Scattering curve sphere

Fig. 4.9 Representative tits of SANS data for PL02<,PPO wPEOi at different temperatures and concentrations.The solid lines are the best tits using a hard sphere Percus-Yevjck approach and including instrumental smearing. The dotted lines represent the same scattering curves with no instrumental smearing (Mortensen and Pedersen 1993).

In a double logarithmic plot of P(q) vs. q, the asymptotic slope at high q of the upper envelope of the scattering curve depends on the ratio of the wall membrane thickness to the outer diameter, e.g -2 for an infinitely thin shell from Eq. (2) and -4 for a solid sphere from Eq. (3). By fitting an experimental scattering curve to the theoretical functional form of Eq. (I). two parameters. 

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-... 

The porosity (1 — < ) is calculated from eq. 4 by using Ip and the value at the plateau of the scattering curve as the cross section for q —> 0. In addition, for small scattering vectors qR < 2 the Guinier approximation S qR < 2) = exp(—[7] was fitted, assuming homogeneous spheres with radius R . This was done in order to check whether this simple approximation is useful for an estimate of the mean micropore radii without assuming a pore size distribution. With the third fit parameter Xp/rn and eq. 2 the mean specific number density Np/rn of micropores with radius Rp is known as well as from... 

Low Temperature and Higher Water Content. A few neutron scattering spectra are shown in figure 4. Experimental (absolute) intensities are represented by open circles, and concern the sample A3 at 20°C. Four deuteration rates (0 %, 40 %, 50 % and 80 % in w/w) of the decane component have been used, which give rise to rather different spectra, both in level of intensity and in shape. The first maximum is due to interparticle effects which can be roughly taken into account by the hard equivalent sphere model (7,8). Here the scattering curves become practically independent of the interparticle interactions for q>qm -0.03 Two theoretical results are shown while... 

FIGURE 14.4 (a) Normalized Mie scattering curves as a function of scattering angle for spheres of refractive index m = 1.05 and a variety of size parameters and (b) same as (a) but plotted vs. qa. Lines with slope -2 and -A are shown. (From Sorensen, C.M. and Fischbach, D.F., Opt. Cormnun., 173, 145, 2000. With permission.)... 

FIGURE 14.7 Schematic diagram of the envelopes of the Mie scattering curves for homogeneous, dielectric spheres (i.e., ignoring the ripple structure). Dashed lines are for the RDG limit at p = 0 with slope -4 and the p —>oo limit with slope -2. Solid line is the envelope for an arbitrary phase shift parameter p. 

FIG. 16 Comparison of the scattering curves of different structure types 1, spheres 2, Gaussian coils 3, stars (10 arms) 4, rods (i, monodisperse polydis-perse systems for details see Ref. 78). (From Ref. 78.)... 

FIG. 17 Scattering curve of a PEC between anionically and cationically modified PNIPAM (symbols) and its interpretation by theoretical curves of polydisperse systems of spheres of different polydispersity (see the inset table). 

These findings confirm the model of polydisperse systems of homogeneous spheres by exact fits of the static light scattering curves, the expected M a3 dependence for PEC homologues prepared at different polymer concentrations, and the asymptotic q 4 behavior of compact spheres. The quantitative information obtained shows that the concentration of the component solutions does not affect the internal structure of the PECs remarkably but controls their level of aggregation to a great extent. 

X-ray and neutron scattering curves measured beyond the Guinier region lead to further shape information in addition to values. Theoretical curves calculated from simple models of uniform density illustrate this (Fig. 7). Thus for a sphere, the scattering curve is given by [6] ... 

The scattering curve of a sphere descends into a series of minima md maxima as Q increasp- i, with the first minimum at QR = 4.493. Thus the Aq of the sphere can be calculated from the minima as a verification of the Guinier analysis. The curves for full and hollow spheres are very similar at low Q. The intensities of the subsidiary... 

Fig. 7. Scattering curves from models of uniform scattering density, (a) A sphere of radius 6.5 nm and Rq 5.0 nm as used in Fig. 4. (b) A hollow sphere with an outer radius of 6.5 nm as in (a) and a ratio of inner/outer radii of 0.5. (c) A straight, cylindrical rod of length 59 nm and diameter 3.4 nm. (d) The cylindrical rod as in (c) is now bent into a circle of radius 9.4 nm.

This expression is an adaptation of the I(Q) calculation for a sphere and the required integration can be performed numerically. Particles that do not have spherical or near-spherical symmetry do not exhibit the minima and maxima noted above, and the scattering curve I(Q) declines more uniformly as Q increases. Other analytical expressions exist for the calculation of I(Q) for ellipsoids, prisms and cylinders and their hollow equivalents [55]. It should be noted, however, that I(Q) for ellipsoids, prisms and cylinders do not differ greatly. For simple models, a first indication of the macromolecular shape in terms of a triaxial body can be extracted by curve-fitting of the calculated scattering to the experimental curve at low Q. 

Sttj and /8 /8a2 subunits [123-127]. The tryptophan synthase scattering curves were modelled using spheres of 1.2 nm diameter for each of the a and the 2 subunits (total Rq of 1.95 and 3.01 nm, respectively). Good I Q) curve fits to 0 = 1.3-1.7 nm and comparable fits to 2 = 4 nm were obtained using shghtly elongated... 

To a first approximation, the sizes of isometric viruses can be estimated by comparing the experimental maxima and minima with the theoretical curves calculated for spheres and hollow spheres [492-494,504]. However, viruses are composed of protein shells and nucleic acid cores (with carbohydrate and lipid in more complex viral structures), so a full analysis requires the explicit consideration of non-uniform scattering densities. In addition, the principle of icosahedral symmetry in the assembly of the protein shell means that, at large Q, deviations from spherical symmetry will influence the scattering curve. The separation of the scattering curve... 

The quantity B(,(q) presents the scattering amplitude of a homogeneous sphere whereas e(q) solely refers to the variation of p inside the sphere. B(,(q) will vanish for tan(q R)=q R and Io(q )= (q ). Hence, in the case of weU-de-fined particles with spherical symmetry the isoscattering points present a prominent feature of the scattering curves as function of contrast and maybe used to determine R. 

The effect of polydispersity can be seen directly when considering first a system of homogeneous spheres having a Gaussian size distribution [46]. Here the scattering curves are fully determined by the form amplitude BQ(q) as defined... 

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