What are Einstein’s A and B coefficients?

Einstein’s A and B coefficients are constants that quantify the probabilities of spontaneous emission, stimulated emission, and absorption of radiation by atoms interacting with electromagnetic fields. :contentReference[oaicite:1]{index=1}

What does the A coefficient represent?

The Einstein A coefficient represents the probability per unit time that an atom in an excited state will undergo spontaneous emission, releasing a photon without external stimulation. :contentReference[oaicite:2]{index=2}

What do the B coefficients represent?

Einstein’s B coefficients quantify the probability of stimulated emission and absorption of photons in the presence of an external electromagnetic field. :contentReference[oaicite:3]{index=3}

How are Einstein’s coefficients related?

Under thermal equilibrium, Einstein’s A and B coefficients follow relationships derived from Planck’s radiation law, showing how spontaneous and stimulated emission rates are connected. :contentReference[oaicite:4]{index=4}

Why are Einstein’s coefficients important in laser physics?

Einstein’s coefficients form the theoretical foundation for understanding how lasers achieve amplification through stimulated emission, which is essential for laser operation. :contentReference[oaicite:5]{index=5}

Einstein Coefficients

Lets consider two levels of an atomic system as shown in Fig. 1 and let N1 and N2 be the number of atoms per unit volume present in the energy levels E₁ and E₂, respectively. An atom in the lower energy level E₁ can absorb the incident radiation at a frequency v and be excited to E₂.

Einstein’s Coefficients: A, B Coefficients in Radiation Theory

Figure.1. Energy diagram of stimulated absorption, spontaneous emission and stimulated emission

The rate (R1) at which absorption takes place from E₁ to E₂ will be proportional to the number of atoms present in the level E₁ and also to the energy density of the radiation and let it be u(v).

\[R_1 \propto N_1 u(v) \rightarrow R_1 = B_{12} N_1 u(v) \tag{1} \]

Where, B₁₂ is a constant of proportionality.

Now during the de-excitation process from E₂ to E₁, Einstein postulated that an atom can make a transition through two distinct processes, namely stimulated emission and spontaneous emission.

In the case of stimulated emission, the radiation which is incident on the atom stimulates it to emit radiation and the rate of transition to the lower energy level is proportional to the energy density of radiation at the frequency v.

Thus, the number of stimulated emissions per unit time per unit volume will be

\[ R_2 = B_{21}N_2u(v) \tag{2}\]

For spontaneous emission of atoms from higher energy levels, the rate will depend on the number of atom in level E₂, Hence

\[ R_3 = A_{21}N_2 \tag{3}\]

At thermal equilibrium, the number of upward transitions must be equal to the number of downward transitions,

\[ R_1 = R_2+R_3 \]

Substitute R₁, R₂ and R₃ values in the above equation,

\[ B_{12} N_1 u(v) = B_{21}N_2u(v) + A_{21}N_2 \ \tag{4} \]

On solving,

\[ u(v) = \frac{A_{21}}{ \frac{N_1}{N_2} B_12 - B_21 } \tag{5}\]

Using Boltzman’s Law, the equilibrium population levels at E₁ and E2 is given by

\[ N_1 = \exp\left(-\frac{E_1}{kT}\right) \quad \text{and} \quad N_2 = \exp\left(-\frac{E_2}{kT}\right) \] \[ \frac{N_1}{N_2} = \exp\left(\frac{E_2 - E_1}{kT}\right) = \exp\left(\frac{h\nu}{kT}\right) \tag{6} \]

substitute the above Eqn in Eqn 6,

\[ u(\nu) = \frac{A_{21}/B_{21}}{B_{21} \exp\left(\frac{h\nu}{kT}\right) - 1} \tag{7} \]

According to Planck’s law, the radiation energy density per unit frequency is

\[ u(\nu) = \frac{8\pi h \nu^3}{c^3 \left(\exp\left(\frac{h\nu}{kT}\right) - 1\right)} \tag{8} \]

Comparing Eqn 7 and Eqn 8,

\[ B_{12} = B_{21} = B \] and \[ \frac{A_{21}}{B_{21}} = \frac{8\pi h \nu^3}{c^3} \tag{9}\]

Thus the stimulated emission rate per atom is the same as the absorption rate per atom. The coefficients A and B are referred to as the Einstein A and B coefficients.

At thermal equilibrium, the ratio of number of spontaneous to stimulated emissions is given by

\[ R = \frac{R_3}{R_2} = \frac{A_{21} N_2}{B_{21} N_2 u(\nu)} \]

Substituting the value of A₂₁/B₂₁ from 9 and u(v) from Eqn 7,

\[ R = \exp\left( \frac{h\nu}{kT} - 1 \right) \]