Surface Rayleigh phonon

Fig. 42. Comparison of experimental and theoretical (solid lines) surface phonon dispersion data (frequency vs. parallel component of wave vector) for an ordered c(2x2)-0 Ni(lOO) surface. The calculations are based on the position of 0-atoms at 0.9 A above the surface and the inclusion of lateral dipole-dipole interactions. For comparison, the experimental data for the clean surface Rayleigh phonon (solid points) are also included [83Leh, 84Rah],...

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. 

Fig. 10. (a) He time-of-flight spectrum taken from a LiF(001) surface along the < 100) azimuth at an incident angle Si = 64.2°. The primary beam energy was 19.2 meV. (After Ref 25.). (b) Measured Rayleigh phonon dispersion curve of LiFfOOl) < 100), including a scan curve (dashed) for the kinematical conditions in (a). (After Ref. 25.)... 

Fig. 22. Phonon dispersion curves for the (26x2) HOC monolayer of Xe on Cu(110) along the FX (a) and FY (b) directions. Notations used R Cu(110) surface Rayleigh wave 1, Ij, -L3 single and multiple excitation of the perpendicular Xe vibration mode H, Hi, H2 branches of the hybridized 1 and R modes. L, L longitudinal Xe modes, G gap mode. Dashed lines are the result of a lattice dynamics calculation [97R].

Fig. 23. Phonon dispersion curves for a Xe monolayer on Cu(lOO) along the [100] substrate direction (after [97G]). Notations used R Cu(lOO) surface Rayleigh wave and SH perpendicular and shear horizontal Xe adlayer modes, respectively. The SH mode was first explained by assigning it to the longitudinal branch [97G,99S2], which required a large softening of the inplane Xe-Xe force constant. Later it was suggested [97B1,00B] that it should be assigned to the SH-polarized mode.

Tamm [3] was first to show that the special states of electrons exist near crystal surface (so-called surface states), which have the discrete energy spectrum and their wave functions decay exponentially on both sides of the surface. Similarly, the vibrations of crystal surface atoms can be considered, which also decay on both sides of the surface. In the long wave limit, one of such surface phonon branches transforms into well-known surface Rayleigh waves, while the others yield special optic branches [4, 5]. 

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. 

The calculated Rayleigh mode (SJ, the lowest lying phonon branch, is in good agreement with the experimental data of Harten et al. for all three metals. Due to symmetry selection rules the shear horizontal mode just below the transverse bulk band edge can not be observed by scattering methods. The mode denoted by Sg is the anomalous acoustic phonon branch discussed above. Jayanthi et al. ascribed this anomalous soft resonance to an increased Coulomb attraction at the surface, reducing the effective ion-ion repulsion of surface atoms. The Coulomb attraction term is similar for all three metals... 

The surface Fuchs-Kliewer modes, like the Rayleigh modes, should be regarded as macroscopic vibrations, and may be predicted from the bulk elastic or dielectric properties of the solid with the imposition of a surface boundary condition. Their projection deep into the bulk makes them insensitive to changes in local surface structure, or the adsorption of molecules at the surface. True localised surface modes are those which depend on details of the lattice dynamics of near surface ions which may be modified by surface reconstruction, relaxation or adsorbate bonding at the surface. Relatively little has been reported on the measurement of such phonon modes, although they have been the subject of lattice dynamical calculations [61-67],... 

The harmonic-oscillator and elastic-continuum models can be used to explain the presence of surface phonons (Rayleigh waves and localized surface modes of vibration) and the larger mean-square displacement of surface atoms compared to that of atoms in the bulk. 

Figure 16. Surface phonon dispersion curves for LiF(OOl). The calculated bulk bands are indicated by the vertical-striped regions. The surface localized modes are shown by heavy solid lines, whereas the resonances lying within bulk bands are given by thinner solid lines. The mode label S refers to the Rayleigh wave, to the longitudinal resonance, Sg to the crossing resonance, and S2, S3, and S4 to optical modes. (Reproduced from Fig. 2 of Ref. 58, with permission of Elsevier Science Publishers.)...

Figure 18. Surface phonon dispersion for RbCl(OOl). The shaded regions correspond to the surface projected density of states, with the darker shades representing higher state densities. The Rayleigh wave, crossing resonance, and optical mode are indicated by RW, CR, and 2, respectively. (Reproduced from Ref. 119.)...

Figure 33. Surface phonon dispersion for W(OOl) in the FM portion of the SBZ showing the measured Rayleigh wave (R) and longitudinal (L) modes. The data in the upper panel were obtained at 1200 K, while in the lower panel the data shown by open circles were obtained at 500 K and those represented by closed circles were obtained at 300 K. The edges of the transverse acoustic (TA) and longitudinal acoustic (LA) bulk bands are given by the hatched lines. The vertical lines in the lower panel denote the widths in the energy transfer distributions of these points. (Reproduced from Figs. 10 and 13 of Ref. 110, with permission.)...

Figure 34. Surface phonon dispersion for 2H-TaSe2. The HAS data are shown as solid circles except for weak points which appear in the TOP spectra as hybridized longitudinal modes that are shown as crosses. All the data were obtained at 60 K, well into the low-temperature phase. The calculated striped and shaded regions, corresponding to transverse and longitudinal polarizations respectively, are the slab-adapted bulk phonon bands, while the solid line is a calculation for the Rayleigh wave based on the Dispersive Linear Chain Model (shown schematically in Fig. 35). The open circles at g = 0 are from Raman scattering experiments. (This figure has been corrected from Fig. 23 in Ref. 54.)...

Fig. 41. Schematic plot of the dispersion of a Rayleigh surface phonon (sohd line) and of a bulk phonon band (hatched area). Atoms or molecules adsorbed on the surface can change the surface phonon dispersion, visible as an increase or decrease of the corresponding frequencies relative to those of the clean surface. These shifts are equivalent to a...

Phonon bands occur in the SBZ, similarly to the surface states discussed in Sect. 5.2.3. When the frequency of a surface mode corresponds to a gap in the bulk spectrum, the mode is localized at the surface and is called a surface phonon. If degeneracy with bulk modes exists, one speaks of surface resonances. Surface phonon modes are labeled Sj ( / = 1, 2, 3,...), and surface resonances by Rj when strong mixing with bulk modes is present, the phonon is labeled MSj. The lowest mode that is desired from the (bulk) acoustic band is often called the Rayleigh mode, after Lord Rayleigh, who first predicted (in 1887) the existence of surface modes at lower frequencies than in the bulk. 

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