Frohlich dipoles

The average contribution of the Frohlich dipoles to the total moment is, analogous to Eq. (3.72), given by... 

At still lower energies, the loss mechanism is the interaction of the electron with the permanent dipoles of the water molecule. Frohlich and Platzman (1953) estimated a constant time rate of energy loss due to this effect at -1013 eV/s. The stopping power in eV/A is then approximately given by (1.7 x 10 3)E 1/2, where the energy E is in eV... 

The solvation time t is considerably shorter than td for many solvents. For example for water ex = 4.84, 0 = 79.2 and tD = 8.72 ps [33]. Thus in water t, = 0.59 ps. Why is the time scale for solvation of a dipole so much shorter than td Why are there apparently two characteristic times (ti and rD) for a dielectric medium Friedman [55] suggested two simple thought experiments to resolve the paradox of two times. The relevant theory of dielectrics was described in the 1940s by Frohlich [89],... 

P. J. W. Debye, Polar Molecules (Dover, New York, reprint of 1929 edition) presents the fundamental theory with stunning clarity. See also, e.g., H. Frohlich, "Theory of dielectrics Dielectric constant and dielectric loss," in Monographs on the Physics and Chemistry of Materials Series, 2nd ed. (Clarendon, Oxford University Press, Oxford, June 1987). Here I have taken the zero-frequency response and multiplied it by the frequency dependence of the simplest dipolar relaxation. I have also put a> = if and taken the sign to follow the convention for poles consistent with the form of derivation of the general Lifshitz formula. This last detail is of no practical importance because in the summation Jf over frequencies fn only the first, n = 0, term counts. The relaxation time r is such that permanent-dipole response is dead by fi anyway. The permanent-dipole response is derived in many standard texts. 

Inside the sphere where the interactions take place, the use of statistical mechanics is required. To represent a dielectric with dielectric constant , consisting of polarizable molecules with a permanent dipole moment, Frohlich [6] introduced a continuum with dielectric constant s X, in which point dipoles with a moment id are embedded. In this model, id has the same nonelectrostatic interactions with the other point dipoles as the molecule had, while the polarizability of the molecules can be imagined to be smeared out to form a continuum with dielectric constant 00 [7]. 

Kirkwood took a more rigorous statistical-mechanical approach in an attempt to incorporate the effect of local ordering. His theory is only valid for rigid dipoles, and it was left to Frohlich to extend the treatment properly to a system of polarisable dipolar molecules. The work is well described in Frohlich s classic text (1949). The final outcome was the following formula,... 

However, the most recent discussions favour these high values of g although values of the order of 20% lower had ori nally been favoured. This is because the Frohlich equation [equation (1)] differs from the earlier version of Kirkwood, and treats the inner field in a more nearly correct manner. It is no longer necessary to make a calculation of the HjO dipole moment in its surroundings in the liquid, as had been necessary in the application of the Kirkwood equation. The dipole moment of the free molecule, /i = 1.84D, is used in equation (1), together with = 1.80 at 293 K. This leads to a value of = 2.82, which is sufficiently close to that calculated from the computer dynamics model to warrant optimism for future calculations. The exact choice of will continue to present difficulties until the far-i.r. data are complete over a wide range of temperature. 

The generalization of the work, by Kirkwood and by Frohlich, on the relation between the mean-square fluctuation in dipole moment, , of a dielectric sphere and its zero-firequency polarizability, o4,j,(0), namely ... 

The static permittivities of mixtures of organic liquids with water may be interpreted in terms of polarization theory. We shall see that water can be used as a solvent for the determination of dipole moments of highly polar molecules, but for less polar molecules the Kirkwood-Frohlich equation has been used as a method of demonstrating molecular interactions between water and solute. 

We shall now almost exclusively concentrate on the fractal time random walk excluding inertial effects and the discrete orientation model of dielectric relaxation. We shall demonstrate how in the diffusion limit this walk will yield a fractional generalization of the Debye-Frohlich model. Just as in the conventional Debye relaxation, a fractional generalization of the Debye-Frohlich model may be derived from a number of very different models of the relaxation process (compare the approach of Refs. 22, 23, 28 and 34—36). The advantage of using an approach based on a kinetic equation such as the fractional Fokker-Planck equation (FFPE) however is that such a method may easily be extended to include the effects of the inertia of the dipoles, external potentials, and so on. Moreover, the FFPE (by use of a theorem of operational calculus generalized to fractional exponents and continued fraction methods) clearly indicates how many existing results of the classical theory of the Brownian motion may be extended to include fractional dynamics. 

