Triaxially deformed relativistic point-coupling model for $Lambda$ hypernuclei: a quantitative analysis of hyperon impurity effect on nuclear collective properties

The impurity effect of hyperon on atomic nuclei has received a renewed interest in nuclear physics since the first experimental observation of appreciable reduction of $E2$ transition strength in low-lying states of hypernucleus $^{7}_Lambda$Li. Many more data on low-lying states of $Lambda$ hypernuclei will be measured soon for $sd$-shell nuclei, providing good opportunities to study the $Lambda$ impurity effect on nuclear low-energy excitations. We carry out a quantitative analysis of $Lambda$ hyperon impurity effect on the low-lying states of $sd$-shell nuclei at the beyond-mean-field level based on a relativistic point-coupling energy density functional (EDF), considering that the $Lambda$ hyperon is injected into the lowest positive-parity ($Lambda_s$) and negative-parity ($Lambda_p$) states. We adopt a triaxially deformed relativistic mean-field (RMF) approach for hypernuclei and calculate the $Lambda$ binding energies of hypernuclei as well as the potential energy surfaces (PESs) in $(beta, gamma)$ deformation plane. We also calculate the PESs for the $Lambda$ hypernuclei with good quantum numbers using a microscopic particle rotor model (PRM) with the same relativistic EDF. The triaxially deformed RMF approach is further applied in order to determine the parameters of a five-dimensional collective Hamiltonian (5DCH) for the collective excitations of triaxially deformed core nuclei. Taking $^{25,27}_{Lambda}$Mg and $^{31}_{Lambda}$Si as examples, we analyse the impurity effects of $Lambda_s$ and $Lambda_p$ on the low-lying states of the core nuclei...

Three-dimensional mesh calculations for covariant density functional theory

In contrast to the non-relativistic approaches, three-dimensional (3D) mesh calculations for the {it relativistic} density functional theory have not been realized because of the challenges of variational collapse and fermion doubling. We overcome these difficulties by developing a novel method based on the ideas of Wilson fermion as well as the variational principle for the inverse Hamiltonian. We demonstrate the applicability of this method by applying it to $^{16}$O, $^{24}$Mg, and $^{28}$Si nuclei, providing detailed explanation on the formalism and verification of numerical implementation.

Quasi-elastic scattering in the 20Ne + 90,92Zr reactions: role of non-collective excitations

Conventional coupled-channels analyses, that take account of only the collective excitations of the colliding nuclei, have failed to reproduce the different behavior of the experimental quasi-elastic barrier distributions for the $^{20}$Ne + $^{90,92}$Zr systems. To clarify the origins of this difference, we investigate the effect of non-collective excitations of the Zr isotopes. Describing these excitations in a random-matrix model, we explicitly take them into account in our coupled-channels calculations. The non-collective excitations are capable of reproducing the observed smearing of the peak structure in the barrier distribution for $^{20}$Ne + $^{92}$Zr, while not significantly altering the structure observed in the $^{20}$Ne + $^{90}$Zr system. The difference is essentially related to the closed neutron shell in $^{90}$Zr.

Sum rule approach to a soft dipole mode in $Lambda$ hypernuclei

Applying the sum rule approach, we investigate the energy of a soft dipole motion in $Lambda$ hypernuclei, which results from a dipole oscillation of a $Lambda$ hyperon against the core nucleus. To this end, we systematically study single-$Lambda$ hypernuclei, from $^{16}_{;,Lambda}$O to $^{208}_{;;;Lambda}$Pb, for which the ground state wave function is obtained in the framework of Hartree-Fock method with several Skyrme-type $Lambda N$ interactions. Our results indicate that the excitation energy of the soft dipole $Lambda$ mode, $E_{sdLambda}$, decreases as the mass number increases. We find that the excitation energy is well parametrized as $E_{sdLambda}=26.6A^{-1/3}+11.2A^{-2/3}$ MeV as a function of mass number $A$.

Application of the inverse Hamiltonian method to Hartree-Fock-Bogoliubov calculations

We solve the Hartree-Fock-Bogoliubov (HFB) equations for a spherical mean field and a pairing potential with the inverse Hamiltonian method, which we have developed for the solution of the Dirac equation. This method is based on the variational principle for the inverse Hamiltonian, and is applicable to Hamiltonians that are bound neither from above nor below. We demonstrate that the method works well not only for the Dirac but also for the HFB equations.

