Single-particle scattering

When a beam of incident radiation strikes a particle, some of the radiation is absorbed by the surface while the remaining portion is scattered out of the surface. The reflection, refraction, diffraction, and transmission depend not only on the optical properties of the particle but also on the particle size dp relative to the wavelength of the incident radiation X. 

It is convenient to introduce the efficiency factors for extinction, scattering, and absorption, which are defined by 

In general, Q s are functions of the orientation of particles and the state of polarization of incident beams. However, for spherical and homogeneous particles, Q s are independent of both. 

Example 4.3 For gas-solid flows, the typical temperature range where thermal radiation should be considered is from 500 K to 2,000 K. Show that for Rayleigh scattering, the particle 

Solution The wavelength at which the monochromatic-emissive power of a blackbody is maximum for a given temperature is given by Eq. (1.63) as 

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Fig. 10-8. Single particle scattering to mass ratio for particles of four different compositions. Carbon particles are also very efficient absorbers of light. Source U.S. Environmental Protection Agency, "Protecting Visibility," EPA-450/5-79-008, Office of Air Quality Planning Standards, Research Triangle Park, NC, 1979.

Bell (1981) (see also Bell and Bickel, 1981) measured all matrix elements for fused quartz fibers of a few micrometers in diameter with a photoelastic polarization modulator similar to that of Hunt and Huffman (1973) the HeCd (441.6 nm) laser beam was normal to the fiber axes. Advantages of fibers as single-particle scattering samples are their orientation is readily fixed and they can easily be manipulated and stored. Two of the four elements for a 0.96-jtim-radius fiber are shown in Fig. 13.16 dots represent measurements and solid lines were calculated using an earlier version of the computer program in Appendix C. Bell was able to determine the fiber radius to within a few tenths of a percent by varying the radius in calculations, assuming a refractive index of 1.446 + iO.O, until an overall best fit to the measured matrix elements was obtained. 

Latimer, P.)A. Brunsting, B. E. Pyle, and C. Moore, 1978. Effects of asphericity on single particle scattering, Appl. Opt., 17, 3152-3158. 

Calculation of the autocorrelation function proceeds by noting that the single particle scattering function, F- (q, t), is simply the Fourier transform of G- (x ), the Van-... 

Cunningham correction factor for, 63 settling velocity of, 81-82 Simple coagulation theory, 308 Simple diffusion, 131-143 Single-particle scattering measurements, 297-299... 

Figure 4.9 (A) Darkfield optical micrograph of four individual Ag nanoparticles that have been hybridized with a 1 1 mixture of the dyes Alexa Fluor 488 and Rhodamine Red. (B) Fluorescence micrograph of the same area collected using Alexa Fluor 488 excitation eind emission. (C) Fluorescence micrograph of the same area collected using Rhodamine Red excitation and emission. (D) Single particle scattering spectra show the LSPR for each particle in (A). Reprinted from reference 9.

The widely used multiple-scattering treatment was developed by Waterman and Truell (1961) and Twersky (1962). The treatment is based on an approximation in which the exciting field seen by a scatteier may be represented by the total field that would exist at the scatterer if the scatterer were not present. Furthermore, it assumes that scatterers are statistically independent, i.e., the probability of finding a scatterer at one point is independent of other scatterers. The treatment yields the following expression for the effective wavenumber in terms of single-particle scattering amplitudes ... 

Figure 5.1 Single particle scattering intensity I (q) for a solid sphere of radius R.

Like in the case of single-particle scattering, there are other possibilities of defining optical potentials. Without going into detail we want to mention that an analogous expression to the Feshbach effective Hamiltonian (109) can be derived for two-particle scattering. The static part has now the matrix elements [H, o ,a, ] o ). For positrons as projectile... 

In interacting systems the optical and orientation factors in a are no longer separable quantities. The induced optical effect is determined both by the single particle scattering [form factor P q)] and by the pair distribution function [structure factor 5( )], the latter being direction-dependent [42]. Since in addition to orientation the electric field causes particle translation, even for spherical particles the radial distribution function g q, /) and the "static" structure factor attain time-dependent induced anisotropy. The deformed surface potential also contributes to this effect. 

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