Predictive theory for elastic scattering and recoil of protons from ${}^4\mathrm{He}$

Low-energy cross sections for elastic scattering and recoil of protons from ${}^4\mathrm{He}$ nuclei (also known as $\alpha$ particles) are calculated directly by solving the Schrödinger equation for five nucleons interacting through accurate two- and three-nucleon forces derived within the framework of chiral effective field theory. Precise knowledge of these processes at various proton backscattering/recoil angles and energies is needed for the ion-beam analysis of numerous materials, from the surface layers of solids, to thin films, to fusion-reactor materials. Indeed, the same elastic scattering process, in two different kinematic configurations, can be used to probe the concentrations and depth profiles of either hydrogen or helium. We compare our results to available experimental data and show that direct calculations with modern nuclear potentials can help to resolve remaining inconsistencies among data sets and can be used to predict these cross sections when measurements are not available.

Figure 1

Calculated $p\text{−}{}^4\mathrm{He}$ phase shifts at $N_{\max }=13$ and $\hslash \Omega =20$ MeV obtained with up to fourteen states of the compound ${}^5\mathrm{Li}$ nucleus as a function of the number of ${}^4\mathrm{He}$ states included in the calculation. Solid (red) lines represent our most complete results. Also shown [dashed (brown) lines] are the results from Ref. [23], i.e., without ${}^5\mathrm{Li}$ square-intregrable eigenstates, as well as the phase shifts from Ref. [38] (crosses), previously shown in Ref. [23] as a term of reference. All values in this and the subsequent figures are in the laboratory frame.

Figure 2

Computed (lines) ${}^4\mathrm{He}{(p,p)}^4\mathrm{H}$ angular differential cross section at forward scattering angle ${\theta }_p={25}^{\circ }$ (a) and backscattering angle ${\theta }_p={141}^{\circ }$ (b) as a function of the proton incident energy compared with measurements (symbols) from Refs. [3, 4, 5, 6] and [10]. The solid (red) line corresponds to the most complete results in Fig. 1. Also shown [dashed (brown) lines] are the results from Ref. [23], i.e., without ${}^5\mathrm{Li}$ square-intregrable eigenstates.

Figure 3

Same as Fig. 2 but at the backscattering angle of ${\theta }_p={169}^{\circ }$ and in the range of proton incident energies near the ${}^5\mathrm{Li}$ resonances. Calculations including five and six ${}^4\mathrm{He}$ states are shown in addition to the most complete results. Experimental data are from Refs. [5, 8, 10], and [40].

Figure 4

Computed (lines) ${}^1\mathrm{H}{(\alpha ,p)}^4\mathrm{He}$ angular differential cross section at proton recoil angles ${\phi }_p=4^{\circ },{15}^{\circ },{20}^{\circ }$, and ${30}^{\circ }$ as a function of the incident ${}^4\mathrm{He}$ energy compared with data (symbols) from Refs. [9, 10, 11, 12, 13, 14, 15, 41], and [42]. In panel (b) we focus on the proton recoil angle ${\phi }_p={30}^{\circ }$, showing, in addition to the most complete results, calculations including five and six ${}^4\mathrm{He}$ states.

Figure 5

Relative difference (as a percentage) between the calculated elastic recoil cross section at $N_{\max }=13$ and that at $N_{\max }=11$ as a function of the proton angle ${\phi }_p$ for the helium incident energies $E_{\alpha }=3.2,6.0$, and $9.5$ MeV. Only the first two ${}^4\mathrm{He}$ states are accounted for in this study.