Spherical scatterers

The oscillating dipole is a source of electromagnetic radiation of the same frequency, polarized in the direction of the oscillations. At large distances, the wave is spherical. According to the electromagnetic theory, the resulting electric vector at a point in the equatorial plane of the dipole is a>2/ r c2 times the moment of the dipole at time t — r /c. The amplitude of the spherically scattered wave at unit distance in the equatorial plane is therefore... 

Modified Spherical Scattering Factor for the Hydrogen Atom... 

A different type of analysis has now provided this information (20) The dimension distributions p(a) of independent spherical scatterers with uniform density and diameter a which produces each of the terms in the sum in Equation (3) can be calculated (19) After obtaining the constants in the sum in Equation (3) by least-squares fits of this equation to the scattering curve measured for Beulah lignite at the University of North Dakota, we used these constants to evaluate the sum of the pore-dimension distribution functions for uniform spheres that are obtained (19) from the terms in the sum in Equation (3) The sum of these pore-dimension distributions was very similar to the power-law distribution given by Equation (4) The fact that we could obtain almost the same power-law dimension distribution by two independent methods suggests that such a distribution may be a good approximation to the pore-... 

In the kinematical theory, we consider the diffraction of a plane wave (of wavelength X) incident upon a three-dimensional lattice array of identical scattering points, each of which consists of a group of atoms and acts as the center of a spherical scattered wave. Our problem is to find the combined effect of the scattered waves at a point outside the crystal, at a distance from the crystal that is large compared with its linear dimensions. In developing the theory, we make several important assumptions ... 

Figure 9. (a) Experimental probability density function of the cross-polarized backscattered intensity produced by interacting spherical scatterers located on a flat substrate, (b) Experimental values of / (/cross = 0) (dots) and their corresponding error bars as a function of the illuminated area. Continuous line corresponds to the fitted function (Eq. 12). 

The reflection matrix for a closely packed medium composed of wavelengthsized scatterers can be represented as a sum of matrices (10) with the coefficients of the addition theorem (7). These coefficients describe all the details of the field in the vicinity of any scatterer, including the near-field effects [26]. We consider the manifestations of these effects quahtatively using the field configuration near a spherical scatterer as the simplest example. 

The differential scattering cross section in the RLM is defined as usual, the ratio of the spherically scattered flux into final state n originating from an incident plane wave in molecular state m,... 

Figure 9.9 shows the theoretical scattering from a system containing a low number density of spherical scattering features of radius R. The size and relative volume fraction of the scattering features can be determined... 

In practice, even at. 00 MHz, the wavelength of an ultrasonic wave Is large compared to a flaw size of 25 im, and thus a more accurate model of the differential scattering cross section is necessary. To address this problem, we have programmed the spherical scattering model of Ying and Truell. with corrections for several errors in the original derivation. This model provides us with an excellent estimate of the scattered field produced by a spherical void (or inciusion) as a function of frequency, flaw size, and flaw depth. 

The former convention was traditionally employed by mathematicians, while the latter is used in physics, in particular for the description of scattering phenomena. This book employs the spherical scattering coordinates. 

These equations can be expressed for different types of coordinates. For spherical scattering coordinates one obtains for Eq. (C.4) ... 

Solutions of this equation can be obtained by separation of variables. For spherical scattering coordinates such a separation may be defined as ... 

In die following sections we desc e the interaction of monochromatic, coherent radiation witti scattering centers, whidi results in spherical scattered waves. Interference among these waves creates the intensity pattern sensed by a detector. The connection between the observed scattered wave intensity and the structure of matter is ultimately sought. We are especially interested to find die spatial periodicities within our material that lead to interference. To this end, the mathematical techniques of Fourier transformation and convolution are presented. We end the chapter with sections on the small angle scattering from lamellar systems, and neutron scattering. 

Figure 3. Geometry of scattering from two point scatterers at A and B. Spherical scattered waves are detected at P.

Figure 3 shows an ordered array of scatterers (6). Two electrons at A and B a vector r apart, have been singled out. Whether or not these two electrons produce constructive interference at the distant location of the detector depends upon the instantaneous phase of the incident electric field at A and B. An approximation is made in considering the scattered wave amplitude. The First Bom Approximation (8), or single-scattering approximatbn, assumes that the amplitude of the spherical scattered wave is veiy small compared to the incident wave. When the scattered wave encounters anodier electron, it is not scattered a second time. Thus, only the incident wave scatters from the electrons. 

For spherical scatterers, the angular intensity in the plane perpendicular to the direction of polarization of the incident light is ... 

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