An atom has a number of possible quantized energy states defined by principal quantum number “n”.

If it is initially in a lower state “1”, it can rise to higher states ‘2’ by absorbing a quanta of radiation (photon) of frequency “ν” given by:

where, E_{1 }and E_{2} are energies of atom in the state “1” and “2”. This is absorption of radiation.This is a stimulated process .

The absorbed photon is the stimulating photon.

The probable rate of occurance of this absorption transition 1 –>2 depends upon the property of states 1 and 2 and is proportional to the energy density u(ν) of the radiation of frequency ν incident on the atom.

The proportionality constant B_{12} is known as Einstein coefficient of absorption of radiation.

**SPONTANEOUS EMISSION :**

Let us now consider an atom initially in higher excited state “Z”.

Observation shows that its life time in higher state is very small (10^{-8}s).

After this ,it jumps to the lower energy side “1” emitting a photon of frequency ν. This is spontaneous emission of radiation.

If there is an assembly of atoms , the radiation emitted spontaneously by each atom has a random direction and a random phases and is therefore incoherent from atom to atom.

The probability of spontaneous emission 2–>1 is determined by the properties of states 2 and 1.

Einstein denoted this probability by A_{21} , which is known as Einstein coefficient of spontaneous emission of radiation.

**STIMULATED EMISSION (INDUCED EMISSION)**

According to Einstein an atom in excited energy state make under the influence of electromagnetic field of a photon of frequency ν incident upon it jump towards lower energy state , emitting an additional photon of same frequency ν.

Thus now two photons , one original and the other emitted move on. This is stimulated emission of radiation.

“*The direction of propogation ,energy, phases of emitted photon is exactly same as that of incident stimulating photon.”*

In other words ,the stimulated radiation is completely coherent with the stimulated radiation.

As a result of this process ,radiation passing through assembly of atom is amplified.

The probability of simulated emission transition 2→1 is proportional to the energy density u(ν)

of stimulating radiation and is written as:

where, B_{21} is Einstein coefficient of stimulated emission of radiation.

The total probability for atom is state 2 to drop to the lower state 1 is

**Relation between spontaneous and stimulated Emission Probabilities:**

Let us consider an assembly of atom in thermal equilibrium. At temperature T with radiation of frequency v and energy density u(v)dv .

Let N_{1} and N_{2} be the number of atoms in states 1 and 2 respectively.

from eq. (1) we have:

For equilibrium ,the objective and emission must occur equally. Thus,

Enstein proved thermodynamically that:

“Probability of absorption (stimulated) is equal to probability of stimulated emission”

Einstein proved thermodynamically that:

According to Boltzmann law the distribution of atoms among different energy states is given by:

where, No = Number of atoms in ground state.

Er = Energy of rth state.

Nr = Number of atoms in rth state.

Now,

This is the formula for the ratio between spontaneous and stimulated emission.