Off-energy-shell $p\text{−}p$ scattering at sub-Coulomb energies via the Trojan horse method

Two-proton scattering at sub-Coulomb energies has been measured indirectly via the Trojan horse method applied to the $p\hspace{0.5em}+\hspace{0.5em}d\to p\hspace{0.5em}+\hspace{0.5em}p\hspace{0.5em}+\hspace{0.5em}n$ reaction to investigate off-energy shell effects for scattering processes. The three-body experiment was performed at 5 and 4.7 MeV corresponding to a $p\text{−}p$ relative energy ranging from 80 to 670 keV. The free $p\text{−}p$ cross section exhibits a deep minimum right within this relative energy region due to Coulomb plus nuclear destructive interference. No minimum occurs instead in the Trojan horse $p\text{−}p$ cross section, which was extracted by employing a simple plane-wave impulse approximation. A detailed formalism was developed to build up the expression of the theoretical half-off-shell $p\text{−}p$ cross section. Its behavior agrees with the Trojan horse data and in turn formally fits the $n\text{−}n$, $n\text{−}p$, and nuclear $p\text{−}p$ cross sections given the fact that in its expression the Coulomb amplitude is negligible with respect to the nuclear one. These results confirm the Trojan horse suppression of the Coulomb amplitude for scattering due to the off-shell character of the process.

Figure 1

Pole diagram describing the QF $p+d\to p+p+n$ process.

Figure 2

Theoretical HOES $p\text{−}p$ scattering cross section from Eq. (22).

Figure 3

Schematic description of the experimental setups for the two experiments. Green and red colors help to visualize the coincidences between the detectors building up the trigger to the acquisition (see text for details).

Figure 4

Schematic description of a PSD layout.

Figure 5

Locus of events in the plane defined by the energies of the coincident particles $E_p$ and $E_p$. The kinematical locus for the ${}^2\mathrm{H}$$(p,\mathit{pp})n$ reaction is apparent.

Figure 6

Angle of one of the two detected protons ${\theta }_p$ vs. the $Q_{\mathrm{value}}$.

Figure 7

Three-body coincidence yield ${\sigma }_3$ projected onto the proton energy axis $E_p$. The dashed line represents the contribution of the $n\text{−}p$ FSI.

Figure 8

Experimental neutron momentum distribution for the ATOMKI (full circles) and Naples runs (open circles). The dashed line represents the shape given by the square of the $n\text{−}p$ Hulthén bound-state wave function in momentum space.

Figure 9

Experimental three-body cross section (filled circles) vs. proton laboratory energy $E_p^{\mathrm{lab}}$ from the Atomki (a) and the Naples runs (b). Solid lines represent: (red) the calculated OES $p\text{−}p$ cross section; (green) the calculated nuclear $p\text{−}p$ cross section; (blue) the calculated HOES $p\text{−}p$ cross section.

Figure 10

(a) THM two-body cross section (black and green dots from present experimental work, red triangles and blue stars from previous work [15]) vs. $E_{\mathit{pp}}$. Solid line represents the theoretical OES $p\text{−}p$ cross section calculated as explained in the text. The red solid line is the HOES cross section calculated using Eq. (22). (b) Weighted average of all the experimental data shown in (a) vs. $E_{\mathit{pp}}$ with the same meaning for the solid lines as above.

Figure 11

(a) THM two-body cross section (black and green dots from present experimental work, red triangles and blue stars from previous work [15]) vs. $p\text{−}p$ relative energy $E$ compared with the on-shell $n\text{−}n$ (dashed-dotted line), $p\text{−}n$ (dashed line), and pure nuclear $p\text{−}p$ (dotted line) ones. The HOES calculated cross section is also reported as red solid line. (b) Weighted average of all the experimental data shown in (a) vs. $E_{\mathit{pp}}$. The blue solid line represents the theoretical HOES $p\text{−}n$ cross section. The other lines have the same meaning as in (a).