Effective Nucleon-nucleon Interaction Research Articles

Pikoe: A computer program for distorted-wave impulse approximation calculation for proton induced nucleon knockout reactions

pikoe, Proton-Induced KnockOut reaction calculation for Exclusive processes, is a Fortran 90 program that calculates triple- and quadruple-differential cross sections, vector analyzing powers, and momentum distributions of reaction residues, for proton-induced nucleon knockout reactions in normal and inverse kinematics. The distorted-wave impulse approximation with the factorization approximation to the nucleon-nucleon transition matrix is adopted, and the distorted waves of the incoming proton and the outgoing two nucleons are calculated quantum mechanically. Kinematics of the reaction particles are treated in a relativistic manner, which gives the proper asymptotics of the three-body scattering wave in the plane-wave limit. Program summaryProgram Title: pikoeCPC Library link to program files:https://doi.org/10.17632/m594h58kck.1Licensing provisions: MITProgramming language: Fortran 90Supplementary material: See https://www.rcnp.osaka-u.ac.jp/~kazuyuki/pikoe/index.phpNature of problem: Calculation of triple- and quadruple-differential cross sections, vector analyzing powers, and momentum distributions of reaction residues, for nucleon knockout reactions with proton beams/targets in normal/inverse kinematics.Solution method: Numerov method for solving single-channel second-order differential equations; multi dimensional integration with the Gauss-Legendre quadrature rule.Additional comments including restrictions and unusual features: The global optical potential by Koning and Deraloche [1] for nucleons at energies from 1 keV to 200 MeV, for (near-)spherical nuclei with mass number between 24 and 209, is implemented in the program. Tables of nucleon-nucleon (NN) elastic differential cross sections and NN transition amplitudes calculated with the NN effective interaction by Franey and Love [2] are provided as external files for pikoe (included in the package). Users are encouraged to prepare tables for the optical potentials and the cross sections of the elementary process according to the reaction system of interest by themselves. With external files thus prepared, pikoe can be applied to various kinds of knockout processes such as α or deuteron knockout by proton, α-induced knockout reactions, etc. Calculation of spin observables, the vector analyzing power in fact, however, is restricted to nucleon-induced nucleon knockout processes with the coplanar kinematics.

APPLICATION OF THE NEW B3Y-FETAL POTENTIAL IN THE SEMI-MICROSCOPIC ANALYSIS OF THE SCATTERING OF ACCELERATED 6LI - LITHIUM AND 16O - OXYGEN NUCLEI FROM THE 12C - CARBON NUCLEUS

The phenomenological and semi-microscopic values of the potentials found in the angular distribution of scattering of ions 6Li -lithium, 16О -oxygen from the target nucleus 12C-carbon accelerated at low energies are determined.The experimental data of elastic scattering were analyzed based on the optical model and the doublefolding model.Based on the folding model, the density-dependent new Fetal potential of Yukawa 3 terms of effective nucleon-nucleon interactions was first used. The density-dependent Fetal, Paris, Reid microfolding potentials were constructed in the doublefolding model as a real part of the optical potential. The efficacy of the new Fetal micropotential at laboratory energies of 28 MeV and 30 Mev for the 16О+12С nuclear system was studied in comparison with the Reid, Paris variants. The efficacy of the new Fetal potential at laboratory energies of 12.3 MeV and 20 MeV for the 6Li+12Csystem has been studied in comparison with Reid and Paris potentials. The relative errors of phenomenological theoretical analysis and experimental cross-sections were determined in the range of 1.1 -3.0. As a result of semimicroscopic analysis, the coefficients of renormalization of folding potentials in the range Nr=0.8-1.0 were determined. The data obtained will be used in various fundamental research, in particular in future thermonuclear installations and nuclear astrophysics

Impact of different components of the Skyrme nucleon–nucleon effective interaction on the nuclear density distribution

