# Excitation energy shift and size difference of low-energy levels in -shell hypernuclei

###### Abstract

We investigated structures of low-lying -orbit states in -shell hypernuclei () by applying microscopic cluster models for nuclear structure and a single-channel folding potential model for a particle. For systems, the size reduction of core nuclei is small, and the core polarization effect is regarded as a higher-order perturbation in the binding. The present calculation qualitatively describes the systematic trend of experimental data for excitation energy change from to , in systems. The energy change shows a clear correlation with the nuclear size difference between the ground and excited states. In and , the significant shrinkage of cluster structures occurs consistently with the prediction of other calculations.

## I Introduction

Owing to high-resolution -ray measurement experiments, spectra of low-lying states of various -shell hypernuclei have been revealed in these years Hashimoto:2006aw ; Tamura:2010zz ; Tamura:2013lwa . Measured energy spectra and electromagnectic transitions are useful information to know properties of -nucleon() interactions and also helpful to investigate impurity effects of a particle on nuclear systems. In order to theoretically study structures of -shell hypernuclei, various calculations have been performed with cluster models Motoba:1984ri ; motoba85 ; Yamada:1985qr ; Yu:1986ip ; Hiyama:1996gv ; Hiyama:1997ub ; Hiyama:1999me ; Hiyama:2000jd ; Hiyama:2002yj ; Hiyama:2006xv ; Hiyama:2010zzc ; Cravo:2002jv ; Suslov:2004ed ; Mohammad:2009zza ; Zhang:2012zzg ; Funaki:2014fba ; Funaki:2017asz , shell models Gal:1971gb ; Gal:1972gd ; Gal:1978jt ; Millener:2008zz ; Millener:2010zz ; Millener:2012zz , mean-field and beyond mean-field models Guleria:2011kk ; Vidana:2001rm ; Zhou:2007zze ; Win:2008vw ; Win:2010tq ; Lu:2011wy ; Mei:2014hya ; Mei:2015pca ; Mei:2016lce ; Schulze:2014oia , hyper antisymmetrized molecular dynamics (HAMD) model Isaka:2011kz ; Isaka:2015xda ; Homma:2015kia ; Isaka:2016apm ; Isaka:2017nuc , and nore-core shell model Wirth:2014apa , and so on.

Since a particle is free from Pauli blocking and - interactions are weaker than - interactions, the spin degree of freedom in hypernuclei more or less weakly couples with core nuclei in general. Therefore, the particle in hypernuclei can be regarded as an impurity of the nuclear system. Indeed, there are many theoretical works discussing impurity effects of on nuclear structures such as shrinkage effects on cluster structures Motoba:1984ri ; motoba85 ; Yu:1986ip ; Hiyama:1996gv ; Hiyama:1997ub ; Hiyama:1999me ; Hiyama:2002yj ; Hiyama:2006xv ; Hiyama:2010zzc ; Isaka:2015xda ; Homma:2015kia ; Isaka:2011zza ; Sakuda:1987pk ; Yamada:1984ii and effects on nuclear deformations Zhou:2007zze ; Win:2008vw ; Win:2010tq ; Lu:2011wy ; Isaka:2011aw ; Isaka:2011kz ; Isaka:2016apm ; Lu:2014wta . One of the famous phenomena is the shrinkage of , which has been theoretically predicted Motoba:1984ri ; motoba85 and later evidenced experimentally through the transition strength measurement Tanida:2000zs . The dynamical effects of on nuclear structures can be significant in the case that core nuclei are fragile systems such as weakly bound systems and shape softness (or coexistence) ones. However, except for such cases, dynamical change of nuclear structure (the core polarization) is expected to be minor in general because of the weaker - interactions and no Pauli blocking. In this context, there might be a chance to probe original properties of core nuclear structures by a particle perturvatively appended to the nuclear system.

