Drude-Lorentz theory

The simple Drude-Lorentz theory described earlier in this chapter pictures the valency electrons in a metal as free to move in a potential well of the form shown in fig. 5.06. Within the metal, from B to C, the potential is uniform, but at the surface a potential difference, V (of the order of 10 V), prevents electron escape. If the electrons are assumed to obey the laws of classical mechanics their energies will correspond to the Boltzmann distribution appropriate to the temperature of the specimen. At room temperatures a quite negligible fraction of the electrons will have energies sufficient to surmount the potential... 

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... 

A metal cluster is excited by the electromagnetic field of a light wave. The theory describing this phenomenon as the dielectric function e(ra) was developed by Drude-Lorentz-Sommerfeld (DLS). The internal field of a metal cluster is calculated by adding the boundaries of the sphere surface. Based on DLS the static electric polarizability of a simple single spherical cluster of radius r is given by ... 

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 electromagnetic theory it is shown that the real and imaginary part of e (co) are not independent of each other, but are connected by a pair of integral equations, the Kramers-Kronig relation (e.g. Bohren Huffman 1983). Equation (A3.10) satisfies these relations, i.e. using a Lorentz-Drude model fitted to the laboratory data automatically guarantees that the optical data satisfy this basic physical requirement. 

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. 

The first successful theory of the metallic state may be said to have arisen from the work of Drude and Lorentz in the early years of the present century. On this theory a metal is to be regarded as an assemblage of positive ions immersed in a gas of free electrons. A potential gradient exists at the surface of the metal to imprison the electrons, but within the metal the potential is uniform.. Attraction between the positive ions and the electron ga gives, the structure its coherence, and the free mobility of this electron gas under the influence... 

The classical theory of absorption in dielectric materials is due to H. A. Lorentz and in metals it is the result of the work of P. K. L. Drude. Both models treat the optically active electrons in a material as classical oscillators. In the Lorentz model the electron is considered to be bound to the nucleus by a harmonic restoring force. In this manner, Lorentz s picture is that of the nonconductive dielectric. Drude considered the electrons to be free and set the restoring force in the Lorentz model equal to zero. Both models include a damping term in the electron s equation of motion which in more modem terms is recognized as a result of electron-phonon collisions. 

The classical theory of the dielectric response in solids is frequently described by the Drude and Lorentz models. The Drude model is applicable to free-eiectron metals its quantum-meehanical analog includes intraband transitions, where intraband transitions are taken to mean all transitions not involving a reciprocal lattice vector. The Lorentz model is applicable to insulators its quantum-mechanieal analog ineludes all direct interband transitions, i.e., all transitions for whieh the final state of an electron lies in a different band but with no change in the k vector in the reduced-zone scheme. In the following discussion, both models will be surveyed and evaluated for real metals. 

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