A comparison of Fraunhofer-type diffraction from an atomic single-slit and a molecular double-slit

We measured the Q-value and the scattering angle distributions for non-dissociative state selective single electron capture in collisions of 7.5 keV H$^+$ and 15 keV H$_2^+$ with He. The experimental data are compared with semiclassical close-coupling calculations and predictions from the classical trajectory Monte Carlo simulations. By analogy with Fraunhofer diffraction, we also developed a toy model to reconstruct an imaginary screen that reflects the reaction impact-parameter dependence, in channels where the magnetic quantum number remains unchanged. It is well established that H$_2^+$ acts as a molecular double-slit in scattering processes. By demodulating the Young’s double-slit-type interference pattern, we extracted the individual slit diffraction pattern of H$_2^+$ and compared it with that of the H$^+$ atomic single-slit. For ground state electron capture, we found that the single and the double slit diffraction patterns have equal fringe width, whereas for excited state electron capture, diffraction patterns are quite different.

💡 Research Summary

In this work the authors investigate non‑dissociative, state‑selective single‑electron capture in collisions of 7.5 keV H⁺ and 15 keV H₂⁺ projectiles with helium atoms, and they interpret the resulting angular‑distribution patterns in terms of Fraunhofer diffraction from an “atomic single slit” and a “molecular double slit”. The experiments were performed at the Tata Institute of Fundamental Research using an electron‑cyclotron‑resonance ion source, a 90° dipole mass selector and a cold‑target recoil‑ion momentum spectroscopy (COLTRIMS) apparatus. Momentum vectors of the recoil He⁺ ion and the neutral projectile after capture were recorded in coincidence, allowing the authors to extract the Q‑value (energy defect) and the laboratory scattering angle θ with high resolution (ΔQ≈3 eV, angular resolution ≈0.1°). The Q‑value spectra reveal two well‑separated peaks for each projectile: a high‑energy peak corresponding to capture into the ground electronic state (H(n = 1) for H⁺ and H₂(X¹Σ⁺_g) for H₂⁺) and a lower‑energy peak corresponding to capture into the first excited manifold (H(n = 2) and various excited H₂ states).

Theoretical analysis is carried out with three complementary approaches. (i) Two‑Center Atomic‑Orbital Close‑Coupling (TC‑AOCC) solves the one‑electron Schrödinger equation in the combined field of projectile and target, providing a fully quantum mechanical transition matrix for each impact parameter. (ii) Semiclassical Asymptotic‑State Close‑Coupling (SCASCC) treats the two‑electron problem (projectile electron plus target electron) semiclassically, retaining electron‑electron correlation. (iii) Classical Trajectory Monte‑Carlo (CTMC) simulations model all three nuclei and the active electron as classical point particles moving under Coulomb forces, with trajectories generated by a random‑sampling of initial conditions and integrated by a Runge‑Kutta scheme. All three methods reproduce the main features of the measured differential cross sections, but only the quantum‑mechanical TC‑AOCC and SCASCC capture the fine interference fringes.

The central conceptual framework is the analogy between the molecular ion H₂⁺ and a double‑slit interferometer. Within the Linear Combination of Atomic Orbitals (LCAO) picture the molecular wavefunction is Ψ = (Ψ_a ± Ψ_b), where Ψ_a and Ψ_b are the atomic orbitals centered on the two nuclei separated by ρ≈2 a.u. The transition matrix for the molecule can be written as T_mol =