Effects of density and parametrization on scattering observables
We calculate the density distribution of protons and neutrons for in the frame-work of relativistic mean field (RMF) theory with NL3 and G2 parameter sets. The microscopic proton-nucleus optical potential for system is evaluted from Dirac NN-scattering amplitude and the density of the target nucleus using Relativistic-Love-Franey and McNeil-Ray-Wallace parametrizations. Then we estimate the scattering observables, such as elastic differential scattering cross-section, analysing power and the spin observables with relativistic impulse approximation. We compare the results with the experimental data for some selective cases and found that the use of density as well as the scattering matrix parametrization is crucial for the theoretical prediction.
\runauthorM. Bhuyan and S.K. Patra
Explaining the nuclear structure by taking the tool of nuclear reaction is one of the most curious and challenging solution for Nuclear Physics both in theory and laboratory. So far the elastic scattering reaction of Neucleon-Nucleus is more interesting than that of Nucleus-Nucleus at laboratory energy 1000 MeV. The Neucleon-Nucleus interaction provides a fruitful source to determine the nuclear structure and a clear path toward the formation of exotic nuclei in laboratory. One of the theoritical method to study such type of reaction is the Relativistic Impulse Approximation (RIA). In a wide range of energy interval, the conventional impulse approximation [1, 2] reproduces quantitatively the main features of quasi-elastic scattering for medium mass nuclei [3, 4]. The observables of the elastic scattering reaction not only depend on the energy of the incident particle but also on the kinematic parameter as well as the density discributions of the target nucleus. In the present letter, our motivation is to calculate the nucleon-nucleus elastic differential scattering cross-section () and other quantities, like optical potential (), analysing power () and spin observables (value) taking input as relativistic mean field (RMF) and recently proposed effective field theory motivated relativistic mean field (E-RMF) density. The RMF and E-RMF densities are obtained from the most successful NL3  and advanced G2  parameter sets, respectively. As representative cases, we used these target densities folded with the NN-aplitude of 1000 MeV energetic proton projectile with Relativistic-Love-Franey (RLF) and McNeil-Ray-Wallace (MRW) parametrizations  for in our calculations.
The RMF and E-RMF theories are well documented [6, 8, 9] and for completeness we outline here very briefly the formalisms for finite nuclei. The energy density functional of the E-RMF model for finite nuclei is written as [10, 11],
where the index runs over all occupied states of the positive energy spectrum, , , and . The terms with , , and take care of the effects related with the electromagnetic structure of the pion and the nucleon (see Ref. ). The energy density contains tensor couplings, and scalar-vector and vector-vector meson interactions, in addition to the standard scalar self interactions and . Thus, the E-RMF formalism can be interpreted as a covariant formulation of density functional theory as it contains all the higher order terms in the Lagrangian, obtained by expanding it in powers of the meson fields. The terms in the Lagrangian are kept finite by adjusting the parameters. Further insight into the concepts of the E-RMF model can be obtained from Ref. . It may be noted that the standard RMF Lagrangian is obtained from that of the E-RMF by ignoring the vector-vector and scalar-vector cross interactions, and hence does not need a separate discussion. In each of the two formalisms (E-RMF and RMF), the set of coupled equations are solved numerically by a self-consistent iteration method and the baryon, scalar, isovector, proton, neutron and tensor densities are calculated.
The numerical procedure of calculation and the detailed equations for the ground state properties of finite nuclei, we refere the reader to Refs. [9, 8]. The densities obtained from RMF (NL3)  and E-RMF (G2)  are used for folding with the NN-sacttering amplitude at , which gives the proton-nucleus complex optical potential for RMF and E-RMF formalisms. RIA involves mainly two steps [12, 13] of calculations for the evaluation of the NN-scattering amplitude. In this case, five Lorentz covariant function  multiply with the so called Fermi invariant Dirac matrix (NN-scattering amplitudes). This NN-amplitudes are folded with the target densities of protons and neutrons to produced a first order complex optical potential . The invariant NN-scattering operater can be written in terms of five complex functions (the five terms involves in the proton-proton pp and neutron-neutron pn scattering) as follows:
where (0) and (1) are the incident and struck nucleons respectively. The amplitude for each is a complex function of the Lorentz invariants T and S with and q is the four momentum. We recommend the redears for detail expressions to Refs. [14, 15, 16, 17, 18, 19, 20, 21, 22]. Then the Dirac optical potential can be written as,
where is the scattering operator, is the momentum of the projectile in the nucleon-nucleus center of mass frame, is the nuclear ground state wave function for A-particle. Finally using the Numerov algorithm the obtained wave function is matched with the coulomb scattering solution for a boundary condition at and we get the scattering observables from the scattering amplitude, which are defined as:
Now we present our calculated results of neutrons and protons density distribution obtained from the RMF and E-RMF formalisms . Then we evaluate the scattering observables using these densities in the relativistic impulse approximation, which involves the following two steps: in the first step we generate the complex NN-interaction from the Lorentz invariant matrix as defined in Eq. (2). Then the interaction is folded with the ground state target nuclear density for both the RLF and MRW parameters  separately and obtained the nucleon-nucleus complex optical potential for the parametrisations. It is to be noted that pairing interaction is taken care using the Pauli blocking approximation. In the second step, we solve the wave function of the scattering state utilising the optical potential prepared in the first step by well known Numerov algorithm . The result approxumated with the non-relativistic Coulomb scattering for a longer range of radial component which results the scattering amplitude and other observables . In thr present paper we calculate the density distribution of protons and neutrons for Ca in NL3 and G2 parameter sets. From the density we evalute the optical potential and other scattering observables and some representative cases are presented in Figures .
