1. Introduction
With the worldwide developments of new generation radioactive-ion-beam (RIB) facilities and advanced nuclear detectors, the research of nuclear physics has been largely extended from a few stable nuclei to several thousand unstable ones, which enrich greatly the knowledge of nuclear physics [1–6]. In contrast to the stable ones, plenty of novelties were successively found in unstable nuclei, such as nuclear halos [7–10], the emergences of new magicity and the disappearances of the traditional ones [11–18], etc. All these largely challenge our conventional understanding on nuclear physics. Moreover, many unstable nuclei are involved in the rapid neutron capture process (r-process), the crucial process of the synthesis of heavy elements, which reveals the special significance of unstable nuclei in understanding the origin of the elements in the universe [19, 20]. It calls for systematic and precise understanding on the structures, decays and reactions of unstable nuclei. However, as limited by the experimental conditions, it is still a long way to synthesize and precisely measure the relevant unstable nuclei, and the pioneering theoretical studies are necessitated.
Although the history of nuclear physics has been more than 100 years, it is still hard to describe precisely the properties of all the nuclides, due to the limitation on the understanding of nuclear force and nuclear many-body method. As generally recognized, nuclear force, a residual strong interaction, contains central, strong spin–orbit (SO) coupling, and non-central tensor force components, etc. Nowadays, the meson-exchange diagram [21] is one of the most successful interpretations of nuclear force. Based on different understanding on nuclear force and many-body method, people developed various nuclear theoretical models, including the ab-initio (see [22] and references therein), the configuration shell model [23–25], the non-relativistic and relativistic energy density functional (EDF) theories [26–34], etc. Restricted on the level of mean field approach, the meson-exchange diagram of nuclear force contains the Hartree and Fock terms, and the significant $\pi $ -meson contributes only via the Fock terms. As ones of the representative EDF models, the relativistic mean field (RMF) theory [30, 31], containing only the Hartree terms of the meson-exchange diagram, owns the advantage of providing self-consistent treatment of strong SO coupling in nucleus, in contrast to the non-relativistic EDF models, and has achieved great success in describing plenty of nuclear phenomena [35–43]. Moreover, incorporating with the Bogoliubov transformation, the RMF extension—the relativistic Hartree–Bogoliubov (RHB) theory [39, 40, 44–47] unifies the descriptions of both mean field and pairing correlations, and the continuum effects appearing in unstable nuclei are taken into account automatically, which promise the extensive reliability on describing unstable nuclei, e.g. the novel halo phenomena [48–52]. Considering the deformation degree of freedom, the RHB models provide satisfied description on the axially deformed nuclei [46, 47, 51–56], under which a global nuclear mass table was obtained recently [57, 58]. However, as limited by the Hartree approach, the significant degrees of freedom associated with the $\pi $ -meson and $\rho $ -tensor couplings cannot be taken into account by the RMF models, in which the important ingredient of nuclear force — the tensor force components are also missing.
Implemented with the Fock terms, the relativistic Hartree–Fock (RHF) theory [32–34] can introduce naturally the $\pi $ -meson and $\rho $ -tensor couplings. While for a rather long time, the RHF models failed to provide reasonable quantitative description of nuclear structure properties, although people tried to introduce the non-linear self coupling of $\sigma $ -meson [59] or high order terms of the scalar field $(\bar{\psi }\psi )$ [60] to evaluate the nuclear in-medium effects. Incorporating the density dependencies of the meson-nucleon coupling strengths for modeling the nuclear in-medium effects, it was eventually achieved to provide satisfactory description on nuclear properties by the RHF theory [61, 62], with similar accuracy as the popular RMF models. Compared to the previous RHF models in [34, 59, 60], it indicates that the density dependencies carried by the meson-nucleon coupling strengths, the modeling of nuclear in-medium effects, are essential to achieve precise description of nuclear structure properties, after overcoming the numerical difficulties induced by the Fock terms. On the other hand, as indicated by recent applications [63], it seems that the completeness of properly evaluating the in-medium effects carried by all the meson-nucleon couplings can be also significant, which were only partly considered in [59, 60].
Not only for implementing the meson-exchange diagram of nuclear force, significant improvements due to the Fock terms were also found on the self-consistent description of nuclear shell evolution [64–66], pseudo-spin symmetry (PSS) restoration [62, 63, 67, 68] and symmetry energy [69, 70], etc. In particular, the important ingredient of nuclear force—tensor force components can be naturally taken into account by the Fock terms [71–74]. Together with the Lorentz covariance of the model itself, unified self-consistent treatments on both SO coupling and tensor force are achieved by the RHF theory, which is significant for the reliable description on the extension from the stable to unstable nuclei. Combined with the Bogoliubov transformation, the RHF extension—relativistic Hartree–Fock–Bogoliubov (RHFB) theory [75] provides reliable description on unstable nuclei, including the novel bubble structures [76, 77], halo phenomena [78, 79] and new magicity [80–82], etc. It is worth noting that the proton bubble structure in 34Si predicted in [76] has been confirmed experimentally [83], and the mechanism responsible for the halo occurrences in Ce isotopes revealed in [78] interprets successfully the experimental results of neutron skin and pseudo-spin orbital splitting in the unstable nucleus 78Ni [84].
Besides the ground state, the Fock terms in the RHF/RHFB models are also essential to provide self-consistent description of the excitation modes of nuclei. Among various nuclear excitation modes, the spin-isospin excitation is of special significance, because it may be tightly related with the intrinsic nuclear degrees of freedom associated with the orbit, spin and isospin, and carry plenty of spin-isospin information of nuclear force [85–88]. Combined with the random phase approximation (RPA) and quasi random phase approximation (QRPA), the RHF and RHFB theories were extended to describe the nuclear spin-isospin excitations and $\beta $ -decays, respectively the RHF + RPA [89–92] and RHFB + QRPA methods [93, 94]. Due to the delicate balance in the Fock diagrams of isoscalar meson-nucleon couplings, both RHF + RPA and RHFB + QRPA methods provide fully self-consistent description of nuclear spin-isospin excitation using the same effective Lagrangian for both mean field and residual interaction [89, 94], particularly for the precise reproduction of the fine structure of 16O [91]. Moreover, the existing experimental data of $\beta $ -decay half-lives were also well reproduced by the RHFB + QRPA method after introducing neutron-proton pairing correlations, and a remarkable speeding up of $r$ -matter flow was predicted, which leads to enhanced $r$ -process abundances of elements with $A\gtrsim 140$ [93]. Recently, the RHF + RPA method was implemented by considering the degrees of freedom associated with the $\rho $ -tensor couplings, and the effects of the tensor force components introduced by the Fock terms are analyzed, which indicates that the tensor forces play the role mainly via the RHF mean field rather than the RPA residual interaction in determining the Gamow-Teller resonances (GTR) [92].
As learned from both experimental and theoretical investigations, most of the nuclei in the nuclear chart, except a few nearby the magic numbers, are of the intrinsic shapes deviating the spherical symmetry, and the single-particle structures of finite nuclei can be notably changed with the development of the nuclear deformation. It also indicates the necessity of considering the deformation degree of freedom for reliable description of unstable nuclei. Recently, applying the expansion upon the spherical Dirac Woods–Saxon base [95], the axially deformed RHF and RHFB models, respectively referred as D-RHF and D-RHFB, have been established for axially deformed nuclei [96, 97]. Thus, a uniform description of SO coupling, tensor force, deformation, pairing correlations and continuum effects is achieved by the D-RHFB model, which provides reliable theoretical tool for exploring the properties of unstable nuclei in a rather wide range. As the first attempt, it was revealed in [96] that the tensor force components carried by the $\pi $ -pseudo-vector couplings can essentially change the shape evolution of the single-particle orbits, showing rather different systematics from the conventional nuclear models [96], which deserves more extensive exploration in unstable nuclei with the newly established D-RHFB model.
As a homage for the 40th anniversary of the Communications in Theoretical Physics, the recent progress on the RHF description of nuclear structure is reviewed in current article, including the extensive development of the RHF theory and the applications in describing the novelty of unstable nuclei. The review is organized as follows. After a brief recall on the RHF scheme in section 2 , the complete RHFB frame and the RHF + RPA method with the $\rho $ -tensor coupling are introduced respectively in sections 3 and 4 . Afterwards, sections 5 and 6 present the review on the novelty of unstable nuclei described by the RHF/RHFB theory, including the mechanism responsible for the PSS restoration, bubble structures and new magicity. In the end, the summary and perspective of the RHF description of unstable nuclei are given in section 7 .
2. RHF approach
Under the meson-exchange diagram [21], nuclear force is assumed to be mediated by massive mesons. In practice, two isoscalar mesons ($\sigma $ and ${\omega }_{\mu }$ ) and two isovector ones (${\vec{\rho }}_{\mu }$ and $\vec{\pi }$ ) of following quantum numbers (${I}^{P},$ $\tau $ ) are considered as the effective fields to propagate the nucleon-nucleon interactions$I,$ $P$ and $\tau $ are respectively the spin, parity and isospin of the mesons. Besides, the photon field ${A}_{\mu }$ is introduced for the Coulomb interactions between protons. In the context, we use arrows to denote the isovectors, and bold types for the space vectors.
