuclear in-medium effects. Furthermore, the implications for the thermal properties of nuclear matter and the development of new effective Lagrangians are briefly mentioned.

    4.1. Elimination of Artificial Shell Closures

    Taking the artificial shell closureZ=92 as an example,Figure 6 presents the proton s.p. energy (SPE) ofPb208, by taking the RHF effective Lagrangians PKA1 [48] and PKO3 [101], and RMF’s one DD-ME2 [142], as the representatives. It can be seen that the splittings between the PS partnerπ1g˜=(π2f7/2,π1h9/2) given by PKO3 and DD-ME2 are rather large, indicating the existence of the artificial shellZ=92 that even compresses the size of the magic shellN=82. In contrast, PKA1 provides consistent results with the experimental data, showing a properly restored PSS that eliminates the artificial shellZ=92. In fact, an experiment on the short-lived isotopeNp223 has already ruled out the possibility of the sub-shellZ=92 [146]. Not only forZ=92, the elimination of the artificial shellZ=58 was also detailed in Ref. [48], and the relevant consequences will be introduced in the following.
    In heavy nuclei, due to the increasing s.p.-level densities, the occurrences of (semi-) magic shells can be essentially related to the PSS restoration. For instance, as shown inFigure 6, there exist two PS doubletsπ1g˜ andπ2p˜=(π3s1/2,π2d3/2), above and below the magic shellN=82, respectively, and it is already seen that the size of the shell can be affected by the PSO splittings. As another typical example, the experimentally confirmed sub-shellZ=64, which shows systematic enhancement towardsZ=64 along the isotonic chain ofN=82 [47], can be treated as the result of the PSS restoration. However, the sub-shellZ=64 is not properly described by the RMF models, which generally give an artificial shellZ=58 due to the obvious PSS breaking [48,130].
    To further understand the relation between the sub-shellZ=64 and artificial shell closureZ=58,Gd6482146 is taken as an example here.Figure 7 presents the proton SPE ofGd146 calculated by the RHF model with PKA1 [48], the Gogny HFB model with D1S [148], the Skyrme HF model with SLy4 [149], and the RMF model with PKDD [115]. In addition to the magic shellZ=50, the RHF, Gogny, and Skyrme models present the evident sub-shellZ=64, consistent with a properly restored PSS for the PS doubletsπ2p˜ andπ1f˜=(π2d5/2,π1g7/2), which are located above and below the sub-shell, respectively. It is obvious as well that small PSO splittings are beneficial for the arising of the sub-shellZ=64. On the contrary, the RMF calculation gives an unphysically large gap between the PS doubletπ1f˜, leading to the artificial shellZ=58, that suppresses the emergence of the sub-shellZ=64; seeFigure 7.
    Following the evolution of theπ2d splittings that gives theZ=64 sub-shell inFigure 8a, one can see a strong enhancement in theπ2d splitting towardsZ=64, and the RHF calculations with PKA1 provide the best agreement with the experimental data. More specifically, the sum of the Fock terms of isoscalarσ- andω-couplings, and the Coulomb field, play a determining role in giving such an enhancement towardsZ=64; seeFigure 8b; while the isovectorπ- andρ-couplings present a tiny contribution for the enhancement in theπ2d splitting towardsZ=64, due to the counteractions between the opposite tensor force contributions carried by theπ-PV andρ-T couplings, respectively. Combining the results inFigure 7 andFigure 8, it can be concluded that the PSS restoration may be essential not only for the occurrence of the sub-shell, but also for the evolution. It is worthwhile to stress that the enhancement in theπ2d splitting cannot be simply attributed to the tensor force, which indeed plays a tiny role in this case.

