Question

Employing Eq. (6.4g), find at `T=0`
(a) the maximum kinetic energy of free electrons in a metal if their concentration is qual to `n`,
(b) the mean kinetic energy of free electrons if their maximum kinetic energy `T_(max)` is known.

Answer 1

(a) From the fromula
`dn=(sqrt(2)m^(3//2))/(pi^(2) ħ^(3))E^(1//2)dE`
the maximum value `E_(max)` of `E` is determined in terms of `n` by
`n=(sqrt(2)m^(3//2))/(pi^(2) ħ^(3)) int_(0)^(E_(max))E^(1//2)dE`
`=(sqrt(2)m^(3//2))/(pi^(2) ħ^(3))(2)/(3)E_(max)^(3//2)`
or `E_(max)^(3//2)=( ħ^(2)/(2m))^(3//2)(3 pi^(2)n)`
`E_(max)=( ħ^(2))/(2m)(3pi^(2)n)^(2//3)`
(b) Mean `K.E lt E gt` is
`lt E geint_(0)^(E_(max))Edn//int_(0)^(E_(max))dn`
`=int_(0)^(E_(max))E^(3//2)//int_(0)^(E_(max))E^1//2dE(2)/(5)E_(max)^(5//2)//(2)/(3)E_(max)^(3//2)=(3)/(5)E_(max)`