$\begin{eqnarray}\sigma ({0}^{+},0),\,{\omega }^{\mu }({1}^{-},0),\,{\vec{\rho }}^{\mu }({1}^{-},1),\,\vec{\pi }({0}^{-},1),\end{eqnarray}$
where For nuclear ground state, the meson-exchange diagram of nuclear force can be divided into two parts, the Hartree and Fock terms as shown in the left two plots of figure 1. In general, the retardation effects, the propagating time of the interaction, are ignored. Notice that the energy transfers involved in nuclear systems are much smaller than the masses of the $\sigma $ -, $\omega $ - and $\rho $ -mesons with the magnitude of several hundred MeV. So that such approximation should be valid, and also to a less extent for the $\pi $ -induced interaction. Following the standard procedure as described in [34, 97], the Hamiltonian of nuclear systems can be obtained as$\phi $ represents various two-body interaction channels, namely the Lorentz scalar ($\sigma $ -S), vector ($\omega $ -V, $\rho $ -V, $A$ -V), tensor ($\rho $ -T), vector-tensor ($\rho $ -VT) and pseudo-vector ($\pi $ -PV)couplings. The interaction vertex ${{\rm{\Gamma }}}_{\phi }({\boldsymbol{r}},{\boldsymbol{r}}^{\prime} )$ reads as
$\begin{eqnarray}\begin{array}{l}H=\displaystyle \int {\rm{d}}{\boldsymbol{r}}\bar{\psi }\left({\boldsymbol{r}}\right)\left(-{\rm{i}}{\boldsymbol{\gamma }}\cdot {\rm{\nabla }}+M\right)\psi \left({\boldsymbol{r}}\right)\\ \,+\,\displaystyle \frac{1}{2}\displaystyle \sum _{\phi }\displaystyle \int {\rm{d}}{\boldsymbol{r}}{\rm{d}}{\boldsymbol{r}}^{\prime} \bar{\psi }({\boldsymbol{r}})\bar{\psi }({\boldsymbol{r}}^{\prime} ){{\rm{\Gamma }}}_{\phi }{D}_{\phi }({\boldsymbol{r}}-{\boldsymbol{r}}^{\prime} )\psi ({\boldsymbol{r}})\psi ({\boldsymbol{r}}^{\prime} ),\end{array}\end{eqnarray}$
where $\begin{eqnarray}\begin{array}{l}{{\rm{\Gamma }}}_{\sigma \text{-}{\rm{S}}}\equiv -{g}_{\sigma }({\boldsymbol{r}}){g}_{\sigma }({\boldsymbol{r}}^{\prime} ),\\ {{\rm{\Gamma }}}_{\omega \text{-}{\rm{V}}}\equiv {\left({g}_{\omega }{\gamma }_{\mu }\right)}_{{\boldsymbol{r}}}{\left({g}_{\omega }{\gamma }^{\mu }\right)}_{{\boldsymbol{r}}^{\prime} },\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{{\rm{\Gamma }}}_{\rho \text{-}{\rm{V}}}\equiv {\left({g}_{\rho }{\gamma }_{\mu }\vec{\tau }\right)}_{{\boldsymbol{r}}}\cdot {\left({g}_{\rho }{\gamma }^{\mu }\vec{\tau }\right)}_{{\boldsymbol{r}}^{\prime} },\\ {{\rm{\Gamma }}}_{\rho \text{-}{\rm{T}}}\equiv \displaystyle \frac{1}{4{M}^{2}}{\left({f}_{\rho }{\sigma }_{\nu k}\vec{\tau }{\partial }^{k}\right)}_{{\boldsymbol{r}}}\cdot {\left({f}_{\rho }{\sigma }^{\nu l}\vec{\tau }{\partial }_{l}\right)}_{{\boldsymbol{r}}^{\prime} },\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{{\rm{\Gamma }}}_{\rho \text{-}\mathrm{VT}}\equiv \displaystyle \frac{1}{2M}{\left({f}_{\rho }{\sigma }^{k\nu }\vec{\tau }{\partial }_{k}\right)}_{{\boldsymbol{r}}}\cdot {\left({g}_{\rho }{\gamma }_{\nu }\vec{\tau }\right)}_{{\boldsymbol{r}}^{\prime} }\\ \,+\ \displaystyle \frac{1}{2M}{\left({g}_{\rho }{\gamma }_{\nu }\vec{\tau }\right)}_{{\boldsymbol{r}}}\cdot {\left({f}_{\rho }{\sigma }^{k\nu }\vec{\tau }{\partial }_{k}\right)}_{{\boldsymbol{r}}^{\prime} },\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{{\rm{\Gamma }}}_{\pi \text{-}\mathrm{PV}}\equiv -\displaystyle \frac{1}{{m}_{\pi }^{2}}{\left({f}_{\pi }\vec{\tau }{\gamma }_{5}{\gamma }_{\mu }{\partial }^{\mu }\right)}_{{\boldsymbol{r}}}\cdot {\left({f}_{\pi }\vec{\tau }{\gamma }_{5}{\gamma }_{\nu }{\partial }^{\nu }\right)}_{{\boldsymbol{r}}^{\prime} },\\ {{\rm{\Gamma }}}_{A\text{-}{\rm{V}}}\equiv \displaystyle \frac{{e}^{2}}{4}{\left({\gamma }_{\mu }(1-\tau )\right)}_{{\boldsymbol{r}}}{\left({\gamma }^{\mu }(1-\tau )\right)}_{{\boldsymbol{r}}^{\prime} }.\end{array}\end{eqnarray}$
Figure 1. Feymann diagrams of the Hartree and Fock terms under the meson-exchange picture of nuclear force, and the deduced Hartree and Fock self-energies (SE). |
For the meson and photon fields, the propagator ${D}_{\phi }$ is of the following form$M$ and ${m}_{\phi }$ are respectively the masses of nucleon and mesons, and ${g}_{\phi }$ ($\phi =\sigma ,\omega ,\rho $ ) and ${f}_{\phi ^{\prime} }$ ($\phi ^{\prime} =\rho ,\pi $ ) are the meson-nucleon coupling strengths.
$\begin{eqnarray}\begin{array}{l}{D}_{\phi }({\boldsymbol{r}}-{\boldsymbol{r}}^{\prime} )=\displaystyle \frac{1}{4\pi }\displaystyle \frac{{e}^{-{m}_{\phi }| {\boldsymbol{r}}-{\boldsymbol{r}}^{\prime} | }}{| {\boldsymbol{r}}-{\boldsymbol{r}}^{\prime} | },\phi =\sigma ,\omega ,\rho ,\pi ;\\ {D}_{A}({\boldsymbol{r}}-{\boldsymbol{r}}^{\prime} )=\displaystyle \frac{1}{4\pi }\displaystyle \frac{1}{| {\boldsymbol{r}}-{\boldsymbol{r}}^{\prime} | },\end{array}\end{eqnarray}$
where the retardation effects have been neglected. In the above expressions, Consistent with the mean field approach, the no-sea approximation is considered as usual, which amounts to neglecting the contributions from the Dirac sea [35]. Thus, in terms of the particle creation and annihilation operators, namely ${c}_{\alpha }$ and ${c}_{\alpha }^{\dagger }$ defined by the positive-energy solutions of the Dirac equation, the Dirac field $\psi $ that describes nucleons can be quantized as${\psi }_{\alpha }$ represents the Dirac spinor of the positive-energy state $\alpha .$ With the no-sea approximation, the Hartree–Fock ground state of nuclear systems can be defined as${\rm{| }}-\unicode{x027E9}$ is the vacuum state, and $A$ is the number of nucleons.