We systematically investigate the impact of the different terms of the Skyrme energy density functional of the effective nucleon-nucleon interaction, and of its associated nuclear matter (NM) properties, on the density distributions of spherical nuclei. Twenty five Skyrme force parameterizations are examined simultaneously, covering a broad range of each characteristic parameter and NM property. The diffuseness and the neutron-skin thickness are found to be the most sensitive density quantities to the force parameterization. The diffuseness is indicated to decrease with increasing the central zero-range and the effective mass terms of the effective force, and the power σ of its density dependent term, as well as with the coefficient of the NM symmetry energy (asym) and its density slope (L) at saturation density, and the incompressibility (Ko). In contrast, the proton and neutron diffuseness tend to increase with increasing the spin–orbit force and the isoscalar effective nucleon-mass (m*), and to increases slightly with the density dependence parameters other than the power σ. Opposite impacts are pointed out for the different parts of the finite-range, and J2 tensor terms on the proton and neutron density. While the neutron-skin thickness tends to increase significantly upon increasing the central zero-range and spin–orbit force terms, asym, L, and Ko, and to increase slightly with the finite-range and J2 tensor terms, and σ, it decreases with the effective-mass term, the density-dependence exchange parameter, and with the indicated isoscalar effective mass. The proton and neutron radii exhibit decreasing behavior with the central zero-range and the spin–orbit terms, and with Ko, and m*. Increasing asym and L indicate slightly less (larger) proton (neutron) radius.

Analysis of α + 16O elastic scattering in the semi-microscopical dispersive optical model

We study the elastic scattering cross sections of the α +16O system at 49.5, 69.5, and 80.7 MeV using the optical potentials calculated in the framework of the semi-microscopical dispersive optical model. In this model, the optical potential contains an energy-independent part calculated in the double folding model with a realistic effective nucleon-nucleon interaction, as well as an energy-dependent dynamic polarization potential, the parameters of which are adjusted to the data. The parameters of the real and imaginary parts of the dynamic polarization potential are connected by the dispersion relations.

Semi-microscopic optical-model with channel coupling analysis to describe neutron-nucleus scattering of light and heavy nuclei

Many studies utilizing various potential models have been conducted on the scattering of nucleons from nuclei. In our study, we added channel coupling to a semi-microscopic optical model. With incident energies between 10 and 30 MeV, we sought to compare the predicted reaction observables for the neutron scattering of the following nuclei: 12C, 16O, 54Fe, 58Ni, 120Sn, and 208Pb. We did this twice, once using the density-dependent M3Y-Reid nucleon-nucleon effective bare interaction and once using the density-independent M3Y-Reid effective bare interaction. The remaining terms of the optical potential model take the form of Woods-Saxon and its derivatives, and the real volume term is obtained by folding the NN effective bare potential over the density of the nuclear target using the density-dependent M3Y-Reid NN effective bare interaction and density-independent M3Y-Reid NN effective bare interaction. To investigate how the potential characteristics depend on energy, the ground state is connected to a select few low-lying collective excited states. The density-dependent M3Y-Reid NN effective bare interaction-based semi-microscopic model reproduces the reaction features and forecasts the experimental total cross sections. The parameters of potential depth exhibit a linear energy dependence. We contrast the outcomes of the M3Y-Reid density-dependent model with those attained using the M3Y-Reid nucleon-nucleon effective bare interaction, which is density-independent. The M3Y Reid density-dependent model performs well when the two models are compared.

Shell evolution and emerging paradigm changes

The shell structure conceived by Mayer and Jensen in 1949 has been shown to be quite appropriate for stable and near-stable nuclei, but substantial deviations from it have also been observed more recently for exotic nuclei with notable neutron excess. Such changes of the basic picture of the nuclear shell structure, called the shell evolution, seem to be the subject studied most extensively and most elaborately by RI beam experiments worldwide. An overview of the shell evolution is presented in this talk, from both theoretical and experimental perspectives. The shell structure is shown to be varied, for instance, from the one presented by Mayer and Jensen, by particular types of the monopole components of the effective nucleon-nucleon interaction in nuclei. Among various contributions, the importance of the tensor force is illuminated here, with an outstanding example: the emergence of new neutron magic number 34 in neutron-rich Ca isotopes. The mechanism of the shell evolution produces significant impacts also on the nuclear shapes. Type II shell evolution shifts the excitation energies of intruder deformed bands, for instance, in some Ni isotopes. In other more general cases, the monopole interaction is shown to produce unexpected crucial effects on the patters of rotational bands of heavy deformed nuclei. In fact, the shape of the ellipsoidal deformation is investigated by large-scale shell model calculations, which is nothing but the Monte Carlo Shell Model. The unique role of the monopole component of the tensor force is clarified: the interplay between this monopole interaction and the quadrupole interaction provides us with various patterns of triaxial shapes for many nuclei, such as 166Er, one of the traditional prolate deformed heavy nuclei. Thus, the prolate preponderance hypothesis by Aage Bohr is investigated for its microscopic validity. Some of the nuclear paradigms are changing now in this way, due to emerging aspects of nuclear-force effects.

