= Quantum degeneracy threshold in a spherical linear trap
{title2=$N\sim Z_1(T,a)=8\pi(k_BT/a)^3/\lambda_T^3$}
For a dilute classical trapped gas, $\mu\simeq k_BT\ln(N/Z_1)$ with energy zero at the potential minimum. Thus $z=e^{\mu/(k_BT)}=N/Z_1$. Quantum occupation effects begin when $z$ is no longer small, giving the displayed order-of-magnitude particle threshold. Equivalently, the central density $n(0)=N/[8\pi(k_BT/a)^3]$ satisfies $n(0)\lambda_T^3\sim1$. This is a degeneracy crossover criterion rather than an exact Bose condensation threshold. In the continuum approximation, spinless bosons have excited-state saturation $N_c=\zeta(9/2)Z_1$, from the trap density of states proportional to $E^{7/2}$. A quantum ground energy should be subtracted consistently when discreteness matters.
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