GATE 2022 PH – Question 29
Potassium has electron concentration $1.4\times10^{28}\ \mathrm{m^{-3}}$ and Fermi-level density of states $6.2\times10^{46}\ \mathrm{J^{-1}m^{-3}}$. Express Pauli susceptibility as $n\times10^{-k}$ in standard scientific form; find k. Given $\mu_B=9.3\times10^{-24}$ J/T and $\mu_0=4\pi\times10^{-7}$ T m/A.
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Correct answer: 6
Explanation
**Pauli (paramagnetic) susceptibility** of a metal:
$$\chi=\mu_0\mu_B^2\,g(E_F),$$
where $g(E_F)$ is the density of states at the Fermi level (per unit volume and per unit energy, counting both spins).
**Numbers.** $\mu_0=4\pi\times10^{-7}$, $\mu_B=9.3\times10^{-24}$ J/T, $g(E_F)=6.2\times10^{46}\ \text{J}^{-1}\text{m}^{-3}$:
$$\chi=4\pi\times10^{-7}\times(9.3\times10^{-24})^2\times6.2\times10^{46}$$
$$=1.2566\times10^{-6}\times8.649\times10^{-47}\times6.2\times10^{46}=6.74\times10^{-6}.$$
Written as $n\times10^{-k}$ in standard form: $6.74\times10^{-6}$, so $k=\mathbf{6}$.