Energy Levels in a Crystalline Solid — Question 9

Question 9

Pauli occupancy changes over an energy range set by thermal energy

Use the Fermi function $f(E)=1/[1+e^{(E-\mu)/(k_BT)}]$. Define $E_{90}$ and $E_{10}$ where f equals .90 and .10 respectively.

1. Find these energies relative to the chemical potential.

2. Find their separation at 300 K using $k_B=8.617\times10^{-5}\,\mathrm{eV/K}$.

Question 9 - Solution

Pauli occupancy changes over an energy range set by thermal energy

(a) Method and calculation

$$f=.90\Rightarrow e^{(E_{90}-\mu)/(k_BT)}=1/9\Rightarrow\boxed{E_{90}=\mu-k_BT\ln9}.$$$$f=.10\Rightarrow\boxed{E_{10}=\mu+k_BT\ln9}.$$

(b) Interpretation and further calculation

$$\Delta E=E_{10}-E_{90}=2k_BT\ln9.$$$$\boxed{\Delta E=2(8.617\times10^{-5})(300)\ln9=.11360\,\mathrm{eV}}.$$

Physical check

The probability is one-half at the chemical potential. These are occupation probabilities per available spin state, not the density of states.