EELS

Once the macroscopic dielectric function is known it is possible to calculate the electron energy loss function (EELS). In transmission electron energy loss spectroscopy one studies the inelastic scattering of a beam of high energy electrons by a target. The scattering rates obtained in these experiments is related to the dynamical structure factor S(q,ω) [1]. In the special case with wavevector q=0, S(q,ω) is related to the longitudinal macroscopic dielectric function. This is the long-wave limit of EELS.
For isotropic system the dielectric function is simply a scalar (1/3 Tr ε_e(ω)). In this case the long-wave limit of the electron energy loss function assumes the trivial form

with ε₁ and ε₂, respectively, the real and imaginary part of the dielectric function.

[1] S. E. Schnatterly, in Solid State Physics Vol.34, edited by H. Ehrenreich, F. Seitz, and D. Turnbull (Academic Press, Inc., New York, 1979).
[2] P. Romaniello, and P. L. de Boeij, Phys. Rev. B (accepted).

References

The three related Ph.D. theses, due to F. Kootstra (on TD-DFT for insulators), P. Romaniello (on TD-CDFT for metals), and A. Berger (on the Vignale-Kohn functional in extended systems) contain much background information, and can be downloaded from the SCM website.

The most relevant publications are
Efficient real-space approach to time-dependent density functional theory for the dielectric response of nonmetallic crystals F. Kootstra, P. L. de Boeij, and J. G. Snijders, J. Chem. Phys. 112, 6517 (2000).

Time-dependent current-density-functional theory for the metallic response of solids, P. Romaniello, and P.L. de Boeij, Phys. Rev. B71, 155108 (2005)

Analysis of the viscoelastic coefficients in the Vignale-Kohn functional: The cases of one- and three-dimensional polyacetylene J.A. Berger, P.L. de Boeij, R. van Leeuwen, Phys. Rev B71 (15): Art. No. 155104 (2005)

A relativistic two-component formulation of time-dependent current-density functional theory: application to the linear response of solids. P. Romaniello, P.L. de Boeij, J. Chem. Phys. (accepted) (2007)