Einstein's co-efficients

EINSTEIN'S CO‒EFFICIENTS

We know that, when light is absorbed by the atoms (or) molecules, then it goes from the lower energy level (E₁) to the higher energy level (E₂) and during the transition from higher energy level (E₂) to lower energy level (E₁), the light is emitted from the atoms (or) molecules.

Let us consider an atom exposed to (light) photons of energy E₂ ‒ E₁=hv, then, three distinct processes takes place.

(i) Absorption.

(ii) Spontaneous emission and

(iii) Stimulated emission.

Process (i) Absorption

An atom in the lower energy level (or) ground state energy level E₁ absorbs the incident photon radiation of energy (hv) and goes to the higher energy level (or) excited energy state E₂ as shown in Fig.10.4

This process is called absorption.

If there are many number of atoms in the ground state then each atom will absorb the energy from the incident photon and goes to the excited state then,

The rate of absorption (R₁₂) is proportional to the following factors.

(i.e) R₁₂ α Energy density of incident radiation (ρ_v)

 α No.of atoms in the ground state (N₁)

(i.e) R₁₂ α ρ_v N₁

 (or)  R₁₂ = B₁₂ ρ_v N₁                 ………..(1)

Where B₁₂ is a constant which gives the probability of absorption transition per unit time.

NOTE: The subscript 12 represents the transition is from energy level E₁ to E₂. Similarly 21 represents the transition is from energy E₂ to E₁.

Normally, the atoms in the excited state will not stay there for a long time, rather it comes to ground state by emitting a photon of energy E=hv. Such an emission takes place by one of the following two methods.

Process (ii) Spontaneous emission

The atom in the excited state returns to the ground state by emitting a photon of energy E=(E₂‒E₁) = hv, spontaneously without any external triggering as shown in Fig.10.5. This process is known spontaneous emission.

Such an emission is random and is independent of incident radiation. If N₁ and N₂ are the numbers of atoms in the ground state (E₁) and excited state (E₂) respectively, then

The rate of spontaneous emission is R₂₁ (Sp) α N₂

(or)

R₂₁ (Sp) = A₂₁N₂

Where A₂₁ is a constant which gives the probability of spontaneous emission transitions per unit time.

Process (iii) Stimulated emission

The atom in the excited state can also return to the ground state by external triggering (or) inducement of photon thereby emitting a photon of energy equal to the energy of the incident photon, known as stimulated emission.

Thus results in two photons of same energy, phase difference and of same directionality as shown in Fig.10.6.

The rate of stimulated emission is given by

R₂₁ (St) α P_v N₂

(or)

R₂₁ (St) = B₂₁ Pv N₂          ………..(3)

where B₂₁ is a constant which gives the probability of stimulated emission transitions per unit time.

Einstein's theory

Einstein's theory of absorption and emission of light by an atom is based on Planck's theory of radiation. Also under thermal equilibrium, the population of energy levels obey the Boltzmann's distribution law.

(i.e) under thermal equilibrium,

The rate of absorption = The rate of emission

 (i.e) Eqn.(1) = Eqn.(2) Eqn. (3)

 B₁₂ ρ_v N₁ = A₂₁ N₂ + B₂₁ ρ_v N₂

 ρ_v [B₁₂N₁ ‒ B₂₁N₂ ] = A₂₁ N₂

……………………….(4)

We know that from Boltzmann distribution law

N₁ = N_o e^‒E1/KBT

Similarly  N₂ = N_o e^‒E2/KBT

Where

K_B is the Boltzmann constant,

T is the absolute temperature and

N_o is the number of atoms at absolute zero.

At equilibrium, we can write the ratio of Population levels as follows,

(i.e) N₁ /N₂ = e^{(E2‒E1)/KBT}

Since E₂‒E₁=hv, we have

N₁/N₂ = e^hv/KBT     …………..(5)

Substituting eqn.(5) in eqn.(4) we have

…………..(6)

This equation has a very good agreement with Planck's energy distribution radiation law.

(ie.)

…………..(7)

Therefore comparing equations (6) and (7), we can write

B₁₂ = B₂₁ = B

A₂₁ / B₂₁ = 8πhv³ / c³

……………………(8)

Taking A₂₁ = A

The constants A and B are called Einstein Coefficients, which accounts for spontaneous and stimulated emission probabilities. It also explains the importance of metastable states.

Ratio of magnitudes of stimulated and spontaneous emission rates

From equations (2) and (3) we have

Substituting eqn.(6) in eqn (9), we can write

 since B₁₂ = B₂₁, we can write

Comparing eqn. (9) and eqn. (10) we get

In a simpler way the ratio can be written as

 R = [B₂₁/A₂₁] ρv

Generally spontaneous emission is more predominant in the optical region (ordinary light). To increase the number of coherent photons, stimulated emission should dominate over spontaneous emission. To achieve this, an artificial condition called population inversion is necessary.