Competition between $T$=1 and $T$=0 pairing in $pf$-shell nuclei with $N=Z$

The pairing correlation energy for two-nucleon configurations with the spin-parity and isospin of $J^pi=0^+$, $T$=1 and $J^pi=1^+$, $T$=0 are calculated with $T$=1 and $T$=0 pairing interactions, respectively. To this end, we consider the $(1f2p)$ shell model space, including single-particle angular momenta of $l=3$ and $l=1$. It is pointed out that a two-body matrix element of the spin-triplet $T$=0 pairing is weakened substantially for the $1f$ orbits, even though the pairing strength is much larger than that for the spin-singlet $T$=1 pairing interaction. In contrast, the spin-triplet pairing correlations overcome the spin-singlet pairing correlations for the $2p$ configuration, for which the spin-orbit splitting is smaller than that for the $1f$ configurations, if the strength for the T=0 pairing is larger than that for the T=1 pairing by 50% or more. Using the Hartree-Fock wave functions, it is also pointed out that the mismatch of proton and neutron radial wave functions is at most a few % level, even if the Fermi energies are largely different in the proton and neutron mean-field potentials. These results imply that the configuration with $J^pi=0^+$, $T$=1 is likely in the ground state of odd-odd $pf$ shell nuclei even under the influence of the strong spin-triplet $T$=0 pairing, except at the middle of the $pf$ shell, in which the odd proton and neutron may occupy the $2p$ orbits. These results are consistent with the observed spin-parity $J^{pi}=0^+$ for all odd-odd $pf$ shell nuclei except for $^{58}_{29}$Cu, which has $J^{pi}=1^+$.

Collective excitations of $Lambda$ hypernuclei

We discuss low-lying collective excitations of $Lambda$ hypernuclei using the self-consistent mean-field approaches. We first discuss the deformation properties of $Lambda$ hypernuclei in the $sd$-shell region. Based on the relativistic mean-field (RMF) approach, we show that the oblate deformation for $^{28}$Si nucleus may disappear when a $Lambda$ particle is added to this nucleus. We then discuss the rotational excitations of $^{25}_{Lambda}$Mg nucleus using the three-dimensional potential energy surface in the deformation plane obtained with the Skyrme-Hartree-Fock method. The deformation of $^{25}_{Lambda}$Mg nucleus is predicted to be slightly reduced due to an addition of $Lambda$ particle. We demonstrate that this leads to a reduction of electromagnetic transition probability, $B(E2)$, in the ground state rotational band. We also present an application of random phase approximation (RPA) to hypernuclei, and show that a new dipole mode, which we call a soft dipole $Lambda$ mode, appears in hypernuclei, which can be interpreted as an oscillation of $Lambda$ particle against the core nucleus.

Impurity effect of Lambda particle on the structure of 18F and 19F_Lambda

We perform three-body model calculations for a $sd$-shell hypernucleus $^{19}_{Lambda}$F ($^{17}_{Lambda}{rm O}+p+n$) and its core nucleus $^{18}$F ($^{16}{rm O}+p+n$), employing a density-dependent contact interaction between the valence proton and neutron. We find that the $B(E2)$ value from the first excited state (with spin and parity of $I^pi=3^+$) to the ground state ($I^pi=1^+$) is slightly decreased by the addition of a $Lambda$ particle, which exhibits the so called shrinkage effect of $Lambda$ particle. We also show that the excitation energy of the $3^+$ state is reduced in $^{19}_{Lambda}$F compared to $^{18}$F, as is observed in a $p$-shell nucleus $^{6}$Li. We discuss the mechanism of this reduction of the excitation energy, pointing out that it is caused by a different mechanism from that in $^{7}_{Lambda}$Li.

Application of random phase approximation to vibrational excitations of double-$Lambda$ hypernuclei

Using the Hartree-Fock plus random-phase-approximation (HF+RPA), we study the impurity effect of $Lambda$ hyperon on the collective vibrational excitations of double-$Lambda$ hypernuclei. To this end, we employ a Skyrme-type $Lambda N$ and $LambdaLambda$ interactions for the HF calculations, and the residual interactions for RPA derived with the same interactions. We find that inclusion of two $Lambda$ hyperons in $^{16}$O shifts the energy of the collective states towards higher energies. In particular, the energy of the giant monopole resonance of $^{,,18}_{LambdaLambda}$O, as well as that of $^{210}_{LambdaLambda}$Pb, becomes larger. This implies that the effective incompressibility modulus increases due to the impurity effect of $Lambda$ particle, if the $beta$-stability condition is not imposed.

Description of single-Lambda hypernuclei with relativistic point coupling model

We extend the relativistic point coupling model to single-$Lambda$ hypernuclei. For this purpose, we add $N$-$Lambda$ effective contact couplings to the model Lagrangian, and determine the parameters by fitting to the experimental data for $Lambda$ binding energies. Our model well reproduces the data over a wide range of mass region although some of our interactions yield the reverse ordering of the spin-orbit partners from that of nucleons for heavy hypernuclei. The consistency of the interaction with the quark model predictions is also discussed.