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Because the nuclear mass is much more than the rest mass of an electron (e.g. the electron has 1823 times less mass than a proton), the energy exchange is small and usually unmeasurable during the interaction of an incident electron with an atomic nucleus, acting as a physical process of elastic scattering. Various relatively simple models of elastic scattering are available in the literature. For instance, a freeatom Thomas–Fermi model with partial wave expansion [1] or Mott crosssections for a screened Coulomb potential [2] have been suggested. For elastic scattering by a single atom the simple atomic Lenz mode of the differential cross section that represents the probability of scattering per unit solid angle Ω is given by [3]
 [4774a]
or,
 [4774b]
Here θ is the
scattering angle of the electron, θ_{0}
the angular width of the scattering distribution, k_{0} is the electron wavenumber (2π/wavelength), E the ‘nonrelativistic’ incident energy , Z the atomic number, and e the elementary charge.
The angular width of the scattering distribution, sometimes called characteristic screening angle, (θ_{0}) is given by,
θ_{0} = (k_{0}r_{0})^{1}  [4774c]
where r_{0} = a_{0}Z^{ 1/3} is the screening radius of
atomic electrons, and a_{0} is the first Bohr radius (53 × 10^{−12} m).
It is necessary to mention that θ_{0} ≈ 100 θ_{E}, where θ_{E}
denotes the characteristic angular range of inelastically scattered
electrons as given by,
θ_{E}=E/γm_{0}v^{2}  [4774d]
where γ is the usual
relativistic factor.
The ‘nonrelativistic’ incident energy (E) is given by
 [4774e]
where m_{0}c^{2} = 511 keV is the electron rest energy. E is less than the actual kinetic energy E_{0} because the relativistic mass of the electron exceeds its rest mass m0 as shown in Table 4774a.
Table 4774a. Parameters E, γE , γ = m/m_{0}, deBroglie
wavelength λ and wavenumber k_{0} = 2π/λ, as a function of the
kinetic energy E_{0} of an electron. [4]
Figure 4774 gives the curves showing the angular distribution of elastic and
inelastic scattering, calculated for 200 keV electrons in amorphous
carbon. The angular distribution
of elastic scattering (θ_{0}) is quite narrow for the incidentelectron
energies used in a TEM (about 20 mrad or 0.35° for incident electrons at 100 keV). To be scattered through
a large angle (θ >> θ_{0}), the electrons must pass very close
to the nucleus.
Figure 4774. Curves show the angular distribution of elastic and
inelastic scattering, calculated for 200 keV electrons in amorphous
carbon. Vertical bars represent the relative intensities of the elastic
Bragg beams in crystalline diamond when the crystal axes are
parallel to the incident beam. [4]
The above theory is good for amorphous materials. Even though crystalline structures scatter electrons over a similar angular range, the scattering is confined to Bragg angles as seen in Figure 4774. The elastic scattering is then called diffraction. The reason of why we have a width for the discrete diffraction spots (black rods) instead of sizeless point is due to electronphonon scattering.
The first peak in EELS spectrum occurs at 0 eV, called the zeroloss peak. This is the most intense peak, especially for a very thin specimen. It represents electrons that did not undergo inelastic scattering but which might have been scattered elastically through interaction with atomic nuclei. The electrons recorded in the zeroloss peak may still have a very small energy loss, which is not measurable.
Table 4774b shows that electrons interact with 1 electron, many electrons, 1 nucleus, and many nuclei in solids.
Table 4774b. Effects of interactions of electrons in solids.

Interaction with electron(s) 
Interaction with nucleus/nuclei 

1 electron 
Many electrons 
1 nucleus 
Many nuclei 
Scattering type 
Inelastic 
Inelastic 
Quasielastic 
Elastic 
Inelastic 
Scattering effect 
Electron Compton effect; electron excitation (from 50 eV to a few keV: EDS and EELS) 
Plasmon excitation (< 50 eV, ~100 nm TEM specimen); Cerenkov effect 
Rutherford scattering; phonon scattering (< 1 eV, heat) 
Bragg scattering 
Bremsstrahlung 
[1] Ichimura, S., Aratama, M., Shimizu, R., 1980. Monte Carlo calculation
approach to quantitative Auger electron spectroscopy. J. Appl. Phys.
51, 2853–2860.
[2] Shimizu, R., Kataoka, Y., Matsukawa, T., 1975. Energy distribution
measurement of transmitted electrons and Monte Carlo simulation for
kilovolt electron. J. Phys. D 8, 820–828.
[3] Lenz, F., 1954. Zur Streuung mittelschneller Elektronen in kleinste Winkel.
Z. Naturf. 9a, 185–204.
[4] Egerton, R. F., 2009. Electron energyloss spectroscopy in the TEM, Rep. Prog. Phys. 72, 016502, 25.
