Electric polarization waves

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. 

H. Frohlich has introduced, in a framework of far-from-equilibrium processes, coherent electric polarization waves as the physical agent able to control the working of distant and separate parts of the system, making them cooperative. On the other hand, it has been shown that biomolecules are able to host on their own chains, in a conservative framework, highly nonlinear subdynamics which give rise to deep structural and conformational changes. 

Another type of mobility is that of electrons in the solid phase this is partly described by the continuum electrostatic approach for dielectrics and conductors, but for metals (the most important conductors) it is better to resort to the jellium model. A QM treatment of jellium models permits us to describe electric polarization waves (polarons, solitons) and their mutual interplay with the liquid across the surface. 

If the plane of polarization is defined as xz, the electric vector can be represented in terms of the unit vector i in the direction of x as E = E0i cos 0. This plane polarized wave may be decomposed into two circularly polarized components propagating in the same direction. A simple description of these waves are given by... 

Suppose that a plane jc-polarized wave is incident on a homogeneous, isotropic sphere of radius a (Fig. 4.1). As we showed in the preceding section, the incident electric field may be expanded in an infinite series of vector spherical harmonics. The corresponding incident magnetic field is obtained from the curl of (4.37) ... 

Consider now the field scattered by an isotropic, optically active sphere of radius a, which is embedded in a nonactive medium with wave number k and illuminated by an x-polarized wave. Most of the groundwork for the solution to this problem has been laid in Chapter 4, where the expansions (4.37) and (4.38) of the incident electric and magnetic fields are given. Equation (8.11) requires that the expansion functions for Q be of the form M N therefore, the vector spherical harmonics expansions of the fields inside the sphere are... 

Let the coordinate system be such as that given in Figure 4. IS. The electric vectors of a plane polarized radiation vibrate along OZ in the ZX plane and OX is the direction of propagation of the plane polarized wave. When a solution of anisotropic molecules is exposed to this plane polarized radiation, the electric vector will find the solute molecules in random orientation. Only those molecules absorb with maximum probability which have their transition moment oriented parallel to OZ (photoselection). Those molecules which are oriented by an angle 6 to this direction will have their absorption probability reduced by a factor cos 6, and the intensity of absorption by cos2 6. Finally, the molecules oriented perpendicular to the electric vector will not absorb at all. These statements are direct consequences of directional nature of light absorption... 

The expectation value of the property A at the space-time point (r, t) depends in general on the perturbing force F at all earlier times t — t and at all other points r in the system. This dependence springs from the fact that it takes the system a certain time to respond to the perturbation that is, there can be a time lag between the imposition of the perturbation and the response of the system. The spatial dependence arises from the fact that if a force is applied at one point of the system it will induce certain properties at this point which will perturb other parts of the system. For example, when a molecule is excited by a weak field its dipole moment may change, thereby changing the electrical polarization at other points in the system. Another simple example of these nonlocal changes is that of a neutron which when introduced into a system produces a density fluctuation. This density fluctuation propagates to other points in the medium in the form of sound waves. 

This is a monochromatic linearly polarized wave with electric field vibrating in the x—z plane, and the magnetic field vibrating in the y—z plane. We have adopted the practice of explicitly identifying the plane of vibration [62, p. 29]. Current density associated with E and B is given by Eq. (16) as J = 0. It is stressed that J = 0 is obtained here from the fields, whereas the conventional approach is to assume the current to be zero on the grounds that pe = 0. 

This example is a nonplanar linearly polarized wave. The direction of vibration of the electric field is still along the x axis, while the magnetic field and the Poynting vector are both contained on the y-z plane. The instantaneous direction of magnetic field is along angle 0 given by... 

Figure 1. (a) Plots of the electric field of the applied light wave (solid) and the induced polarization wave (dotted) as a function of time for a linear material (b) cartoon depicting the polarization of the material as a function of time (c) plot of induced polarization vs. applied field. 

Just as linear polarization leads to linear optical effects, such as refractive index and birefringence, nonlinear polarization leads to other and usually more subtle (nonlinear) effects. It is precisely these effects we hope to understand and exploit. In Figure 14, application of a symmetric field (i.e., the electric field associated with the light wave) to the anharmonic potential leads to an asymmetric polarization response. This polarization wave shows diminished maxima in one direction and accentuated... 

In Equations 6.1 and 6.2, sp is the electric permittivity and /10 is the magnetic permeability. The solutions of Equations 6.1 and 6.2 are vectors vibrating in coordinate planes perpendicular to each other (Figure 6.1). E oscillates in the xz plane and H in the yz plane when a linearly polarized wave is propagating in the positive direction of the z-axis. 

A linearly vibrating vector can be considered as made up of two equal vectors that rotate at the same rate in opposite sense, as shown schematically in Figure 6.3. The resultant electric vector for two circularly polarized waves with 9r = 6t is of the form E = E0 cos 9. More generally, in complex... 

Table G Definitions of the Electric Field E, the (Di)electric Polarization P, the Electric Displacement D, the Magnetic Field H, the Magnetization M, the Magnetic induction or flux density B, statement of the Maxwell equations, and of the Lorentz Force Equation in Various Systems of Units rat. = rationalized (no 477-), unrat. = the explicit factor 477- is used in the definition of dielectric polarization and magnetization c = speed of light) (using SI values for e, me, h, c) [J.D. Jackson, Classical Electrodynamics, 3rd edition, Wiley, New York, 1999.]. For Hartree atomic u nits of mag netism, two conventions exist (1) the "Gauss" or wave convention, which requires that E and H have the same magnitude for electromagnetic waves in vacuo (2) the Lorentz convention, which derives the magnetic field from the Lorentz force equation the ratio between these two sets of units is the Sommerfeld fine-structure constant a = 1/137.0359895...

An applied electric field can be the electric held component of an electromagnetic wave, in which case electronic excitations or other optical responses may ensue. These are the topic of the next chapter. Here, the concern is with electrostatics, specihcally, the dielectric, or insulative, properties of materials. In an electrical conductor, an applied electric held, E, produces an electric current - ions, in the case of an ionic conductor, or electrons, in the case of an electronic conductor. Electrical conductivity has already been examined in earlier chapters. In insulating solids, the topic of the current discussion, the response to an applied electric held is a static spatial displacement of the bound ions or electrons, resulting in an electrical polarization, P, or net dipole moment (charge separahon) per unit volume, which is a vector quantity. In a homogeneous linear and isotropic medium, the polarization and electric held are aligned. In an anisotropic medium, this need not be so. The fth component of the polarization is related to the jth component of the electric held by ... 

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