Phonons amplitude

Isotope superlattices of nonpolar semiconductors gave an insight on how the coherent optical phonon wavepackets are created [49]. High-order coherent confined optical phonons were observed in 70Ge/74Ge isotope superlattices. Comparison with the calculated spectrum based on a planar force-constant model and a bond polarizability approach indicated that the coherent phonon amplitudes are determined solely by the degree of the atomic displacement, and that only the Raman active odd-number-order modes are observable. 

Coherent optical phonons can couple with localized excitations such as excitons and defect centers. For example, strong exciton-phonon coupling was demonstrated for lead phtalocyanine (PbPc) [79] and Cul [80] as an intense enhancement of the coherent phonon amplitude at the excitonic resonances. In alkali halides [81-83], nuclear wave-packets localized near F centers were observed as periodic modulations of the luminescence spectra. 

Fig. 3.14. Left transient reflectivity change of Te obtained with transform limited, negatively chirped, and positively chirped pulses. Right coherent phonon amplitude as a function of the pulse chirp. Adapted from [42]...

Inelastic scattering of radiation in solids is typified by the Raman effect, which involves the creation or annihilation of phonons or magnons. If a single phonon is involved, the scattering event is referred to as the first-order Raman effect in second-order Raman effect two phonons are involved. The polarizability associated with a phonon mode can be represented as a power series of the phonon amplitude, u, as follows ... 

The e term refers to a unit polarization vector, is the first-order derivative of the susceptibility tensor with respect to the phonon amplitude, also known as the Raman tensor, and rij is the phonon occupation number for the /th mode, given by ... 

Sq Si absorption band, shifting of the probe wavelength affects probe transmission just as does shifting of the crystalline absorption spectrum. In Figure 15, the major contribution is that due to coherent scattering. Quantitative determination of the phonon-induced spectral shift is impossible, but an approximate upper limit can be set. Its value ( 1 cm" for a 10" -A phonon amplitude) indicates that the initial slope of the Sj potential with respect to intermolecular separation is rather gradual and that the slope must increase as displacement increases [98]. Thus, in addition to information about excimer formation dynamics, partial (and at this point, qualitative) information about the reactive potential surface has been elucidated. 

In real space the correlation QiQj) of phonon amplitudes at the sites i and j is described by the Ornstein-Zernike function... 

We perform a numerical calculation of dipole-active phonon modes in spherical QDs made of a polar material based on the approach proposed in [6,7]. Analytical expressions for the wave vector of the longitudinal (LO) and transverse (TO) phonon components were obtained by Roca, Trallero-Giner and Cardona [1] by imposing the condition that the phonon amplitude vanishes at the QD/medium... 

We shall always be interested in longitudinal vibrations of quasi-one-dimensional systems, so the vector notation can mostly be omitted in subsequent discussions if one refers to the component of a vector parallel to the chain axis. The phonon amplitudes m, can be expressed in terms of phonon creation and annihilation operators as ... 

We will take the deviations from ideal positions to be small quantities and the corresponding phonon amplitudes to be small. To simplify the calculations, we define the vectors as... 

This expression can be interpreted as the scattering from two phonons, one of wave-vector q, the other of wave-vector q — q the corresponding phonon amplitudes and projections of the scattering wave-vector onto the phonon polarizations are also involved. 

Phonons are nomial modes of vibration of a low-temperatnre solid, where the atomic motions around the equilibrium lattice can be approximated by hannonic vibrations. The coupled atomic vibrations can be diagonalized into uncoupled nonnal modes (phonons) if a hannonic approximation is made. In the simplest analysis of the contribution of phonons to the average internal energy and heat capacity one makes two assumptions (i) the frequency of an elastic wave is independent of the strain amplitude and (ii) the velocities of all elastic waves are equal and independent of the frequency, direction of propagation and the direction of polarization. These two assumptions are used below for all the modes and leads to the famous Debye model. 

Thermal properties of overlayer atoms. Measurement of the intensity of any diffracted beam with temperature and its angular profile can be interpreted in terms of a surface-atom Debye-Waller factor and phonon scattering. Mean-square vibrational amplitudes of surfece atoms can be extracted. The measurement must be made away from the parameter space at which phase transitions occur. 

