The relativistic distorted-wave Born approximation is used to calculate differential and total cross sections for inner shell ionization of neutral atoms by electron and positron impact. The target atom is described within the independent-electron approximation using the self-consistent Dirac-Fock-Slater potential. The distorting potential for the projectile is also set equal to the Dirac-Fock-Slater potential. For electrons, this guarantees orthogonality of all the orbitals involved and simplifies the calculation of exchange T-matrix elements. The interaction between the projectile and the target electrons is assumed to reduce to the instantaneous Coulomb interaction. The adopted numerical algorithm allows the calculation of differential and total cross sections for projectiles with kinetic energies ranging from the ionization threshold up to about ten times this value. Algorithm accuracy and stability are demonstrated by comparing differential cross sections calculated by our code with the distorting potential set to zero with equivalent results generated by a more robust code that uses the conventional plane-wave Born approximation. Sample calculation results are presented for ionization of K-and L-shells of various elements and compared with the available experimental data.

We present a fully relativistic formulation of the energy loss of a charged particle traversing a conductive monoatomic layer and apply it to the case of graphene in a transmission electron microscope (TEM). We use two models of conductivity appropriate for different frequency regimes: (a) THz (terahertz) frequency range and (b) optical range. In each range we distinguish two types of contributions to the electron energy loss: the energy deposited in graphene in the form of electronic excitations (Ohm losses), and the energy that is emitted in the form of radiation. We find strong relativistic effects in the electron energy loss spectra, which are manifested, e.g., in the increased heights of the principal π and σ + π peaks that may be observed in TEM in the optical range. While the radiative energy losses are suppressed in the optical range in comparison to the Ohmic losses, we find that these two contributions are comparable in magnitude in the THz range, where the response of doped graphene is dominated by the Dirac plasmon polariton (DPP). In particular, relative contributions of the Ohmic and radiative energy losses are strongly affected by the damping of DPP. In the case of a clean graphene with low damping, the angular distribution of the radiated spectra at the sub-THz frequencies exhibit strong and possibly observable skewing towards graphene.

We present a fully relativistic formulation of the energy loss of a charged particle traversing a number of graphene layers and apply it to the case of two spatially separated layers probed by an energetic electron. We focus on the THz frequency range, using a Drude model to describe the conductivity of graphene and allowing for different doping density in each layer. We distinguish two types of contributions to the electron energy loss: the energy deposited in graphene layers in the form of electronic excitations (Ohm losses), which include the excitation of Dirac plasmon polaritons (DPP), and the energy that is emitted in the form of transition radiation. We study in detail the contribution of each layer to the ohmic losses and analyze the directional decomposition of the radiation emitted in the half-spaces defined by the graphene planes. By increasing the interlayer distance and changing the relative doping densities in graphene layers, we find surprisingly strong asymmetries in both the directional and layer-wise decompositions with respect to the direction of motion of the external electron. A modal decomposition is also performed in the limit of vanishing damping in graphene, exposing quite intricate roles of bonding and antibonding hybridization between DPPs in ohmic losses.