$\begin{eqnarray}\psi (x)=\displaystyle \sum _{\alpha }{\psi }_{\alpha }({\boldsymbol{r}}){{\rm{e}}}^{-{\rm{i}}{\varepsilon }_{\alpha }t}{c}_{\alpha },{\psi }^{\dagger }(x)=\displaystyle \sum _{\alpha }{\psi }_{\alpha }^{\dagger }({\boldsymbol{r}}){{\rm{e}}}^{+{\rm{i}}{\varepsilon }_{\alpha }t}{c}_{\alpha }^{\dagger },\end{eqnarray}$
where $\begin{eqnarray}{\rm{| HF}}\gt =\,\displaystyle \prod _{l}^{A}{c}_{l}^{\dagger }{\rm{| }}-,\unicode{x027E9},\end{eqnarray}$
where Eventually, starting from the meson-exchange picture of nuclear force, combined with the quantization of nucleon field (5), the expectation of the Hamiltonian (2) with respect to the Hartree–Fock ground state ${\rm{| }}{\rm{HF}}\unicode{x027E9}$ leads to the energy functional $E$ of nuclear systems [34, 42]${E}_{{\rm{k}}},$ and the Hartree (Direct) and Fock (Exchange) terms of the potential energies from various coupling channel $\phi ,$ respectively ${E}_{\phi }^{D}$ and ${E}_{\phi }^{E}.$
$\begin{eqnarray}E=\left\langle {\rm{HF| }}H{\rm{| HF}}\right\rangle ={E}_{{\rm{k}}}+\displaystyle \sum _{\phi }({E}_{\phi }^{D}+{E}_{\phi }^{E}),\end{eqnarray}$
including the kinetic energy From the variation of the energy functional (7), the single-particle Dirac equations of nucleons can be derived, in which one can obtain two types of self-energies (SE), the local Hartree and non-local Fock ones; see the right two plots of figure 1. Obviously, it is not an easy task to handle with the Fock terms due to the non-locality shown in figure 1. It shall be reminded that in the RMF models, the so-called covariant density functional theory (CDFT), only the Hartree terms ${E}_{\phi }^{D}$ are considered, and the degrees of freedom associated with the $\pi $ -PV and $\rho $ -T couplings, which contribute mainly via the Fock diagrams, cannot be taken into account efficiently.
3. RHF Bogoliubov scheme
For unstable nuclei, whose nucleon separation energies can be of the magnitude of less than $1$ MeV, the weak binding mechanism related to the pairing correlations and the continuum shall be treated carefully. Compared to the traditional BCS method, the Bogoliubov scheme has the advantage of uniformly dealing with both the mean fields and pairing correlations, as illustrated by the RHB models and the applications [40–42]. As aforementioned, it is rather straightforward to derive the general formalism of the RHF theory using the concept of the HF single particle, that is defined by the solutions of the Dirac equation. However, it is not an apparent task to obtain a complete RHFB energy functional, or to quantize the Dirac spinor field $\psi $ in the Bogoliubov quasi-particle space. Fortunately, a proper way was proposed recently in [97] to build a full RHFB energy functional by introducing new quantization form of the Dirac spinor field.
As well known, the Bogoliubov transformation defines the link from the HF particle to the Bogoliubov quasi-particle spaces, which reads as${\beta }_{k}$ and ${\beta }_{k}^{\dagger }$ are respectively the annihilation and creation operators of the Bogoliubov quasi-particle. To avoid misleading, the indices $l$ and $k$ are set to denote the HF single-particle states and Bogoliubov quasi-particle states, respectively. Considering the requirement of establishing the Cooper pairs, the following specific relationships between the HF and HFB wave functions were suggested in [97]$\bar{l}$ and $\bar{k}$ hold for the time-reversal partners of the single-particle and quasi-particle states, respectively. Subsequently, the quantization of the Dirac spinor field $\psi $ in the Bogoliubov quasi-particle space is suggested as the following form${\varepsilon }_{k}$ is the quasi-particle energy, and ${\psi }^{U}$ and ${\psi }^{V}$ are respectively the $U$ and $V$ components of the quasi-particle spinors. For the HFB ground state ${\rm{| }}{\rm{HFB}}\unicode{x027E9},$ it shall fulfill the following condition [98]$\psi .$
$\begin{eqnarray}\left(\begin{array}{c}{\beta }_{k}\\ {\beta }_{k}^{\dagger }\end{array}\right)=\displaystyle \sum _{l}\left(\begin{array}{cc}{U}_{{lk}}^{* } & {V}_{{lk}}^{* }\\ {V}_{{lk}} & {U}_{{lk}}\end{array}\right)\left(\begin{array}{c}{c}_{l}\\ {c}_{l}^{\dagger }\end{array}\right),\end{eqnarray}$
where $\begin{eqnarray}{\psi }_{k}^{V}(x)=\displaystyle \sum _{l}{V}_{{lk}}{\psi }_{l}(x),{\bar{\psi }}_{k}^{V}(x)\,=\,\displaystyle \sum _{l}{V}_{{lk}}^{* }{\bar{\psi }}_{l}(x),\end{eqnarray}$
$\begin{eqnarray}{\psi }_{\bar{k}}^{U}(x)=\displaystyle \sum _{l}{U}_{{lk}}{\psi }_{\bar{l}}(x),{\bar{\psi }}_{\bar{k}}^{U}(x)\,=\,\displaystyle \sum _{l}{U}_{{lk}}^{* }{\bar{\psi }}_{\bar{l}}(x),\end{eqnarray}$
to keep consistence with the expansions (5) and the Bogoliubov transformation (8), where $\begin{eqnarray}\begin{array}{l}\psi (x)=\displaystyle \sum _{k}\left[{\psi }_{\bar{k}}^{U}({\boldsymbol{r}}){{\rm{e}}}^{-i{\varepsilon }_{k}t}{\beta }_{k}+{\psi }_{k}^{V}({\boldsymbol{r}}){{\rm{e}}}^{+i{\varepsilon }_{k}t}{\beta }_{k}^{\dagger }\right],\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}\bar{\psi }(x)=\displaystyle \sum _{k}\left[{\bar{\psi }}_{k}^{V}({\boldsymbol{r}}){{\rm{e}}}^{-i{\varepsilon }_{k}t}{\beta }_{k}+{\bar{\psi }}_{\bar{k}}^{U}({\boldsymbol{r}}){{\rm{e}}}^{+i{\varepsilon }_{k}t}{\beta }_{k}^{\dagger }\right],\end{array}\end{eqnarray}$
where $\begin{eqnarray}{\beta }_{k}{\rm{| HFB}}\gt =0,\end{eqnarray}$
which is rather different from the HF ground state (6). From this condition, one may understand why it is not a trivial task to quantize the Dirac spinor field Applying the suggested quantization (11), the RHF Hamiltonian (2) of nuclear systems can be expressed as${\rm{| HFB}}\gt $ are omitted. Obviously, the first two terms are respectively the kinetic and potential energy terms, and the last one accounts for the pairing correlations. Thus, without giving an explicit form of the HFB ground state, the full energy functional that contains both mean field and pairing correlations can be obtained from the expectation of the Hamiltonian (13) with respect to ${\rm{| }}{\rm{HFB}}\unicode{x027E9}.$ It gives${E}_{{\rm{k}}},$ the Hartree potential energy ${E}_{\phi }^{D}$ and Fock one ${E}_{\phi }^{E},$ and the pairing energy ${E}_{\phi }^{{pp}}$ read as
$\begin{eqnarray}\begin{array}{l}H=\displaystyle \sum _{kk^{\prime} }\displaystyle \int {\rm{d}}{\boldsymbol{r}}{\bar{\psi }}_{k}^{V}\left({\boldsymbol{r}}\right)\left(-{\rm{i}}{\boldsymbol{\gamma }}\cdot {\rm{\nabla }}+M\right){\psi }_{k^{\prime} }^{V}\left({\boldsymbol{r}}\right){\beta }_{k}{\beta }_{k^{\prime} }^{\dagger }\\ \,+\displaystyle \frac{1}{2}\displaystyle \sum _{\phi }\displaystyle \sum _{{k}_{1}{k}_{2}{k}_{2^{\prime} }{k}_{1^{\prime} }}\displaystyle \int {\rm{d}}{\boldsymbol{r}}{\rm{d}}{\boldsymbol{r}}^{\prime} \left[{\bar{\psi }}_{{k}_{1}}^{V}({\boldsymbol{r}}){\bar{\psi }}_{{k}_{2}}^{V}({\boldsymbol{r}}^{\prime} ){{\rm{\Gamma }}}_{\phi }({\boldsymbol{r}},{\boldsymbol{r}}^{\prime} )\right.\\ \times \,{D}_{\phi }({\boldsymbol{r}}-{\boldsymbol{r}}^{\prime} ){\psi }_{{k}_{2^{\prime} }}^{V}({\boldsymbol{r}}^{\prime} ){\psi }_{{k}_{1^{\prime} }}^{V}({\boldsymbol{r}}){\beta }_{{k}_{1}}{\beta }_{{k}_{2}}{\beta }_{{k}_{2^{\prime} }}^{\dagger }{\beta }_{{k}_{1^{\prime} }}^{\dagger }+{\bar{\psi }}_{{k}_{1}}^{V}({\boldsymbol{r}}){\bar{\psi }}_{{\bar{k}}_{2}}^{U}({\boldsymbol{r}}^{\prime} )\\ \times \,\left.{{\rm{\Gamma }}}_{\phi }({\boldsymbol{r}},{\boldsymbol{r}}^{\prime} ){D}_{\phi }({\boldsymbol{r}}-{\boldsymbol{r}}^{\prime} ){\psi }_{{\bar{k}}_{2^{\prime} }}^{U}({\boldsymbol{r}}^{\prime} ){\psi }_{{k}_{1^{\prime} }}^{V}({\boldsymbol{r}}){\beta }_{{k}_{1}}{\beta }_{{k}_{2}}^{\dagger }{\beta }_{{k}_{2^{\prime} }}{\beta }_{{k}_{1^{\prime} }}^{\dagger }\right],\end{array}\end{eqnarray}$