The Coulomb interaction between the re-electrons is neglected. The standard tra/is-polyacetylene parameters are ta=2.5 eV for the hopping amplitude in the undimcrizcd chain, u-4. cV/A for the electron-phonon coupling, and K= 21 eV/A2 for the spring constant [1,4, 8]. 

In order to estimate the phonon scattering strength and thus the heat conductivity, we need to know the effective scattering density of states, the transition amplitudes, and the coupling of these transitions to the phonons. 

We do not possess detailed information on the transition amplitudes however, they should be on the order of the transition frequencies themselves, just as is the case for those two-level systems that are primarily responsible for the phonon... 

We now calculate the density of the phonon scattering states. Since we have effectively isolated the transition amplitude issue, the fact of equally strong coupling of all transitions to the lattice means that the scattering density should directly follow from the partition function of a domain via the... 

In the molecular approximation used in (14) only the L = 3W — 6 (W is the number of atoms) discrete intramolecular vibrations of the molecular complex in vacuo are considered. In general these vibrations correspond to the L highest optical branches of the phonon spectrum. The intermolecular vibrations, which correspond to the three acoustical branches and to the three lowest optical branches are disregarded, i.e., the center of mass and - in case of small amplitudes - the inertial tensor of the complex are assumed to be fixed in space... 

Fig. 9. Incidence energy dependence of the vibrational state population distribution resulting when NO(u = 12) is scattered from LiF(OOl) at a surface temperature of (a) 480 K, and (b) 290 K. Relaxation of large amplitude vibrational motion to phonons is weak compared to what is possible on metals. Increased relaxation at the lowest incidence energies and surface temperatures are indicators of a trapping/desorption mechanism for vibrational energy transfer. Angular and rotational population distributions support this conclusion. Estimations of the residence times suggest that coupling to phonons is significant when residence times are only as long as ps. (See Ref. 58.)...

The above picture points to the very interesting possibility of selectively inducing or enhancing the polymerisation process, at a temperature where this is unlikely, by resonantly driving with an intense laser beam in the infrared the vibrational modes and wc that are involved in the polymerisation. As a consequence of their anharmonicity (45) these modes, when driven near resonance by an electromagnetic field, beyond a certain critical value of the later, can reach amplitudes comparable to the critical ones required for the polymerisation to be initiated or proceed the anharmonicity in the presence of the intense laser beam acts as a defect and localizes the phonons creating thus a critical distorsion. 

The key requirements for ISRS excitation are the existence of Raman active phonons in the crystal, and the pulse duration shorter than the phonon period loq1 [19]. The resulting nuclear oscillation follows a sine function of time (i.e., minimum amplitude at t=0), as shown in Fig. 2.2e. ISRS occurs both under nonresonant and resonant excitations. As the Raman scattering cross section is enhanced under resonant excitation, so is the amplitude of the ISRS-generated coherent phonons. 

The coherent oscillation of the A g mode was a cosine function of time (Fig. 2.8) [21,24-26]. Recent X-ray measurements demonstrated a clear shift in the equilibrium position at photoexcitation [35], as we will see in the next chapter. These results confirmed the displacive generation of coherent A g phonons, as discussed in Sect. 2.2.2. In contrast, the coherent oscillation of the Eg mode was a sine function of time, and its amplitude exhibited a cos 2ip dependence on the pump polarization angle ip. Both features indicated the ISRS generation of the coherent Eg phonons [21,25],... 

Fig. 2.8. Left oscillatory part of the reflectivity change of Bi (0001) surface at 8K (open circles). Fit to the double damped harmonic function (solid curve) shows that the Aig and Eg components (broken and dotted curves) are a sine and a cosine functions of time, respectively. Right pump polarization dependence of the amplitudes of coherent Aig and Eg phonons of Bi (0001). Adapted from [25]...

Under even more intense photoexcitation ( 10mJ/cm2), the coherent A g and Eg phonons of Bi and Sb exhibit a collapse-revival in their amplitudes (Fig. 2.10) [42,43], This phenomenon has a clear threshold in the pump density, which is common for the two phonon modes but depends on temperature and the crystal (Bi or Sb). At first glance, the amplitude collapse-revival appears to be analogous to the fractional revival in nuclear wavepackets in molecules [44,45]. However, the pump power dependence may be an indication of a polarization, not quantum, beating between different spatial components of the coherent response within the laser spot [46],... 

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