Читать книгу Spectroscopy for Materials Characterization - Группа авторов - Страница 19

In the previous paragraph, the simplest molecular model using the Born–Oppenheimer approximation enabled to determine the dipole moment matrix element (1.92). The first factor is related to the electronic wavefunction and the second factor is due to the nuclear wavefunction. It is usual to consider two contributions in the electronic wavefunction, the first due to the orbital motion and the second due to the spin degrees of freedom. The dipole moment matrix element can then be written

(1.100)

where the first factor accounts for the spatial dependence of the electron motion (orbital contribution), the second factor for the spin contribution, and the third factor for the nuclear vibration (Franck–Condon factor). Each of these factors contributes to the evaluation of the dipole moment matrix element, and they give rise to the selection rules for the vibronic transition [5, 8]. A given transition is usually called spin forbidden if

(1.101)

this is typically the most limiting rule. The dipole moment matrix element is related to the oscillator strength by Eq. (1.57) and to the experimental absorption through (1.56) and the molar extinction coefficient through (1.8); so, it is observed that the latter parameter is in the range (10⁻⁵ < ε < 10⁰) M⁻¹ cm⁻¹ for spin forbidden transitions. When the orbital factor is null

(1.102)

the transition is called orbitally forbidden and the range (10⁰ < ε < 10³) M⁻¹ cm⁻¹ is found for the molar extinction coefficient. Finally, values (10³ < ε < 10⁵) M⁻¹ cm⁻¹ typically pertain to allowed transitions. It is worth observing that these rules could not be strictly respected since in some cases the vibronic states could have a not pure spin or orbital angular momentum contribution, being instead a mixture of states [5, 8, 9].

Figure 1.6 Jablonski diagram for the transition processes of electrons among vibronic states. The electronic levels are labeled by S ₀, S ₁, T ₁ and thick horizontal lines, thinner horizontal lines mark the vibrational levels. Continuous arrows represent photon‐related (radiative) transitions; white arrows mark relaxation transitions among vibrational levels; short‐dashed arrows represent the intersystem crossing process (ISC); dashed line the internal conversion process. Typical times of the processes of absorption (Abs) fluorescence (Fluo), phosphorescence (Phos), and vibrational relaxation (R) are inserted.

The overall sequence of transitions occurring among the energy states of a molecule can now be described in more detail. Figure 1.6 shows a schematic representation of the processes connecting the ground state S ₀ and the excited state S ₁, assumed to be spin singlet states, and the excited spin triplet state T ₁.

This scheme is known as the Jablonski diagram [2, 5]. The system is assumed to be at a temperature T characterized by a thermal energy much lower than the vibrational energy of nuclei: kT ≪ ℏω, so only the lowest vibrational levels could be populated in thermal equilibrium; as an extreme case, it could be assumed that T = 0 K. The interaction with an electromagnetic field of opportune frequency gives rise to the absorption process in which a photon is lost by the field and the electron is promoted from the S ₀ to the S ₁ state. This effect is in a typical timescale much faster than any nuclear motion and can be assumed to occur in 10⁻¹⁵ s [5]. As stated above, the nuclei are in a fixed position and with given momentum during this process; as a consequence, if the starting vibrational state at low T is the ground vibrational state, the arrival vibrational state, in the electron excited state S ₁, is not necessarily the lowest vibrational energy level (this depends on the Franck–Condon factor). This is sketched by the tip of the absorption arrow pointing to a high vibrational level in S ₁. Successively, the nuclei relax and release their vibrational energy, reaching the lowest vibrational level (marked by the white arrow R), in a time typical of nuclear vibration: 10⁻¹² s [7, 14]. On reaching the lowest vibrational 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.