Explicit inclusion of nonlocality in $(d,p)$ transfer reactions

Background: Traditionally, nucleon-nucleus optical potentials are made local for convenience. In recent work we studied the effects of including nonlocal interactions explicitly in the final state for $(d,p)$ reactions, within the distorted wave Born approximation.

Purpose: Our goal in this work is to develop an improved formalism for nonlocal interactions that includes deuteron breakup and to use it to study the effects of including nonlocal interactions in transfer $(d,p)$ reactions, in both the deuteron and the proton channel.

Method: We extend the finite-range adiabatic distorted wave approximation to include nonlocal nucleon optical potentials. We apply our method to $(d,p)$ reactions on ${}^{16}\mathrm{O}$, ${}^{40}\mathrm{Ca}$, ${}^{48}\mathrm{Ca}$, ${}^{126}\mathrm{Sn}$, ${}^{132}\mathrm{Sn}$, and ${}^{208}\mathrm{Pb}$ at 10, 20, and 50 MeV.

Results: We find that nonlocality in the deuteron scattering state reduces the amplitude of the wave function in the nuclear interior, and shifts the wave function outward. In many cases, this has the effect of increasing the transfer cross section at the first peak of the angular distributions. This increase was most significant for heavy targets and for reactions at high energies.

Conclusions: Our systematic study shows that, if only local optical potentials are used in the analysis of experimental $(d,p)$ transfer cross sections, the extracted spectroscopic factors may be incorrect by up to $40\%$ due to the local approximation.

Figure 1

Absolute value of the $d+A$ source term when nonlocal and local potentials are used. (a) $d+{}^{48}\mathrm{Ca}$ at $E_d=50\mathrm{MeV}$. (b) $d+{}^{208}\mathrm{Pb}$ at $E_d=50\mathrm{MeV}$. Both are for the $L=1$ and $J=0$ partial wave.

Figure 2

Absolute value of the $d+A$ scattering wave function when nonlocal and local potentials are used. (a) $d+{}^{48}\mathrm{Ca}$ at $E_d=50\mathrm{MeV}$. (b) $d+{}^{208}\mathrm{Pb}$ at $E_d=50\mathrm{MeV}$. Both are for the $L=1$ and $J=0$ partial wave.

Figure 3

Angular distributions for $(d,p)$ transfer cross sections as a function of scattering angle. The insets are the theoretical distributions normalized to the peak of the data distribution. (a) ${}^{48}\mathrm{Ca}(d,p){}^{49}\mathrm{Ca}$ at $E_d=10\mathrm{MeV}$ with data [41] at 10 MeV in arbitrary units. (b) ${}^{132}\mathrm{Sn}(d,p){}^{133}\mathrm{Sn}$ at $E_d=10\mathrm{MeV}$ with data [2] at $E_d=9.4\mathrm{MeV}$. (c) ${}^{208}\mathrm{Pb}(d,p){}^{209}\mathrm{Pb}$ at 20 MeV with data [42] (circles) and [43] (squares) at $E_d=22\mathrm{MeV}$.

Figure 4

Angular distributions for $(d,p)$ transfer cross sections as a function of scattering angle. The insets are the theoretical distributions normalized to the peak of the data distribution. (a) ${}^{48}\mathrm{Ca}(d,p){}^{49}\mathrm{Ca}$ at $E_d=50\mathrm{MeV}$ with data [44] at 56 MeV. (b) ${}^{132}\mathrm{Sn}(d,p){}^{133}\mathrm{Sn}$ at $E_d=50\mathrm{MeV}$. (c) ${}^{208}\mathrm{Pb}(d,p){}^{209}\mathrm{Pb}$ at 50 MeV.

Figure 5

Comparison of $(d,p)$ transfer cross sections when using ADWA as compared to DWBA. (a) ${}^{16}\mathrm{O}(d,p){}^{17}\mathrm{O}$ (b) ${}^{48}\mathrm{Ca}(d,p){}^{49}\mathrm{Ca}$ with data [44] at 56 MeV. (c) ${}^{132}\mathrm{Sn}(d,p){}^{133}\mathrm{Sn}$. All distribution at $E_d=50\mathrm{MeV}$.

Figure 6

Comparison of $(d,p)$ transfer cross sections with the explicit inclusion of nonlocality in the deuteron scattering state, and when using the proposed energy shift to take nonlocality in the deuteron scattering state into account. (a) ${}^{16}\mathrm{O}(d,p){}^{17}\mathrm{O}$ at $E_d=10\mathrm{MeV}$ (b) ${}^{40}\mathrm{Ca}(d,p){}^{41}\mathrm{Ca}$ at $E_d=10\mathrm{MeV}$ (c) ${}^{208}\mathrm{Pb}(d,p){}^{209}\mathrm{Pb}$ at $E_d=20\mathrm{MeV}$.