Oriented particles

As illustrated by Eig. 4.13, an electron microscope offers additional possibilities for analyzing the sample. Diffraction patterns (spots from a single-crystal particle and rings from a collection of randomly oriented particles) enable one to identify crystallographic phases as in XRD. Emitted X-rays are characteristic for an element and allow for a determination of the chemical composition of a selected part of the sample. This technique is referred to as energy-dispersive X-ray analysis (EDX). 

The dimensionless proportionality factor % is the magnetic susceptibility. The magnetization and consequently also the susceptibility depend on the number of orientable particles in a given volume. A volume-independent, material-specific magnitude is the molar susceptibility xmoi... 

The heavy-end portions (usually called heavy fractions) of bitumen (e.g. asphaltenes, preasphaltenes) can exist both in a random oriented particle aggregate form or in an ordered micelle form, peptized with resin molecules (16.17). In their natural state, asphaltenes exists in an oil-external (Winsor s terminology) or reversed micelle. The polar groups are oriented toward the center, which can be water, silica (or clay), or metals (V, Ni, Fe, etc.). The driving force of the polar groups... 

For a system of uncorrelated but highly oriented particles in a sample (e.g., oriented needle-shaped voids in a fiber) it may be possible to factorize the particle density, e.g.,... 

Turner (1973) and McKellar (1976) applied RG theory to ensembles of randomly oriented particles of arbitrary shape the former author included spheres with anisotropic optical constants. Optically active particles have been treated within the framework of the RG approximation by Bohren (1977). 

Figure 11.22 Schematic diagram of the microwave analog technique for measuring extinction by single oriented particles.

Extinction is easy to measure in principle but may be difficult in practice, especially for large particles where it becomes difficult to discriminate between incident and forward-scattered light. Spheres and ensembles of randomly oriented particles do not linearly polarize unpolarized light upon transmission. But single elongated particles or oriented ensembles of such particles can polarize unpolarized light by differential extinction. 

Further discussion of extinction by oriented particles is given in van de Hulst (1957, Chaps. 15 and 16). 

The maximum amount of information about scattering by any particle or collection of particles is contained in all the elements of the 4x4 scattering matrix (3.16), which will be treated in more generality later in this chapter. Most measurements and calculations, however, are restricted to unpolarized or linearly polarized light incident on a collection of randomly oriented particles with an internal plane of symmetry (no optical activity, for example). In such instances, the relevant matrix elements are those in the upper left-hand 2x2 block of the scattering matrix, which has the symmetry shown below (see, e.g.,... 

For an isotropic medium such as a collection of many randomly oriented particles, which may themselves be anisotropic, the scattered irradiance and hence the differential scattering cross section is independent of . 

This is the form of the scattering matrix for any medium with rotational symmetry even if all the particles are not identical in shape and composition. A collection of optically active spheres is perhaps the simplest example of a particulate medium which is symmetric under all rotations but not under reflection. Mirror asymmetry in a collection of randomly oriented particles can arise either from the shape of the particles (corkscrews, for example) or from optical activity (circular birefringence and circular dichroism). 

Few measurements or calculations of all 16 scattering matrix elements have been reported. There are only four nonzero independent elements for spherical particles and six for a collection of randomly oriented particles with mirror symmetry (Section 13.6). It is sometimes worth the effort, however, to determine if the expected equalities and zeros really occur. If they do not, this may signal interesting properties such as deviations from sphericity, unexpected asymmetry, or partial alignment some examples are given in this section. But we begin with spherical particles. 

Scattering matrix elements off the block diagonal are zero for mirror-symmetric collections of randomly oriented particles, as they are for spheres. 

For incident unpolarized light to be (partially) circularly polarized upon scattering by a collection of particles, the scattering matrix element S4l must not be zero. It was shown in Section 13.6 that the scattering matrix for a collection (with mirror symmetry) of randomly oriented particles has the form... 

In Section IV.B a procedure of numerical solution for Eq. (4.329) is described and enables us to obtain the linear and cubic dynamic susceptibilities for a solid system of uniaxial fine particles. Then, with allowance for the polydispersity of real samples, the model is applied for interpreting the magnetodynamic measurements done on Co-Cu composites [64], and a fairly good agreement is demonstrated. In our work we have proposed for the low-frequency cubic susceptibility of a randomly oriented particle assembly an interpolation (appropriate in the whole temperature range) formula... 

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