Einstein’s Theory of Matter-Radiation Interaction and A and B Coefficients

Definition

Einstein’s theory of matter-radiation interaction is a quantum mechanical model that describes the exchange of energy between atoms and radiation through three processes: absorption, spontaneous emission, and stimulated emission. The A coefficient represents the probability per unit time of spontaneous emission, while the B coefficients represent the probability per unit time for absorption and stimulated emission in the presence of radiation.

Main Content

1. Interaction of Radiation with Atomic Energy Levels

• Atoms exist in discrete energy states, and when radiation of suitable frequency interacts with an atom, transitions occur between these energy levels.
• If an atom in a lower energy state absorbs a photon whose energy exactly matches the energy gap between two levels, it moves to the higher state. Similarly, atoms in excited states can return to lower states by emitting radiation.

In a two-level atomic system, let the lower energy level be $E_1$ and the upper energy level be $E_2$, with

$$E_2 - E_1 = h\nu$$

where $h$ is Planck’s constant and $\nu$ is the frequency of radiation. Only radiation with this exact energy difference can induce transitions, which shows the quantized nature of matter-radiation interaction.

The transition rates depend not only on the energy difference but also on the number of atoms in each state and the intensity of the radiation field. This is why statistical ideas are essential in Einstein’s model.

2. Einstein A and B Coefficients

A coefficient

• : The probability per unit time that an atom in the excited state $E_2$ will spontaneously emit a photon and fall to the lower state $E_1$.

B coefficients

• : Probabilities per unit time for radiation-induced transitions:
• $B_{12}$: absorption from $E_1$ to $E_2$
• $B_{21}$: stimulated emission from $E_2$ to $E_1$

If $N_1$ and $N_2$ are the number of atoms in the lower and upper levels respectively, and $\rho(\nu)$ is the radiation energy density at frequency $\nu$, then the transition rates are:

Absorption rate

• = $N_1 B_{12}\rho(\nu)$

Stimulated emission rate

• = $N_2 B_{21}\rho(\nu)$

Spontaneous emission rate

• = $N_2 A_{21}$

Here:

• $A_{21}$ is independent of external radiation.
• $B_{12}$ and $B_{21}$ depend on the radiation field.
• These coefficients are fundamental because they connect atomic transition probabilities with macroscopic radiation properties.

Einstein also showed that the coefficients are related through equilibrium conditions and the Planck blackbody radiation law. This was a major success because it connected quantum theory with thermal radiation.

3. Absorption, Spontaneous Emission, and Stimulated Emission

Absorption

• occurs when an atom in the lower state absorbs a photon and moves to a higher energy state.

Spontaneous emission

• occurs when an excited atom returns to a lower state on its own, emitting a photon randomly in direction, phase, and polarization.

Stimulated emission

• occurs when an incident photon of the right frequency induces an excited atom to emit another photon identical to the incident one.

Absorption: When radiation of frequency $\nu$ interacts with atoms in the lower state, some atoms absorb photons and move upward. This reduces the radiation intensity.

Spontaneous emission: This is a natural decay process. Even without external radiation, an excited atom eventually emits a photon. The emitted photon is random in direction and phase, so spontaneous emission produces incoherent light.

Stimulated emission: This is the most important process in lasers. A photon striking an excited atom can trigger the atom to emit a second photon that is exactly the same as the first one in frequency, phase, direction, and polarization. This leads to amplification of light.

The dominance of stimulated emission over absorption is the condition for laser operation, which requires population inversion, meaning more atoms in the excited state than in the lower state.

Working / Process

1. Establish energy levels and radiation interaction

• Consider atoms with two energy levels $E_1$ and $E_2$, separated by $h\nu$.
• When radiation of frequency $\nu$ is incident on the medium, atoms may absorb energy or emit energy depending on their initial state.

2. Calculate transition probabilities using Einstein coefficients

• The rate of upward transitions is $N_1 B_{12}\rho(\nu)$.
• The rate of downward transitions has two parts: spontaneous emission $N_2 A_{21}$ and stimulated emission $N_2 B_{21}\rho(\nu)$.
• These rates determine how atoms distribute themselves among the two levels.

3. Apply thermal equilibrium conditions

• In equilibrium, upward and downward transition rates must balance.
• This balance leads to relationships among $A_{21}$, $B_{12}$, $B_{21}$, and $\rho(\nu)$.
• By comparing the result with Planck’s radiation law, Einstein derived the coefficient relations: $$\frac{A_{21}}{B_{21}} = \frac{8\pi h\nu^3}{c^3}$$ and $$B_{12} = B_{21}$$ for non-degenerate levels, showing the fundamental symmetry between absorption and stimulated emission probabilities.

Advantages / Applications

• It provides the theoretical foundation for laser action, especially the role of stimulated emission and population inversion.
• It explains blackbody radiation and successfully connects atomic processes with the Planck distribution.
• It is widely used in spectroscopy, optical amplifiers, masers, and photonics to analyze emission and absorption processes in atoms and molecules.

Summary

• Einstein’s theory describes how atoms interact with radiation through absorption, spontaneous emission, and stimulated emission.
• The A coefficient gives spontaneous emission probability, while the B coefficients describe absorption and stimulated emission probabilities.
• The theory is essential for understanding laser operation and the quantum basis of light amplification.
• Important terms to remember: absorption, spontaneous emission, stimulated emission, Einstein A coefficient, Einstein B coefficient, population inversion, radiation density