Guinier function

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)). 

This last result is known as the Guinier function and is the basis for determining the radius of gyration from small-angle scattering experiments for objects with unknown form factor. 

At these small angles, P(6) can be approximated by the Guinier function [l-q ((.y )/3)], which allows the determination of the radius of gyration of macromolecules independently of their shape the higher limit for measurements by SAXS, SANS, and light scattering is 5nm, 20 nm, and 200 nm, respectively. 

These complex CSAS profiles Ik q) (k = s and d for the swollen and dried pellicles, respeaivdy) will be well approximated by a linear combination of the power laws having the eiqronents and the Guinier functions with the radii of gyration... 

For hierarchical systems composed of n stmctural levels, eqn [7] is extended using the Guinier function as described by Beaucage, ... 

Notice the change of variables Guinier s law is applied to the Fourier transform of the peak as a function of the order (h) of the peak at fixed positions r. 

Step 3 Guinier Plot Separation of Size and Distortion Effects. The inner part of the correlation functions y0bs(h) (T) s readily expanded into a power series. For this purpose we resort to Eq. (8.14). Assuming that instrumental broadening is already eliminated we have... 

Figure 8.35 shows for homogeneous identical spheres the radial correlation function (Guinier and Fournet [65] p. 12-19 Letcher and Schmidt [192])... 

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... 

Recently, a detailed study4,5 of AB02 compounds confirms that AgFe02 has the CuFe02 (delafossite) structure. The cell dimensions, determined with a Guinier camera, are found to be a = 3.0391 2 A. and c = 18.590 2 A. Single crystals are used to measure resistivity as a function of temperature.6 Silver fer-rate(III) exhibits semiconductor behavior with an activation energy of 0.7 e.V. An unusual anisotropy in resistivity is found withp(j c) = 3 X 107 S2-cm. and P( c) = 2 X 1010 J2-cm. 

Figures 45 and 46 shows the logarithm of intensity count as a function of scattering angle (2 theta in minutes) for as-cast and as-postcured epoxies respectively. Specimens of the same thickness were used (0.825 mm). Guinier and Debye analysis were then carried in the regions indicated.

Strictly speaking, A/ depends also on the atomic number of the scattering element, which means that a different correction curve is required for every element. But the variation of A/with Z is not very large, and Fig. 13-8, which is computed for an element of medium atomic number (about 50), can be used with fairly good accuracy as a master correction curve for any element. Figure 13-8 was calculated from data in James [G.7, p. 608]. Anomalous dispersion is also discussed by Guinier [G.21]. Values of A/ as a function of Z at constant 2, for five characteristic wavelengths, can be calculated from data in [G.ll, Vol. 4]. 

Assuming that no correlation among particles is present, the meanings of the structure related parameters of Guinier s law are readily established by application of the mathematical tools of scattering (cf. Chap. 2). Further assumptions state that each particle k is considered to be immersed in a matrix of constant electron density and that its correlation function is monotonously decaying. Thus, the particle is discriminated from the matrix by its number nik of electrons that it has more (respectively, less) than the homogeneous matrix, nik is called the number of excess electrons of the particle k. 

The region of crossover between the functionalities of Equations 14.25a and b is called the Guinier regime and for p 0 is described by... 

In summary, return to Figure 14.10, which schematically illustrates all the quantitative aspects of the structure factor. In htting data from a real system it is best to analyze the Guinier regime hrst. Equation I4.47a implies that the inverse normalized (to q = 0) scattered intensity is a linear function of q, i.e.. 

Information about both the size and the geometry of the scattering particles can be obtained from different scattering domains. In the Guinier range with QR < < 1 the scattering function can be written... 

The function p(r) represents the frequency of vector length r connecting small volume elements within the volume of the scattering particle, that is, the protein molecule, with maximum dimension Dmax- As the use of the Guinier approximation for the determination of the radius of gyration, Rg, might lead to errors caused by concentration or aggregation effects, we used the normalized second moment of the pair distance distribution function for calculation of R, ... 

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