The Lorentz Model

While the Lorentz model only allows for a restoring force that is linear in the displacement of an electron from its equilibrium position, the anliannonic oscillator model includes the more general case of a force that varies in a nonlinear fashion with displacement. This is relevant when tire displacement of the electron becomes significant under strong drivmg fields, the regime of nonlinear optics. Treating this problem in one dimension, we may write an appropriate classical equation of motion for the displacement, v, of the electron from equilibrium as... 

The classical picture that describes these anisotropic effects based on the Lorentz model is illustrated in Fig. 9.8, which is a generalization of the spring model of Fig. 9.1 note that the spring stiffness depends on direction. 

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. 

We start our discussion of laser-controlled electron dynamics in an intuitive classical picture. Reminiscent of the Lorentz model [90, 91], which describes the electron dynamics with respect to the nuclei of a molecule as simple harmonic oscillations, we consider the electron system bound to the nuclei as a classical harmonic oscillator of resonance frequency co. Because the energies ha>r of electronic resonances in molecules are typically of the order 1-10 eV, the natural timescale of the electron dynamics is a few femtoseconds to several hundred attoseconds. The oscillator is driven by a linearly polarized shaped femtosecond... 

Just as the linear response of the medium displays a resonance in the vicinity of an electronic absorption, the non-linear properties also show resonant behaviour. These can be understood by extending the Lorentz model to the case of an anharmonic oscillator where the restoring force contains terms proportional to the square, cube, etc. of the displacement. Hence Equation (3.63) becomes ... 

Figure 37. Variation of the dielectric constant s/s2 of a metal-insulator colloid solution as a function of the metallic phase (silver) p (a) calculation based on the Lorentz model (b) calculation based on the fractal model.

It is clearly seen that the Bloch equation in the weak intensity limit, Eq. (A.30), has the same structure as the Lorentz equation, Eq. (A.29). Note that two parameters, and m, only appear in the Lorentz model while two other parameters, 2 = —d -E/h and h, appear in the Bloch equation. To make the equivalence between these two approaches complete, the following relations can be deduced by comparing Eq. (A.29) with Eq. (A.30), and Eq. (A.27) with Eq. (4.11),... 

The Lorentz model becomes more useful and applicable over a wide frequency range if the effect of oscillators of different types is taken into account. In the simplest harmonic approximation, the oscillators representing the lattice... 

Metals are denoted as fi ee-electron metals if most of the electronic and optical properties are due to the conduction electrons alone. Examples are Al, Mg, and alkali metals. The dispersion of the optical constants of the fi ee-electron metals is described by the Drude model [72], which can be regarded as a special case of the Lorentz model (1.3.3°) when the restoring (bounding) force is equal to zero, and hence the resonance frequency of free carriers, a>o = -JWfm (here, m is electron mass), is also equal to zero, damping force y results... 

The local field can be found, e.g., from some models, particularly the Lorentz model [3]. For its discussion we select a single molecule and surround it by a fictitious spherical cavity shown in Fig. 7.4b. Then Eioc is a sum of four fields ... 

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. 

These models solve for the electron s motion in the presence of the electromagnetic field as a driving force. From this, it is possible to write an expression for the polarization induced in the medium and from that to derive the dielectric constant. The Lorentz model for dielectrics gives the relative real and imaginary parts of the dielectric constant as... 

The Drude model for metals assumes that the electrons are free to move. This means that it is identical to the Lorentz model except that coq is set equal to zero. The real and imaginary parts of the metal s dielectric constant are then given by... 

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. 

The Lorentz and Drude models can be explained rigorously in relation to electronic band structure. Indeed, both models have quantum-mechanical analogs intraband transitions for the Drude model and direct interband transitions for the Lorentz model. To see the role of both models in describing real metals, consider the schematic band diagram as shown in Figure 149. TWo typical transitions are illustrated in Figure 149. The first of these, called an intraband transition, corresponds to the optical excitation of an electron from below the Fermi level (Ep) to another state above the Ef within the same band. There is no threshold energr for such transitions, and they can occur only in metals. 

In the Lorentz model,the medium is not polarizable. Then the reaction field is... 

Fig. 3.6 The effective electric field acting on a molecule in a polarizable medium (shaded rectangles) is Eiq fErtj d where is the field in the medium and / is the local-field correction factor, hi the cavity-field model (A) Ei c is the field that would be present if the molecule were replaced by an empty cavity (Ecav), in the Lorentz model (B) Ei c is the sum of E av and the reaction field (Ereaa) resulting from polarization of the medium by induced dipoles within the molecule (P)...

The Lorentz-Drude model is largely ad hoc [14,15] but still useful as starting points and for developing feelings for optical properties. Consider an atom with electrons boiuid to the nucleus in the same way as a small mass bound to a large mass by a spring. This is the Lorentz model and is applicable to a wide variety of materials (i.e., metals, semiconductors and insulators). The motion of an electron boiuid to the nnclens is then written as... 

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