# Derive **Relation between Einstein`s Coefficients A & B in LASER.**

**Relation between ***Einstein`s Coefficients A & B*

*Einstein`s Coefficients A & B*

Let us consider an atomic system in thermal equilibrium at absolute temperature *T*. Let *N*_{1} and *N*_{2} be the number of atoms per unit volume in the ground energy state *E*_{1} and excited energy state *E*_{2} respectively. Let *J** _{u}* is the energy density of the incident radiation corresponding to frequency

*u*

*.*

Energy density J is defined as the incident energy on an atom as per unit volume in a state._{u} |

According to Einstein,

1) The rate of absorption of light (*R*_{1}) is proportional to the number of atoms *N*_{1} per unit volume in the ground energy state *E*_{1} and energy density *J** _{u}*, of the incident radiation corresponding to frequency

*u*.

That is *R*_{1} *N*_{1} *J*_{u}

or *R*_{1} = *B*_{12}*N*_{1}*J** _{u}* (1)

Where *B*_{12} is known as the *Einstein’s coefficient of stimulated absorption* and it represents the probability of absorption of radiation.

2) The rate of spontaneous emission (*R*_{2}) is independent of energy density *J** _{u}* of the incident radiation and is proportional to number of atoms

*N*

_{2}in the excited state E

_{2}thus

*R*_{2} *N*_{2}

or

*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.

3) The rate of stimulated emission (R_{3}) is proportional to the energy density *J** _{u}*, of the incident radiation corresponding to frequency

*u*and number of atoms

*N*

_{2}in the excited energy state

*E*

_{2}, thus

*R*_{3} *N*_{2} *J*_{u}

or

*R*_{1= }*B*_{21}*N*_{2}*J** _{u}* (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

* B*_{12}*N*_{1}*J** _{u}* =

*A*

_{21}

*N*

_{2}+

*B*

_{21}

*N*

_{2}

*J*

_{u}or *B*_{12}*N*_{1}*J** _{u}* –

*B*

_{21}

*N*

_{2}

*J*

*=*

_{u}*A*

_{21}

*N*

_{2}

or (*B*_{12}*N*_{1}– *B*_{21}*N*_{2}) *J** _{u}* =

*A*

_{21}

*N*

_{2}

*J** _{u}* =

*A*

_{21}

*N*

_{2 }/ (

*B*

_{12}

*N*

_{1}–

*B*

_{21}

*N*

_{2}) (4)

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

*B*_{12} = *B*_{21}

Then from equation (4),

*J** _{u}* =

*A*

_{21}

*N*

_{2 }/

*B*

_{12}

*(N*

_{1}–

*N*

_{2})

*J** _{u}* =

*A*

_{21}

_{ }/ {

*B*

_{12}

*(N*

_{1}/

*N*

_{2})-1}

According to Boltzman’s distribution law, at absolute temperature *T *the probability that an atom is occur in an energy state *E *is proportional to *e*^{–E/KT} , where *K *is the Boltzman`s constant. Thus the ratio of populations of two energy levels (*E*_{1} and *E*_{2}) at temperature T can be expressed as

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

**Significance of relation between Einstein`s coefficient**:

It is clear from the above relation that the ratio of Einstein’s coefficient of spontaneous emission to the Einstein’s coefficient of stimulated absorption is proportional to cube of frequency *u*. It means that at thermal equilibrium, the probability of spontaneous emission increases rapidly with the energy difference between two states.

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