Einstein coefficients

Quantities describing probability of absorption or emission of light

In atomic, molecular, and optical physics, the Einstein coefficients are quantities describing the probability of absorption or emission of a photon by an atom or molecule. The Einstein A coefficients are related to the rate of spontaneous emission of light, and the Einstein B coefficients are related to the absorption and stimulated emission of light. Throughout this article, "light" refers to any electromagnetic radiation, not necessarily in the visible spectrum. These coefficients are named after Albert Einstein, who proposed them in 1916.

In physics, one thinks of a spectral line from two viewpoints. An emission line is formed when an atom or molecule makes a transition from a particular discrete energy level E2 of an atom, to a lower energy level E1, emitting a photon of a particular energy and wavelength. A spectrum of many such photons will show an emission spike at the wavelength associated with these photons. An absorption line is formed when an atom or molecule makes a transition from a lower, E1, to a higher discrete energy state, E2, with a photon being absorbed in the process. These absorbed photons generally come from background continuum radiation (the full spectrum of electromagnetic radiation) and a spectrum will show a drop in the continuum radiation at the wavelength associated with the absorbed photons. The two states must be bound states in which the electron is bound to the atom or molecule, so the transition is sometimes referred to as a "bound–bound" transition, as opposed to a transition in which the electron is ejected out of the atom completely ("bound–free" transition) into a continuum state, leaving an ionized atom, and generating continuum radiation. A photon with an energy equal to the difference E2 − E1 between the energy levels is released or absorbed in the process. The frequency ν at which the spectral line occurs is related to the photon energy by Bohr's frequency condition where h denotes the Planck constant.

An atomic spectral line refers to emission and absorption events in a gas in which is the density of atoms in the upper energy state for the line, and is the density of atoms in the lower energy state for the line. The emission of atomic line radiation at frequency ν may be described by an emission coefficient with units of energy/(time × volume × solid angle). ε dt dV dΩ is then the energy emitted by a volume element in time into solid angle For atomic line radiation,

where is the Einstein coefficient for spontaneous emission, which is fixed by the intrinsic properties of the relevant atom for the two relevant energy levels. The absorption of atomic line radiation may be described by an absorption coefficient with units of 1/length. The expression κ' dx gives the fraction of intensity absorbed for a light beam at frequency ν while traveling distance dx. The absorption coefficient is given by

where and are the Einstein coefficients for photon absorption and induced emission respectively. Like the coefficient , these are also fixed by the intrinsic properties of the relevant atom for the two relevant energy levels. For thermodynamics and for the application of Kirchhoff's law, it is necessary that the total absorption be expressed as the algebraic sum of two components, described respectively by and , which may be regarded as positive and negative absorption, which are, respectively, the direct photon absorption, and what is commonly called stimulated or induced emission. The above equations have ignored the influence of the spectroscopic line shape. To be accurate, the above equations need to be multiplied by the (normalized) spectral line shape, in which case the units will change to include a 1/Hz term. Under conditions of thermodynamic equilibrium, the number densities and , the Einstein coefficients, and the spectral energy density provide sufficient information to determine the absorption and emission rates.

The number densities and are set by the physical state of the gas in which the spectral line occurs, including the local spectral radiance (or, in some presentations, the local spectral radiant energy density). When that state is either one of strict thermodynamic equilibrium, or one of so called "local thermodynamic equilibrium", then the distribution of atomic states of excitation (which includes and ) determines the rates of atomic emissions and absorptions to be such that Kirchhoff's law of equality of radiative absorptivity and emissivity holds. In strict thermodynamic equilibrium, the radiation field is said to be black body radiation and is described by Planck's law. For local thermodynamic equilibrium, the radiation field does not have to be a black body field, but the rate of interatomic collisions must vastly exceed the rates of absorption and emission of quanta of light, so that the interatomic collisions entirely dominate the distribution of states of atomic excitation. Circumstances occur in which local thermodynamic equilibrium does not prevail, because the strong radiative effects overwhelm the tendency to the Maxwell–Boltzmann distribution of molecular velocities. For example, in the atmosphere of the Sun, the great strength of the radiation dominates. In the upper atmosphere of the Earth, at altitudes over 100 km, the rarity of intermolecular collisions is decisive. In the cases of thermodynamic equilibrium and of local thermodynamic equilibrium, the number densities of the atoms, both excited and unexcited, may be calculated from the Maxwell–Boltzmann distribution, but for other cases, (e. g. lasers) the calculation is more complicated.

In 1916, Albert Einstein proposed that there are three processes occurring in the formation of an atomic spectral line. The three processes are referred to as spontaneous emission, stimulated emission, and absorption. With each is associated an Einstein coefficient, which is a measure of the probability of that particular process occurring. Einstein considered the case of isotropic radiation of frequency ν and spectral energy density ρ(ν). Paul Dirac derived the coefficients in a 1927 paper titled "The Quantum Theory of the Emission and Absorption of Radiation".

Hilborn has compared various formulations for derivations for the Einstein coefficients, by various authors. For example, Herzberg works with irradiance and wavenumber; Yariv works with energy per unit volume per unit frequency interval, as is the case in the more recent (2008) formulation. Mihalas & Weibel Mihalas work with radiance and frequency; also Goody & Yung; Loudon uses angular frequency and radiance.

Stimulated emission (also known as induced emission) is the process by which an electron is induced to jump from a higher energy level to a lower one by the presence of electromagnetic radiation at (or near) the frequency of the transition. From the thermodynamic viewpoint, this process must be regarded as negative absorption. The process is described by the Einstein coefficient (m3 J−1 s−2), which gives the probability per unit time per unit energy density of the radiation field per unit frequency that an electron in state 2 with energy will decay to state 1 with energy , emitting a photon with an energy The change in the number density of atoms in state 1 per unit time due to induced emission will be

where denotes the spectral energy density of the isotropic radiation field at the frequency of the transition (see Planck's law). Stimulated emission is one of the fundamental processes that led to the development of the laser. Laser radiation is, however, very far from the present case of isotropic radiation.

Absorption is the process by which a photon is absorbed by the atom, causing an electron to jump from a lower energy level to a higher one. The process is described by the Einstein coefficient (m3 J−1 s−2), which gives the probability per unit time per unit energy density of the radiation field per unit frequency that an electron in state 1 with energy will absorb a photon with an energy and jump to state 2 with energy The change in the number density of atoms in state 1 per unit time due to absorption will be