Model Drude

Soon after the electron was discovered in 1897, there appeared a model for metallic condnctivity, developed by the German physicist Panl Drnde. In this model, the electrons are moving as free particles. The positive ions left behind are treated as a homogeneous attractive field. 

The most important property of a metal, that an electric field, no matter how weak, always causes conductivity, follows in the Drude model. However, there are other features of metals, particularly visible at a low temperature, where the Drude model fails. For example, resistivity as a function of temperature is incorrectly described. 

In an electric field (E), the electrons move with an average velocity called the drift velocity (v ). The electron mobility p is defined as 

The conductivity is proportional to the product of mobility and carrier concentration. If we have n charge carriers per unit volume, the net flux of charge carriers per unit time is given by n Vd and the current density J = I/A by 

I = J A is the current measured in amperes and V is the voltage over the conductor of length , measured in volts. 

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Fig. 4, 33 The Drude model for dispersive interactions. (Figure adapted from Rigby M, E B Smith, W A Wakeham and G C Maitland 1986. The Forces Between Molecules. Oxford, Clarendon Press.)...

Irude model thus predicts that the dispersion interaction varies as 1//. wo-dimensional Drude model can be extended to three dimensions, the result being ... 

Irude model only considers the dipole-dipole interaction if higher-order terms, due to e-quadrupole, quadrupole-quadrupole, etc., interactions are included as well as other i in the binomial expansion, then the energy of the Drude model is more properly an as a series expansion ... 

The Lennard-Jones potential is characterised by an attractive part that varies as r ° and a repulsive part that varies as These two components are drawn in Figure 4.35. The r ° variation is of course the same power-law relationship foimd for the leading term in theoretical treatments of the dispersion energy such as the Drude model. There are no... 

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)). 

Figure 12.1 The Drude model small circles are electrons, large circles cations...

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. 

While nonbonded atom pairs will typically not come within 1A of each other, it is possible for covalently bound pairs, either directly bounds, as in 1-2 pairs, or at the vertices of an angle, as in 1-3 pairs. Accordingly it may be considered desirable to omit the 1-2 and 1-3 dipole-dipole interactions as is commonly performed on additive force fields for the Coulombic and van der Waals terms. However, it has been shown that inclusion of the 1-2 and 1-3 dipole-dipole interactions is required to achieve anistropic molecular polarizabilites when using isotropic atomic polariz-abilites [50], For example, in a Drude model of benzene in which isotropic polarization was included on the carbons only inclusion of the 1-2 and 1-3 dipole-dipole interactions along with the appropriate damping of those interactions allowed for reproduction of the anisotropic molecular polarizability of the molecule [64], Thus, it may be considered desirable to include these short range interactions in a polarizable force field. 

An important alternative to SCF is to extend the Lagrangian of the system to consider dipoles as additional dynamical degrees of freedom as discussed above for the induced dipole model. In the Drude model the additional degrees of freedom are the positions of the moving Drude particles. All Drude particles are assigned a small mass mo,i, taken from the atomic masses, m, of their parent atoms and both the motions of atoms and Drude particles (at positions r, and rdj = r, + d, ) are propagated... 

Anisimov VM, Vorobyov IV, Roux B, MacKerell AD (2007) Polarizable empirical force field for the primary and secondary alcohol series based on the classical drude model. J Chem Theory Comput 3(6) 1927-1946... 

Bade WL (1957) Drude-model calculation of dispersion forces. I. General theory. J Chem Phys 27(6) 1280-1284... 

Amos AT (1996) Bond properties using a modern version of the Drude model. Int J Quant Chem 60(l) 67-74... 

Wang F, Jordan KD (2002) Application of a Drude model to the binding of excess electrons to water clusters. J Chem Phys 116(16) 6973-6981... 

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 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 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. 

By a careful inspection of Figure 4.17, we see how further transitions between bands below and above the Fermi level can also occur at energies higher than 1.5 eV. However, as these bands are not parallel, the density of states at these energies is lower than at 1.5 eV. In any case, the absorption probability is still significant, and it acconnts for the experimentally observed redaction in the reflectivity of Al in respect to the predictions from the Drude model (see Figure 4.5). 

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

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