Dielectric Lorentz model

We shall call the frequency at which t = —2em and t" — 0 the Frohlich frequency coF the corresponding normal mode—the mode of uniform polarization—is sometimes called the Frohlich mode. In his excellent book on dielectrics, Frohlich (1949) obtained an expression for the frequency of polarization oscillation due to lattice vibrations in small dielectric crystals. His expression, based on a one-oscillator Lorentz model, is similar to (12.20). The frequency that Frohlich derived occurs where t = —2tm. Although he did not explicitly point out this condition, the frequency at which (12.6) is satisfied has generally become known as the Frohlich frequency. The oscillation mode associated with it, which is in fact the lowest-order surface mode, has likewise become known as the Frohlich mode. Whether or not Frohlich s name should be attached to these quantities could be debated we shall not do so, however. It is sufficient for us to have convenient labels without worrying about completely justifying them. 

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... 

An increase in fractional free volume will reduce the number of polarisable groups per unit volume, and thereby reduce the relative permittivity of the polymer. Quantitatively, the effect may be estimated by means of the Clausius-Mossotti/Lorenz-Lorentz model for dielectric mixing (Bottcher, 1978) ... 

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 a well-known result that,for a dielectric molecular material, the dielectric function can be written using a Lorentz model ... 

Lorentz Model this model treats electrons as bound strongly to an atom, subject to dissipative, inertial, electromagnetic, as well as restoring forces. In this model, the dielectric function s(o)) is given by s(w) =... 

Fig. 7.4 Lorentz model for the local field. Polarization of an ellipsoidal form dielectric sample and appearance of depolarizing field Ei (a), Lorentz cavity field E2 and the field of individual molecules within the cavity E3 (b)...

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. 

Fig. 7. Model calculations for the reflectivity (a) and the optical conductivity (b) for a simple (bulk) Drude metal and an effective medium of small metallic spherical particles in a dielectric host within the MG approach. The (bulk) Drude and the metallic particles are defined by the same parameters set the plasma frequency = 2 eV, the scattering rate hr = 0.2 eV. A filling factor/ = 0.5 and a dielectric host-medium represented by a Lorentz harmonic oscillator with mode strength fttOy, 1 = 10 eV, damping ftF] = I eV and resonance frequency h(H = 15 eV were considered for the calculations.

Continuum models have a long and honorable tradition in solvation modeling they ultimately have their roots in the classical formulas of Mossotti (1850), Clausius (1879), Lorentz (1880), and Lorenz (1881), based on the polarization fields in condensed media [32, 57], Chemical thermodynamics is based on free energies [58], and the modem theory of free energies in solution is traceable to Bom s derivation (1920) of the electrostatic free energy of insertion of a monatomic ion in a continuum dielectric [59], and Kirkwood and Onsager s... 

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... 

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. 

In order to formulate a theory for the evaluation of vibrational intensities within the framework of continuum solvation models, it is necessary to consider that formally the radiation electric field (static, Eloc and optical E[jc) acting on the molecule in the cavity differ from the corresponding Maxwell fields in the medium, E and Em. However, the response of the molecule to the external perturbation depends on the field locally acting on it. This problem, usually referred to as the local field effect, is normally solved by resorting to the Onsager-Lorentz theory of dielectric polarization [21,44], In such an approach the macroscopic quantities are related to the microscopic electric response of... 

For vapors, the local field vanishes because of the spherical symmetry and eqs A.5 and A.6 provide good agreement with experiment. However, for liquids one can no longer use eq A.6 for the polarizability in the Lorentz— Debye model. Indeed, for liquid water, eq A.5 diverges for values of y about 4 times smaller than the value of y for its vapor, which at 300 K is y = ye 4 yd = 32.3 x 10 40 C2 m2 J "1. One can regard eq A.5 as the definition of the molecular polarizability and calculate y in terms of the macroscopic dielectric constant e. The lower value of the polarizability in liquid than in vapor can be explained in the framework of the Lorentz—Debye model by the hindered rotation of the permanent dipole moment by the neighboring molecules in the condensed state. 

Empirical models for the induced trace have also been obtained from (nonspectroscopic) measurements of the second virial dielectric coefficient of the Clausius-Mosotti and Lorentz-Lorenz expansions [30]. Excellent surveys with numerous references to the historical as well as the modern dielectric research activities were given by Buckingham [27], Kielich [89], and Sutter [143] in 1972 see also a recent review with a somewhat more spectroscopic emphasis [11]. 

For larger particle concentrations, the interactions between particles influence the electromagnetic properties. For interparticle distances much smaller than the wavelength, the Maxwell-Gamett model applies and leads to the Lorentz-Lorenz relation for the effective dielectric constant eff of the composite medium, which takes the form... 

One of the presented structures is a monodispersion of subwavelength inclusions i (spheres) in dielectric host h. Fig. 2.22a. The other is polydispersion. Fig. 2.22b. The first situation can be described by the well-known MaxweU-Gamett model [171], the oldest effective medium model, obtained by the use of Clausius-Mossotti/ Lorenz-Lorentz equation. The other case is polydispersion, described by the implicit Bruggeman expression [172, 173]. 

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