The Lorentz Oscillator

In the previous section we have demonstrated how the measurable optical magnitudes are related to the dielectric constants (si, 2)- Now we need to establish how these 

Obviously, we need to start from microscopic (classic and quantum) models. These models require some knowledge about the nature of the interatomic (or interionic) bonding forces in our sohd and whether or not the valence electrons are free to move inside the solid. 

In metals, valence electrons are conduction electrons, so they are free to move along the solid. On the contrary, valence electrons in insulators are located around fixed sites for instance, in an ionic solid they are bound to specific ions. Semiconductors can be regarded as an intermediate case between metals and insulators valence electrons can be of both types, free or bound. 

The most simple, but general, model to describe the interaction of optical radiation with solids is a classical model, due to Lorentz, in which it is assumed that the valence electrons are bound to specific atoms in the solid by harmonic forces. These harmonic forces are the Coulomb forces that tend to restore the valence electrons into specific orbits around the atomic nuclei. Therefore, the solid is considered as a collection of atomic oscillators, each one with its characteristic natural frequency. We presume that if we excite one of these atomic oscillators with its natural frequency (the resonance frequency), a resonant process will be produced. From the quantum viewpoint, these frequencies correspond to those needed to produce valence band to conduction band transitions. In the first approach we consider only a unique resonant frequency, o in other words, the solid consists of a collection of equivalent atomic oscillators. In this approach, coq would correspond to the gap frequency. 

This model of atomic oscillators, in which we assume bound valence electrons, is also perfectly valid for metals, except that in this case we must set coq = 0. 

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In Chapter 4 we will consider the so-called classical approximation, in which the electromagnetic radiation is considered as a classical electromagnetic wave and the solid is described as a continnous medium, characterized by its relative dielectric constant e or its magnetic permeability fx. The interaction will then be described by the classical oscillator (the Lorentz oscillator). 

Let us now check the validity of the simple Lorentz model in order to explain the spectra of real solids. Figure 4.2 shows the dependence of the reflectivity on photon energy for a typical semiconductor. Si (Figure 4.2(a)), and for a typical insulator, KCl (Figure 4.2(b)). The Lorentz oscillator cannot quantitatively explain both spectra. In fact, we have supposed a single resonance frequency >o, but in the most general case a... 

Let us now imagine the motion of the free electrons just after the driving external local field is eliminated. Then Equation (4.12) for the Lorentz oscillator appears in a simplified form, as... 

Our derivation of (9.41) follows closely that of Gevers (1946) and is similar to that of Brown (1967, pp. 248-255). Because of the nature of this derivation it should hardly be necessary to do so, but it can be shown directly by integration—more easily than for the Lorentz oscillator—that the real and imaginary parts of the Debye susceptibility satisfy the Kramers-Kronig relations (2.36) and (2.37). 

The imaginary part of the dielectric function (9.41) is a maximum at co = 1/r and behaves similarly to e" for the Lorentz oscillator. The real part,... 

The microscopic characteristics of a real adsorbate layer have been considered [33, 34] by separating the x, y and z components of the electric field at the interface (Fig.4), and applying the Lorentz oscillator model to microscopically represent the adsorbate in the three-layer model. For the case of external reflection at a vacuum/semi-conducting, where Ss is real (no absorption) and isotropic, we can write ... 

We would like to emphasize that the photon statistics we consider is classical, while the Bloch equation describing dynamics of the SM has quantum mechanical elements in it (i.e., the coherence). In the weak laser intensity case, the Bloch equation approach allows a classical interpretation based on the Lorentz oscillator model as presented in Appendix B.3. 

In Appendix B, we find expressions for the photon current using the perturbation expansion. In Section B.3, we use the Lorentz oscillator model to derive similar results based on a classical picture. 

The Lorentz oscillator model is a classical model which is usually used to describe the dispersion relation of semiconductor and crystalline materials. [6, 7, 67, 68] On the basis of this model, the dielectric function is usually expressed as... 

In order to illustrate some of the basic aspects of the nonlinear optical response of materials, we first discuss the anliannonic oscillator model. This treatment may be viewed as the extension of the classical Lorentz model of the response of an atom or molecule to include nonlinear effects. In such models, the medium is treated as a collection of electrons bound about ion cores. Under the influence of the electric field associated with an optical wave, the ion cores move in the direction of the applied field, while the electrons are displaced in the opposite direction. These motions induce an oscillating dipole moment, which then couples back to the radiation fields. Since the ions are significantly more massive than the electrons, their motion is of secondary importance for optical frequencies and is neglected. 

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... 

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)). 

