level in S 1, the system could relax to the S 0 state through coupling to its highly excited vibrational levels and the internal conversion (IC, dashed arrow) process and without energy release to the electromagnetic field (non‐radiative relaxation, heating of the molecule). It is also possible that the system relaxes to the S 0 state by emitting a photon. This process is called fluorescence and the overall permanence in the S 1 state of an ensemble of similar molecules gives the lifetime of this emission process. A typical timescale starting from 10−9 s characterizes this process. This transition is spin allowed and the short lifetime is a characteristic feature. It is worth noting that also in this case the arrival vibrational state in S 0 is determined by selection rules and could not be the ground vibrational level. Another pathway from the S 1 state involves the molecular vibrations and the spin–orbit coupling [2, 5]. This process enables to change the spin state of the electrons, giving a change from singlet (S 1) to triplet (T 1) state. This interaction process is known as intersystem crossing (ISC, short‐dashed arrow) and gives rise to a non‐radiative relaxation between excited states. Once in T 1, where the vibrational state could be different from the ground state, a vibrational relaxation occurs to the ground vibrational level (white arrow). Successively, the system could reach the S 0 state through ISC relaxation to its high‐energy vibrational states (short‐dashed arrow). Furthermore, a transition to S 0 could occur with the emission of a photon by the phosphorescence phenomenon. In this case, the permanence in the excited state T 1 determines the lifetime of this emission process that typically is in a timescale larger than 10−4 s. It is observed that this transition is spin forbidden, since a passage from triplet to singlet state occurs, and the related lifetime is much longer than for fluorescence. Overall, the ISC and IC processes are related to vibrations of the molecule and are known as phonon‐assisted processes. The presence of vibrations is related to the temperature of the system and an Arrhenius law is assumed for these processes [2, 5]. In detail, the rate of the intersystem crossing process, K ISC, is given by
(1.103)
where K 0 is a pre‐exponential factor taking into account entropic‐statistical factors, ΔE is the activation energy of the process, and k is the Boltzmann constant.
The Jablonski diagram is useful to describe the overall emission features of a system and to schematize the energy levels distribution and their dynamics aspects.
1.2.5 Excited States Rate Equations
To go deeper in the emission features, a simplified Jablonski diagram for the transition processes of electrons among vibronic states can be considered, joining the representations reported in Figure 1.4 (bottom) and Figure 1.6. The singlet energy levels S 0 and S 1 are connected by the absorption process and by the radiative fluorescence emission from S 1 with rate
where the absorbed light through a sample of thickness L, giving transitions from S 0 to S 1, has been introduced based on Eq. (1.5). It is observed that the emission from the excited states depends on their population; so, it can be stated that for the fluorescence the emitted light is given by
where the excitation, E exc, and the emission, E em, energies have been introduced together with the temperature dependence and a lineshape f(E em) including the homogeneous and inhomogeneous distributions of levels [18]. Analogously, for phosphorescence, it is found that
where the lineshape for the triplet to singlet emission of phosphorescence, g(E em), has been inserted. In the stationary state (ss), it is found that
(1.108)
(1.109)
which can be substituted into (1.106) and (1.107) to obtain
Apart from the dependence on the rate of transitions, these equations show that the emission intensities of both fluorescence and phosphorescence depend on the absorption process. In this context, in the case of low absorption, it is found that (1 – e −αL ) ~ αL. Since the absorption coefficient is a function of the excitation energy, both Eqs. (1.110) and (1.111) enable to determine α(E exc) when the E em is fixed and to reconstruct the absorption profile. This kind of measurement is known as excitation spectrum [2, 18] and enables to determine the connection between the spectra of phosphorescence and fluorescence, relating them to the same absorption pathway, and to reconstruct the Jablonski diagram.
Finally, in analogy to Eq.