in which the terms with zero expectation referring to $\begin{eqnarray}E=\left\langle {\rm{HFB| }}H{\rm{| HFB}}\right\rangle ={E}_{{\rm{k}}}+\displaystyle \sum _{\phi }({E}_{\phi }^{D}+{E}_{\phi }^{E}+{E}_{\phi }^{{pp}}),\end{eqnarray}$
where the kinetic energy $\begin{eqnarray}{E}^{\mathrm{kin}.}=\displaystyle \sum _{k}\int {\rm{d}}{\boldsymbol{r}}{\bar{\psi }}_{k}^{V}({\boldsymbol{r}})(-{\rm{i}}{\boldsymbol{\gamma }}\cdot {\boldsymbol{\nabla }}+M){\psi }_{k}^{V}({\boldsymbol{r}}),\end{eqnarray}$
$\begin{eqnarray}{E}_{\phi }^{{\rm{D}}}=+\displaystyle \frac{1}{2}\displaystyle \sum _{{kk}^{\prime} }\int {\rm{d}}{\boldsymbol{r}}{\rm{d}}{\boldsymbol{r}}^{\prime} {\bar{\psi }}_{k}^{V}({\boldsymbol{r}}){\bar{\psi }}_{k^{\prime} }^{V}({\boldsymbol{r}}^{\prime} ){{\rm{\Gamma }}}_{\phi }{D}_{\phi }{\psi }_{k^{\prime} }^{V}({\boldsymbol{r}}^{\prime} ){\psi }_{k}^{V}({\boldsymbol{r}}),\end{eqnarray}$
$\begin{eqnarray}{E}_{\phi }^{{\rm{E}}}=-\displaystyle \frac{1}{2}\displaystyle \sum _{{kk}^{\prime} }\int {\rm{d}}{\boldsymbol{r}}{\rm{d}}{\boldsymbol{r}}^{\prime} {\bar{\psi }}_{k}^{V}({\boldsymbol{r}}){\bar{\psi }}_{k^{\prime} }^{V}({\boldsymbol{r}}^{\prime} ){{\rm{\Gamma }}}_{\phi }{D}_{\phi }{\psi }_{k}^{V}({\boldsymbol{r}}^{\prime} ){\psi }_{k^{\prime} }^{V}({\boldsymbol{r}}),\end{eqnarray}$
$\begin{eqnarray}{E}_{\phi }^{{pp}}=+\displaystyle \frac{1}{2}\displaystyle \sum _{{kk}^{\prime} }\int {\rm{d}}{\boldsymbol{r}}{\rm{d}}{\boldsymbol{r}}^{\prime} {\bar{\psi }}_{k}^{V}({\boldsymbol{r}}){\bar{\psi }}_{\bar{k}}^{U}({\boldsymbol{r}}^{\prime} ){{\rm{\Gamma }}}_{\phi }{D}_{\phi }{\psi }_{\bar{k}^{\prime} }^{U}({\boldsymbol{r}}^{\prime} ){\psi }_{k^{\prime} }^{V}({\boldsymbol{r}}).\end{eqnarray}$
It shall be stressed that the potential energies, the two-body interactions, do give the Hartree and Fock potential terms, namely equations (15b ) and (15c ), while the pairing energy (15d ) does not contain commutative antisymmetric contributions. In the RHFB and D-RHFB models, the finite range Gogny force D1S [99] is utilized as the pairing force, regarding the advantage of finite range of natural convergence with the configuration space. That is to say, the term ${{\rm{\Gamma }}}_{\phi }{D}_{\phi }({\boldsymbol{r}}-{\boldsymbol{r}}^{\prime} )$ in ${E}_{\phi }^{{pp}}$ is replaced by $({\gamma }_{0}{)}_{x}({\gamma }_{0}{)}_{x^{\prime} }{V}_{{\rm{Gogny}}}^{{pp}}({\boldsymbol{r}}-{\boldsymbol{r}}^{\prime} ),$ which is detailed in [97].
Besides the quantization of the Dirac spinor field, the variation of the RHFB energy functional (14) is also different from the RHF one (7), which is performed in general referring to the single-particle spinor ${\psi }_{\alpha }.$ In contrast, one has to perform the variation with respect to the generalized density matrix ${\mathscr{R}}$ to get the full RHFB equation. Following the procedure detailed in [100], the density matrix ${\mathscr{R}}$ and derived quasi-particle Hamiltonian read as$\rho $ and $\kappa $ are respectively the density and pairing tensor, and $h$ and ${\rm{\Delta }}$ are respectively the single-particle Hamiltonian and pairing potential. Notice that there only exist $V$ components in the kinetic and potential energies (15a –15c ). If performing the variation with respect to the quasi-particle spinors, only the equations of the $V$ component can be derived. More seriously, one may deduce an additional factor $1/2$ in the pairing potentials ${\rm{\Delta }},$ which artificially reduces the pairing strength by half. In fact, some unitary condition of the Bogoliubov transformation, such as $U{V}^{\dagger }+{V}^{* }{U}^{T}=0,$ cannot be properly taken into account when performing the variation with respect to the quasi-particle spinors. Previously, in order to get reasonable description of the pairing effects, people introduced the so-called exchange terms in ${\rm{\Delta }}$ to implement such artificial reduction.
$\begin{eqnarray}{\mathscr{R}}\equiv \left(\begin{array}{cc}\rho & \kappa \\ -{\kappa }^{* } & 1-{\rho }^{* }\end{array}\right),{\mathscr{H}}=\frac{\partial E}{\partial {\mathscr{R}}}=\left(\begin{array}{cc}h & {\rm{\Delta }}\\ -{{\rm{\Delta }}}^{* } & -{h}^{* }\end{array}\right),\end{eqnarray}$
where Similar to the RHF approach recalled in section 2 , one can now derive the full RHFB scheme by cooperating with the Bogoiubov quasi-particle space, which shares the starting point with the RHF scheme. It provides a standard and apparent procedure for people to understand the Bogoliubov scheme which is very efficient and reliable in dealing with the unstable nuclear systems. Moreover, as the natural extension of the D-RHF model, the D-RHFB model for deformed unstable nuclei is presented in [97], in which a qualitative analysis on the nature of $\pi $ -PV and $\rho $ -T couplings are presented. With such analysis, it becomes also apparent to understand the role played by the $\pi $ -PV and $\rho $ -T couplings in deciding the structures of deformed unstable nuclei.
4. RPA method based on the RHF approach
Among various excitation modes of nuclei, the spin-isospin excitation is of special significance, because it carries rich orbital, spin and isospin information of nuclear force. Combined with the (quasi) random phase approximation [(Q)RPA], the RHF and RHFB theories provide unified precise description of both ground state and spin-isospin excitation using the same effective Lagrangians for both mean field and residual interactions [89–91, 93, 94]. As revealed from the non-relativistic investigations, the tensor force may play significant role in determining the excitation properties of nuclei and further the $\beta $ -decay [101–105]. Even though, it still remains as an open question on the strength and even the sign of tensor force within the non-relativistic models. In contrast, it is demonstrated that not only the $\pi $ -PV and $\rho $ -T couplings, the Fock terms arising from the other meson-nucleon coupling channels also contain the tensor-force type contributions which exhibit characteristic spin dependence [71–74].
Aiming at the role of tensor force in determining the spin-isospin excitations, the RPA based on the RHF theory was extended recently by implementing the degree of freedom associated with the $\rho $ -T coupling [92]. From PKO1 to PKA1, the self-consistency of RHF + RPA maintains well after implementing the $\rho $ -T coupling, which was demonstrated through the description of isobaric analog state [92]. Taking the GTR as examples, it is illustrated that the $\rho $ -T and $\rho $ -VT couplings play significant role in maintaining the self-consistency. Referring to the unperturbed results in figure 2, it is clear that the isoscalar $\sigma $ -S and $\omega $ -V couplings fully via the Fock terms present substantial contributions to the ${ph}$ residual interactions, consistent with the previous conclusions in [89, 91]. Moreover, the isovector $\rho $ -T and $\rho $ -VT couplings show more pronounced contributions than the isovector $\rho $ -V and $\pi $ -PV ones, and the contributions from the $\sigma $ -S and $\omega $ -V channels become less remarkable. This is due to the fact that the balance between nuclear attractions and repulsion is notably changed by the strong $\rho $ -T coupling in PKA1 [63].
