Scattering spheres

These maxima and minima or Tyndall colours can only be observed if all of the scattering spheres are of closely the same size. If there is heterogeniety in size the maxima and minima arising from different size spheres occur at different angles so that only a monotonic decrease of intensity with angle is observed. 

FIGURE 5.19 Crystalline powder diffraction in left, the scattering spheres in the reciprocal space in right, the associated Ewald s sphere of diffraction for any of them after (Matter Diffraction, 2003 X-Rays, 2003 Putz Lacrama, 2005 HyperPhysics, 2010). 

Four volumetric defects are also included a spherical cavity, a sphere of a different material, a spheroidal cavity and a cylinderical cavity (a side-drilled hole). Except for the spheroid, the scattering problems are solved exactly by separation-of-variables. The spheroid (a cigar- or oblate-shaped defect) is solved by the null field approach and this limits the radio between the two axes to be smaller than five. 

Micellar structure has been a subject of much discussion [104]. Early proposals for spherical [159] and lamellar [160] micelles may both have merit. A schematic of a spherical micelle and a unilamellar vesicle is shown in Fig. Xni-11. In addition to the most common spherical micelles, scattering and microscopy experiments have shown the existence of rodlike [161, 162], disklike [163], threadlike [132] and even quadmple-helix [164] structures. Lattice models (see Fig. XIII-12) by Leermakers and Scheutjens have confirmed and characterized the properties of spherical and membrane like micelles [165]. Similar analyses exist for micelles formed by diblock copolymers in a selective solvent [166]. Other shapes proposed include ellipsoidal [167] and a sphere-to-cylinder transition [168]. Fluorescence depolarization and NMR studies both point to a rather fluid micellar core consistent with the disorder implied by Fig. Xm-12. 

The physical situation of interest m a scattering problem is pictured in figure A3.11.3. We assume that the initial particle velocity v is comcident with the z axis and that the particle starts at z = -co, witli x = b = impact parameter, andy = 0. In this case, L = pvh. Subsequently, the particle moves in the v, z plane in a trajectory that might be as pictured in figure A3.11.4 (liere shown for a hard sphere potential). There is a point of closest approach, i.e., r = (iimer turning point for r motions) where... 

The final scattering angle 0 is defined rising 0 = 0(t = oo). There will be a correspondence between b and 0 that will tend to look like what is shown in figure A3.11.5 for a repulsive potential (liere given for the special case of a hard sphere potential). 

Figure A3.11.4. Trajectory associated with a particle scattering ofT a hard sphere potential.

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... 

This method, introduced originally in an analysis of nuclear resonance reactions, has been extensively developed [H, 16 and F7] over the past 20 years as a powerful ab initio calculational tool. It partitions configuration space into two regions by a sphere of radius r = a, where r is the scattered electron coordinate. 

Like e, t is the product of two contributions the concentration N/V of the centers responsible for the effect and the contribution per particle to the attenuation. It may help us to become oriented with the latter to think of the scattering centers as opaque spheres of radius R. These project opaque cross sections of area ttR in the light path. The actual cross section is then multiplied by the scattering efficiency factor optical cross... 

Two physically reasonable but quite different models have been used to describe the internal motions of lipid molecules observed by neutron scattering. In the first the protons are assumed to undergo diffusion in a sphere [63]. The radius of the sphere is allowed to be different for different protons. Although the results do not seem to be sensitive to the details of the variation in the sphere radii, it is necessary to have a range of sphere volumes, with the largest volume for methylene groups near the ends of the hydrocarbon chains in the middle of the bilayer and the smallest for the methylenes at the tops of the chains, closest to the bilayer surface. This is consistent with the behavior of the carbon-deuterium order parameters,. S cd, measured by deuterium NMR ... 

Diffraction is usefiil whenever there is a distinct phase relationship between scattering units. The greater the order, the better defined are the diffraction features. For example, the reciprocal lattice of a 3D crystal is a set of points, because three Laue conditions have to be exactly satisfied. The diffraction pattern is a set of sharp spots. If disorder is introduced into the structure, the spots broaden and weaken. Two-dimensional structures give diffraction rods, because only two Laue conditions have to be satisfied. The diffraction pattern is again a set of sharp spots, because the Ewald sphere cuts these rods at precise places. Disorder in the plane broadens the rods and, hence, the diffraction spots in x and y. The existence of streaks, broad spots, and additional diffuse intensity in the pattern is a common... 

---