Quadratic response theory of the energy loss of charged particles in an electron gas

A quadratic response theory of the energy loss of charged particles in matter is developed, following a diagrammatic analysis of the many-body interactions between a moving charge and the electron gas. The linear and quadratic density response functions of the medium are evaluated in the random-phase approximation (RPA), and a result for the stopping power is obtained, up to third order in the ion charge, Z₁e, for a wide range of particle velocities. The low- and high-velocity limits of the full RPA result are also studied, and a local plasma approximation is used to show that our calculations are in good agreement with measurements of the energy loss of protons and antiprotons in silicon.