Drude-model metal

Figure 7. Field distribution for the n = / and n -2 TE waveguide mode for an air-PDA film-silver system for npda = 1.51, d = 5500 A and a Drude-model metal.

We now want to study the consequences of such a model with respect to the optical properties of a composite medium. For such a purpose, we will consider the phenomenological Lorentz-Drude model, based on the classical dispersion theory, in order to describe qualitatively the various components [20]. Therefore, a Drude term defined by the plasma frequency and scattering rate, will describe the optical response of the bulk metal or will define the intrinsic metallic properties (i.e., Zm((a) in Eq.(6)) of the small particles, while a harmonic Lorentz oscillator, defined by the resonance frequency, the damping and the mode strength parameters, will describe the insulating host (i.e., /((0) in Eq.(6)). 

In Pauli s model, we still envisage a core of rigid cations (metal atoms that have lost electrons), surrounded by a sea of electrons. The electrons are treated as non-interacting particles just as in the Drude model, but the analysis is done according to the rules of quantum mechanics. 

We will now analyze the general optical behavior of a metal using the simple Lorentz model developed in the previous section. Assuming that the restoring force on the valence electrons is equal to zero, these electrons become free and we can consider that Drude model, which was proposed by R Drude in 1900. We will see how this model successfully explains a number of important optical properties, such as the fact that metals are excellent reflectors in the visible while they become transparent in the ultraviolet. 

Thus, the Drude model predicts that ideal metals are 100 % reflectors for frequencies up to cop and highly transparent for higher frequencies. This result is in rather good agreement with the experimental spectra observed for several metals. In fact, the plasma frequency cop defines the region of transparency of a metal. It is important to realize that, according to Equation (4.20), this frequency only depends on the density of the conduction electrons N, which is equal to the density of the metal atoms multiplied by their valency. This allows us to determine the region of transparency of a metal provided that N is known, as in the next example. 

The simple free electron model (the Drude model) developed in Section 4.4 for metals successfully explains some general properties, such as the filter action for UV radiation and their high reflectivity in the visible. However, in spite of the fact that metals are generally good mirrors, we perceive visually that gold has a yellowish color and copper has a reddish aspect, while silver does not present any particular color that is it has a similarly high reflectivity across the whole visible spectrum. In order to account for some of these spectral differences, we have to discuss the nature of interband transitions in metals. 

This is the Drude model for the optical properties of a free-electron metal. The... 

The optical constants of a metal are determined to a large degree by the free electrons. According to the Drude model, the contribution of the free electrons to the frequency-dependent dielectric function is expressed as follows (16) ... 

The Drude model applies the kinetic theory of gases to metal conduction. It describes valence electrons as charged spheres that move through a soup of stationary metallic ions with finite chance for scattering. 

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. 

Assume atoms of atomic number Z, with Z electrons of charge — e each, and nuclei with charge +Z e in the Drude model a subset of z electrons per atom join the electron gas, leaving (Z — z) core electrons to surround the nucleus and form with the nucleus an "ionic core" of charge +z e (in 1900, protons and neutrons had not yet been identified ). Given a metal of volume V containing N "free" electrons, the electron density n (= number of free electrons per cm3) is given by... 

Another success story for the Drude model was the explanation for the Wiedemann19 -Franz20 law of 1858, which stated the empirical observation that the ratio of the thermal conductivity k to the electrical conductivity absolute temperature—that is, that the so-called Lorentz number k/oT was independent of metal and temperature. The thermal conductivity k (> 0) is defined by assuming that the heat flow JH is due to the negative gradient of the absolute temperature T (Fourier s law) ... 

Figure 3. Energy diagram for free electrons in a metal. The positive background charge of the core ions leads to a potential energy well with respect to the energy of the electron in vacuum vac- The averaged kinetic energy of the free electrons is indicated with dashed lines 3/2 k%T according to the Drude model, and 3/5 according to the Sommerfeld model. The electrochemical potential of the electrons in the metal [Fermi level] is also indicated.

Prior to quantum mechanics, bonding in metals was described by the Drude model, named for the German physicist Paul Drude. The solid was viewed as a... 

Before the quantum theory of solids (see description in Chapter 21), microscopic descriptions of metals were based on the Drude model, named for the German physicist Paul Drude. The solid was viewed as a fixed array of positively charged metal ions, each localized to a site on the solid lattice. These fixed ions were surrounded by a sea of mobile electrons, one contributed by each of the atoms in the solid. The number density of the electrons, is then equal to the number density of atoms in the solid. As the electrons move through the ions in response to an applied electric field, they can be scattered away from their straight-line motions by collisions with the fixed ions this influences the mobility of the electrons. As temperature increases, the electrons move more rapidly and the number of their collisions with the ions increases therefore, the mobility of the electrons decreases as temperature increases. Equation 22.7 applied to the electrons in the Drude model gives... 

The electrical conductivity for copper is given in Table 22.6. The electron mobility in copper at room temperature is 3.0 X 10 m s . Using the Drude model for metallic conductivity, calculate the number of free electrons per Cu atom. The density of copper is 8.9 g cm. ... 

Whereas Eq. (8) succeeds in explaining qualitatively the broadening and damping of the SPR absorption band with decreasing nanoparticle size, it presents some major drawbacks. First, the parameter A takes different values, from tenths to few units, depending on the theory. The value A = 1 is arbitrarily the most often used. Secondly, the introduction of such a 1/7 dependence in the Drude model results in the red-shift of the SPR with decreasing size, whereas a blue-shift is observed for noble metal nanoparticles [19]. This is due to the influence of bound d electrons which is ignored in the size-dependence considerations that we have described until now [24-27]. However - and even if it cannot of course explain on its own all the size effects - the 1 /7 dependence of different factors is an attractive intuitive... 

The simplest electronic theory of metals regards a metallic object as abox filled with noninteracting electrons. (A slightly more elaborate picture is the jellium model in which the free electrons are moving on the backgroimd of a continuous positive uniform charge distribution that represents the nuclei.) The Drude model, built on this picture, is characterized by two parameters The density of electrons n (number per unit volume) and the relaxation time r. The density n is sometimes expressed in terms of the radius of a sphere whose volume is the volume per electron in the metal... 

Moreover, the metallic nature already observed in the case of polyacetylene [58,59] has been found in other polymers such as camphor sulphonic acid (CSA) doped polyaniline [60] or polypyrrole [61]. In these materials, a negative dielectric constant [53], and a cross-over from the metallic to the non-metallic state between 20 and 50 K has been revealed in the case of polyaniline [62]. Polyorthotoluidine fibres doped with CSA in m cresol exhibit similar behaviour [63-65], PFg doped polypyrrole also exhibits such behaviour and the Drude model has been used for a description of dielectric constant evolution up to the infrared range [66], This model has also been applied to polyaniline [67],... 

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