Scattering centres

Patterson function The Fourier transform of observed intensities after diffraction (e.g. of X-rays). From the Patterson map it is possible to determine the positions of scattering centres (atoms, electrons). 

Figure Bl.23.9. Scattering intensity of 4 keV Ne versus azimuthal angle 8 for a Ni 110] surface in the clean (1 X 1), (1 X 2)-H missing row, and (2 x l)-0 missing row phases. The hydrogen atoms are not shown. The oxygen atoms are shown as small open circles. 0-Ni and Ni-Ni denote the directions along which O and Ni atoms, respectively, shadow the Ni scattering centre.

Each atom in the lattice acts as a scattering centre, which means that the total intensity of the diffracted beam in a given direction depends on the extent to which contributions from individual atoms are in phase. Relating the underlying structure to the observed diffraction pattern is not straightforward, but is essentially a trial-and-error search involving extensive computer-based calculations. 

Borrion, H., Griffiths, H. Money, D., Tait P. and Baker C., Scattering centre extraction for Extended targets , Proc. RADAR 2005 Conference, Washington DC, IEEE Publ., pp 173-178, 9-12 May 2005. 

The beams of reactant molecules A and B intersect in a small scattering volume V. The product molecule C is collected in the detector. The detector can be rotated around the scattering centre. Various devices may be inserted in the beam path, i.e. between reactants and scattering volume and between scattering volume and product species to measure velocity or other properties. The angular distribution of the scattered product can be measured by rotating the detector in the plane defined by two molecular beams. The mass spectrophotometer can also be set to measure a specific molecular mass so that the individual product molecules are detected. 

In direct methods calculations we use normalised structure factors E(hkl), which are the structure factors compensated for the fall-off of the atomic scattering factors f hkl). In fact this procedure tries to simulate point-like scattering centres. 

The damping factors take into account 1) the mean free path k(k) of the photoelectron the exponential factor selects the contributions due to those photoelectron waves which make the round trip from the central atom to the scatterer and back without energy losses 2) the mean square value of the relative displacements of the central atom and of the scatterer. This is called Debye-Waller like term since it is not referred to the laboratory frame, but it is a relative value, and it is temperature dependent, of course It is important to remember the peculiar way of probing the matter that EXAFS does the source of the probe is the excited atom which sends off a photoelectron spherical wave, the detector of the distribution of the scattering centres in the environment is again the same central atom that receives the back-diffused photoelectron amplitude. This is a unique feature since all other crystallographic probes are totally (source and detector) or partially (source or detector) external probes , i.e. the measured quantities are referred to the laboratory reference system. 

If V(R) is small with respect to T in (11) the periodic potential V(R) of the ionic cores is that of a lattice of scattering centres. The electronic excitation, described by is a scattered electron wave of high kinetic energy, traveling through the crystal. Many... 

More recent work has shown, however, that an exponential decay of the screening potential as in (21) is not correct, and that round any scattering centre the charge density falls off as r 3 cos 2kFr. This we shall now show, by introducing the phase shifts t/, defined as follows (cf. (13)). Consider the wave functions Fx of a free electron in the field of an impurity. These behave at large distances from the impurity as (Mott and Massey 1965)... 

It remains to calculate l We consider the case when the resistivity is due to impurities. Limiting ourselves to the case of a spherical Fermi surface, we suppose that there are N0 scattering centres per unit volume, and that for each of these the differential cross-section for the scattering of an electron through an angle 0 into the solid angle do is 1(8) do. Then... 

For large values of the mean free path and a spherical Fermi surface, (34) should lead to the same value (31) as the Boltzmann formulation. Edwards (1958) was the first to show that this was so, for the special case of weak scattering centres distributed at random in space in this paper formula (33) for l was deduced. Mott and Davis (1979), taking l as the distance in which phase coherence is lost, give a proof that the Kubo-Greenwood formulation reduces to that of Boltzmann when l > a. 

As the reflected radiation is emitted from the sample in a random direction, diffusely reflected radiation can be separated from, potentially sensor-blinding, specular reflections. Common techniques are off-angle positioning of the sensor with respect to the position(s) of the illumination source(s) and the use of polarisation filters. Application restrictions apply to optically clear samples with little to no scattering centres, thin samples on an absorbing background and dark samples. In either of these cases, the intensity of radiation diffusely reflected off such samples is frequently insufficient for spectral analysis. While dark objectives remain a problem, thin and/or transparent samples can be measured in transmission or in transflectance. 

The symbol n0 refers to the number density of scatter centres. 

Interference between X-rays scattered at different atomic centres occurs in exactly the same way as for an atom. The scattered amplitude becomes a function of an atomic distribution function. In an amorphous fluid, a gas or non-crystalline solid the function is spherically symmetrical and the scattering independent of sample orientation. It only depends on the radial distribution of scattering centres (atoms). 

In most organic semiconductors the presence of charges modifies the local structure of the network by deformation of the particular site. This so-called polaron formation thus creates scattering centres for other charges. Moreover these locally trapped carriers commonly alter the energy conditions because of their Coulomb interaction. In combination with the polaron energy, the latter may be attractive or repulsive. These effects, as they involve more than one electron, force us to give up the one-electron picture and hence to use the correlated-electron description. 

Fig. 3. A schematic view of a crossed-molecular beam apparatus used for studying the reactions of chlorine atoms with halogen molecules. The mass spectrometer detector is rotatable about the scattering centre for measuring the angular distributions of the reaction products whose recoil velocities are determined by time-of-flight analysis. (Reproduced from ref. 558 by permission of the authors and the American Institute of Physics.)...

Some supplementary remarks to the theory of Penn might be appropriate here. There are additional effects which are of relevance if a more quantitative theory of the photoemission process from an adsorbate-covered surface is envisaged. The first point is that the Anderson model as applied to chemisorption is a clearly oversimplified model to describe real metal-adsorbate systems. Besides overlap effects due to the nonorthogonality of the states k) and a), there are several interaction effects which are neglected in the Hamiltonian, Eq.(5). The adsorbed atom, for instance, may act as a scattering centre for the metal electrons and thus modify the Bloch wave functions characteristic of the free substrate. This can be accounted for by adding a term... 

Dopants and other impurities and lattice defects will also affect the mobility by acting as scattering centres for electrons and holes. In high purity materials... 

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