    4.2. New Mechanism of PSS Restoration

    As previously, there is a systematic difference between PKA1 and other RHF/RMF effective Lagrangians in the description of the PSS restoration. This difference can be closely linked to the magic or semi-magic shells. Taking the doubly magic nucleiCa48,Zr90,Sn132, andPb208 and the predicted superheavy one126310 [110] as examples,Figure 9a,b illustrate the relevant proton shell gaps and the splittings of the PS doublets with highl˜ neighboring the shells, as predicted by the RHF and RMF models. When compared to experimental data [147], PKA1 shows appropriate agreements, whereas the others distinctly overestimate the PSO splittings. This has implications for the emergence of the neighboring shells in heavy nuclei. For example, less pronounced proton shellsZ=82 andZ=50 are typically accompanied by a larger value ofΔEPSOπ for the PS doubletsπ1g˜ inPb208 andπ1f˜ inSn132; seeFigure 9. Furthermore, the close correlation between the emergence of shell closures and the PSS is also predicted for SHN, which will be discussed in detail inSection 5.3.
    Conceptually, the conditionV+S=0 ord(V+S)/dr=0 can preserve the PSS precisely, which cannot be satisfied in realistic nuclei. On the other hand, such a condition indeed indicates certain in-medium balance between nuclear attractions and repulsions. TakingPb208 as an example,Table 2 shows the contributions to the energy functional from the sum of kinetic energy and the isoscalarσ-S andω-V couplings, that of the isovectorρ andπ fields, and that of the electromagnetic field. Following the implementation of the Fock terms and further theρ-T coupling, one can observe changes from the RMF to RHF approaches on modeling the in-medium balance between nuclear attractions and repulsions.
    As demonstrated inTable 2, all the effective Lagrangians give comparable total binding energies forPb208, whereas the specific contributions, exceptEcou., exhibit notable difference. In the RMF approach, the binding of the nucleus is primarily influenced by the counteraction between the strong attractiveσ-S and repulsiveω-V couplings. The isovector contributionEρ+π is notably enhanced by the Fock terms when moving from the RMF to the RHF approach. In contrast, the dominant isoscalar termEkin.+σ+ω is reduced in order to maintain the proper binding ofPb208. Within the RHF approach, the isovector termEρ+π is considerably enhanced from PKO3 to PKA1 due to theρ-T coupling in PKA1, which contributes a fairly strong attraction. In contrast, the isoscalar termEkin.+σ+ω is significantly reduced [104].
    In fact, the systematic differences in the modeling of the binding of the nucleus can be manifested on the PSS restoration, not only in several doubly magic nuclei, shown inFigure 9b, but also in a single nucleus.Figure 10a shows the proton PSO splittings,ΔEPSOπ, inPb208 with respect to the pseudo-orbitl˜ calculated by RHF Lagrangians PKA1 and PKO3, and the RMF one DD-ME2. In contrast to the results obtained with the PKO3 and DD-ME2, the PKA1 results show a rapid reduction with respect tol˜ and well-restored PSS for the PS doubletl˜=4, which is in good agreement with the experimental data. Moreover, from a comparison of the results presented inFigure 10a,b it is evident that the sum contributionEkin.+σ+ω, which contains the kinetic term, and the dominantσ-S andω-V couplings, has a significant impact on the systematic behavior ofΔEPSOπ. It is thus revealed that the discrepancies in the PSS restoration are primarily due to the in-medium balance between the attractiveσ-S and repulsiveω-V couplings. In particular, these balances are systematically altered from the RMF Lagrangian DD-ME2 to the RHF one PKO3 and subsequently to PKA1. This is evidenced by the reduction in the sum contributionsEkin.+σ+ω, as shown inFigure 10b, which is consistent with the contribution of the binding energy, as shown inTable 2. It is noteworthy that the dependencies ofEkin.+σ+ω on thel˜ exhibit a nearly parallel shift from DD-ME2 to PKO3, but undergo a significant change from PKO3 to PKA1.
    Combining the results inTable 2 andFigure 10, the in-medium balance between nuclear attractions and repulsions can be significantly changed by the inclusion of the Fock terms, particularly in the context of theρ-T coupling. Consequently, it can be reasonably anticipated that these findings will have implications for the modeling of nuclear in-medium effects, as evaluated by the density dependencies of the meson–nucleon coupling strengths. As shown inFigure 10c, both the values and density dependencies of the coupling strengthsgσ andgω, associated with the degrees of freedom of theσ-S andω-V couplings, respectively, are reduced from DD-ME2 to PKO3, due to the substantial contributions from the isovector channel, as evidenced inTable 2. This may also indicate that the dominant isoscalar channels (σ-S andω-V) do not require the same strength as in the RMF approach. On the other hand, the isovector channels can be significantly enhanced by the Fock terms, which can incorporate more nuclear in-medium effects. Consequently, the density dependencies ofgσ andgω are reduced as well from DD-ME2 to PKO3. Even though,gσ andgω still exhibit nearly parallel density-dependent behaviors from DD-ME2 to PKO3, which is common for the popular RMF Lagrangians and RHF ones PKOi (i=1,2,3). However, this situation dramatically changes from PKO3 to PKA1. As revealed inTable 2 andFigure 10e, theρ-T coupling in PKA1 not only contributes fairly strong attraction, which can even affect the balance between the dominantσ-S andω-V channels, but also takes into account much more nuclear in-medium effects than PKO3, as shown inFigure 10e. Consequently, the implementation of theρ-T coupling results in a further reduction in the coupling strengthsgσ andgω in PKA1, which no longer maintain parallel density-dependent behaviors. Further, combined with the schematic pictureFigure 11, the consequence of the in-medium balance on the microscopic PSS restoration becomes more apparent.
    Schematically,Figure 11 shows the profiles of the matter density (ρb) and the probability densities of thes-,p-,d-, andf-orbits. Combined with the results inFigure 10c–e,Figure 11 can help us to understand thel˜-dependence of the PSO splittingΔEPSOπ inFigure 10. As illustrated inFigure 11, the centrifugal potentials drive nucleons outwards as the angular momentuml increases. From the central region to the surface of the nucleus, the matter density gradually decreases from a near-saturated value to zero. This may result in a variation in the in-medium balance of nuclear attractions and repulsions, given that the meson–nucleon coupling strengths are density-dependent. As seen fromFigure 10c, the coupling strengthsgσ andgω in both DD-ME2 and PKO3 exhibit near parallel density-dependent behaviors, whereas PKA1 provides a rather different modeling on the in-medium balance between nuclear attractions and repulsions. In accordance with this, the PSO splittings provided by DD-ME2 and PKO3, as well as the sum contributionsEkin.+σ+ω, show a relatively weak angular momentum dependence. However, the PKA1 results represent a pronouncedl˜-dependence. Consequently, by combining the schematic systematics of the orbits with differentl quantities inFigure 11, it is evident that the in-medium balance of nuclear attractions and repulsions can be manifested as thel˜-dependence of theΔEPSOπ values inFigure 10, which provides a qualitative guidance to understand the nuclear in-medium effects.

