Kinematical scattering

RHEED is a powerful tool for studying the surface structure of crystalline samples in vacuum. Information on the surface symmetry, atomic-row spacing, and evidence of surfece roughness are contained in the RHEED pattern. The appearance of the RHEED pattern can be understood qualitatively using simple kinematic scattering theory. When used in concert with MBE, a great deal of information on film growth can be obtained. 

According to the Fraunhofer approximation of kinematic scattering theory the real space and the reciprocal space are related to each other by an integral transform known by the name Fourier transform, which shall be indicated by the operator The n-dimensional (nD) Fourier transform of h (r) is defined by... 

Structure refinement based on kinematical scattering was already applied by the Russian scientist 60 years ago. Weirich et al. (1996) first solved the structure of an unknown TinSe4by HREM combined with crystallographic image processing. Then they used intensities extracted from selected area electron diffraction patterns of a very thin crystal and refined the structure to a precision of 0.02 A for all the atoms. Wagner and Terasaki et al. (1999) determined the 3D structure of a new zeolite from selected area electron diffraction, based on kinematical approach. 

BLANK DISC CBED, AND STRUCTURE-INDEPENDENT TESTS FOR KINEMATIC SCATTERING... 

The second advantage is the small size of the micro-crystallites which form the thin film. Accordingly kinematical or quasi-kinematical scattering are more likely and effects of diffuse scattering are small (important for background substruction). 

Aside a few specimens with predominantly kinematical scattering, many specimens investigated by EDSA show pronounced d5mamical scattering. In these cases suitable corrections must be applied to link lOhki I and observed Ihki- For the latter case one has to use successive approximations, i.e. evaluation of parameters from weak kinematical reflections which are then used to apply dynamical corrections to strong dynamical reflections. Without such corrections the residual / -factor of the structure amplitudes is usually about 20 % or more, while suitable corrections lead to / -factors in the range of 5 - 2 %. 

Measurement of kinematical scattering from a defective surface 162... 

Extinction, which is the failure of the kinematic scattering theory (Ihki hki) is only a minor problem in X-ray diffraction. In neutron diffraction, extinction is serious and pervasive throughout the whole data, as shown by the examples in Thble 3.2. The best methods available for extinction correction require careful measurement of crystal dimensions. Although somewhat empirical, it has proved to be very effective [184, 185]. At least one, and sometimes six, additional extinction parameters, gis0 or gij, have to be added to the variable parameters. Uncertainty in the validity of these extinction parameters appears to have very little effect on atomic positional coordinates, but may influence the absolute values of the atomic temperature factors. This is important in charge density or electrostatic potential... 

It has been shown [1] that the criterion for changing from kinematic scattering to d3mamic scattering is given by the size of the crystallites A, determined from the equation... 

The sizes of NaBr crystallites, determined from the line widths, were approximately 40 to 60 A, which made it possible, in chai ng from the detected intensities to structure amplitudes hkl > kinematic scattering approximation, according to which, for poly-... 

Since the interaction of hard X-rays with matter is weak, provided that the exact Bragg conditions for a perfect crystal are not met, the kinematical scattering approximation, in which scattering is treated as a single event, can be used. For a crystal volume defined by A i, N2, and A 3 unit cells along the crystal axes defined by the vectors ai, 82, and 33, the scattered intensity can be written as the product of two scattering amplitudes, one from the unit cell and one from the lattice of unit... 

In the second step, the arrangement of the electrons (. e., the structure of matter) is added, and the Fourier relation of the structure to the scattered X-ray intensity, I (s), is established by the so-called kinematic scattering theory. 

As stated by the kinematic scattering theory the relation between the structure of matter and the scattered intensity can be best understood [60], if the latter is treated in the so-called reciprocal space s = (i l, S2, S3), i.e. 

---