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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 J_u 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 J_u, of the incident radiation corresponding to frequency u.

That is                R₁  N₁ J_u

or                         R₁ = B₁₂N₁J_u                                                                                                            (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 J_u 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 J_u, of the incident radiation corresponding to frequency u and number of atoms N₂ in the excited energy state E₂, thus

R₃  N₂ J_u

or

R₁₌ B₂₁N₂J_u                                   (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

                            B₁₂N₁J_u = A₂₁N₂ + B₂₁N₂J_u

or                         B₁₂N₁J_u – B₂₁N₂J_u = A₂₁N₂

or                         (B₁₂N₁– B₂₁N₂) J_u = A₂₁N₂

J_u = A₂₁N₂ / (B₁₂N₁– B₂₁N₂)                                                            (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),

J_u = A₂₁N₂ / B₁₂(N₁– N₂)

J_u = 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 e^–E/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.