Quantum treatment of the capture of an atom by a fast nucleus incident on a molecule

The classical double-scattering model of Thomas for the capture of electrons from atoms by fast ions yields a cross section $\sigma$ which dominates over the single scattering contribution for sufficiently fast ions. The magnitude of the classical double-scattering $\sigma$ differs, however, from its quantum-mechanical (second-Born) analog by an order of magnitude. Further, a "fast ion" means an ion of some MeV, and at those energies the cross sections are very low. On the other hand, as noted by Bates, Cook, and Smith, the double-scattering cross section for the capture of atoms from molecules by fast ions dominates over the single-scattering contribution for incident ions of very much lower energy; roughly, one must have the velocity of the incident projectile much larger than a characteristic internal velocity of the particles in the target. It follows that we are in the asymptotic domain not at about 10 MeV but at about 100 eV. For the reaction ${\mathrm{H}}^{+}$ + C${\mathrm{H}}_4$→${\mathrm{H}}_2^{+}$ + C${\mathrm{H}}_3$ with incident proton energies of 70 to 150 eV, the peak in the angular distribution as determined experimentally is at almost precisely the value predicted by the classical model, but the theoretical total cross section is about 30 times too large. Using a quantum version of the classical model, which involves the same kinematics and therefore preserves the agreement with the angular distribution, we obtain somewhat better agreement with the experimental total cross section, by a factor of about 5. (To obtain very good agreement, one may have to perform a really accurate calculation of large-angle elastic scattering of protons and H atoms by C${\mathrm{H}}_3$, and take into account interference effects.) In the center-of-mass frame, for sufficiently high incident energy, the first of the two scatterings involves the scattering of ${\mathrm{H}}^{+}$ by H through an angle of very close to 90°, and it follows that the nuclei of the emergent ${\mathrm{H}}_2^{+}$ ion will almost all be in the singlet state. We have also calculated the cross section for the reaction ${\mathrm{D}}^{+}$ + C${\mathrm{H}}_4$→${\mathrm{}\left(\mathrm{HD}\right)\mathrm{}}^{+}$ + C${\mathrm{H}}_3$.