Drude approximation

Fig. 11.6 Absorption spectra K restored from diffuse reflection R by using the Kubelka-Munk equation. Spectra 1, 2, and 3 are initial nanostructures, hydrogenated and annealed at 700°C during 6 h, respectively, (a) Spectra of NFs, (b) SWNTs. Dashed curves are Drude approximation of the absorption spectra, K = A. v0 5...

Fig. 11.7 Optical density spectra of thin films of the initial, hydrogenated and outgassed SWNTs (spectra 1, 2 and 3, respectively). The dashed curves in the low-energy part of the spectra represent the Drude approximation of the absorption spectra. The inset in Fig. 11.7 shows the absorption lines of the C-H bonds in a thin film of the hydrogenated SWNTs...

Hamada et al. 1992 Saito et al. 1992), this increase of absorption is caused by high-frequency conductivity of the free carriers in metallic nanotubes. Relative intensities of the spectra of Fig. 11.7 we have found as a result of the diffuse reflection measurements of powders at low wavenumbers. The discussed above Drude approximations of the low-energy part of the absorption spectra are shown by dashed curves in Fig. 11.7. Comparison of the spectra 1 and 2 shows that hydrogenation decreases high-frequency conductivity of the SWNTs by one order of magnitude. 

Free-charge-carrier contributions fcc(o ) are typically described by the classical Drude approximation... 

Ellipsometric Determination of Film Thickness. The principal use of the ellipsometer was to study the upper edge of the films on vertical plates where the thickness became less than 1000 A. and could not be observed with the interference microscope. The ellipsometric measurement of film thickness is described in the literature [4,6]. For the purposes to which the technique was applied here, the approximate equation given by Drude [6] relating the film thickness and refractive index with the optical parameters of the incident and reflected light was entirely adequate. In order to use the Drude approximations it was necessary to assume the refractive index of the liquid film to be the same as that of bulk liquid. These approximate equations are highly inaccurate for film thickness greater than 100 A. 

In this case, the optics of the interface is slightly complicated in view of the birefringence of the surface induced pre-nematic phase. The interface is now consisting of (i)the isotropic bulk liquid crystal, (ii) the surface induced, weakly optically uniaxial layer with a thickness of the nematic correlation length and with the optical axis perpendicular to the surface, and, (iii) an isotropic substrate (such as glass). If the optical anisotropy and the thickness of the anisotropic layer are small, the eUipticity coefficient at the Brewster angle> PB, can be calculated in the Drude approximation [1]... 

Consequently, in the Drude approximation, for the usual case when cOp 1/t, one expects the following behaviors ... 

When the Drude particles are treated adiabatically, a SCF method must be used to solve for the displacements of the Drude particle, d, similarly to the dipoles Jtj in the induced dipole model. The implementation of the SCF condition corresponding to the Born-Oppenheimer approximation is straightforward and the real forces acting on the nuclei must be determined after the Drude particles have attained the energy minimum for a particular nuclear configuration. In the case of N polarizable atoms with positions r, the relaxed Drude particle positions r + d5CF are found by solving... 

From this point of view it is of interest to examine the consequences of full ther-malization of the classical Drude oscillators on the properties of the system. This is particularly important given the fact that any classical fluctuations of the Drude oscillators are a priori unphysical according to the Bom-Oppenheimer approximation upon which electronic induction models are based. It has been shown [12] that under the influence of thermalized (hot) fluctuating Drude oscillators the corrected effective energy of the system, truncated to two-body interactions is... 

An elementary treatment of the free-electron motion (see, e.g., Kittel, 1962, pp. 107-109) shows that the damping constant is related to the average time t between collisions by y = 1 /t. Collision times may be determined by impurities and imperfections at low temperatures but at ordinary temperatures are usually dominated by interaction of the electrons with lattice vibrations electron-phonon scattering. For most metals at room temperature y is much less than oip. Plasma frequencies of metals are in the visible and ultraviolet hu>p ranges from about 3 to 20 eV. Therefore, a good approximation to the Drude dielectric functions at visible and ultraviolet frequencies is... 

Near the plasma frequency in metals 2 y2 therefore, to good approximation, the imaginary part of the Drude dielectric function (9.26) is... 