    4.3. The Thermal Nuclear Matter and Development of New Effective Lagrangians

    Theρ-T coupling, which may play an essential role in determining the nuclear in-medium balance, is significant not only for microscopic nuclear structure, but also for the liquid–gas (LG) phase transition of thermal nuclear matter. The LG phase transition of thermal nuclear matter has important implications for heavy-ion collisions [151,152,153,154], nuclear astrophysics [155,156,157,158,159], and so on. Numerous investigations have been devoted to the LG phase transition of thermal nuclear matter [160,161,162,163,164,165] under the RMF and RHF approaches. As illustrated inTable 3 andFigure 12, the effective Lagrangian PKA1, which exhibits a robustρ-T coupling, predicts distinctly different critical properties and LG phase diagrams from the other effective Lagrangians.
    In order to figure out the physics behind the difference between PKA1 and other effective Lagrangians, particularly for the role ofρ-T coupling and nuclear in-medium effects, two series of temporary parameterizations were proposed, namedxκρ andxκρ* [166], wherex=1.0,0.9,… means the scaling factor of theρ-T coupling strengthκρ=fρ/gρ. In addition, the seriesxκρ* maintains the density dependencies ofgσ(ρb) andgω(ρb) unchanged from PKA1, whereas the seriesxκρ,gσ, andgω share the same density dependencies, which show similar systematics as the other density-dependent effective Lagrangians. As shown inFigure 12a, nice linear correlations between the critical temperatureTC and pressurePC are preserved along thexκρ series, being coincident with the selected effective Lagrangians except PKA1. On the contrary, PKA1 and the seriesxκρ* present different linear correlations due to their different in-medium balances from the other effective Lagrangian and the seriesxκρ. Moreover, as shown inFigure 12b, a large expansion in the LG phase diagrams from the original PKA1 to the seriesxκρ is observed, and the results of set0.8κρ are already rather close to PKO3, that does not contain theρ-T coupling. From the above discussions, it is not hard to deduce that the in-medium balance of nuclear attraction and repulsion is essential for the van der Waals-like behaviors of thermal nuclear matter, in which the residual nuclear in-medium effects embedded in the balance between the dominant attractive and repulsive channels play a significant role [166].
    In Refs. [143,167], new RMF Lagrangians DD-LZ1 and PCF-PK1 were proposed, guided by the consistent relation between the PSS restoration and nuclear in-medium effects. In contrast to other popular RMF effective Lagrangians, the density dependences ofgσ andgω do not parallel each other in the case of DD-LZ1, as shown inFigure 10c. As with PKA1, DD-LZ1 addresses the common issue in previous RMF calculations, namely, the occurrence of artificial shell closures atN/Z=58 andN/Z=92, as illustrated in the left panel ofFigure 13. Similar to DD-LZ1, the new RMF Lagrangian PCF-PK1, which incorporates local exchange terms for the four-fermion interactions by utilizing the Fierz transformation, effectively eliminates the artificial shell closuresZ=58 andZ=92 as well, as shown in the right panel ofFigure 13, for which the unparalleled density dependences ofgS andgV are essential. Moreover, DD-LZ1 and PCF-PK1 restore the PSS for the high-l˜ pseudo-spin doublets near the Fermi levels, which further corroborates the consistent relation between the PSS restoration and the nuclear in-medium balance between nuclear attraction and repulsion.

    5. The PSS and Novel Nuclear Phenomena

    InSection 4, the mechanism of the PSS restoration for highl˜ are briefly recalled, in which the in-medium balance between nuclear attraction and repulsion plays an important role. As aforementioned, unstable nuclei exhibit plenty of novel nuclear phenomena, such as the halo and bubble-like structures, which reveal the characteristics of the microscopic structure of unstable nuclei. As one of the hottest topics in nuclear physics, the SHN are essentially stabilized by the shell effects, which may be closely related to the PSS restoration as discussed before. Therefore, in this section, the influence of PSS restoration and its breaking on novel nuclear phenomena, as well as on the emergence of superheavy magic shells, will be reviewed.

    5.1. Halo Occurrence and PSS Restoration

    As aforementioned, following the PSS restoration, the artificial shell closureZ=58 given by the previous RMF and RHF calculations was eliminated by the RHF Lagrangian PKA1. Extending to the drip-line region, it can have a strong impact on the halo occurrence, for instance, in cerium (Ce) isotopes [109]. As one of the typical novel phenomena, a halo is characterized by an extensive matter distribution [7,8,9,10], arising from nucleons populating weakly bound low-l orbits. Due to the centrifugal barrier, the nucleons populating the neighboring high-l orbits do not contribute to the halo, but play an important role in stabilizing the dilute nucleons via the pairing correlations [64,65,109,171].
    As revealed in Ref. [109], the neutron halo occurrence in the drip-line Ce isotopes is tightly connected with the PSS restoration of valence proton orbits, and such a relation is enhanced strongly by theρ-T coupling, which presents substantial contributions to the neutron–proton interactions via the Fock terms. FromFigure 14a, the two-body interaction matrix elementsVab between the proton PS partnersπ1f˜ and neutron orbits(ν1i13/2,ν2g9/2), one can see distinct nodal effects, i.e., that orbits with same nodal numbers present stronger couplings than those with different nodal numbers, which can be understood well following the principle of quantum mechanics.
    As aforementioned, large PSO splitting ofπ1f˜ gives rise to the artificial shellZ=58, and thus, the valence protons may occupy the orbitπ1g7/2 mainly, namely, the proton (I) configuration inFigure 14b. In contrast to that, following the elimination of the artificial shellZ=58 with properly restored PSS, valence protons can spread over the PS partnersπ1g7/2 andπ2d5/2, which give the proton (II) configuration inFigure 14b. Regarding the fact that the shell gapN=126 is determined by the neutron orbits(ν1i13/2,ν2g9/2), it can be deduced fromFigure 14 that the valence protons populating the orbitπ1g7/2 (orπ2d5/2) tend to enlarge (or squeeze) the shell gapZ=126 due to the nodal effects. Following the PSS restoration of proton PS doubletπ1f˜, the shell gapN=126 is largely reduced, and thus, the Ce isotopic chain can be extended toN=140, as predicted by PKA1, in contrast to the drip lineN=126 given by other effective Lagrangians.
    Due to the extension of the neutron drip line toN=140, the occurrence of halos becomes possible in the Ce isotopes. As shown inFigure 15, when valence neutrons gradually occupy the low-l orbits above theN=126 shell, the neutron density distributions become more and more diffuse after the isotopeCe186 (N=128), indicating the emergence of ordinary and giant halos [109]. Moreover, as seen fromFigure 15b, the continuum plays a dominant role in the formation and stabilization of the neutron halos, which are, respectively, contributed by thesd- andg-orbits. CombiningFigure 14 andFigure 15, the squeezed shell gapN=126 following the proton PSS restoration is essential for the halo occurrence, in which theρ-T-enhanced neutron–proton interactions play the bridge role. It is worth mentioning that the mechanism responsible for the halo occurrences in Ce isotopes successfully interprets the experimental results of reduced PSO splittingπ1d˜=(π1f5/2,π1p3/2) approaching theN=50 shell for Cu isotopes and the neutron skin of the unstable nucleusNi78 [172].