Level structures of 56,58Ca cast doubt on a doubly magic 60Ca

Gamma decays were observed in 56Ca and 58Ca following quasi-free one-proton knockout reactions from 57,59Sc beams at ≈200 MeV/nucleon. For 56Ca, a γ ray transition was measured to be 1456(12) keV, while for 58Ca an indication for a transition was observed at 1115(34) keV. Both transitions were tentatively assigned as the 21+→0gs+ decays, and were compared to results from ab initio and conventional shell-model approaches. A shell-model calculation in a wide model space with a marginally modified effective nucleon-nucleon interaction depicts excellent agreement with experiment for 21+ level energies, two-neutron separation energies, and reaction cross sections, corroborating the formation of a new nuclear shell above the N = 34 shell. Its constituents, the 0f5/2 and 0g9/2 orbitals, are almost degenerate. This degeneracy precludes the possibility for a doubly magic 60Ca and potentially drives the dripline of Ca isotopes to 70Ca or even beyond.

Systematic study of the relationship between nuclear structure and reactions for the 10Be nucleus

Abstract We systematically investigate the relationship between the nuclear structure and reaction in the 10Be nucleus using a theoretical framework. The structure of the 10Be nucleus is constructed with a cluster model based on a microscopic viewpoint. In this paper, the 10Be nucleus is prepared with different structures by manipulating the parameters of an effective nucleon–nucleon interaction. The nuclear structure and expectation values of physical quantities are drastically changed by the modification. We summarize such changes and show the effects on the elastic and inelastic scatterings for proton and 12C targets in the microscopic coupled-channel calculation. Of especial interest, we recently reported the visualization of the dineutron correlation in 10Be on proton inelastic scattering in [Phys. Rev. C 104, 034613 (2021)]. In this preceding work, we found that changing the degree of dineutron correlation in 10Be leads to drastic changes of the inelastic cross section for the 2$_2^+$ state. The development (or breaking) of the dineutron correlation is governed by the strength of the spin–orbit interaction of the structure calculation. However, in the previous work, some of the realistic physical points were missing, for example, the binding energy (BE). Therefore, we reconstruct the 10Be nucleus by adjusting the effective nucleon–nucleon interaction to obtain a reasonable BE of the ground state. With this improvement, we again discuss the dineutron correlation in the 10Be nucleus. We reconfirm the way to measure the degree of development (or breaking) of the dineutron cluster structure: its sensitivity to the inelastic cross section of the 2$_2^+$ state of 10Be. Subject Index cluster model, folding model, dineutron correlation, effective nucleon-nucleon interaction.

Chiral uncertainties in ab initio elastic nucleon-nucleus scattering

The effective interaction between a nucleon and a nucleus is one of the most important ingredients for reaction theories. Theoretical formulations were introduced early by Feshbach and Watson, and efforts of deriving and computing those ‘optical potentials’ in a microscopic fashion have a long tradition. However, only recently the leading order term in the Watson multiple scattering approach could be calculated fully ab initio, meaning that the same nucleon-nucleon (NN) interaction enters both the structure as well as the reaction pieces on equal footing. This allows the uncertainties from the underlying chiral effective NN interaction to be systematically explored in nucleon-nucleus elastic scattering observables. In this contribution the main ingredients for arriving at the ab initio leading order of the effective nucleon-nucleus interaction in the Watson approach will be reviewed. Concentrating on one specific chiral NN interaction from the LENPIC collaboration and light nuclei with a 0+ ground state, the leading order nucleon-nucleus interaction is calculated using up to the third chiral order (N2LO) in the nucleon-nucleon potential, and elastic scattering observables are extracted. Then pointwise as well as correlated uncertainty quantification is used for the estimation of the chiral truncation error. Elastic scattering observables for 4He, 12C, and 16O for between 65 and 200 MeV projectile energy will be analyzed.

