GATE 2026 PH – Question 63
Copper has an electron number density of $8.3\times10^{28}\ \mathrm{m^{-3}}$. Its Fermi energy in eV (rounded off to one decimal place) is _____ ($\hbar=1.06\times10^{-34}$ J s, mass of electron $m_e=9.10\times10^{-31}$ kg, charge of electron $=1.60\times10^{-19}$ C)
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Correct answer: 6.96 to 7.04
Explanation
For a free-electron gas, the **Fermi energy** is
$$E_F=\frac{\hbar^2}{2m_e}\left(3\pi^2n\right)^{2/3}.$$
**Numbers.** $n=8.3\times10^{28}\text{ m}^{-3}$:
- $3\pi^2n=3\times9.8696\times8.3\times10^{28}=2.457\times10^{30}\text{ m}^{-3}$
- $(2.457\times10^{30})^{2/3}=1.821\times10^{20}\text{ m}^{-2}$
- $\hbar^2=(1.06\times10^{-34})^2=1.1236\times10^{-68}$
$$E_F=\frac{1.1236\times10^{-68}\times1.821\times10^{20}}{2\times9.10\times10^{-31}}=\frac{2.046\times10^{-48}}{1.82\times10^{-30}}=1.124\times10^{-18}\text{ J}.$$
**In electron-volts:**
$$E_F=\frac{1.124\times10^{-18}}{1.60\times10^{-19}}=\mathbf{7.0\ eV}.$$