Question 8.7: Calculate the average energy of conduction electrons near ab...
Calculate the average energy of conduction electrons near absolute zero.
The average energy of the electrons may be obtained by multiplying each value of the energy \epsilon by the probability p(\epsilon) d \epsilon and integrating over all possible energies. We have
\epsilon _{av}=\int_{0}^{\infty }\epsilon P(\epsilon )d \epsilon .Substituting Eq. (8.73) into the above equation gives
P(\epsilon )d \epsilon =\frac{3}{2}\epsilon _{F}^{-3/2}\epsilon ^{1/2}d\epsilon \frac{1}{e^{(\epsilon -\epsilon _{F})/k_{B}T}+1}. (8.73)
\epsilon _{av}=\frac{3}{2}\epsilon _{F}^{-3/2} \int_{0}^{\infty }\epsilon ^{3/2}d\epsilon \frac{1}{e^{(\epsilon -\epsilon _{F})/k_{B}T}+1}.This last equation may be simplified by again using the fact that near absolute zero the function 1/(e^{(\epsilon−\epsilon_{F} )/k_{B}T} + 1) is equal to one for \epsilon<\epsilon_{F} and equal to zero for \epsilon>\epsilon_{F} . We thus obtain
\epsilon _{av}=\frac{3}{2}\epsilon _{F}^{- 3/2} \int_{0}^{\epsilon _{F} }\epsilon ^{3/2}d\epsilon.Evaluating the integral, we obtain
\epsilon _{av}=\frac{3}{5}\epsilon _{F}.