Question

Use Eq. 41-9 to verify 7.0eV as copper’s Fermi energy.

Short answer

The Fermi energy of the copper metal is 7.0eV .

Step-by-step solution

The given data

The number of conduction electrons per unit volume of copper, $n=8.43\times {10}^{28}{\mathrm{m}}^{-3}$

the concept of Fermi energy

The highest value of energy that an electron can have at absolute zero temperature, is known as Fermi energy $\left(E_F\right)$. The energy level that is occupied by such electrons is called the Fermi level. The formula for Fermi energy is given below-

${\mathbf{E}}_{\mathbf{F}}\mathbf{=}{\left[\frac{3}{16\sqrt{2}\mathrm{\tau \tau }}\right]}^{\mathbf{2}\mathbf{/}\mathbf{3}}\frac{{\mathbf{h}}^{\mathbf{2}}}{\mathbf{m}}{\mathbf{n}}^{\mathbf{2}\mathbf{/}\mathbf{3}}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\left(1\right)$.

where n is the number of conduction electrons per unit volume, m is the mass of an electron and h is Planck’s constant.

the Fermi energy of Copper

Using the given values in equation (1) , the Fermi energy of the Copper metal can be given as follows:

$$\begin{gathered} {\mathrm{E}}_{\mathrm{F}}={\left[\frac{3}{16\sqrt{2}\mathrm{\pi }}\right]}^{2/3}\frac{{\left(\mathrm{hc}\right)}^2}{\left({\mathrm{m}}_{\mathrm{e}}{\mathrm{c}}^2\right)}{\mathrm{n}}^{2/3} \\ =\frac{0.12\times {\left(1240\mathrm{eV}.\mathrm{nm}\right)}^2}{511\times {10}^3\mathrm{eV}}\times \left(8.43{\mathrm{nm}}^{-3}\right) \\ =7.0\mathrm{eV} \\ \left(\because hc=1240eV.nm,K=m_e{c}^2=511\times {10}^3\mathrm{eV}\right) \end{gathered}$$

Hence, it is verified that the Fermi energy of copper is 7.0 eV .