Figure 2. Transition strength distributions |
Applying the implemented RHF + RPA method, the tensor force effects on the spin-isospin excitations were also explored in [92]. Taking 208Pb as an example, figure 3 shows the GTR transition strengths ${R}^{-}$ (MeV−1) as functions of the excitation energy $E$ (MeV) calculated by RHF + RPA method using PKA1 and PKO $i$ ($i=1,2,3$ ) [64], where $^{\prime\prime} 11^{\prime\prime} ,$ $^{\prime\prime} 10^{\prime\prime} ,$ and $^{\prime\prime} 00^{\prime\prime} $ represent the full calculations, the ones excluding the tensor force components in residual interaction, and the ones excluding the tensor force components in both mean field and residual interaction, respectively. As seen from figure 3, the RHF Lagrangians PKA1 and PKO series present similar description on the GTR of 208Pb, and reproduce well the main peak (denoted by arrows), which further demonstrates the self-consistent persistence of the RHF + RPA method. More specifically, PKA1 slightly overestimates the peak energy, which is underestimated by all PKO Lagrangians.
Figure 3. Transition strength distributions |
Comparing the $^{\prime\prime} 10^{\prime\prime} $ results to the full $^{\prime\prime} 11^{\prime\prime} $ ones, it can be seen that the main peaks given by the selected RHF Lagrangians remain almost unchanged, and the low-energy peaks are solely moved towards the low-energy region slightly. This indicates that the tensor force components introduced via the Fock terms do not make significant contribution to the ${ph}$ residual interactions. Further to the $^{\prime\prime} 00^{\prime\prime} $ results, which neglect the tensor force components on both RHF mean field and RPA levels, the main peak given by PKA1 still remains unchanged, while PKO1 and PKO3 present upward shift of about 0.3 and 0.6 MeV, respectively, and a downward shift of about 0.4 MeV is obtained by PKO2. Such differences can be understood from the nature of tensor force components carried by the RHF Lagrangians PKA1 and PKO $i$ ($i=1,2,3$ ). Specifically, PKA1 contains all the meson-nucleon couplings as described in equation (2 ), i.e., the $\sigma $ -S, $\omega $ -V, $\rho $ -V, $\pi $ -PV, $\rho $ -VT and $\rho $ -T couplings, whereas the PKO series do not contain the $\rho $ -VT and $\rho $ -T ones, and PKO2 even does not contain the $\pi $ -PV one. In fact, the signs of the tensor force components carried by various meson-nucleon couplings are not all the same [71, 73, 74, 106]. In short, the sign of tensor force in $\pi $ -PV coupling is opposite to those in the $\omega $ -V, $\rho $ -V, $\rho $ -VT and $\rho $ -T channels.
For PKO1 and PKO3, the tensor force in $\pi $ -PV coupling dominates over those from all the other couplings, and it determines the whole sign of the tensor force [73]. While for PKO2, only the $\omega $ -V and $\rho $ -V couplings contain tensor force components. This means that the sign of the net tensor force in PKO2 is opposite to those in PKO1 and PKO3, which explains why the tensor-force effects in PKO2 are opposite to those in PKO1 and PKO3. In addition, it is known that the coupling strength of $\pi $ -PV is larger in PKO3 than in PKO1, due to relatively weaker density dependence in PKO3 [61, 64]. As a result, the tensor force components in PKO3 affect the GTR properties more remarkably than those in PKO1. For PKA1, the situation is more complicated. Compared with the PKO1 and PKO3, the tensor force arising from the $\pi $ -PV coupling in PKA1 is further cancelled by those from the $\rho $ -T and $\rho $ -VT ones. Thus, the tensor force components carried by PKA1 show almost invisible influence on the main peak of the GTR in 208Pb. However, as revealed by the RPA calculations based on Skyrme Hartree–Fock (SHF + RPA) theory, the effects of tensor force in the ${ph}$ residual interactions are rather considerable. In contrast to the results in figure 3, the calculation with the effective interactions SIII and SIII + T shows that the tensor force in the residual interaction can bring downwards the main peaks by about 4.1 MeV [101]. If only neglecting the tensor force in the SHF mean field, the main peak is shifted downwards by about 0.8 MeV [101], which is qualitatively consistent with the tensor effects given by PKO2, but opposite to those in PKO1 and PKO3.
Different from the fully self-consistent RHF + RPA method, the tensor force in the SHF + RPA frame is treated as an individual degree of freedom, and the strength and even the sign of tensor force still remain as an open question. As aforementioned, the tensor force components are naturally introduced via the Fock diagrams under the RHF approach. Thus, after implementing the $\rho $ -T coupling, the fully self-consistent RHF + RPA method lay the foundation of exploring the tensor-force effects in the spin-isospin excitations. In terms of those, it is expected to provide the subtle spin and isospin information of the in-medium interactions in nuclear systems. Moreover, the comparison and the further cross reference between the relativistic and non-relativistic researches shall promote our understanding not only on the nature of nuclear force, but also on the nuclear many-body method.
As mentioned in the introduction, the tensor force can play a critical role in determining the shell structure and the evolution [107–109]. Although the Fock terms can get the tensor force component into account in a natural way, the strength of the tensor force is still not totally constrained in RHF theory. As stressed in a series of previous references, the key point to constrain the tensor force is to find uniquely sensitive observable [108]. However, the single-particle energies, which are frequently adopted in the traditional strategy, may not be well justified, because the effects of particle-vibration coupling (PVC) also play crucial roles in the single-particle properties. In this situation, the ab initio calculation is promising to serve as reliable benchmarks and provide a bridge between the realistic nuclear forces and the effective ones [22, 110]. For example, the results obtained by the Brueckner Hartree–Fock calculation contain no beyond-mean-field effects, and thus can be used as pseudo data to benchmark the DFT calculations [111].
Recently, the self-consistent relativistic Brueckner-Hartree–Fock (RBHF) theory for finite nuclei has been established [112, 113]. Compared with the nonrelativistick BHF calculation using two-body interaction only, the RBHF achieved much better agreement with experimental data. The RBHF calculation was performed for the neutron drop [111, 114], which is a collective of neutrons confined by an external field. Clear tensor-force effects were revealed on the evolution of SO splitting in neutron drops. Adopting these results as pseudo data, it was argued that the tensor force in the current RHF effective interactions seems too weak [111]. Meanwhile, using the same pseudo data, weaker density dependence of the $\pi $ -PV coupling was suggested [115], which is related to the ‘tensor renormalization persistency' [116] supported by the shell-model study. Within the non-relativistic DFT, a new Skyrme functional, namely SAMi-T, was developed under the guidance of RBHF calculations of neutron-proton drops [117]. Similar progress was also made for the RHF effective interaction [118], which may guide the development of new RHF Lagrangian.
5. PSS in RHF
According to the effective field theory, nuclear force contains strong attractive and repulsive ingredients, which are described respectively by the exchanges of the scalar and vector mesons [119], and the delicate balance of the attractions and repulsions guarantees the binding of nuclear systems. Practically, such balance can be manifested as the conservation of the so-called PSS in nuclear structure, which is recognized as a relativistic symmetry [120–122]. In the energy spectrum, the PSS corresponds to the quasi degeneracy of two single-particle (s.p.) states with the quantum numbers ($n,$ $l,$ $j$ = $l$ +1/2) and ($n-$ 1, $l$ +2, $j$ =$l$ +3/2), which form the pseudo-spin (PS) doublet ($n^{\prime} $ =$n\,-$ 1, $l^{\prime} $ =${l}$ + 1, $j^{\prime} $ =$j$ =$l^{\prime} \pm $ 1/2) [123, 124], and the pseudo orbit $l^{\prime} $ is nothing but the orbital angular momentum of the lower component of Dirac spinor [120].
Under the RMF scheme, the condition of the PSS conservation is verified as $S(r)+V(r)=0$ [120] or ${\rm{d}}\left[S(r)+V(r)\right]/{\rm{d}}{r}=0$ [125] with $S$ and $V$ respectively for the attractive scalar and repulsive vector potentials. Obviously both conditions imply certain balance of strong attraction S($r$ ) and repulsion V($r$ ) in nuclear medium. However, in the RMF calculations and the RHF ones with PKO $i$ ($i=1,2,3$ ), the PSS is often broken for the high-$l^{\prime} $ PS doublet nearby the Fermi levels, inducing the spurious shell closures $N/Z$ =58 and 92 [62, 78] and largely overestimated binding energies around ${}^{140}$ Ce ($Z=58$ ) and ${}^{218}$ U ($Z=92$ ) [126]. Implemented with the degree of freedom of the $\rho $ -T coupling, such spurious shell closures were eliminated eventually by the RHF Lagrangian PKA1 that properly restores the PSS for the high-$l^{\prime} $ PS doublets [65, 78, 80].
