Transverse-optic frequency

For frequencies low compared with the transverse optical frequency dielectric function (9.21) approaches the limiting value 0v ... 

We could have used particular frequencies instead of the elastic constants to determine the force constants in the model that alternative is of some interest, We have used, for reasons to be discussed, the measured transverse optical frequency at /( = 0 and the transverse acoustical mode at k — Inja to obtain alternate values of C,) and C, which are listed in Table 9-1. The differences from the values given... 

When metals have Raman active phonons, optical pump-probe techniques can be applied to study their coherent dynamics. Hase and coworkers observed a periodic oscillation in the reflectivity of Zn and Cd due to the coherent E2g phonons (Fig. 2.17) [56]. The amplitude of the coherent phonons of Zn decreased with raising temperature, in accordance with the photo-induced quasi-particle density n.p, which is proportional to the difference in the electronic temperature before and after the photoexcitation (Fig. 2.17). The result indicated the resonant nature of the ISRS generation of coherent phonons. Under intense (mJ/cm2) photoexcitation, the coherent Eg phonons of Zn exhibited a transient frequency shift similar to that of Bi (Fig. 2.9), which can be understood as the Fano interference [57], A transient frequency shift was aslo observed for the coherent transverse optical (TO) phonon in polycrystalline Zr film, in spite of much weaker photoexcitation [58],... 

At high temperatures above Tb 617 K PMN behaves Hke all other simple perovskites. The dynamics of the system is determined by the soft transverse optical (TO) phonon which exhibits a normal dispersion and is imderdamped at all wave vectors. Below Tb, in addition to the soft mode—which becomes overdamped—a new dielectric dispersion mechanism appears at lower frequencies which can be described by a correlation time distribution function /(t). 

I he notation 0e indicates that this is the dielectric function at frequencies low i ompared with electronic excitation frequencies. We have also replaced co0 with l (, the frequency of the transverse optical mode in an ionic crystal microscopic theory shows that only this type of traveling wave will be readily excited bv a photon. Note that co2 in (9.20) corresponds to 01 e2/me0 for the lattice vibrations (ionic oscillators) rather than for the electrons. The mass of an electron is some thousands of times less than that of an ion thus, the plasma liequency for lattice vibrations is correspondingly reduced compared with that lor electrons. 

At infrared wavelengths extinction by the MgO particles of Fig. 11.2, including those with radius 1 jam, which can be made by grinding, is dominated by absorption. This is why the KBr pellet technique is commonly used for infrared absorption spectroscopy of powders. A small amount of the sample dispersed in KBr powder is pressed into a pellet, the transmission spectrum of which is readily obtained. Because extinction is dominated by absorption, this transmission spectrum should follow the undulations of the intrinsic absorption spectrum—but not always. Comparison of Figs. 10.1 and 11.2 reveals an interesting discrepancy calculated peak extinction occurs at 0.075 eV, whereas absorption in bulk MgO peaks at the transverse optic mode frequency, which is about 0.05 eV. This is a large discrepancy in light of the precision of modern infrared spectroscopy and could cause serious error if the extinction peak were assumed to lie at the position of a bulk absorption band. This is the first instance we have encountered where the properties of small particles deviate appreciably from those of the bulk solid. It is the result of surface mode excitation, which is such a dominant effect in small particles of some solids that we have devoted Chapter 12 to its fuller discussion. 

Comparison of measurements for particles dispersed on and in KBr is quite revealing. The extinction curve for particles on a KBr substrate shows a peak at approximately 400 cm-1, the transverse optical mode frequency for bulk MgO. This feature has been observed a number of times and it is discussed in some of the references already cited. Its explanation now appears to be the tendency of MgO cubes to link together into chains, which more closely... 

From the lattice dynamics viewpoint a transition to the ferroelectric state is seen as a limiting case of a transverse optical mode, the frequency of which is temperature dependent. If, as the temperature falls, the force constant controlling a transverse optical mode decreases, a temperature may be reached when the frequency of the mode approaches zero. The transition to the ferroelectric state occurs at the temperature at which the frequency is zero. Such a vibrational mode is referred to as a soft mode . 

Hereby, the branches with E - and / -symmetry are twofold degenerated. Both A - and / d-modes are polar, and split into transverse optical (TO) and longitudinal optical (LO) phonons with different frequencies wto and wlo, respectively, because of the macroscopic electric fields associated with the LO phonons. The short-range interatomic forces cause anisotropy, and A - and / d-modcs possess, therefore, different frequencies. The electrostatic forces dominate the anisotropy in the short-range forces in ZnO, such that the TO-LO splitting is larger than the A -E splitting. For the lattice vibrations with Ai- and F -symmetry, the atoms move parallel and perpendicular to the c-axis, respectively (Fig. 3.2). 

