Clausius-Mossotti function

It is well known that for atomic gases at low densities the Clausius-Mossotti function can be related to the atomic polarizability via the following virial relation ... 

Vidal D, Lallemand M (1976) Evolution of the Clausius-Mossotti function of noble gases and nitrogen, at moderate and high density, near room temperature. J Chem Phys 64 4293-4302... 

The Clausius-Mossotti function or total polarization, of a gas can be expanded in inverse powers of the molar volume, V ... 

It is seen that the effect of the variation of the polarizability is of the same order of magnitude as the effect of statistical fluctuations in the dipole moments / described by the functions S2. For the highest frequencies which may be considered within the approximation introduced into the calculation, the correction to the Lorentz-Lorenz function is about 15 per cent larger than the corresponding correction for the static case, (i.e., to the Clausius-Mossotti function). A similar qualitative behavior may be expected for other noble gases under the same conditions. 

Experimentally Uhlig, Kirkwood, and Keyes30 observed an increase of the Clausius-Mossotti function for compressed C02 up to 200 atm. The experiments for the same gas by Michels and Michels81 showed that at higher densities (corresponding to pressures of about 1000 atm.), the Clausius-Mossotti function... 

Note that this interaction energy is even more favorable since (1-fa)results presented here are discussed by Bottcher (11). The dielectric constant of a fluid may be represented by the Clausius-Mossotti function (12-13). ... 

The vector function/i(e- , e ) appearing above is a generalized Clausius-Mossotti function. Combining equations (6.189), (6.202), and (6.203), we arrive at... 

Capacitance. The determination of density through capacitance measurements depends upon the fact that the dielectric constant and density are related for simple fluids, such as hydrogen, by the well-known Clausius-Mossotti function... 

A particle is subdivided into a small number of identical elements, perhaps 100 or more, each of which contains many atoms but is still sufficiently small to be represented as a dipole oscillator. These elements are arranged on a cubic lattice and their polarizability is such that when inserted into the Clausius-Mossotti relation the bulk dielectric function of the particle material is obtained. The vector amplitude of the field scattered by each dipole oscillator, driven by the incident field and that of all the other oscillators, is determined iteratively. The total scattered field, from which cross sections and scattering diagrams can be calculated, is the sum of all these dipolar fields. 

At low atomic densities N, the dielectric response of a medium can be written in the popular Clausius-Mossotti or Lorentz-Lorenz form as a function of N and a coefficient a that includes atomic or molecular polarizability ... 

The dipole moment of a selected functional group in the polymer can also be calculated using the Clausius-Mossotti equation. According to Van Krevelan, (15) the "effective polarizability" of a functional group in the polymer is calculated from the measured dielectric constant, the polymer density and the number of moles of that group in the polymer repeat unit. Using the data for the six polyamide-imides listed in Table I, the effective polarizability of the amide group can be determined from... 

Figure 1. Plot of the variation of the real and imaginary parts of the Clausius-Mossotti as a function of frequency.

In practice, it is difficult to measure the DEP force due to the effects of Brownian motion and electrical field-induced fluid flow [3]. Instead, the DEP crossover frequency can be measured as a function of medium conductivity and provides sufficient information to determine the dielectric properties of the suspended particles. The DEP crossover frequency,is the transition frequency point where the DEP force switches from pDEP to nDEP or vice versa. According to Eq. (6), the crossover frequency is defined to be the frequency point where the real part of the Clausius-Mossotti factor equals zero ... 

Experimental data for ej of mercury at the constant photon energy of 1.27 eV are shown in Fig. 4.4 as a function of density. In the lower density region where mercury is most highly insulating, ej is only weakly dependent on temperature and follows roughly the Clausius-Mossotti model of the dielectric constant due to induced dipoles (Herzfeld, 1927). 

This is the Langevin equation which describes the degree of polarization in a sample when an electric field, E, is applied at temperature T. Experimentally, a poling temperature in the vicinity of Tg is used to maximize dipole motion. The maximum electric field which may be applied, typically 100 MV/m, is determined by the dielectric breakdown strength of the polymer. For amorphous polymers p E / kT 1, which places these systems well within the linear region of the Langevin function. The following linear equation for the remanent polarization results when the Clausius Mossotti equation is used to relate the dielectric constant to the dipole moment 41). 

Application of high pressure changes the position of the electronic conduction level, Vq (see Section 6.9), and it increases the dielectric constant of the liquid. Both effects have an influence on the ionization energy of a solute. The dependence of Vo(p) is complicated and experimental data must be used. The effect of pressure on the dielectric constant is due to the increase in density and it is well described by the Clausius-Mossotti equation (see Section 1.6). In Figure 8a the photoconductivity spectrum of TMPD in neohexane is shown as a function of pressure. The variation of the photoconductivity threshold with pressure is depicted in Figure 8b. Evaluation of the data by means of Bom s formula (Chapter 7, Equation 94) led to the hypothesis that an additional increase of liquid density around the solute molecule due to fluctuations is responsible for the observed shifts (Katoh et al., 1995). 

Equation (1.193) represents the complex polarizability of a small sphere of sub-wavelength diameter in the quasi-static approximation. Note that it shows the same functional form as the Clausius-Mossotti polarizability [1]. 

Semiconductor cluster polarizabilities have been the subject of some very important experimental studies via beam-deflection techniques (Backer 1997 Schlecht et al. 1995 Schnell et al. 2003 Schafer et al. 1996 Kim et al. 2005) while they have been extensively studied using quantum chemical and density functional theory. In this research realm, one of the areas intensively discussed is the evolution of the cluster s polarizabilities per atom (PPA) with the cluster size. The PPA is obtained by dividing the mean polarizability of a given system by the number of its atoms. Such property offers a straightforward tool to compare the microscopic polarizability of a given cluster with the polarizability of the bulk (see O Fig. 20-16) as the latter is obtained by the hard sphere model with the bulk dielectric constant via the Clausius-Mossotti relation ... 

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