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Similar to incident xrays, energetic incident electrons can also interact with electrons in materials. This process results in an amount ħq of momentum being transferred to the sample. In EELS measurements, the electron interaction is due to coupling between the electrons which may have the form, [1]
 [3375a]
where,
ν  A normalization volume for the probe electron [2];
 The density
fluctuation operator, given by,
 [3375b]
 [3375c]
Therefore, the double differential scattering crosssection (DDSCS) for inelastic electrons from the ground state ψ_{i}> of the Hamiltonian H_{0} by applying Fermi's golden rule can be given by,
 [3375d]
where,
ħω  The energy lost by the probe electron scattered into solid angle;
m  The mass of the electron;
Finally, the EEL spectrum can be described in a dielectric formulation [3] by,
 [3375e]
where,
v  The speed of the incident electron.
n_{a}  The number
of atoms per unit volume.
θ_{E}  The characteristic scattering angle (θ_{E} = E/γm_{0}v^{2}).
Im(−1/ε)  The energy loss function.
a_{0}  Bohr radius.
σ  The total scattering cross section.
Ω  The solid angle.
Equation 3375e fundamentally presents the probability of a specific scattering event in materials, expressed by the scattering crosssection. This also is the most fundamental equation of EELS. Based on Equation 3375e and a Kramers–Kronig analysis, the complex dielectric function ε = ε_{1} + iε_{2} can be obtained from the lowloss EEL spectrum.
[1] H. J. Hagemann, W. Gudat, and C. Kunz, Optical Constants from the Far Infrared to the Xray Region: Mg, Al, Cu, Ag, Au, Bi, C and Al2O3, DESY SR7417. Desy, Hamburg, W. Germany, 1974.
[2] W. S. M. Werner Surf. Interface Anal., vol. 31, p. 141, 2001.
[3] R. F. Egerton, Electron Energy Loss Spectroscopy in the Electron Microscope, Plenum Press, New York, 1996.
