Ratio of Spontaneous and Stimulated Transition Probabilities in Thermal Equilibrium

Ratio of Spontaneous and Stimulated Transition Probabilities in Thermal Equilibrium

The probability per unit time of spontaneous emission is

$$P_{nm}^{\mathrm{sp}}=A_{nm}.$$

The probability per unit time of stimulated emission is

$$P_{nm}^{\mathrm{st}}=B_{nm}u(\nu).$$

Therefore, the ratio of spontaneous to stimulated emission probabilities is

$$\boxed{ \frac{P_{nm}^{\mathrm{sp}}}{P_{nm}^{\mathrm{st}}} = \frac{A_{nm}}{B_{nm}u(\nu)} } \tag{1}$$

In thermal equilibrium, the spectral energy density of blackbody radiation is given by Planck's radiation law:

$$u(\nu)= \frac{8\pi h\nu^3}{c^3} \frac{1}{e^{h\nu/k_BT}-1}.$$

Using this expression in equation (1),

$$\begin{array}{c}\end{array}$$

where $k_B$ is Boltzmann's constant and $\mathrm{T\; is\; the\; absolute\; temperature.}$

Case I: 

$h\nu\gg k_BT$

When the photon energy is much greater than the thermal energy,

$$e^{h\nu \mathrm{/}{k}_{B}T}\gg 1.$$

Therefore,

$$\frac{P_{nm}^{\mathrm{sp}}}{P_{nm}^{\mathrm{st}}}\gg1.$$

Thus, spontaneous emission is much more probable than stimulated emission. This condition is generally satisfied for many optical and higher-energy electronic transitions at ordinary temperatures.

Case II: $h\nu\ll k_BT$

For low-frequency radiation, using the approximation $e^x-1\approx x$ for $x\ll \mathrm{1,}$

$$e^{h\nu/k_BT}-1 \approx\frac{h\nu}{k_BT}.$$

Hence,

$$\frac{P_{nm}^{\mathrm{sp}}}{P_{nm}^{\mathrm{st}}} \approx\frac{h\nu}{k_BT}\ll1.$$

Therefore,

$$\boxed{ P_{nm}^{\mathrm{sp}}\ll P_{nm}^{\mathrm{st}} }$$

Stimulated emission becomes the dominant emission process in this limit.