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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} = E/mv^{2}/2E_{0}).
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.
For small scattering angles (q → 0), the energy differential crosssection, up to a collection angle β, can be given by, [7]
 [3375f]
where,
v  the velocity of the incident electron,
m  the mass of the incident electron,
R  the Rydberg energy (see page4433),
E_{0}  the incident electron energy.
Equation 3375f is valid within the dipole region, where the GOS (generalised oscillator
strength) may be approximated to the optical value f(0,E).
In EELS experimental measurements, the number of atoms in the beam corresponding to, as a function of the coreloss inelastic crosssection, can be given by, [4]
 [3375g]
where,
Δ  a same energy window above the zeroloss peak and coreloss,
I_{c}(Δ,β)  the number of
counts into a coreloss edge, i.e. the number of counts
above the edge after background is subtracted,
I_{0}(Δ,β)  the number of counts in the energy window Δ above the zeroloss peak,
σ_{c}(Δ,β)  an appropriate inelastic crosssection obtainable either from theory or from a standard,
β  the collection semiangle of EELS,
B  the effective area of the electron beam.
Table 3375 lists the energy losses and scattering angles of various inelastic electron scatterings in electron interaction with materials.
Table 3375. Energy losses and scattering angles of various inelastic electron scatterings in electron interaction with materials. 
Process 
Phonon excitation 
Plasmon excitation (oscillation): (Introduction)
(Theory) 
Direct radiation losses 
Inner shell ionization 
Electrons in outermost shell: only weakly
bound to atoms but are coupled to each other by electrostatic
forces 
Intraband transitions 
Bremsstrahlung radiation due to deceleration of the electron beam in the Coulomb field of an atom 
Coreloss 
Electrons in valence band 
Electrons in conduction band 
Interband transitions 
Excitation of loosely bound electrons in conduction band, leading to secondary electrons (SEs) emissions 
Thermal diffuse scattering, heating 
Main transitions of valence electrons [5] 
Special note 
It normally cannot be resolved in EELS due to the energy spread of the electron
source 
~20% contribution of peak broadening in EELS at large scattering angles (> 20 mrad) [5] 
Most SEs have
a kinetic energy of <50 eV; The resulting SE distribution is peaked at 3 to 5 eV, with the
distribution decreasing sharply as the energy
increases above 5 eV 



Region of
energy loss 
Low loss
(< 50 eV) 
Low loss
(< 50 eV): Energy loss is low because these electrons require only small amounts of energy for excitation due to loosely bound 


High loss
(> 50 eV) 
Energy loss E [eV] 
20 meV–1 eV 
3  25 


10  2000 


Controlled by the density of loosely bound states 



Materials 

Predominant in insulators and semiconductors 
Predominant in metals 



EELS background 

Predominated by interband transitions of the valence electrons in the absence of
plural plasmon scattering, i.e. in sufficiently thin samples [5] 




EELS broadening 

Main origin of peak broadening in EELS [5] 



Interaction 

Most common inelastic interaction due to high free electron density;
Inelastic scattering is mainly brought
about by the collisions with the loosely bound solidstate electrons 



Scattering angle θ_{E} [mrad] 
5  15 
< 10 


0.1  10 
Oscillation 
Collective oscillations of atoms (e.g. lattice vibrations) 
Collective oscillations of free electrons, a quantum
of a collective longitudinal
wave in the electron gas of a
solid 



Detectability by EELS 
Is not resolved 
Yes 


Yes 
Effects 
Causes specimen to heat up 





Time 

Damped out in < 10^{−15} s 



Localization 

Localized to < 10 nm 



Cross sections 

Relatively large 


Relatively small 
Meanfree
paths 

Relatively short 


Relatively large 
Intensity 

Much intense 


Much smaller 
Modification 
Can reduce the number
of phonons by cooling the
specimen 




Kikuchi patterns 

May be important in the formation
of Kikuchi patterns from light elements [6] 



Characteristics 
Diffuse background, don’t carry any useful information 
Signature of the structure 


Elemental information 
[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 Al_{2}O_{3}, 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.
[4] A. P. Stephens, Quantitative microanalysis by electron energyloss spectroscopy: Two corrections, 5 (1–3) (1980), 343349.
[5] R. F. Egerton, Inelastic scattering of 80 keV electrons in amorphous carbon, 31(1), 1975, 199215.
[6]
Philip, J. G., Whelan, M. J., And Ecerton, R. F.. 1974, Proc. 8th Int. Congress on
Electron Microscopy, Canberra. Vol. 1, p. 276.
[7] P.J. Thomas and P.A. Midgley, An introduction to energyfiltered transmission electron microscopy, Topics in Catalysis, 21 (4), (2002), 109.