Taking the doubly magic nuclei ${}^{48}$ Ca, ${}^{90}$ Zr, ${}^{132}$ Sn, and 208Pb, and the predicted superheavy one ${}^{310}$ 126 as examples, the proton shells and the splittings of the PS doublets neighboring the shells given by the RHF and RMF models were analysed in [63] as shown in figures 4(a) and (b). It can be seen that the pseudo-spin orbital (PSO) splittings ${\rm{\Delta }}{E}_{{\rm{PSO}}}$ show nearly parallelled trends from the light to heavy nuclei between the models. Referring to the experimental data [130], only PKA1 shows appropriate agreements, whereas the others distinctly overestimate the PSO splittings. This brings essential effects on the emergences of the neighboring shells in heavy nuclei. As a general trend of the models, less pronounced proton shells $Z=82$ and $Z=50$ usually accompany with larger ${\rm{\Delta }}{E}_{{\rm{PSO}}}^{\pi }$ of the PS doublet ($2{f}_{7/2},$ $1{h}_{9/2}$ ) in 208Pb and the one ($2{d}_{5/2},$ $1{g}_{7/2}$ ) in 132Sn, respectively. Notice that the current analyses are restricted on the level of the mean-field approach. If going beyond the mean-field approach by considering the effects of the particle vibration coupling (PVC) [131], the single-particle levels may be shifted systematically toward the Fermi levels, and such effects are reduced in general when the distances to the Fermi levels are enlarged [132]. Since the s.p. states that determine the shell gaps are either below or above the Fermi levels, the shell gaps in figure 4(a) will be squeezed by the PVC effects, which leads to further underestimated shell $Z=82$ for PKO3, DD-ME2, PK1 and NL3*. In contrast, because the relevant PS doublet orbits are either both above or below the Fermi levels, tiny PVC effects can be deduced for the PSO splittings in figure 4(b), which would not change the final conclusions. Extending to heavier nuclear systems, such as super-heavy nucleus ${}^{310}126$ predicted by PKA1 [80], it can be seen from figure 4(c) that well preserved PSS for the PS doublet ($2{g}_{9/2},$ $1{i}_{11/2}$ ) seems to approve the shell $Z=126$ in the PKA1 results. However for the other models, the large PSO splitting between the PS partners $2{g}_{9/2}$ and $1{i}_{11/2}$ gives massive shell $Z=138.$ It indicates that the emergences of shell closures in superheavy nuclei may be essentially related to the PSS restoration.
Figure 4. Proton shell gaps (MeV) [plot (a)] and the splittings of neighboring PS partners |
As aforementioned, the conservation condition of the PSS indicates an in-medium balance of nuclear attractions and repulsions. Taking 208Pb as an example, nuclear in-medium effects on the PSS restoration are demonstrated in [63], particularly for the high-$l^{\prime} $ PS doublets. Figure 5 shows the proton ($\pi $ ) PSO splittings ${\rm{\Delta }}{E}_{{\rm{PSO}}}^{\pi }$ in 208Pb as functions of pseudo orbit $l^{\prime} $ given by the RHF Lagrangians PKA1 and PKO3, and the RMF one DD-ME2. As shown in figure 5, the splittings ${\rm{\Delta }}{E}_{{\rm{PSO}}}^{\pi }$ given by PKA1 decrease quickly with respect to pseudo orbit $l^{\prime} ,$ leading to well conserved PSS at $l^{\prime} =4$ which is consistent with the experimental data. In contrast, PKO3 and DD-ME2 present rather large ${\rm{\Delta }}{E}_{{\rm{PSO}}}^{\pi }$ values, inducing a spurious shell $Z=92$ [62]. Such model deviations can be well interpreted by the sum contributions ${E}_{{\rm{kin}}.+\sigma +\omega }$ of the splitting ${\rm{\Delta }}{E}_{{\rm{PSO}}}^{\pi }$ from the kinetic term, and the dominant $\sigma $ -S and $\omega $ -V channels; see figure 5(b). It can be seen that the systematics of ${\rm{\Delta }}{E}_{{\rm{PSO}}}^{\pi }$ are almost fully determined by the sum contributions ${E}_{{\rm{kin}}.+\sigma +\omega }$ which manifest the balance between the attractive $\sigma $ -S and repulsive $\omega $ -V couplings indeed.
Figure 5. Proton ( |
Moreover, as shown in figure 5(b), such balances are systematically changed from the RMF Lagrangian DD-ME2 to the RHF one PKO3 and further to PKA1, i.e., the sum contributions ${E}_{{\rm{kin}}.+\sigma +\omega }$ are successively reduced. In particular, the $l^{\prime} $ dependencies are shifted nearly in parallel from DD-ME2 to PKO3. As demonstrated in figure 6(a) that shows the contributions of the binding energy of 208Pb, all these imply the systematical changes on the balance between nuclear attractions and repulsions from the RMF to RHF approaches. In figure 6(a), all the effective Lagrangians give similar total binding energy for 208Pb, while the specific contributions except ${E}_{{\rm{cou}}.}$ are quite different. From the RMF to RHF approaches, the isovector $\rho $ - and $\pi $ -couplings are notably enhanced by the Fock terms, leading to negatively large ${E}_{\rho +\pi }$ values in figure 6(a). Consistently, the dominant isoscalar term ${E}_{{\rm{kin}}.+\sigma +\omega }$ becomes less negative to maintain the proper binding of 208Pb. Within the RHF approach, due to the $\rho $ -T coupling in PKA1 that contributes a fairly strong attraction, the isovector term ${E}_{\rho +\pi }$ becomes much more negative from PKO3 to PKA1, and the negative isoscalar term ${E}_{{\rm{kin}}.+\sigma +\omega }$ is significantly reduced [63]. All these indicate that the in-medium balance of nuclear attractions and repulsions is substantially changed by the Fock terms, particularly by the strong $\rho $ -T coupling in PKA1.
Figure 6. Plot (a) shows the contributions to the binding energy (MeV) of 208Pb from the kinetic and isoscalar potential energies ( |
In fact, the presence of the Fock terms not only changes the balance of nuclear attractions and repulsions, but also bring notable consequences on the modeling of the nuclear in-medium effects, here evaluated by the density dependencies of the meson-nucleon coupling strengths in PKA1, PKO3 and DD-ME2; see figures 6(b)–(d). In figure 6(b), both the values and density dependencies of the coupling strengths ${g}_{\sigma }$ and ${g}_{\omega }$ are reduced from DD-ME2 to PKO3. As aforementioned, the Fock terms can considerably enhance the isovector channels, and thus the dominant channels $\sigma $ -S and $\omega $ -V are not necessitated as strong as the ones in the RMF approach. Meanwhile, enhanced isovector channels can also take more in-medium effects into account. To provide proper modeling of nuclear in-medium effects, the density dependencies of the isoscalar ${g}_{\sigma }$ and ${g}_{\omega }$ should be reduced as well. It is also interesting to see from figure 6(b) that ${g}_{\sigma }$ and ${g}_{\omega }$ maintain nearly parallel density-dependencies for both DD-ME2 to PKO3. Such situation is indeed common for the popular RMF Lagrangians and RHF ones PKO $i$ ($i=1,2,3$ ). Further to PKA1, as revealed in figures 6(a) and (d), the $\rho $ -T coupling not only contributes fairly strong attraction, which can even shake the balance between the dominant $\sigma $ -S and $\omega $ -V channels, but also carries notable nuclear in-medium effects due to the remarkable density dependence. Thus, after implementing the $\rho $ -T coupling, the coupling strengths ${g}_{\sigma }$ and ${g}_{\omega }$ in PKA1 are further reduced, which even do not maintain the parallel density-dependent behaviors any more.
To qualitatively understand the consequence of the in-medium balance on microscopic nuclear structure, here on the PSS restoration, figure 7 shows schematic profiles of the matter density (${\rho }_{b}$ ) and probability densities of the $s$ -, $p$ -, $d$ - and $f$ -orbits. Combined with the results in figures 6, 7 can help us to understand the $l^{\prime} $ -dependence of the PSO splitting ${\rm{\Delta }}{E}_{{\rm{PSO}}}^{\pi }$ in figure 5. From the central region to the surface of nucleus, the matter density varies from near saturated value to zero, and changes as well the in-medium balance of nuclear attractions and repulsions. Simultaneously, nucleons are driven outwards by the repulsive centrifugal potentials when the angular momentum $l$ increases. As described in figure 6(b), different density-dependent behaviors of ${g}_{\sigma }$ and ${g}_{\omega }$ indicate different balance between $\sigma $ -S and $\omega $ -V couplings from the center to the surface regions of nucleus. Following the schematic behaviors of the orbits in figure 7, it is not hard to deduce the fact that the in-medium balance of nuclear attractions and repulsions is manifested as the $l^{\prime} $ -dependence of the ${\rm{\Delta }}{E}_{{\rm{PSO}}}^{\pi }$ values in figure 5, according to the conservation condition of the PSS.
Figure 7. Schematic diagrams of the matter density |
The $\rho $ -T coupling, which changes distinctly the in-medium balance, influence not only the structure of nucleus, but also the liquid-gas (LG) phase transition of thermal nuclear matter that has important impact on the heavy-ion collisions [134–137], nuclear astrophysics [138–142], and so on. Under the RMF and RHF approaches, there are also lots of investigations on the LG phase transition [143–148]. Different from those, the consequences of the in-medium balance were clarified recently in [133] by applying the RHF model with PKA1. As shown in table 1 and figure 8, PKA1 that contains the $\rho $ -T coupling predicts rather different critical properties and LG phase diagrams from the other effective Lagrangians. As demonstrated in [133], the in-medium balance of nuclear attraction and repulsion, manifested as various modeling of the nuclear in-medium effects, is essential for the van der Waals-like behaviors of thermal nuclear matter, in which the residual nuclear in-medium effects carried by the balance of the dominant channels play a significant role [133].
