Homogeneous Sphere

The form factor for a particle of arbitrary shape can be calculated by numerical integration of (6.10). However, for certain regular geometrical shapes, it is possible to obtain analytical expressions for /. In this section we consider one such particle, a homogeneous sphere. 

The vector ez — e is normal to planes R (ez — er) = constant, over which the phase S is constant we can write the phase as 

Note that / vanishes at those values of 6 for which 

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First, consider the field of a homogeneous sphere with radius a. Taking into account the spherical symmetry of the mass distribution, it is natural to introduce a spherical system of coordinates with its origin at the center of the sphere, Fig. 1.5c. Then, the vector g p) is in general characterized by three components ... 

Certainly, Equations (1.127 and 1.128) can be derived directly from Equation (1.6), but it requires a rather cumbersome integration. This example also allows us to illustrate the fact that the gravitational field has a finite value inside any mass. With this purpose in mind, imagine that the observation point p is located at the center of a small and homogeneous sphere. Fig. 1.12a. Then, the total field can be represented as a sum ... 

Thus, for the earth and a homogeneous sphere we obtain the following values ... 

Figure 8.35. The homogeneous sphere of radius R. Radial correlation function, ys (r), distance distribution function (DDF) ps (r) and chord length distribution (CLD) gs (r)...

Fet us now consider the 3D equivalent of the aforementioned example an ensemble of uncorrelated homogeneous spheres - with polydispersity, meaning that the observed CLD... 

Initial conditions We mentioned that we need to fix 3 initial conditions in the center one for the central density of matter (n(0) = no or e(0) = eo), and two for the metric, either for A(0) and A (0), or, equivalently, for A(0) and (0). But we cannot properly impose these conditions in r = 0, and these conditions are somehow not independent. However the technical problem is well known already in 4 dimensions [1]. First, the proper way is to approximate the innermost core of radius 6 with a homogeneous sphere of density no, where the exact value of 6 is irrelevant if small enough. Then u no al r 6, and e2A = 1 — 87r[Pg.305]

The dashed line give the scattering function calculated for a homogeneous sphere. The experimental data can only be described at small q by this model at... 

Eq. (4) calculated for the highest contrast possible. The solid line gives the best fit of the latter term by an empirical expression whereas the inset displays T r) obtained from T(q) by Fou-rier-inversion. The dashed line in Fig. 7 is the scattering function of a homogeneous sphere of same ... 

The preceding sections have demonstrated that dendrimers of lower generation are akin to branched polymeric structures. It is therefore to be expected that their flow behavior in dilute solution may be described in terms of the well-known concepts of dilute polymer solutions [14, 15]. Hence, dissolved dendrimers should behave like non-draining spheres. From an experimental comparison of and the immobilization of solvent inside the den-drimer can be compared directly since in this case the dendrimer may be approximated by a homogeneous sphere. Therefore R = 3/5 Rl where Ry, denotes the hydrodynamic radius of the dendrimer. This has been found experimentally [19]. 

Note For homogeneous spheres, this phenomenon is rigorously described by the theory developed by Mie. 

Appendix A. Homogeneous Sphere, 477 Appendix B. Coated Sphere, 483 Appendix C. Normally Illuminated Infinite Cylinder, 491 References, 499 V... 

I he field scattered by any spherically symmetrical particle composed of materials described by the constitutive relations (2.7)-(2.9) has the same form as that scattered by the homogeneous sphere considered in Chapter 4. However, the functional form of the coefficients an and bn depends on the radial variation of e and ju. In this section we consider the problem of scattering by a homogeneous sphere coated with a homogeneous layer of uniform thickness, the solution to which was first obtained by Aden and Kerker (1951). This is one of the simplest examples of a particle with a spatially variable refractive index, and it can readily be generalized to a multilayered sphere. 

If mx = m2, then An = B = 0 and the coefficients (8.2) reduce to those for a homogeneous sphere. We also have ima 0An = lima >0 = 0 therefore, in the limit of zero core radius the coefficients (8.2) reduce to those for a homogeneous sphere of radius b and relative refractive index m2, as required. When m2= 1, the coefficients reduce to those for a sphere of radius a and relative refractive index mx this gives us yet another check on the correctness of our solution. 

If the particles are homogeneous spheres, then from the results of the preceding section we have... 

The classical method of solving scattering problems, separation of variables, has been applied previously in this book to a homogeneous sphere, a coated sphere (a simple example of an inhomogeneous particle), and an infinite right circular cylinder. It is applicable to particles with boundaries coinciding with coordinate surfaces of coordinate systems in which the wave equation is separable. By this method Asano and Yamamoto (1975) obtained an exact solution to the problem of scattering by an arbitrary spheroid (prolate or oblate) and numerical results have been obtained for spheroids of various shape, orientation, and refractive index (Asano, 1979 Asano and Sato, 1980). 

The conditions for the vanishing of the denominators of the scattering coefficients an and bn for a homogeneous sphere are (4.54) and (4.55). We now consider these conditions in the limit of vanishingly small x. From the series expansions (5.1) and (5.2) of the spherical Bessel functions of order n, together with a bit of algebra, we can show that the denominator of an vanishes in the limit x -> 0 (finite m ) provided that... 

In the preceding paragraphs we considered a homogeneous sphere. Let us now examine what happens when a homogeneous core sphere is uniformly coated with a mantle of different composition. Again, the condition for excitation of the first-order surface mode can be obtained from electrostatics. In Section 5.4 we derived an expression for the polarizability of a small coated sphere the condition for excitation of the Frohlich mode follows by setting the denominator of (5.36) equal to zero ... 

The mathematical form of all the scattering functions for a coated sphere—efficiencies and matrix elements—have the same form as those for a homogeneous sphere. Only the scattering coefficients (8.2) are different these may be written in a form more suitable for computations ... 

Theoretical Feasibility. Earlier work (5) had established the general domain of size parameters and refractive index expected. This allowed a number of model calculations to be carried out to clearly establish the theoretical feasibility of the method. Computations were carried out for homogeneous spheres and for concentric core-shell spheres. 

Figure 1. Theoretical calculations for (A) homogeneous sphere with a = 3.71, m = 1.20 and (B) homogeneous sphere with a = 5.74, m = 1.09, which is the volume-weighted refractive index for the core-shell structure of (C) with a = 3.71, V = 5.75, m, = 1.20, m2 = 1.05.

Since the core-shell fits were better than the homogeneous sphere fits there might be some material in a layer around the core other than pure solvent. The data for the shell, however, must be close to the uncertainty of the fit obtained with the crude step-function model so that at best the arguments for a shell are tenuous. Clearly, a further investigation is required in order to test the statistical significance of the fits and to consider other possible models. 

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