Dispersion of Surface Phonons

In recent years there is a growing interest in the study of vibrational properties of both clean and adsorbate covered surfaces of metals. For several years two complementary experimental methods have been used to measure the dispersion relations of surface phonons on different crystal faces. These are the scattering of thermal helium beams" and the high-resolution electron-energy-loss-spectroscopy. ... 

The dispersion curves of surface phonons of short wavelength are calculated by lattice dynamical methods. First, the equations of motion of the lattice atoms are set up in terms of the potential energy of the lattice. We assume that thejxitential energy (p can be expressed as a function of the atomic positions 5( I y in the semi-infinite crystal. The location of the nth atom can be... 

Kinematics of surface phonon He spectroscopy. The thick lines correspond to scan curves of a 18 meV He beam. The thin lines display the Rayleigh phonon dispersion curve of Pt(lll) along the f M azimuth. 

The first successful measurement of surface phonons by means of inelastic He scattering was performed in Gottingen in 1980. By using a highly monochromatic He beam (Av/t 1%) Brusdeylins et al. were able to measure the dispersion of the Rayleigh wave of the LiF(001) crystal surfae. In earlier attempts the inelastic events could not be resolved satisfactorily due to the low beam monochromaticity. In Fig. 10a we show a typical TOF spectrum. 

When the frequency of the surface biphonon lies within the band of the surface polariton, Fermi resonance occurs and the dispersion curve of the po-lariton is subject to a number of essential changes (gaps appear, etc. (86)). Consequently, experimental research of surface polariton dispersion under these conditions could yield, like similar investigations of bulk polaritons, a great deal of interesting information, not only about the surface biphonons themselves, but about the density of states of surface phonons and the magnitude of their anharmonicity constants as well. 

Theoretical calculations of surface phonon dispersion have been carried out in two ways. One method is to use a Green s function technique which treats the surface as a perturbation of the bulk periodicity in the z-direction [34, 35]. The other is a slab dynamics calculation in which the crystal is represented by a slab of typically 15-30 layers thick, and periodic boundary conditions are employed to treat interactions outside the unit cell as the equations of motion for each atom are solved [28, 33, 35, 37]. In the latter both the bulk and the surface modes are found and the surface localized modes are identified by the decay of the vibrational amplitudes into the bulk in the former the surface modes can be obtained directly. When the frequency of a surface mode lies within a bulk band of the same symmetry, then hybridization can take place. In this event the mode can no longer be regarded as strictly surface localized and is referred to as a surface resonance [24]. Figure 8, adapted from Benedek and Toennies [24], shows how the bulk and surface modes develop as more and more layers are taken in a slab dynamics calculation. 

Figure 8. The evolution of surface phonon dispersion curves for a monatomic fee (111) surface in slab dynamics calculations as a function of the number of layers in the slab. The surface localized modes, marked by arrows in the last panel (iV = 15), lie below the bulk bands Mross the entire surface Brillouin zone and appear between the bands in the small gap near K in the TK region and in the larger gap in the MK region. (Reproduced from Fig. 1 of Ref. 24, with permission of Elsevier Science Publishers.)...

To date, the phonon confinement effects have not been explicitly detected for CVD diamond films and results remained unsatisfactory in the case of DND. To improve the agreement between the predictions of the model and experimental Raman spectra of DND, effects such as crystal size distribution, lattice defects, and the energy dispersion of the phonon modes were taken into consideration and incorporated into the PCM. This work has shown that phonon wave vectors from small vibration domains lead to a broad shoulder peak at 1250 cm", that is often observed in the Raman spectrum of DND. Although the agreement between experimentally obtained and calculated Raman spectra has been significantly improved, some limitations remain, as was pointed ont in Ref. 98. The limitations imposed by the small ND size on the applicability of the PCM arise from the assumption that nanocrystals of 3-20 nm in size, showing extensive surface reconstruction and lattice defects, are assumed to have the phonon density of states of bulk diamond. 

Problem 2.3. The dispersion curves of surface phonons are shown in Fig. 2.27 by thick solid and dashed lines. Those which are indicated by dashed lines have the character of surface resonances. The branch of acoustical surface phonons (Si) originates at the T-point (ky = 0) where its energy is equal to zero. The dispersion curves S2 and S3 correspond to optical surface phonons. One cannot classify the curve S4 based on this figure. 

WIT Witte, G., Toennies, J. P., Well, C. Comparison of surface phonon dispersion for the clean and hydrogen covered Rh(l 11) surface Surf. Sci. 323 (1995) 228. 

Fig. 63. C(IOO) (2x1) H. Calculated surface phonon dispersion at 1 ML coverage [96San]. Shaded area projection of bulk phonon bands solid curves dispersion of surface modes and resonances.

Dipole scattering is the dominant mechanism in this scattering geometry. However, it is restricted to a very small parallel momentiun transfer and, therefore, inadequate for studies of surface phonon dispersions [Pg.318]

C. (1985) Helium time-of-flight spectroscopy of surface-phonon dispersion curves of the noble metals. Faraday Discuss. Chem. Soc., 80, 137. 

Doak, R. B. (1981) PhD thesis. Measurement of surface phonon dispersion relations for LiF, NaF, and KQ through energy-analysed inelastic scattering of... 

Fig. 5. Schematics of the formation of the surface phonon dispersion of a (111) f.c.c. crystal.

Recently, we hav measured the surface phonon dispersion of Cu(l 10) along the rx, rF, and F5 azimuth of the surface Brillouin zone (Fig. 13) and analyzed the data with a lattice dynamical slab calculation. As an example we will discuss here the results along the TX-direction, i.e. the direction along the close-packed Cu atom rows. 

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