Surface mode

For simplicity s sake, consider p-polarized radiation incident from a nonabsorbing medium with i( ) = const 1 onto a second medium with 62(0)). Assume that the dielectric function of the second medium is real and expressed by Eq. (1.40). The coordinate axes are defined with respect to the interface as shown in Fig. 1.9. 

The wave equations (1.5) for the electric and magnetic fields at the interface have the forms 

From the boundary conditions (1.60), it follows that the tangential components of E and H must be equal at the interface, where z = 0, 

From Eqs. (3.5) and (3.6), it follows that a nontrivial solution only exists if 

On the basis of Eqs. (3.7) and (3.9), the dispersion relation for electromagnetic waves at the interface of two isotropic nonconducting semi-infinite media can be written as 

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Under Httle or no illumination,/ must be minimized for optimum performance. The factor B is 1.0 for pure diffusion current and approaches 2.0 as depletion and surface-mode currents become important. Generally, high crystal quality for long minority carrier lifetime and low surface-state density reduce the dark current density which is the sum of the diffusion, depletion, tunneling, and surface currents. The ZM product is typically measured at zero bias and is expressed as RM. The ideal photodiode noise current can be expressed as follows ... 

HRELS = high-resolution, electron-energy-loss spectroscopy. " Surf. Sci. (in press). Ref. (123). Ref (101). Softened pCHj surface-mode. Weak band observed around 1500 cm could be a surface-dipole-forbidden, Pfc mode. Hidden under intense SCHj mode of free C2H4 in the matrix. " One of these bands belongs to Ni2(C2H4)2. 

The eigenfrequency spectrum of the surface modes of a hollow sphere with gas inside is well known (e.g., see Ref. [109] as well as our Appendix A). If we pretend for a moment that the surface tension coefficient a is curvature independent, the possible values of the eigenfrequency oo are found by solving the following equation ... 

Condition (2) is also quite common. For instance, in crystals it results in a reduced sound velocity, v q) when q approaches a boundary of the Brillouin zone [93,96], a direct result of the periodicity of a crystal lattice. In addition, interaction between modes can lead to creation of soft mode with qi O and corresponding structural transitions [97,98]. The importance of nonlocality at fluid interfaces and the corresponding softening of surface modes has been demonstrated recently, both theoretically [99] and experimentally [100]. 

Here we briefly present the relevant theory of capillary waves. The thermally excited displacement (r, t) of the free surface of a liquid from the equilibrium position normal to the surface can be Fourier-decomposed into a complete set of surface modes as... 

Boardman, A.D., editor, 1982, Electromagnetic Surface Modes, John Wiley Sons, Chichester. 

Fig. 8. Lck SH2 domain-peptide complex (Ac-cmF-Glu-Glu-Ile-OH, 12) revealing the twopronged plug engaging a two-holed socket 1 binding mode, reminiscent of the majority of SH2 domains (Protein Databank entry code 1BHF.PDB [118]). The protein is depicted in a Connolly surface mode, the ligand is given in a ball-and-stick representation. The cmF residue is deeply buried in its binding pocket (left)...

The highest phonon frequency measured so far has been reported for NaF(lOO). By using primary beam energies of 90meV, Brusdeylins et al. have detected optical surface modes with frequencies of 40meV. The intensities, however, were rather low and the multiphonon background dominated the energy loss spectra. 

The thermal properties of AU55 are treated in Sect. 3, using especially the results of MES measurements [24,25,42]. These are discussed in connection with the concept of bulk versus surface modes in small particles. An explanation of the temperature dependence of the MES [42] absorption intensities and the Cv results [25] on the basis of a model using the site coordination and the center-of-mass motion are briefly reviewed. The consequences of the Mossbauer results for surface Debye temperatures and for the melting temperature of small gold particles are also discussed. 

The experimental observation that one has different Debye temperatures for the three distinct surface sites of the AU55 cluster makes the use of a continuum-model picture for discussing the thermal behavior questionable. Indeed, for such small particle sizes, where the surface structure is so manifest, the use of the concept of surface modes becomes dubious, and is certainly inadequate to explain the observed temperature dependence of the f-factors. None the less, it has proven possible to describe the low temperature specific heat of AU55 quite well using such a continuum-model, when the center-of-mass motion is taken into account [99],... 

Vp(fO is peaked at the surface. Many collective oscillations manifest themselves as predominantly surface modes. As a result, already one separable term generating by (74) usually delivers a quite good description of collective excitations like plasmons in atomic clusters and giant resonances in atomic nuclei. The detailed distributions depends on a subtle interplay of surface and volume vibrations. This can be resolved by taking into account the nuclear interior. For this aim, the radial parts with larger powers and spherical Bessel functions can be used, much similar as in the local RPA [24]. This results in the shift of the maxima of the operators (If), (12) and (65) to the interior. Exploring different conceivable combinations, one may found a most efficient set of the initial operators. 

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. 

The extinction curves for magnesium oxide particles (Fig. 11.2) and aluminum particles (Fig. 11.4) show the dominance of surface modes. The strong extinction by MgO particles near 0.07 eV( - 17 ju.m) is a surface mode associated with lattice vibrations. Even more striking is the extinction feature in aluminum that dominates the ultraviolet region near 8 eV no corresponding feature exists in the bulk solid. Magnesium oxide and aluminum particles will be treated in more detail, both theoretically and experimentally, in this chapter. 

In Sections 12.1 and 12.2 we discuss the theory of surface modes in spherical and nonspherical particles, respectively in Sections 12.3 and 12.4 comparisons between theory and experiment are given, first for insulators and then for metals and metal-like materials. 

The greater the order of the normal mode, the more the field is localized near the surface of the sphere, hence the designation surface modes. The lowest-order mode (n = 1) is uniform throughout the sphere, and this mode is sometimes called the mode of uniform polarization. 

We shall call the frequency at which t = —2em and t" — 0 the Frohlich frequency coF the corresponding normal mode—the mode of uniform polarization—is sometimes called the Frohlich mode. In his excellent book on dielectrics, Frohlich (1949) obtained an expression for the frequency of polarization oscillation due to lattice vibrations in small dielectric crystals. His expression, based on a one-oscillator Lorentz model, is similar to (12.20). The frequency that Frohlich derived occurs where t = —2tm. Although he did not explicitly point out this condition, the frequency at which (12.6) is satisfied has generally become known as the Frohlich frequency. The oscillation mode associated with it, which is in fact the lowest-order surface mode, has likewise become known as the Frohlich mode. Whether or not Frohlich s name should be attached to these quantities could be debated we shall not do so, however. It is sufficient for us to have convenient labels without worrying about completely justifying them. 

In the preceding paragraphs we considered a homogeneous sphere. Let us now examine what happens when a homogeneous core sphere is uniformly coated with a mantle of different composition. Again, the condition for excitation of the first-order surface mode can be obtained from electrostatics. In Section 5.4 we derived an expression for the polarizability of a small coated sphere the condition for excitation of the Frohlich mode follows by setting the denominator of (5.36) equal to zero ... 

A consequence of (9.23) and (9.24) is that to, < toF < to,. This is no more than a statement that (12.6) is satisfied only in the region where c is negative, which, for simple one-oscillator materials, lies between to, and to/ we shall call this the surface mode region. 

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