Importance of deuteron breakup in the deuteron knockout reaction

Background: An isoscalar $pn$ pair is expected to emerge in nuclei that have similar proton and neutron numbers and it may be a candidate for a deuteron ``cluster.'' There is, however, no experimental evidence for it.Purpose: The purpose of this paper is to construct a new reaction model for the $(p,pd)$ reaction including the deuteron breakup in the elementary process and the deuteron reformation by the final-state interactions. How these processes contribute to the observables of the reaction is investigated.Methods: The distorted wave impulse approximation is extended twofold. The elementary processes of the $(p,pd)$, i.e., the $p\text{\ensuremath{-}}d$ elastic scattering and the $d(p,p)pn$ reaction, are described with an impulse picture employing a nucleon-nucleon effective interaction. The three-body scattering waves in the final state of the $(p,pd)$ reaction are calculated with the continuum-discretized coupled-channels method. The triple-differential cross section (TDX) of the $(p,pd)$ reaction is calculated with the new model.Results: The elementary processes are described reasonably well with the present model. As for the $(p,pd)$ reaction, the deuteron reformation can either increase or decrease the TDX height depending on the interference between the elastic and breakup channels of deuteron, while the back-coupling effect always decreases it.Conclusions: It is shown that the deuteron reformation significantly changes the TDX of the $(p,pd)$ reaction through the interference. It is important to include this process to quantitatively discuss the $(p,pd)$ cross sections in view of the deuteron formation in nuclei. For more quantitative discussion regarding the experimental data, further improvement will be necessary.

Structure and α decay for the neutron-deficient nuclei with 89≤Z≤94 in the density-dependent cluster model combined with a relativistic mean-field approach

The density-dependent cluster model combined with relativistic mean-field theory is used to explore the structure and $\ensuremath{\alpha}$ decay for the neutron-deficient nuclei with $89\ensuremath{\le}Z\ensuremath{\le}94$, including two newly discovered nuclei $^{207}\mathrm{Th}$ [Phys. Rev. C 105, L051302 (2022)] and $^{214}\mathrm{U}$ [Phys. Rev. Lett. 126, 152502 (2021)]. The effective nucleon-nucleon interactions and matter density distributions from the relativistic mean field are employed to construct the $\ensuremath{\alpha}$-daughter potential with a double-folding model. The Pauli blocking effect is considered by normalizing the strength of the $\ensuremath{\alpha}$-daughter potential with the Bohr-Sommerfeld quantization condition. The $\ensuremath{\alpha}$-preformation factor is calculated with the cluster formation model. The calculated $\ensuremath{\alpha}$-decay half-lives for the 106 observed nuclei with $89\ensuremath{\le}Z\ensuremath{\le}94$ are in excellent agreement with experimental data. Extending this model to the unknown nuclei $^{201--204}\mathrm{Ac}$, $^{205--206}\mathrm{Th}$, $^{209--210}\mathrm{Pa}$, $^{212--213,220}\mathrm{U}$, $^{215--218,221}\mathrm{Np}$, and $^{220--227}\mathrm{Pu}$, the evolution of the $N=126$ shell closure with neutron number is explored for the high-$Z$ isotopes. The available $\ensuremath{\alpha}$-decay energies, preformation factors, and $\ensuremath{\alpha}$-decay half-lives show a regular change with increasing neutron number. Especially, the robustness of the $N=126$ shell closure is shown up to the Pu isotopes.

Be10 -nucleus optical potentials developed from chiral effective field theory NN interactions

We present a determination of optical potentials for $^{10}\mathrm{Be}$-nucleus collisions using the double-folding method to compute the real part and Kramers-Kronig dispersion relations to derive the imaginary part. As microscopic inputs we use chiral effective field theory nucleon-nucleon interactions at next-to-next-to-leading order combined with state-of-the-art nucleonic densities. With these potentials, we compute elastic-scattering cross sections for the exotic nucleus $^{10}\mathrm{Be}$ off various targets, and compare them to experiment. Without any fitting parameter, we obtain good agreement with data. For collisions on light targets, we observe significant uncertainty related to the short-range physics, whereas for heavy targets that uncertainty remains small.

