GATE 2022 PH – Question 36
A noninteracting spin-one gas has moment mu and $\mu B/(k_BT)\ll1$. Find its mean magnetic moment.
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Correct answer: (A) $2\mu^2B/(3k_BT)$
Explanation
A spin-1 particle with magnetic moment $\mu$ in a field $B$ has three states with $m=+1,0,-1$ and energies $E=-\mu Bm$. The magnetic moment components are $\mu m$ with $m=+1,0,-1$.
**Mean moment** with Boltzmann weights ($x=\mu B/k_BT$):
$$\langle\mu_z\rangle=\mu\,\frac{e^{x}-e^{-x}}{e^{x}+1+e^{-x}} .$$
**For $x\ll1$:** numerator $\approx2x$, denominator $\approx3$:
$$\langle\mu_z\rangle\approx\mu\,\frac{2x}{3}=\frac{2\mu^2B}{3k_BT}\quad(\text{option A}).$$