Coulomb problem in momentum space without screening

Background: The repulsive Coulomb force poses severe challenges when solving the three-body problem for ($d,p$) reactions on intermediate mass and heavy nuclei. Recently, a new approach based on the Coulomb-distorted basis in momentum space was proposed.

Purpose: In this work, we demonstrate the feasibility of using the Coulomb-distorted basis in momentum space for calculating matrix elements expected in a wide range of nuclear reactions.

Method: We discuss the analytic forms of the Coulomb wave function in momentum space. We analyze the singularities in the Coulomb-distorted form factors and the required regularization techniques. Employing a separable interaction derived from a realistic nucleon-nucleus optical potential, we compute and study the Coulomb-distorted form factors for a wide range of cases, including charge, angular momentum, and energy dependence. We also investigate in detail the precision of our calculations.

Results: The Coulomb-distorted form factors differ significantly from the nuclear form factors except for the very highest momenta. Typically, the structure of the form factor is shifted away from zero momentum due to the Coulomb interaction. Unlike the Yamaguchi forms typically used in three-body methods, our realistic form factors have a short high-momentum tail, which allows for a safe and efficient truncation of the momentum grid.

Conclusions: Our results show that the Coulomb-distorted basis can be effectively implemented.

Figure 1

The partial-wave Coulomb form factors $u_l^C\left(p\right)$ obtained with a Yamaguchi interaction as a function of the external momentum $p$ for selected angular momenta $l$. Comparison between our numerical evaluation (solid lines) and the mathematica® [26] results (symbols): (a) real part $u_l^C\left(p\right)$ for $p+{}^{12}\text{C}$; (b) imaginary part of $u_l^C\left(p\right)$ for $p+{}^{12}\text{C}$; (c) real part $u_l^C\left(p\right)$ for $p+{}^{208}\text{Pb}$; (d) imaginary part of $u_l^C\left(p\right)$ for $p+{}^{208}\text{Pb}$.

Figure 2

The fractional contribution of the pole $\mathbf{D}\left(\Delta \right)$ for the Yamaguchi form factors of Fig. 1 at specific momenta: (a) $p+{}^{12}\text{C}$ at $p=0.6$ fm${}^{-1}$ and (b) $p+{}^{208}\text{Pb}$ at $p=1.1$ fm${}^{-1}$. Shown are $l=0$ (dot-dashed line) and $l=4$ (dashed line) partial waves.

Figure 3

The real parts of the partial-wave nuclear form factors $u_l\left(p\right)$ (left panels) and the Coulomb-distorted nuclear form factors $u_l^C\left(p\right)$ (right panels) as function of the the external momentum $p$ for selected angular momenta $l$: (a) $\Re e{u}_l\left(p\right)$ for $n+{}^{12}\text{C}$; (b) $\Re e{u}_l^C\left(p\right)$ for $p+{}^{12}\text{C}$; (c) $\Re e{u}_l\left(p\right)$ for $n+{}^{48}\mathrm{Ca}$; (d) $\Re e{u}_l^C\left(p\right)$ for $p+{}^{48}\mathrm{Ca}$; (e) $\Re e{u}_l\left(p\right)$ for $n+{}^{208}\text{Pb}$; (f) $\Re e{u}_l^C\left(p\right)$ for $p+{}^{208}\text{Pb}$. The form factors for ${}^{12}\mathrm{C}$ correspond to the fixed support point $E_{\mathrm{cm}}=30$ MeV, that for ${}^{48}\mathrm{Ca}$ is at a fixed support point $E_{\mathrm{cm}}=36$ MeV, while the nuclear form factors for ${}^{208}\mathrm{Pb}$ are at a fixed support point $E_{\mathrm{cm}}=36$ MeV for $l=0,4$ and $E_{\mathrm{cm}}=39$ MeV for $l=8$.

Figure 4

Same as Fig. 3, but for the imaginary parts of the nuclear form factors $u_l\left(p\right)$ (left panels) and Coulomb-distorted nuclear form factors $u_l^C\left(p\right)$ (right panels).

Figure 5

The real part of the $l=0$ Coulomb-distorted nuclear form factors $u_0^C\left(p\right)$ as function of the external momentum $p$ for ${}^{12}\mathrm{C}$ at $E_{\mathrm{cm}}=30$ MeV. The solid (black) line shows the full results, while for all other curves an interval of the size $\Delta$ has been cut out left and right of the pole $p$ while performing the integration.

Figure 6

The real part of the $l=0$ Coulomb-distorted nuclear form factors $u_0^C\left(p\right)$ as function of the external momentum $p$ for ${}^{208}\mathrm{Pb}$ at the fixed support point $E_{\mathrm{cm}}=36$ MeV. The solid (black) line shows the full results, while for all other curves an interval of the size $\Delta$ has been cut out left and right of the pole $p$ while performing the integration.

Figure 7

The real (a) and imaginary (b) parts of the partial-wave $S$ matrix $S_{L+1/2}$, for the $p+{}^{48}\mathrm{Ca}$ system obtained from the CH89 [24] phenomenological optical potential as a function of the angular momentum $l$ at a projectile incident kinetic energy of 38 MeV. The solid (green) circles indicate the coordinate-space calculation for which the contribution due to the Coulomb force is omitted. The (black) X symbols correspond to the coordinate-space calculation including the Coulomb force, The (red) + symbols give the momentum-space calculation based on the separable representation of the CH89 potential [19] including the Coulomb force.

Figure 8

Comparison of two expansions of the spherical function $Q_l^{i\eta }\left(\zeta \right)$ as a function of momentum. Results are shown for the real part of $Q_l^{i\eta }\left(\zeta \right)$ with parameters evaluated for ${}^{12}\text{C}$ at $p$ = 0.67 fm${}^{-1}$, for $l$ = 0 (top panel) and $l$ = 4 (lower panel).

Figure 9

The singularity $\left|\Im m\left[J_{a,b,c}\left(y\right)\right]\right|$ according to Eq. (B2) near the on-shell point $y=0$: (a) unregularized, (b) with principal value regularization, and (c) with Gel'fand-Shilov regularization. The precise definition of the function $J_{a,b,c}\left(y\right)$ in each panel is defined in the text.