    5.2. Bubble-like Structure and the PSS

    Since finite nuclei are complicated many-body systems; the PSS cannot be always conserved, even for the PS doublets near the Fermi surface. For example, the large PSO splitting ofπ1p˜=(π2s1/2,π1d3/2) inCa40 has been verified experimentally [147]; seeFigure 16. This may show certain impacts on the emergence of bubble-like structure, which is characterized by distinct central depressions of the matter distributions [19,20,21,22,23,24,25,26,27,28,29,30,31,32,173]. Generally speaking, thes-orbits play a decisive role in determining the central density profiles due to the vanishing of the centrifugal barrier; seeFigure 11. Therefore, the absence of nucleon occupation in thes-orbits near the Fermi surface may lead to evident depression in the interior region of nuclear density profiles, giving rise to the bubble-like structure.
    Figure 16 shows the PSO splittings between the proton PS partnersπ1d3/2 andπ2s1/2 along the isotopic chain of calcium (Ca). It can be seen that only PKA1 can properly reproduce the experimental values atCa40 andCa48, while the others show systematical deviations. As revealed in Ref. [31], such model deviations can be found consistently in describing the emergence of bubble-like structure in theN=28 isotones and Ca isotopes [111].
    Figure 17 shows the proton SPE and charge density distributions ofN=28 isotones. It can be seen fromFigure 17c thatAr46 andS44 are predicted as the candidates with bubble-like structure by various models except PKA1. Such a model discrepancy can be traced back to the ordering ofπ2s1/2 andπ1d3/2 in the proton SPE shown inFigure 17a,b. Due to large PSO splitting for the PS doubletπ1p˜, the RMF calculations with DD-ME2, as well as the RHF ones with PKO3, predict bubble-like structures for the isotonesAr46 andS44, since valence protons populate mainly the orbitπ1d3/2, instead of2s1/2. However, PKA1 predicts a fairly large amount of occupation on the proton orbit2s1/2, which washes out the bubble-like structures inAr46 andS44; seeFigure 17a,c. Combining the results inFigure 16 andFigure 17, one can see a certain relation between the PSS and the emergence of bubble-like structures inAr46 andS44.
    ForSi42, due to the existence of a large shell gapZ=14, all the models predict evident bubble-like structure. Similarly, bothSi34 andCa34 are predicted by all the selected models to have distinct proton and neutron bubble-like structures, respectively, which is attributed to the presence of sub-shellZ/N=14 [31]. In fact, the proton bubble-like structure inSi34 has been confirmed experimentally [177], which gives an empty proton orbitπ2s1/2 and distinct sub-shellZ=14. Moreover, the emergence of the sub-shellN=34 inCa54 is also accompanied by a neutron bubble-like structure, but due to a different mechanism from general cases [111]. Further extending to the drip line along the isotopic chain of Si and isotonic chain ofN=34 simultaneously, it is interestingly to see thatSi48 is predicted as a new doubly magic nucleus (N=34 andZ=14) with dual (neutron and proton) bubble-like structures by the RHFB calculations with PKA1 [32].

    5.3. Superheavy Magicity and the PSS

    As aforementioned, the appearance of magic and semi-magic shells may accompany well-restored PSS, which is also predicted for the occurrence of superheavy magicity. Taking doubly magic SHN120184304 as the example, predicted by the RHFB calculations with PKA1, one can see clear relations between the emergence of superheavy magic shells and the PSS [51]. As shown inFigure 18, the s.p. spectra of120184304, the proton shellZ=120, originating from large splitting of the PS doubletπ2d˜=(π3p3/2,π2f5/2), is commonly supported by the models. In addition, due to small PSO splitting ofπ1h˜=(π2g9/2,π1i11/2) given by PKA1, a more pronounced superheavy shellZ=126 appears, in contrast to the other model predictions. Moreover, such a model discrepancy can be also found in predicting the neutron magic shells. In the right panel ofFigure 18, PKA1 predicts the most evident neutron shellN=184. Consistent with that, both PS doubletsν1i˜=(2h11/2,1j13/2) andν3p˜=(ν4s1/2,ν3d3/2), above and below the shell, show well-restored PSS. However, the other selected models present large splitting for the PS doubletν1i˜, which gives a fairly large shell gapN=198. In fact, such a situation is rather similar to the results shown inFigure 6, in which the magic shellZ=82 can also be squeezed by the artificial oneZ=92, as well as the one inFigure 7.
    To further explore the relationship between superheavy shells (Z=120 andN=184) and the PSS,Figure 19 presents the proton and neutron density profiles with126⩽N⩽258 and82⩽Z⩽138 calculated by PKA1 and PKO3. With the numbers of neutrons and protons increasing, the high-l orbits, such asν1i11/2,ν2g9/2,ν1j15/2, andπ1h9/2,π1i13/2, are occupied. Since the nucleons populating the high-l orbits are driven towards the surface of the nucleus due to the strong centrifugal potential and large Coulomb repulsion [31,178], an obvious bubble-like structure can be commonly found in SHN, as shown inFigure 19, in particular for the ones marked in green. Following the appearance of the bubble-like structure, which can become even more extensive in SHN than in ordinary ones, the spin-orbit splittings of low-l orbits are largely squeezed. For instance, as shown in the left panel ofFigure 18, fairly small splittings between the spin partnersπ3p andπ2f lead to large PSO splitting for the proton PS doubletπ2d˜, that gives rise to the shellZ=120. The impact of the bubble-like structure on SO splitting has been introduced in Ref. [32]. Combined with the mechanism behind the PSS restoration of high-l˜ orbits, it can be concluded that reliable modeling of the in-medium balance between nuclear attractions and repulsions is essentially significant not only for ordinary nuclei but also for superheavy ones.

