ρ(ν) is the density of energy for unit volume and unit frequency interval, h the Planck’s constant, ν the radiation frequency, c the speed of light, k the Boltzmann’s constant (1.38 ⋅ 10−23 J K−1), and T the absolute temperature.
Figure 1.4 Top: Schematic representation of two energy‐level atom. The arrows represent the absorption and emission transition processes whose probability is given by Einstein’s coefficients A 21, B 12, B 21 and the density of radiation ρ(ν). Bottom: Schematic representation of two energy levels of an atom. The arrows represent the excitation and relaxation transitions. The absorption rate is R abs, the radiative emission rate is k r, and the non‐radiative transition rate is k nr.
The interaction between a density N of atoms for unit of volume and the radiation field causes an exchange of energy with those electromagnetic waves’ modes having frequency related to the atom’s energy levels’ separation by the equation E 2 − E 1 = hν. As a consequence, based on Einstein’s treatment, the following processes can occur [8, 13]:
Transition from the state E 1 to the state E 2, stimulated by the absorption of a photon; based on Einstein’s theory, the rate of this process is given by(1.25)
where N 1 is the density of atoms (population) in the lower energy state.
Transition from the state E 2 to the state E 1, stimulated by the emission of a photon; the rate is given by(1.26)
where N 2 is the density of atoms in the upper energy state.
Spontaneous transition from the state E 2 to the state E 1 with a rate(1.27)
The Einstein’s coefficients A 21, B 12, and B 21 have been used. In particular, it is worth observing that B 12 and B 21 are related to the presence of the field (stimulated processes of absorption and emission, respectively), whereas A 21 is present also without electromagnetic field and is related to spontaneous emission. This term is related to the radiative emission lifetime introduced in the previous paragraph and, in detail, it is the reciprocal of the lifetime at low temperature, A 21 = 1/τ [13]. At thermal equilibrium, the population of atomic states should reach a stationary condition and it is expected that
(1.28)
and, based on the above reported processes, one obtains
and the relation
The Boltzmann distribution at thermal equilibrium in a two‐level system without degeneracy predicts that [14]
Equating (1.30) and (1.31) and solving with respect to ρ(ν), it is found that
This distribution of energy density in the electromagnetic field should coincide with the Planck’s law at thermal equilibrium. As a consequence, by equating (1.32) and (1.24), it is found that
(1.33)
and, finally
In the case of degenerate energy levels with degeneracy g 1 and g 2, it is shown that (1.35) transforms into
(1.36)
whereas (1.34) remains unchanged [8, 13].
Using the quantum mechanical treatment of the interaction between radiation and matter and, in particular, neglecting any magnetic contribution and considering the electric dipole approximation, the atom can be described by a dipole moment
(1.37)
where e is the electron charge (1.602 ⋅ 10−19 C) and r is its position vector with respect to the atomic nucleus. The time‐dependent perturbation theory enables to show that the probability to populate the higher energy level of the atom E 2 (multiplied by unit frequency interval), starting from the level with energy E 1, is given by [8, 9, 13]:
(1.38)
where V is the interaction energy between the electric field and the electric dipole moment:
(1.39)
and ℏ = h/2π. Considering a linearly polarized lightwave with electric field of amplitude
(1.40)
the
