Separable representation of proton-nucleus optical potentials

Recently, a new approach for solving the three-body problem for (d,p) reactions in the Coulomb-distorted basis in momentum space was proposed. Important input quantities for such calculations are the scattering matrix elements for proton- (neutron-) nucleus scattering. We present a generalization of the the Ernst-Shakin-Thaler scheme in which a momentum space separable representation of proton-nucleus scattering matrix elements in the Coulomb basis can be calculated. The success of this method is demonstrated by comparing $S$-matrix elements and cross sections for proton scattering from ${}^{12}\mathrm{C},{}^{48}\mathrm{Ca}$, and ${}^{208}\mathrm{Pb}$ with the corresponding coordinate space calculations.

Figure 1

The partial-wave $S$ matrix for $\mathrm{p}+{}^{208}\mathrm{Pb}$ elastic scattering obtained from the CH89 [19] global optical potential as a function of the angular momentum. (a) and (b) show the real and imaginary parts of the $S$ matrix at 10 MeV laboratory kinetic energy; (c) and (d) give the real and imaginary parts of the $S$-matrix elements at 45 MeV laboratory kinetic energy. The total angular momentum is $j=$ $l+1/2$. The cross symbols (red) $\left(i\right)$ give the $S$-matrix elements calculated from the momentum space separable representation of the CH89 global optical potential for $\mathrm{p}+{}^{208}\mathrm{Pb}$ elastic scattering, while open circles (black) $\left(ii\right)$ depict the corresponding coordinate space calculation. The open diamonds (blue) $\left(iii\right)$ show the calculation in which the short-range Coulomb potential is omitted. The solid circles (green) $\left(iv\right)$ indicate the $S$-matrix elements for $\mathrm{n}+{}^{208}\mathrm{Pb}$ elastic scattering.

Figure 2

The unpolarized differential cross section for elastic scattering of protons from ${}^{12}\mathrm{C}$ (upper) and ${}^{48}\mathrm{Ca}$ (lower) divided by the Rutherford cross section as a function of the c.m. angle calculated for a laboratory kinetic energy of 38 MeV. The ${}^{12}\mathrm{C}$ cross section is scaled by a factor 1.5. The solid lines $\left(i\right)$ depict the cross section calculated in momentum space based on the rank-4 separable representation of the CH89 [19] phenomenological optical potential, while the dotted lines $\left(ii\right)$ represent the corresponding coordinate space calculations. The dashed lines $\left(iii\right)$ show the results in which the short-ranged Coulomb potential is omitted.

Figure 3

The real parts of the partial-wave proton-nucleus form factor for ${}^{48}\mathrm{Ca}$ as a function of the momentum $p$ for selected angular momenta $l$: (a) $l=0$, (b) $l=3$, and (c) $l=6$. The form factors are calculated at $E_{\mathrm{c}.\mathrm{m}.}=36\mathrm{MeV}$ and based on the CH89 global optical potential: The full calculations $\left(i\right)$ are compared to those omitting the short-range Coulomb $\left(ii\right)$, the proton-nucleus form factor in the plane wave basis $\left(iii\right)$, and the form factor obtained using the techniques introduced in [9] $\left(iv\right)$.