Corrections to the eikonal approximation for nuclear scattering at medium energies

The upcoming Facility for Rare Isotope Beams (FRIB) at the National Superconducting Cyclotron Laboratory (NSCL) at Michigan State University has reemphasized the importance of accurate modeling of low energy nucleus-nucleus scattering. Such calculations have been simplified by using the eikonal approximation. As a high energy approximation, however, its accuracy suffers for the medium energy beams that are of current experimental interest. A prescription developed by Wallace [Phys. Rev. Lett. 27, 622 (1971) and Ann. Phys. (NY) 78, 190 (1973)] that obtains the scattering propagator as an expansion around the eikonal propagator (Glauber approach) has the potential to extend the range of validity of the approximation to lower energies. Here we examine the properties of this expansion, and calculate the first-, second-, and third-order corrections for the scattering of a spinless particle off of a ${}^{40}\mathrm{Ca}$ nucleus, and for nuclear breakup reactions involving ${}^{11}\mathrm{Be}$. We find that, including these corrections extends the lower bound of the range of validity down to energies as low as about 45 MeV. At that energy the corrections provide as much as a 15% correction to certain processes.

Figure 1

Total nuclear cross section $\sigma$ for a proton incident on ${}^{40}\mathrm{Ca}$ as a function of beam energy. The exact partial wave result is the thick (blue) line, the zeroth-order eikonal approximation is the thin (magenta) line, the first-order eikonal approximation is the (beige) dashed line, the second-order eikonal approximation is the (green) dot-dashed line, and the third-order eikonal approximation is the (red) dotted line.

Figure 2

Differential elastic cross section $d\sigma /d\Omega$ for p + ${}^{40}\mathrm{Ca}$ at a beam energy of 40 MeV in logarithmic scale. The angle $\theta$ is the scattering angle from the forward direction. The designations for the lines are the same as in Fig. 1.

Figure 3

Same as Fig. 2, but zoomed in on the forward scattering region, and with a linear scale.

Figure 4

Real and imaginary parts of the transition matrix elements $T^{\left(n\right)}\left(b\right)$ for successive orders $n$ in the eikonal expansion. The designations for the lines are the same as in Fig. 1. (a), (b) Beam energy at 20 MeV. (c), (d) Beam energy at 40 MeV. (e), (f) Beam energy at 98 MeV.

Figure 5

(a) Imaginary surface (solid blue) $W_{\mathrm{s}}\left(E\right)$ and volume (dashed magenta) $W_{\mathrm{v}}\left(E\right)$ terms of Varner potential as a function of beam energy. (b) Real (solid blue) and imaginary (dashed magenta) parts of ${\tau }_3\left(b\right)$ at a beam energy of 20 MeV.

Figure 6

Real and imaginary parts of $ln\left[T^{\left(n\right)}\left(b\right)\right]$ for $n=0$ (thin magenta), $n=1$ (dashed beige), $n=2$ (dot-dashed green), and $n=3$ (dotted red) calculated at a beam energy of 25 MeV.

Figure 7

Coordinates used in this calculation. $\mathbf{R}$ is the coordinate of the center of mass of the halo nucleus, and ${\mathbf{b}}_c$ and ${\mathbf{b}}_n$ denote the components of ${\mathbf{R}}_c$ and ${\mathbf{R}}_n$ that are transverse to the beam direction.

Figure 8

Comparison of first-order corrections with standard (zeroth-order) eikonal approximation with a beam energy of 40 MeV/nucleon. The zeroth-order terms are shown with solid markers, and the first-order terms are with outlined markers. The solid (blue) line with circles is diffractive scattering, the dashed (magenta) line with squares is core stripping, the dotted (beige) line with diamonds is neutron stripping, and the dash-dotted (green) line with triangles is total absorption of the core and neutron.

Figure 9

Comparison of corrections to scattering data at 41 MeV/nucleon from [21]. The designations for the lines are the same as in Fig. 4, with the Coulomb breakup cross section as a dashed (black) line. Because we did not calculate corrections to the Coulomb cross section, the corrections appear smaller on this plot at high energies where the Coulomb term is larger.

Figure 10

Fractional corrections at various orders and beam energies. The designations for the lines are the same as in Fig. 8. (a) Effect of first-order corrections at beam energy of 40 MeV/nucleon. (b) Effect of second-order corrections at beam energy of 40 MeV/nucleon. (c) Effect of third-order corrections at beam energy of 40 MeV/nucleon. (d) Effect of first-order corrections at a beam energy of 100 MeV/nucleon.

Figure 11

The integrand of Eq. (31) (solid blue) for a ${}^{208}\mathrm{Pb}$ target, and the second term in the same integrand, which is the elastic scattering for the system (dashed magenta). Both are given as functions of $R_{\perp }$ (see Fig. 7) with all other variables integrated out.