Anomalous rotational diffusion in a potential may be treated by using the fractional equivalent of the diffusion equation in a potential [7], This diffusion equation allows one to include explicitly in Frohlich s model as generalized to fractional dynamics (i) the influence of the dissipative coupling to the heat bath on the Arrhenius (overbarrier) process and (ii) the influence of the fast (high-frequency) intrawell relaxation modes on the relaxation process. The fractional translational diffusion in a potential is discussed in detail in Refs. 7 and 31. Here, just as the fractional translational diffusion treated in Refs. 7 and 31, we consider fractional rotational subdiffusion (0rotation about fixed axis in a potential Vo(< >)- We suppose that a uniform field Fi (having been applied to the assembly of dipoles at a time t = oo so that equilibrium conditions prevail by the time t = 0) is switched off at t = 0. In addition, we suppose that the field is weak (i.e., pFj linear response condition). 

Further developments in the theory of the structure of polar liquids included estimates of the correlation of a given dipole to its neighbors. Important contributions were made in this direction by Kirkwood [22] and Frohlich [23]. In Kirkwood s model, the field Ej is calculated by considering all possible orientations of surrounding dipoles in a spherical cavity for a fixed orientation of the central dipole. By averaging over these orientations, Kirkwood obtained an improved estimate of the polarization of the medium. For the case of non-polarizable dipoles the result is... 

Onsager s equation has been used for slightly polar solvents such as toluene. With strongly polar solvents, chloroform for instance, Kirkwood s or Frohlich s theories must be resorted to, and no value of the dipole moment can be obtained unless the correlation factor g is known by independent data concerning the structure of the solution. 

The influence of temperature on the mean dipole moment of polybutyl methacrylate dissolved in carbon tetrachloride is shown in Table II. The average moments were calculated from Frohlich s equation (see Section VI) taking unity as the most probable value of the correlation factor. Since the Onsager theory makes use of the refractive index of the solute, for which only approximate values can be found, results obtained by Onsager s and FrOhlich s theories for solutions are not identical even in a nonpolar solvent like carbon tetrachloride. The moments given in Table II are not comparable to those given in Table I, especially as they are not extrapolated to infinite dilution. 

An analysis of the previously defined function B(T) for dilute solutions of polymers in polar solvents may be helpful for the understanding of the interaction between polymer and these solvents.33 Applying Frohlich s theory, in which deformation polarization is treated macroscopically, we consider the solution as a cpntinuous medium containing polar units. The dielectric constant of the continuous medium is taken as equal to the square of the refractive index of the solution n0. This value is very close to that of the solvent. Each polar unit is represented as a sphere of dielectric constant 0a, haying a point dipole located at its center. It must be stressed that polar units may be either whole... 

These difficulties have been avoided by Frohlich whose Reasoning is very similar to Kirkwood s but who has chosen his model in such a manner that he need consider no boundary effect. He has treated the deformation polarization as a macroscopic phenomenon. Molecules are replaced by a set of nondeformable point dipoles, having a moment p and placed in a continuous medirnn of dielectric constant= refractive index), accounting for deformation effects. The moment of a spherical molecule is given by... 

This very general result is due to Frohlich and may be specialized, as we shall see presently, to give other well-known expressions. Before we do this, however, let us look briefly at the quantity m. If there is no interaction between molecular dipoles then m = m and indeed this can also be shown to hold if each molecule can be treated as a point dipole or as a uniformly polarized sphere. The deviation of [Pg.204]

Here, p and v denote the magnitude of molecular dipole and the number density of those dipoles, respectively is the Boltzmann constant and T is the absolute temperature, f is a correction factor for a difference between the applied and internal electric fields (f = (Sq + 2) /9 in the Onsager form for nonpolar low-M molecules), and g is the Kirkwood-Frohlich factor that represents the magnitude of the motional correlation of the dipoles (i.e., of the dipole-carrying molecules). 

According to Frohlich, a pure condensed dielectric consisting of polarizable molecules with a permanent dipole moment p may be formally represented by a continuum permittivity accounting for the molecular polarizability, embedded in the bulk continuum with the effective permittivity 8. The fundamental polarization equation for such a polar dielectrics is... 

Note that the induced part is formally separated from the permanent dipole moment contribution in a particular manner. In Frohlich s version of the Onsager model the spherical dipoles have effective dipole moments... 

Chemical equilibria such as ion-pair formation of electrolytes in aqueous solution where the hydrated ion pairs behave as polarizable dipoles, may be treated in terms of the Frohlich formalism. 

ABSTRACT We present a dynamical scheme for biological systems. We use methods and techniques of quantum field theory since our analysis is at a microscopic molecular level. Davydov solitons on biomolecular chains and coherent electric dipole waves are described as collective dynamical modes. Electric polarization waves predicted by Frohlich are identified with the Goldstone massless modes of the theory with spontaneous breakdown of the dipole-rotational symmetry. Self-organization, dissipativity, and stability of biological systems appear as observable manifestations of the microscopic quantum dynamics. 

---