    6. Summary

    In this paper, studies on the pseudo-spin symmetry (PSS) under the relativistic Hartree–Fock (RHF) theory were reviewed, including the effects of Fock terms on the PSS restoration, especially the role ofρ-T coupling, and the relations to novel nuclear phenomena such as halo and bubble-like structures, and superheavy magicity.
    Utilizing the Schrödinger-type equation, the influences of Fock terms are manifested as the cancellation within the Fock contributions, like between pseudo-spin orbit potentialsVPSOE andV1E, and the RHF models with PKOi (i=1,2,3) provide a similar restoration picture as the popular RMF ones. However, when theρ-T coupling is incorporated into the effective Lagrangian PKA1, the PSS can also be well conserved for the high-l˜ pseudo-spin (PS) doublets, which eliminates the artificial shellsZ=58 and 92 appearing in the previous RMF and RHF calculations. It is found that the PSS restoration for high-l˜ PS doublets can be essentially connected with the modeling of the in-medium balance of nuclear attractions and repulsions, whereρ-T coupling, that contributes mainly via the Fock diagram, plays an important role. This indicates that PSS restoration can provide a qualitative guidance on understanding the in-medium effects of nuclear force.
    Furthermore, the impacts of PSS restoration and breaking on various nuclear phenomena are reviewed, including halos, bubble-like structure, and superheavy magicity. It was revealed that PSS restoration, that eliminates the artificial shellZ=58, can essentially be connected with the emergence of halo structures in drip-line Ce isotopes, in which the enhanced neutron–proton interactions by theρ-T coupling play the role of a bridge. Moreover, the PSO splitting inπ1p˜ and the corresponding order betweenπ2s1/2 andπ1d3/2 can determine the prediction of bubble-like structures forS44 andAr46. For the superheavy nuclei, the large PSO splitting ofπ2d˜ and small one ofν1i˜ are found to determine the emergence of superheavy magicityZ=120 andN=184, respectively.
    In fact, PSS restoration or breaking has manifested the characteristics of the microscopic nuclear structure, which corresponds to the consequences of the nuclear force in complicated nuclear many-body systems. Even though, the PSS itself and the mechanism therein can provide us with a different insight to understand not only the nature of the nuclear force, that accounts for the binding of nucleus, but also various nuclear phenomena, including the shells, novel halo and bubble-like structures, superheavy magicity, etc. All these together may deepen our knowledge of nuclear physics.

    Author Contributions

    Conceptualization, W.L.; writing—original draft preparation, J.G. and Z.W.; writing—review and editing, J.G., Z.W., J.L. (Jia Liu), J.L. (Jiajie Li), and W.L.; investigation, J.G., J.L. (Jia Liu), J.L. (Jiajie Li), and W.L. All authors have read and agreed to the published version of the manuscript.

    Funding

    This research was funded by the Strategic Priority Research Program of Chinese Academy of Sciences under Grant No. XDB34000000, the National Natural Science Foundation of China under Grant Nos. 12275111 and 12075104, the National Key Research and Development (R&D) Program under Grant No. 2021YFA1601500, the Fundamental Research Funds for the Central Universities (lzujbky-2022-sp02), and the Supercomputing Center of Lanzhou University.

    Data Availability Statement

    Not applicable.

    Conflicts of Interest

    The authors declare no conflicts of interest.

    Abbreviations

    The following abbreviations are used in this manuscript:
    RHFRelativistic Hartree–Fock
    RMFRelativistic mean field
    RHBRelativistic Hartree–Bogoliubov
    DDRHFDensity-dependent relativistic Hartree–Fock
    RHFBRelativistic Hartree–Fock–Bogoliubov
    ρ-Tρ-tensor
    SHESuperheavy element
    SHNSuperheavy nuclei
    PSSPseudo-spin symmetry
    PSPseudo-spin
    SOSpin orbit
    PSOPseudo-spin orbit
    PCBPseudo-centrifugal barrier
    PSOPPseudo-spin orbital potential
    s.p.Single particle
    SPESingle-particle energy
    LGLiquid–gas

