# Einsteins Coefficients

**Expression of Einstein’s coefficient: **Consider a system of atoms with two energy states E_{1} and E_{2} with N_{1} and N_{2} number of atoms per unit volume in each energy states respectively. The N_{1} and N_{2} 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 E_{1} goes to an excited state E_{2} by absorbing a suitable photon of energy hν = E_{2} − E_{1}. The number of such absorptions per unit time, per unit volume is called the rate of induced absorption. This depends on the number density N_{1} of the ground state and the energy density of the incident radiation Eν. That is,

Rate of induced absorption ∝N_{1}E_{ν}.

By introducing the constant of proportionality B_{12}, we get

Rate of induced absorption = B_{12}N_{1}E_{ν}.

In the case of spontaneous emission, an atom in the excited stateE_{2} makes a transition to the ground stateE_{1} by emitting a photon of appropriate energy hν = E_{2 }− E_{1}. 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 N_{2} of the excited state. That is,

Rate of spontaneous emission ∝ N_{2}.

By introducing the constant of proportionality A_{21}, we get

Rate of spontaneous emission = A_{21}N_{2}.

In the case of stimulated emission, an atom in the excited state E_{2} makes a transition to the ground state E_{1} upon incidence of a photon of suitable energy hν = E_{2} − E_{1}, 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 N_{2} of the excited state and the energy density of the incident radiation Eν. That is,

Rate of stimulated emission ∝N_{2}E_{ν}.

By introducing the constant of proportionality B_{21}, we get

Rate of stimulated emission = B_{21}N_{2}E_{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,

B_{12}N_{1}E_{ν} = A_{21}N_{2} B_{21}N_{2}E_{ν}.

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

B_{12}N_{1}E_{ν} − B_{21}N_{2}E_{ν} = A_{21}N_{2},

(B_{12}N_{1} − B_{21}N_{2})E_{ν} = A_{21}N_{2},