Integral equations structure factor

Equation 1 is a discrete Fourier transform, it is discrete rather than continuous because the crystalline lattice allows us to sum over a limited set of indices, rather than integrate over structure factor space. The discrete Fourier transform is of fundamental importance in crystallography - it is the mathematical relationship that allows us to convert structure factors (i.e. amplitudes and phases) into the electron density of the crystal, and (through its inverse) to convert periodic electron density into a discrete set of structure factors. 

Equations (125) and (126) explicitly show that in the initial slope approximation the elements of the generalized mobility matrix can be expressed only in terms of integrals over the corresponding partial static structure factor. Both equations are valid as long as one assumes a Gaussian distance distribution of the distances r between the monomers i on arm a and monomers j on arm p. 

To analyze this phenomenon further, 2D numerical simulations of (49) and (50) were performed using a central finite difference approximation of the spatial derivatives and a fourth order Runge-Kutta integration of the resulting ordinary differential equations in time. Details of the simulation technique can be found in [114, 119]. The material parameters of the polymer blend PDMS/PEMS were used and the spatial scale = (K/ b )ll2 and time scale r = 2/D were established from the experimental measurements of the structure factor evolution under a homogeneous temperature quench. 

In equation (4.51) of Chapter 4, the equal time structure factor, C (x), was defined. For light scattered from fluctuations in the dielectric properties of a material, it was shown that the light intensity was proportional to this quantity. In the problem of total intensity light scattering discussed in that chapter, the measurement is integrated over time and time-dependent fluctuations are not directly observed. When time-dependent fluctuations are... 

Olvera de la Cruz and Sanchez [76] were first to report theoretical calculations concerning the phase stability of graft and miktoarm AnBn star copolymers with equal numbers of A and B branches. The static structure factor S(q) was calculated for the disordered phase (melt) by expanding the free energy, in terms of the Fourier transform of the order parameter. They applied path integral methods which are equivalent to the random phase approximation method used by Leibler. For the copolymers considered S(q) had the functional form S(q) 1 = (Q(q)/N)-2% where N is the total number of units of the copolymer chain, % the Flory interaction parameter and Q a function that depends specifically on the copolymer type. S(q) has a maximum at q which is determined by the equation dQ/dQ=0. 

To elucidate the structure of a solution of flexible polyelectrolytes, we again use the integral equation theory approach of Sect. 2.2. The necessary structure factor is determined self-consistently using the reference chain (9) of the last section. The intermolecular interactions are taken into account by a medium-induced intramolecular potential [35, 47,48]... 

These expressions are formally exact and the first equality in Eq. (123) comes from Euler s theorem stating that the AT potential u3(rn, r23) is a homogeneous function of order -9 of the variables r12, r13, and r23. Note that Eq. (123) is very convenient to realize the thermodynamic consistency of the integral equation, which is based on the equality between both expressions of the isothermal compressibility stemmed, respectively, from the virial pressure, It = 2 (dp/dE).,., and from the long-wavelength limit S 0) of the structure factor, %T = p[.S (0)/p]. The integral in Eq. (123) explicitly contains the tripledipole interaction and the triplet correlation function g (r12, r13, r23) that is unknown and, according to Kirkwood [86], has to be approximated by the superposition approximation, with the result... 

Figure 19. Static structure factor S(q) in the small- region, at T = 297.6 K, for p = 0.95, 1.37, 1.69, and 1.93 nm-3 (from the bottom to the top), calculated with the ODS integral equation. The dotted lines correspond to the calculations with the twobody potential alone and the solid lines correspond to those that combine the two- and three-body interactions. The squares are for the experimental data of Formisano et al. [13] and the filled circles, at zero-q value, for the PVT data of Michels et al. [115]. Taken from Ref. [129].

Static Structure Factor S(q) at q — 0, Calculated with the HMSA+ODS Integral Equation Scheme by the Using Two-Body Potential Alone and the Two- plus Three Body Potentials0... 

As mentioned above, careful numerical inspection of the k integral in Equation (21) shows that all of the contribution to the integral occurs for k sufficiently large that Equation (26) is the appropriate form for the dynamic structure factor for evaluating the integral. Equation (23) is never used in the calculations, but it was presented to motivate the form of Equation (26). 

The first system we consider is the solute iodine in liquid and supercritical xenon (1). In this case there is clearly no IVR, and presumably the predominant pathway involves transfer of energy from the excited iodine vibration to translations of both the solute and solvent. We introduce a breathing sphere model of the solute, and with this model calculate the required classical time-correlation function analytically (2). Information about solute-solvent structure is obtained from integral equation theories. In this case the issue of the quantum correction factor is not really important because the iodine vibrational frequency is comparable to thermal energies and so the system is nearly classical. 

The model is formulated in terms of an integral equation which is solved with the condition at the boundary of the open crack and the bridging zone (.x — 0) that 6(0) = 5C. It is interesting to note that the structure of the rate-dependent problem is such that, aside from the material parameters, the solution is completely determined for a given crack velocity. For a given velocity, the value of the applied stress intensity factor, K, and the length of the cohesive zone, L, that maintains this condition is determined. Selected results are presented below. 

K. S. Schweizer and J. G. Curro, Macromolecules, 21, 3082 (1988). Integral Equation Theory of Polymer Melts Density Fluctuations, Static Structure Factor, and Comparison with Incompressible and Continuum Limit Models. 

For each Bragg reflection, the raw data normally consist of the Miller indices (h,k,l), the integrated intensity I(hkl), and its standard deviation [ a[I) ]. In Equation 7.2 (earlier), the relationship between the measured intensity / [hkl] and the required structure factor amplitude F[hkl) is shown. This conversion of I hkl) to F hkl) involves the application of corrections for X-ray background intensity, Lorentz and polarization factors, absorption effects, and radiation damage. This process is known as data reduction.The corrections for photographic and diffractometer data are slightly different, but the principles behind the application of these corrections are the same for both. 

The Fourier transform of the y-th TS unit layer, Gj(HK.Lr), is two-dimensionally periodic and the reciprocal lattice coordinate in the direction lacking periodicity is not restricted to integral values but is a real variable, labeled Lr. In Equation (22), Gj plays a role analogous to that of the atomic scattering factor in the expression of the structure factor. 

Figure 5.2 exhibits the site-site static structure factors, Eq. (5.90), calculated from the extended version of the RISM integral-equation theory [58, 59]. The peak positions are fcmax = 1-69 and 1.65 A for A-A and B-B pairs, respectively. Note that in the k 0 limit all the site-site structure factors coincide [40], and we define... 

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