Question

Show that the probability P(E) that an energy level having energy Eis not occupied is$\mathbf{P}\mathbf{\left(}\mathbf{E}\mathbf{\right)}\mathbf{=}\frac{\mathbf{1}}{{\mathbf{e}}^{\mathbf{-}\mathbf{∆}\mathbf{EIkT}}\mathbf{}\mathbf{+}\mathbf{1}}\mathbf{where}\mathbf{}\mathbf{∆}\mathbf{E}\mathbf{=}\mathbf{E}\mathbf{-}{\mathbf{E}}_{\mathbf{F}}$where .

Short answer

The probability P(E) that an energy level having energy E is not occupied is$\frac{\mathbf{1}}{{\mathbf{e}}^{\mathbf{-}\mathbf{∆}\mathbf{EIkT}}\mathbf{}\mathbf{+}\mathbf{1}}$.

Step-by-step solution

Understanding the concept of probability of unoccupied state

The probability of an electron occupying a certain energy state is called the occupancy probability of that state. The total probability, of an electron, occupying or not occupying a state is 1. Thus the probability of not occupying a state is equal to the difference between 1 and the probability of occupying a state.

Calculation of the probability that the energy level is not occupied

The probability that a state is occupied by an electron is P and the probability that the state is unoccupied by an electron is P' . Thus, we can write-

$$\begin{gathered} P\text{'}+P=1 \\ P\text{'}=1-P.........................................\left(1\right) \\ \mathrm{The}\mathrm{occupancy}\mathrm{probability}\mathrm{of}\mathrm{a}\mathrm{state}\mathrm{is}\mathrm{given}\mathrm{as}- \\ P=\frac{1}{e^{∆EIkT}+1}.......................................\left(2\right) \\ \mathrm{Using}\mathrm{equation}\left(1\right)\mathrm{and}\left(2\right),\mathrm{we}\mathrm{get}- \\ P\text{'}=\frac{1}{e^{∆EIkT}+1} \\ =\frac{e^{∆EIkT}}{e^{∆EIkT}+1} \\ =\frac{1}{{\displaystyle \frac{e^{∆EIkT}}{e^{∆EIkT}+1}}} \\ =\frac{1}{e^{-∆EIkT}+1} \end{gathered}$$

Hence, the probability of an unoccupied state is$\frac{1}{e^{-∆EIkT}+1},\mathrm{where}∆\mathrm{E}=\mathrm{E}-{\mathrm{E}}_{\mathrm{F}}$, where .