Lorentz oscillators

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 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 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 ... 

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... 

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 ... 

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 ... 

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,... 

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... 

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

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 ... 

To introduce changes in the dielectric constant related to phonon modes in compound crystals, it is relevant to consider the classical interaction between an atomic system with resonant frequency uyi and an electromagnetic field E = Eoexp [iu>t]. The 1-D equation of motion for such a system, also known as a Lorentz oscillator, is ... 

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 stochastic Bloch equation is a semiphenomenological equation with some elements of quantum mechanics in it. To understand better whether our results are quantum mechanical in origin, we analyze a classical model. Lorentz invented the theory of classical, linear interaction of light with matter. Here, we investigate a stochastic Lorentz oscillator model. We follow Allen and Eberley [108] who considered the deterministic model in detail. The classical model is also helpful because its physical interpretation is clear. We show that for weak laser intensity, the stochastic Bloch equations are equivalent to classical Lorentz approach. 

Method A uses bulk ATR spectra of thick polycrystalline samples of PEG with a random orientation of helical coils Using the SpectraRay 2 software package (SENTECH Instruments GmbH, Berlin, Germany), Drude-Lorentz oscillator parameters [15] were determined by spectral line fits to selected vibrational bands of an ATR spectrum, taken from... 

Fig. 2 Bold line FTIR-spectrum of a thick layer of polycrystalline PEG symbols are representing the TDM orientations of the corresponding bands. Faint line Calculated spectrum between 1200 and 1400 cm consisting of five Drude-Lorentz oscillators with parameters given in Table 1 (spectrum is offset by + 0.5 absorbance units)...

Fig. 134. Near-nonnal incidence reflectivity of UPtj at 5K. and 300 K. The circles correspond to the computed reflectivity from the fit with Lorentz oscillators. (After Marabelli et al. 1986a.)...

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... 

Another model for optical functions of amorphous materials is Tauc-Lorentz (T-L) model which is first proposed by G. E. JeUison et al. in 19% [68]. The parameterization is obtained as a combination of Tauc expression and Lorentz oscillator model for 2 of a collection of non-interacting atoms, and 2(8) is given by... 

GITT galvanostatic intermittent titration LO Lorentz oscillator... 

---