Rayleigh mode

The first mention of surface phonons is due to Lord Rayleigh (1885), who predicted the existence of a surface acoustic mode with a sound velocity lower than in the bulk. He proved this result, using elasticity theory, by representing the semi-infinite sofid by a continuous and isotropic medium (Landau and Lifehitz, 1967). Considering an infinitesimal volume element, he wrote a Fourier component of its displacement u q, co), in the following form  

The u vector has transverse (V iit = 0) and longitudinal (Vxm/ = 0) components, associated with sound velocities Ct and q, respectively. Boundary conditions require that all the volume elements are in equilibrium, especially at the surface (z = 0)  

The tensor of stresses (Xy is related to the tensor of deformations, to lowest order, by Hooke s law. The relevant elastic constants are the Young modulus Y and the Poisson s coefficient P. 

A surface mode, with a polarization parallel to the saggital plane xz, obeys the boundary conditions. The longitudinal and transverse components of the u vector fulfil the following relations  

It depends upon the ratio ct/c = (1 — 2P)/2(1 — P). Only one root of (4.1.6) fulfils the condition required for Kt and Kf to be real. Since cfcf is larger than zero and smaller than 1/ (0 P 1/2), belongs to the interval [0.874, 0.955]. The sound velocity of the Rayleigh mode is thus lower than the lowest sound velocity in the bulk. Close to the zone centre, the mode is located below the bulk acoustic modes (Si mode in Fig. 4.1). 

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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],... 

These tests demonstrate the sensitivity of the higher-mode data to the upper-mantle structure. Figure 4a shows the displacement amplitude v. depth for the fundamental and first eight higher Rayleigh modes at 15 s period for the southern African model shown at the left in the figure. The fundamental mode at this period is sensitive to the velocity structure in the top c. 50 km of the model, whereas the higher modes are... 

Cara, M. 1979. Lateral variation of S velocity in the upper mantle from higher Rayleigh modes. Geophysical Journal of the Royal Astronomical Society, 57, 646-670. 

The appearances of these devices are very similar to that of Rayleigh mode devices, but a thin solid film or grating is added to prevent wave diffraction into the bulk. 

The liquid jet cannot be atomized at low gas velocities and breaks into droplets mainly through capillary action. This is the Rayleigh mode. At higher velocities, the... 

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. 

Fig. 4.1. Phonon dispersion curves in MgO(lOO) (according to Chen et ai, 1977). Hatched zones are the projection of the bulk modes. Surface modes S are indexed by n (1 < n < 7) the Rayleigh mode is Si the Fuchs and Kliewer modes have a frequency close to 12x10 rad s S3 is an example of a microscopic mode.

The macroscopic modes have an attenuation length which varies as the inverse of the normal component of their wave vector. They penetrate deeply inside the crystal. On this length scale, the precise atomic structure is unimportant and the elasticity theory of continuous media or dielectric theories may be used. Depending upon whether they are acoustic or optic, one distinguishes the Rayleigh mode (1885) and the Fuchs and Kliewer modes (1965). 

As in the case of metals and semi-conductors, there exist specific surface excitations in insulating oxides. Three types of surface phonon modes may be distinguished the Rayleigh mode, the Fuchs and Kliewer modes and the microscopic surface modes. The first two modes have a long penetration length into the crystal. They are located below the bulk acoustic branches and in the optical modes, respectively. The latter are generally found in the gap of the bulk phonon spectrum. 

Figure 6.25 Linewidth broadening of the Cu(lll) surface hole state as a function of binding energy, Pe e + Pe ph (solid line), Pe-ph (dotted line), the Rayleigh mode contribution to Pe-ph (dashed line), and photoemission data (diamonds). Adapted from Ref. [67].

Figure 9.47 Surface phonon dispersion on Si(l 11 )-(2x 1) obtained by He-atom energy-loss spectroscopy. Two modes are visible the Rayleigh mode and a second flat mode, which arises from backfolding of the Rayleigh mode because of the twofold periodicity of the reconstruction. (Figure adapted from Ref [85].)...

The surface-localized vibrations are low-frequency acoustical modes (Rayleigh modes) which are easily excited by collisions with atoms and molecules. The bulk modes are often approximated by the Debye model where the distribution is given as [98]... 

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