Crystals structure factor

To a first order approximation, the scattering potential of a crystal may be represented as a sum of contributions from isolated atoms, having charge distributions of spherical symmetry around their nuclei. In a real crystal the charge distribution deviates from the spherical symmetry around the nucleus and the difference reflects the charge redistribution or bonding in the crystal. The problem of experimental measurement of crystal bonding is therefore a problem of structure factor refinement, i.e. accurate determination of the difference between the true crystal structure factors... 

In this section we will discuss perturbation methods suitable for high-energy electron diffraction. For simplicity, in this section we will be concerned with only periodic structures and a transmission diffraction geometry. In the context of electron diffraction theory, the perturbation method has been extensively used and developed. Applications have been made to take into account the effects of weak beams [44, 45] inelastic scattering [46] higher-order Laue zone diffraction [47] crystal structure determination [48] and crystal structure factors refinement [38, 49]. A formal mathematical expression for the first order partial derivatives of the scattering matrix has been derived by Speer et al. [50], and a formal second order perturbation theory has been developed by Peng [22,34],... 

A major application of QED is the accurate determination of crystal charge density. The scientific question here is how atoms bond to form crystals, which can be addressed by accurate measurement of crystal structure factors (Fourier transform of charge density) and from that to map electron distributions in crystals. 

D reconstruction can be performed by restoring the 3D Fourier space of the object from a series of 2D Fourier transforms of the projections. Then the 3D object can be reconstructed by inverse Fourier transformation of the 3D Fourier space. For crystalline objects, the Fourier transforms are discrete spots, i.e. reflections. In electron microscopy, the Fourier transform of the projection of the 3D electrostatic potential distribution inside a crystal, or crystal structure factors, can be obtained from HREM images of thin crystals. So one can obtain the 3D electrostatic potential distribution (p(r) inside a crystal from a series of projections by... 

The mean thermal electron density in the unit cell can be computed by Fourier summation, over the reciprocal lattice vectors S, of the X-ray crystal structure factors ... 

The scattering amplitude is given by the absolute value of the following expression, which is a Fourier transform of the total electron-density distribution in a unit cell and is called a crystal structure factor ... 

Box 3.3. Electrostatic Properties of Molecules Derived from Fourier Summations of Crystal Structure Factors, Fh [211]... 

Because of the small size of the configuration (even if it contains many thousands of atoms) the statistical accuracy of the three-dimensional distribution g(r) is too poor to enable direct transformation to the single crystal structure factor T(Q). In addition there are truncation problems due to long range order, as mentioned earlier for powders. We therefore use essentially the same method as in Section 2.5.4, e.g. equation (24). It is, if possible, sensible to choose the Q points at which the measurement is to be made to be appropriate to the model, otherwise it is necessary to interpolate the data onto these points. For example, if the model is cubic and contains 10 x 10 x 10 unit cells then Q = 0.1 (/iA, B, /C) where A, B and C are the reciprocal lattice vectors and h, k, l are integers. 

To verify and understand how these absences result due to the presence of these microsymmetries, let us consider the following derivation of crystal structure factor in the case of their presence. 

In this equation, Fhki is the crystal structure factor and p r) is the electron density expressed in electron per unit volume. The structure factor Fhki is given in terms of the fractional coordinates x, y, and 2 of the atoms in the unit cell and diffraction vector S in terms of reciprocal vector as... 

It is a simple matter to show that a crystal structure is greatly overdetermined by the measured intensities. This does not tell one how to find a solution, but it makes the search rational. A system of simultaneous equations is formed by the definition of the crystal structure factors given by equation (15). since the values of the scattered intensities are measured for a laige number of h. The unknown quantities in equation (15) are the phases 0 and the atomic positions ry. The known quantities are the F obtained from the measur intensities and the /), which differ little from the theoretically calculated atomic scattering factors for free atoms. Since each equation in (15) involves complex quantities, there are really two equations one for the real and one for the imaginary part. In order to determine the the overdeterminacy, a comparison is made of the number of unknown quantities with the number of independent data available. With the use of CuX<, radiation, the overdeterminacy can be as great as a factor of about 50 for crystals that have a center of symmetry and about 25 for those that do not. In practice, somewhat fewer than the maximum available data are measured, but the overdeterminacy is still quite high. 

The fundamental result was that the necessary and sufficient condition for the electron density distribution in a crystal to be non-negative is that an infinite system of determinants involving the crystal structure factors be non-negative. A typical determinant is ... 

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