GATE 2024 PH – Question 62
Three-dimensional noninteracting bosons have zero chemical potential and $\epsilon\propto k^2$. Their low-temperature $C_V\propto T^{n/2}$. The integer n is _____
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Correct answer: 3
Explanation
For non-interacting bosons with zero chemical potential (as for photons or the condensed phase below $T_c$) in three dimensions with $\epsilon\propto k^2$:
**Density of states.** The number of states with wave vector below $k$ is $\propto k^3\propto\epsilon^{3/2}$, so $g(\epsilon)\propto\epsilon^{1/2}$.
**Internal energy:**
$$U=\int_0^\infty\frac{\epsilon\,g(\epsilon)}{e^{\epsilon/k_BT}-1}\,d\epsilon\propto\int\frac{\epsilon^{3/2}}{e^{\epsilon/k_BT}-1}d\epsilon\propto T^{5/2}.$$
(The substitution $x=\epsilon/k_BT$ gives a factor $(k_BT)^{5/2}$ times a constant integral.)
**Heat capacity:**
$$C_V=\frac{\partial U}{\partial T}\propto T^{3/2}=T^{n/2}\;\Rightarrow\;n=\mathbf{3}.$$