Form factor sphere

The parameters reported by Zintl Hauke were taken as the starting point of the parameter determination. Using these parameters, structure factors were calculated for all of the planes in the sphere of reflection. The atomic form factors of James Brindley (1935) were used. (Subsequent calculations made with two... 

Figure 3 Example of SANS curves at two times of the reaction. The lines are calculations of the form factor. (A) prior to TEOS addition, the micelles are well described by core-shell spheres, with an external radius of 7.1 ran. ( ) 15 minutes after the beginning of the reaction, the micelles can be viewed as cylinders of length 50 nm and radius 6.9 nm.

In the case of finite star chains with very high functionality, the units are concentrated near and in the star core. Therefore, their theoretical behavior can approximately be described by a rigid sphere [2]. The form factor of a sphere presents a series of oscillations. The experimental data of stars with 128 arms [67] show a smooth function covering the first two oscillations of the sphere, followed by a peak coincident with the third oscillation and the asymptotic behavior for high q previously described for stars of lower functionalities. It seems that the chain resembles a soft spherical core with a peripheral region of considerably smaller density. 

The form factor depends only on intraparticle interferences and is independent of concentration as long as the particles remain unchanged. For example, for a uniform sphere of radius a comparable to Xq ... 

The form factor for a particle of arbitrary shape can be calculated by numerical integration of (6.10). However, for certain regular geometrical shapes, it is possible to obtain analytical expressions for /. In this section we consider one such particle, a homogeneous sphere. 

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. 

To allow for the influence of various particle shapes and size distributions within a defined sieve fraction, in lay-out calculations it is customary to employ an effective particle diameter, deff, as nominal size [32]. The diameter de is defined as the ratio of equivalent diameter A and a form factor j/. A is equal to the diameter of a sphere with a volume equal to the (average) volume of the particles, and ijj is the average ratio of the particle surface to the surface of a sphere of equal volume. 

Therefore, the three significant parameters for the evaluation of the structure factors of glasses are a) porosity, b) polydispersity, and c) the size of solid entities. Fig. 3a illustrates the calculated averaged structure factors. Note that as the size of the hard spheres is increased the maxima are shifted to the lower Q region corresponding to the increasing distance between pore centroids. Based on the setting parameters for the structure factors, the form factor is calculated for a polydisperse pore size distribution ftmction, D (Rp), that states the number of pores defined by the radius parameter Rp (Fig. 3b). 

The mechanisms and rates of metal-catalyzed initiation operative in individual reaction systems are determined by a complex mixture of factors the metal and type of complexes it forms (inner sphere or outer sphere), the chelator or complex-ing agent, redox potential of the metal and its complexes, solvents, phase localization of the metal, and availability of oxygen or preformed hydroperoxides. The reactions outlined below show the multiplicity of mechanisms possible. 

We should now consider the factor 6 in the numerator of the pre-exponential in Eq. (14) which has the physical meaning of either a form factor, packing factor or coordination number of the closest spherical (cubic) sphere packing. The points of Fig. 3 are somewhat scattered probably because of the equal proportions of open cells (9J in samples of different volume weights. Eq. (14) has a maximum when 9(,/9p = 1 (Fig. 3), i.e. when the volume ratio of polymer in a sample is equal to that in a plastic foam with closest spherical packing the gas phase volume is then 74% which, for polyurethane foam, corresponds to a volumetric weight of 315 kg/m. ... 

The reciprocal form factor /P(q) for a ideal linear chain [Eq. (2.160)] is shown in Fig. 2.21 as a function of q iR ) (medium line) and is compared with the reciprocal form factors for a rigid rod (thin line) and for a solid sphere (thick line). The form factors of a rod,... 

Calculate the form factor of a uniform sphere of radius R. 

Figured displays simulated form factors for a multiarm star polymer of varying functionality and a hard sphere [41], The high-g asymptotic behavior, characteristic of the coil structure, is absent in the latter case. A handicap in the experimental determination of P(g) is often the narrow-g range accessible by the scattering techniques that can be overcome through the combination of low-g light scattering and high-g X-ray and/or neutron scattering (utilized on the same system). Size and shape also determine the translational diffusion Dq of the nanoparticles in dilute solution, and hence Dq can prove the consistency of the scattering results.

Fig. 6 Computer simulations results of the scattering wavevector dependence of the form factors P( ) of a star with / = 24 arms (solid line) and a (compact lattice) hard sphere with nearly the same number of beads (dotted line). Taken from [41]... 

The experimental form factor P( ) shown in Fig. 12a can be expressed as P q) = [bcFc q,rc) + bsFs q,rc,R )], where bc,bs are the contrast factors for the core (c) and shell (5) with core radius and overall micelle radius R, whereas Fc q), F q) aiQ the scattering amplitudes of the core and shell, respectively. Under core contrast conditions (Z s 0), the expected first minimum for the compact sphere at high q values falls outside the ("/-range, whereas under shell contrast conditions the power-law behavior arising from blob (swollen PEO shell) scattering is observed. Hence, the dual colloid-polymer character of the particle is clearly reflected in Fig. 12a. 

Fig. 10 Glass form factors fq as function of wavevector q in a colloidal glass of hard spheres for packing fractions as labeled. Data obtained by van Megen and coworkers by dynamic light scattering are qualitatively compared to MCT computations using the PY-5j at values chosen ad hoc to match the experimental data from [12]. The PY structure factor at the glass transition density = 0.58 is shown as broken line, rescaled by a factor 1/10...

K )dP ( ) hI (s ) is the form factor envelop of the scattering, made from the form factor of ideal spheres and the attenuation term describing a smooth transition of the density at the phase boundary (cf. p. 124). 

Aluminium oxide (AI2O3) or zirconium oxide (Zr02) are also used as supports of reticulated deposits based upon polymers of butadiene or styrene-divinylbenzene or hydroxymethylstyrene. Porous graphite, in the form of spheres whose surface is 100 per cent carbon and therefore completely hydrophobic, has been used in applications with compounds possessing atoms carrying lone pairs of electrons thus having high retention factors. 

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