Global analysis of isospin dependent microscopic nucleon-nucleus optical potentials in a Dirac-Brueckner-Hartree-Fock approach

Background: For the study of exotic nuclei it is important to have an optical model potential that is reliable not only for stable nuclei but can also be extrapolated to nuclear systems with exotic numbers of protons and neutrons. An efficient way to obtain such a potential is to develop a microscopic optical potential (MOP) based on a fundamental theory with a minimal number of free parameters, which are adjusted to describe stable nuclei all over the nuclide chart.

Purpose: The choice adopted in the present work is to develop the MOP within a relativistic scheme which provides a natural and consistent relation between the spin-orbit part and the central part of the potential. The Dirac-Brueckner-Hartree-Fock (DBHF) approach provides such a microscopic relativistic scheme, which is based on a realistic nucleon-nucleon interaction and reproduces the saturation properties of symmetric nuclear matter without any adjustable parameter. Its solution using the projection technique within the subtracted $T$-matrix representation provides a reliable extension to asymmetric nuclear matter, which is important to describe the features of isospin asymmetric nuclei. The present work performs a global analysis of the isospin dependent nucleon-nucleus MOP based on the DBHF calculation in symmetric and asymmetric nuclear matter.

Methods: The DBHF approach is used to evaluate the relativistic structure of the nucleon self-energies in nuclear matter at various densities and asymmetries. The Schrödinger equivalent potentials of finite nuclei are derived from these Dirac components by a local density approximation (LDA). The density distributions of finite nuclei are taken from the Hartree-Fock-Bogoliubov approach with Gogny D1S force. An improved LDA approach (ILDA) is employed to get a better prediction of the scattering observables. A ${\chi }^2$ assessment system based on the global simulated annealing algorithm is developed to optimize the very few free components in this study.

Results: The nucleon-nucleus scattering calculations are carried out for a broad spectrum of $n$ and $p$ scattering experiments below 200 MeV with targets ranging from ${}^{12}\mathrm{C}$ to ${}^{208}\mathrm{Pb}$. The scattering observables including the neutron total cross section, proton reaction cross section, elastic scattering angular distribution, analyzing power, and spin rotation are evaluated and compared with the experimental data, as well as with results derived from the widely used phenomenological Koning-Delaroche global potential.

Conclusions: Results with the present relativistic MOP reproduce the $n,p+A$ scattering observables with good accuracy over a broad range of targets and a large region of energies fitting only the free-range factor $t$ in ILDA and minor adjustments of the scalar and vector potentials in the low-density region.

Figure 1

The proton and neutron radial densities for ${}^{208}\mathrm{Pb}$. The solid and dashed lines indicate the calculated results from HFB and the Negele's empirical formula, respectively.

Figure 2

Comparisons of angular distributions for $n+{}^{40}\mathrm{Ca}$ and ${}^{208}\mathrm{Pb}$ at incident neutron energy around 7.0, 13.0, 30.0, and 65.0 MeV. The dashed line indicates the results based on a linear assumption of $\left(f_1,f_2\right)$ and the solid line denotes the results with the values of (0.86,1.14), and the experimental data are listed in Table 1.

Figure 3

Example for the real and imaginary part of the scalar $\left(U_s\right)$ and vector $\left(U_0\right)$ components of the Dirac potential as a function of density for nucleons with an incident particle energy of 90 MeV. The circles, triangles, x marks, and squares represent the calculated (adjusted) values for isospin asymmetries $\beta =0.0$, 0.2, 0.6, and 1.0, respectively. The connecting solid line shows the polynomial interpolation in the case of symmetric matter, while the dashed and dotted lines visualize the corresponding interpolations for the neutron- and proton-potentials, respectively.

Figure 4

Comparison of predicted neutron total cross section (solid line) and experimental data (points) and KD calculation (dashed line) for $n+{}^{12}\mathrm{C}$. The experimental data are measured for natural carbon.

Figure 5

Comparison of predicted neutron total cross section (solid line) and experimental data (points) and KD calculation (dashed line) for $n+{}^{56}\mathrm{Fe}$. The experimental data are measured for natural iron.

Figure 6

Comparison of predicted neutron total cross section (solid line) and experimental data (points) and KD calculation (dashed line) for $n+{}^{208}\mathrm{Pb}$. The experimental data are measured for natural lead.

Figure 7

Comparisons of angular distributions for $n+{}^{12}\mathrm{C},{}^{27}\mathrm{Al},{}^{40}\mathrm{Ca},{}^{56}\mathrm{Fe},{}^{98}\mathrm{Mo}$, and ${}^{208}\mathrm{Pb}$ at incident neutron energy around 30 MeV. The dashed lines indicate the results from KD potential and the solid lines denote the present prediction.

