Direct momentum matching

The genuine four-body nature of the basic one-electron transfer H⁺ + He(1s²) → H(1s) → He⁺(1s) offers the intriguing task of establishing the relative importance of the inter-electron and electron–projectile potentials. In principle, both interactions can lead to one-electron capture but the invoked mechanisms differ considerably from each other. Let _1,2 and x_1,2 be the relative vectors of e _1,2 with respect to Z_P,T. A large momentum m_ev_p could be transferred to e₁ only if the impinging proton p comes close to the electron e₁, which possesses a very small initial momentum in the target. In order to conserve the total momentum of the whole four-body collision system, the residual He⁺ ion in the exit channel must recoil in the backward direction.