Nonlocal nucleon-nucleus interactions in $(d,p)$ reactions: Role of the deuteron $D$ state

Theoretical models of the $(d,p)$ reaction are exploited for both nuclear astrophysics and spectroscopic studies in nuclear physics. Usually, these reaction models use local optical model potentials to describe the nucleon- and deuteron-target interactions. Within such a framework, the importance of the deuteron $D$ state in low-energy reactions is normally associated with spin observables and tensor polarization effects, with very minimal influence on differential cross sections. In contrast, recent work that includes the inherent nonlocality of the nucleon optical model potentials in the Johnson-Tandy adiabatic-model description of the $(d,p)$ transition amplitude, which accounts for deuteron break-up effects, shows sensitivity of the reaction to the large $n-p$ relative momentum content of the deuteron wave function. The dominance of the deuteron $D$-state component at such high momenta leads to significant sensitivity of calculated $(d,p)$ cross sections and deduced spectroscopic factors to the choice of deuteron wave function [Phys. Rev. Lett. 117, 162502 (2016)]. We present details of the Johnson-Tandy adiabatic model of the $(d,p)$ transfer reaction generalized to include the deuteron $D$ state in the presence of nonlocal nucleon-target interactions. We present exact calculations in this model and compare these to approximate (leading-order) solutions. The latter, approximate solutions can be interpreted in terms of local optical potentials, but evaluated at a shifted value of the energy in the nucleon-target system. This energy shift is increased when including the $D$-state contribution. We also study the expected dependence of the $D$-state effects on the separation energy and orbital angular momentum of the transferred nucleon. Their influence on the spectroscopic information extracted from $(d,p)$ reactions is quantified for a particular case of astrophysical significance.

Figure 1

(a) Adiabatic and Watanabe potentials for $d-{}^{40}\mathrm{Ca}$ at $E_d=10$ MeV and (b) ${}^{40}\mathrm{Ca}(d,p){}^{41}\mathrm{Ca}$ differential cross sections for $E_d=10$ MeV, using the Chapel Hill 89 phenomenological local optical potential for two NN potential models: (i) the central Hulthén potential with an $S$-state deuteron (dots) and (ii) the realistic AV18 potential and deuteron with both the $S$ and $D$ states (solid lines).

Figure 2

(a) Trivially equivalent local adiabatic potentials for $d-{}^{40}\mathrm{Ca}$, and (b) the ${}^{40}\mathrm{Ca}(d,p){}^{41}\mathrm{Ca}$ differential cross sections obtained from full (solid lines) and lowest-order (dashed lines) calculations at $E_d=11.8$ MeV. Calculations assume the $S$-wave Hulthén potential (blue) and the more realistic AV18 NN potential with both $S$ and $D$ states (black). These are compared to the lowest-order TJ results for the same NN potentials. For the Hulthén potential, the TJ cross sections in next-to-leading order are also shown by the dashed-dotted (magenta) curve.

Figure 3

The $E_d=11.8$ MeV, $d-{}^{40}\mathrm{Ca}$ distorted waves in $L=0$ (a,b) and $L=4$ (c,d) partial waves, calculated using the AV18 (a,c) and Hulthén (b,d) NN model potentials. The exact nonlocal calculations are shown by the solid lines while the approximate, lowest-order (LO) calculations are shown as dashed lines.

Figure 4

Difference between the ${}^{40}\mathrm{Ca}(d,p){}^{41}\mathrm{Ca}$ differential cross sections, for $E_d=11.8\mathrm{MeV}$, calculated with only the central terms ($a=0$) and with the full ($a=0,2$) nonlocal adiabatic potential that includes the $D$-state generated tensor terms (solid line). The corresponding leading order (LO) potential results are shown by the dashed line.

Figure 5

The ${}^{28}\mathrm{Si}(d,p){}^{29}\mathrm{Si}$ (g.s.) reaction differential cross sections, at $E_d=10$ MeV, obtained using the ${\mathcal{U}}_{dA}$ calculated with the $S$-state Hulthén and $S+D$-state AV18 deuteron wave functions.

Figure 6

Fractional changes (as $\%$) in the differential cross sections (at their first peak) of calculations with the Hulthén and AV18 deuteron wave functions (closed circles and solid lines). Results are for the ${}^{28}\mathrm{Si}(d,p){}^{29}\mathrm{Si}$ reaction. The changes are shown as functions of the assumed separation energy of the transferred neutron with orbital angular momenta $l=0,1,2,3$. The changes in the same cross sections, but at a fixed center-of-mass angle, are also shown (open circles connected by dashed lines). The fixed angles used were 0, 16, 31 and 44 degrees for $l=0,1,2,3$, respectively.

Figure 7

Calculated $l=0$ (dashed lines) and $l=2$ (dot-dashed lines) differential cross sections and their sums (solid lines) for the ${}^{26}\mathrm{Al}{(d,p)}^{27}\mathrm{Al}$ (7806 keV) reaction at 12 MeV. The red and orange curves result from nonlocal potential analyses using the Hulthén and AV18 NN wave function models. These calculations have been scaled by the spectroscopic factors deduced from the data using the local potential analysis of Ref. [25], shown by the black lines.