A semi-microscopic approach to transfer reactions

We develop a semi-microscopic model to describe transfer reactions. For the sake of simplicity, we focus on nucleon-transfer reactions, but the theory can be generalized to other transfer processes. The model makes use of overlap integrals computed in the microscopic Resonating Group Method (RGM). This technique is based on an effective nucleon-nucleon interaction and on the cluster approximation. It avoids the choice of a nucleon-core potential for the residual nucleus, and does not require any fit of spectroscopic factors. The model is therefore free of parameter. For the entrance channel, we use the Continuum Discretized Coupled Channel (CDCC) method where the breakup of the projectile is simulated by approximations of the continuum. This technique is well known for elastic scattering and for breakup reactions. It is well adapted to weakly bound nuclei such as the deuteron. The model is applied to the \(^{14}\mathrm{C}(d,p)^{15}C\) and \(^{6}\mathrm{He}(d,n)^{7}Li\) reactions with RGM wave functions of \(^{15}\)C and of \(^7\)Li. The \(^{15}\)C nucleus is described by \(^{14}\mathrm{C}(0^+,2^+)+n\) configurations whereas \(^7\)Li contains the \(\alpha +t\), \(^{6}\mathrm{He}(0^+)+p\) and \(^{6}\mathrm{Li}(1^+,0^+)+n\) configurations. The spectroscopic factors are in fair agreement with the literature. We show that the model provides an excellent description of the transfer cross sections considering that no parameter is adjusted.

Medium-dependent relativistic NN potential: Application to fusion dynamics

In-medium effects are introduced in the microscopic description of the effective nucleon-nucleon (NN) interaction potential entitled DDR3Y in terms of the density-dependent nucleon-meson couplings within the Relativistic-Hartree-Bogoliubov (RHB) approach. The nuclear densities of the interacting target and projectile nuclei and NN potentials are obtained for non-linear NL3$^*$ and TM1 parameter sets within the relativistic mean-field approach and density-dependent DDME1 and DDME2 parameter sets within the Relativistic-Hartree-Bogoliubov (RHB) formalism. The DDR3Y NN potential and the densities are used to obtain the nuclear potential by adopting the double folding approach. This nuclear potential is further used to probe the fusion dynamics within the $\ell-$summed Wong model for a few {\it even-even} systems leading to the formation of light, heavy and superheavy nuclei. The calculations are also performed for the relativistic R3Y, density-dependent and independent M3Y interaction potentials for the comparison. We observed that the DDR3Y NN potential gives a better overlap with the experimental data as compared to non-relativistic M3Y and DDM3Y NN potentials. From the comparison of R3Y and DDR3Y interactions, it is manifested that the inclusion of in-medium effects in terms of density-dependent nucleon-meson couplings raises the fusion barrier and consequently decreases the fusion and/or capture cross-section. Moreover, the nuclear densities, as well as the relativistic R3Y NN potential obtained for the NL3$^*$ parameter set, are observed to give a comparatively better fit to the experimental data.

INVESTIGATION OF THEENERGY DEPENDENCE OF THE INTERACTION POTENTIALS OF THE 16O+12С NUCLEAR SYSTEM WITH A SEMI-MICROSCOPIC METHOD

The study of the collision of heavy ions with light nuclei at low energy is important in nuclear physics, thermonuclear energy,and astrophysics. The high-precision values of the nuclear system described at an energy close to the Coulomb barrier are used to control the synthesis of light nuclei inside thermosynthesis. For cross sections of reactions of light nuclei on the sun, plasma,and stars, we can use the parameters we have set. The article presents a microscopic approach to describing the process of nuclear-nuclear interaction.In the phenomenological approach, which determines empirical values based on comparison with experimental data, it is possible to find many sets of parameters with potentials that characterize the cross-section well. But the question arises which of them are real.Therefore, it is necessary to additionally describe microscopic potentials. For the same reason, a semi-microscopic analysis was carried out, which describes the imaginary part of the nuclear potential on the basis of an optical model, and the real part on a double-folding model.The folding potential is constructed depending on the effective nucleon-nucleon interaction based on the matrix element of two nucleons and the density of the nucleon distribution.As a result of the analysis, differential cross sections and optimal parameters were determined that well characterize the experimental cross sections of the 16О+12C nuclear system at energies ELab=20, 24, 36 MeV. The coefficients of normalization of differential cross sections, described on the basis of real microfolding potentials, were determined in the range N=0.85-1.0.

Optical model and coupled reaction channels analyses of the 9Be+12C elastic scattering

This work analyzes the elastic scattering angular distribution data of the 9Be+12C system in the energy range of 13–50 MeV using the optical model (OM). The real and imaginary parts of the optical potential (OP) are generated using the double-folding (DF) model. In the DF model, three different effective nucleon-nucleon (NN) interactions are used. These effective NN interactions are the density-dependent CDM3Y6, the latest São Paulo Potential version 2 (SPP2), and the Brazilian Nuclear Potential (BNP). In the DF calculations, the renormalization factors NR and NI of real and imaginary parts of the OP were allowed to freely vary. We considered the elastic transfer of the 3He-cluster in our calculations using the coupled reaction channels (CRC) method. We extracted the optimum values of the spectroscopic amplitudes (SA) for the dissociation 12C→9Be+3He during the CRC analysis. The energy dependencies of real and imaginary potential depths, potential volume integrals, and reaction cross sections have been studied.

