Measuring Structure Factors

In an experiment, only the diffracted intensity is directly determined, and not the structure factor. More precisely, one normally determines the integrated intensity by integrating the intensity over a weU-defined section in reciprocal space. In this way, the result does not depend on details such as the divergence of the X-ray beam or the width of the reflection. In bulk diffraction, the integration is over the entire point in reciprocal space. In SXRD, the scattering is along a rod, and thus the 

The integrated intensity is not only proportional to the square of the structure factor amplitude but also contains several other factors that need to be taken into account and that depend on the scattering geometry and the type of scan. In order to find the relation between the structure factor and the integrated intensity, the experimental geometry needs to be analyzed. For surface diffraction, the differential scattering cross section can be written as [29] 

When evaluating this integral, one encounters several geometrical factors, whose precise expression depends on the type of diffractometer. We only mention the most important factors here for a z-axis diffractometer more details for several types of diffractometers can be found in the literature [28, 29]. The conversion of angular space in Eq. (3.4.2.32) to reciprocal space (using the diffraction indices h, k, and 1) gives the Lorentz factor 

The amount of rod that is intercepted depends on the outgoing angle, leading to a rod interception correction factor 

Finally, we need an expression for the polarization factor P. For a fully horizontally polarized beam and a vertical scattering geometry, this is 

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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... 

In this work I choose a different constraint function. Instead of working with the charge density in real space, I prefer to work directly with the experimentally measured structure factors, Ft. These structure factors are directly related to the charge density by a Fourier transform, as will be shown in the next section. To constrain the calculated cell charge density to be the same as experiment, a Lagrange multiplier technique is used to minimise the x2 statistic,... 

There are two approaches to map crystal charge density from the measured structure factors by inverse Fourier transform or by the multipole method [32]. Direct Fourier transform of experimental structure factors was not useful due to the missing reflections in the collected data set, so a multipole refinement is a better approach to map charge density from the measured structure factors. In the multipole method, the crystal charge density is expanded as a sum of non-spherical pseudo-atomic densities. These consist of a spherical-atom (or ion) charge density obtained from multi-configuration Dirac-Fock (MCDF) calculations [33] with variable orbital occupation factors to allow for charge transfer, and a small non-spherical part in which local symmetry-adapted spherical harmonic functions were used. 

We want to know the electron density that is determined by the measured structure factor amplitudes and their phases. The electron density at point x is calculated by a Fourier summation ... 

The progress of iterative real- and reciprocal-space refinement is monitored by comparing the measured structure-factor amplitudes IFobsl (which are proportional to (/obs ) /2) with amplitudes IFca(c I from the current model. In calculating the new phases at each stage, we learn what intensities our current model, if correct, would yield. As we converge to the correct structure, the measured Fs and the calculated Fs should also converge. The most widely used measure of convergence is the residual index, or R-factor (Chapter 6, Section V.E). 

Fourier representation of electron density suggests the possibility of direct structure analysis. If all structure factors, F(hkl), are known, p(xyz) can be computed at a large number of points in the unit cell and local maxima in the electron-density function are interpreted to occur at the atomic sites. A typical single-crystal diffraction pattern of the type used for measuring structure factor amplitudes is shown in Figure 6.12. 

The Fourier coefficients in crystallographic analysis are the measured structure factor amplitudes of diffraction maxima and correspond to the Fourier transform of the periodic density. Numerical solution of the phase problem enables the Fourier transformation that synthesizes the unit-cell electron-density function and hence the three-dimensional molecular structure. Quantum-chemical computations assume the molecular structure and calculate Fourier coefficients for a limited basis set to redefine the electron density. 

Figure 1. Comparison of measured structure factors of x-ray diffraction lines with those calculated on basis of octahedral and tetrahedral...

