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

TL;DR

This work presents a relativistic, QED-based framework for core-loss EELS, deriving a fully general inelastic differential cross section expressed as a multipole expansion in terms of reduced transition matrix elements. By solving Dirac-Hartree-Fock for the atomic structure and explicitly including core-hole relaxation via a relaxed final state (with connections to RPAE), the authors capture both near-threshold spectral shaping and high-energy behavior, validated through Ru/W ionization edges and DyScO$_3$ excitation spectra described by crystal-field multiplet theory. The study demonstrates that core-hole relaxation substantially influences edge shapes and that a fully relativistic treatment, including transverse photon contributions and Breit-type QED corrections, provides improved agreement with experimental EELS data and supports enhanced elemental quantification. These results establish a rigorous, ab initio pathway to interpret core-loss EELS across complex materials, with clear avenues for incorporating decay processes, discrete-continuum coupling, and oriented media in future work.

Abstract

The elemental composition and electronic-structure of materials analyzed by Electron Energy Loss Spectroscopy (EELS) are probed by the inner-shell ionization of atoms. 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. This is a localized process that can be approximated by the scattering of an electron beam from a free atom. 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.

Figures (12)

• Figure 1: The tree level diagram for the inelastic scattering of an incoming electron ($| i \rangle \to | f \rangle$) from a heavy target which is excited from the initial multi-particle state $| \Phi_\alpha \rangle$ to the final $| \Phi_\beta \rangle$. In our case $| \Phi_\beta \rangle$ will be either an ionized or an excited state.
• Figure 2: The coordinate system of the expansion of the vector current operator is defined with respect to the momentum transfer vector.
• Figure 3: The large component of the Dirac wavefunctions of the Ru-$3d_{5/2}$ orbital and the ejected electron in terms of the radial distance $r$ at energy 30eV (a) and 200eV (b) above the ionization threshold for the relaxed and frozen DHF approximations. The wave function of the vacancy orbital in the frozen and relaxed cases are denoted as VF : vacancy frozen and VR : vacancy relaxed. The continuum wavefunction for $\kappa =-2,-4$ in the frozen and relaxed cases are denoted CF: continuum frozen, CR : continuum relaxed.
• Figure 4: Ru-$\mathrm{M}_5$ (a) and W-$\mathrm{M}_5$ (b) differential cross section in terms of the energy loss of the incoming electron Eq. (\ref{['QEDDCSFinal']}). For each atom, the ground state is calculated within the DHF framework nd the final state is calculated in the relaxed (orange curves) and frozen (blue curves) approximations. Correlation effects such as the orbital relaxation due to the core hole make the spectrum shape smoother close to the ionization threshold compared to the frozen core approximation. The beam accelaration voltage is $300$ keV and the collection angle $100$ mrad.
• Figure 5: The comparison of the differential ionization cross sections for O-K (a) and Si-K (b). Relaxed core-hole: the atomic orbitals relax under the influence of the core-hole potential. Decayed core-hole the core-hole decays to be filled in by a valence electron. Frozen N+1: the final atomic bound states are the same as in the neutral atom. Frozen core-hole the final state has one vacancy in the ionized shell, but all the orbitals are the same as in a neutral atom. The beam acceleration voltage is $V_{beam}=300$ kV and the maximum scattering angle is $\theta = 100 \,mrad$.