The Einstein coefficients equations describe the rates of absorption, spontaneous emission, and stimulated emission of radiation by atoms or molecules.In 1916, Albert Einstein published the paper 'Quantum Theory of Radiation', where he introduced the A and B coefficients describing the probabilities of transitions between energy levels in atoms. Before that, physicists only knew about absorption and spontaneous emission; Einstein predicted a third process — stimulated emission. He performed an ingenious thermodynamic analysis: in a box with atoms at thermal equilibrium, the number of upward and downward transitions must balance. By substituting Planck's radiation distribution, he found that without stimulated emission, the balance does not work out. Moreover, it turned out that the coefficients for stimulated emission and absorption are proportional (taking into account level degeneracy), and the coefficient for spontaneous emission is expressed through them and the frequency. This work laid the theoretical foundation for the creation of lasers forty years later.
Albert Einstein introduced the A and B coefficients in 1916, generalizing Planck's radiation law. Before that, Bohr proposed a model of the atom with quantum transitions, but the emission mechanism remained semi-classical. Einstein, applying statistical mechanics and the principle of detailed balance, showed that compatibility with classical electrodynamics and the Planck spectrum required the existence of stimulated emission. Later, in 1927, Paul Dirac provided a strict quantum electrodynamic justification of these coefficients, linking them to matrix elements of the dipole moment. Einstein's work provided the impetus for the creation of masers (1954, Charles Townes) and lasers (1960, Theodore Maiman).
Einstein coefficients are used to calculate the intensity of spectral lines, population inversion, and gain in active media. For example, the condition for laser generation: gain per pass must exceed losses — a direct consequence of the equations. Limits of applicability: valid for dipole transitions in rarefied gases and weak fields; in strong fields or for forbidden transitions, quantum electrodynamics corrections are required.
The operation of all lasers (laser pointer, CD drive) and masers is based on stimulated emission. Spectroscopists calculate atomic level populations in stars.
Einstein called stimulated emission 'induced radiation', not suspecting that it would become the basis of a technology. The term 'laser' itself — an acronym for Light Amplification by Stimulated Emission of Radiation — directly refers to his prediction.
| \dot{n}_{\text{ion}} | variable | ionization rate per unit volume — ionization rate metre per second per cubic metre |
| L | variable | line luminosity — luminosity watt |
| R | variable | emission coefficient — emission coefficient watt per cubic metre per steradian |
| c | variable | speed of light — speed of light metre per second |
| f_{\text{esc}} | variable | escape fraction of ionizing photons — escape fraction |
| E(B-V) | variable | Color excess between B and V bands — color excess |
| k(λ) | variable | Extinction curve at wavelength λ — extinction curve |
| F_λ | variable | Flux at wavelength λ — flux watt per metre squared |
| λ | variable | Wavelength — wavelength metre |
| log10 | operator | Base-10 logarithm — logarithm base 10 |
| \varepsilon(q) | variable | Excitation energy as a function of wavevector q — energy hartree |
| q | variable | Wavevector magnitude — wavevector inverse bohr |
| S(q) | variable | Static structure factor — structure factor |
| J | variable | photolysis rate constant — rate constant per second |
| \sigma(\lambda) | variable | absorption cross-section as function of wavelength — cross section square metre |
| \phi(\lambda) | variable | quantum yield as function of wavelength — quantum yield |
| F(\lambda) | variable | actinic flux as function of wavelength — photon flux per square metre per second per metre |
| \lambda | variable | wavelength — wavelength metre |
| \int | operator | integration over wavelength — integral |
