Phonon acoustic

Cerullo G, De Silverstri S and Banin U 1999 Size-dependent dynamics of coherent acoustic phonons in nanocrystal quantum dots Phys. Rev. B 60 1928... 

Thus far we have discussed the direct mechanism of dissipation, when the reaction coordinate is coupled directly to the continuous spectrum of the bath degrees of freedom. For chemical reactions this situation is rather rare, since low-frequency acoustic phonon modes have much larger wavelengths than the size of the reaction complex, and so they cannot cause a considerable relative displacement of the reactants. The direct mechanism may play an essential role in long-distance electron transfer in dielectric media, when the reorganization energy is created by displacement of equilibrium positions of low-frequency polarization phonons. Another cause of friction may be anharmonicity of solids which leads to multiphonon processes. In particular, the Raman processes may provide small energy losses. 

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 ... 

In the crystal, the total number of vibrations is determined by the number of atoms per molecule, N, and the nmnber of molecules per primitive cell, Z, multiplied by the degrees of freedom of each atom 3ZN. In the case of a-Sg (Z =4, N =8) this gives a total of 96 vibrations ( ) which can be separated in (3N-6)—Z = 72 intramolecular or "internal" vibrations and 6Z = 24 intermo-lecular vibrations or lattice phonons ("external" vibrations). The total of the external vibrations consists of 3Z = 12 librational modes due to the molecular rotations, 3Z-3 = 9 translational modes, and 3 acoustic phonons, respectively. 

TRXRD detects the propagation of coherent acoustic phonons as a transient change in the diffraction angles. In contrast, the atomic motions associated with coherent optical phonons modify only the Bragg peak intensity, because they do not change the barycentric positions of the crystal lattice. The Bragg peak intensity is proportional to the squared modulus of the structure factor [1,3,4] ... 

For mi = m2, the expression reduces to that obtained for a monoatomic chain (eq. 8.18). When q approaches zero, the amplitudes of the two types of atom become equal and the two types of atom vibrate in phase, as depicted in the upper part of Figure 8.10. Two neighbouring atoms vibrate together without an appreciable variation in their interatomic distance. The waves are termed acoustic vibrations, acoustic vibrational modes or acoustic phonons. When q is increased, the unit cell, which consists of one atom of each type, becomes increasingly deformed. At < max the heavier atoms vibrate in phase while the lighter atoms are stationary. 

Note that dra(t)/dt = [H,ra]=(l/ma)[pa-qaA(ra)] and, consequently, the first term in (69) represents the kinetic energy of the system of particles in the presence of the transverse electromagnetic field. Note the analogy between this representation and the dynamical solute-solvent coupling of section 2.6 where the optical phonons are equivalent to electromagnetic photons of low frequency (the acoustical phonons are related to sound waves). 

The application of an electric field E to a conducting material results in an average velocity v of free charge carriers parallel to the field superimposed on their random thermal motion. The motion of charge carriers is retarded by scattering events, for example with acoustic phonons or ionized impurities. From the mean time t between such events, the effective mass m of the relevant charge carrier and the elementary charge e, the velocity v can be calculated ... 

Bulk silicon is a semiconductor with an indirect band structure, as schematically shown in Fig. 7.12 c. The top of the VB is located at the center of the Brillouin zone, while the CB has six minima at the equivalent (100) directions. The only allowed optical transition is a vertical transition of a photon with a subsequent electron-phonon scattering process which is needed to conserve the crystal momentum, as indicated by arrows in Fig. 7.12 c. The relevant phonon modes include transverse optical phonons (TO 56 meV), longitudinal optical phonons (LO 53.5 meV) and transverse acoustic phonons (TA 18.7 meV). At very low temperature a splitting (2.5 meV) of the main free exciton line in TO and LO replicas can be observed [Kol5]. 

The calculated Rayleigh mode (SJ, the lowest lying phonon branch, is in good agreement with the experimental data of Harten et al. for all three metals. Due to symmetry selection rules the shear horizontal mode just below the transverse bulk band edge can not be observed by scattering methods. The mode denoted by Sg is the anomalous acoustic phonon branch discussed above. Jayanthi et al. ascribed this anomalous soft resonance to an increased Coulomb attraction at the surface, reducing the effective ion-ion repulsion of surface atoms. The Coulomb attraction term is similar for all three metals... 

An interesting aspect of many structural phase transitions is the coupling of the primary order parameter to a secondary order parameter. In transitions of molecular crystals, the order parameter is coupled with reorientational or libration modes. In Jahn-Teller as well as ferroelastic transitions, an optical phonon or an electronic excitation is coupled with strain (acoustic phonon). In antiferrodistortive transitions, a zone-boundary phonon (primary order parameter) can induce spontaneous polarization (secondary order parameter). Magnetic resonance and vibrational spectroscopic methods provide valuable information on static as well as dynamic processes occurring during a transition (Owens et ai, 1979 Iqbal Owens, 1984 Rao, 1993). Complementary information is provided by diffraction methods. 

Nonradiative transitions can also occur between 4/ rare-earth levels. Orbit-lattice interaction may induce these between two stark-split components of the same 4fN term or between stark-split components of different 4fN terms. One assumes that the crystal field the ion sees is modulated by the vibrations of the surrounding ions. If the spacing between the two 4fN levels is less than the Debye cut-off frequency, there will be acoustical phonons capable of inducing direct transitions between the levels. The theoretical treatment of this problem is quite complex (36). 

A. Long-Wave Optical Phonons and Polaritons 1. Longitudinal Optical and Acoustical Phonons... 

So far only one degree of freedom of the vibration has been considered, namely, in the direction of the wave vector. The removal of this restriction gives transverse optical and acoustical phonons. For these, the atoms or ions move perpendicular to the direction of wave propagation. Again, there are two possibilities. When A and B atoms vibrate in phase, there is no change of dipole moment and one speaks of a transverse acoustical phonon (TA). However, for a vibration with opposite phases in the A and B atoms, the electric dipole moment changes so that we have a transverse optical phonon (TO). 

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