Wave vector structure factors

Fig. 3.19 Single chain dynamic structure factor Schain(Q>0/5(Q)chain from a M =80 kg/mol PEP melt at T=492 K for the following scattering wave vectors Q=0.03 A Q=0.05 A Q=0.068 A" 0=0.077 A (from above), Q=0.09 A Q=0.115 A"LThe solid lines represents a fit with Eq. 3.39. (Reprinted with permission from [43]. Copyright 2003 The American Physical Society)...

This representation of a structure factor is equivalent to thinking of a wave as a complex vector spinning around its axis as it travels thorough space (Fig. 6.2b). If its line of travel is perpendicular to the tail of the vector, then a projection of the head of the vector along the line of travel is the familiar sine wave. The phase of a structure factor tells us the position of the vector at some arbitrary origin, and to know the phase of all reflections means to know all their phase angles with respect to a common origin. 

Figure 5. (a) The dynamic structure factor vs. wave vector along the edge of the... 

Figure 19 Static structure factor (minus the trivial self-scattering term) versus wave vector. Circles are X-ray diffraction results for polyethylene at 1 atm and 430 K. Curves are predictions of RISM theory for a RIS model with hard site diameters of 3.70 A (dashed line) and 3.90 A (solid line). The solid square is experimental S(0) - 1. An expanded view of the first peak is shown in the inset. (From Ref. 123.)...

To take into account the disparity in electrons, and hence scatting power, for the various atoms in the unit cell, the atomic number Zj has been introduced as a means of defining amplitudes for the component waves. The total diffracted wave for the entire crystal will be the product of the equation above with the total number of unit cells in the crystal.1 For a single unit cell, like that in Figures 5.12a and 5.12b, an atom s contribution to the total structure factor of one unit cell can also be illustrated in vector terms as in Figure 5.13. To put the expression in the correct units, it is necessary to multiply the summation by a constant, the volume of the unit cell. Here, that is simply V = a x b x c. Thus... 

FIGURE 5.13 The structure factor Fhki is a wave, and it therefore has an amplitude and phase. It can also be described as a vector in the complex plane, as was seen in Chapter 4. The individual wave contributions to F%ki by each atom in the molecule can also be described by vectors. The sum of these vectors for all atoms yields Fhki The vectors added here correspond to contributions from the five atoms of the molecule filling the unit cell in Figure 5.12. 

The function F is called the structure factor for the hkl family of planes in the crystal. Because it is a wave, or vector, it has both a magnitude and a phase. The set of all F for all values of h comprise the set of diffracted rays resulting from all possible families of planes in the crystal and thereby constitutes the diffraction spectrum of the crystal. The magnitude of Fis readily measured as the square root of the observed diffracted intensity, that is, /lh = Fh, but there is no experimental means presently available to directly measure its phase 4 ... 

The summation of exponential terms on the right is a Dirac delta function, a discrete function, which is everywhere zero except when the argument is zero or integral. The summation on the left is a continuous function, which determines the value of the entire transform at those nonzero points. Now d ki is normal to the set of planes of a particular family, and d ki I is the interplanar spacing. In order for dhu s = 1, s must be parallel with dhki and have magnitude 1/ Smreciprocal lattice vector. If s h, then there is destructive interference of the waves diffracted by different unit cells, and the resultant wave from the crystal is zero. The elements of the diffraction spectra, the structure factors, for the crystal can therefore be written as... 

The structure factor Fkki for any hkl is a wave which can be represented as a vector composed of scattering contributions fj for every atom j in the unit cell, where the individual fj are also vectors. In Figure 8.1, the vector sum of the fj for all of the light atoms, J2j fj is usually a resultant vector of modest length. This is because the sum of any collection of small, essentially arbitrary vectors tends to be small (waves of random phase tend to destructively interfere rather than constructively interfere for vectors, a random walk never... 

Let us consider the simplest case. Suppose that we have a pure sinusoidal (Cq = 0) wave modulation with a single modulation vector q. The expression of the structure factor for the satellite reflection h = H + mq is reduced to ... 

For crystalline samples under plane-wave illumination, a diffraction pattern is observed as a spot pattern. The individual spots depend on the crystal orientation, its structure factor, and obey Bragg s law. Bragg s law states that the difference between the scattered k and the incident ko wave vector is equal to a vector g of the reciprocal lattice ... 

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