Dielectric ellipsoid

Dielectric Ellipsoid. The general formulae (328) and (336) permit the calculation of electrostrictive and electrocaloric changes in electric permittivity of a dielectric of arbitrary geometrical shape. To this aim, one has only to resort to the relation between the macroscopic field E existing in the anisotropic dielectric and the electric field in vacuum ... 

Fig. 6. The main features of these theories are seen by considering a single dielectric ellipsoid with a laser excitation field l( l) directed along the principal axis of the ellipsoid. Then the field inside the particle is uniform and given by...

An important fact that emerged from this study is that the observed features were best reproduced when the local dielectric ellipsoid was taken to be a prolate spheroid, with the principal axis Oc parallel to z and e = e—(x. Thus the assumption that is generally made that the local dielectric ellipsoid is uniaxial would appear to be valid to a very good approximation as far as optical calculations are concerned (see, however, 4.10). 

The SmA phase has the same symmetry and the same dielectric ellipsoid as in nematics, therefore, everything said above about the birefringence and dichroism is valid for the SmA phase. However, due to specific elastic properties of the layered structure, the director fluctuations are strongly quenched, and the SmA preparations are much more transparent than the nematic ones. This is related to specific elastic properties of the lamellar SmA phase [14]. 

Due to the linear profile of 9(z) it is very easy to calculate the phase retardation of the initially homeotropic cell for the normal light incidence, kHz. Without electric field, the longest axis of the dielectric ellipsoid coincides with the director axis z. Therefore, refraction index for any polarization is o = With increasing field E, due to deflection of the director within plane xz, the y- and -components of the refraction index will correspond to the ordinary and extraordinary rays, Uy = no = n , rix(z) = rig(z). Integration provides us with the average extraordinary index ... 

As a result of numerical calculations [30], five phases shown in Fig. 13.21 have been found. In the first two rows we find the symbols and types of the phases whereas the third colunrn represents the corresponding unit cells for the first four phases in terms of smectic layer numbers (ni) per one period of the structure. The SmC o phase is incommensurate in the sense that it has a short-pitch helical structure with a period not coinciding with integer number of the smectic layers. In the fourth colunrn, a top view of the dielectric ellipsoid is presented for different layers within each unit cell. All these phases are in agreement with sequence... 

Fig. 13.21 Classification and structure of ferro-, ferri and antiferroelectiic phases. The third column represents the number (m) of the smectic layers / in a unit cell (for SmC abbreviation IC means incommensurate). In the right column the orientation of the dielectric ellipsoid is presented for different layers within the unit cell viewed along the z-axis. The long-pitch helical structure due to the molecular chirality is igntued for clarity, although it slightly influences the value of angle (p for the ellipsoids in the xy plane for each structure, see the next figure...

Figure 13.21 presents the picture of the dielectric ellipsoid orientation within each unit cell that is at the nanometer scale. The weak molecular chirality results in additional weak twisting of all structures with characteristic pitch of about Po 0.1-1 pm. An example of a such twisted structure is shown in Fig. 13.22 it is an antiferroelectric double-layer cell describing two geared helices upon rotation about z-axis. The helices are shifted in phase by (p = 7t and have the same handedness. On the molecular scale, due to molecular chirality, the c-director turns from layer to layer by a small angle 5cp = 2nllPo, therefore, for / Inm,... 

It can also be expressed graphically using a dielectric ellipsoid that represents the magnitude of the dielectric constant in any direction. The ellipse has three principal axes equal to the eigenvalues of and the effective dielectric constant is simply the radius of the ellipse along the electric field direction. This ellipsoid... 

In this model, dielectric ellipsoids spiral about the z axis with pitch 27r/j3, but the period of the periodic z) matrix is I = ttIP (see Fig. 2). If the ellipsoids were tilted, would have no zeros and would contain sin (Pz) and cos (Pz) terms, so that the period would be I = 2ttIP, We found solutions for this case. Additional strong Bragg reflection bands corresponding to the longer fundamental period appear in this case. These additional bands were not observed... 

Figure 2. Spiraling dielectric ellipsoids in Oseen s optical model of a cholesteric liquid crystal.

In [56, 58] the researchers investigated the biaxiality of the FLC dielectric tensor (Fig. 7.9). Indeed, if we suppose that the dielectric ellipsoid has three components, e (along the director L), Sg (along the C2 axis), and et (in the direction perpendicular to both C2 and L), this should be taken into account when considering the FLC interaction with the external field. 

J. B. Schneider, I.C. Peden, Differential cross section of a dielectric ellipsoid by the T-matrix extended bormdaxy condition method, IEEE Trans. Antennas Propagat. 36, 1317 (1988)... 

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