Let us focus on energy spectra of -shell hypernuclei. The low-energy levels are understood as core excited states with a -orbit ( states). When we consider the particle as an impurity giving a perturbation to the core nuclear system, the first-order perturbation on the energy spectra, that is, change of excitation energies by the particle, comes from structure difference between the ground and excited states through the - interactions, whereas dynamical structure change gives second-order perturbation effects on the energy spectra. For excited states with structures much different from that of the ground state, the particle can give significant effect on energy spectra as discussed by Isaka et al. for Be isotopes Isaka:2015xda ; Homma:2015kia . In this concern, it is meaningful to look at excitation energy shifts, that is, excitation energy changes from to , in available data. For simplicity, we here ignore the intrinsic spin degree of freedom because spin dependence of the - interactions is weak. In the observed energy spectra of -, -, -, and - systems, one can see that the excitation energies for , , and are significantly raised by the particle in systems compared with those in systems. On the other hand, the situation is opposite in - systems. the is decreased by the particle. To systematically comprehend the energy spectra of -shell hypernuclei, it is worth to examine the excitation energy shifts and their link with the structure difference between the ground and excited states.

Precise data of spectroscopy in various hypernuclei are becoming available and they provide fascinating physics in nuclear many-body systems consisting of protons, neutrons, and s. Sophisticated calculations have been achieved mainly in light nuclei and greatly contributed to progress of physics of hypernuclei. Nevertheless, systematic studies for energy spectra of hypernuclei in a wide mass-number region are still limited compared with those for ordinary nuclei, for which various structure models have been developed and used for intensive and extensive studies. It is time to extend application of such structure models developed for ordinary nuclei to hypernuclei. To this end, it might be helpful to propose a handy and economical treatment of a particle and core polarization in hyper nuclei that can be applied to general structure models.

Our first aim, in this paper, is to investigate energy spectra of low-lying states in -shell hypernuclei. A particular attention is paid to the excitation energy shifts by the and their link with structures of core nuclei. The second aim is that we are to propose a handy treatment of the particle in hypernuclei and to check its phenomenological applicability. To describe detailed structures of the ground and excited states of core nuclei, we apply the generator coordinate method (GCM) GCM1 ; GCM2 of microscopic , , and cluster models for , , and , respectively, and that of extended and cluster models with the cluster breaking for , , and . For description of states in hypernuclei, a single -wave channel calculation with a folding potential model is performed. Namely, the -nucleus potentials are constructed by folding - interactions with the nuclear density calculated by the microscopic cluster models. As a core polarization effect, the core size reduction is taken into account in a simple way.

## Ii formulation

### ii.1 microscopic cluster model for core nuclei

Structures of core nuclei are calculated by the microscopic cluster models with the GCM using the Brink-Bloch cluster wave functions Brink66 . In the cluster GCM calculations, we superpose the microscopic , , and , , and wave functions for , , , , and .

For a system consisting of clusters ( is the number of clusters), the Brink-Bloch cluster wave function is given as

(1) |

where indicates the position parameter of the cluster, and indicate the coordinate and spin-isospin configuration of the th nucleon, is the antisymmetrizer of all nucleons, is the mass number, and is the mass number of the cluster. The -nucleon wave function for the cluster is written by the harmonic oscillator shell model wave function with the center shifted to the position . The intrinsic spin configurations of , , and clusters are , 1/2, and 0 states, respectively. The width parameter ( is the size parameter) of the harmonic oscillator is set to be a common value so that the center of mass (cm) motion can be removed exactly. In the present work, we use the same parameter fm as that used in Ref. Suhara:2014wua which reasonably reproduces the ground-state sizes of -shell nuclei. The Brink-Bloch cluster wave function is a fully microscopic -nucleon wave function, in which the degrees of freedom and antisymmetrization of nucleons are taken into account, differently from non-microscopic cluster models (simple -body potential models) and such semi-microscopic cluster models as the orthogonal condition model (OCM) Saito:1969zz .

To take into account inter-cluster motion, the GCM is performed with respect to the cluster center parameters . Namely, the GCM wave function for the state is expressed by linear combination of the spin-parity projected Brink-Bloch wave functions with various configurations of as

(2) |

where is the spin-parity projection operator. The coefficients are determined by diagonalization of the Hamiltonian and norm matrices. In the present calculation, for two-cluster systems of and , is chosen to be with fm}. For three-cluster systems of , , , is chosen to be

(3) | |||

(4) |

with fm}, fm}, .

