For a two-level atom the Einstein relations are and

PHYSICS · Subjective · Fermi University Physics · Atomic, Molecular & Optical Physics · Astrophysics & Cosmology · Radiation, radiative transfer and stellar atmospheres

Stars from the Inside, Ch Radiative transfer solved: the grey atmosphere, the diffusion limit and the curve of growth

Problem

For a two-level atom the Einstein relations are g1B12=g2B21g_1B_{12} = g_2B_{21} and A21=(2hν3/c2)B21A_{21} = (2h\nu^3/c^2)B_{21}. The absorption coefficient built from level 1 alone, ignoring stimulated emission, is αν\alpha_\nu. (a) Starting from the net rate of photon removal, n1B12Jν−n2B21Jνn_1B_{12}J_\nu - n_2B_{21}J_\nu, and assuming the level populations n1n_1, n2n_2 obey the Boltzmann distribution at temperature TT, derive the stimulated-emission-corrected absorption coefficient ανcorr=αν(1−e−hν/kBT)\alpha_\nu^{\rm corr} = \alpha_\nu\left(1 - e^{-h\nu/k_BT}\right). (b) Evaluate hν/kBTh\nu/k_BT and the correction factor 1−e−hν/kBT1-e^{-h\nu/k_BT} for the hydrogen Balmer-alpha line (λ=656.3 nm\lambda = 656.3\,\mathrm{nm}) in a stellar photosphere at T=10 000 KT = 10\,000\,\mathrm{K}, and state what fraction of naive absorption events this correction removes.

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