Группа авторов

Spectroscopy for Materials Characterization


Скачать книгу

target="_blank" rel="nofollow" href="#ulink_1d451ca0-a32e-5d9c-9715-ecd46ce6b650">(2.6)equation

      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)equation

      The total energy is:

      (2.8)equation

      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.

      2.1.2 Optical Transitions: The Franck–Condon Principle

      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)equation

      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:

Schematic illustration of configuration coordinate diagram.

      (2.11)equation

      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, images.

      Thus, the absorption transition probability is given by:

      The