In a long history of structure study of and , the and GCM calculations have been performed in many works since 1970’s (see Ref. Fujiwara-supp ; horiuchi-rev and references therein), and successfully described cluster structures except for the ground state of . For the ground state of C, the traditional models are not sufficient because the cluster breaking component, in particular, the -closed configuration is significantly mixed in it, and therefore, they usually fail to reproduce the - energy spacing and . Suhara and the author have proposed an extended cluster model by adding the -closed configuration in the GCM calculation, which we call the model Suhara:2014wua . In the present calculation of , we apply the model and take into account the cluster breaking component. We also apply the extended version, the model to by taking account the cluster breaking component by adding the configuration in the GCM calculation (the model to with configuration in the GCM calculation).

The nuclear density in the core nuclei is calculated for the obtained GCM wave function . The is the -dependent spherical density of the state after extraction of the cm motion.

### ii.2 Hamiltonian of nuclear part

Hamiltonian of the nuclear part consists of the kinetic term, effective nuclear interactions, and Coulomb interactions as follows,

(5) | |||||

(6) | |||||

(7) | |||||

(8) | |||||

(9) |

where is the kinetic term of the cm motion, and and are the effective - central and spin-orbit interactions. The energy of the core nucleus is given as (the nuclear energy). In the GCM calculation, the coefficients in (2) are determined so as to minimize .

### ii.3 Hamiltonian and folding potential of -nucleus system

states of hypernuclei are calculated with a folding potential model by solving the following single -wave channel problem within local density approximations,

(10) | |||||

(11) | |||||

(12) | |||||

(13) | |||||

(14) | |||||

(15) | |||||

(16) | |||||

(17) |

where , , and are defined with respect to the relative coordinate of the from the cm of the core nucleus. and are the even and odd parts of the effective - central interactions, respectively, where is the parameter for density dependence of the effective - interactions.

The nuclear density matrix in the exchange potential is approximated with the density matrix expansion (DME) using the LDA Negele:1975zz ,

(18) | |||

(19) | |||

(20) | |||

(21) |

To see ambiguity of choice of local density and Fermi momentum in the DME approximation we also used the second choice (LDA2),

(22) | |||||

(23) |

and found that the first and the second choices give qualitatively similar results. In this paper, we use the DME approximation with the first choice in the calculation of the exchange folding potential .

For a given nuclear density , the -core wave function and energy are calculated by solving the one-body potential problem with the Gaussian expansion method Kamimura:1988zz ; Hiyama:2003cu . The rms radius () measured from the core nucleus and the averaged nuclear density () for the distribution are calculated with the obtained -core wave function ,

(24) | |||||

(25) |

### ii.4 core polarization effect

We take into account the core polarization, which is the structure change of core nuclei caused by the impurity, the particle, in hypernuclei as follows. In the present folding potential model, the binding reflects the core nuclear structure only through the nuclear density . When the -orbit particle is regarded as an impurity of the nuclear system, the - interactions may act as an additional attraction to the nuclear system and make the nuclear size slightly small. To simulate the nuclear structure change induced by the -orbit , we add artificial nuclear interactions by slightly enhancing the central part by hand and perform the GCM calculation of the nuclear system for the modified Hamiltonian,

(26) |

with the additional term , where is the enhancement factor and taken to be . For the GCM wave function of the state obtained with , we calculate the nuclear energy and the nuclear density . Then we calculate the wave function () and energy () for the obtained -dependent nuclear density . Finally, we search for the optimum value so as to minimize the energy of the total system,

(27) | |||||

(28) |

The binding energy () is calculated as for the optimized value, where is the unperturbative nuclear energy without the particle.

We vary only the GCM coefficients for the fixed basis cluster wave functions corresponding to the inert cluster ansatz. In this assumption, the enhancement of the effective central nuclear interactions acts like an enhancement of the inter-cluster potentials.

