Einsteins Coefficients

Expression of Einstein’s coefficient: Consider a system of atoms with two energy states E₁ and E₂ with N₁ and N₂ number of atoms per unit volume in each energy states respectively. The N₁ and N₂ are called the number densities of the atoms. Let a radiation of energy density Eν of frequency ν be incident on the system. In the case of induced absorption, an atom in the ground state E₁ goes to an excited state E₂ by absorbing a suitable photon of energy hν = E₂ − E₁. The number of such absorptions per unit time, per unit volume is called the rate of induced absorption. This depends on the number density N₁ of the ground state and the energy density of the incident radiation Eν. That is,

Rate of induced absorption ∝N₁E_ν.

 

By introducing the constant of proportionality B₁₂, we get

Rate of induced absorption = B₁₂N₁E_ν.

 

In the case of spontaneous emission, an atom in the excited stateE₂ makes a transition to the ground stateE₁ by  emitting a photon of appropriate energy hν = E₂ − E₁. The number of such spontaneous emissions per unit time, per unit volume is called the rate of spontaneous emission. This depends only on the number density N₂ of the excited state. That is,

Rate of spontaneous emission ∝ N₂.

 

By introducing the constant of proportionality A₂₁, we get

Rate of spontaneous emission = A₂₁N₂.

 

In the case of stimulated emission, an atom in the excited state E₂ makes a transition to the ground state E₁ upon incidence of a photon of suitable energy hν = E₂ − E₁, by emitting a photon of same energy. The number of such stimulated emissions per unit time, per unit volume is called the rate of stimulated emission. This depends on the number density N₂ of the excited state and the energy density of the incident radiation Eν. That is,

Rate of stimulated emission ∝N₂E_ν.

 

By introducing the constant of proportionality B₂₁, we get

Rate of stimulated emission = B₂₁N₂E_ν.

 

In the above discussion, the constants of proportionality are called the Einstein’s coefficients. Under thermal equilibrium,

 

Rate of induced absorption = Rate of spontaneous emission Rate of stimulated emission

 

that is,

B₁₂N₁E_ν = A₂₁N₂ B₂₁N₂E_ν.

 

Taking Eν terms on to the left hand side, we get

 

B₁₂N₁E_ν − B₂₁N₂E_ν = A₂₁N₂,

 

(B₁₂N₁ − B₂₁N₂)E_ν = A₂₁N₂,