Bragg peaks structure factors

The Bragg peak intensity reduction due to atomic displacements is described by the well-known temperature factors. Assuming that the position can be decomposed into an average position, ,) and an infinitesimal displacement, M = 8R = Ri — (R,) then the X-ray structure factors can be expressed as follows ... 

The diffraction lines due to the crystalline phases in the samples are modeled using the unit cell symmetry and size, in order to determine the Bragg peak positions 0q. Peak intensities (peak areas) are calculated according to the structure factors Fo (which depend on the unit cell composition, the atomic positions and the thermal factors). Peak shapes are described by some profile functions 0(2fi—2fio) (usually pseudo-Voigt and Pearson VII). Effects due to instrumental aberrations, uniform strain and preferred orientations and anisotropic broadening can be taken into account. 

The used S5mbols are K, scale factor n, number of Bragg peaks A, correction factor for absorption P, polarization factor Jk, multiplicity factor Lk, Lorentz factor Ok, preferred orientation correction Fk squared structure factor for the kth reflection, including the Debye-Waller factor profile function describing the profile of the k h reflection. 

TRXRD detects the propagation of coherent acoustic phonons as a transient change in the diffraction angles. In contrast, the atomic motions associated with coherent optical phonons modify only the Bragg peak intensity, because they do not change the barycentric positions of the crystal lattice. The Bragg peak intensity is proportional to the squared modulus of the structure factor [1,3,4] ... 

Molecular orientational order in adsorbed monolayers can be inferred indirectly from elastic neutron diffraction experiments if it results in a structural phase transition which alters the translational symmetry of the 2D lattice. In such cases, Bragg reflections appear which are not present in the orientationally disordered state. Experiments of this type have inferred orientational order in monolayers of oxygen (41) and nitrogen (42) adsorbed on graphite. However, these experiments have not observed a sufficient number of Bragg reflections to determine the molecular orientation by comparing relative Bragg peak intensities with a model structure factor. 

The factors that are included when calculating the intensity of a powder diffraction peak in a Bragg-Brentano geometry for a pure sample, composed of three-dimensional crystallites with a parallelepiped form, are the structure factor Fhkl 2=l/ TS )l2, the multiplicity factor, mm, the Lorentz polarization factor, LP(0), the absorption factor, A, the temperature factor, D(0), and the particle-size broadening factor, Bp(0). Then, the line intensity of a powder x-ray diffraction pattern is given by [20-22,24-26]... 

The spin of the neutron interacts with the spins in the sample, and this is responsible for a magnetic contribution to the structure factors of Bragg peaks, the sign of which changes on reversal of the neutron spin direction ... 

In this equation f is the atomic scattering factor or atomic form factor, h,k,l are the Miller indices of the reflecting plane, and x ,y ,z are the coordinates of the scattering atom in decimal fractions of unit cell parameters, a,b,c, respectively. For simple structures the structure factor indicates what types of Bragg planes in a given kind of structure can produce a diffraction peak, i.e., have non-cancelled, coherent scattering, and also indicates the relative intensity of the allowed peaks (Warren 1969). A few structure factors for simple crystal structures are shown below. 

The two latter groups of factors may be viewed as secondary, so to say, they are less critical than the principal part defining the intensities of the individual diffraction peaks, which is the structural part. Structural factors depend on the internal (or atomic) structure of the crystal, which is described by relative positions of atoms in the imit cell, their types and other characteristics, such as thermal motion and population parameters. In this section, we will consider secondary factors in addition to introducing the concept of the integrated intensity, while the next section is devoted to the major component of Bragg peak intensity - the structure factor. 

Earlier, we considered the effect of lattice symmetry on the multiplicity factors in powder diffraction only in terms of the number of completely overlapping equivalent Bragg peaks (section 2.10.3). Because Friedel s law is violated in non-centrosymmetric structures in the presence of anomalously scattering chemical elements, the evaluation of multiplicity factors should also include Eqs. 2.114 and 2.115. Thus, the multiplicity factor,maybe comprehensively defined as the number of equivalent reflections ihkT), which satisfy Eq. 2.114 in addition to having identical lengths of the... 

As noted at the beginning of this section, crystallographic symmetry has an effect on the structure amplitude, and therefore, it affects the intensities of Bragg peaks. The presence of translational symmetry causes certain combinations of Miller indices to become extinct because symmetrical contributions into Eq. 2.110 result in the cancellation of relevant trigonometric factors in Eq. 2.107. It is also said that some combinations of indices are forbidden due to the occurrence of translational symmetry. 

Solving the crystal structure using either heavy atom or direct techniques does not always work in a straightforward fashion even when the well-resolved and highly accurate diffraction data from a single crystal are available. The complicating factor in powder diffraction is borne by the intrinsic overlap of multiple Bragg peaks. The latter may become especially severe when the unit cell volume and complexity of the structure increase. 

There is a variety of freely available software, which enables one to deconvolute a powder diffraction pattern and determine either or all individual intensities, lattice and peak shape function parameters, and observed structure factors of all possible Bragg reflections. Freeware codes include EXPO, FullProf, GSAS, LHPM-Rietica, and others. In addition to free programs, nearly all manufacturers of commercial powder diffractometers offer software for sale either as a package with the sale of the equipment or as stand-alone products. ... 

Considering Eqs. 7.6 and 7.7, it is clear that each additional crystalline phase adds multiple Bragg peaks plus a new seale factor along with a set of eorresponding peak shape and structural parameters into the non-linear least squares. Even though mathematically they are easily aceounted for, the finite accuracy of measurements as well as the limited resolution of even the most advanced powder diffractometer, usually result in lowering the quality and stability of the Rietveld refinement in the case of multiple phase samples. Thus, when the precision of structural parameters is of concern, it is best to work with single-phase materials, where Eqs. 7.3 and 7.4 are applicable. On the other hand, since individual scale factors may be independently... 

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