Validity of the relativistic impulse approximation for elastic proton-nucleus scattering at energies lower than 200 MeV

We present the first study to examine the validity of the relativistic impulse approximation (RIA) for describing elastic proton-nucleus scattering at incident laboratory kinetic energies lower than 200 MeV. For simplicity we choose a ${}^{208}\mathrm{Pb}$ target, which is a spin-saturated spherical nucleus for which reliable nuclear structure models exist. Microscopic scalar and vector optical potentials are generated by folding invariant scalar and vector scattering nucleon-nucleon ($\mathit{NN}$) amplitudes, based on our recently developed relativistic meson-exchange model, with Lorentz scalar and vector densities resulting from the accurately calibrated PK1 relativistic mean field model of nuclear structure. It is seen that phenomenological Pauli blocking (PB) effects and density-dependent corrections to $\sigma N$ and $\omega N$ meson-nucleon coupling constants modify the RIA microscopic scalar and vector optical potentials so as to provide a consistent and quantitative description of all elastic scattering observables, namely, total reaction cross sections, differential cross sections, analyzing powers and spin rotation functions. In particular, the effect of PB becomes more significant at energies lower than 200 MeV, whereas phenomenological density-dependent corrections to the $\mathit{NN}$ interaction also play an increasingly important role at energies lower than 100 MeV.

Figure 1

Scalar and vector (baryon) density (in units of fm${}^{-3}$) distribution in radial direction $r$ (in fm) for proton and neutron of ${}^{208}\mathrm{Pb}$, respectively. The proton and neutron densities are denoted by solid and dashed curves, respectively. The dash dotted curve corresponds to the proton scalar density, and dotted curve indicates the neutron scalar density. The PK1 effective interaction is used.

Figure 2

Differential cross sections $d\sigma /d\Omega$ (in units of mb/sr), analyzing powers $A_y$ and spin rotation functions $Q$, plotted as a function of center-of-mass scattering angles ${\theta }_{\mathrm{c}.\mathrm{m}.}$ (in degrees), for elastic proton scattering from ${}^{208}\mathrm{Pb}$ at incident laboratory kinetic energy of 200 MeV. The experimental data [24, 25] are denoted by open circles. Uncorrected RIA predictions are represented by black dashed curves (legend: RIA; color red online), the black dotted curves correspond to predictions based on the EDAI Dirac global optical potentials (GOP) [2] (legend: GOP; color olive online), and RIA calculations including phenomenological Pauli blocking (PB) corrections are indicated by the black solid curves (legend: PB).

Figure 3

Same as Fig. 2, except for incident laboratory kinetic energies of 121 or 98 MeV. The experimental data at 121 [23] and 98 [22] are denoted by open circles.

Figure 4

Same as Fig. 2, except for an incident laboratory kinetic energy of 65 MeV. The gray dashed curve (color cyan online) corresponds to RIA predictions based on the renormalized medium-modified $\sigma N$ and $\omega N$ coupling constants (denoted by the legend: $g^{*}$), and the gray solid lines (color cyan online) denote the combined effect of medium modified coupling constants together with PB corrections (legend: $g^{*}$ with PB). The experimental data [21] are denoted by open circles. The analyzing powers $A_y$ and spin rotation functions $Q$ are multiplied by 3 at the ${\theta }_{\mathrm{c}.\mathrm{m}.}$ angles from 0 to 25 degrees.

Figure 5

Linear interpolation and extrapolation of PB correction factors calculated from a relativistic Dirac-Brueckner approach. The four curves from top to bottom indicate the PB correction factors corresponding to the imaginary (Im) and real (Re) parts of the vector $V$ and scalar $S$ optical potentials. In the code corresponding to Ref. 8, the values at $T_{\mathrm{lab}}=170$ MeV are chosen same as $T_{\mathrm{lab}}=200$ MeV, which causes the flat section in the curves. The linear extrapolation at $T_{\mathrm{lab}}=98$ MeV is emphasized.

Figure 6

(a) Analyzing power $A_y$ for elastic proton scattering from ${}^{208}\mathrm{Pb}$ at 98 MeV, plotted as a function of center-of-mass scattering angles ${\theta }_{\mathrm{c}.\mathrm{m}.}$ (in degrees). The experimental data are taken from Ref. 22. Uncorrected RIA predictions are represented by black dashed curves (legend: RIA; color red online), the black dotted curves (color olive online) correspond to predictions based on the EDAI Dirac global optical potentials (GOP) [2] (legend: GOP), and RIA calculations including extrapolated Dirac-Brueckner and phenomenological PB corrections are denoted by gray [legend: PB (DB); color cyan online] and black solid curves [legend: PB (this work)], respectively. (b) The corresponding real (Re) and imaginary (Im) scalar $S$ and vector $V$ optical potentials (in units of MeV) for elastic proton scattering from ${}^{208}\mathrm{Pb}$ at 98 MeV, plotted as a function of the nuclear radius $r$ (in units of fm). In particular, the two black solid curves corresponding to the real parts of the scalar and vector optical potentials are covered by the gray curves.

Figure 7

Same as Fig. 2, except for elastic proton scattering from ${}^{40}\mathrm{Ca}$ at incident laboratory kinetic energy of 152 MeV. The experimental data at 152 MeV are taken from Ref. 29.

Figure 8

Strengths of the real (Re) and imaginary (Im) scalar $S$ and vector $V$ optical potentials (in units of MeV) at the center (radius $r=0$) of a ${}^{208}\mathrm{Pb}$ nucleus as a function of incident laboratory kinetic energies $T_{\mathrm{lab}}$ (in units of MeV). The identification of the curves is the same as in Fig. 4.

Figure 9

Same as Fig. 4, except for an incident laboratory kinetic energy of 30 MeV. The experimental data are taken from Refs. [36, 37].

Figure 10

The total reaction cross section ${\sigma }_R$ (in units of b) for elastic proton scattering from ${}^{208}\mathrm{Pb}$ versus the incident laboratory kinetic energy $T_{\mathrm{lab}}$ (in units of MeV). The identification of the curves is the same as in Fig. 4. The experimental data are taken from Refs. [39, 40, 41, 42, 43, 44, 45, 46].