Vibrational mode transverse optical

Figure 8.11 (a) Dispersion curve for CuCl(s) along [110] of the cubic unit cell, (b) Density of vibrational modes [3], Here L, T, A and O denote longitudinal, transverse, acoustic and optic. Reproduced by permission of B. Hennion and The Institute of Physics. 

In three dimensions, transverse and longitudinal optic and acoustic modes result. The dispersion curve for CuCl along [100] of the cubic unit cell [3] is shown in Figure 8.11(a) as an example. The number of discrete modes with frequencies in a defined interval can be displayed as a function of the frequency. This gives what is termed the density of vibrational modes or the vibrational density of states (DoS). The vibrational DoS of CuCl is given in Figure 8.11(b). 

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. 

We did not differentiate between the various modes of vibration (longitudinal, transversal, acoustical, optical) for the sake of simplicity. The vibrational states in a crystal are called phonons. Figure 5-2 illustrates the collective, correlated transversal vibrational motion of a linear elastic chain of particles. 

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). 

For the particular case of longitudinal optical modes, we found in Eq. (9-27) the electrostatic electron-phonon interaction, which turns out to be the dominant interaction with these modes in polar crystals. Interaction with transverse optical modes is much weaker. There is also an electrostatic interaction with acoustic modes -both longitudinal and transverse which may be calculated in terms of the polarization generated through the piezoelectric effect. (The piezoelectric electron phonon interaction was first treated by Meijer and Polder, 1953, and subsequently, it was treated more completely by Harrison, 1956). Clearly this interaction potential is proportional to the strain that is due to the vibration, and it also contains a factor of l/k obtained by using the Poisson equation to go from polarizations to potentials. The piezoelectric contribution to the coupling tends to be dominated by other contributions to the electron -phonon interaction in semiconductors at ordinary temperatures but, as we shall see, these other contribu-... 

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 ... 

In the solid state, the polar phonons (those that are IR active) split into two components, the transverse optical mode (TO) and the longitudinal optical mode (LO). This TO/LO splitting occurs because the electric field associated with the transverse wave = 0 while that associated with the longitudinal wave is 0. The coupling of these modes with the electric fields associated with the vibration gives rise to Vlo > Vto- This factor is relevant in relation to the shape and interpretation of the IR spectra of solid materials and will be further considered below. 

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 the case of strongly polar or concentrated species (which is typically the case for an oxide), the vibrational contribution to e(co) may become larger than Boo in the region of the resonance. The shapes of the functions -lm[g(< )] and lm[l/g(< )] then become different, the former exhibiting its maximum for (o=(Oo, whereas for the latter the maximum turns out to be shifted to a o=(coo+Ne lBoBcoR). If one considers the phonon modes in the infinite 3D material, the two modes o coq appear as the zero-wavevector limit of the transverse-optical (TO) and longitudinal-optical (LO) phonon branches, and for that reason are generally termed TO mode and LO mode [94]. 

Several theoretical and experimental studies assess the vibrational properties of the high-pressure phases of silicon. A group-theoretical analysis of lattice vibrations in the -tin structure has been made by Chen [98]. In the vicinity of the F point, the optical modes consist of one longitudinal optical (LO) branch and at higher frequencies of a doubly degenerate transverse optical (TO) branch, both of which are Raman active. Zone-center phonon frequencies of Si-11 have been calculated as a function of pressure using the ab initio pseudopotential method... 

Boron nitride films containing both the hBN and cBN phase cause strong infrared absorption bands at about 1050 cm owing to the presence of cBN (transversal optical mode) [42] as well as at about 800 and 1400 cm corresponding to B—N bending (out-of-plane vibration) and stretching (in-plane vibrations) modes, respectively [43]. 

A qualitative, physical description of the longitudinal and transverse optical phonons is illustrated in Fig. 1.6. Kittel [22] phenomenologically explains the fact that ft)Lo > oio as follows. The local electric field induces polarization of the surrounding atoms in the opposite direction to that of the longitudinal mode but in the same direction as the transverse mode. This polarization causes an increasing resistance to the longitudinal vibration relative to the transverse one (see also Ref. [33]). 

The skeletal IR spectra indeed show a well-defined maximum at 430-420 cm that can be confidently assigned to vto (the only IR-active transverse optical mode) of the Ni-Mg oxide rock salt-t3q)e solid solution. The absence of bands in the far infrared (i.e. below 400 cm ) points to the absence of a spinel phase. However, the additional features, found in the region 900-600 cm , are typically associated with M-0 vibrations of tetrahedrally coordinated cations. Indeed, two absorptions are found in the spectra, near 800 cm and 650 cm . In the case of the Mg-free Ni-Al compound, a similar band was observed at 850 cm This suggests that the band at 800-850 cm is associated with M-0 stretchings of tetrahedrally coordinated... 

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