Observed and calculated intensities of reflections on two oscillation photographs, one of which is reproduced in Fig. 5, are given in Table III. The first number below each set of indices (hkl) is the visually estimated observed intensity, and the second the intensity calculated by the usual Bade-methode formula with the use of the Pauling-Sherman /0-values1), the Lorentz and polarization factors being included and the temperature factor omitted. No correction for position on the film has been made. It is seen that the agreement is satisfactory for most of the... 

Let us now analyze the interaction of a light wave with our collection of oscillators at frequency two- In this case, the general motion of a valence electron bound to a nucleus is a damped oscillator, which is forced by the oscillating electric field of the light wave. This atomic oscillator is called a Lorentz oscillator. The motion of such a valence electron is then described by the following differential equation ... 

Considering our single two energy level center, it is easy to understand that the area under the absorption spectrum, /a co) dco, must be proportional to both /x and the density of absorbing centers, N. In order to build up this proportionality relationship, it is very common to use a dimensionless quantity, called the oscillator strength, f. This magnitude has already been introduced in the previous chapter (Section 4.3), when treating the classical Lorentz oscillator. It is defined as follows ... 

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... 

The return to equilibrium of a polarized region is quite different in the Debye and Lorentz models. Suppose that a material composed of Lorentz oscillators is electrically polarized and the static electric field is suddenly removed. The charges equilibrate by executing damped harmonic motion about their equilibrium positions. This can be seen by setting the right side of (9.3) equal to zero and solving the homogeneous differential equation with the initial conditions x = x0 and x = 0 at t = 0 the result is the damped harmonic oscillator equation ... 

On physical grounds, relaxation of permanent dipoles is expected to be highly dependent on temperature this is in contrast with Lorentz oscillators, the dielectric behavior of which is relatively insensitive to changes in temperature. Debye (1929, Chap. 5) derived a simple classical expression for the relaxation time of a sphere of radius a in a fluid of viscosity tj ... 

It is not difficult to show that the emissivity of small spherical particles, composed of both insulating and metallic crystalline solids, is expected to vary as 1/A2 in the far infrared. For example, if the low-frequency limit of the dielectric function for a single Lorentz oscillator (9.16) is combined with (5.11), the resulting emissivity is... 

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... 

From Eq, (1) it is clear that a model of crystal polarization that is adequate for the description of the piezoelectric and pyroelectric properties of the P-phase of PVDF must include an accurate description of both the dipole moment of the repeat unit and the unit cell volume as functions of temperature and applied mechanical stress or strain. The dipole moment of the repeat unit includes contributions from the intrinsic polarity of chemical bonds (primarily carbon-fluorine) owing to differences in electron affinity, induced dipole moments owing to atomic and electronic polarizability, and attenuation owing to the thermal oscillations of the dipole. Previous modeling efforts have emphasized the importance of one more of these effects electronic polarizability based on continuum dielectric theory" or Lorentz field sums of dipole lattices" static, atomic level modeling of the intrinsic bond polarity" atomic level modeling of bond polarity and electronic and atomic polarizability in the absence of thermal motion. " The unit cell volume is responsive to the effects of temperature and stress and therefore requires a model based on an expression of the free energy of the crystal. 

This amplitude is found from the equation of motion of a harmonic oscillator affected by an a.c. field E(t). This approach yields the Lorentz and Van Vleck-Weisskopf lines, respectively, for a homogeneous and Boltzmann distributions of the initial a.c. displacements x(l0) established after instant to of a strong collision. The susceptibility corresponding to the Van Vleck-Weisskopf line in terms of our parameters is given by [66]... 

On the other hand, the application of a static or slowly varying electric field will be able to displace ions and electrons away from their equilibrium positions and, as a consequence, the polarizability of the electrons will be modified. In the description of H. A. Lorentz s electronic oscillators, the small shifts in the ionic positions modify the spring constants and restoring forces of the electronic oscillators. 

Some applications of the method will be shown for the ternary salt trimethylammonium (TMA)-TCNQ-I. Tanner et al. [43] have obtained a best fit with a Drude-Lorentz dielectric function with a>p = 5290 cm-1, -y = 2560 cm"1, ec = 2.65, , is the frequency and T is the relaxation rate of the second oscillator. The average value of the frequency-dependent conductivity below 200 cm"1 should be 19 IS- cm"1. The energy gap of 0.10 to 0.14 eV, the effective quantity of electrons participating in the electric transport Ncfi = 0.67 per molecule, and the effective mass of the carriers m = 5.2 m were found from the plasma frequency. 

Fig. 3.19 Variation in the real and imaginary parts of the refractive index near resonance for a Lorentz oscillator.

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