Dipole approximation in electron-energy-loss spectroscopy:K-shell excitations

We investigate the conditions for the validity of the ‘‘dipole approximation’’ in electron-energy-loss spectroscopy (EELS) of atomic K-shell excitations on an analytic model that takes account of the wave function of the ejected core electron, as well as that of the core state. We derive, for the first time, a closed-form expression for the limiting magnitude $q_d$ of the momentum-transfer vector, as a function of both the atomic number and the energy ɛ of the ejected core electrons. We find that the value currently assumed, namely $q_d$=$Z^{\mathrm{*}}$, where $Z^{\mathrm{*}}$ is the effective nuclear charge, is only strictly valid in the limit of low energies of the ejected core electron. In fact $q_d$ decreases noticeably with increasing ɛ. Matrix elements with many different values of q contribute to a typical EELS signal, but the dominant ones lie close to the minimum value $q_{\min }$. The increase of $q_{\min }$ with ɛ coupled with the decrease of $q_d$ with the same quantity makes it easier to satisfy the conditions for the dipole approximation in the near edge rather than the extended-fine-structure region of the energy-loss spectrum. Our analytic approach is well suited for extension to the cases of other absorption edges (e.g., L, M, edges, etc.).