The density of states of a system with a periodic potential can be split up into each band,
Sn(ε) -- A surface of constant energy.
For instance, the density of states at the Fermi energy is obtained by an integral over the Fermi surface.
By solving the Schrödinger equation for the potential in a solid, the electron states in the solid can be expressed by the wave function, ψ(r). During the scattering process induced by the incident electrons in TEM-EELS measurements, each ejected electron makes a transition from an initial state, ψi(r), to an originally unoccupied and final state, ψf(r). Based on Fermi's golden rule, the intensity (IE, θ) in the ionization edge at a given energy loss (E) in EELS profile and scattering angle (θ) of the incident electrons, reflecting the probability of the transition (Ti→f) is given by,
a0 -- The Bohr radius;
ρ(E) -- The density of states (DOS) at a given energy loss.