Figure 8

Comparisons of angular distributions for $n+{}^{28}\mathrm{Si},{}^{40}\mathrm{Ca},{}^{56}\mathrm{Fe},{}^{90}\mathrm{Zr},{}^{120}\mathrm{Sn}$, and ${}^{208}\mathrm{Pb}$ at incident neutron energy around 65 MeV. The dashed line indicates the results from KD potential and the solid line denotes the present prediction.

Figure 9

Comparison of predicted $d\sigma /d\mathrm{\Omega }$ (solid lines) and experimental data (points) and KD calculation (dashed lines) for $n+{}^{12}\mathrm{C}$.

Figure 10

Comparison of predicted $d\sigma /d\mathrm{\Omega }$ (solid lines) and experimental data (points) and KD calculation (dashed lines) for $n+{}^{27}\mathrm{Al}$.

Figure 11

Comparison of predicted $d\sigma /d\mathrm{\Omega }$ (solid lines) and experimental data (points) and KD calculation (dashed lines) for $n+{}^{40}\mathrm{Ca}$.

Figure 12

Comparison of predicted $d\sigma /d\mathrm{\Omega }$ (solid lines) and experimental data (points) and KD calculation (dashed lines) for $n+{}^{56}\mathrm{Fe}$.

Figure 13

Comparison of predicted $d\sigma /d\mathrm{\Omega }$ (solid lines) and experimental data (points) and KD calculation (dashed lines) for $n+{}^{98}\mathrm{Mo}$ and ${}^{103}\mathrm{Rh}$.

Figure 14

Comparison of predicted $d\sigma /d\mathrm{\Omega }$ (solid lines) and experimental data (points) and KD calculation (dashed lines) for $n+{}^{208}\mathrm{Pb}$.

Figure 15

Comparisons of analyzing power for $n+{}^{12}\mathrm{C},{}^{40}\mathrm{Ca},{}^{58}\mathrm{Ni}$, and ${}^{208}\mathrm{Pb}$ at incident neutron energy around 10 MeV. The dashed line indicates the results from KD potential and the solid line denotes the present work.

Figure 16

Comparison of predicted reaction cross section (solid lines) and experimental data (points) and KD calculation (dashed lines) for $p+{}^{40}\mathrm{Ca}$.

Figure 17

Comparison of predicted reaction cross section (solid lines) and experimental data (points) and KD calculation (dashed lines) for $p+{}^{120}\mathrm{Sn}$.

Figure 18

Comparison of predicted reaction cross section (solid lines) and experimental data (points) and KD calculation (dashed lines) for $p+{}^{208}\mathrm{Pb}$.

Figure 19

Comparison of predicted angular distribution (solid lines) and experimental data (points) and KD calculation (dashed lines) for $p+{}^{28}\mathrm{Si},{}^{40}\mathrm{Ca},{}^{56}\mathrm{Fe},{}^{90}\mathrm{Zr}$, and ${}^{208}\mathrm{Pb}$ at incident neutron energy 65 MeV and 61.5 MeV for $p+{}^{120}\mathrm{Sn}$.

Figure 20

Comparison of predicted $d\sigma /d\mathrm{\Omega }$ (solid lines) and experimental data (points) and KD calculation (dashed lines) for $p+{}^{28}\mathrm{Si}$.

Figure 21

Comparison of predicted $d\sigma /d\mathrm{\Omega }$ (solid lines) and experimental data (points) and KD calculation (dashed lines) for $p+{}^{58}\mathrm{Ni}$.

Figure 22

Comparison of predicted $d\sigma /d\mathrm{\Omega }$ (solid lines) and experimental data (points) and KD calculation (dashed lines) for $p+{}^{90}\mathrm{Zr}$.

Figure 23

Comparison of predicted $d\sigma /d\mathrm{\Omega }$ (solid lines) and experimental data (points) and KD calculation (dashed lines) for $p+{}^{208}\mathrm{Pb}$.

Figure 24

Comparison of predicted $A_y$ and $Q$ (solid lines) and experimental data (points) for $p+{}^{208}\mathrm{Pb}$ at $E_p=80$ and 200 MeV. The experimental data of $A_y$ are taken from the EXFOR library, while the data of $Q$ are read from Ref. [33].

Figure 25

Comparison of predicted $A_y$ (solid lines) and experimental data (points) and KD calculation (dashed lines) for $p+{}^{56}\mathrm{Fe}$ and ${}^{58}\mathrm{Ni}$. The curves and data points at the top represent true values; the others are offset by factors of 2, 4, 8, etc.