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    Symmetry 16 00631 g001
    Figure 1. Feynman diagrams of the Hartree and Fock terms under the meson-exchange picture of the nuclear force, where the solid lines with arrows represent the interacting nucleons, and the blue and red colors are used to identify the nucleons dressed by different quantum numbers.
    Figure 1. Feynman diagrams of the Hartree and Fock terms under the meson-exchange picture of the nuclear force, where the solid lines with arrows represent the interacting nucleons, and the blue and red colors are used to identify the nucleons dressed by different quantum numbers.
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    Figure 2. The PSO splittingΔEPSO=(El˜j=l˜−1/2−El˜j=l˜+1/2)/(2l˜+1) versus the average s.p. energyE¯PSO=(El˜j=l˜−1/2+El˜j=l˜+1/2)/2 for neutron PS doubletsν1p˜,ν1d˜,ν1f˜, andν2p˜ forSn132. The SO splittingΔESO=(Elj=l−1/2−Elj=l+1/2)/(2l+1) is also given forν1p,ν1d,ν1f,ν1g, andν2p ν2d pairs as a function ofE¯SO=(Elj=l−1/2+Elj=l+1/2)/2. The results are obtained by the RHF with PKO1 (filled symbols) and the RMF with PKDD (open symbols), respectively. The figure is taken from Ref. [100].
    Figure 2. The PSO splittingΔEPSO=(El˜j=l˜−1/2−El˜j=l˜+1/2)/(2l˜+1) versus the average s.p. energyE¯PSO=(El˜j=l˜−1/2+El˜j=l˜+1/2)/2 for neutron PS doubletsν1p˜,ν1d˜,ν1f˜, andν2p˜ forSn132. The SO splittingΔESO=(Elj=l−1/2−Elj=l+1/2)/(2l+1) is also given forν1p,ν1d,ν1f,ν1g, andν2p ν2d pairs as a function ofE¯SO=(Elj=l−1/2+Elj=l+1/2)/2. The results are obtained by the RHF with PKO1 (filled symbols) and the RMF with PKDD (open symbols), respectively. The figure is taken from Ref. [100].
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    Figure 3. The PCB and PSOP scaled by the factorF2/(VD−E) for the PS partnerν2p˜ inSn132. The PCB contributions are shown by the red lines, while the PSOP are shown by the blue shadows. The RHF with PKO1 is used for the calculations. The figure is taken from Ref. [100].
    Figure 3. The PCB and PSOP scaled by the factorF2/(VD−E) for the PS partnerν2p˜ inSn132. The PCB contributions are shown by the red lines, while the PSOP are shown by the blue shadows. The RHF with PKO1 is used for the calculations. The figure is taken from Ref. [100].
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    Figure 4. The functionsVPSOEF2/(VD−E) andV1EFF′/(VD−E) given by the exchange (Fock) terms of the RHF with PKO1 for the PS partnerν2p˜ ofSn132. The figure is taken from Ref. [100].
    Figure 4. The functionsVPSOEF2/(VD−E) andV1EFF′/(VD−E) given by the exchange (Fock) terms of the RHF with PKO1 for the PS partnerν2p˜ ofSn132. The figure is taken from Ref. [100].
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    Figure 5. The radial wave functionsG andF (left panels), and the non-local termsX andY (right panels) given by the RHF with PKO1 for the PS partnerν2p˜ inSn132. The figure is taken from Ref. [100].
    Figure 5. The radial wave functionsG andF (left panels), and the non-local termsX andY (right panels) given by the RHF with PKO1 for the PS partnerν2p˜ inSn132. The figure is taken from Ref. [100].
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    Symmetry 16 00631 g006
    Figure 6. Proton SPE ofPb208. The results are calculated by RHF with PKA1 and PKO3, and RMF with DD-ME2. The experimental data are from Ref. [147].
    Figure 6. Proton SPE ofPb208. The results are calculated by RHF with PKA1 and PKO3, and RMF with DD-ME2. The experimental data are from Ref. [147].
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    Figure 7. Proton SPE forGd146, calculated by RHF with PKA1 [48], Gogny HFB with D1S [148], Skyrme HF with SLy4 [149], and RMF with PKDD [115]. The figure is taken from Ref. [49]. Reprinted with permission from Ref. [49]. Copyright 2009 Elsevier.
    Figure 7. Proton SPE forGd146, calculated by RHF with PKA1 [48], Gogny HFB with D1S [148], Skyrme HF with SLy4 [149], and RMF with PKDD [115]. The figure is taken from Ref. [49]. Reprinted with permission from Ref. [49]. Copyright 2009 Elsevier.
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    Figure 8. (a) SO splittingsΔE2d of protonπ2d partner extracted from the calculations of RHF with PKA1, Gogny HFB with D1S, Skyrme HF with SLy4, and RMF with PKDD, in comparison with experimental data [47]. (b) Detailed contributions ofΔE2d from different channels given by the RHF calculations with PKA1. The figure is taken from Ref. [49]. Reprinted with permission from Ref. [49]. Copyright 2009 Elsevier.
    Figure 8. (a) SO splittingsΔE2d of protonπ2d partner extracted from the calculations of RHF with PKA1, Gogny HFB with D1S, Skyrme HF with SLy4, and RMF with PKDD, in comparison with experimental data [47]. (b) Detailed contributions ofΔE2d from different channels given by the RHF calculations with PKA1. The figure is taken from Ref. [49]. Reprinted with permission from Ref. [49]. Copyright 2009 Elsevier.
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    Symmetry 16 00631 g009
    Figure 9. Proton shell gaps (MeV) (a) and the splittings of neighboring PS partnersEPSOπ (MeV) (b) for the traditional magic nucleiCa48,Zr90,Sn132, andPb208, and the superheavy one126310. The results are calculated by PKA1, PKO3, DD-ME2, PK1 [115], and NL3* [150]. The shadow areas denote the spreading of the PSO splittings given by PKA1 and the other selected models. Experimental data are taken from Ref. [147]. The figure is taken from Ref. [104]. Reprinted with permission from Ref. [104]. Copyright 2019 American Physical Society.
    Figure 9. Proton shell gaps (MeV) (a) and the splittings of neighboring PS partnersEPSOπ (MeV) (b) for the traditional magic nucleiCa48,Zr90,Sn132, andPb208, and the superheavy one126310. The results are calculated by PKA1, PKO3, DD-ME2, PK1 [115], and NL3* [150]. The shadow areas denote the spreading of the PSO splittings given by PKA1 and the other selected models. Experimental data are taken from Ref. [147]. The figure is taken from Ref. [104]. Reprinted with permission from Ref. [104]. Copyright 2019 American Physical Society.
