Absorption and Scattering Cross Sections

In this section we derive an approximate expression for the absorption cross section of a large weakly absorbing sphere. We assume that the incident plane wave can be subdivided into a large number of rays the behavior of which at interfaces is governed by the Fresnel equations and Snell s law (Section 2.7). A representative ray incident on the sphere at an angle 0, is shown in Fig. 7.1. At point 1 on the surface of the sphere the incident ray is divided into externally reflected and internally transmitted rays these lie in the plane of incidence, which is determined by the normal to the sphere and the direction of the incident ray. If the polar coordinates of point 1 are (a, 0f, J ), the plane of incidence is the plane f = constant. At point 2 the transmitted ray encounters another boundary and therefore is partially reflected and partially transmitted. In a like manner we can follow the path of the rays within the sphere, a path that does not deviate outside the plane of incidence. At each point where a ray encounters a boundary it is partially reflected internally and partially transmitted into the surrounding medium. On physical grounds we know that the absorption cross section cannot depend on the polarization of the incident 

A fraction 1 — e of the transmitted energy at 1 is absorbed as the ray traverses a path length = 2a)]n2 — sin2 ,/n between points 1 and 2, where a is the absorption coefficient of the sphere. The amplitude of the reflected field at 2 is 

If the incident field component is perpendicular to the plane of incidence, all the expressions for reflected, transmitted, and absorbed light are identical in form with those in the preceding paragraph we need merely substitute R and T for / n and 7j(. 

The total energy lTabs absorbed in the sphere is obtained by summing the energy deposited by all internal rays for both components of polarization and a given incident ray and then integrating over all incident rays (i.e., all angles of incidence between 0 and it/2). The result is 

Up to this point we have only assumed that k n subject to this restriction and, of course, the assumption that geometrical optics combined with the Fresnel formulas is a good approximation, (7.1) is completely general. Let us further assume that the sphere is sufficiently weakly absorbing that 2 aa 1 with this assumption 

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Similarly, absorption and scattering cross-sections are defined to give... 

The mathematical basis of the Mie theory is the subject of this chapter. Expressions for absorption and scattering cross sections and angle-dependent scattering functions are derived reference is then made to the computer program in Appendix A, which provides for numerical calculations of these quantities. This is the point of departure for a host of applications in several fields of applied science, which are covered in more detail in Part 3. The mathematics, divorced from physical phenomena, can be somewhat boring. For this reason, a few illustrative examples are sprinkled throughout the chapter. These are just appetizers to help maintain the reader s interest a fuller meal will be served in Part 3. 

A brief treatment of scattering by large, absorbing particles and the concept of absorption and scattering cross sections are presented in Section 5.7 along with two examples of applications of the Mie theory (to absorbing, but small, particles) and a discussion of Tyndall spectra. 

The small particle limit x 1. In electromagnetic theory it is shown that the absorption and scattering cross-sections of small dielectric particles are (e.g. Bohren Huffman 1983) ... 

The profiles of the actinic flux are computed at each grid point of the model domain. To determine the absorption and scattering cross sections needed by the radiative transfer model, predicted values of temperature, ozone, and cloud liquid water content are used below the upper boundary of WRF. Above the upper boundary of WRF, fixed typical temperamre and ozone profiles are used to determine the absorption and scattering cross sections. These ozone profiles are scaled... 

Recently, Lee and Pilon (2013) demonstrated that the absorption and scattering cross-sections per unit length of randomly oriented linear chains of spheres, representative of filamentous cyanobacteria (Fig. ID), can be approximated as those of randomly oriented infinitely long cylinders with equivalent volume per unit length. Then, for linear chains of monodisperse cells of diameter d, the diameter 4, v of the volume-equivalent infinitely... 

The average absorption and scattering cross-sections Cabs,x and Csca, i of microorganisms suspensions can be experimentally measured using a spectrometer equipped with an integrating sphere. Pint, the spectral normal-normal T j and normal—hemispherical T f, transmissions of several dilute suspensions with different known concentrations are measured, as illustrated in Fig. 5. Here, the scattering phase function Or (0) previously measured for the same suspension is used to correct for various optical effects. 

Similarly, Kx and can be divided by the samples respective dry mass concentration X to obtain the average mass absorption and scattering cross-sections Aahs,x and Ssca,i-... 

The most often used optical non-invasive techniques for the detection of anomalies in living tissue are optical coherence tomography (OCT) and measurements of backscattered light. Any local anomaly of the tissue causes a change of the absorption- and scattering cross sections. In OCT the different layers of the tissue are inspected and anomalies show up as a change in the interference pattern. There are three different techniques of tomography [1549] ... 

For highly diluted randomly oriented ellipsoids, van de Hulst [233] obtained the following expressions for the absorption and scattering cross sections ... 

For a smaller nanoparticle, the scattering cross-section is much smaller than the absorption cross-section, and thus Cabs Cext. From (4.2)-(4.4), it is revealed analytically that the absorption and scattering cross-sections are proportional to the particle volume and its square, respectively. 

Absorption and scattering cross sections for the elements of importance in reactor physics are summarized in Table A.4. For reasons to be discussed later, the values given are for a neutron energy of 0.0253 eV, corresponding to a neutron velocity of 2200 ms Attention may be drawn to some of the particularly low absorption cross sections, such as that of zirconium, which is consequently valuable as a structural material in reactors. The reason for zirconium, and also lead, having such a low cross section is that the main isotope in both cases has a nucleus with a magic number of neutrons or protons, or both (see Section 1.4). 

Table A.4. Microscopic and Macroscopic Absorption and Scattering Cross Sections...

Collective oscillations of conduction electrons known as plasmons, have a characteristic resonance frequency cop which depends on the mass, density and charge of the carriers. If the incident light frequency matches the plasmon resonance frequency, a strong absorption and scattering cross-section is obtained. Metal... 

Figure 1 Absorption and scattering cross-sections for (A) platinum, and (B) oxygen.

The total cross section will consist of two components, scattering and absorption. In the case of gold and indium, the beam will be attenuated primarily by neutron absorption in the case of aluminum and iron, scattering will furnish the main contribution. Listed below are the absorption and scattering cross sections of the materials in question (taken from BNL-325) ... 

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