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Interband transitions are excited from occupied valence bands and shallow core levels to the unoccupied conduction bands. The electronic and optical interband transitions are categorized by direct and indirect transitions because the conduction band bottom and the valance band top contain more than one extrema. The electronic and optical band gaps can be calculated by the predominant mechanism of band to band transitions and are normally less than several volts. In the allowed direct transition, the electrons in the valence band transit vertically to the conduction band under the excitation of incident electrons or photons. The absorption coefficient as a function of photon energy in the allowed direct transition can be given by [1],
 [3952a]
where,
A  A constant
v  The frequency
h  The Planck constant
E_{gd}  The allowed direct energy gap.
The interband transitions correspond to the absorption peaks in the imaginary part of dielectric function ε_{2}. On the other hand, it is strongly modulated by ε^{2}_{1} + ε^{2}_{2} in energy loss function, which dumps the peak strength of interband transitions.
In the indirect transition, the electrons transit from the valence band top to the conduction band bottom with participation of photons and appropriate phonons. In this case, the absorption coefficient can be given by [2],
 [3952b]
where,
B  A constant
E_{gi}  The indirect energy gap
E_{p}  The energy of the absorbed (+) or emitted () phonons.
Interband transitions are an effect of the lattice. They shift the plasmon energy and can be included via the Drude–Lorentz theory. [3  5] The energy region of the EEL spectrum (EELS) up to the energy loss of ∼50 eV is dominated by the collective excitations of valence electrons (plasmon) and by interband transitions. In order to include interband transitions, we need to have a good understanding of the electronic structure. The EELS of the lowenergy loss region less than 50 eV is particularly called valence electron energy loss spectroscopy (VEELS).
Table 3952. Energy losses and scattering angles of various inelastic electron scatterings in electron interaction with materials. 
Process 
Phonon excitation 
Inter of intra band transitions 
Plasmon excitation 
Inner shell ionization 
Region of
energy loss 
Low loss
(< 50 eV) 
High loss
(> 50 eV) 
Energy loss E [eV] 
~0.02 
3  25 
5  25 
10  2000 
Scattering angle θ_{E} [mrad] 
5  15 
5  10 
< 0.1 
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 
Yes 
Effects 
Causes specimen to heat up 



Time 


Damped out in < 10^{−15} s 

Localization 


Localized to < 10 nm 

Interaction 


Most common inelastic interaction due to high free electron density 

Cross sections 


Relatively large 
Relatively small 
Meanfree
paths 


Relatively short 
Relatively large 
Intensity 


Much intense 
Much smaller 
Materials 


Predominant in metals 

Modification 


Can reduce the number
of phonons by cooling the
specimen. 

Other names 
Thermal diffuse scattering 



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

Elemental information 
Figure 3952a shows the EEL spectra of plasmon region (zeropeak is not shown) for crystalline and amorphous diamond. The loss function (Im[l/ε]) can then be obtained by removing contribution from multiple scattering using the Fourierlog deconvolution method.
Figure 3952a. EEL spectra of plasmon region for (a) crystalline and (b) amorphous diamond.
An optical absorption method can directly provide the imaginary part of the dielectric function, ε_{2}, associated with a single electron excitation of an interband transition, while EELS cannot directly give ε_{2}. Based on the obtained loss function, the real part (ε_{1}), and the imaginary part (ε_{2}) of the dielectric function for the crystalline and the amorphous diamond are extracted using KramersKronig analysis as shown in Figure 3952b. Table 3952 lists the onset energy (indicating the band gap energy) and the peaks in the imaginary spectrum obtained from Figure 3952b and the original interband transitions.
Figure 3952b. Real part (ε_{1}), and the imaginary part (ε_{2}) of the dielectric function for (a) crystalline and (b) amorphous diamond.
Table 3952. Onset energies (band gap energies), peaks in the imaginary spectrum in Figure 3952b and the original interband transitions.

Onset energy (band gap energy) 
Peaks in imaginary spectrum 
Interband transition 
Crystal diamond 
5.5 eV 
8.2 
Γ point 
12.7 
X and L points 
Amorphous diamond 
4.0 eV 
7.2 eV 
Γ point 
Note if the interband transitions are in the energy range less than ∼ 2 eV, these spectral features are very close to the intense zeroloss peak from elastic scattering, so that it is difficult to resolve them.
[1] Tauc JC (1972) Optical properties of solids. NorthHolland,
Amsterdam.
[2] ElKorashy A, ElFadl AA (1999) Phys B 271:205.
[3] R.F. Egerton, Electron Energy Loss Spectroscopy in the Electron
Microscope, Plenum Press, New York, 1996.
[4] P. Schattschneider, B. Jouffrey, in: L. Reimer (Ed.), Energyfiltering
Transmission Electron Microscopy, Springer, 1995, p. 151.
[5] H. Raether, Excitation of Plasmons and Interband Transitions by
Electrons, SpringerVerlag, Berlin, 1980.
