A fully relativistic description of valence energy losses suffered by fast electrons passing near finite clusters of arbitrarily disposed dielectric objects is presented using an accurate technique suited to solve Maxwell’s equations. The method is based upon an expansion of the electromagnetic field in terms of multipoles around each of the objects of the cluster. Multiple elastic scattering of those multipole expansions is then performed until convergence is achieved. The energy loss, obtained from the induced electric field acting back on the electron, is computed in a time proportional to the square of the number of objects in the cluster, $N^2.$ Numerical examples are presented for various clusters formed by $N=1\mathrm{}–198$ homogeneous spheres made of ${\mathrm{SiO}}_2$ and Al, and also for clusters of Si spheres coated with ${\mathrm{SiO}}_2.$ Both relativistic effects and the interaction between the constituents of the cluster are shown to be of primary importance in the understanding of the position and magnitude of the features exhibited by the calculated electron-energy-loss spectra.

• Received 26 January 1999

DOI:https://doi.org/10.1103/PhysRevB.60.6103