Spectroscopy for Materials Characterization. Группа авторов
target="_blank" rel="nofollow" href="#ulink_1d451ca0-a32e-5d9c-9715-ecd46ce6b650">(2.6)
The vibration frequency in the lth electronic state is related to the coefficient of quadratic terms in expansion (2.6) by ω l = (a l /μ)1/2, where μ is the reduced mass of the system. The substitution of Eq. (2.6) into (2.5) transforms it into the Schrödinger equation of a harmonic oscillator. Its solution is well known: the energy levels are E l, n = W l (R l0) + ℏω l (n + 1/2), where n is the vibrational quantum number and φ l, n (R) are Hermite polynomials multiplied by Gaussian functions [3, 4].
Most generally, the defect‐matrix complex consists of N nuclei with f = 3N − 6 degrees of freedom and the function W l (R) can be expanded in a series analogous to Eq. (2.6). In this case, it is appropriate to introduce new variables (normal coordinates), q s (s = 1,2,…,f), so that the problem reduces to find the energy levels and wave functions for the stationary states of a set of f independent harmonic oscillators (normal modes). For each normal coordinate, the nuclear potential curve takes the form of a parabola centered at q l, s0:
(2.7)
The total energy is:
(2.8)
where the index s indicates the configurational coordinate with vibrational frequency ω l, s , and n denotes the set of vibrational quantum numbers n s . The total wave function is therefore the product of the individual normal oscillators:
The interplay between the normal modes of the defect and those of the whole solid is crucial to determine the optical lineshape. Indeed, it is known that the eigenfunction spectrum for the normal modes of a solid consists of alternating allowed and forbidden bands. When a defect, composed of a finite number of atoms undergoing vibrations, is introduced into the solid, we can distinguish two different cases [2].
Band vibrations: The vibrational frequency of the atoms included in the defect lies in one of the allowed bands of the solid matrix. The defect is in resonance with the eigenfrequencies of the host network and radiates elastic waves, thus losing energy. In this case, all N atoms, both belonging to the defect and to the host matrix, participate to the motion and share the finite energy of the normal mode. Then, each atom has energy depending on N −1 and its displacement is related to N −1/2. Vibrations of this kind are called band vibrations.
Localized vibrations: The vibrational frequency of the atoms included in the defect lies in one of the forbidden bands of the solid. In this case, since the defect is not in resonance with any eigenfrequency of the unperturbed host matrix, it does not radiate elastic waves. The vibration amplitudes of the atoms in the solid drop off rapidly with increasing the distance from the defect; only the atoms of the defect environment participate in the vibration and their displacements are independent of N. Such vibrations are called localized vibrations.
2.1.2 Optical Transitions: The Franck–Condon Principle
To describe the optical transition between two electronic states, Figure 2.1 shows a configuration coordinate (q s ) diagram where the potential energy curves of the ground, W I, s (q s )), and excited, W II,s (q s ), states are represented together with the vibrational levels, and ε 0 is the energy difference between them. Since the electronic state changes in a time (∼10−15 s) much shorter than the nuclear vibration (∼10−12 s), it can be assumed that the nuclei do not move nor change their momenta during the electronic transition (Franck–Condon Principle) [8, 9]. Therefore, both the absorption (from the ground to the excited state) and the luminescence (from the excited to the ground state) are represented by vertical arrows.
According to quantum mechanics, optical absorption and luminescence processes are quite well described by the first‐order time‐dependent perturbation theory. Let us consider the electronic state I, ψ I, n (r, q) = ϕ I (r, q)φ I, n (q), with the set of vibrational levels n ≡ {n 1, n 2, …, n f } thermally populated in accordance with the Boltzmann distribution; the absorption transition probability W(I, n → II, m) to the set of vibrational levels m ≡ {m 1, m 2, …, m f } of the electronic state II, ψ II, m (r, q) = ϕ II(r, q)φ II, m (q), is proportional to the absolute square of the matrix element of the perturbation operator. Since the light wavelength is much greater than the size of the defect (electric dipole approximation), the perturbation operator will be the dipole moment due to the electronic and nuclear charges:
(2.10)
It is worth noting that in the description of electronic transitions, the contribution D nucl, involving purely vibrational transitions within a single electronic state, can be neglected. The matrix element of P is therefore:
Figure 2.1 Configuration coordinate diagram. The potential energy of the ground W I, s and the excited W II, s electronic states are depicted together with the associated vibrational levels. For simplicity sake, the electronic transitions (vertical arrows) are supposed to take place from the lower vibrational level.
(2.11)
Hereafter, we will use the notation:
to indicate the electronic matrix element. D I → II(q) can be argued to be only weakly dependent on the nuclear coordinates, and in agreement with the Condon Approximation, it can be replaced by its value at the nuclear equilibrium position,
Thus, the absorption transition probability is given by:
The