rel="nofollow" href="#ulink_1d073c30-2bb3-58a5-924c-29f780cd5c9e">Eq. (1.71) with (1.23), it is found that
(1.72)
and finally, it is found that
It is worth considering that, typically, the non‐radiative processes are active at high temperature when the two‐level system is coupled to other degrees of freedom of the atomic system. At variance, at very low temperature, k nr = 0. On this basis, it is found that at T = 0 K (or at low enough temperature)
(1.74)
(1.75)
and a measure of the emission lifetime at low temperature, τ 0, is a direct estimate of k r. It is expected that the quantum efficiency decreases on increasing the temperature due to the activation of the non‐radiative relaxation processes [2, 5, 10]. In addition, the intrinsic or natural lifetime, τ 0 = 1/k r, can in general be determined at any temperature by the measure of the quantum efficiency and of the emission lifetime using (1.73)
(1.76)
this shows that typically the lifetime decreases on increasing the temperature. In general, the lifetime concept and the quantum efficiency are relevant aspects in the dynamic study of emitting centers and in the characterization of optoelectronic devices. They give useful information on the coupling of the system to its neighborhood.
1.2.3 Vibronic States: Homogeneous and Inhomogeneous Lineshape
The absorption and emission processes have usually a dependence on the frequency ν of the exciting electromagnetic field. In particular, the amplitude of the absorption coefficient or the intensity of the emitted radiation changes by changing ν. This effect is related to the specific features of the centers interacting with the electromagnetic field through the distribution of the density of their electronic states as a function of ν or of the electron’s energy [8, 9]. In a given experiment, where the intensity of transmitted or emitted light is measured as a function of the frequency, a profile of I(ν) is recorded. This profile is usually called lineshape of the spectrum. In this context, it is possible to distinguish two contributions to the lineshape: homogeneous and inhomogeneous. This classification is related to the origin of the physical process contributing to the spectral width of the lineshape [2, 6, 11, 13]. It is now useful to consider a wider class of systems interacting with the radiation: the molecules. In this case, in addition to the degrees of freedom of the electrons, the degrees of freedom of the nuclei are present. In particular, not only the electrons can move and change their energy, but also the nuclei can move relative to each other, adding a contribution to the total energy of the system. The features affecting the lineshape can now be listed, collecting them in the above‐cited two classes [8, 13].
Homogeneous contributions
1 Once in the excited state, the electron should return to the starting state in a time that corresponds to the emission lifetime. This time, based on the Heisenberg uncertainty principle, is linked to an indetermination in the energy value of the excited state. Such effect gives rise to a spread in the energy of the transition and, as a consequence, a width of the lineshape. In addition, in the case of a gas of atoms or molecules, the collisions between the particles give rise to changes in the permanence time in the excited state and, therefore, impose changes to the lifetime and width of the lineshape.
2 When the system is a molecule, the vibrational degrees of freedom affect the distribution of energy levels and give rise to specific lineshapes (see Eq. 1.92).
Inhomogeneous contributions
1 When the atoms or molecules of a given species are embedded in an environment, the different spatial distributions of the other atoms and molecules originate local electric fields that cause a different energy levels’ separation and distribution among the centers under investigation.
2 When atoms or molecules move in space, their thermal velocity with respect to an observer fixed in space is randomly spread according to the Maxwell–Boltzmann distribution, giving rise to frequency variation of the absorbed or emitted light due to the Doppler effect.
Typically, the homogeneous effect is “intrinsic” to a given lineshape and the inhomogeneous effect induces a replica of the intrinsic lineshape, overall giving a broader lineshape that is the envelope of the many replicas. Neglecting the vibrations, the more common homogeneous lineshape is the Lorentzian, whereas the inhomogeneous lineshape is the Gaussian. The intermediate lineshape that is a Gaussian convolution of Lorentzian lineshapes is known as Voigt lineshape. The analytic forms of these functions are:
(1.77)
this lineshape is centered at ν 0 and is characterized by a full width at half maximum (FWHM) of the amplitude equal to 2A;
this is centered at ν 0 and has FWHM
(1.79)
where the Gaussian and Lorentzian contributions have been introduced. The final shape of the Voigt function depends on the balance between the FWHM of the two composing contributions.
To go deeper into the homogeneous lineshape features, the case of a molecular system has to be considered. It is then necessary to determine its electronic states in detail. First of all, the molecule is a many‐body system constituted by the electrons, the nuclei (it is useful to consider this unit for spectroscopic aims, associating the constituent parts: protons and neutrons, motion) and their motion and interaction. The more general Hamiltonian for this system is
(1.80)
where the subscripts label electrons’ (e) and nuclei’s (n) kinetic and potential energies, in order: the electrons kinetic energy; the nuclei kinetic energy; the electron–electron interaction potential energy; the electron–nuclei interaction potential energy; the nuclei–nuclei interaction potential energy. This Hamiltonian is really complex and it can be simplified considering that the proton mass is >103 times larger than the electron mass. As a consequence, it can be
