Sphere scattering function

The sphere scattering function shows a damped response with maxima and minima (Figure 4.10). Equations 4.53 - 4.55 are valid if the sphere is surrounded by non-scattering material. For the case where the... 

WHEN THE SOLVENT SLD MATCHES THE CORE, SANS GIVES THE SHELL MORPHOLOGY VIA THE HOLLOW-SPHERE SCATTERING FUNCTION... 

Single stmctural levels for three-dimensional (3D) objerts have been obtained by dirert Fourier transform of calculated correlation functions. The simplest and most widely used of these is the sphere scattering function given by. 

The dashed line give the scattering function calculated for a homogeneous sphere. The experimental data can only be described at small q by this model at... 

Eq. (4) calculated for the highest contrast possible. The solid line gives the best fit of the latter term by an empirical expression whereas the inset displays T r) obtained from T(q) by Fou-rier-inversion. The dashed line in Fig. 7 is the scattering function of a homogeneous sphere of same ... 

The mathematical form of all the scattering functions for a coated sphere—efficiencies and matrix elements—have the same form as those for a homogeneous sphere. Only the scattering coefficients (8.2) are different these may be written in a form more suitable for computations ... 

The method of Simpson and Steinfink (4, 5) which uses liquid scattering functions was employed to take into account the unlocated atoms, assuming that they are uniformly distributed throughout a sphere. Atomic parameters were refined with 235 structure factors corresponding to all reflections with h2 + k2 + l2 <396 except the 111 line. A Guinier-type camera with monochromatized Cu K i radiation was used because of its low background diffusion, to detect the broad diffraction fines of external... 

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. 

H. H. Denman, W. Heller and W. J. Pangonls, "Angular Scattering Functions for Spheres", Wayne State University Press, Detroit, 1966. 

Denman, H.H. Heller, W. Pangonis, W.J. "Angular Scattering Functions for Spheres" Wayne State Univ. Press Detroit, 1966. 

The evalution of the three A, leads to a construction in three-dimensionals space. The solutions are found as the intersection line of a plane with the surface of a sphere. With increasing index 1 of the multipoles the correlation between the resonant structure and the whole structure through the basic scattering function luv(h) usually gets weaker and weaker. 

Particle-scattering functions for random coils and spheres are indistinguishable at values of Z near unity (Figure 2). Likewise, the radii of random coils and spheres are also similar in the range of Z values less than 1.2 (Figure 3). This is reflected in the similarity of radii calculated for random coils and spheres (columns 7 and 8 of Table II). The largest difference between radii calculated for these two shapes is 8.5% for... 

Figure 2. Reciprocal of particle-scattering functions for random coils (-------) and spheres (------).

Taking the hard sphere colloids as a reference state, the mean-square displacement (MSD) in dilute suspensions is associated with the particle self-diffusion whereas at finite volume fractions the onset of interactions marks the alteration of the dynamics. The latter can be probed by the intermediate scattering function C(, t) which measures the spatiotemporal correlations of the thermal volume fraction fluctuations [91]. Figure depicts two representations (lower inset and main plot) of the non-exponential for a nondilute hard sphere colloidal... 

Fig- 7 Intermediate scattering function C q,t) for a suspension of PMMA hard spheres (radius R = 118 nm) recorded at R = 2.95 and volume fraction (/> = 0.42 in a semi-log and log-log representation (lower inset). The static structure factor is shown in the upper inset where the vertical line indicates the value qR = 2.95 at which the function C(g,/) was recorded. The short-time collective diffusion is obtained from the initial plot of C(, i) (tetl line)... 

Fig. 19 (a) Intermediate scattering function C(, r) of hard sphere PMMA (7 h = 205 nm) suspension (in cz5-decalin) at qR = 2.68, from the dilute to the glassy regimes (volume fraction (+) 0.494, open circle 0.528, open triangle 0.535, open square 0.558, (x) 0.567, diamond 0.574, solid triangle 0.581, solid circle 0.587). Solid lines represent MCT fits. Taken from [237]. (b) Respective response of a colloidal polybutadiene star suspension (in good solvent cyclohexane, at 20°C) at different values of the effective hydrodynamic volume fraction c/c open circle 0.06, open square 0.72, asterisk 1.02, open triangle 1.48. Taken from [248]... 

Fig. 21 Steady state incoherent intermediate scattering functions d> (r) as functions of accumulated strain yt for various shear rates y the data were obtained in a col loidal hard sphere dispersion at packing fraction

In Fig. 4.16 the particle scattering function P(0) is plotted for random coils, spheres, and rods as a function of respective parameters. 

For low polydispersities, the parameter cra corresponds to the relative standard deviation. The choice of the type of distribution function is of minor importance [78,79]. The intraparticular scattering function of spheres with the radius a is given as [42]... 

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