explain the absorption and emission transitions

Question

Explain the absorption and emission transitions of A and B coefficients of Einstein.

Assume that N₁ and N₂ be the number of atoms per unit volume along with energy E₁ and E₂ correspondingly. Assume that 'n' be the number of photons per unit volume at frequency ν therefore hν= E₂- E₁. In that case energy density of interacting photons ρ(ν) is specified as ρ(ν) = nhν

Upward transition:

The rate of stimulated absorption depends upon the number of atoms obtainable into the lower energy state as well as the energy density of the interacting photons.

Stimulated absorption rate α N₁∝ ρ(ν) = N₁ρ(v) B₁₂

Now B₁₂ is the constant of proportionality termed as Einstein coefficient of stimulated absorption.

Downward transition

While the atoms are excited through stimulated absorption, they keep in the excited state for a short duration of time termed as the life time of the excited position. After their life time they fall to the lower energy level spontaneously through emitting photons. The spontaneous emission rate depends onto the number of atoms into the excited energy state.

Spontaneous emission rate α N₂= N₂ A₂₁

Here the constant of proportionality A₂₁ is the Einstein coefficient of spontaneous emission.

Before the atoms within the excited state de-excite to their lower energy states through spontaneous emission, they may interact along with photons resulting into stimulated emission of photons. Therefore rate of stimulated emission depends onto the number of atoms obtainable into the excited state and the energy density of interacting photons. 

Stimulated emission rate α N₂ ∝ ρ(ν) = N₂ ρ(ν) B₂₁

Here the constant of proportionality B₂₁ is the Einstein coefficient of stimulated emission.

Throughout stimulated emission, the stimulating photon and the stimulated photon are within phase along with each other.

Throughout stimulated absorption, the photon density reduces but throughout stimulated emission, the photon density raises.