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Einstein analyzes the nature of Kirchhoff’s law discussed in the previous lecture. Consider a population of atoms. The number density $n_1$ in energy state 1 and the number density in a higher energy state 2 with number density $n_2$ are related because the atoms are in one energy level or the other.
Spontaneous emission will result in a decrease in the number of atoms in energy state 2. Also, it happens more often if there are more atoms in energy state 2. Therefore, the Spontaneous emission corresponds to
\[\frac{\partial n_2}{\partial t} \propto - n_2\]Stimulated emission is proportional to the number density of atoms in state 2 and the local radiation field with mean intensity $J_\nu$. This gives the dependency
\[\frac{\partial n_2}{\partial t} \propto -n_2 J_{\nu}\]Similarly, absorption of light will move atoms in state 1 up to state 2. Therefore, absoprtion increases the number of atoms in energy state two depending on the radiation field and the number of atoms in state 1:
\[\frac{\partial n_2}{\partial t} \propto n_1 J_{\nu}\]Einstein made the simple (and genius) model complete by introductin coefficiencts $B_{21}$ (for stimulated emission), $B_{12}$ (for absorption), and $A_{21}$ (for spontaneous emission) to make a differential equation
\[\frac{ \partial n_2}{\partial t} = n_1 J_{\nu} B_{12} - n_2 J_{\nu} B_{21} - n_2 A_{21}\]Additionally, note that there is one transition possible, from energy $E_1$ to energy $E_2$. Therefore, all of these atoms will only interact with light with energy $\Delta E = h \nu_0 \approx E_2 - E_1$. Therefore, the equation is
\[\frac{ \partial n_2}{\partial t} = n_1 J_{\nu_0} B_{12} - n_2 J_{\nu} B_{21} - n_2 A_{21}\]In thermal equilibrium, $n_2$ and $n_1$ are constant. Therefore, we get
\[J_{\nu_0} = \frac{ A_{21}}{B_{21}} \frac{1}{\frac{n_1 B_{12}}{n_2 B_{21}} - 1}\]Additionally, in thermal equilibrium the densities follow the Boltzmann equation
\[\frac{n_1}{n_2} = \frac{g_1}{g_2} e^{\frac{h\nu_0}{k_B T}}\]Plugging this in and noting that $J_{\nu_0} \rightarrow B_{\nu_0}$ for thermal equilibrium implies
\[\frac{2h \nu_0^3}{c^2} \frac{1}{e^{\frac{h\nu_0}{k_B T}} - 1} = \frac{ A_{21}}{B_{21}} \frac{1}{\frac{g_1 B_{12}}{g_2 B_{21}} e^{\frac{h\nu_0}{k_B T}}- 1}\]This equality provides the Einstein relations for the Einstein coefficients:
\[\frac{ A_{21}}{B_{21}} = \frac{2h \nu_0^3}{c^2}\] \[\frac{g_1 B_{12}}{g_2 B_{21}} = 1\]These Einstein relations are an example of detailed balance relations for bound-bound transitions. Note with 3 coefficients and 2 equations, we have on free parameter. Quantum mechanical calculations provide one parameter (often $A_{21}$) and that determines the others. Additionally, note the temperature $T$ does not factor in. As long as a system in thermal equilibrium has 2 bound states, these Einstein relations work. Additionally, the methodology is generalizable to systems with a higher number of energy states. The coefficients of that system will look like $B_{ij}$ but they are still called Einstein coefficients.
Detailed Balance
This concept requires that the flux of every reaction is net zero. That requirement is easily generalizable to systems with more energy states and reactons.
Tying back to Radiative Transfer
For sponetaneous emission, we know only $A_{21}$ is involved, and it has units such that
\[\frac{dE}{dV \, dt} = h\nu \cdot n_2 \cdot A_{21}\]adding in solid angle and frequency dependency gives the emissivity
\[j_\nu = \frac{d E_\nu}{dV \, dt \, d\nu \, d\Omega} = \frac{1}{4\pi} h\nu n_2 A_{21} \phi_\nu\]where $\phi_\nu \approx \delta(\nu - \nu_0)$ captures the frequency dependance of the interaction.
For absorption and stimulated emission, the coefficients are
\[\alpha_{\nu, \mathrm{abs}} = \frac{h\nu}{4\pi} n_1 B_{12} \phi_\nu\] \[\alpha_{\nu, \mathrm{stim}} = - \frac{h\nu}{4\pi} n_2 B_{21} \phi_\nu\]giving a combined absoprtion coefficient
\[\alpha_\nu = \frac{h\nu}{4\pi} n_1 B_{12} \phi_\nu \left(1 - \frac{g_1 n_2}{g_2 n_1} \right)\]The importan result from the above is that the sign of $\alpha_{\nu}$ depends on the inequality
\[\frac{g_1 n_2}{g_2 n_1} < 1\]if this inequality is not satisfied, then we get a population inversion. This drives LASER and MASER activity (Light Amplified by Stimulated Emission of Radiation).