Semiclassical calculations for the ${}^{156}\mathrm{Gd}$ (p,d) reaction

Numerical semiclassical calculations are carried out to study the angular distribution of deuterons from the p,d pickup reaction of 25 MeV protons incident on the nucleus ${}^{156}\mathrm{Gd}$ and also its proton elastic scattering. It is found that, due to the rapid fall of the real optical potential in the vicinity of the target nucleus, the classical trajectories are very sensitive to the proton impact parameters. A selection of 276,983 trajectories is used for protons with impact parameters $b_{\mathrm{p}}$ satisfying $7.23018\mathrm{fm}\le b_{\mathrm{p}}\le 10\mathrm{fm}$ with steps of ${10}^{-5}\mathrm{fm}$. Using the imaginary part of the optical potential for protons, a simple quantum approach is constructed to evaluate the probability of a surviving proton throughout its path. In addition, a simple three-body quantum approach is developed to calculate the probability of a neutron transfer by a surviving proton at closest approach. The formed deuteron is then allowed to start its trajectory while keeping its identity until detected. Throughout this journey, the deuteron trajectory is under the influence of its Coulomb and real optical potential, while its absorption is determined by the imaginary optical potential component. Within estimated uncertainties, the resulting theoretical angular distribution achieves a comparable fit with experimental results for the angular momentum transfer $\mathrm{L}=0$ compared to other theoretical models, and concludes that the strong p,d cross sections are due to the dominant ${\mathrm{s}}_{1/2}$ component of the Nilsson ${\frac{1}{2}}^{+}\left[400\right]$ level in ${}^{155}\mathrm{Gd}$.

Figure 1

Relative cross sections for p,d reactions on three even-even Gd targets (from Ross et al. [2]). The excitation energy of the Nilsson band-head energies for ${}^{155}\mathrm{Gd}$ are also taken from data in Allmond et al. [10] work.

Figure 2

(Top) The experimental angular momentum transfer $\mathrm{L}=0$ data of Allmond et al. [10] and $\mathrm{L}=0$ DWBA [11] angular distributions for the direct population of the 367 keV, ${\frac{1}{2}}^{+}\left[400\right]$. (Bottom) The experimental data of the level 729 keV are the colored dots with error bars and are taken from Fig. 7(e) of Ross et al. [4]. The graph shows the DWBA angular distributions fit which is best characterized by $\mathrm{L}=0,1,4$. The three different L values are: solid black for $\mathrm{L}=0$; colored line running through the highest point for $\mathrm{L}=1$, and a dot-dash black line for $\mathrm{L}=4$.

Figure 3

Schematic diagram for the scattering trajectory of a proton on a deformed Gd nucleus (exaggerated scale).

Figure 4

(a) Variation of the nuclear density of a ${}^{156}\mathrm{Gd}$ nucleus along the major axis and (b) along the minor axis.

Figure 5

(a) Values of the real optical potential (central + spin-orbit) for protons (red) and deuterons (black) [26]. The assumed sharp maximum and minimum radii of the target and residual nuclei and their corresponding potential values are displayed. (b) The small contribution of the spin-orbit term to the total real optical potential for protons is displayed in a smaller window.

Figure 6

The figure shows five incoming proton trajectories with a difference of 0.02 fm between each value of $b_{\mathrm{p}}$. After the CA, the outgoing proton trajectories acquire a small range of deflection $\mathrm{\Delta }{\theta }_{\mathrm{p}}$. If the proton picks up a neutron at CA, the range of deuteron deflection $\mathrm{\Delta }{\theta }_{\mathrm{d}}$ is much bigger than $\mathrm{\Delta }{\theta }_{\mathrm{p}}$ due to the greater attraction of the nuclear potential at the CA.

Figure 7

Final deflection angles of 276,983 protons and deuterons with respect to the proton impact parameter.

Figure 8

For $0<r\le 12\mathrm{f}\mathrm{m}$, the negative values of the imaginary optical potential $W$ are displayed for protons (red), Eq. (9), and with similar equation for the deuteron (black) [26].

Figure 9

Variation with time $t$ when $550\mathrm{ys}\le t\le 950\mathrm{ys}$ for (a) $r$, (b) $W_{\mathrm{pT}}^{\mathrm{Opt}}$, (c) ${\int }_0^t{W}_{\mathrm{pT}}^{\mathrm{Opt}}dt$, and (d) $|a_{\mathrm{p}}\left(t\right){|}^2$. For the regions $t<630\mathrm{ys}$ and $t>875\mathrm{ys}$, almost no proton absorption occurs.

Figure 10

At time $t$ near the point of CA, and with respect to an arbitrary origin o, the positions of the proton and the loosely- bound neutron are ${\vec{r}}_2$ and ${\vec{r}}_{\mathrm{n}}$, respectively. The figure displays an exaggerated deuteron size.

Figure 11

Graph of the variation of $|a_2\left(t\right){|}^2,(0\le t\le t_{\mathrm{CA}})$ and $a_{\mathrm{d}}\left(t\right),(t\ge t_{\mathrm{CA}})$ for three selected impact parameters.

Figure 12

The survival probability as a function of the impact parameter is plotted for protons (a) and deuterons (b).

Figure 13

Each point on the abscissa represents a bin size of $\mathrm{\Delta }{\theta }_{\mathrm{p}}=\mathrm{\Delta }{\theta }_{\mathrm{d}}=1.5^{\circ }$. (a) Angular distribution weighted by a proton's survival probability. (b) The same as (a), but for deuterons weighted by their creation probability at CA and then surviving from being absorbed by the imaginary optical potential after the CA and then detected. A smaller window is added to show the number of deuterons in the experimental range.

Figure 14

Comparison between the deuterons angular distributions of the present calculations with the experimental angular momentum transfer $\mathrm{L}=0$ data for the direct population of the ground state and the level 367-keV, ${\frac{1}{2}}^{+}\left[400\right]$ of Allmond et al. [10] and $\mathrm{L}=0$ DWBA calculations [11].

Figure 15

Comparison between the experimental results of T. Ross et al. [4] and the present calculations for deuterons.