Frohlich mode

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 ... 

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... 

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. 

Far-infrared absorption measurements gave an independent determination of the electron density from the position of the Frohlich mode near 9 meV ( 140 jum) a density of 2.3 X 1017 cm-3 was inferred. Other experiments on Sb-doped Ge and pure germanium irradiated at different powers showed appreciable changes in absorption band positions and shapes. Rose et al. 

Another example listed in the table is graphite, the Frohlich mode of which is near 5.5 eV (2200 A) the boundaries of the negative e region are about 4 and 6.5 eV. The graphite surface plasmon has been tentatively identified as responsible for a feature in the interstellar extinction spectrum (see Section 14.5). 

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]. 

Frohlich mode fluctuations with some ordering at low temperatures. Neither model is as yet quantitative in its fit of the experimental data although the presence of the Frohlich mode fluctuations is increasingly supported by other data (526), vide infra. 

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).

In a similar way, if one wants the electrons in a solid to hop from TCNQ to TCNQ to TCNQO, say, the intermediate state TCNQ " must be relatively easy to form. K these states are not accessible, then the motion of metallic electrons can only occur in a correlated, "Frohlich" mode. 

Frohlich was the first to describe in 1954, theoretically, this type of collective electron motion. Bardeen suggested twenty years later that it may contribute to the conductivity of the organic charge transfer salt TTF-TCNQ above Tp/ and later proposed that this "Frohlich mode" gives rise to the non-linearities of the conductivity in NbSe3 observed by Monceau et al. below T. ... 

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... 

The spherical modes (/ = 0) of the dot are excited with parallel polarization of the exciting laser light and have no surface contribution. The three degenerate modes for / = 1 are called Frohlich modes and correspond to an uniform polarization of the sphere. For I = 2, one gets the so-called spheroidal quadrupolar modes. The modes with / > 1 are usually called surface modes. Theoretical calculations of the one-phonon scattering in nanocrystallites predict that the surface mode is only allowed for / = 1, whereas the 1 = 2 modes are forbidden within the dipole approximation [178]. 

The extinction spectrum of Ag core/poly(ADA)-shell hybridized NCs dispersion liquid is shown in Fig. 6 [50]. The main peak at ca. 450 nm is assigned to ESP of Ag core [108], and was red-shifted and broadened because of the large size and its distribution of Ag core [109-111]. The small shoulder peak at 540 nm may come from the EAP of poly(ADA) shell [112], and the inset in Fig. 6 also displays the differential spectrum before and after solid-state polymerization. The obtained extinction spectrum could be reproduced by spectrum simulation on the basis of Mie scattering theory [113], assuming spherical Ag core (radius = 45 nm)/poly(ADA) shell (thickness = 5 nm) type hybridized NC. In Fig. 7, both main peaks around 450-500nm is attribute to ESP (Frohlich mode) of Ag core, and the two sharp peaks at 380 and 400 nm are the quadrupole mode of ESP [108]. 

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