Energy Transfers and Stopping Power

In CasP fast numerical calculations of the mean electronic energy transfer Q_e (due to excitation and ionization of target atoms) are performed for each individual impact parameter b in a collision. The total electronic energy-loss cross-section S_e (equivalent to the stopping power) is subsequently calculated from Q_e(b). The computation of Q_e and S_e accounts for a selected predefined projectile-screening function. By selecting a proper screening function, it is possible to treat non-equilibrium energy-loss phenomena. Furthermore, the unitary convolution approximation UCA (default selection) is a non-linear theory, as it includes the Bloch terms. Therefore, even for bare projectiles, the results do not scale with the square of the projectile charge (as most other quantum theories do).

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An ion may loose its kinetic energy by

(a) exciting or ionizing target atoms (including the collective plasmon excitations in solids)
(b) excitation or ionization of projectile electrons (projectile-electron loss), due to the interaction with screened target atoms
(c) capture of target electrons
(d) the kinematic energy transfer to the target nuclei, due to (quasi-) elastic collisions (the so-called nuclear energy loss)

The so-called electron-capture and loss cycles may be important for energies close to or below the matching velocity, where the projectile speed is equal to the orbital velocity of the target-electron shell under consideration (roughly 25 keV/u for the valence bands). At higher projectile velocities the energy transfer to target electrons becomes by far the dominant contribution to the slowing down of ions. For most screening options, only this electronic energy transfer to neutral target atoms - mechanism (a) - is included in the current version of the program CasP. A specific screening option, however, allows to calculate the contribution to equilibrium stopping powers due to projectile-electron loss plus electron capture.

The program CasP makes use of the convolution approximation (either the perturbative convolution approximation PCA or the more advanced unitary convolution approximation UCA). The physical inputs of the program are the projectile velocity, the projectile-screening potential, the target-electron density distribution (which we have tabulated using results from our Hartree-Fock-Slater code) and the oscillator strengths for the target electrons. The code is based on an exact matching of the quantum mechanical mean electronic energy transfers for the asymptotic region of very low and very high energy transfers to the target electron, similar as in the Bethe or in the Bloch theory. 

It  includes a simplified impact-parameter dependent relativistic correction  (no radiation energy loss and no relativistic density effect) and thus, it becomes inaccurate for kinetic energies per nucleon exceeding several 100 MeV/u.

The PCA is an impact-parameter dependent approximation to the first Born approximation (1st order perturbation theory, often also denoted semi-classical approximation SCA or, for total cross sections, Bethe theory). It is based the assumption of a classical straight-line motion of the projectile, which is well justified as long as the projectile is faster than a few keV/u (if projectile scattering angles are detected or energy loss due to inner-shell electrons is considered, the range of validity of this approximation shifts to higher energies). The PCA approximation converges slightly more rapidly towards the exact 1st order results for the total stopping cross section than the simple Bethe formula.

The UCA includes an approximation to the wavepacket formalism by Bloch, and thus each electron is counted only once in a collision and ionization probabilities are restricted to a maximum of 100% (unitarity). This is not the case for the PCA, which correspondingly is less suited for the prediction of heavy-ion stopping powers. Note, that the current version of the UCA should be accurate for very light and very heavy ions. Futhermore, polarization effects (e.g., the Barkas effect) due to close collisions are now  included in the current version of CasP.