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### Fundamentals Of Physics

Book edition 10th Edition
Author(s) David Halliday
Pages 1328 pages
ISBN 9781118230718

# Show that P(E), the occupancy probability in Eq. 41-6, is symmetrical about the value of the Fermi energy; that is, show that ${\mathbit{P}}{\mathbf{\left(}}{{\mathbit{E}}}_{{\mathbf{F}}}{\mathbf{+}}{\mathbit{\Delta }}{\mathbit{E}}{\mathbf{\right)}}{\mathbf{+}}{\mathbit{P}}{\mathbf{\left(}}{{\mathbit{E}}}_{{\mathbf{F}}}{\mathbf{-}}{\mathbit{\Delta }}{\mathbit{E}}{\mathbf{\right)}}{\mathbf{=}}{\mathbf{1}}$.

It is shown that $P\left({E}_{F}+\Delta E\right)+P\left({E}_{F}-\Delta E\right)=1$ that is the occupancy probability is symmetrical about the value of the Fermi energy.

See the step by step solution

## Step 1: Understanding the concept of occupancy probability

The probability of a state to be occupied by and electron is referred to as the occupancy probability. At 0 K temperature, the states below the Fermi level have occupancy probability equal to and for the states above the Fermi level, its value is .

Formula:

The occupancy probability of the state with energy E is-

$P\left(E\right)=\frac{1}{{e}^{\left(E-{E}_{F}\right)/KT}+1}$ ( i )

Here ${E}_{F}$ is the Fermi energy, $k=8.62×{10}^{-5}eV/K$ and T is the absolute temperature.

## Step 2: Calculation of the given symmetrical condition of probability

Upon expansion in view of equation (i), the LHS value of the given equation can be solved as follows:

$LHS=P\left({E}_{F}+∆E\right)+P\left({E}_{F}-∆E\right)\phantom{\rule{0ex}{0ex}}=\frac{1}{{e}^{\left({E}_{F}+∆E-{E}_{F}\right)/KT}+1}+\frac{1}{{e}^{\left({E}_{F}-∆E-{E}_{F}\right)/KT}+1}\phantom{\rule{0ex}{0ex}}=\frac{1}{{e}^{∆E/KT}+1}+\frac{1}{{e}^{-∆E//KT}+1}\phantom{\rule{0ex}{0ex}}=\frac{{e}^{∆E/KT}+1+{e}^{∆E/KT}+1}{\left({e}^{∆E/KT}+1\right)\left({e}^{-∆E/KT}+1\right)}\phantom{\rule{0ex}{0ex}}$

On further solving,

$L.H.S=\frac{{e}^{∆E/KT}+1+{e}^{∆E/KT}+2}{{e}^{∆E/KT}+{e}^{∆E/KT}+2}\phantom{\rule{0ex}{0ex}}=1\phantom{\rule{0ex}{0ex}}=R.H.S$

Hence, the given condition is proved and this implies the symmetrical condition for occupancy probability.