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The loosely bound electrons in the valence and
conduction bands can oscillate collectively around the quasistationer set of nuclei, producing
oscillating polarization that can be described by a quasiparticle, called plasmon. As listed in Table 3417a, the bulk plasmon is a collective oscillation of the
loosely bound electrons. In EEL spectrum (EELS), the peak next to the zero loss peak is caused by excitation of surface and/or bulk plasmons. The energy region of the EEL spectrum up to the energy loss of ~50 eV is dominated by the collective excitations of valence electrons (plasmon) and by interband transitions. The plasmon peak is the second most dominate feature after the zeroloss peak. Typical values of the plasmon energies of materials are between 5 eV and 30 eV and it can vary due to some mechanisms (refer to Origin of Change of Plasmon Energy). The plasmon peaks are thus in the low energy loss region. The plasmons can provide information about the dielectric function [1], valence electron densities, and, in some cases, the phases presented in alloys.
Table 3417a. Main physical mechanisms which the energy and intensity of the plasmon peak represents. 
Materials

Main mechanisms 
General 
Oscillation of loosely bound electrons (conduction and valenceband electrons) 
Metals 
Excitation of freeelectron gas (electrons in conductionband) 
Semiconductor 
Resonance of valence electrons 
Plasmon consists of longitudinal wavelike oscillations of weakly bonded electrons. The collective oscillations are rapidly damped with a typical lifetime of ~10^{15} seconds and thus are localized to <10 nm. For many materials the plasmon energy, based on freeelectron theory (Drude model [2]), E_{plasmon}, is proportional to the square root of the density (n) of valence electrons (approximately equal to freeelectron density):
 [3417a]
= ħω(q)  [3417b]
where,
ħ  The reduced Planck constant.
n  The density of valence electrons.
e  The charge of the electron.
m_{e}  The effective mass of the electron.
ε_{0}  The permittivity of vacuum (or called the dielectric constant of free space).
For instance, the density (n) of valence electrons in Si_{3}N_{4} is given by,
 [3417c]
where,
ρ  The Si_{3}N_{4} atomic density (3.0 g/cm^{3}),
N_{A}  The Avogadro number,
A_{Si}
and A_{N}  The atomic weights of silicon and nitrogen, respectively,
n_{Si} and n_{N}  The numbers of valence electrons per silicon and per
nitrogen atom taking part in the plasmon oscillation, respectively. n_{Si}= 4 and n_{N} = 3~5.
Since the density of valance electrons in different materials such as Si_{3}N_{4}, SiO_{2}, and Si are different, the bulk plasmon energies are also different in these materials.
The plasmon
energy can be changed by:
i) the density of valence electrons in insulators and semiconductors as shown in Equation 3417a.
ii) the density of loosely bound states in metals. For instance, when pure aluminum (Al) is alloyed with another
element, the density of loosely bound states invariably
decreases, with a resulting change in the plasmon
energy.
Table 3417b shows the properties of surface and bulk plasmons. For instance, at the energy of 2000 eV of an incident electron beam, the Si 2p electrons with a binding energy of about 100 eV will have a kinetic energy of 1900 eV, while at the energy of 150 eV, the same Si 2p electrons will have a kinetic energy of about 50 eV. Therefore, bulk plasmon can be investigated with high energy EELS and plasmon in a thin surface layer can be studied with low excitation energy.
Table 3417b. Surface and bulk plasmons.

Comments 
Surface plasmon 
Transverse waves, half the energy of
bulk plasmons 
Bulk plasmon 
Longitudinal waves 
For thick TEM specimens, a significant portion of the electron beam has undergone inelastic scattering many times. In each scattering event, the incident electrons lose 16.7 eV [page4623] so that those electrons that have
scattered twice present an energy peak at 2 x 16.7 = 33.4 eV, those that scattered 3 times add a peak at 3x16.7=50.1 eV, and those that scattered 4 times at 4x16.7 = 66.8 eV, etc (see Figure 3417a).
Figure 3417a. Low energy loss of silicon (Si) TEM film with specimen thickness of 20 and 250 nm. 
The plasmon energy loss can experimentally provide indirect semiquantitative information because the freeelectron density n changes with the chemistry or bonding state of the TEM specimen. The empirical plasmon peak position (E_{p}) can be given by,
E_{p} = E_{p}(0) ± C(dE_{p}/dC)  [3417c]
where,
E_{p}(0)  The plasmon energy loss for the pure component (See the table on page4623).
C  The composition.
Therefore, once E_{p} is obtained from EELS measurement, then C can be evaluated. For instance, as shown in Figure 3417b, empirical sp^{2}/sp^{3} ratios in amorphous carbons were evaluated based on Equation 3417c and the relative intensity of kedge 1s to π* transition [48].
Figure 3417b. Fraction of sp^{3} bonding as a function of the bulk plasmon energy. [4]
The DrudeLorentz model presents the maximum intensity of the plasmon energy loss peak, given by, [3]
 [3417d]
where,
Γ  A constant describing the damping of the oscillation (which is full width at half maximum (FWHM) of the plasmon energy
loss peak)
And,  [3417e]
where,
 Energy of single damped harmonic
oscillator;
ε_{c}  Dielectric constant.
The dispersion relation of bulk plasmon for small wavevectors q is given by,
 [3417f]
where,
E_{P,0}  The plasmon energy at q = 0.
γ  An experimental coefficient.
Note that Table 4623 lists bulk plasmon energies, fullwidthathalfmaximum of bulk plasmon energies, bulk plasmon mean free path, and inelastic mean free path of some common elements and compounds, as well as their crystal structure.
The interpretation of the features of lowloss regions in EELS from freeelectron metals is fairly straightforward because the signals arise mainly from bulk and surface plasmon excitations and the energies of the peaks from bulk (E_{bulk}) and surface (E_{surface}) plasmon excitations have the simple relationship given by,
E_{bulk} = √2E_{surface}  [3417g]
For thick TEM specimens, e.g. ≥80 nm for Si, the surfaceplasmon effects becomes negligible, while the beam spreading becomes significant because more electrons suffer inelastic collision in larger scattering angles. In this case, more electrons scatter outside the finite collection aperture.
On the other hand, the interpretation of the features of the lowloss regions from transition metals is more difficult because of some factors. In transition metals, both plasmon losses and interband transitions induce energy loss and the plamon losses generally do not occur at the energies theoretically predicted by freeelectron model. Different from the case presented in Equation 3417g for freeelectron metals, there is no simple relationship between the bulk and surface plasmon loss energies for a transition metal due to too much dependence on the band structure. Moreover, some electrons in certain dbands may not be able to participate in collective plamon oscillations.
Note that the meanfreepath is also slightly dependent on the electron optical conditions of the microscopes, such as objective and collection aperture sizes. However, in most EELS measurements, this dependence is negligible.
Page3375 lists the behaviors and properties of various inelastic electron scatterings in electron interaction with materials, including inter and intraband transitions, inner shell ionization, phonon excitation and plasmon excitation.
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