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Q2P

Expert-verifiedFound in: Page 1272

Book edition
10th Edition

Author(s)
David Halliday

Pages
1328 pages

ISBN
9781118230718

**Calculate the density of states ${\mathit{N}}{\mathbf{\left(}}{\mathit{E}}{\mathbf{\right)}}$ for metal at energy ${\mathit{E}}{\mathbf{=}}{\mathbf{8}}{\mathbf{.}}{\mathbf{0}}{\mathit{e}}{\mathit{V}}$and show that your result is consistent with the curve of Fig. 41-6.**

The density of states $N\left(E\right)$ for metal is $1.9\times {10}^{28}{\mathrm{m}}^{-3}.{\left(\mathrm{eV}\right)}^{-1}$ and it is consistent with the curve of figure 41-6.

Energy of the metal, $E=8\mathrm{eV}$

**The number of states per unit energy range per unit volume ${\left(N\left(E\right)\right)}$, present in a sample of the material at a particular energy ${\left(E\right)}$, is known as density of states. The formula for density of states is given as-**

$N\left(E\right)=\frac{8\sqrt{2}{\mathrm{\pi m}}^{3/2}}{{h}^{3}}{E}^{1/2}.............................\left(1\right)\phantom{\rule{0ex}{0ex}}\mathrm{where}\mathrm{h}=6.63\times {10}^{-34}\mathrm{J}.\mathrm{s}\mathrm{and}\mathrm{m}=9.1\times {10}^{-31}\mathrm{kg}$

We can write equation (1) as follows:

$N\left(E\right)=C{E}^{1/2}$

In the above equation, the value of C is -

$C=\frac{8\sqrt{2}{\mathrm{\pi m}}^{3/2}}{{h}^{3}}\phantom{\rule{0ex}{0ex}}=\frac{8\sqrt{2}\mathrm{\pi}{\left(9.1\times {10}^{-31}\mathrm{kg}\right)}^{3/2}}{\left(6.63\times {10}^{-34}\mathrm{J}.\mathrm{s}\right)}\phantom{\rule{0ex}{0ex}}=1.062\times {10}^{56}{\mathrm{kg}}^{3/2}/{\mathrm{J}}^{3}.{\mathrm{s}}^{3}\phantom{\rule{0ex}{0ex}}=6.81\times {10}^{27}{\mathrm{m}}^{-3}.{\left(\mathrm{eV}\right)}^{-2/3}$

Using the given data in equation (1), the density of states for the metal with energy $E=8eV$can be calculated as follows:

$N\left(E\right)=\left[6.81\times {10}^{27}{\mathrm{m}}^{-3}.{\left(\mathrm{eV}\right)}^{-2/3}\right]{\left(8\mathrm{eV}\right)}^{1/2}\phantom{\rule{0ex}{0ex}}=1.9\times {10}^{28}{\mathrm{m}}^{-3}.{\left(\mathrm{eV}\right)}^{-1}$

This value of density of state is consistent with the given figure 41-6.

Hence, the value of the density of states is $1.9\times {10}^{28}{\mathrm{m}}^{-3}.{\left(\mathrm{eV}\right)}^{-1}$.

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