Phonons: Harmonic Approximation
As illustrated in Sect. 3.1, in the Born-Oppenheimer (BO) approximation (also known as the adiabatic approximation), the equations which determine the electronic state are decoupled from those related to the ionic dynamics. In other words, the atomic motion of the system (molecule, cluster, or solid) is studied under the hypothesis that the electronic system always remains in the electronic ground state associated with the instantaneous geometrical configuration. We have seen in Sect. 3.1 the conditions of applicability of the BO scheme. In Sect. 6.1 we analyze the solution of the problem of atomic motion in the case of small oscillations near the equilibrium configuration (i.e. when every atom stays always near to its equilibrium position, and there is no atomic diffusion). In that case, writing down a second order expansion of the total potential felt by the ions (harmonic approximation), we show that it is possible to reduce the problem to that of a collection of independent harmonic oscillators. The general solution is then a superposition of 3N at normal modes of vibration, each of them having its own frequency and its own eigenvector, obtained by diagonalizing the dynamical matrix. The quantum description of this set of independent harmonic oscillators leads to the concept of phonons.
KeywordsGraphite Cage Hexagonal Fullerene Rium
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