Question

Calculate the iterval (in `eV` units) between neighbouring levels of free electrons in a metal at `T=0` near the Fermi level, if the concentration of free electrons is `n= 2.0.10^(22)cm^(-3)` and the volume of the metal is `V=1.0cm^(3)`.

Answer 1

We write expression for the number of electrons as
`dN=(Vsqrt(2)m^(3//2))/(pi^(2)ħ^(3))E^(1//2)dE`
Hence if `DeltaE` is the spacing between neighbouring levels near the Fermi level we must have
`2=(Vsqrt(2)m^(3//2))/(pi^(2)ħ^(3))E_(F)^(1//2)DeltaE`
(`2` on the `RHS` is to take care of both spins `f` electrons). Thus
`DeltaE=(sqrt(2)pi^(2)ħ^(3))/(Vm^(3//2)E_(F)^(1//2))`
But `E_(F)^(1//2)=(ħ)/(sqrt(2)m^(1//2))(3 pi^(2)n)^(1//3)`
So `DeltaE=(2 pi^(2)ħ^(2))/(mV(3pi^(2)n)^(1//3))`
Substituting the data we get
`DeltaE= 1.79xx10^(-22)eV`