Ellipsoid equation

Further experimental evidence of shape effects in absorption spectra of SiC particles is found in the data of Pultz and Herd (1966), who investigated infrared absorption by SiC fibers with and without Si02 coatings. Although these measurements were not mass-normalized, they show a strong absorption band at 795 cm-1 and a weaker band at 941 cm-1. If the fibers are approximated as ellipsoids with L2 = L3 = and Lx = 0 (i.e., a cylinder), then the ellipsoid equation (12.27) predicts absorption peaks for particles in air at frequencies where c = -1 and c = — oo. This corresponds to absorption bands at 797 and 945 cm-1 for the dielectric function of isotropic SiC, in excellent agreement with the experimental peak positions for the fibers. 

Important examples are the trigonal crystals LiNb03 and LiTa03. For E = (0,0, E) the ellipsoid equation changes to the following form ... 

At the extensions of the Maxwell Garnett theory for parallel-oriented ellipsoids, equation (13) [23, 24], all particles are ellipsoids with parallel the mean axis. Only one depolarization factor L is necessary. L describes the ratio between the axes of the ellipsoids, and values for L between 0spherical particles (L =i), equation (13) gives the same result as the Maxwell Garnett theory, equation (12). For particles embedded in one plane, the orientation of the ellipsoids in ratio to the substrate is parallel or perpendicular to the plane of the incident light. An orientation of the ellipsoidal particles diagonally to the substrate cannot be considered. 

The extensions of the Maxwell Garnett theory for random-oriented ellipsoids, equation (14) [25], needs three depolarization factors Li, Li, L3 with 2L, = 1 to describe the embedded ellipsoids. Frequently, ellipsoids with a symmetrical axis of rotation are assumed with Li = L3. Extreme geometries are rods with Li L2 = L3 and disks with Li L2 = L3. For Li =y, the extensions of the Maxwell Garnett theory for random-oriented ellipsoids, equation (14), give the same result as the Maxwell Garnett theory, equation (12). 

This simplified formula corresponds to the formula [4] in (5). The angle dependent absorption A q,P) describes here an ellipsoid equation. 

The tilt angle profile G z) can be used to calculate the average refractive index, n, of the liquid crystal at different applied voltages. For a given tilt angle, the refractive index is given by the index ellipsoid equation [54] ... 

Analysis of the results on DPH and DAPH are based on a model which considers DPH to be a prolate ellipsoid. Equation (4) then becomes... 

Here, r is the aspect ratio of the ellipsoid. Equation 13.2 follows from the requirement that the ellipsoid rotate with the local fluid but that it retain a unit length. 

Finally, the effect of reorientation on the striation thickness reduction can be assessed in the example of an initially spherical particle that is deformed into an ellipsoid. Equation 6.105 gives the reduction in striation thickness as a function of the shear strain. If the ellipsoid is entering the next step with its long axis perpendicular to the shearing planes, the new striation thickness is given by the same equation. Thus, after n steps the striation thickness, S, with reorientation is given by... 

In the last section we noted that Simha and others have derived theoretical expressions for q pl(p for rigid ellipsoids of revolution. Solving the equation of motion for this case is even more involved than for spherical particles, so we simply present the final result. Several comments are necessary to appreciate these results ... 

Cone-bottom vertical vessels are sometimes used where solids are anticipated to be a problem. Most cones have either a 90 apex (a = 45 ) or a 60 apex ia = 30 ). These are referred to respectively as a 45 or 60 cone because of the angle each makes with the horizontal. Equation 12-4 is for the thickness of a conical head that contains pressure. Some operators use internal cones within vertical vessels with standard ellipsoidal heads as shown in Figure 12-2. The ellipsoidal heads contain the pressure, and thus the internal cone can be made of very thin steel. 

The weight of nozzles and internals can be estimated at 5 to 10% of the sum of the shell and head weights. The weight of a skirt can be estimated as the same weight per foot as the shell with a length given by Equation 12-8 for an ellipsoidal head and Equation 12-9 for a conical head. 

Symmetry 50. Intercepts 50. Asymptotes 50. Equations of Slope 51. Tangents 51. Equations of a Straight Line 52. Equations of a Circle 53. Equations of a Parabola 53. Equations of an Ellipse of Eccentricity e 54. Equations of a Hyperbola 55. Equations of Three-Dimensional Coordinate Systems 56. Equations of a Plane 56. Equations of a Line 57. Equations of Angles 57. Equation of a Sphere 57. Equation of an Ellipsoid 57. Equations of Hyperboloids and Paraboloids 58. Equation of an Elliptic Cone 59. Equation of an Elliptic Cylinder 59. 

