The intensities of diffracted beams

The experimental techniques outlined in the previous sections allow the lattice parameters of a crystal to be determined. However, the determination of the appropriate crystal lattice, face-centred cubic as against body-centred, for example, requires information on the intensities of the diffracted beams. More importantly, in order to proceed with a determination of the complete crystal structure, it is vital to understand the relationship between the intensity of a beam diffracted from a set of (hkl) planes and the atoms that make up the planes themselves. 

The intensities of diffracted beams vary from one radiation type to another, and are found to depend upon the following factors  

To explain these factors it is convenient to describe the determination of intensities with respect to X-ray diffraction. The differences that arise with other types of radiation are outlined below (Sections 6.12, 6.14, 6.17). 

In the following four sections the important contribution made by the arrangement and types of atoms in crystals is considered. The other factors, which can be regarded as correction terms, are described later in this Chapter. In the case of X-rays, the initial beam is almost undiminished on passing through a small crystal, and the intensities of the diffracted beams are a very small percentage of the incident beam intensity. For this reason, it is reasonable to assume that each diffracted 

X-ray photon is scattered only once. Scattering of diffracted beams back into the incident beam direction is ignored. This reasonable approximation is the basis of the kinematical theory of diffraction, which underpins the theoretical calculation of the intensity of a diffracted beam of X-rays. 

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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... 

The present chapter deals first with all the preliminary steps which must be taken to obtain suitable data for structure determination (whether by direct or indirect methods)—the measurement of the intensities of diffracted beams, and the application of the corrections necessary to isolate the factors due solely to the crystal structure from those associated with camera conditions. It then goes on to deal with the effect of atomic arrangement on the intensities of diffracted beams, the procedure in deducing the general arrangement, and finally the methods of determining actual atomic coordinates by trial. It follows from what has been said that, as soon as atomic positions have been found to a sufficient degree of approximation to settle the phases of the diffracted beams, then the direct method can be used this, in fact, is the normal procedure in the determination of costal structures. 

In a diffraction experiment, the angle of incidence is varied (scanned) either step by step or continuously, and the intensities of diffraction beams detected and recorded as a function of angle. Some diffractometers are designed to scan in both SAXD and WAXD modes simultaneously. 

The intensities of diffracted beams are modified by crystal quality, disorder, temperature, and any anomalous scattering caused by the proximity of the incident X-ray wavelength to the absorption edge of an atom in the crystal. 

The graphical method gives a lucid picture of scattering from a unit cell, but is impractical as a method for calculation of the intensities of diffracted beams. The pictorial summation must be expressed algebraically for this purpose. The simplest way of carrying this out is to express the scattered wave as a complex amplitude, (see Appendices 5 and 6) ... 

Not all materials are so well behaved. For example, many metal alloys have considerable composition ranges and a correct calculation of the intensities of diffracted beams needs inclusion of a site occupancy factor. For example, the disordered gold-copper alloy Au Cu, is able to take compositions with x varying from 1, pure gold, to 0, pure copper. The structure of the alloy is the copper (Al) structure, (see Chapter 1), but in the alloy the sites occupied by the metal atoms contain a mixture of Cu and Au, (Figure 8.1). This situation can be described by giving a site occupancy factor to each type of atom. For example,... 

As discussed earlier, the symmetry of the structure plays an important part in modifying the intensity of diffracted beams, one consequence of which is that the intensities of a pair of reflections hkl and hlcl are equal in magnitude. This will cause the diffraction pattern from a crystal to appear centrosymmetric even for crystals that lack a centre of symmetry and the point symmetry of any sharp diffraction pattern will belong to one of the 11 Laue classes, (see Section 4.7 and Chapter 6, especially Section 6.9). 

One final point concerns thermal motion. Atoms are never at rest, and thermal motion causes a further decay in the intensity of diffracted beams. The atomic structure... 