## Iii Effective interactions

### iii.1 Effective nuclear interactions

As for the effective two-body nuclear interactions, we use the finite-range central interactions of the Volkov No.2 parametrization VOLKOV and the spin-orbit interactions of the G3RS parametrization LS ,

(29) | |||||

(30) | |||||

(31) | |||||

(32) | |||||

(33) | |||||

(34) | |||||

(35) |

where () is the spin(isospin) exchange operator, is the relative distance for the relative coordinate , is the angular momentum for , and is the sum of nucleon spins .

We use , , and for the central interactions, and MeV for the spin-orbit interactions. These parameters reproduce the deuteron binding energy, the - scattering phase shift, and properties of the ground and excited states of C Suhara:2014wua ; uegaki1 ; uegaki3 . For Li, we use a modified values , , , and MeV to reproduce the and energies relative to the threshold energy. Note that this modification gives no effect on -shell nuclei, , , , and .

### iii.2 Effective -nucleon interactions

For the effective - central interactions, we use the -matrix interactions derived from - interactions of the one-boson-exchange model, which we denote as the interactions Yamamoto:2010zzn ; Rijken:2010zzb . In this paper, we adopt the central part of the interactions with the ESC08a parametrization,

(36) | |||||

(37) | |||||

(38) | |||||

(39) |

with fm, fm, and fm. Values of the parameters are listed in Table 1. Note that, in the present -wave calculation, the effective - interactions are spin-independent central interactions, as the singlet and triplet parts are averaged with the factors 1/4 and 3/4, respectively, and the spin-orbit interactions are dropped off.

As for the parameter of the interactions, we adopt two treatments. One is the density-dependent interactions with , where is the averaged Fermi momentum for the particle,

(40) |

and self-consistently determined for each state. This choice of the interactions is the so-called “averaged density approximation (ADA)” used in Refs. Yamamoto:2010zzn ; Isaka:2016apm ; Isaka:2017nuc . The other is the density-independent interaction with a fixed value, . Here, the input parameter is chosen for each system. It means that the is system dependent but “state independent”. In this paper, we use the mean value of of low-energy states obtained by the former treatment (ADA) as the input of . These choices reasonably reproduce the binding energies of , , , , , and . We label the first treatment, density-dependent interactions with , as ESC08a(DD), and the second one, the density-independent interactions with ESC08a(DI). Note that the former is state-dependent (structure-dependent) and the latter is state-independent (structure-independent), but the system-dependent is used in both cases.

The interactions have been applied to various structure model calculations of hypernuclei such as cluster model, mean-field, and HAMD calculations. In the application of the interactions to cluster model and HAMD calculations, the parameter of the density-independent interactions is usually adjusted to fit the binding energy for each (sub)system. In applications of the effective interactions to in a wide mas number region, the density-dependent interactions have been used, for instance, in the mean-field calculations and recent HAMD calculations Isaka:2016apm ; Isaka:2017nuc , because they were originally designed in the density-dependent form to reproduce systematics of binding energy Yamamoto:2010zzn . In Refs. Yamamoto:2010zzn ; Isaka:2016apm , they also showed the results with another choice of in addition to the ADA results. In the present calculation for states, the results obtained with show similar results to the present ESC08a(DD) ones.

## Iv Results

By applying the , , , , and GCM to core nuclei, , , , , and , we calculate states in with the single-channel folding potential model by taking account the core polarization effect.

In the present calculation, the particle around the state of the core nucleus feels the spin-independent potentials, and therefore the spin partner states in completely degenerate. We denote the spin partner states in hypernuclei by . We calculate low-lying states with dominant configurations in , and the corresponding states in .

The and states, which are strictly speaking quasi-bound states, are calculated in the bound state approximation with the boundary condition fm of the GCM model space. The present GCM calculation gives stable results for these states. The state is a broad resonance state, for which we can not obtain a stable result in the bound state approximation. Instead, we calculate the excitation energy of from the scattering phase shifts with the resonating group method (RGM).

To see the effect of the cluster breaking component in and , we also show some results for and obtained by the traditional GCM calculation without the cluster breaking () component and compare them with those obtained by the present model. Note that, in the present model, the cluster breaking components contribute only to and but do not affect other spin-parity states.

