Frohlich frequency/mode

The dielectric function of a metal can be decomposed into a free-electron term and an interband, or bound-electron term, as was done for silver in Fig. 9.12. This separation of terms is important in the mean free path limitation because only the free-electron term is modified. For metals such as gold and copper there is a large interband contribution near the Frohlich mode frequency, but for metals such as silver and aluminum the free-electron term dominates. A good discussion of the mean free path limitation has been given by Kreibig (1974), who applied his results to interpreting absorption by small silver particles. The basic idea is simple the damping constant in the Drude theory, which is the inverse of the collision time for conduction electrons, is increased because of additional collisions with the boundary of the particle. Under the assumption that the electrons are diffusely reflected at the boundary, y can be written... 

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. 

Absorption resonances resulting from excitation of surface modes are accompanied by scattering resonances at approximately the same frequencies this was pointed out following (12.26). In most experiments transmission is measured to determine extinction, which is nearly equal to absorption for sufficiently small particles. However, surface mode resonances have been observed in spectra of light scattered at 90° by very small particles of silver, copper, and gold produced by nucleation of vapor in an inert gas stream (Eversole and Broida, 1977). The scattering resonance peak was at 3670 A, near the expected position of the Frohlich mode, for the smallest silver particles. Although peak positions were predictable, differences in widths and shapes of the bands were concluded to be the result of nonsphericity. 

Two-peak behavior of the IR spectrum shown in Fig. 2 may be due to the electrostatic Frohlich mode (cof) corresponding to a uniform polarization of the ZnSe sphere. The peak near cop (the theoretically predicted frequency is 229 cm ) is formed by the modes whose frequencies are slightly higher than cop [8]. 

Fig. 33. Far infrared to uv reflectivity of K2Pt(CN)4Bro.3o(H20)8 for light polarized parallel to the conducting axis. The dashed line is for the sample at 300°K, the solid line 40 K. The low frequency structure (50cm i) at 40 °K is assigned to the response of a pinned charge density wave (pinned Frohlich mode) (72).

The frequencies at which the condition (1.117) is met are referred to as Frohlich frequencies cof, and the corresponding modes of a sphere that is small compared to k are referred to as the Frohlich modes, giving credit to the pioneering work of Frohlich [13], who determined theoretically that an ensemble of small spherical particles absorbs at (see the discussion in Section 3.9). When the imaginary part of s o)) vanishes, Eq. (1.117) is reduced to the Mie condition... 

Sphere. A complete description of the coupling of an electromagnetic wave and the eigenmodes of an isolated sphere of any size, given by polariton theory based on Mie s formalism (Section 1.10), indicates that all modes of a sphere-shaped crystal are radiative [293, 298], These modes are called surface modes since their origin lies in the finite size of the sample [297]. For very small spheres, there is only the lowest order surface mode (the Frohlich mode), which is neither transverse nor longitudinal [293]. Its frequency (the Frohlich frequency) is given by... 

For oxidized metal spheres, there is only one absorption band, near the vlo frequency of the coating material, as shown by the example of a Mg sphere with a MgO coating [298]. This is similar to the case of an ultrathin film on a metal plane substrate (Section 3.2). It follows that the usual SSR (Section 1.8.2) may also be applied to the surfaces of metal powders (see also the discussion in Section 3.9.4). Bamickel and Wokaun [314] reported that a dielectric coating shifts the resonance frequency of a metallic particle toward the red. Applying the MGEM dielectric function, Martin [315] calculated the reflection spectrum of an ensemble of S-pm Zn spheres coated with ZnO of variable thickness. He obtained the same qualitative result as Ruppin for the isolated particle [298], that in the limiting case of the absence of the core, the absorption maximum is at the Frohlich frequency. As the thickness of the coating decreases relative to the core radius, the surface-mode frequency increases and approaches vlo monotonicaUy. 

For any real material, the frequency at which (12.27) is satisfied is complex—the surface modes are virtual. However, its real part is approximately the frequency where the cross sections have maxima, provided that the imaginary part is small compared with the real part. We shall denote this frequency by us. For a sphere, o>s is the Frohlich frequency wF. If used intelligently, always keeping in mind its limitations, (12.27) is a guide to the whereabouts of peaks in extinction spectra of small ellipsoidal particles but it will not necessarily lead to the exact frequency. 

Several predicted features of infrared surface mode absorption by small spheres are verified by the experimental results shown in Fig. 12.13. The frequency of peak absorption by spheres is shifted an appreciable amount from what it is in the bulk solid the e" curve peaks at 1070 cm", whereas the peak of the small-sphere absorption is at 1111 cm-1, very close to the frequency where e is — 2em (— 4.6 for a KBr matrix). The absorption maximum (absorption is nearly equal to extinction for these small particles) is very strong Qabs for a 0.1-jum particle is about 7 at the Frohlich frequency. 

The curves of Fig. 12.17 nicely illustrate the varied optical effects exhibited by small metallic particles in the surface mode region, both those explained by Mie theory with bulk optical constants and those requiring modification of the electron mean free path (see Section 12.1). Absorption by particles with radii between about 26 and 100 A peaks near the Frohlich frequency (XF — 5200 A), which is independent of size. Absorption decreases markedly at longer... 

Optical study indicates that at low temperatures the low-energy electronic properties of some organic metal-like conductors (e.g., TTF-TCNQ) are dominated by charge density wave (CDW) effects. Frequency-dependent conductivity of TTF-TCNQ, obtained from the IR reflectance, at 25 K displays a double-peak structure with a low-frequency band near 35 cm-1 and a very intense band near 300 cm-1 [45]. The intense band may be ascribed to single-particle transitions across the gap in a 2kF (Peierls) semiconducting state, while the 35-cm-1 band is assigned to the Frohlich (i.e., CDW) pinned mode. Low-temperature results based on the bolometric technique [72,73] (Fig. 15) confirm the IR reflectance data. Such a con-... 

Anomalous rotational diffusion in a potential may be treated by using the fractional equivalent of the diffusion equation in a potential [7], This diffusion equation allows one to include explicitly in Frohlich s model as generalized to fractional dynamics (i) the influence of the dissipative coupling to the heat bath on the Arrhenius (overbarrier) process and (ii) the influence of the fast (high-frequency) intrawell relaxation modes on the relaxation process. The fractional translational diffusion in a potential is discussed in detail in Refs. 7 and 31. Here, just as the fractional translational diffusion treated in Refs. 7 and 31, we consider fractional rotational subdiffusion (0rotation about fixed axis in a potential Vo(< >)- We suppose that a uniform field Fi (having been applied to the assembly of dipoles at a time t = oo so that equilibrium conditions prevail by the time t = 0) is switched off at t = 0. In addition, we suppose that the field is weak (i.e., pFj linear response condition). 

The wavenumber at which the Ti02 absorption band occurs in the measured infrared spectra (see Figure 7) is shifted from the wavenumber predicted by theory. The absorption maximum, which is usually coincident with the Frohlich frequency (Vp, the lowest-order surface mode) 20), can be calculated from the transverse optical mode of the material (Vyo), as long as the dielectric function is known for the material at zero (Cq) and infinite frequency and the dielectric function for the matrix (b ), according to 20) ... 

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