# Spontaneous emission and induced radiation

In Einstein A., Zur Quantentheorie der Strahlung, Phys.ZS., 18, 121-137 (1917) spontaneous emission is considered to occur together with induced radiation so that one can write the following condition for equilibrium (which also includes induced absorption): $$p_n e^{-\frac{\varepsilon_n} {kT}} B_n^m \rho = p_m e^{-\frac{\varepsilon_m} {kT}} \left( B_m^n \rho + A_m^n \right)$$ where $p_n$ and $p_m$ are statistical weights of the states $n$ and $m$, $\rho$ is radiation density of frequency $\nu$, $A_m^n$ is a constant characteristic of the spontaneous $m \rightarrow n$ transition (spontaneous emission), $B_m^n$ and $B_n^m$ are constants expressing the change of state under induced emission and absorption.

To arrive at Planck's radiation density law it is considered that at high temperatures the above equation becomes $$p_n B_n^m = p_m B_m^n .$$ What is the justification, however, to substitute $B_n^m$ expressed through the latter equality (valid for extreme temperatures) into the initial equation above (valid for lower temperatures)?

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