= Solution
At $7000\,\mathrm K$, $k_BT=0.603\,\mathrm{eV}$ and
$$
C_{12}=1.415\times10^{-13}\,\mathrm{m^3\,s^{-1}},\qquad
\boxed{n_e=\frac{0.007A_{21}}{C_{12}}=1.13\times10^5\,\mathrm{m^{-3}}=0.113\,\mathrm{cm^{-3}}.}
$$
The $2p$ ground term has fine-structure levels $J=1/2$ and $J=3/2$, with <energy-level degeneracy> $g_1=2$ and $g_2=4$. <Detailed balance> gives $C_{21}=C_{12}(g_1/g_2)e^{\Delta E/(k_BT)}=7.17\times10^{-14}\,\mathrm{m^3\,s^{-1}}$. The <critical density> is therefore
$$
n_{\rm crit}=A_{21}/C_{21}=3.19\times10^7\,\mathrm{m^{-3}}\simeq32\,\mathrm{cm^{-3}}.
$$
With collisions and spontaneous decay both retained, $N_2/N_1=n_eC_{12}/(A_{21}+n_eC_{21})$. A <Boltzmann distribution>, $N_2/N_1\to2e^{-\Delta E/(k_BT)}\simeq1.97$, requires $n_e\gg n_{\rm crit}$, not merely equality. For example $n_e\gtrsim3\times10^8\,\mathrm{m^{-3}}$ brings the ratio within about ten per cent of that limit.
Back to article page