Impact of tensor force on quantum shell effects in quasifission reactions

Quantum shell effects drive many aspects of many-body quantal systems and their interactions. Among these are the quasifission reactions that impede the formation of a compound nucleus in superheavy element (SHE) searches. Fragment production in quasifission is influenced by shell effects as a nontrivial manifestation of microscopic dynamics hindering the full equilibration of the composite system to form the compound nucleus. In this Letter, we use the microscopic time-dependent Hartree-Fock (TDHF) theory to study 48Ca+249Bk collisions to investigate the influence of the tensor component of the effective nucleon-nucleon interaction. The results show that the inclusion of the tensor force causes the spherical shell effect to become more prominent, particularly for the neutron number yield whose peak is exactly at magic number N=126. This suggests that the tensor force plays a compelling role in the evolution of dynamical shell effects in nuclear reactions, influencing the competition between spherical and deformed shell gaps.

Fusion cross section of α -induced reactions for heavy target nuclei

The fusion reactions induced by $^{4}\mathrm{He}$ projectiles on heavy target nuclei provide the reference to study the reaction dynamics of exotic halo nuclei. The present analysis aims to probe the effects of different nuclear density distributions and the effective nucleon-nucleon ($NN$) interaction potentials on the fusion cross section of $^{4}\mathrm{He}$-induced reactions for heavy target nuclei. The nuclear densities for the heavy nuclei are obtained within the well-known Skyrme-Hartree-Fock model and the relativistic mean-field (RMF) approach for the $\mathrm{NL}{3}^{*}$ parameter set. These densities are integrated with the density-dependent M3Y and the relativistic R3Y nucleon-nucleon ($NN$) potentials to obtain the nuclear interaction potential. The fusion cross section and the astrophysical $S$ factor are obtained for $^{4}\mathrm{He}+^{208}\mathrm{Pb}, ^{4}\mathrm{He}+^{209}\mathrm{Bi}, ^{4}\mathrm{He}+^{235}\mathrm{U}$, and $^{4}\mathrm{He}+^{238}\mathrm{U}$ systems within the $\ensuremath{\ell}$-summed Wong model. The relativistic R3Y $NN$ potential folded with RMF densities are observed to give a better fit to the available experimental data. Furthermore, a comparison is made for the results obtained using Hill-Wheeler and Wentzel-Kramers-Brillouin transmission coefficients for $^{4}\mathrm{He}+^{208}\mathrm{Pb}$ system. We found a reasonable fit in the cross section at around the Coulomb-barrier energies for both the approximations and also similar predictions can be drawn for all considered systems.

Microscopic spin orbit analysis for proton+9Be scattering

microscopic folded potentials for both the real central and the spin-orbit (SO). For the imaginary central, we used surface Woods-Saxon (WS) potential. We aimed to test the microscopic SO potential based on the M3Y effective nucleon-nucleon (NN) interaction for the light system p+$^9$Be. The present calculation showed that the microscopic SO potential is satisfactory reproduce $A_y$ above 8 MeV and qualitatively reproduced $A_y$ below 8 MeV. In addition, we found that the calculated real central potentials are successfully reproduced the $d\sigma/d\Omega$ for all the considered energies. From the present analysis, we excepted that the present microscopic SO potential could reproduce successfully the $A_y$ for p+nucleus as the incident proton energy increases above 10 MeV.

Charge radii of Ca isotopes and correlations

We study the effects of short- and long-range correlations on the charge radii of Ca isotopes. We start our investigation with an independent particle model consisting in Hartree-Fock plus Bardeen-Cooper-Schrieffer calculations with finite-range effective nucleon-nucleon interactions of Gogny type. The short-range correlations effects are evaluated by considering all the terms of a cluster expansion containing a single correlation line. The long-range correlations are taken into account by including the coupling with the quasi-particle random phase approximation phonons. While the effects of the short-range correlations are negligible, those of the long-range correlations largely modify the independent particle model results and improve the agreement with the experimental data.