# Fraction of electrons within kT of fermi energy?

#### Dawei

I was reading this website, and I don't understand this last statement.
http://hyperphysics.phy-astr.gsu.edu/hbase/solids/fermi.html

It reads: "Since only a tiny fraction of the electrons in a metal are within the thermal energy kT of the Fermi energy, they are "frozen out" of the heat capacity by the Pauli principle."

Does anyone know how I could derive the general equation for that fraction that is within kT of the Fermi energy?

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#### Physics Monkey

The basic idea is to consider the difference between the Fermi function $$f(E) = (\exp{(\beta(E - \mu))} + 1)^{-1}$$ at finite temperature and zero temperature. The quantity $$f_{T \neq 0 } - f_{T = 0}$$ is only non-zero as a function of energy within a window of size kT around the chemical potential. So only electrons within an energy window of size kT around the Fermi energy are perturbed by a finite temperature. The energy window kT times the density of states at the Fermi energy gives a rough estimate of the number of electrons active at finite temperature. The most precise formulation is just in terms of the difference between the distribution functions that you can use to compute all physical observables as a function of temperature.