Spheres, scattering from

Figure 3.12. Inelastic scattering of Ar from Pt(lll) at the various input energies listed in the figure and for an initial angle of incidence 0, = 45° and Ts = 800 K. Results are plotted as EfIE vs. the final scattered angle . Points are the experimental results and the lines marked adjacently in the label are results of molecular dynamics simulations on an empirical PES. The long dot-dashed curve is the prediction of a cube model of energy transfer, while the dashed curve is the prediction from hard sphere scattering. From Ref. [135].

For some typical modes of scattering from large spherical particles (f >5), simple formulations of phase functions can be obtained. These modes include scattering from a specularly reflecting sphere, scattering from a diffuse reflection sphere, and scattering by diffraction from a sphere. 

Dadap J I, Shan J, Eisenthal K B and Heinz T F 1999 Second-harmonic Rayleigh scattering from a sphere of centrosymmetric material Phys. Rev. Lett. 83 4045-8... 

The scattering from an isolated sphere may be calculated from equation (B1.9.3 2). This derivation assumes that the sphere is iinifonn, with its density profile p(r) = Pq if r < rg and p(r) = 0 if r > Tq (surrounded by a non-scattering material). With this assumption, equation (BE9.32) becomes... 

As shown in Ref. (Bulgac and Wirzba., 2001) the semiclassical result is a very good approximation of the full quantum mechanical result calculated from the exact expression (18) of the two-sphere scattering matrix when plugged into the modified Krein formula (15). 

Guinier function which describes the scattering from sphere-like objects (equation (3b)), and the Sphere function which gives the exact scattering from a perfect sphere, where R = (5/3) iRg (equation (3c)). 

Dendrigraft molecules are star-like in the Oth generation, but have a uniform interior with an exterior transition zone for G3 [17,34], Random hyperbranched molecules have broad distributions both in molecular mass and in shape [17], Scattering from hyperbranched is closer to that of linear polymers than to spheres, indicating that there is a gradually tapering distribution on units from the center to the exterior. 

Our experiments are typically carried out at DNA concentrations of 20-50 /ig/ml with 1 ethidium per 300 bp, so that depolarization by excitation transfer is negligible.(18) The sample is excited with 575-nm light, and the fluorescence is detected at 630, 640, or 645 nm. Less than one fluorescent photon is detected for every 100 laser shots. The instrument response function e(t) is determined using 575-nm incident light scattered from a suspension of polystyrene latex spheres. 

Aden, A. L., 1951. Electromagnetic scattering from spheres with sizes comparable to the wavelength, J. Appl. Phys., 22, 601-605. 

Foley, J. T., and D. N. Pattanayak, 1974. Electromagnetic scattering from a spatially dispersive sphere, Opt. Commun., 12, 113-117. 

Kattawar, G. W., and G. N. Plass, 1967. Electromagnetic scattering from absorbing spheres, Appl. Opt., 6, 1377-1382. 

To account for the total directional scattering from a large sphere, effects of both diffraction and reflection must be considered. When a spherical particle is in the path of incident radiation, the diffracted intensity may be obtained from Babinet s principle, which states that the diffracted intensity is the same as that for a hole of the same diameter. The phase function for diffraction by a large sphere is given by [Van de Hulst, 1957]... 

FIG. 12 Pattern of light scattered from a single layer of colloidal particles in the disordered phase. The particles are polystyrene spheres, of diameter 2 /glass plates. Except for the contribution of the form factor P(k), which depends on the scattering angle, and normalization and geometrical factors, this picture shows directly the static structure factor of the system. 

In what follows, the word particle will be used to describe the source of the scattering. The word will encompass rigid structures such as latex spheres as well as macromolecules such as linear polymer chains. The particle will be viewed as a collection of smaller components each of which scatters light so that the total scattering from a particle will be written as a sum of scattered waves from the components. Such an approach permits the convenient calculation of angular dependence of scattering as well as the description of internal dynamics of linear polymer chains. 

Because the viscosity of water changes with temperature, the samples should ideally be in a thermostatted cell at 25°C but, for convenience, the experiment can be done at room temperature, which should be recorded. The sample cuvette can be glass or plastic, and care should be taken to avoid scattering by scratches, smudges, or fingerprints on the cell walls. Again, it is important to eliminate all traces of dust particles and any air bubbles in the sample since scattering from these can overwhelm that from the polystyrene spheres. 

Three of the experiments are completely new, and all make use of optical measurements. One involves a temperature study of the birefringence in a liquid crystal to determine the evolution of nematic order as one approaches the transition to an isotropic phase. The second uses dynamic laser light scattering from an aqueous dispersion of polystyrene spheres to determine the autocorrelation function that characterizes the size of these particles. The third is a study of the absorption and fluorescence spectra of CdSe nanocrystals (quantum dots) and involves modeling of these in terms of simple quantum mechanical concepts. 

Mie, G. Ann. Phys. 1908, 25, 377. This theory represents the exact solution to Maxwell s equations for a sphere. For details on recent theories to describe scattering from nonspherical nanostructures,... 

Figure 2. Scattering from silica particles bound with strongly cationic polymers ( ), compared with that from free silica particles (+). The particles are spheres of precipitated silica with a radius of 19 nm In water at pH near 7 they bear 0.3 negative charge per nm of surface, most of which Is compensated by adsorbed counterions (15). The polymers are AM-CH copolymers with a ratio of cationic to total monomers equal to 0.3 the total amount of polymer In the floe approximately compensates the chemical charge borne by the silica particles (9).

Figure 3. Scattering from silica spheres bound with PEO macromolecules In water (/, compared with free silica spheres (O) and with a concentrated suspension where the spheres repel each other (+). The solvent Is water at pH = 8 In this solvent the spheres bear about 0.3 S10 group per nm of surface, and the macromolecules bind to the remaining SlOH groups (1 ). Very long macromolecules (M 2E6) are used to promote bridging and flocculation with shorter ones no floes are obtained unless the surface charges are neutralized or screened (16).

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