How to Play with Springs and Pulses in a Classical Harmonic Crystal
The classical dynamics of a harmonic lattice is presented. The aim of these lectures is not that of explaining once more the elements of the lattice dynamics. On the subject there is infact a large number of excellent treatises, either extended or short, starting from the milestone book of Born and Huang.
Here we examine how pulses, locally generated inside a crystal, propagate into the surrounding harmonic crystal, when the whole system of dispersion curves is taken into account. We focus attention on the process of propagation and not on that of pulse generation. A pulse is assumed to be already present at t=0 with a given shape and very localized. At t>0 it relaxes into the sourrounding threedimensional crystal with which it interacts through harmonic springs. This is the socalled regime of ballistic propagation when any source of scattering is disregarded.
We consider the propagation inside both a perfect and an imperfect crystal. For the last case we examine the effects due to a very localized change of force constants induced by a substitutional imperfection.
How the pulse is generated  is not considered here. Actually, it results from the interaction of the crystal with an external source, here assumed classical. Classical means that the source is able to perturbe the lattice by generating elastic pulses, but not to generate a single or few phonons, which are quantum objects. It describes, for instance, the semiclassical approach in the study of the electron-phonon interaction of bound electrons, where the electron is treated by quantum mechanics and the lattice classically. It describes also the behavior of the average values of position and momentum quantum variables or the atoms when the phonon field is in a coherent state (such a state is also called semiclassical just because of this property). When the source generates phonons in a coherent state, a first description of the phonon propagation can be done via the expectation values of position and momenta of the atoms, i.e. through the study of the dynamics of the corresponding classical values. This relatively easier problem is here treated. It is however a first approximation of the whole quantum problem. The complete quantum behavior of the phonon wavepacket propagation is recovered when also the fluctuations at all the orders are taken into account.
The plan of these lectures is the following. First, the lattice dynamics of a harmonic nonconducting crystal is briefly reviewed, mainly in order to define useful quantities. The equation of motion is discussed in terms of normal modes. Both the perfect and the imperfect crystal, containing a low concentration of substitutional impurities, are considered (Section I, II).
In order to follow the propagation of a pulse inside the crystal, a specific case must be studied, by solving numerically the equation of motion inside a specific crystal. We have chosen the diatomic KI and LiF ionic crystals as our model crystals, and the very reliable Breathing Shell Model as our lattice dynamics model.
We have considered a very localized initial pulse, as that generated, within the limits of the classical approximation, by excitating a bound electron in the Condon approximation. The relaxation of the lattice around the excited electron (i.e. the time dipendent dynamics of the n.n. ions), is numerically evaluated, for the cases of the thallium impurity in KI and the F 2 + color center in LiF (Section III, IV). The relaxation time deduced for the last case is compared with the experiments reported in the literature (Knox, Fork, Mollenauer ‘86:) a good agreement is found.
Here, we do not report and discuss the propagation of the pulse inside the crystal, described by the time dependent dynamics of the distant ions. From one hand it is a problem beyond the limits of these lectures. On the other hand, the linear and harmonic approximations, here adopted, correspond to the pulse dynamics in the socalled ballistic regime, when scattering and damping processes are still irrelevant, as in the first transient. Care must be taken in extending this approach to larger times and to further ions, reached by the pulse at delayed times, depending on its group velocity. However, we believe that, with all the necessary caution, this classical approach could be useful for a first undestanding of the socalled photoacoustics, i.e. the ways of propagation of lattice pulses induced by sudden optic transitions.
KeywordsCoherent State Defect Model Perfect Crystal Harmonic Approximation Projected Density
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