The diffractometer is an apparatus used to determine the angles at which diffraction occurs for powdered specimens its featmes are represented schematically in Figure 3.23. A specimen S in the form of a flat plate is snpported so that rotations about the axis labeled O are possible this axis is perpendicular to the plane of the page. The monochromatic x-ray beam is generated at point T, and the intensities of diffracted beams are detected with a counter labeled C in the figure. The specimen, x-ray source, and counter are coplanar. 

It is relatively straightforward to detemiine the size and shape of the three- or two-dimensional unit cell of a periodic bulk or surface structure, respectively. This infonnation follows from the exit directions of diffracted beams relative to an incident beam, for a given crystal orientation measuring those exit angles detennines the unit cell quite easily. But no relative positions of atoms within the unit cell can be obtained in this maimer. To achieve that, one must measure intensities of diffracted beams and then computationally analyse those intensities in tenns of atomic positions. 

Chemical reactions of surfeces. Diffraction can be used qualitatively to identify different surface phases resulting from adsorption and chemical reaction at surfaces. Reaction rates can be investigated by following the evolution of diffracted beam intensities. 

Another major difference between the use of X rays and neutrons used as solid state probes is the difference in their penetration depths. This is illustrated by the thickness of materials required to reduce the intensity of a beam by 50%. For an aluminum absorber and wavelengths of about 1.5 A (a common laboratory X-ray wavelength), the figures are 0.02 mm for X rays and 55 mm for neutrons. An obvious consequence of the difference in absorbance is the depth of analysis of bulk materials. X-ray diffraction analysis of materials thicker than 20—50 pm will yield results that are severely surface weighted unless special conditions are employed, whereas internal characteristics of physically large pieces are routinely probed with neutrons. The greater penetration of neutrons also allows one to use thick ancillary devices, such as furnaces or pressure cells, without seriously affecting the quality of diffraction data. Thick-walled devices will absorb most of the X-ray flux, while neutron fluxes hardly will be affected. For this reason, neutron diffraction is better suited than X-ray diffraction for in-situ studies. 

Atoms are not rigidly bound to the lattice, but vibrate around their equilibrium positions. If we were able to look at the crystal with a very short observation time, we would see a slightly disordered lattice. Incident electrons see these deviations, and this, for example, is the reason that in LEED the spot intensities of diffracted beams depend on temperature at high temperatures the atoms deviate more from their equilibrium position than at low temperatures, and a considerable number of atoms are not at the equilibrium position necessary for diffraction. Thus, spot intensities are low and the diffuse background high. Similar considerations apply in other scattering techniques, as well as in EXAFS and in Mossbauer spectroscopy. 

According to Friedel s law, a diffracted X-ray beam from the (010) side of the R crystal will have the same intensity as that from the opposite (010) side. Moreover, the intensity of this beam will be equal in magnitude to those of the diffracted beams from the (010) and (010) planes of the S crystal. On such a basis one cannot distinguish between the R and S structures. 

Hitherto only the positions of the X-ray beams diffracted by crystals have been considered unit cell dimensions are determined from the positions of diffracted beams without reference to their intensities. To discover the arrangement and positions of the atoms in the unit cell it is necessary to consider the intensities of the diffracted beams. 

General expression for the structure amplitude. We are interested primarily in the arrangement of the atoms in crystals and the effect of the arrangement on the intensities of diffracted X-ray beams. 

LEED does not only reveal the relative periodicities of the adsorbate mesh with respect to the substrate lattice. Applying dynamical scattering theory, i.e., modeling the scattering intensity of diffracted beams versus electron energy (so-called I-V curves), allows determination of absolute positions of atoms on the surface [20]. Unfortunately, the complexity of the method limits the number of atoms per unit cell and makes it applicable only to atomic or small-molecule lattices. 

The effect of surface atom vibrations is seen clearly in low energy electron diffraction (EEED). Experimentally, an exponential decrease in the intensity of scattered beams and an increase in background intensity are observed with increasing temperature. This arises as a result of the increased vibrational amplitude of the surface atoms that occurs at... 

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