Phonon frequencies

The dynamics of ion surface scattering at energies exceeding several hundred electronvolts can be described by a series of binary collision approximations (BCAs) in which only the interaction of one energetic particle with a solid atom is considered at a time [25]. This model is reasonable because the interaction time for the collision is short compared witii the period of phonon frequencies in solids, and the interaction distance is shorter tlian the interatomic distances in solids. The BCA simplifies the many-body interactions between a projectile and solid atoms to a series of two-body collisions of the projectile and individual solid atoms. This can be described with results from the well known two-body central force problem [26]. 

Iditional importance is that the vibrational modes are dependent upon the reciprocal e vector k. As with calculations of the electronic structure of periodic lattices these cal-ions are usually performed by selecting a suitable set of points from within the Brillouin. For periodic solids it is necessary to take this periodicity into account the effect on the id-derivative matrix is that each element x] needs to be multiplied by the phase factor k-r y). A phonon dispersion curve indicates how the phonon frequencies vary over tlie luin zone, an example being shown in Figure 5.37. The phonon density of states is ariation in the number of frequencies as a function of frequency. A purely transverse ition is one where the displacement of the atoms is perpendicular to the direction of on of the wave in a pmely longitudinal vibration tlie atomic displacements are in the ition of the wave motion. Such motions can be observed in simple systems (e.g. those contain just one or two atoms per unit cell) but for general three-dimensional lattices of the vibrations are a mixture of transverse and longitudinal motions, the exceptions... 

Once the phonon frequencies are known it becomes possible to determine various thermodynamic quantities using statistical mechanics (see Appendix 6.1). Here again some slight modifications are required to the standard formulae. These modifications are usually a consequence of the need to sum over the points sampled in the Brillouin zone. For example, the zero-point energy is ... 

In Equation (5.58) the outer summation is over the p points q which are used to sample the Brillouin zone, is the fractional weight associated with each point (related to the volume of Brillouin zone space surrounding q) and vi are the phonon frequencies. In addition to the internal energy due to the vibrational modes it is also possible to calculate the vibrational entropy, and hence the free energy. The Helmholtz free energy at a temperature... 

In the analysis of crystal growth, one is mainly interested in macroscopic features like crystal morphology and growth rate. Therefore, the time scale in question is rather slower than the time scale of phonon frequencies, and the deviation of atomic positions from the average crystalline lattice position can be neglected. A lattice model gives a sufiicient description for the crystal shapes and growth [3,34,35]. 

Momentum conservation implies that the wave vectors of the phonons, interacting with the electrons close to the Fermi surface, are either small (forward scattering) or close to 2kp=7i/a (backward scattering). In Eq. (3.10) forward scattering is neglected, as the electron interaction with the acoustic phonons is weak. Neglecting also the weak (/-dependence of the optical phonon frequency, the lattice energy reads ... 

At very low temperatures, Holstein predicted that the small polaron would move in delocalized levels, the so-called small polaron band. In that case, mobility is expected to increase when temperature decreases. The transition between the hopping and band regimes would occur at a critical temperature T, 0.40. We note, however, that the polaron bandwidth is predicted to be very narrow ( IO Viojo, or lO 4 eV for a typical phonon frequency of 1000 cm-1). It is therefore expected that this band transport mechanism would be easily disturbed by crystal defects. 

In this equation v is a phonon frequency, such that hv is approximately k, with the Debye characteristic temperature of the metal. The quantity p is the product of the density of electrons in energy at the Fermi surface, N(0), and the electron-phonon interaction energy, V. 

An accurate calculation of the heat conductivity requires solving a kinetic equation for the phonons coupled with the multilevel systems, which would account for thermal saturation effects and so on. We encountered one example of such saturation in the expression (21) for the scattering strength by a two-level system, where the factor of tanh((3co/2) reflected the difference between thermal populations of the two states. Neglecting these effects should lead to an error on the order of unity for the thermal frequencies. Within this single relaxation time approximation for each phonon frequency, the Fermi golden rule yields, for the scattering rate of a phonon with Ha kgT,... 

In the Ising-type model, the change of molecular volume AV due to the LS<->HS transformation leads to a change of phonon frequencies of the lattice. The effect may be treated within the Debye approximation which requires that the interaction parameters and J2 are replaced by J and J 2 where ... 

Table 4.1. Various processes contributing to the spectral line broadening for local vibrations. Frequencies of collectivized local vibrations QK (solid arrows) are supposed to exceed phonon frequencies oiRq (dashed arrows) Ok > max oncq. For an extremely narrow band of local vibrations, diagrams A and B respectively refer to relaxation and dephasing processes, whereas diagrams C account for the case realizable only at the nonzero band width for local vibrations.

Fig. 2.13. (a) Anisotropic reflectivity change of graphite measured with sub-10 fs pulses at 3.1eV. (b) Time evolution of the E2g2 phonon frequency obtained from time-windowed FT. From [51]... 

In combination with DFT calculations, the time- and depth-dependent phonon frequency allows to estimate the effective diffusion rate of 2.3 cm2 s 1 and the electron-hole thermalization time of 260 fs for highly excited carriers. A recent experiment by the same group looked at the (101) and (112) diffractions in search of the coherent Eg phonons. They observed a periodic modulation at 1.3 THz, which was much slower than that expected for the Eg mode, and attributed the oscillation to the squeezed phonon states [9]. 

Fig. 3.11. Left spectrum of the tailored pump beam for different spectral separations Aw CC2 — oji between two spectral packets (oJi and Lof). The bottom case is resonant to the E2 phonon frequency l o- Right transient transmittance of GaN excited with the tailored pump pulses, showing the enhancement of coherent oscillation of the E2 phonon for u) — u)2 — Oo- From [28]...

The Kieffer approach uses a harmonic description of the lattice dynamics in which the phonon frequencies are independent of temperature and pressure. A further improvement of the accuracy of the model is achieved by taking the effect of temperature and pressure on the vibrational frequencies explicitly into account. This gives better agreement with experimental heat capacity data that usually are collected at constant pressure [9],... 

The first term on the right-hand side of Equation (1.24) accounts for the generated intensity due to Rayleigh scattered light, while the second term is related to the intensity of the Raman scattered light. For visible light coo 10 Hz, while the characteristic phonon frequencies are much shorter, typically 12 10 Hz. Then coq and the intensity of Raman scattering varies as coq, as stated in point (iv) above. 

When the chemisorbed molecule is vibrationally excited this influences not only the metal electrons but also the ion cores in the neighbourhood. The vibrating ion cores can then in turn couple to other molecules and give rise to a short range interaction mediated via the substrate lattice. However, as Cl is much larger than the highest substrate phonon frequency the effect of this interaction is very small , but it can be important for low frequency modes . 

Fig. 12. Inelastic one-phonon He scattering cross-section (He beam energy 20 meV and 100 meV) as a function of phonon frequency. (After Ibach )...

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