In Fig. 1, the protons and neutrons density distribution for using NL3 and G2 parameter sets (upper panel) and the optical potential obtained with RLF and MRW parametrisation for at 1000 MeV proton energy (lower panel) are shown. From the figure, it is noticed that, there is no significant difference in desities for RMF and E-RMF parameter sets. However, a careful inspection shows a small enhancement in central density (0-1.6 fm) for NL3 set. On the otherhand the densities obtained from G2 elongated to a larger distance towards the tail part of the density distribution. As the optical potential is a complex function which constitute both real and imaginary part for both scalar and vector, we have displyed those values in the lower panel of Fig. 1. Unlike to the (upper panel) of protons and neutrons density distribution, here we find a large difference of between the RLF and MRW parametrisation. Further, the value of either RLF or MRW differs significantly depending on the NL3 or G2 force parameters. That means, the optical potential not only sensitive to RLF or MRW but also to the use of NL3 or G2 parameter sets. Investigating the figure it is clear that, the extrimum magnitude of real and imaginary part of the scalar potential are -442.2 and 113.6 MeV for RLF (G2) and -372.4 and 109.1 MeV for RLF (NL3). The same values for the MRW parametrisation are -219.8 and 32.8 MeV with G2 and -175.1 and 33.2 MeV with NL3 sets. In case of vector potential, the extremum values for real and imaginary parts are 361.3 and -179.2 MeV for RLF (G2) and 279.2 and -164.8 MeV for RLF (NL3) but with MRW parametrisation these are appeared at 128.1 and -87.4 MeV in G2 and 99.2 and -76.6 MeV in NL3. From these large variation in magnitude of scalar and vector potentials, it is clear that the predicted results not only depend on the input target density, but also highly sensitive with the kinematic of the reaction dynamics. A further analysis of the results for the optical potential with NL3 and G2, it suggest that the value extends for a larger distance in NL3 than G2. For example, with RLF the central part of with G2 is more expanded than with NL3 and ended at , whereas the optical potential persists till in NL3. Similar situation is also valid in MRW parametrisation. This nature of the potential suggests the applicability of NL3 over G2 force parameter. This is because in case of NL3 the soft-core interaction between the projectile and the target nucleon is more effective.
In Fig. 2., we have plotted the elastic scattering cross-section of the proton with at laboratory energy 1000 MeV using both densities obtained in the NL3 and G2 parameter sets with RLF and MRW parametrisations. The experimental data  are also given for comparison. It is reported in Refs. [7, 26] the superiority of RLF over MRW for lower energy ( MeV), however the MRW shows better results at energy MeV. In the present case, our incident energy is 1000 MeV which matches better (MRW) with experimental values. This is consistent with the optical potential also (see Fig. 1). From the differential cross-section for both NL3 and G2 densities with MRW parametrization, it is clearly seen that with NL3 desity is more closer to experimental data which insist not only the importance of parametrization (RLF or MRW) but also to choose proper density input for the reaction dynamics. Analysing the elastic differential cross-section along the isotopic chain of Ca from A=40 to 48, the calculated results improve with increasing mass number of the target.
The analysing power for composite system is calculated from the general formulae given in eqns. (4) and (5) and are shown in Fig. 3 with RLF and MRW. The and values obtained by NL3 and G2 sets almost matches with each other both in RLF and MRW. But if we compare the results with RLF and MRW it differs significantly. Again, we get a small oscillation of and in G2 set with increasing scattering angle for RLF which does not appear in NL3 set. There is a rotation of value from positive to negative direction when we calculate with MRW parametrization, which does not appear in case of RLF parametrization. This rotation shows a shining path towards the formation of exotic nuclei in the laboratory.
In summary, we calculate the density distribution of protons and neutrons for by using RMF (NL3) and E-RMF (G2) parameter sets. We found similar density distribution for protons and neutrons in both the sets with a small difference at the central region. This small difference in densities make a significant influence in the prediction of optical potential, elastic differential cross-section, analysing power and the spin observable for systems. The effect of kinematic parameters for reaction dynamics, RLF and MRW, are also highly sensitive to the predicted results. That means, the differential scattering cross-section and scattering observables are highly depent on the input density and the choice of parametrisation.
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