Patterns diffraction

The final grid is positively charged to accelerate the accepted electrons onto the fluorescent screen. The diffraction pattern may then be photographed. 

The diffraction pattern consists of a small number of spots whose symmetry of arrangement is that of the surface grid of atoms (see Fig. IV-10). The pattern is due primarily to the first layer of atoms because of the small penetrating power of the low-energy electrons (or, in HEED, because of the grazing angle of incidence used) there may, however, be weak indications of scattering from a second or third layer. 

Electrons interact with solid surfaces by elastic and inelastic scattering, and these interactions are employed in electron spectroscopy. For example, electrons that elastically scatter will diffract from a single-crystal lattice. The diffraction pattern can be used as a means of stnictural detenuination, as in FEED. Electrons scatter inelastically by inducing electronic and vibrational excitations in the surface region. These losses fonu the basis of electron energy loss spectroscopy (EELS). An incident electron can also knock out an iimer-shell, or core, electron from an atom in the solid that will, in turn, initiate an Auger process. Electrons can also be used to induce stimulated desorption, as described in section Al.7.5.6. 

Another mode of electron diffraction, low energy electron diffraction or FEED [13], uses incident beams of electrons with energies below about 100 eV, with corresponding wavelengths of the order of 1 A. Because of the very strong interactions between the incident electrons and tlie atoms in tlie crystal, there is very little penetration of the electron waves into the crystal, so that the diffraction pattern is detemiined entirely by the... 

Although the structure of the surface that produces the diffraction pattern must be periodic in two dimensions, it need not be the same substance as the bulk material. Thus LEED is a particularly sensitive tool for studying the structures and properties of thin layers adsorbed epitaxially on the surfaces of crystals. 

The otiier type of noncrystalline solid was discovered in the 1980s in certain rapidly cooled alloy systems. D Shechtman and coworkers [15] observed electron diffraction patterns with sharp spots with fivefold rotational synnnetry, a syimnetry that had been, until that time, assumed to be impossible. It is easy to show that it is impossible to fill two- or tliree-dimensional space with identical objects that have rotational symmetries of orders other than two, tliree, four or six, and it had been assumed that the long-range periodicity necessary to produce a diffraction pattern with sharp spots could only exist in materials made by the stacking of identical unit cells. The materials that produced these diffraction patterns, but clearly could not be crystals, became known as quasicrystals. 

We have thus far discussed the diffraction patterns produced by x-rays, neutrons and electrons incident on materials of various kinds. The experimentally interesting problem is, of course, the inverse one given an observed diffraction pattern, what can we infer about the stmctirre of the object that produced it Diffraction patterns depend on the Fourier transfonn of a density distribution, but computing the inverse Fourier transfomi in order to detemiine the density distribution is difficult for two reasons. First, as can be seen from equation (B 1.8.1), the Fourier transfonn is... 

It has been shown that spherical particles with a distribution of sizes produce diffraction patterns that are indistingiushable from those produced by triaxial ellipsoids. It is therefore possible to assume a shape and detemiine a size distribution, or to assume a size distribution and detemiine a shape, but not both simultaneously. 

Figure Bl.8.6. An electron diffraction pattern looking down the fivefold synnnetry axis of a quasicrystal. Because Friedel s law introduces a centre of synnnetry, the synnnetry of the pattern is tenfold. (Courtesy of L Bendersky.)...

Prince E 1987 Diffraction patterns from tilings with fivefold symmetry Acfa Crystaiiogr.A AZ 393-400... 

It will also be useful to introduce concepts of two-dimensional ordering and the corresponding nomenclature used to characterize specific stmctiires. We can then describe how the surface diffraction pattern relates to the ordering and, thus, provides important two-dimensional stmctiiral infonnation. 

In LEED experunents, the matrix M is detennined by visual inspection of the diffraction pattern, thereby defining the periodicity of the surface structure the relationship between surface lattice and diffraction pattern will be described in more detail in the next section. 

The diffraction pattern observed in LEED is one of the most connnonly used fingerprints of a surface structure. Witii XRD or other non-electron diffraction methods, there is no convenient detector tliat images in real time the corresponding diffraction pattern. Point-source methods, like PD, do not produce a convenient spot pattern, but a diffrise diffraction pattern that does not simply reflect the long-range ordermg. 

So it is essential to relate the LEED pattern to the surface structure itself As mentioned earlier, the diffraction pattern does not indicate relative atomic positions within the structural unit cell, but only the size and shape of that unit cell. However, since experiments are mostly perfonned on surfaces of materials with a known crystallographic bulk structure, it is often a good starting point to assume an ideally tenuinated bulk lattice the actual surface structure will often be related to that ideal structure in a simple maimer, e.g. tluough the creation of a superlattice that is directly related to the bulk lattice. 

In this section, we concentrate on the relationship between diffraction pattern and surface lattice [5], In direct analogy with the tln-ee-dimensional bulk case, the surface lattice is defined by two vectors a and b parallel to the surface (defined already above), subtended by an angle y a and b together specify one unit cell, as illustrated in figure B1.21.4. Withm that unit cell atoms are arranged according to a basis, which is the list of atomic coordinates within drat unit cell we need not know these positions for the purposes of this discussion. Note that this unit cell can be viewed as being infinitely deep in the third dimension (perpendicular to the surface), so as to include all atoms below the surface to arbitrary depth. 

Figure Bl.21.4. Direct lattices (at left) and reciprocal lattices (middle) for the five two-dimensional Bravais lattices. The reciprocal lattice corresponds directly to the diffraction pattern observed on a standard LEED display. Note that other choices of unit cells are possible e.g., for hexagonal lattices, one often chooses vectors a and b that are subtended by an angle y of 120° rather than 60°. Then the reciprocal unit cell vectors also change in the hexagonal case, the angle between a and b becomes 60° rather than 120°.

The reciprocal lattices shown in figure B 1.21.3 and figure B 1.21.4 correspond directly to the diffraction patterns observed in FEED experiments each reciprocal-lattice vector produces one and only one diffraction spot on the FEED display. It is very convenient that the hemispherical geometry of the typical FEED screen images the reciprocal lattice without distortion for instance, for the square lattice one observes a simple square array of spots on the FEED display. 

One of the spots in such a diffraction pattern represents the specularly reflected beam, usually labelled (00). Each other spot corresponds to another reciprocal-lattice vector = ha + kb and is thus labelled (hk), witli integer h and k. 

Jamieson J C, Lawson A W and Nachtreib N D 1959 New device for obtaining X-ray diffraction patterns from substances exposed to high pressures Rev. Sc/, instrum. 30 1016... 

Figure C2.17.7. Selected area electron diffraction pattern from TiC nanocrystals. Electron diffraction from fields of nanocrystals is used to detennine tire crystal stmcture of an ensemble of nanocrystals [119]. In tliis case, tliis infonnation was used to evaluate the phase of titanium carbide nanocrystals [217].

Idistribution functions can be measured experimentally using X-ray diffraction. The regular arrangement of the atoms in a crystal gives the characteristic X-ray diffraction pattern with bright, sharp spots. For liquids, the diffraction pattern has regions of high and low intensity but no sharp spots. The X-ray diffraction pattern can be analysed to calculate an experimental distribution function, which can then be compared with that obtained from the simulation. 

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