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Transition Moments

Radiation interaction processes and selection rules and Einstein coefficients

Atoms are excited to higher energy levels by collisions or by the absorption of radiation and can de-excite by ejecting electrons or by emitting energy as discrete quanta called photons. These photons have a characteristic wavelengths which are the inverse of the energy emitted. The optical range of wavelengths of interest in chemical analysis is 100 nm to 900 nm, from the extreme ultraviolet (UV) to the near infrared (IR). The visible light region is approximately 400 nm (blue) to 700 nm (red).. In the thermodynamic equilibrium between matter and interacting electromagnetic radiation, three basic processes have been described by A. Einstein. For electric dipole interaction the following selection rules apply for transitions between the energy levels 0 and 1.

Δ n = 0, ± 1, ± 2, ± 3,...
Δ L = ± 1
Δ J = 0, ± 1
Δ mj = 0, ± 1

Max Planck, summarizing Rutherfords and Bohrs theories, came to the conclusion that electrons turn around their nuclei, following a planetary model, in well defined orbits with discrete energy levels without emitting radiation. But they may pass from an orbit to another one emit light with frequency:

eq 1.0 


Einstein, in 1917, gave a great contribution to “quantum optics”, deriving Plancks blackbody law from thermo dynamical considerations started by Planck. He described the number of transitions the atoms make per second between an excited state and an another state with less of energy. He also introduced rate constants, now called Einstein coefficients related to three different processes: spontaneous emission, photoabsorption and stimulated emission.

Spontaneous emission: this process occur when the electron is in its upper state (E2) and no photon present: it can emit a photon spontaneously.

Spontaneous Emission
  • Scheme of atomic spontaneous emission –

Let us consider an atom in an excited state 2 of energy E2 can in general make a spontaneous radiative transition to a state 1 of lower energy E1, with emission of a photon of energy:



corresponding to a spectrum line of wavenumber:



We shall denote by a21 the probability per unit time that an atom in state 2 will make such a transition to the state 1.
For an isolated, field-free atom in a state with total angular momentum Ji, there are:



degenerate quantum states of energy Ei , corresponding to the 2Ji + 1 possible values of the magnetic quantum number Mi.
The Einstein spontaneous emission transition probability rate is defined to be the total probability per unit time of an atom in a specific state j making a transition to any of the gi states of energy level i:



If at time t there are N2(t) atoms in state 2, the rate of change of N2 due to spontaneous transitions to all states of the level 1 is :



Atomic absorption or Photoabsorption: this process occurs when the electron is in its lower level (E1) and n photons are present: it can absorb a photon and make a transition to its upper level (E2).

    • Scheme of photoabsorption-

We can say that transitions may not only occur spontaneously, but may also be induced by the presence of a radiation field. We assume this radiation field to be isotropic and unpolarized and to have energy per unit volume of r(σ) dσ in the wavenumber range . The Einstein coefficients of absorption B12 and of stimulated emission B21 are defined as follow: if r(σ) is essentially constant over the profile of the spectrum line, then absorption by atoms in a state 1 results in transition to states of the level 2 at a rate:



Stimulated emission: this process was invented by A. Einstein for symmetry reasons and to satisfy Plancks blackbody equation. It occur when the electron is in its upper level (E2) and an electromagnetic radiation at (or near) the same frequency is present: it can emit an additional photon by stimulated emission decaying to the lower energy level (E1).

Stimulated Emission

- Scheme of atomic stimulated emission-

and atoms in a state 2 are stimulated (or induced) to make radiative transitions to state of the level 1 at the rate.



Values of the three Einstein coefficients are not independent and their mutual relationship may be inferred as follow. We suppose the radiation field and the atoms to be in mutual thermodynamic equilibrium at temperature T. The radiation energy density per unit wavenumber interval is given by Planck’s law



and the relative numbers of atoms in different quantum states are given by the Maxwell-Boltzmann law



According to the law of Detailed Balance, the rate of transition from all states of level 1 to all states of level 2 due to absorption from the radiation field must be equal to the rate of spontaneous plus induced emission from level 2 to level 1:



dividing by N2 and using (1.9) we obtain



which by comparison with (1.8) implies

g1B12 = g2B21




Thanks to relation (1.12), we shall drop the subscripts and refer simply to gA and gB because g is always the statistical weight of the initial level, i.e., the upper level for emission and the lower level for absorption.
The Einstein transition probabilities are intrinsic physical properties of the atom depending only on the initial and final states, on the intensity of any incident light, how strongly is the interaction between light and atoms and are independent of whether or not a state of thermodynamic equilibrium actually exists.



  • H. Haken, Light , North-Holland Physics Publishing, 1981.
  • Robert D. Cowan, The Theory of Atomic Structure and Spectra, University of California Press, 1981.
  • B. L. van der Waerden, Sources of Quantum Mechanics, Dover Publications, New York, 1968.

First published on the web: 15 November 1999 by Richard Payling

Author of the latest version (May 2008): Elisa Barisone  GLADNET ESR at EMPA in Thun, Switzerland.