The Drude Model

The simplest picture of a metallic conductor is one where we have a rigid lattice of metal (M) atoms, each of which has lost one or more electrons to form a surrounding sea of electrons. 

Secondly, the influence of the cations on their surrounding space is not negligible. The cations produce strong non-uniform electrostatic fields and the electron density in such a model is far from constant, or even slowly varying. The free electrons experience a very strong attraction when they approach the cations. 

The dielectric function of a metal can be obtained considering the dielectric response of a plasma sea of electrons with electron concentration N. The optical properties of metals are in fact determined mainly by the response of free electrons. The role of the crystal lattice can be reduced to the modification of the electron mass to give the effective mass m instead of the free electron mass me, and to the development of states of high energy for optical transitions. 

A particular solution of this equation, describing the oscillation of the electron, isx (t) = Re[xoe ] with [5]  

The amplitude xo is complex to account for possible phase shift between the driving field and the medium response. 

The displaced electrons contribute to a macroscopic polarization Px given by  

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]  

Electrons in metals at the top of the energy distribution (near the Fermi level) can be excited into other energy and momentum states by photons with very small energies thus, they are essentially free electrons. The optical response of a collection of free electrons can be obtained from the Lorentz harmonic oscillator model by simply clipping the springs, that is, by setting the spring constant K in (9.3) equal to zero. Therefore, it follows from (9.7) with co0 = 0 that the dielectric function for free electrons is 

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

If we consider only small departures from equilibrium ( 50l/0l 1), the 

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 

These equations are identical with the high-frequency limit (9.13) of the Lorentz model this indicates that at high frequencies all nonconductors behave like metals. The interband transitions that give rise to structure in optical properties at lower frequencies become mere perturbations on the free-electron type of behavior of the electrons under the action of an electromagnetic field of sufficiently high frequency. 

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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 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... 

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. 

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... 

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

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). 

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... 

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) ... 

Figure 12.1 shows a slice through such a solid the cations are to be thought of as a rigid lattice, and the electrons form a gas. I have deliberately drawn the cations as large objects for two reasons. First, the very early models such as that due to Drude tried to treat the electron sea as a perfect gas. It was eventually recognized that the electrons would collide with the cations and with each other an uncomfortable number of times. In any case, many of the predictions of the Drude model turned out to be demonstrably flawed. 

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. 

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