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 deexcite 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:

(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 (E_{2}) and no photon present: it can emit a photon spontaneously.
 Scheme of atomic spontaneous emission –
Let us consider an atom in an excited state 2 of energy E_{2} can in general make a spontaneous radiative transition to a state 1 of lower energy E_{1}, with emission of a photon of energy:

(1.1) 
corresponding to a spectrum line of wavenumber:

(1.2) 
We shall denote by a_{21} the probability per unit time that an atom in state 2 will make such a transition to the state 1.
For an isolated, fieldfree atom in a state with total angular momentum J_{i}, there are:

(1.3) 
degenerate quantum states of energy E_{i} , corresponding to the 2J_{i} + 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 g_{i} states of energy level i:

(1.4) 
If at time t there are N_{2}(t) atoms in state 2, the rate of change of N2 due to spontaneous transitions to all states of the level 1 is :

(1.5) 
Atomic absorption or Photoabsorption: this process occurs when the electron is in its lower level (E_{1}) and n photons are present: it can absorb a photon and make a transition to its upper level (E_{2}).
 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 dσ. The Einstein coefficients of absorption B_{12} and of stimulated emission B_{21} 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:

(1.6) 
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 (E_{2}) 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 (E_{1}).
 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.

(1.7) 
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

(1.8) 
and the relative numbers of atoms in different quantum states are given by the MaxwellBoltzmann law

(1.9) 
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:

(1.10) 
dividing by N_{2} and using (1.9) we obtain

(1.11) 
which by comparison with (1.8) implies
g_{1}B_{12} = g_{2}B_{21} 
(1.12) 

(1.13) 
Thanks to relation (1.12), we shall drop the subscripts and refer simply to g_{A} and g_{B }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.
References:

H. Haken, Light , NorthHolland 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.