The polarizability tensor may therefore be defined by a set of nine components which reduce in number to six because the tensor is symmetric. The physical significance of molecular polarizability is often explained in terms of the polarizability ellipsoid which is defined by the equation ... 

An ellipsoidal nucleus with two spherons in the inner core has major radius greater than the minor radii by the radius of a spheron. about 1.5 f, which is about 25 percent of the mean radius. The amount of deformation given by this model is accordingly in rough agreement with that observed (18). In a detailed treatment it would be necessary to take into account the effect of electrostatic repulsion in causing the helions to tend to occupy the poles of the prolate mantle, with tritons tending to the equator. 

The quantity riV/RT is equal to six times the rotational period. The rotational relaxation time, p, should he shorter than the fluorescence lifetime, t, for these equations to apply. It is possible to perform calculations for nonspherical molecules such as prolate and oblate ellipsoids of revolution, but in such cases, there are different rotational rates about the different principal axes. 

Scheraga-Mandelkern equations (1953), for effective hydrodynamic ellipsoid factor p (Sun 2004), suggested that [rj] is the function of two independent variables p, the axial ratio, which is a measure of shape, and Ve, the effective volume. To relate [r ] to p and Ve, introduced f, the frictional coefficient, which is known to be a direct function of p and Ve. Thus, for a sphere we have... 

This function is a solution of Laplace s equation regardless of the values of constants, and our goal is to find such of them that the potential satisfies the boundary condition on the surface of the given ellipsoid of rotation and at infinity. In order to solve this problem we have to discuss some features of Legendre s functions. First of all, as was shown in Chapter 1, the Legendre s function of the first kind P (t]) has everywhere finite values and varies within the range... 

Thus, we have demonstrated that the potential C/g is a solution of Laplace s equation and satisfies the boundary condition at the surface of the ellipsoid of rotation and at infinity. In other words, we have solved the Dirichlet s boundary value problem and, in accordance with the theorem of uniqueness, there is only one function satisfying all these conditions. 

The latter allows us to determine the potential of the gravitational field outside the ellipsoid of rotation, including points of its surface, provided that the mass, geometry, and angular frequency are given. Making use of Equation (2.148), we can represent the potential of the gravitational field in the form kM 1 1 2 2 SiC/s) - 1... 

Before we continue, let us relate the coordinate o with the semi-axes of the given ellipsoid. As follows from Equation (2.121) we have... 

With a decrease of sg the ellipsoid approaches a disk on the contrary, when 8q oo, it tends to a sphere. In order to find the gravitational field of the rotating ellipsoid, it is convenient to represent Equation (2.149) in a slightly different form. For this purpose introduce notations rj — sin and, therefore. 

Proceeding from Equation (2.153) we will derive the expressions for the normal gravitational field and the main attention is paid to the field on the surface of the ellipsoid. To emphasize the fact that we deal with the normal field it is conventional to use the letter y instead of g. Then, at each point outside the mass we have... 

Taking into account the fact that on the surface of the ellipsoid e = eq and d — dg we arrive at the obvious result, namely, at the level surface the tangential component of the gravitational field vanishes, — 0. Next, consider the component Again differentiating Equation (2.153) we obtain... 

We have derived Equation (2.164), which shows how the field varies with the reduced latitude p on the surface of the spheroid. The reduced latitude is the angle between the radius vector and the equatorial plane. Fig. 2.7c. Also, it is useful to study the function y — y q>), where tp is the geographical latitude. This angle is formed by the normal to the ellipsoid at the given point p and the equatorial plane. Fig. 2.7b. First, we find expressions for coordinates v, y of the meridian ellipse. Its equation is... 

Now we arrive at an important result, namely, with an accuracy of the square of the flattening this equation characterizes the ellipsoid of rotation. From the last equation we have... 

Here N is the distance between points p and q measured along the perpendicular from the point q to the geoid. Fig. 2.9a. The linear behavior of the normal potential implies that the field y is constant between the geoid and the reference ellipsoid. The change of sign in Equation (2.260) is related to the fact that the field has a direction, which is opposite to the direction of differentiation. As follows from the first equation of the set (2.258 and 2.260) we have... 

Here Wq and Uq are the total and normal potentials on the surface of the geoid and on the surface of the reference ellipsoid, respectively. By definition, y — —dUjdz is the magnitude of the normal gravitational field. Thus, Equation (2.292) becomes... 

First of all, we choose the parameters of the ellipsoid in such a way that the normal potential on its surface, Uq, is equal to the potential of the total field at points of the geoid, Wq. Then, Equation (2.294) is greatly simplified and we obtain... 

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