The determination of the atomic structure of a reconstruction requires the quantitative measurement of as many allowed reflections as possible. Given the structure factors, standard Fourier methods of crystallography, such as Patterson function or electron-density difference function, are used. The experimental Patterson function is the Fourier transform of the experimental intensities, which is directly the electron density-density autocorrelation function within the unit cell. Practically, a peak in the Patterson map means that the vector joining the origin to this peak is an interatomic vector of the atomic structure. Different techniques may be combined to analyse the Patterson map. On the basis of a set of interatomic vectors obtained from the Patterson map, a trial structure can be derived and model stracture factor amplitudes calculated and compared with experiment. This is in general followed by a least-squares minimisation of the difference between the calculated and measured structure factors. Of help in the process of structure determination may be the difference Fourier map, which is... 

A, C.,s=-0.09 0.02A, Ci=0.02 0.01A and C4s=-0.005 0.003A. For the 138 measured structure factors a global of 1.5 is obtained with 9 structural parameters, the roughness and a scale factor. The agreement is good (Fig. 11) and the relaxations extend deeply into the crystal. The smallest bond length in the refined model is 1.9A, that is a 10% contraction, and is located between the last complete layer and the 25% vacant layer of the octopole. 

In direct methods, the measured structure factors are modified so that the maximum information on atomic position can be extracted from them. Other effects, such as the falloff of intensity at high scattering angles due to atomic size and atomic vibrations (see Figures 3.12 and 3.13. Chapter 3) are eliminated, to be considered when the structure has been... 

As we do not know where our model is in the unit cell, any set of structure factors Fm that we calculate (called Gilt with phase ctcalc) will be wrong. However, if the model (which we have) and the unknown structure are the same, we do not actually need to find xy yy and zyfor each and every atom of our model in the unit cell of the unknown structure. We need to find where and how the model should sit in the unit cell of the unknown structure. Once we know that, the model diffraction pattern and the unknown structure diffraction pattern (for which we have measured structure factors, called -Fobs, but not phases) will be the same - to the extent that the model and the unknown structure are the same. 

Fig. 13. Neutron scattering results for liquid Rb and RbgO at 320 K. S t) (a) measured structure factors S (Q) for Rb (dashed line) and RbgO (full line), (b) pair correlation functions g(r) for Rb (dashed line) and Rb60 (full line)...

The radial deformation of the valence density is accounted for by the expansion-contraction variables (k and k ). The ED parameters P, Pim , k, and k are optimized, along with conventional crystallographic variables (Ra and Ua for each atom), in an LS refinement against a set of measured structure factor amplitudes. The use of individual atomic coordinate systems provides a convenient way to constrain multipole populations according to chemical and local symmetries. Superposition of pseudoatoms (15) yields an efficient and relatively simple analytic representation of the molecular and crystalline ED. Density-related properties, such as electric moments electrostatic potential and energy, can readily be obtained from the pseudoatomic properties [53]. 

In Fig. 2.40 a few representative measured structure factors S(Q) of colloidal spheres at a colloid volume fraction (f> = 0.086 are plotted at a few PEP concentrations. Clearly, the measured stracture factor increases upon adding more free polymer at (2 <0.2 nm, corresponding to an increase of the attraction between the colloids. This increase of S(Q) at small Q has been found also in a few other studies [97-99]. Mutch et al. [99] showed it is possible to rescale structure factors at high q (relatively large polymers) to obtain a universal S Q) behaviour. 

In any case, in order the molecular stmctures (through the associated molecular crystals) be assessed, the experiment for recording the difi ac-tion pattern is indispensable, together with an optimal rationalization of the measured structure factors. The following sections present experimental diffraction techniques and the rationalization of their interpretation, in order to determine the maps of electronic localization. 

T. Arai and I. Yokoyama, Effecting pair potential reproducing the measured structure factor of hquid Cu near the melting point, J. Phys. Condens. Matter, 3, 7475,1991. 

Figure 6 Development of structure in aqueous dispersion of polystyrene latex particles (diameter 50 nm) on removing counter ions by ion-exchange resin, (a) The measured structure factor S(K) is plotted against the wave-number K after (i) 2 h, (ii) 48 h, and (iii) over 300 h. (b) The function ph(r) obtained by Fourier transformation is plotted against separation r (Redrawn from ref. 69).

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