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**Einsteins A and B Coefficients and Expression for energy density**

Let E_{1} and E_{2} be the two energy states of the system of atoms. Let N_{1} and N_{2} be the number of atoms with energy states E_{1} and E_{2} respectively. Let Uν represent the energy density of incident photon having frequency (ν). Hence the system absorbs and emits energy through the following processes

(a)**Stimulated Absorption:**

In case of Stimulated absorption, the rate of absorption is given by number of absorptions per second per unit volume. Rate of absorption depends upon, (1) Number of atoms in the lower energy state. (N_{1})

(2) The energy density (Uν).

Hence, the Rate of absorption ∝ N_{1}Uν and Rate of absorption = B_{12}N_{1}Uν, where B_{12} is proportionality constant called Einsteins coefficient of induced absorption.

(b)**Spontaneous emission:**

In case of Spontaneous emission, the rate of spontaneous emission is given by number of spontaneous emissions per second per unit volume. Rate of Spontaneous emission depends upon,

(1) Number of atoms in the higher energy state (E_{2}). i.e. (N_{2})

Hence, the Rate of Spontaneous emission ∝ N_{2}

Rate of Spontaneous emission = A_{21}N_{2}.

Where A_{21} is proportionality constant Called Einsteins coefficient of Spontaneous emission.

(c)**Stimulated Emission:**

In case of Stimulated Emission The rate of Stimulated emission is given by number of Stimulated emissions per second per unit volume. Rate of Stimulated Emission depends upon,

(1) Number of atoms in the higher energy state (E_{2}). i.e. (N_{2})

(2) The energy density (Uν)

Hence, the Rate of Stimulated emission ∝ N_{2} Uν

Rate of Stimulated emission = B_{21} N_{2} U_{ν}, where B_{21} is proportionality constant Called Einsteins coefficient of Stimulated Emission.

Under Thermal Equilibrium the total Energy of the System remains unchanged. Hence number of photons absorbed is equal to number of photons emitted. Rate of Induced Absorption = Rate of Spontaneous emission + Rate of Stimulated Emission. Therefore

After simplification

But according to the Boltzman’s law of distribution the number of atoms in the higher energy state E_{2} at any temperature “T” is given by

Substituting for N_{1}/N_{2} in equation (5.1)

According to Planck’s Law of Radiation. The equation for Energy density is given by

Comparing (5.2) and (5.3) we have

(i.e. Probability of Stimulated absorption is equal to Probability of Stimulated emission.) Hence A_{21} and B_{21} can be replaced by A and B and the expression for the Energy density is given by,

Where h is Plancks constant, k is Boltzmann Constant, T is Absolute Temperature, and c is the velocity of light.

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