Lorentz electron model

An almost forgotten issue is the proposed relativistic nature of an electron as elucidated by Lorentz. The electron was seen as a flexible spherical unit of charge which distorts as it contracts in the direction of any motion. To account for the relativistic contraction of macroscopic bodies Lorentz further assumed that the electrical forces which bind atoms together were essentially states of stress and strain in the aether. Countless prominent scientists have expressed similar views without trying to develop a coherent theory of matter. The Lorentz electron model antedates de Broglie s postulate of matter waves and the development... 

Appendix A The Lorentz Condition Appendix B Electron Model of Present Theory B.l. General Equations of the Equilibrium State The Charged-Particle State... 

This result may be expressed in the more customary units of cubic meters or cubic nanometers by dividing by dTiSp. Thus, Op is equal to 1.472 x 10 nm at 25°C. When the calculation is repeated at 50°C the result is p = 1.471 x 10 nm. One expects the polarizability to be independent of temperature in a range where the electrons in the molecule remain in the same molecular orbitals. The small change in the polarizability reflects the weakness of the Lorentz-Lorenz model, which is based on continuum concepts. However, the estimated change is small, so that one may assume that the model is reasonably good. 

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... 

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... 

Simple Spectral Method [23] In the simple spectral method, a model dielectric response function is used. It combines a Debye relaxation term to describe the response at microwave frequencies with a sum of terms of classical form of Lorentz electron dispersion (corresponding to a damped harmonic oscillator model) for the frequencies from IR to UV ... 

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... 

Third, a further simplification of the Boltzmann equation is the use of the two-term spherical harmonic expansion [231 ] for the EEDF (also known as the Lorentz approximation), both in the calculations and in the analysis in the literature of experimental data. This two-term approximation has also been used by Kurachi and Nakamura [212] to determine the cross section for vibrational excitation of SiHj (see Table II). Due to the magnitude of the vibrational cross section at certain electron energies relative to the elastic cross sections and the steep dependence of the vibrational cross section, the use of this two-term approximation is of variable accuracy [240]. A Monte Carlo calculation is in principle more accurate, because in such a model the spatial and temporal behavior of the EEDF can be included. However, a Monte Carlo calculation has its own problems, such as the large computational effort needed to reduce statistical fluctuations. 

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. 

We will now analyze the general optical behavior of a metal using the simple Lorentz model developed in the previous section. Assuming that the restoring force on the valence electrons is equal to zero, these electrons become free and we can consider that Drude model, which was proposed by R Drude in 1900. We will see how this model successfully explains a number of important optical properties, such as the fact that metals are excellent reflectors in the visible while they become transparent in the ultraviolet. 

Finally, a special type of primary bond known as a metallic bond is found in an assembly of homonuclear atoms, such as copper or sodium. Here the bonding electrons become decentralized and are shared by the core of positive nuclei. Metallic bonds occur when elements of low electronegativity (usually found in the lower left region of the periodic table) bond with each other to form a class of materials we call metals. Metals tend to have common characteristics such as ductility, luster, and high thermal and electrical conductivity. All of these characteristics can to some degree be accounted for by the nature of the metallic bond. The model of a metallic bond, first proposed by Lorentz, consists of an assembly of positively charged ion cores surrounded by free electrons or an electron gas. We will see later on, when we... 

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... 

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... 

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. 

To describe the optical properties of the crystals over a wide temperature range and to derive the basic phenomenological parameters of the electronic transition, we fitted the R(co) and a(co) spectra using the Drude-Lorentz model. The Drude part describes the intraband transition of free carriers,... 

Abstract. Muonium is a hydrogen-like system which in many respects may be viewed as an ideal atom. Due to the close confinement of the bound state of the two pointlike leptons it can serve as a test object for Quantum Electrodynamics. The nature of the muon as a heavy copy of the electron can be verified. Furthermore, searches for additional, yet unknown interactions between leptons can be carried out. Recently completed experimental projects cover the ground state hyperfine structure, the ls-2s energy interval, a search for spontaneous conversion of muonium into antimuonium and a test of CPT and Lorentz invariance. Precision experiments allow the extraction of accurate values for the electromagnetic fine structure constant, the muon magnetic moment and the muon mass. Most stringent limits on speculative models beyond the standard theory have been set. 

Hall Effect in the Drude Model. The Drude treatment of the Hall effect starts from the Lorentz force on an electron ... 

Theory of the dielectric function. The discussion of absorption properties of astrophysically relevant solids is frequently based on the classical Lorentz model for dielectric materials. This assumes that the electrons and ions forming the solid matter are located at fixed equilibrium positions in the solid, determined by internal forces. An applied electromagnetic field shifts the charged particles, labeled by... 

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