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

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

Let us consider an atomic system in thermal equilibrium at absolute temperature T. Let N1 and N2 be the number of atoms per unit volume in the ground energy state E1 and excited energy state E2 respectively. Let Ju is the energy density of the incident radiation corresponding to frequency u.

 Energy density Ju is defined as the incident energy on an atom as per unit volume in a state.

According to Einstein,

1) The rate of absorption of light (R1) is proportional to the number of atoms N1 per unit volume in the ground energy state E1 and energy density Ju, of the incident radiation corresponding to frequency u.

That is                R1  N1 Ju

or                         R1 = B12N1Ju                                                                                                            (1)

Where B12 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 (R2) is independent of energy density Ju of the incident radiation and is proportional to number of atoms N2 in the excited state E2 thus

R2  N2

or                         R2 = A21N2                                                                                                               (2)

Where A21 is known as Einstein’s coefficient for spontaneous emission and it represents the probability of spontaneous emission.

3) The rate of stimulated emission (R3) is proportional to the energy density Ju, of the incident radiation corresponding to frequency u and number of atoms N2 in the excited energy state E2, thus

R3  N2 Ju

or

R1 = B21N2Ju                                                                                                            (3)

Where B21 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

R1 = R2 + R3

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

B12N1Ju = A21N2 + B21N2Ju

or                         B12N1JuB21N2Ju = A21N2

or                         (B12N1B21N2) Ju = A21N2

Ju = A21N/ (B12N1B21N2)                                                            (4)

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

B12 = B21

Then from equation (4),

Ju = A21N/ B12(N1N2)

Ju = A21 / {B12(N1/N2)-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 eE/KT , where K is the Boltzman`s constant. Thus the ratio of populations of two energy levels (E1 and E2) 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.