### iv.1 Properties of core nuclei

cal | exp | cal | exp | cal | exp | |

0.43 | 2.224 | 0.98 | 1.941 | |||

6.9 | 8.481 | 1.23 | 1.504 | |||

27.6 | 28.296 | 1.55 | 1.410 | |||

29.5 | 31.995 | 1.48 | 1.48 | 2.56 | 2.426 | |

55.0 | 56.499 | 0.21 | 0.09 | 3.37 | ||

60.5 | 64.75 | 4.88 | 5.93 | 2.39 | 2.253 | |

71.8 | 76.203 | 9.66 | 11.13 | 2.33 | 2.229 | |

69.2 | 73.439 | 7.05 | 8.37 | 2.34 | ||

90.2 | 92.16 | 7.37 | 7.27 | 2.35 | 2.298 | |

w/o | 88.1 | 3.22 | 2.52 | |||

cal | exp | |||||

11.3 | 10.7(8) | |||||

5.2 | 4.15(2) | |||||

9.5 | 8.9(3.2) | |||||

7.3 | 7.6(4) | |||||

w/o | 10.6 | 7.6(4) |

(B.E) | ||||||
---|---|---|---|---|---|---|

cal | exp | |||||

2.33 | 2.60 | 2.764 | ||||

- | ||||||

cal | exp | cal | exp | |||

2.50 | 0.18 | 2.79 | 2.125 | 0.17 | 0.13 | |

2.58 | 0.26 | 5.57 | 5.020 | 0.22 | 0.22 | |

2.51 | 0.19 | 4.66 | 4.445 | 0.16 | 0.13 |

Nuclear properties of isolate core nuclei without the particle are shown in Tables 2 and 3. The calculated values of the binding energies , relative energies () measured from cluster break-up threshold energies, root-mean-square (rms) radii of nuclear matter (), and transition strengths to the ground states are listed compared with experimental data in Table 2. For the experimental data of nuclear radii, the rms radii of point-proton distribution () reduced from the charge radii are shown. We also show the results for , , and clusters of configurations with fm. The energies and sizes are reasonably reproduced by the calculation except for the deuteron and triton. The deuteron size is much underestimated, because the fixed-width configuration is assumed in the present cluster model. The calculated are in agreement with the experimental data without using any effective charges. For C, the GCM calculation without the cluster breaking () gives a larger size and than those of the present calculation, meaning that slightly shrinks because of the cluster breaking effect as discussed in Ref. Suhara:2014wua .

In Table 3, we show the Coulomb shift , which is defined by the excitation energy difference between mirror nuclei, for nuclei together with calculated radii. Since the Coulomb interactions give only minor change of nuclear structure, and therefore the Coulomb shift sensitively probes the size difference between the ground and excited states except for weakly bound or resonance states. The calculated Coulomb shifts for agree well with the experimental data indicating that the size differences of these states are reasonably described by the present calculation.

### iv.2 Ground states of hypernuclei

We here discuss the ground state properties of . In , the nuclear size ( is the rms nuclear matter radius measured from the cm of the core nucleus) slightly decreases and the nuclear energy slightly increases from the original size () and energy () of unperturbative core nuclei without the . We calculate the nuclear size change and the nuclear energy change caused by the particle in . To see the core polarization effect, we also calculate the energy gain, , defined by the energy difference between the calculations with and without the core polarization. Here is the energy without the core polarization, that is the energy in the -( system with the unperturbative core nucleus.

In Table 4, we show the calculated results of the ground state properties of together with the experimental . As reference data, we also show the results for obtained by the - calculation with the inert core assumption. Systematics of binding energies in this mass-number region is reasonably reproduced in both ESC08a(DI) and ESC08a(DD) interactions, though the reproduction is not perfect.

For systems, the nuclear size change is less than 5%. The small size change of the core nucleus in the ground state of is consistent with the prediction of other calculations Motoba:1984ri ; motoba85 ; Hiyama:1997ub ; Hiyama:2010zzc ; Funaki:2017asz . Moreover, the nuclear energy change and energy gain by the core polarization are also small and compensate each other. It indicates that the core polarization effect is minor and regarded as a higher-order perturbation in the binding except for systems. The core polarization effects in the ESC08a(DD) results for systems are particularly small, because the ESC08a(DD) interactions become weak as the nuclear density increases because of the dependence.

