**Einstein Coefficient Relation derivation and discussion:
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Einstein showed the interaction of radiation with matter with the help of three processes called stimulated absorption, spontaneous emission and stimulated emission. He showed in 1917 that for proper description of radiation with matter,the process of stimulated emission is essential.Let us first derive the Einstein coefficient relation on the basis of above theory:

Let N_{1} be the number of atoms per unit volume in the ground state E_{1} and these atoms exist in the radiation field of photons of energy E_{2}-E_{1} =h v such that energy density of the field is E.

Let R_{1} be the rate of absorption of light by E_{1} -> E_{2} transitions by the process called stimulated absorption

This rate of absorption R_{1} is proportional to the number of atoms N_{1} per unit volume in the ground state and proportional to the energy density E of radiations.

That is R_{1}∞ N_{1} E

Or R_{1} = B_{12}N_{1} E (1)

Where B_{12} is known as the Einstein’s coefficient of stimulated absorption and it represents the probability of absorption of radiation. Energy density e is defined as the incident energy on an atom as per unit volume in a state.

Now atoms in the higher energy level E_{2} can fall to the ground state E_{1} automatically after 10^{-8} sec by the process called spontaneous emission.

The rate R_{2} of spontaneous emission E_{2}-> E_{1} is independent of energy density E of the radiation field.

R_{2} is proportional to number of atoms N_{2} in the excited state E_{2} thus

R_{2}∞ N_{2}

R_{2}=A_{21} N_{2} (2)

Where A_{21} is known as Einstein’s coefficient for spontaneous emission and it represents the probability of spontaneous emission.

Atoms can also fall back to the ground state E_{1} under the influence of electromagnetic field of incident photon of energy E_{2}-E_{1 }=hv by the process called stimulated emission

Rate R_{3} for stimulated emission E_{2}-> E_{1} is proportional to energy density E of the radiation field and proportional to the number of atoms N_{2} in the excited state,thus

R_{3}α N_{2} E

Or R_{3}=B_{21}N_{2} E (3)

Where B_{21} is known as the Einstein coefficient for stimulated emission and it represents the probability of stimulated emission.

In steady state (at thermal equilibrium), the two emission rates (spontaneous and stimulated) must balance the rate of absorption.

Thus R_{1}=R_{2}+R_{3}

Using equations (1,2, and 3) ,we get

N_{1}B_{12}E=N_{2}A_{21}+N_{2}B_{21}E

Or N_{1}B_{12}E –N_{2}B_{21}E=N_{2}A_{21}

Or (N_{1}B_{12}-N_{2}B_{21}) E =N_{2}A_{21}

Or E= N_{2}A_{21}/N_{1}B_{12}-N_{2}B_{21}

= N_{2}A_{21}/N_{2}B_{21}[N_{1}B_{12}/N_{2}B_{21} -1]

[by taking out common N_{2}B_{21}from the denominator]

Or E=A_{21}/B_{21} {1/N_{1}/N_{2}(B_{12}/B_{21}-1)) (4)

Einstein proved thermodynamically,that the probability of stimulated absorption is equal to the probability of stimulated emission.thus

B_{12}=B_{21}

Then equation(4) becomes

E=A_{21}/B_{21}(1/N_{1}/N_{2}-1) (5)

From Boltzman’s distribution law, the ratio of populations of two levels at temperature T is expressed as

N_{1}/N_{2}=e^{(E}_{2}–^{E}_{1}^{)/KT}

N_{1}/N_{2}=e^{hv/KT}

Where K is the Boltzman’s constant and h is the Planck’s constant.

Substituting value of N_{1}/N_{2}in equation (5) we get

E= A_{21}/B_{21}(1/e^{hv/KT}-1) (6)

Now according to Planck’s radiation law, the energy density of the black body radiation of frequency v at temperature T is given as

E = 8πhv^{3}/c^{3}(1/e^{hv/KT}) (7)

By comparing equations (6 and 7),we get

A_{21}/B_{21}=8πhv^{3}/c^{3}

This is the relation between Einstein’s coefficients in laser.

**Significance of Einstein coefficient relation**: This shows that the ratio of Einstein’s coefficient of spontaneous emission to the Einstein’s coefficient of stimulated absorption is proportional to cube of frequency v. It means that at thermal equilibrium, the probability of spontaneous emission increases rapidly with the energy difference between two states.

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