We turn now to a matter of more direct physical relevance, the local electric dipole moment induced when a particular ion is displaced by some vector u. The transverse charge is defined to be the magnitude of that dipole moment divided by the displacement (and by the magnitude of the electronic charge). We saw in Eq. (9-22) that is directly related to an observable splitting between the longitudinal and transverse optical-mode frequencies, so that this is a quantity that can be compared with experiment. 

From the frequency of the transverse optical mode in a simple AB lattice with k = Q a force constant can be derived which is a measure of the restoring forces experienced by the atoms as they are distorted from the equilibrium position. This force constant, Fflattice), is a linear combination of internal force constants, since in a lattice a linear combination of equilibrium distances and angles yields a coordinate of this vibration. Based on this assumption, the GF method (Wilson et al, 1955) can be applied. For diamond (or zinc blende), the following relation is obtained ... 

Wurtzite ZnO structure with four atoms in the unit cell has a total of 12 phonon modes (one longitudinal acoustic (LA), two transverse acoustic (TA), three longitudinal optical (LO), and six transverse optical (TO) branches). The optical phonons at the r point of the Brillouin zone in their irreducible representation belong to Ai and Ei branches that are both Raman and infrared active, the two nonpolar 2 branches are only Raman active, and the Bi branches are inactive (silent modes). Furthermore, the Ai and Ei modes are each spht into LO and TO components with different frequencies. For the Ai and Ei mode lattice vibrations, the atoms move parallel and perpendicular to the c-axis, respectively. On the other hand, 2 modes are due to the vibration of only the Zn sublattice ( 2-low) or O sublattice ( 2-high). The expected Raman peaks for bulk ZnO are at 101 cm ( 2-low), 380 cm (Ai-TO), 407 cm ( i-TO), 437 cm ( 2-high), and 583 cm ( j-LO). 

Of central importance for understanding the fundamental properties of ferroelec-trics is dynamics of the crystal lattice, which is closely related to the phenomenon of ferroelectricity [1]. The soft-mode theory of displacive ferroelectrics [65] has established the relationship between the polar optical vibrational modes and the spontaneous polarization. The lowest-frequency transverse optical phonon, called the soft mode, involves the same atomic displacements as those responsible for the appearance of spontaneous polarization, and the soft mode instability at Curie temperature causes the ferroelectric phase transition. The soft-mode behavior is also related to such properties of ferroelectric materials as high dielectric constant, large piezoelectric coefficients, and dielectric nonlinearity, which are extremely important for technological applications. The Lyddane-Sachs-Teller (LST) relation connects the macroscopic dielectric constants of a material with its microscopic properties - optical phonon frequencies ... 

Conventional infrared spectra of powdery materials are very often used for studying solid hydrates in terms of sample characterization (fingerprints), phase transitions, and both structural and bonding features. For the latter objects mostly deuteration experiments are included. However, it must be born in mind that the band frequencies observed (except those of isotopically dilute samples (see Sect. 2.6)) are those of surface modes rather than due to bulk vibrations, i.e., the transverse optical phonon modes, and, hence, not favorably appropriate for molecular and lattice dynamic calculations. 

The weaker the interaction between the metal and the hydrogen the more important are the inter-hydrogen forces in determining their dynamics. This leads to dispersion, a good example of which is that found in PdH, measured as its deuteride PdDo.es [58]. This dispersion is shown in Fig. 6.21 The low frequency acoustic modes, involving the Pd vibrations, have little hydrogen displacement and show only weakly in the INS spectrum of powdered PdH however, the optic modes appear strongly, see Fig. 6.22 The relatively undispersed transverse optic modes,... 

In compound crystals, the ujn values considered are wlo, the frequency of the longitudinal optical phonons on the high-energy (h-e) side, and wto, the frequency of the transverse optical phonons, on the low-energy side. The dielectric constant at frequencies above c lo is denoted as while that below wto is denoted as s (the index s represents static, despite the fact that s shows a small dispersion between the value just below ujto and the one at radiofrequencies1). It can be seen from expressions (3.14) and (3.15) that above ujo, the ionic contribution decreases such that qo is smaller than s. Typical values are given in Table 3.1. 

Now let us estimate (see Fig 25) the so-called transverse optic-longitudinal optic splitting [8,50] characteristic for ice at v 230 cm-1. Namely, the loss curve e"(v) is shifted on the frequency scale with respect to the energy loss function ... 

Table 8.3 The lattice constant (a), bulk modulus (B) and frequency of the transverse optic phonon at the zone centre (TO-T) for MgO from PW-LDA and LCAO-HF calculations compared to observations extrapolated to the...

EqCOS Sit and Qj the optical frequency of the molecule excited. T, and Tj are the longitudinal and transverse optical relaxation constants also... 

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