Scattering vector calculation

Figure 6. A ngular dependence of apparent structure factor S(K) for PCS Al-Dl in water (0) C = 0J2 mg/mL fVl C = 0.46 mg/mL C ) C = 0.71 mg/mL (O) C -- 1.1 mg/mL and (X) C = 2.2 mg/mL ( ) values at small scattering vector calculated using Equation 5. Identical representation for S(K) is obtained by assuming P(K) for PCS to be that of a star polymer with polydisperse rays fsee Equation 6).

Fig. 9a,b. The intermoleailar scattering patterns calculated for a the smectic A b the smectic B phases of the mesogen GB(4.4, 20.0, 1, 1) with the scattering vector parallel to the layer normal. The scaled scattering vectors Q and Q range from -8n to 8n... 

Using the valence profiles of the 10 measured directions per sample it is now possible to reconstruct as a first step the Ml three-dimensional momentum space density. According to the Fourier Bessel method [8] one starts with the calculation of the Fourier transform of the Compton profiles which is the reciprocal form factor B(z) in the direction of the scattering vector q. The Ml B(r) function is then expanded in terms of cubic lattice harmonics up to the 12th order, which is to take into account the first 6 terms in the series expansion. These expansion coefficients can be determined by a least square fit to the 10 experimental B(z) curves. Then the inverse Fourier transform of the expanded B(r) function corresponds to a series expansion of the momentum density, whose coefficients can be calculated from the coefficients of the B(r) expansion. 

Here P(q) is the particle scattering factor and q = (47i/Z0) sin (0/2) is the scattering vector. The value of P reflects the specific size and shape of the polymer particle. This parameter has been calculated and tabulated for many different kinds of idealized colloidal and macromolecular structures (Burchard, 1994 Evans, 1972 Tanford, 1961). 

Fresnel s equations can be used to calculate the dependence of the reflectivity [(reflected beam intensity)/(incident beam intensity)] on the incident angle expressed in terms of the scattering vector Q (Fig. 5),... 

From specimens cut in the x-z and y-z shearing planes, mean square chain dimensions similar to Rg j and Rg q can also be determined in the principal shear directions x, y and z. Typical isointensity patterns in the three shearing planes are shown in Fig. 28 for sample B. It can be seen that the shape of these curves in the x-z and y-z planes is nearly elliptical like in the x-y plane. However, due to the symmetry of the shear flow, the z direction is a principal direction for molecular orientation at all scales and the major and minor axes of the isointensity curves are indeed found to be parallel to the x,z and y,z directions. Therefore, the analysis of the scattering data in the x-z and y-z planes has been carried out by fixing the direction of the axes of the ellipse (to 0° and 90°) in the least-squares fit program. The major and minor axes of these ellipses then define the scattering vectors q, q and q in the principal shear directions and allow us to calculate the mean square chain dimensions Rg, Rg y and Rg 2-... 

Yoon and Flory > have compared the experimental data of Gawrisch et al. to the scattering of unperturbed chains calculated on the basis of the rotational isomeric state theory. They found good agreement up to the highest values of the scattering vector corresponding to distances of approximately one chain diameter... 

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... 

It should be noted, that the applied spherical model is just an approximation and does not necessarily represent the real structure of the micropores. However, the structural resolving power of this method is improved by assuming a pore size distribution. The additional structural parameters obtained are the distribution width and the number density of pores, which allows a calculation of [)ore volume and surface area, even for ultra-microporous materials, where no Porod decay daldQ(qR > 4..5) oc q [6] is observed within the measured scattering vector range (here q ai-R 4). 

The refractive index of the ethane/propane mixture, needed for calculation of the scattering vector, was determined from experimental density measurements and an empirical Lorentz-Lorenz relationship (12). Incorporating the errors from the viscosity and refractive index calculations into the Stokes-Einstein relation results in a maximum error in the hydrodynamic radius of approximately 5%. 

Similar expressions can be written for any strain or stress component in any reference system. In Equation (79) a,- is in fact a placeholder for a,-, e,-, cr,-, Si and also for 8h that gives the diffraction peak shift caused by the strain in a crystallite. To calculate the peak shift for a polycrystalline sample, 8h (R + r, g) given by Equation (79) must be averaged over r, k and g, where g represents those crystallite orientations for which h is parallel to y, the direction in the sample of the scattering vector. Taking account that the sample could be textured this multiple average is the following ... 

From this relation, we will calculate the expression of this distribution. In order to simplify the calculations, we will consider a peak of the form (001). The tip of the scattering vector, in this case, describes a reciprocal lattice point with coordinates (hjt,l), the extent of which increases when the mean ciystal size decreases. The diffiacted intensity is different from zero only when the values of h and k are very small (see Figure 5.8) and the norm of the scattering vector can be expressed by the relation ... 

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