Einstein coefficients

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.^[1] 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.

Spectral lines[edit]

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 $E 2$ of an atom, to a lower energy level $E 1$ , 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, $E 1$ , to a higher discrete energy state, $E 2$ , 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 $E 2 - E 1$ 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 $E 2 - E 1 = hν$ where $h$ denotes the Planck constant.^[2]^[3]^[4]^[5]^[6]^[7]

Detailed balancing[edit]

The Einstein coefficients are fixed probabilities per time associated with each atom, and do not depend on the state of the gas of which the atoms are a part. Therefore, any relationship that we can derive between the coefficients at, say, thermodynamic equilibrium will be valid universally.

At thermodynamic equilibrium, we will have a simple balancing, in which the net change in the number of any excited atoms is zero, being balanced by loss and gain due to all processes. With respect to bound-bound transitions, we will have detailed balancing as well, which states that the net exchange between any two levels will be balanced. This is because the probabilities of transition cannot be affected by the presence or absence of other excited atoms. Detailed balance (valid only at equilibrium) requires that the change in time of the number of atoms in level 1 due to the above three processes be zero: $0=A_{21}n_{2}+B_{21}n_{2}\rho (\nu )-B_{12}n_{1}\rho (\nu ).$

Along with detailed balancing, at temperature $T$ we may use our knowledge of the equilibrium energy distribution of the atoms, as stated in the Maxwell–Boltzmann distribution, and the equilibrium distribution of the photons, as stated in Planck's law of black body radiation to derive universal relationships between the Einstein coefficients.

From Boltzmann distribution we have for the number of excited atomic species i: ${\frac {n_{i}}{n}}={\frac {g_{i}e^{-E_{i}/kT}}{Z}},$ where n is the total number density of the atomic species, excited and unexcited, k is the Boltzmann constant, T is the temperature, $g_{i}$ is the degeneracy (also called the multiplicity) of state i, and Z is the partition function. From Planck's law of black-body radiation at temperature $T$ we have for the spectral radiance (radiance is energy per unit time per unit solid angle per unit projected area, when integrated over an appropriate spectral interval)^[26] at frequency $ν$ $\rho _{\nu }(\nu ,T)=F(\nu ){\frac {1}{e^{h\nu /kT}-1}},$ where^[27] $F(\nu )={\frac {2h\nu ^{3}}{c^{2}}},$ where $c$ is the speed of light and $h$ is the Planck constant.

Substituting these expressions into the equation of detailed balancing and remembering that $E 2 - E 1 = hν$ yields $A_{21}g_{2}e^{-h\nu /kT}+B_{21}g_{2}e^{-h\nu /kT}{\frac {F(\nu )}{e^{h\nu /kT}-1}}=B_{12}g_{1}{\frac {F(\nu )}{e^{h\nu /kT}-1}},$ or $A_{21}g_{2}(1-e^{-h\nu /kT})+B_{21}g_{2}F(\nu )e^{-h\nu /kT}=B_{12}g_{1}F(\nu ).$

The above equation must hold at any temperature, so from $T\to \infty$ one gets $B_{21}g_{2}=B_{12}g_{1},$ and from $T\to 0$ $A_{21}g_{2}=B_{21}g_{2}F(\nu ).$

Therefore, the three Einstein coefficients are interrelated by ${\frac {A_{21}}{B_{21}}}=F(\nu )$ and ${\frac {B_{21}}{B_{12}}}={\frac {g_{1}}{g_{2}}}.$

When this relation is inserted into the original equation, one can also find a relation between $A_{21}$ and $B_{12}$ , involving Planck's law.

Oscillator strengths[edit]

The oscillator strength $f_{12}$ is defined by the following relation to the cross section $\sigma$ for absorption:^[19] $\sigma ={\frac {e^{2}}{4\varepsilon _{0}m_{e}c}}\,f_{12}\,\phi _{\nu }={\frac {\pi e^{2}}{2\varepsilon _{0}m_{e}c}}\,f_{12}\,\phi _{\omega },$

where $e$ is the electron charge, $m_{e}$ is the electron mass, and $\phi _{\nu }$ and $\phi _{\omega }$ are normalized distribution functions in frequency and angular frequency respectively. This allows all three Einstein coefficients to be expressed in terms of the single oscillator strength associated with the particular atomic spectral line: ${\begin{aligned}B_{12}&={\frac {e^{2}}{4\varepsilon _{0}m_{e}h\nu }}f_{12},\\[1ex]B_{21}&={\frac {e^{2}}{4\varepsilon _{0}m_{e}h\nu }}{\frac {g_{1}}{g_{2}}}f_{12},\\[1ex]A_{21}&={\frac {2\pi \nu ^{2}e^{2}}{\varepsilon _{0}m_{e}c^{3}}}{\frac {g_{1}}{g_{2}}}f_{12}.\end{aligned}}$

Einstein coefficients

Spectral lines[edit]

Detailed balancing[edit]

Oscillator strengths[edit]

Emission Spectra from various light sources