Prediction of reaction cross section for pCr

Nuclear matter densities and charge radii for even 46−62Cr isotopes have been calculated using relativistic Hartree-Bogoliubov model based on density-dependent meson-exchange relativistic energy density functional. The calculated root-mean-square charge radii agree well with corresponding data and the kink at N=28 is reproduced by the calculation. The calculated target matter densities are folded with the Jeukenne, Lejeune, and Mahaux-Bruyéres inter-nucleon interaction to obtain semi-microscopic optical model potentials for incident protons of 65 MeV on even 46−62Cr isotopes. The elastic scattering differential cross sections calculated using the proton optical potentials reproduce corresponding data for stable isotopes. The optical model potential parameters required for prediction of differential and total reaction cross sections for unstable even isotopes have been obtained.


Introduction
The structure and reactions of nuclei close to drip lines and their role in the understanding of nuclear ground state properties has been a topic of experimental and theoretical interest.The total reaction cross section (σ R ) is an important observable and plays a crucial role in both optical and statistical model calculations.A definite correlation exists between nuclear charge radii ( r c 2 1/2 ) and σ R .Knowing r c 2 1/2 , this correlation can be employed to make predictions of σ R for nuclei, especially those that lie far from the stability line [1,2].Limited experimental information about both r c 2 1/2 [3] and σ R [4] for neutron-deficient and neutron-rich nuclei in the region Z=20 − 28 are available, with most studies having been performed on stable nuclei.If the structure of nuclei involved in the reaction is accurately known, the reactions can be analyzed with scattering theory.The elastic scattering calculation using folding optical model has been known to be successful when radial matter densities are obtained from well-established structure models (relativistic or nonrelativistic) and microscopic, spherical nucleon-nucleus optical model potentials based on different approximations or effective forces are robust [5,6].For the predictions to hold, there must be no or few adjustable parameters in the model.Such an analysis would be able to predict the reaction observables successfully.In the present work, nuclear ground-state properties have been calculated for even 46−62 Cr isotopes.The σ R has been predicted by studying the p-Cr elastic scattering using folding optical model.
The nuclear ground state properties are calculated in the framework of relativistic Hartree-Bogoliubov (RHB) model [7] based on density-dependent meson-exchange a e-mail: hema@cbs.ac.in (DD-ME2) relativistic energy density functional [8] for even Cr isotopes.The ground state properties are compared to the corresponding available data.The target radial matter densities calculated in RHB framework, have been used in the semi-microscopic optical model to obtain the proton optical potentials for even Cr isotopes.The Jeukenne-Lejeune-Mahaux-Bruyeŕes (JLMB) energy-and density-dependent nucleon-nucleon interaction [9] are folded with the target radial matter densities.The resulting real and imaginary parts of the folded optical potential are used to compute the differential and reaction cross sections for 65 MeV-proton elastic scattering off even 46−62 Cr isotopes.
The RHB model and ingredients required for calculation of nuclear ground state properties as well as results of calculation for even Cr isotopes are included in Sec. 2. A brief description of semi-microscopic folded optical model are presented in Sec. 3. The results of calculation of cross section observables for (p , p) scattering at 65 MeV off even Cr isotopes are also given in the same section.

Relativistic Hartree-Bogoliubov model
The RHB model based on self-consistent mean-field and relativistic (covariant) energy density functionals has been effective in explaining nuclear structure over a range of isotopes: stable to drip-line [10,11].In RHB model, an effective Lagrangian describes the nucleus as a system of Dirac nucleons which interact by means of electromagnetic fields and meson (isoscalar scalar σ, isoscalar vector ω, and isovector vector ρ) exchange.In the meanfield approximation, meson field operators are replaced by their expectation values in the nuclear ground state.A medium dependence required for quantitative treatment of nuclear matter and finite nuclei, has been introduced by assuming a density-dependence for the meson-nucleon couplings.The relativistic density-dependent energy density functional (DD-ME2) [8] that provides a good description has been considered here.The energy density functional parameters for the DD-ME2 set have been obtained by treating the pairing correlations in the Bardeen-Cooper-Schrieffer (BCS) constant-gap approximation with empirical pairing gaps (5-point formula) [7].The particleparticle channel of the effective nucleon-nucleon interaction is described by a separable finite-range pairing force [7].The RHB equation is solved in the configuration space of harmonic oscillator wave functions with appropriate symmetry, while the densities are obtained in coordinate space.The wave functions in configuration space are generated by diagonalization of the RHB matrix equation.The density matrix obtained is then transformed to coordinate space, and the resulting vector and scalar densities are used to calculate the potentials.In the case of nuclei with axial symmetry, the solution of the Helmholtz equations for the meson fields is obtained by expanding in a harmonic oscillator basis.The solution of the relativistic mean-field equations are described in Ref. [12].The calculation has been carried out for describing ground state properties of nuclei and the details are reported in Ref. [1].In the present work, RHB model [7] combined with the successful DD-ME2 [8] has been employed for the calculation of ground state properties for Cr nuclei, as described below.The calculation is denoted as DIRHB in this article.