Table 1. Critical parameters of LG phase transition for symmetric nuclear matter, i.e., the critical temperature |
| | | | | | |
|---|---|---|---|---|---|
| PKA1 | 11.55 | 0.050 | 0.114 | −40.69 | 0.196 |
| PKO3 | 14.57 | 0.048 | 0.198 | −75.03 | 0.286 |
| DD-ME2 | 13.11 | 0.044 | 0.155 | −62.92 | 0.267 |
| PK1 | 15.11 | 0.049 | 0.223 | −82.83 | 0.305 |
Figure 8. LG phase diagrams of thermal nuclear matter at the temperatures T = |
Enlightened by the PSS restoration and nuclear in-medium effects, the new relativistic mean-field Lagrangian DD-LZ1 is proposed in [149]. For the new effective Lagrangian DD-LZ1, the density dependencies of ${g}_{\sigma }$ and ${g}_{\omega }$ do not maintain parallel, which is deduced by ignoring the condition ${g}_{\sigma ^{\prime\prime} }(1)={g}_{\omega ^{\prime\prime} }(1)$ in the parametrization of DD-LZ1. Benefited from that, DD-LZ1 cures the common disease in previous RMF calculations, namely the occurrences of the spurious shell closures $N/Z$ = 58 and 92, as shown in figure 9. Consistently, DD-LZ1 also restores the PSS for the high-$l^{\prime} $ pseudo-spin doublets nearby the Fermi levels, which indeed approves the effectiveness of the analysis for the PSS restoration and nuclear in-medium balance between nuclear attraction and repulsion. Moreover, better accuracy obtained by DD-LZ1 in describing nuclear mass is quite desirable for further applications [149].
Figure 9. Two-proton shell gap |
6. New magicity $N=32$ and $34$ and Novelty in 48Si
As one of the milestones in nuclear physics, the traditional magic numbers and shell structures were explained by introducing the empirical spin-obit coupling in atomic nuclei [151, 152]. With the development of RIB facilities, the new magicity were found in unstable nuclei such as $N=16$ in drip-line nucleus ${}^{24}$ O [13, 153], which attracted intensive interests. As another typical example, the magic nature of $N=32$ and $34$ has been confirmed experimentally by the enhanced ${2}_{1}^{+}$ excitation energy and the relatively reduced $B(E2{\rm{;}}{0}^{+}\to {2}_{1}^{+})$ transition probabilities [18, 154–158]. Theoretically, the tensor force is recognized to play an important role in determining the shell evolution and the formation of new magicity due to its spin-dependent feature [72, 73, 107].
Compared to the other popular models, the RHF and RHFB models provide reliable tools to verify the underlying mechanism of the emergence of new magicity, since both SO coupling and tensor force, the important ingredients of nuclear force, can be uniformly treated under the RHF approach [72–74]. As indicated by the study along the isotonic chains [81], which shows that the new magicity $N=32$ and $34$ can be uniformly described by the RHFB theory with PKA1, the Lorentz $\pi $ -PV and $\rho $ -T couplings seem to play a determinant role in the formation of the subshell $N=32,$ but negligible for the one $N=34.$
Inspired by later high precision mass measurements [160–162], particularly the first direct mass measurements of neutron-rich calcium isotopes beyond neutron number 34 [159], the mechanism accounting for the occurrence of new magicity $N=32$ and $34$ was verified by applying the RHF model with PKA1 [82]. As shown in figure 10(a), the significant drops of the two-neutron separation energies ${S}_{2n}$ (MeV) across traditional magic number $N=28$ and the new ones $N=32$ and 34 can be well reproduced by PKA1, although the values are systematically overestimated. Moreover, for the empirical energy gap ${\delta }_{e}$ and two-neutron gap ${{\rm{\Delta }}}_{2n}$ respectively shown in figures 10(b) and (c), the PKA1 calculations also show precise agreement with the data at $N=28,$ 32 and 34. In contrast, the S${}_{2n}$ values calculated by PKO3 and DD-ME2 decrease smoothly from $N=30$ to ${N}$ = 40, and the ${\delta }_{e}$ and ${{\rm{\Delta }}}_{2n}$ values are also distinctly underestimated.
Figure 10. Plot (a) shows two-neutron separation energies S |
Not only the precise mass measurement, the quasi free neutron knockout experiment from ${}^{54}$ Ca also corroborates the shell closure at ${N}$ = 34 [163]. To clarify the mechanism for the successive magicity ${N}$ = 32 and 34, figure 11 gives the single particle energy referring to the neutron ($\nu $ ) orbit ν1f7/2 and the density distribution of neutron and proton for ${}^{52,54}$ Ca calculated by PKA1. It is seen that both the neutron and proton densities of ${}^{52}$ Ca show distinct central-bumped structures, being consistent with the large SO splitting of $\nu 2p$ orbits, which leads to the $N=32$ shell closure. When the $\nu 2{p}_{1/2}$ orbit is populated, the density profiles of ${}^{54}$ Ca become central-depressed, even showing a neutron semi-bubble structure, and consistently the $\nu 2p$ splitting is reduced. Combined with the unchanged $\nu 1f$ one, this reduction gives the subshell $N=34.$
Figure 11. 3D Neutron/proton densities (left panels) and neutron ( |
Considering the fact that the SO potential can be expressed in terms of the derivative of nuclear densities [151], it is not difficult to understand the evolution of the $\nu 2p$ splitting that depends sensitively on central density profiles [164]. Experimentally, a reduced $\nu 2p$ splitting, attributed to the proton bubble structure, has been observed by comparing the spectroscopic information on the ${}^{35}$ Si and ${}^{37}$ S isotones [165]. However, it should be noted that the bubble structure arise generally from a reduced occupation of the $s$ -orbits. For example, the one-proton removed reaction experiment shows that the proton $(\pi )$ orbit $\pi 2{s}_{1/2}$ is essentially empty to give the proton bubble structure in ${}^{34}$ Si [76, 83]. However from ${}^{52}$ Ca to ${}^{54}$ Ca, the occupations of $s$ -orbits do not change due to the robust magic core ${}^{48}$ Ca.
As the further verification, the interaction matrix elements between the valence orbit $\nu 2{p}_{1/2}$ and the others were analyzed in [82] by using the RHF Lagrangians PKA1, PKO3, and the RMF one DD-ME2, see figures 12(a)–(c). It is interesting to see that the interaction between $\nu 2{p}_{1/2}$ and $s$ -orbits are generally repulsive, and PKA1 presents the strongest repulsive results. Thus, when the $\nu 2{p}_{1/2}$ orbit is populated in ${}^{54}$ Ca, such strong repulsive interactions given by PKA1 drive the $s$ -orbital nucleons away from the center to reduce the central densities as shown in figure 11, and consistently the $\nu 2p$ splitting is reduced to form the subshell $N=34$ in ${}^{54}$ Ca, since the $\nu 1f$ orbits are not sensitive to the alteration of central densities.
Figure 12. Interacting matrix elements between the valence neutron |
It shall be mentioned that within the relativistic scheme, e.g. the RMF and RHF approaches, the Dirac spinors of nucleons contain the upper ($U$ ) and lower ($L$ ) components. In particular, the $L$ -components of the ${p}_{1/2}$ orbits share the same angular wave functions with the $U$ -ones of the ${s}_{1/2}$ orbits, and vice versa. Such doublets are named as the Dirac inversion partners (DIPs) in [82], which are of the same total angular momentum but opposite parity. To understand the repulsive couplings between the DIPs, here ($\nu 2{p}_{1/2},$ $\nu $ s${}_{1/2}$ ), the total interaction matrix elements are decomposed into two parts, the ${UL}$ -terms and ${UU}$ -terms as shown respectively in figures 12(b) and (c). For the ${UU}$ -term, it contains the contributions only from the $U$ -components of Dirac spinors, while the $L$ -components contribute the ${UL}$ -terms, namely the ones which exclude the ${UU}$ -terms from the total.
As shown in figure 12(b), the ${UL}$ -terms present repulsive contributions, which are notably enhanced for the DIPs due to the inversion similarity, i.e., the $U$ -/$L$ -components of $\nu 2{p}_{1/2}$ orbit and the $L$ -/$U$ -ones of its DIPs ${s}_{1/2}$ share the same angular momentum. Such that the interactions between the DIPs ($\nu 2{p}_{1/2},$ ${\nu s}_{1/2}$ ) are repulsively enhanced by the strong ${UL}$ -terms, eventually giving the strong repulsive ${V}_{{s}_{1/2},\nu 2{p}_{1/2}}$ in figure 12(a), the key mechanism accounting for the shell closure ${N}$ = 34 in ${}^{54}$ Ca. As an implemented illustration, the repulsive ${UL}$ -terms in ${V}_{\nu 2{p}_{1/2},\nu {s}_{1/2}}$ are dropped for the calculation of ${}^{54}$ Ca, as shown in the last row of figure 11 (marked as ${}^{54}$ Ca*). It can be seen that without the repulsive effects between the DIPs, the central-bumped structures similar as ${}^{52}$ Ca appear in both neutron and proton densities, and consistently the $\nu 2p$ splitting becomes large enough to eliminate the $N=34$ subshell.
