Calculated diffracted beam intensities 001 beams

Table I shows the calculated diffracted beam amplitudes for the (000), (002), (004) and (006) beams for both topologies, for a crystal 60 A thick. The principal difference between the intensities for the two structures lies in the intensity of the (004) beam relative to (006). [Note that in both cases the intensity of the central (000) beam is approximately the same and hence the intensities are approximately normalised with respect to each other thus allowing direct comparison]. In the case of the Imma structure the (004) beam is weak with respect to the (006) and hence the (006) must be included to resolve the satellite channels clearly. At a limited resolution of 3.5 A the (006) beam lies outside the objective aperture and hence the satellite channels are only very faint in the image. However, for the Cmcm model the relative intensities are reversed with the (004) beam stronger than the (006) and hence even if the (006) beam is excluded from the aperture the satellite channels will be clearly resolved.

There are many examples in the literature of ordered structures observed by LEED, but only a few of these structures have been calculated from the diffraction beam intensities. However, the ordering characteristics and size and orientation of the unit cells have been determined from the geometry of the LEED patterns. By studying the systematic variation of their shape and bonding characteristics correlations can be made between these properties and their interactions with the metal surfaces. 

How do we find phase differences between diffracted spots from intensity changes following heavy-metal substitution We first use the intensity differences to deduce the positions of the heavy atoms in the crystal unit cell. Fourier summations of these intensity differences give maps of the vectors between the heavy atoms, the so-called Patterson maps (Figure 18.9). From these vector maps it is relatively easy to deduce the atomic arrangement of the heavy atoms, so long as there are not too many of them. From the positions of the heavy metals in the unit cell, one can calculate the amplitudes and phases of their contribution to the diffracted beams of the protein crystals containing heavy metals. 

The simplest diffraction measurement is the determination of the surface or overlayer unit mesh size and shape. This can be performed by inspection of the diffraction pattern at any energy of the incident beam (see Figure 4). The determination is simplest if the electron beam is incident normal to the surface, because the symmetry of the pattern is then preserved. The diffraction pattern determines only the size and shape of the unit mesh. The positions of atoms in the surface cannot be determined from visual inspection of the diffraction pattern, but must be obtained from an analysis of the intensities of the diffracted beams. Generally, the intensity in a diffracted beam is measured as a fimction of the incident-beam energy at several diffraction geometries. These intensity-versus-energy curves are then compared to model calculations. ... 

Surface atomic structure. The integrated intensity of several diffracted beams is measured as a fimction of electron beam energy for different angles of incidence. The measurements are fitted with a model calculation that includes multiple scattering. The atomic coordinates of the surfiice atoms are extracted. (See also the article on EXAFS.)... 

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. 

In crystallography, the difiiraction of the individual atoms within the crystal interacts with the diffracted waves from the crystal, or reciprocal lattice. This lattice represents all the points in the crystal (x,y,z) as points in the reciprocal lattice (h,k,l). The result is that a crystal gives a diffraction pattern only at the lattice points of the crystal (actually the reciprocal lattice points) (O Figure 22-2). The positions of the spots or reflections on the image are determined hy the dimensions of the crystal lattice. The intensity of each spot is determined hy the nature and arrangement of the atoms with the smallest unit, the unit cell. Every diffracted beam that results in a reflection is made up of beams diffracted from all the atoms within the unit cell, and the intensity of each spot can be calculated from the sum of all the waves diffracted from all the atoms. Therefore, the intensity of each reflection contains information about the entire atomic structure within the unit cell. 

The intensities of diffracted beams, or reflections as they ate commonly called, depend upon the strength of the scattering that the material inflicts upon the radiation. Electrons are scattered strongly, neutrons weakly and X-rays moderately. The basic scattering nnit of a crystal is its unit cell, and we may calculate the scattering at any angle by mnltiplying... 

In reflection, the intensity of the X-ray wavefield inside the crystal falls off very rapidly away from the surface, due to transfer of energy to the diffracted beam. Absorption also becomes important at low incident angles to the surface. By choosing the radiation and the reflection (inclnding its symmetry), the penetration may be varied between about 0.05 and 10 micrometres. This is ideally matched to device stmctures. This is quantified by the extinction distance g, defined as the depth at which the incident intensity has decreased to 1/e of its value at the surface. This may be calculated from diffraction theory, and some examples, for GaAs with CuK radiation, are shown in Table 3.2. It is assumed that the wafer surface is (001), hence the 004 reflection is symmetric and the others asymmetric. 

LEED studies have revealed that the atoms in this platinum surface are in the positions expected from the projection of the X-ray unit cell to the surface (5). The diffraction pattern that is exhibited (Fig. 4) clearly indicates a sixfold rotational symmetry that is expected in such a surface. Calculations of surface structure from LEED beam intensities indicate that atoms are in those positions in the surface layer (with respect to the second layer) as indicated by the X-ray unit cell within 5% of the interlayer distance (6,7). 

From the unit cell dimension one can calculate the directions of a diffracted beam, but not its intensity, which depends on the types and positions of atoms in the cell. Inversely, these positions cannot be determined without the knowledge of intensities, except in a few simplest structmes (e.g. NaCl), where all atoms occupy the lattice points. The intensity hu of a reflection depends on many factors. Thus, for X-ray diffraction. 

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