Ground state properties
The nuclear ground state properties such as binding energies, two neutron separation energies, proton, neutron and charge radii as well as matter densities have been calculated for the even 46−62 Cr isotopes and compared with experimental values, where available.The difference between the DIRHB calculated binding energies for even Cr isotopes and the corresponding experimental data [13] is plotted in Fig. 1.The calculated binding energies for even Cr isotopes are found to agree within 1% of the corresponding data.Further, experimental values of two neutron separation energies [13] are reproduced by DIRHB calculations.
The r c 2 1/2 calculated by DIRHB are shown in Fig. 2 for even isotopes of Cr.For comparison, the corresponding spherical droplet model [14] estimates for r c 2 1/2 (normalized to N=28 data) are also shown in Fig. 2. The DIRHB calculated r c 2 1/2 agrees well with the corresponding data [3].Data for r c 2 1/2 are available only for the stable even isotopes 50,52,54 Cr [3].Both DIRHB calculated and experimental r c 2 1/2 show lower values for 52 Cr (N=28), as expected.Calculations have also been carried out for neutron-deficient ( 46,48 Cr) and neutron-rich ( 56,58,60,62 Cr) nuclei.The DIRHB calculation for r c shows a change of slope at N=30 ( 54 Cr) and then increases smoothly for neutron-rich isotopes.In the neutron-  deficient region, both calculation and experiment indicate an increase in r c 2 1/2 as neutrons are removed from N=28, contrary to the decrease expected from the droplet model.It is also seen that as neutrons are removed from closed-shell nuclei, 52 Cr, the calculated r c 2 1/2 increases smoothly up to 48 Cr followed by a change in slope at N=24.The behavior of r c 2 1/2 in Cr isotopes is similar to that in the Ti isotopic chain [1] and is correlated with change in quadrupole deformation [15].As expected, DIRHB shows maximum deformation for the mid-shell nucleus, 48 Cr.It would be interesting to verify the calculation of nuclear charge radii by performing isotope shift and hyperfine structure measurements using laser spectroscopy for unstable Cr isotopes.
Further, the L=0 projected and renormalised DIRHB point proton and neutron density distributions for even EPJ Web of Conferences 08006-p.2 46−62 Cr are calculated and used as inputs to the semimicroscopic folded optical model calculation as described below.

Semi-microscopic Optical Model
It is known that the folding model associates the elastic scattering cross section with the structure of nuclei.There is a definite correlation between r c 2 1/2 and σ R .Once r c 2 1/2 and nuclear matter density distributions are accurately calculated, the correlation can be utilized to make predictions of σ R for proton scattering from target nuclei.This is particularly useful for unstable nuclei for which very little information on cross sections are available.For the calculation of cross section for nuclei that lie far from stability, microscopically calculated potentials are effective.An extended semi-microscopic JLM potential [9] derived from Brückner-Hartree-Fock approximation based on Reid's hard core nucleon-nucleon interaction has become available recently.A new parameterization has been obtained and the resulting potential is referred to as JLMB [9].The JLM interaction is energyand density-dependent as well as spin-independent in nuclear matter.An improved local density approximation is obtained to make JLM interaction, that is applicable for nuclear matter, usable for finite nuclei.Both the real and imaginary parts of the central potential are obtained in this approach.In JLMB, the deformed complex spin-orbit potential has been calculated phenomenologically [9] in the full Thomas form [9]. Other nucleon-nucleon interactions [16,17] that are successful and based on different approximations or effective forces can also be used.In the present work, the optical model potential (OMP) is calculated by folding the DIRHB densities with the JLMB interaction using the code MOM (Microscopic Optical Model) [9].The OMP obtained has the real and imaginary parts of the microscopic central as well as phenomenological spin-orbit potential.These are used in the code ECIS [18] to estimate the differential (dσ/dΩ) and reaction cross sections for elastic scattering of protons of 65 MeV incident on even Cr isotopes.The detailed description of the analysis is given in Ref. [1].In spite of the fact that most of the Cr isotopes considered here, except N=28, are deformed and with evidence of collectivity [19,20], the coupled channel effects were not required to be considered.This is because these effects are not important for higher proton energies (65 MeV) considered in this work.