As seen from figure 12(b), the ${UL}$ -terms are in general repulsive, and such systematics can be qualitatively explained by the nature of the coupling channels. As an illustration, table 2 shows the Hartree (H) and Fock (F) contributions to the interaction matrix element ${V}_{\nu 2{p}_{1/2},\nu 1{s}_{1/2}}$ (MeV) for various channels, including the ${UU}$ -, ${UL}$ -terms and the total ($V$ ). For the selected mesonic degrees of freedom, the isoscalar $\sigma $ -S and $\omega $ -V couplings dominate respectively the attraction and repulsion, and the cancellation between each another gives residual attraction. However, due to the specific nature of Lorentz scalar coupling, the $\sigma $ -S coupling shows similar repulsive ${UL}$ -terms as the $\omega $ -V one, instead of counteracting each another, seeing ${UL}$ -H terms in table 2. It shall be emphasized that repulsive ${UL}$ -terms are obtained for both the neutron-neutron and proton-proton couplings, as both are dominated by the $\sigma $ -S and $\omega $ -V couplings.
Table 2. Contributions from the Hartree (H) and Fock (F) terms of the |
| | | | | | | |
|---|---|---|---|---|---|---|
| | −3.935 | 2.894 | 0.101 | | | |
| | 1.617 | −1.328 | −0.046 | −0.089 | | −0.007 |
| | 0.359 | 0.269 | 0.009 | 0.000 | 0.001 | |
| | 0.027 | 0.368 | 0.013 | 0.006 | 0.030 | −0.001 |
| | −3.576 | 3.163 | 0.110 | 0.000 | 0.001 | |
| | 1.645 | −0.961 | −0.033 | −0.084 | 0.030 | −0.008 |
Compared to the ordinary cases, the repulsive ${UL}$ -terms are strongly enhanced for the DIPs $(\nu 2{p}_{1/2},{s}_{1/2}),$ which can be understood well from the inversion similarity between the DIPs, i.e, the $U$ -/$L$ -components of $\nu 2{p}_{1/2}$ orbit and the $L$ -/$U$ -ones of its DIPs ${s}_{1/2}$ share the same angular momentum. From the RMF to RHF approaches, new couplings between the $U$ - and $L$ -components of Dirac spinors are introduced by the Fock terms, such as the space parts of the vector couplings ($\omega $ -V and $\rho $ -V), and the $\rho $ -VT one and the time component of $\rho $ -T one in PKA1, which enhance the repulsive ${UL}$ -terms by different extent, see table 2. Thus, the RHF models present more repulsive ${UL}$ -terms for the DIPs than the RMF ones. However, for the enhanced repulsion between the DIPs $(\nu 2{p}_{1/2},\nu {s}_{1/2})$ from PKO3 to PKA1 in figure 12(b), one cannot simply attribute to the effects of the additional $\rho $ -T and $\rho $ -VT couplings, and it is indeed due to different in-medium balance between nuclear attractions and repulsions given by PKA1 from PKO3 as aforementioned and clarified in [63].
Qualitatively, the connection between bubble structure and SO splitting could be understood well in terms of SO potential. As indicated from figures 12(a)–(c), the ${UL}$ -terms of the DIPs' couplings, here the doublets (${s}_{1/2},$ $\nu 2{p}_{1/2}$ ), are largely enhanced, compared to the others, e.g. the $({s}_{1/2},\nu 2{p}_{3/2})$ couplings. Thus, it is appealing to understand the effect of strong coupling of DIPs in determining the SO splitting of $\nu 2p$ states, e.g. the subshell ${N}$ = 32 from the angle of two-body interaction. Figure 12(d) shows the evolution of ${\rm{\Delta }}{E}_{\nu 2p}$ values (MeV), as well as the contributions from the couplings with the $s$ -orbits (${V}_{{s}_{1/2},\nu 2p}$ ) and their ${UL}$ -terms. It can be seen that the $s$ -orbital contributions, which are dominated by the ${UL}$ -terms, play a determinant role for giving notable $\nu 2p$ splitting, namely the subshell $N=32.$ On the proton-deficient side, the subshell ${N}$ = 32 maintains until ${}^{48}$ S due to the $s$ -orbital contributions. However, it should be noted that the proton magic shell ${Z}$ = 16, shown in figure 12(e), is the key to assure the full occupation of the proton ($\pi $ ) orbit $\pi 2{s}_{1/2}$ orbit in both ${}^{50}$ Ar and ${}^{48}$ S.
In fact, the experimental low-lying structure of ${}^{50}$ Ar has been investigated and the calculation of modified SDPF-MU effective interaction indicated that the magnitude of the ${N}$ = 32 subshell gap in ${}^{50}$ Ar is similar to those in ${}^{52}$ Ca [18]. From ${}^{52}$ Ca to ${}^{54}$ Ca, the occupation of valence neutron on the $\nu 2{p}_{1/2}$ orbit drives the $s$ -orbital neutrons far away from the center and then weaken its contribution to the splitting of $\nu 2p$ states, which eventually lead to a prominent subshell at ${N}$ = 34. Combined with the results in figures 11 and 12, it can be seen that either the emergence of new magicity ${N}$ = 34 in ${}^{54}$ Ca or the persistence of the ${N}$ = 32 subshell from ${}^{48}$ S to ${}^{52}$ Ca, as well as its quenching in ${}^{46}$ Si, are originated from the presence/vanishing of the strong repulsive effect between the DIPs (${s}_{1/2},$ $\nu 2{p}_{1/2}$ ).
As mentioned above, the bubble structure could arise from the reduced occupation of $s$ -orbits or $s$ -orbital nucleons driven away from the center of nucleus. Extending along the $N=34$ isotonic chain by reducing the proton number, it is found that the drip-line isotone ${}^{48}$ Si shows several novelties, including the doubly magicity $N=34$ and $Z=14,$ dual (neutron and proton) bubble structures, and the reentrance of pairing correlations if thermally excited [77]. As shown in figure 13, the atypical nucleus ${}^{48}$ Si exhibits dual semi-bubble structures according to the calculation of PKA1. As confirmed by the extensive precision electron scattering, the central densities of stable nuclei are almost constant as ${\rho }_{0}$ ≈ 0.16 fm3, independent of the mass number [166]. However, as shown in figure 13(a) the central densities of ${}^{48}$ Si are much lower than those of ${}^{54}$ Ca, which refresh our understanding to the incompressibility of atomic nuclei and the saturation of nuclear force. On the other hand, it should be noted that such dual semi-bubble structure cloud weaken substantially the splitting of $\nu 2p$ states and further stabilize this extreme neutron-rich nucleus at the drip line [77]. Taking ${}^{48}$ Si as an example, the theoretical study of proton-induced reactions shows that the emitted protons are distinctly enhanced along the beam direction with a bubble configuration in the target [167].
Figure 13. Neutron( |
7. Summary and perspective
In this paper, we review some recent progress on the RHF description of nuclear structure, including the extension for axially deformed nuclei and nuclear excitations, and the applications in describing the restoration of PSS and the novelty in unstable nuclei. Basically the complete frameworks, such as D-RHF and D-RHFB models, and RHF + RPA method with $\rho $ -T couplings, for describing both unstable nuclei and nuclear spin-isospin excitation are available now, which have the advantages of full self-consistence. Taking the PSS restoration and the new magicity as examples of the applications, it is shown that the $\rho $ -T coupling that contributes mainly via the Fock diagram plays an important role in maintaining the delicate balance of nuclear attractions and repulsions, which are manifested as the proper restoration of the PSS for the high-$l^{\prime} $ PS doublet and uniform interpretation of the new magicity $N=32$ and $34$ in ${}^{52,54}$ Ca. To a certain extent, the revealed mechanisms help us much to understand the nature of nuclear force under the relativistic scheme, which is of special significance to guide the further development of the effective nuclear force.
Since the full scheme of the RHF models, including the extensions associated with the Bogoliubov transformation, RPA method and the deformation, is valid as proved from the pioneer applications, it is also expected for the extensive explorations such as the role of tensor force components carried by the Fock terms in determining the structure of deformed unstable nuclei and superheavy ones, as well as the spin-isospin excitations and nuclear $\beta $ -decays. On the other hand, benefiting from the experiences in developing the D-RHF and D-RHFB models, it also becomes natural to consider the degree of freedom of the octuple deformation as the further development of the models. All these may help us to understand more deeply the nature of nuclear force and the properties of unstable nuclei in a wide range.