probability of population of the excited state per unit of time, coinciding with the transition rate, is then given by
in which the electric dipole matrix element μ 12 relative to the considered atomic states in the direction of the external field has been introduced. Using (1.29), it is possible to find that
This result shows a connection between the macroscopic empiric quantities and the microscopic ones related to the quantum mechanical states of the electron in the atom. In particular, it is shown that the transition probability is related to the electric dipole matrix element μ 12.
Considering that in vacuum [8, 13]
(1.43)
ε 0 being the permittivity of free space (8.854 × 10−12 kg−1 m−3 s4 A2), it is possible to find that
Furthermore, since the intensity of radiation and the energy density are related by [1, 8, 13]
(1.45)
using (1.42), it is found that the transition rate between the atom’s energy states is given by
(1.46)
a connection with the intensity of radiation is made explicit now. The rate of energy absorbed per unit of volume by the atom from the electromagnetic field can then be written as
(1.47)
By assuming that all the atoms reside in the N 1 state, this is the energy lost by the radiation field. If a sample of thickness dx is considered, the energy lost for unit area by the electromagnetic wave is then
(1.48)
By recalling the Lambert–Beer law in differential form from (1.1), it is shown that
(1.49)
where the frequency dependence has been inserted, and finally one obtains
(1.50)
This is a more direct connection between experimental parameters and the microscopic ones. In fact, by using (1.44), it is found that
Integrating (1.51) over the entire frequency range pertaining to the given atomic (or molecular) species gives
where the central absorption frequency
1.2.2 Oscillator Strength, Lifetime, Quantum Yield
In the previous paragraph, we have determined theoretical quantities relating the atomic wavefunctions of the energy levels to spectroscopic observables. In the simplified model of the atom with a single electron, it is considered that this latter can oscillate in a harmonic potential well. The atomic system is then a charged harmonic oscillator. The wavefunctions of this system enable to evaluate the electric dipole matrix element μ 12 and to determine the theoretical integrated absorption reported by (1.52). It can be shown that the expected value of integrated absorption is [5, 11]:
where N A is Avogadro’s number, c is the speed of light, and m is the electron mass (9.109 ⋅ 10−31 kg). Equation (1.53) gives a numerical value that can be compared with experimental results. This comparison gives origin to the quantity called oscillator strength and usually given by f:
(1.54)
The oscillator strength is a dimensionless quantity characterizing the transition between the two considered energy levels E 1 and E 2. By introducing [5, 11]
(1.55)
it is shown that
Expected values of f are lower equal than unity and on decreasing of the probability of the absorption process decreases too. This feature is linked to the selection rules that highlight those quantum transitions between energy levels giving an electric dipole matrix element different from zero [5].
Another form of the oscillator strength is [8, 15]
