Cross-section (Probability) of Inelastic Scattering in EELS Measurements
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Similar to incident x-rays, 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]

               Cross-section for Inelastic Scattering in EELS Measurements ------------------------------ [3375a]

where,
          ν -- A normalization volume for the probe electron [2];
         Cross-section for Inelastic Scattering in EELS Measurements -- The density fluctuation operator, given by,

          Cross-section for Inelastic Scattering in EELS Measurements ------------------------------ [3375b]
          Cross-section for Inelastic Scattering in EELS Measurements ----------------------------------------- [3375c]

Therefore, the double differential scattering cross-section (DDSCS) for inelastic electrons from the ground state |ψi> of the Hamiltonian H0 by applying Fermi's golden rule can be given by,

               Cross-section for Inelastic Scattering in EELS Measurements -------- [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,

               Cross-section for Inelastic Scattering in EELS Measurements -------- [3375e]

where,
          v -- The speed of the incident electron.
          na -- The number of atoms per unit volume.
          θE -- The characteristic scattering angle (θE = E/γm0v2).
          Im(−1/ε) -- The energy loss function.
          a0 -- 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 cross-section. This also is the most fundamental equation of EELS. Based on Equation 3375e and a Kramers–Kronig analysis, the complex dielectric function ε = ε1 + iε2 can be obtained from the low-loss EEL spectrum.
          
          



 

 

 

 

 

[1] H. J. Hagemann, W. Gudat, and C. Kunz, Optical Constants from the Far Infrared to the X-ray Region: Mg, Al, Cu, Ag, Au, Bi, C and Al2O3, DESY SR-7417. 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.

 

 

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