By means of this combination of the cross section for an ellipsoid with the Drude dielectric function we arrive at resonance absorption where there is no comparable structure in the bulk metal absorption. The absorption cross section is a maximum at co = ojs and falls to approximately one-half its maximum value at the frequencies = us y/2 (provided that v2 ). That is, the surface mode frequency is us or, in quantum-mechanical language, the surface plasmon energy is hcos. We have assumed that the dielectric function of the surrounding medium is constant or weakly dependent on frequency. 

Extinction calculations for aluminum spheres and a continuous distribution of ellipsoids (CDE) are compared in Fig. 12.6 the dielectric function was approximated by the Drude formula. The sum rule (12.32) implies that integrated absorption by an aluminum particle in air is nearly independent of its shape a change of shape merely shifts the resonance to another frequency between 0 and 15 eV, the region over which e for aluminum is negative. Thus, a distribution of shapes causes the surface plasmon band to be broadened, the... 

The observed darkening of the indium slides results from a shift of the absorption peak because of the coating on the particles. Because of the cumbersomeness of the expressions for coated ellipsoids (Section 5.4) this shift can be understood most easily by appealing to (12.15), the condition for surface mode excitation in a coated sphere. For a small metallic sphere with dielectric function given by the Drude formula (9.26) and coated with a nonabsorbing material with dielectric function c2, the wavelength of maximum absorption is approximately... 

A common alternative is to synthesize approximate state functions by linear combination of algebraic forms that resemble hydrogenic wave functions. Another strategy is to solve one-particle problems on assuming model potentials parametrically related to molecular size. This approach, known as free-electron simulation, is widely used in solid-state and semiconductor physics. It is the quantum-mechanical extension of the classic (1900) Drude model that pictures a metal as a regular array of cations, immersed in a sea of electrons. Another way to deal with problems of chemical interaction is to describe them as quantum effects, presumably too subtle for the ininitiated to ponder. Two prime examples are, the so-called dispersion interaction that explains van der Waals attraction, and Born repulsion, assumed to occur in ionic crystals. Most chemists are in fact sufficiently intimidated by such claims to consider the problem solved, although not understood. 

Drude obtained the expression k/oT (3/2) (kB/e)2 = l.ll x 10 8 watt Q/K2, which is 50% of the approximate experimental k/itT values. This good fit was made possible by the fortuitous cancellation of two errors, both factors of about 100 in particular, a large value for the electronic specific heat Cv was used, which by classical equipartition arguments would equal (3/2) kB, but is 100 times smaller experimentally. Drude also estimated the thermopower as Q = —kB/2e. 

The optical properties of organic conductors may be described by the simplest model, which assumes noninteracting electrons (one-electron model). In this approximation the infrared (IR) properties may be derived in the self-consistent field approximation. Assuming a frequency-independent relaxation rate, y, and a background dielectric constant arising from virtual high-frequency transitions, e0, the result takes the Drude form [12] ... 

In the electron theory of metals no accurate allowance has hitherto been made for the effects of the conduction electrons on each other. Drude and Lorentz even went the length of assuming that to a first approximation the mutual action of the electrons and ions may be neglected, and accordingly spoke of a gas of free electrons. 

Various other electronic transitions are possible upon light excitation. Besides the band-band transitions, an excitation of an electron from a donor state or an impurity level into the conduction band is feasible (transition 2 in Fig. 1.9). However, since the impurity concentration is very small, the absorption cross-section and therefore the corresponding absorption coefficient will be smaller by many orders of magnitude than that for a band-band transition. At lower photon energies, i.e. at ph < g, an absorption increase with decreasing ph has frequently been observed for heavily doped semiconductors. This absorption has been related to an intraband transition (transition 4 in Fig. 1.9), and is approximately described by the Drude theory [4]. This free carrier absorption increases with the carrier density. It is negligible for carrier densities below about 10 cm ... 

Finally, the dispersion term between molecules A and B can be approximated by the following expression due to Drude [29] ... 

The region near the plasma edge in Id conductors /any gap uj < interband region/ is known to approximately obey the Drude formula for the dielectric function ... 

We assume that the incident EW has a linear polarization so that the electric field vector resides in the plane of incidence. We describe the response of 2D electron plasma on segments 0local approximation (Drude model) by the surface conductivities in the form ox,B=(e AA,Bi/w ) /(l-i r), where e and m are the electron charge and effective mass, r is the phenomenological relaxation time. 

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