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    Figure 10. Proton (π) PSO splittingsΔEPSOπ (MeV) (a) inPb208 as functions of pseudo-orbitl′, and the sum contributions from kinetic energy,σ, andω potential energiesEkin.+σ+ω (b). The results are extracted from the calculations with PKA1, PKO3, and DD-ME2. Meson–nucleon coupling constants, namely, (c) the isoscalargσ andgω, (d) the isovectorgρ, and (e)κρ [κρ(0)=fρ(0)/gρ(0)] andfπ, as functions of the densityρb (fm−3) for PKA1, PKO3, DD-ME2, and DD-LZ1. The figure is taken from Refs. [104,143]. Reprinted with permission from Ref. [104]. Copyright 2019 American Physical Society. Reprinted with permission from Ref. [143]. Copyright 2020 Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Sciences and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Publishing Ltd.
    Figure 10. Proton (π) PSO splittingsΔEPSOπ (MeV) (a) inPb208 as functions of pseudo-orbitl′, and the sum contributions from kinetic energy,σ, andω potential energiesEkin.+σ+ω (b). The results are extracted from the calculations with PKA1, PKO3, and DD-ME2. Meson–nucleon coupling constants, namely, (c) the isoscalargσ andgω, (d) the isovectorgρ, and (e)κρ [κρ(0)=fρ(0)/gρ(0)] andfπ, as functions of the densityρb (fm−3) for PKA1, PKO3, DD-ME2, and DD-LZ1. The figure is taken from Refs. [104,143]. Reprinted with permission from Ref. [104]. Copyright 2019 American Physical Society. Reprinted with permission from Ref. [143]. Copyright 2020 Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Sciences and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Publishing Ltd.
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    Symmetry 16 00631 g011
    Figure 11. Schematic diagrams of the matter densityρb and probability densities of thes-,p-,d-, andf-orbits.
    Figure 11. Schematic diagrams of the matter densityρb and probability densities of thes-,p-,d-, andf-orbits.
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    Symmetry 16 00631 g012
    Figure 12. Panel (a) shows the correlation between the critical parametersTC (MeV) andPC (MeVfm−3). The solid circles correspond to 20 selected relativistic Lagrangians from Ref. [165] and PKA1, and the open circles correspond to the testing setsxκρ, withx = 1.0 to 0.7 (in red) andxκρ* withx = 0.9, 0.8, and 0.7 (in blue). As the references, the solid and dashed lines represent the linear fittings with the Pearson correlation coefficientr. Panel (b) presents LG phase diagrams of thermal nuclear matter at temperatureT=10 MeV given by the setsxκρ (x = 1.0, 0.9, 0.8, and 0.7), as compared to the ones given by RHF Lagrangians PKA1 and PKO3. The figure is taken from Ref. [166]. Reprinted with permission from Ref. [166]. Copyright 2021 American Physical Society.
    Figure 12. Panel (a) shows the correlation between the critical parametersTC (MeV) andPC (MeVfm−3). The solid circles correspond to 20 selected relativistic Lagrangians from Ref. [165] and PKA1, and the open circles correspond to the testing setsxκρ, withx = 1.0 to 0.7 (in red) andxκρ* withx = 0.9, 0.8, and 0.7 (in blue). As the references, the solid and dashed lines represent the linear fittings with the Pearson correlation coefficientr. Panel (b) presents LG phase diagrams of thermal nuclear matter at temperatureT=10 MeV given by the setsxκρ (x = 1.0, 0.9, 0.8, and 0.7), as compared to the ones given by RHF Lagrangians PKA1 and PKO3. The figure is taken from Ref. [166]. Reprinted with permission from Ref. [166]. Copyright 2021 American Physical Society.
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    Symmetry 16 00631 g013
    Figure 13. Two-proton shell gapδ2p forN = 82 and 126 isotonic chains obtained from RH(F)B calculations with the new effective interaction DD-LZ1 (left panel) and PCF-PK1 (right panel). For comparison, the experimental data [168] and calculated results of PKA1, PKO1, DD-ME2, PC-PK1 [169], DD-PC1 [170], and DD-MEδ [123] are also given. The results of DD-LZ1 are taken from Ref. [143], and those of PCF-PK1 are taken from Ref. [167]. Reprinted with permission from Ref. [143]. Copyright 2020 Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Sciences and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Publishing Ltd. Reprinted with permission from Ref. [167]. Copyright 2022 American Physical Society.
    Figure 13. Two-proton shell gapδ2p forN = 82 and 126 isotonic chains obtained from RH(F)B calculations with the new effective interaction DD-LZ1 (left panel) and PCF-PK1 (right panel). For comparison, the experimental data [168] and calculated results of PKA1, PKO1, DD-ME2, PC-PK1 [169], DD-PC1 [170], and DD-MEδ [123] are also given. The results of DD-LZ1 are taken from Ref. [143], and those of PCF-PK1 are taken from Ref. [167]. Reprinted with permission from Ref. [143]. Copyright 2020 Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Sciences and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Publishing Ltd. Reprinted with permission from Ref. [167]. Copyright 2022 American Physical Society.
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    Symmetry 16 00631 g014
    Figure 14. (a) Two-body interaction matrix elementsVab, wherea=π2d5/2 (filled symbols) andπ1g7/2 (open symbols), andb=ν2g9/2 andν1i13/2. The figure is taken from Ref. [109]. Reprinted with permission from Ref. [109]. Copyright 2010 American Physical Society. (b) Schematic diagrams of the proton–neutron configurations for the existence (I) and elimination (II) ofZ=58 artificial shell.
    Figure 14. (a) Two-body interaction matrix elementsVab, wherea=π2d5/2 (filled symbols) andπ1g7/2 (open symbols), andb=ν2g9/2 andν1i13/2. The figure is taken from Ref. [109]. Reprinted with permission from Ref. [109]. Copyright 2010 American Physical Society. (b) Schematic diagrams of the proton–neutron configurations for the existence (I) and elimination (II) ofZ=58 artificial shell.
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    Symmetry 16 00631 g015
    Figure 15. (a) Neutron and proton densities; (b) neutron canonical SPE, occupation probability (x-axis error bars), and Fermi energyEF (open circles). Results are calculated by RHFB with PKA1 plus the Gogny pairing force D1S. The figure is taken from Ref. [109]. Reprinted with permission from Ref. [109]. Copyright 2010 American Physical Society.
    Figure 15. (a) Neutron and proton densities; (b) neutron canonical SPE, occupation probability (x-axis error bars), and Fermi energyEF (open circles). Results are calculated by RHFB with PKA1 plus the Gogny pairing force D1S. The figure is taken from Ref. [109]. Reprinted with permission from Ref. [109]. Copyright 2010 American Physical Society.
