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Q10P

Expert-verifiedFound in: Page 1273

Book edition
10th Edition

Author(s)
David Halliday

Pages
1328 pages

ISBN
9781118230718

**Show that the probability P(E) that an energy level having energy E is not occupied is${\mathbf{P}}\mathbf{\left(}\mathbf{E}\mathbf{\right)}{\mathbf{=}}\frac{\mathbf{1}}{{\mathbf{e}}^{\mathbf{-}\mathbf{\u2206}\mathbf{EIkT}}\mathbf{}\mathbf{+}\mathbf{1}}{\mathbf{where}}{\mathbf{}}{\mathbf{\u2206}}{\mathbf{E}}{\mathbf{=}}{\mathbf{E}}{\mathbf{-}}{{\mathbf{E}}}_{{\mathbf{F}}}$ where .**

The probability P(E) that an energy level having energy E is not occupied is$\frac{\mathbf{1}}{{\mathbf{e}}^{\mathbf{-}\mathbf{\u2206}\mathbf{EIkT}}\mathbf{}\mathbf{+}\mathbf{1}}$.

**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.**

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-

$P\text{'}+P=1\phantom{\rule{0ex}{0ex}}P\text{'}=1-P.........................................\left(1\right)\phantom{\rule{0ex}{0ex}}\mathrm{The}\mathrm{occupancy}\mathrm{probability}\mathrm{of}\mathrm{a}\mathrm{state}\mathrm{is}\mathrm{given}\mathrm{as}-\phantom{\rule{0ex}{0ex}}P=\frac{1}{{e}^{\u2206EIkT}+1}.......................................\left(2\right)\phantom{\rule{0ex}{0ex}}\mathrm{Using}\mathrm{equation}\left(1\right)\mathrm{and}\left(2\right),\mathrm{we}\mathrm{get}-\phantom{\rule{0ex}{0ex}}P\text{'}=\frac{1}{{e}^{\u2206EIkT}+1}\phantom{\rule{0ex}{0ex}}=\frac{{e}^{\u2206EIkT}}{{e}^{\u2206EIkT}+1}\phantom{\rule{0ex}{0ex}}=\frac{1}{\frac{{e}^{\u2206EIkT}}{{e}^{\u2206EIkT}+1}}\phantom{\rule{0ex}{0ex}}=\frac{1}{{e}^{-\u2206EIkT}+1}$

Hence, the probability of an unoccupied state is$\frac{1}{{e}^{-\u2206EIkT}+1},\mathrm{where}\u2206\mathrm{E}=\mathrm{E}-{\mathrm{E}}_{\mathrm{F}}$, where .

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