The structure factor

Electrons also scatter radiation by another effect, the Compton effect. Compton scattering adds to the general background of scattered X-rays, and is usually ignored in structure determination. 

To obtain the total intensity of radiation scattered by a unit cell, the scattering of all of the atoms in the unit cell must be combined. This is carried out by adding together the waves scattered from each set of (hkl) planes independently, to obtain a value called the structure factor, F(hkl), for each hkl plane. It is calculated in the following way. 

The amplitude (i.e. the strength ) of the diffracted radiation scattered by an atom in a plane (hkl) will be given by the value of fa appropriate to the correct value of sin9/X (=1/2dhlci) for the (hkl) plane. However, because the scattering atoms are at various locations in the unit cell, the waves scattered by each atom are out of step with each other as they leave the unit cell. The difference by which the waves are out of step is called the phase difference between the waves. 

The phase difference between the waves scattered by two atoms will depend upon their relative positions in the unit cell and the directions along which the waves are superimposed. The directions of importance are those specified by the Bragg equation, equation (6.1), which, for simplicity, are denoted by the indices of the (hkl) planes involved in the scattering, rather than the angle itself. The phase of the wave (in radians) scattered from an atom A at a position xA, yA, zA, into the (hkl) reflected beam is  

When the scattered waves from all of the atoms in a unit cell are added, the form of the resultant wave will depend, therefore, upon both the scattering power of the atoms involved and the individual phases of all of the separate waves. A clear picture of the relative importance of the scattering from each of the atoms in a unit cell, including the phase information, and how they add together to give a final value of F(hkl) can be 

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If the structure factor vanishes, the corresponding fonn factor is irrelevant as it is multiplied by a zero... 

The correlation fiinction G(/) quantifies the density fluctuations in a fluid. Characteristically, density fluctuations scatter light (or any radiation, like neutrons, with which they can couple). Then, if a radiation of wavelength X is incident on the fluid, the intensity of radiation scattered through an angle 0 is proportional to the structure factor... 

Karle J and Flauptman FI 1950 The phases and magnitudes of the structure factors Acfa Crystaiiogr.Z 181-7... 

This treatment may be extended to spheres core-shell structure. If the core density is p 0 to fp the shell density is p2 in the range o density of the surrounding medium is Pq, th of the structure factor becomes... 

As we have introduced the structure factor S(q) (B1.9.113), it is usefiil to separate this factor into two categories of interferences for a system containing A scattering particles [9] ... 

In the procedure of X-ray refinement, the positions of the atoms and their fluctuations appear as parameters in the structure factor. These parameters are varied to match the experimentally determined strucmre factor. The term pertaining to the fluctuations is the Debye-Waller factor in which the atomic fluctuations are represented by the atomic distribution tensor ... 

The Q and ft) dependence of neutron scattering structure factors contains infonnation on the geometry, amplitudes, and time scales of all the motions in which the scatterers participate that are resolved by the instrument. Motions that are slow relative to the time scale of the measurement give rise to a 8-function elastic peak at ft) = 0, whereas diffusive motions lead to quasielastic broadening of the central peak and vibrational motions attenuate the intensity of the spectrum. It is useful to express the structure factors in a form that permits the contributions from vibrational and diffusive motions to be isolated. Assuming that vibrational and diffusive motions are decoupled, we can write the measured structure factor as... 

The (5-part in (2.50) is responsible for elastic scattering, while the second term, proportional to the Fourier transform of C(t), leads to the gain and loss spectral lines. When the system exercises undamped oscillations with frequency Aq, this leads to two delta peaks in the structure factor. 

Nevertheless, Leggett et al. [1987] have argued, with some provisos, that [with the initial condition (O) = 1] and C(t) may be practically taken the same. If C(t) then obeys the damped oscillator equation (2.41), then the inelastic part of the structure factor has the Lorentzian form with the peaks at [Pg.25]

Here Pyj is the structure factor for the (hkl) diffiaction peak and is related to the atomic arrangements in the material. Specifically, Fjjj is the Fourier transform of the positions of the atoms in one unit cell. Each atom is weighted by its form factor, which is equal to its atomic number Z for small 26, but which decreases as 2d increases. Thus, XRD is more sensitive to high-Z materials, and for low-Z materials, neutron or electron diffraction may be more suitable. The faaor e (called the Debye-Waller factor) accounts for the reduction in intensity due to the disorder in the crystal, and the diffracting volume V depends on p and on the film thickness. For epitaxial thin films and films with preferred orientations, the integrated intensity depends on the orientation of the specimen. 

These results indicate that in these new ester mydriatics, the structural factors, which influence the development of this type of pharmacological action are similar to those made evident by the chemical investigations of Jowett and Pyman and the pharmacological work of Marshall, Dale, Laidlaw and Cushny on the tropeines. The nature of the basic component is obviously important since mydriasis is produced by simple bases such as ephedrine. As regards the nature of the esterifying acid, Jowett and Pyman drew the following conclusions — ... 

For comparison with Gobs- values, calculated values were determined as follows Gcaic. = Zm.Fi where F, is the structure factor and wt- the... 

For this range of values of u the structure factor for (154) is much greater than that for (037), if aluminum atoms are at (a) and (b) the observation that the latter plane reflects much more strongly than the former despite its smaller interplanar distance accordingly eliminates this arrangement. 

For arrangements (a) and (b) the structure factor in the first order is 4Ba for planes with hSG even and kSG + lSG even, and 0 for all other planes. These arrangements are definitely eliminated by the experimental data for example, (411)SG is absent, and (521)sg> with smaller interplanar distance, reflects very strongly at the same wave length. Such wide discrepancies cannot be explained as due to the effect of sulfur and oxygen atoms. The barium atoms are, therefore, located as in (c). Because of the presence of other atoms no attempt was made to determine the two parameters involved. 

With 24 0 in the general case strong reflections would be expected from some planes with h -f- k -f- l odd unless two of the parameters were very nearly equal, in which case the structure factor would become very small. This limits considerably the region of parameter values to be discussed but not sufficiently to permit the deduction of a single set of values from X-ray date alone. 

There accordingly remains only structure 3. We may take 8(Mn,Fe) in 8e rather than 8i, which leads to the same arrangements. The structure factor for various orders from (100) is then... 

Now there are two physically distinct arrangements of the metal atoms corresponding to w = 0.030, the first with u = 0.030, and the second with u = — 0.030 and it is not possible to distinguish between them with the aid of the intensities of reflection of X-rays which they give. Let us consider the positions 24e. The structure factor for 24e is ... 

The intensity of the third order (111) reflection is greater than that of the second order, and the structure factor, 5 = VA2 + B2, must be greater for the third order. For Arrangements II and III, the corresponding values are... 

On the previously made assumption regarding relative reflecting powers, the values for the structure factor of the classes of planes increase in this... 

However, an error was made in the application of the temperature factor, which resulted in incorrect weights therefore the structure factors and their derivatives were recalculated on the basis of these parameters and a second least-squares treatment was carried out as described below. 

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