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    Symmetry 16 00631 g016
    Figure 16. Energy differenceΔEsdπ=επ2s1/2−επ1d3/2 of Ca isotopes. The experimental values ofCa40 andCa48 are taken from Ref. [147] (up triangle), Ref. [174] (down triangle), Ref. [175] (left triangle), and Ref. [176] (right triangle). The figure is taken from Ref. [110].
    Figure 16. Energy differenceΔEsdπ=επ2s1/2−επ1d3/2 of Ca isotopes. The experimental values ofCa40 andCa48 are taken from Ref. [147] (up triangle), Ref. [174] (down triangle), Ref. [175] (left triangle), and Ref. [176] (right triangle). The figure is taken from Ref. [110].
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    Symmetry 16 00631 g017
    Figure 17. Canonical proton SPE along theN=28 isotonic chain calculated by RHFB with PKA1 (a) and RHB with DD-ME2 (b). The lengths of thick bars correspond with the occupation probabilities of the proton orbits and the filled stars denote the experimental data. Charge distributions ofN=28 isotones calculated by RHFB with PKA1 and PKO3, and by RHB with DD-ME2 (c). The figure is taken from Ref. [31]. Reprinted with permission from Ref. [31]. Copyright 2016 American Physical Society.
    Figure 17. Canonical proton SPE along theN=28 isotonic chain calculated by RHFB with PKA1 (a) and RHB with DD-ME2 (b). The lengths of thick bars correspond with the occupation probabilities of the proton orbits and the filled stars denote the experimental data. Charge distributions ofN=28 isotones calculated by RHFB with PKA1 and PKO3, and by RHB with DD-ME2 (c). The figure is taken from Ref. [31]. Reprinted with permission from Ref. [31]. Copyright 2016 American Physical Society.
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    Symmetry 16 00631 g018
    Figure 18. Proton (left panel) and neutron (right panel) SPE of SHN120184304. The results are extracted from RHFB calculations with PKOi (i=1,2,3) and PKA1, and compared to the RHB ones with PKDD and DD-ME2. In all cases, the pairing force is derived from the finite-range Gogny force D1S with the strength factorf = 0.9. The figure is taken from Ref. [51].
    Figure 18. Proton (left panel) and neutron (right panel) SPE of SHN120184304. The results are extracted from RHFB calculations with PKOi (i=1,2,3) and PKA1, and compared to the RHB ones with PKDD and DD-ME2. In all cases, the pairing force is derived from the finite-range Gogny force D1S with the strength factorf = 0.9. The figure is taken from Ref. [51].
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    Symmetry 16 00631 g019
    Figure 19. The neutron (ν) and proton (π) density distributions of superheavy nuclei calculated by the RHFB model with PKA1 (red line) and PKO3 (blue line).
    Figure 19. The neutron (ν) and proton (π) density distributions of superheavy nuclei calculated by the RHFB model with PKA1 (red line) and PKO3 (blue line).
    Symmetry 16 00631 g019
    Table 1. Details for the density-dependent effective Lagrangians PKA1 [48], PKO1 [122], PKO2 [101], PKO3 [101], DD-ME2 [142], DD-LZ1 [143], and PKDD [115] which are used in this review.
    Table 1. Details for the density-dependent effective Lagrangians PKA1 [48], PKO1 [122], PKO2 [101], PKO3 [101], DD-ME2 [142], DD-LZ1 [143], and PKDD [115] which are used in this review.
    HartreeFockσ-Sω-Vρ-Vπ-PVρ-T (VT)
    PKA1✓✓✓✓✓✓✓
    PKO1✓✓✓✓✓✓×
    PKO2✓✓✓✓✓××
    PKO3✓✓✓✓✓✓×
    DD-ME2✓×✓✓✓××
    PKDD✓×✓✓✓××
    DD-LZ1✓×✓✓✓××
    Table 2. Contributions to the total energy functionalETotal (MeV) ofPb208 from the kinetic and isoscalar potential energies (Ekin.+σ+ω), the isovector potential energies (Eρ+π), and the Coulomb ones (Ecou.). The results are calculated by PKA1, PKO3, and DD-ME2. The table is taken from Ref. [104]. Reprinted with permission from Ref. [104]. Copyright 2019 American Physical Society.
    Table 2. Contributions to the total energy functionalETotal (MeV) ofPb208 from the kinetic and isoscalar potential energies (Ekin.+σ+ω), the isovector potential energies (Eρ+π), and the Coulomb ones (Ecou.). The results are calculated by PKA1, PKO3, and DD-ME2. The table is taken from Ref. [104]. Reprinted with permission from Ref. [104]. Copyright 2019 American Physical Society.
    Pb208Ekin.+σ+ωEρ+πEcou.ETotal
    DD-ME2−2559.81100.27827.23−1637.39
    PKO3−1781.19−648.75798.38−1636.80
    PKA1−795.09−1634.84798.62−1636.27
    Table 3. Critical parameters of LG phase transition of symmetric nuclear matter, i.e., the critical temperatureTC (MeV), critical densityρC (fm−3), the critical pressurePC (MeVfm−3), the critical incompressibilityKC (MeV), and the compressibility factorZC. The results are calculated by the RHF Lagrangians with PKA1 and PKO3, and the RMF ones with DD-ME2, PK1. The table is taken from Ref. [166]. Reprinted with permission from Ref. [166]. Copyright 2021 American Physical Society.
    Table 3. Critical parameters of LG phase transition of symmetric nuclear matter, i.e., the critical temperatureTC (MeV), critical densityρC (fm−3), the critical pressurePC (MeVfm−3), the critical incompressibilityKC (MeV), and the compressibility factorZC. The results are calculated by the RHF Lagrangians with PKA1 and PKO3, and the RMF ones with DD-ME2, PK1. The table is taken from Ref. [166]. Reprinted with permission from Ref. [166]. Copyright 2021 American Physical Society.
    TCρCPCKCZC
    PKA111.550.0500.114−40.690.196
    PKO314.570.0480.198−75.030.286
    DD-ME213.110.0440.155−62.920.267
    PK115.110.0490.223−82.830.305
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    Geng, J.; Wang, Z.; Liu, J.; Li, J.; Long, W. Pseudo-Spin Symmetry and the Hints for Unstable and Superheavy Nuclei.Symmetry2024,16, 631. https://doi.org/10.3390/sym16050631

    AMA Style

    Geng J, Wang Z, Liu J, Li J, Long W. Pseudo-Spin Symmetry and the Hints for Unstable and Superheavy Nuclei.Symmetry. 2024; 16(5):631. https://doi.org/10.3390/sym16050631

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    Geng, Jing, Zhiheng Wang, Jia Liu, Jiajie Li, and Wenhui Long. 2024. "Pseudo-Spin Symmetry and the Hints for Unstable and Superheavy Nuclei"Symmetry 16, no. 5: 631. https://doi.org/10.3390/sym16050631

    APA Style

    Geng, J., Wang, Z., Liu, J., Li, J., & Long, W. (2024). Pseudo-Spin Symmetry and the Hints for Unstable and Superheavy Nuclei.Symmetry,16(5), 631. https://doi.org/10.3390/sym16050631

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