Drude free-electron model

The optical data will not be discussed here. Beckman and Pitzer analyzed their reflectance data in terms of thermally activated electrons the Hall effect data are inconsistent with this interpretation. An effective mass could be introduced to bring their data into concert with the Drude (free electron) model, but would add little to our understanding. 

From Eqs. (1.116,1.12,1.14) we can arrive at the expression for the dielectric function for the Drude free-electron model [6] ... 

Apart from tabulated experimental data, there exist analj ical models for the refractive indices. The most widely used is the Drude free-electron model [37] (see Sec. 1.2.1) ... 

This partition has been used for example by Zeman and Schatz [76]. In the classical Drude free-electron model (see Sec. 1.2) I/ can be expressed as ... 

Similarly, the reflectance closely follows the Drude free electron model except for the region in the vicinity of the interband transition as seen in Figure 24.12. 

The sharp transition from transparency to reflection at the plasma frequency predicted by the Drude free electron model can be better understood by considering a free electron gas. The undamped motion of an electron re onding to a time varying electric field is mx = —eEoe . The solution is x (o) = eEofmdipole moment of a single... 

For spherical, metal particles having a diameter much smaller than the incident wavelength, the electric field intensity is uniformly distributed across the particle surface, such that all conduction band electrons are equally excited. In this case, electron movement can be well approximated by the Drude free-electron model,... 

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. 

As an example of extinction by spherical particles in the surface plasmon region, Fig. 12.3 shows calculated results for aluminum spheres using optical constants from the Drude model taking into account the variation of the mean free path with radius by means of (12.23). Figure 9.11 and the attendant discussion have shown that the free-electron model accurately represents the bulk dielectric function of aluminum in the ultraviolet. In contrast with the Qext plot for SiC (Fig. 12.1), we now plot volume-normalized extinction. Because this measure of extinction is independent of radius in the small size... 

Optical properties of metal nanoparticles embedded in dielectric media can be derived from the electrodynamic calculations within solid state theory. A simple model of electrons in metals, based on the gas kinetic theory, was presented by Drude in 1900 [9]. It assumes independent and free electrons with a common relaxation time. The theory was further corrected by Sommerfeld [10], who incorporated corrections originating from the Pauli exclusion principle (Fermi-Dirac velocity distribution). This so-called free-electron model was later modified to include minor corrections from the band structure of matter (effective mass) and termed quasi-free-electron model. Within this simple model electrons in metals are described as... 

The DOS at 0 K is shown in Figure 11.18 using the free electron model. However, the Drude-Lorentz model employs the classical equipartition of energy and does not take into account the fact that quantum mechanics places restrictions on the placement of the electrons (as a result of the Pauli exclusion principle). A revised theory, known as the Sommerfield model, allows for this modification. At temperatures above 0 K, the fraction, f(E), of allowed energy levels with energy E follows the... 

Despite of its simplicity the Drude model correctly describes the optical properties of simple metals. In Fig. 1.4 we report the experimental dielectric constant, the refractive index and the reflectance for aluminum. Figure 1.4 closely resembles Fig. 1.3 with (Op 15 eV. The best agreement of the dielectric function of metal with that obtained in the framework of the free electron model can be obtained for the alkali metals (Li, Na, K, Cs, Rb), whose response seems to be weakly affected by the contribution from the core electrons. Notably, alkali metals, such as sodium, have an almost free-electron-like response and thus, in accordance... 

The normal spectral emissivity Sn was measured by Lange and Schenck at 650 nm for the first time [39]. Since then, over ten measurements have been reported [40-50], as shown in Fig. 4.12. Shvarev et al. [40] reported that thermal emission from molten silicon can be explained by Drude s free-electron model. Pulse lasers have been used to melt silicon. This method has advantages, in that measurement can be carried out in a very short time and a furnace is not required [42-44]. However, it is difficult to measure the sample temperature accurately as long as laser heating is employed. Recently, measurements using a cold crucible or levitator have been attempted these techniques assure measurement conditions without optical contamination, because there is no crucible wall at high temperature, which causes disturbing emission and reflection [47-50]. 

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

A theory for the metallic state proposed by Drude at the turn of this century explained many characteristic features of metals. In this model, called the free-electron theory, all the atoms in a metallic crystal are assumed to take part collectively in bonding, each atom providing a certain number of (valence) electrons to the bond. These free electrons belong to the crystal as a whole. The crystal is considered to be... 

The sp-valent metals such as sodium, magnesium and aluminium constitute the simplest form of condensed matter. They are archetypal of the textbook metallic bond in which the outer shell of electrons form a gas of free particles that are only very weakly perturbed by the underlying ionic lattice. The classical free-electron gas model of Drude accounted very well for the electrical and thermal conductivities of metals, linking their ratio in the very simple form of the Wiedemann-Franz law. However, we shall now see that a proper quantum mechanical treatment is required in order to explain not only the binding properties of a free-electron gas at zero temperature but also the observed linear temperature dependence of its heat capacity. According to classical mechanics the heat capacity should be temperature-independent, taking the constant value of kB per free particle. 

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) ... 

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... 

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.

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. ... 

Ey stands for the free-electron contribution, which can be described in a classical way by the Drude model [9] ... 

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... 

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