Cross sections
The dσ/dΩ and σ R values for proton scattering from even Cr isotopes are calculated using the semi-microscopic optical model potential.An overall renormalization is required for the calculated optical model potential to provide a good description of data.Searches on renormalization constants for real and imaginary parts of the central (λ V and λ W ) and spin-orbit (λ V so and λ W so ) potentials were performed to obtain minimum χ 2 values in fitting dσ/dΩ data [21] for stable isotopes.The λ W so obtained was negligible for stable isotopes and hence not used in the analysis.To keep the number of parameters in the prediction of σ R to a minimum, λ V so was fixed at 88 (average value for stable isotopes) for all isotopes.Thus, the number of parameters in the prediction of σ R was reduced to two viz., λ V and λ W .A two-parameter search on λ V and λ W was carried out with λ V so =88 fixed, which gave a good fit to the dσ/dΩ data [21].These values of λ V and λ W , with λ V so =88, are referred to as best fit values and are given in Table 1.The λ V values obtained for all even stable isotopes have a small A dependence.For the closed-shell nucleus 52 Cr, λ V and λ W are close to unity.The calculated elastic scattering differential cross section for protons of 65 MeV incident on stable even Cr nuclei are plotted in Fig. 3.It is seen clearly from the figure that the calculated dσ/dΩ agree quite well with the data [21].The calculated dσ/dΩ shapes have similar behavior for all stable isotopes.The minima in dσ/dΩ are shifted slightly to lower angles with the increase in neutron number.The corresponding calculated σ R obtained from the best fit analysis for stable Cr isotopes are given in Table 1.It is seen that the calculated σ R increases as a function of mass number, as expected.
There are no measured values of σ R for protons of incident energy 65 MeV scattering off stable and unstable Cr nuclei.The present prediction can be verified by performing cross-section measurements in inverse kinematics for unstable nuclei with Cr isotopes as projectiles and an hydrogen target in available radioactive ion-beam facilities.
To make predictions of dσ/dΩ and σ R for unstable isotopes, a least-squares fit needs to be carried out for λ V and λ W as a function of A for stable isotopes.The variation of λ V and λ W obtained from best fit for stable isotopes exhibit a dependence on A. These renormalization constants are then extrapolated and are used for calculation of dσ/dΩ and σ R for neutron-deficient and neutron-rich isotopes.

Summary
Nuclear ground state properties such as binding energies, two neutron separation energies, proton, neutron and charge radii as well as matter densities have been calculated within RHB approach using DD-ME2 interaction, for the even 46−62 Cr.The DIRHB calculations agree well with data, where available.Both calculation and experiment show a kink in r c 2 1/2 at the N=28 shell closure.Prediction of r c 2 1/2 are made for neutron-deficient and neutron-rich Cr isotopes.Using calculated target matter densities and JLMB interaction, a folding model analysis has been carried out.Resulting OMPs are used to calculate cross sections for 65 MeV-protons scattering from even 46−62 Cr isotopes.The elastic scattering dσ/dΩ show good agreement with data.The OMP parameters that are required for prediction of dσ/dΩ and σ R for unstable even isotopes have been obtained.The correlation between nuclear charge radius and total reaction cross section has been utilized to make predictions of σ R for stable and unstable Cr isotopes.

DOI: 10
.1051/ C Owned by the authors, published by EDP Sciences, 201

Figure 1 .
Figure1.The difference between DIRHB calculated and the corresponding experimental[13] binding energies for even Cr isotopes.

Figure 2 .
Figure 2. The DIRHB calculated root-mean-square charge radii ( r c 2 1/2 ) for even Cr isotopes.The corresponding data [3] are shown by solid circles.The spherical droplet model estimate (normalized to N=28 data) for even Cr isotopes are shown by dashed lines.

Table 1 .
Best fit values of renormalization constants for real and imaginary parts of the central (λ V and λ W ) and real spin-orbit (λ Vso ) potentials, for protons of 65 MeV elastic scattering from stable even Cr isotopes.The corresponding total reaction cross sections (σ R ) are also given.