Scattering and diffraction. Kinematic model

Kinematic model for  X-ray diffraction

A material sample is primarily regarded as a distribution of electrons which is peculiar to each case. The interaction of electromagnetic radiations with matter is modeled considering radiation as particles (photons), or as waves. We are dealing here with a typical ondulatory phenomenon. In fact, in addition to the well known experiment of Max von Laue, the original experiments (Friedrich, Knipping and Laue, 1912, in CuSO4 crystals) were the proof that X-rays are waves and that crystals are structured in three-dimensional lattices with periodic distances in the range of X-ray wavelengths.

Waves scatter whenever there is a change in the incident wave front due to a discontinuity in the medium in which they propagate (an astronaut does not see the "blue sky"). If the phase relationship between waves scattered by the discontinuity remain constant, the waves combine in a cooperative and coherent manner, producing interferences, known as diffraction. However, in the scattering process the phase relations occur randomly and are not maintained over time.

The theoretical model of matter-wave interaction is provided by the four Maxwell equations, the equation of charge continuity and the two equations that characterize the materials (the electric and magnetic polarization and the dielectric and permeability constants).

Although this model describes the phenomenon macroscopically, it also applies to the atomic scale for electronic distributions weakly bound in the atom and which move at speeds lower than light. In addition, the nuclei are regarded as massive, and therefore fixed (Born approximation). Furthermore, it is supposed that the material sample extends indefinitely, that its dielectric constant is independent of time, that the permeability of the material is close to the unit, and that both constants are homogeneous and isotropic.

Basically, the kinematic conditions are summarized considering that the incident wave is hardly modified by the material as it passes through. Therefore, the scattered wave is seen as a small perturbation due to an interaction of a very weak magnitude. More specifically:

• The incident beam is considered homogeneous, covering the sample completely and crossing it straight and unchanged, so that its incident intensity is constant. This incident intensity only decreases slowly, and isotropically, by a regular absorption mechanism when crossing the sample. This condition is violated if the crystal sample is perfect, because there would be interactions between the diffracted and incident beams, leading to extinction, multiple diffraction, anomalous absorption ...
• The scattering occurs at discrete points of the sample, producing a diffracted beam at an angle 2θ with respect to the incident. The diffracted intensity is a small proportion of the incident intensity (Kirchoff approximations for visible light, and Born approximations for the phenomenon of scattering). The small magnitude of the energy and momentum transfers from the incident beam to to the diffracted ones implies that coherence is maintained throughout the sample and that the thermal diffuse scattering (TDS) is not considered.
• The scattering is considered elastic (no change in frequency) and coherent, and therefore the phase relationships due to different crossed paths are maintained (diffraction), and the whole sample diffracts in phase in the direction of the incident beam. Therefore, Compton scattering and the possible excitation of electrons to other energy levels (which would lead to effects of fluorescence and anomalous scattering) are ignored. The interactions between X-rays and nuclei are also ignored.
• Coherence means that the detected intensity is the result of composing different diffracted beams, taking into account their respective phases. In the kinematic model, with the detector far from the sample (in relation to its size: Fraunhoffer diffraction model), the diffracted spherical wave can be seen as a plane wave at the detector, with all rays being parallel.
• All these features have to be met and implemented into the experimental devices. If these conditions are not met, we have to neglect the dynamic effects, or we have to apply the corresponding corrections. A critical feature in the fulfillment of the kinematic conditions is the mosaicity, which takes into account that the sample deviates from a perfect single crystal, and that it is formed by single crystal micro-blocks distributed randomly and not being highly distorted.

Distribution of single crystal micro-blocks in a crystal sample suitable for implementing the kinematic model. As in many other aspects of Crystallography, we must reach a compromise... The orientations must not be totally random, but without reaching the perfect alignment of the micro-blocks. This is why we say that the sample must be "perfectly imperfect" .

Kinematic model for a single crystal

Continuing with the above, for the structural analysis of samples using kinematic X-ray diffraction, we consider that:

• An atom consists of a "point" nucleus, inert to X-rays. As the duration time of the experiment and of the interaction is very large, as compared with the frequency of the incident X-rays, it is considered that the distribution of the Z electrons around the nucleus is continuous. And this is why we can talk about distribution of electron density per unit of volumeρ.
• This electron distribution can be considered spherical (where the atomic scattering factors do not depend on the orientation of the diffracted beam, but only on the angle θ), or non-spherical (where the atomic scattering factors do depend on the orientation of the diffracted beam) when we take into account the distortions which occur in the electron density of the valence electrons due to the interatomic bonds, the free-electron pairs, non-spherical orbitals, intermolecular interactions... In fact, X-ray diffraction experiments (sometimes together with the neutron diffraction) can detect these density deformations due to the electronic interactions.
• The electrons distributed around the nucleus are considered weakly linked to the atom, so their vibrational frequencies are very different from the incident radiation. If this is not assumed, several corrections must be applied to the atomic scattering factors to take into account the intrinsic phase changes produced by the interaction with the linked electrons.
• In a single crystal, the distribution of the electron density is periodic in the three dimensions of space. Therefore, one only needs to know the electron density located inside the unit cell. This model of periodic structure explains the correlations that define the crystalline order.
• The unit cell (a simple mathematical representation of the independent repeating blocks of the structure) contains the electron density with concentrated peaks in a finite set of positions where it is assumed that the atoms are located. Subsidiary maxima occur due to the density distortions mentioned above. This electron density distribution can also show different types of symmetry: exact (crystallographic) or approximate (non-crystallographic). The crystallographic symmetry, together with the translations that define the lattice, define the so-called space group of the crystal structure. All this means that the entire crystal sample can be described by repeating the independent part using the lattice and the symmetry of the space group. The independent part is known as the asymmetric unit of the structure, and contains the motif that is being repeated within the crystal.
• The experimental diffraction pattern obtained, and therefore the structure that can be derived, represent an average over the duration time of the experiment and over all the unit cells in the sample. That is:

<structure>  (over time and unit-cells)   =
<lattice & symmetry> (according to distortions, vibrations ...)   +
<motif>  (according to vibrations, orientations, disorder ...)
(symbols < > mean "average")
• All the components producing the averages arise from the fact that crystal samples are not perfectly ordered; they contain a certain degree of disorder. The disorder can be static or dynamic. Static disorder is due to the fact that repetitions are not always exact; there is some mosaicity as well as modulations and some inconmensurate crystal structures may occur as a consequence of small shifts in the atomic positions. Dynamic disorder comes from the different modes of thermal vibration of atoms around their equilibrium positions; lattice phonons and harmonic or anharmonic vibrations of the atoms or of the molecule as a whole.
• Finally, it must be remembered that the sample has a finite size and a certain shape. Both characteristics have to be taken into account for some corrections not considered in the kinematic model.

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