Transversal optical mode

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... 

To identify c-BN, the characteristic transverse optical mode (TO) at 1065 cm-1 and longitudinal optical mode (LO) at 1340 cm-1 have been described [56]. When investigating commercial c-BN, commonly only one IR-peak between 1050 and 1100 cm-1 is observed. 

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 . 

Indium nitride has twelve phonon modes at the zone centre (symmetry group Cev), three acoustic and nine optical with the acoustic branches essentially zero at k = 0. The infrared active modes are Ei(LO), Ei(TO), Ai(LO) and Ai(TO). A transverse optical mode has been identified at 478 cm 1 (59.3 meV) by reflectance [6] and 460 cm 1 (57.1 meV) by transmission [24], In both reports the location of a longitudinal optical mode is inferred from the Brout sum rule, giving respective values of 694 cm 1 (86.1 meV) and 719 cm 1 (89.2 meV). Raman scattering of single crystalline wurtzite InN reveals Ai(LO) and E22 peaks at 596 cm 1 and at 495 cm 1 respectively [25],... 

Transverse acoustical mode, /< = In/a Transverse optical mode, /c = 0 0 C ... 

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-... 

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 ... 

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. 

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,... 

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]. 

It follows that the polarization and the internal electric field show resonance behavior at the frequency of the LO mode of the film, vlo, where e(v) =0 (1.3.7°). Since the dielectric function of a cubic ionic crystal is a scalar, the polarization is parallel to the external electric field and, according to the general selection rule in Eq. (1.27), strong absorption occurs at vlq. When the external electric field is parallel to the film surface ( -polarization and E from p-polarization. Fig. 1.9), the additional polarization due to the surface charges is absent, E = Eq, and the absorption resonance position coincides with the frequency of the transverse optical mode of the cubic ionic crystal. 

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... 

---