As explained previously, the core polarization effect is taken into account by changing the enhancement factor , which can be regarded as a control parameter of the nuclear size . In Fig. 1, we show the nuclear size dependence of , , and in obtained by varying the enhancement factor . The energies are plotted as functions of the nuclear size . The -dependence of is also shown. In the ESC08a(DI) result, the energy () gradually goes down with the nuclear size reduction because the higher nuclear density gives larger attraction to the potentials. As a result, the particle slightly reduces the core nuclear size. In contrast, in the ESC08a(DD) result, the energy has almost no dependence on the nuclear size, because the density-dependence of the interactions compensates the energy gain in the higher nuclear density. As a result, the particle hardly changes the core nucleus size. Namely, the density dependence of the ESC08a(DD) interactions suppresses the size reduction of the core nuclei.

Let us turn to systems, and . Differently from systems, rather significant core size reduction occurs, because and have spatially developed - and -cluster structures, respectively, and they are rather fragile (soft) against the size reduction. This is consistent with the size shrinkage predicted by pioneering works in Refs. Motoba:1984ri ; motoba85 followed by many works (see a review paper Hiyama:2010zzc and references therein). Particularly remarkable core polarization effects are found in , because is a very fragile system of the loosely bound (strictly speaking, quasi-bound) state. The core polarization effects are seen in the nuclear size change and also the energy changes, and , in both ESC08a(DI) and ESC08a(DD) calculations. For , the core size reduction is 13% in the ESC08a(DI) result, whereas it is 6% in the ESC08a(DD) result. It should be commented that the size reduction discussed here is the reduction of nuclear matter radii of core nuclei. Detailed discussions of the shrinkage of the inter-cluster distance in and are given later.

ESC08a(DI) | ||||||||||

0.95 | 2.84 | 0.95 | 1.55 | 3.6 | 3.12(2) | 3.12(2) | ||||

0.93 | 2.57 | 0.95 | 2.22 | 0.59 | 5.4 | 5.58(3) | 5.12(3) | |||

0.90 | 2.44 | 0.98 | 2.44 | 1.69 | 7.0 | 6.71(4) | 6.71(4) | |||

1.03 | 2.36 | 1.10 | 2.29 | 0.37 | 10.0 | 10.24(5) | 10.09(5) | |||

1.07 | 2.33 | 1.16 | 2.24 | 0.29 | 10.9 | 11.37(6) | 11.27(6) | |||

1.06 | 2.32 | 1.16 | 2.25 | 0.31 | 11.1 | 10.76(19) | 10.65(19) | |||

1.11 | 2.35 | 1.18 | 2.26 | 0.27 | 11.1 | 11.69(12) | 11.69(12) | |||

w/o | 1.11 | 2.45 | 1.11 | 2.41 | 0.35 | 9.9 | 11.69(12) | 11.69(12) | ||

ESC08a(DD) | ||||||||||

2.83 | 0.95 | 1.55 | 3.6 | 3.12(2) | 3.12(2) | |||||

2.66 | 0.91 | 2.40 | 0.08 | 5.4 | 5.58(3) | 5.12(3) | ||||

2.67 | 0.90 | 2.69 | 0.44 | 6.4 | 6.71(4) | 6.71(4) | ||||

2.48 | 1.06 | 2.38 | 0.00 | 9.0 | 10.24(5) | 10.09(5) | ||||

2.45 | 1.11 | 2.33 | 0.00 | 9.6 | 11.37(6) | 11.27(6) | ||||

2.45 | 1.11 | 2.34 | 0.00 | 9.6 | 10.76(19) | 10.65(19) | ||||

2.44 | 1.13 | 2.35 | 0.00 | 10.1 | 11.69(12) | 11.69(12) | ||||

w/o | 2.47 | 1.08 | 2.51 | 0.01 | 10.1 | 11.69(12) | 11.69(12) |