Many-body and QED effects in electron-atom inelastic scattering in EELS

The elemental composition and electronic structure of materials analyzed by electron energy loss spectroscopy (EELS) are probed by the inner-shell ionization of atoms. This is a localized process that can be approximated by the scattering of an electron beam from a free atom. We calculate the inelastic differential cross section perturbatively within the framework of quantum electrodynamics (QED). The interaction between the incoming electron and the atom factorizes into a high-energy electron term and the atomic transition current. The matrix elements of the transition current are computed within the relaxed Dirac Hartree Fock method. We analyze the correlation effects arising from the relaxation of the atomic orbitals induced by the creation of a core hole. These effects are particularly relevant in quantum many-body systems and have a significant impact on the shape of the differential cross section near the ionization threshold in EELS spectra. In addition to the continuum, we calculate the discrete excitation spectrum of $\mathrm{DyScO_3}$ using crystal-field multiplet theory. The calculated spectrum shows very good agreement with experimental EELS data.

💡 Research Summary

The paper presents a comprehensive quantum‑electrodynamical (QED) treatment of electron‑atom inelastic scattering as it applies to core‑loss electron energy‑loss spectroscopy (EELS). Starting from the minimal‑coupling QED Lagrangian, the authors derive the S‑matrix for the scattering of a relativistic electron from a heavy atom and, after averaging over the unpolarized electron spin, factorize the transition amplitude into a high‑energy electron kinematic factor and a target transition current. By exploiting current conservation, the transition current is split into a longitudinal (charge) component and a transverse (current) component, each weighted by kinematic coefficients that depend on the momentum transfer and electron energies.

A multipole expansion of the charge density and transverse current in spherical harmonics is performed, yielding irreducible SO(3) tensor operators for Coulomb (scalar), electric (vector), and magnetic (axial) transitions. The differential cross‑section is expressed as a sum over multipole order J of reduced matrix elements ⟨Φβ‖T^{coul}_J‖Φα⟩, ⟨Φβ‖T^{el}_J‖Φα⟩, and ⟨Φβ‖T^{mag}_J‖Φα⟩, multiplied by longitudinal and transverse kinematic factors C_L(q) and C_T(q). This formulation is analogous to the description of photo‑ionization, highlighting the deep connection between inelastic electron scattering and photon absorption.

For the atomic structure, the authors employ a fully relativistic, self‑consistent Dirac‑Hartree‑Fock (DHF) calculation using the AMBIT code. Crucially, they include the core‑hole created by inner‑shell ionization explicitly, allowing the remaining electrons to relax around the vacancy. This “relaxed DHF” final state is shown to be equivalent to the time‑forward diagrams of the Random Phase Approximation with Exchange (RPAE), and it is contrasted with the frozen‑core and Z + 1 approximations. The inclusion of the core‑hole dramatically reshapes the differential cross‑section near the ionization threshold, reproducing the characteristic edge fine structure observed in experimental EELS spectra.

Beyond the continuum, the paper tackles the discrete excitation spectrum of the complex oxide DyScO₃. Using crystal‑field multiplet theory, the authors construct a many‑electron Hamiltonian that incorporates the full multiplet interaction, spin‑orbit coupling, and the crystal‑field potential for Dy 4f and Sc 3d electrons. The core‑hole is again treated self‑consistently within the Hamiltonian, ensuring that excitonic effects and charge‑transfer contributions are captured. Calculated total excitation and ionization cross‑sections for the Dy M‑edges and Sc L‑edges show excellent agreement with measured EELS data, reproducing both the overall intensity distribution and the subtle near‑edge fine structure.

The study’s key contributions are: (1) a rigorous QED‑based derivation of the inelastic electron‑atom scattering cross‑section, including a clear multipole decomposition that separates longitudinal and transverse channels; (2) the implementation of a relaxed DHF atomic model that self‑consistently accounts for core‑hole relaxation, providing a more accurate description of edge shapes than traditional frozen‑core or Z + 1 models; (3) the successful application of crystal‑field multiplet theory to a real material (DyScO₃), demonstrating that the same formalism can handle both continuum ionization and discrete multiplet excitations within a unified framework.

These results have immediate practical implications for quantitative EELS analysis in transmission electron microscopy. By incorporating the full QED kinematics and realistic atomic final states, the method enables more reliable extraction of elemental composition and electronic structure from core‑loss edges, especially for heavy elements where relativistic and many‑body effects are pronounced. The authors suggest future extensions to include higher‑order transverse photon corrections (Breit interaction, vacuum polarization), time‑dependent many‑body perturbation theory for dynamic correlation, and application to low‑dimensional or amorphous systems where the spherical‑symmetry assumption breaks down. Overall, the paper provides a solid theoretical foundation that bridges high‑energy scattering theory, atomic many‑body physics, and practical EELS spectroscopy.