Intensities of Diffracted Beams

Let us recapitulate the fundamental theorem of diffraction. A crystal structure is characterized by  

As a general rule, the measurements yield relative intensities, i.e. integrated intensities with an arbitrary scale X, because it is difficult to know what part of the intensity of the primary beam passes through the crystal. The constant X is thus an unknown. The function g 6) is analytic and its values can be easily calculated. The calculation of the absorption factor A is carried out with a computer and, today, poses no major problem. The theory of extinction is still poorly understood, but the factor y is often close to 1. Thus, from the intensity measurements, structure amplitudes F hkl) are obtained on a relative scale, typically with a precision of the order of 1-5%. The values of F hkl) represent the experimental information about the distribution of the atoms in the unit cell. A discussion of this information forms the subject of this section. However, we will discuss neither the theory nor the practice of structure determination by diffraction. 

Equation (3.53) is valid for all periodic electron density distributions p(xyz) ,. Equation (3.54) is valid for the atomistic model (Section 3.3) and can be used to calculate the structure factor if the atomic coordinates and the thermal displacements are known. Remembering that the structure factor represents a wave, it is, in general, a complex number  

Equation (3.56) expresses the observations F hkl) as a function of the atomic coordinates In order to determine an unknown structure, 

The Lorentz factor L 9) is contained in the term g(6) of equation (3.52). The lower the rate of passage of the point through the sphere, the higher is the integrated intensity. The point (000) always lies on the sphere, independent of the position of the crystal. Consequently, L(0) = oo. If the rotation axis does not lie in the reflecting plane, we obtain a speed v which is lower than that in equation (3.58). Hence L 6) depends on the experimental technique used. For the rotating crystal method (Section 3.5.2), with the rotation axis parallel to c and the primary beam perpendicular to c, the result for the nth layer is  

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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. 

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. 

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. 

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. 

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. 

Monochromatic beam of electrons diffracted on passing through thin and very small fragment of crystalline material (in an electron microscope). Angles and intensities of diffracted beam record photographically... 

Monochromatic beam of neutrons is diffracted by a crystal (or powdered crystalline material). Angles and intensities of diffracted beam recorded by a detector... 

Phase problem The problem of determining the phase angle (relative to a chosen origin) that is to be associated with each diffracted wave that is combined to give an electron-density map. The measured intensities of diffracted beams give only the squares of the amplitudes, but the relative phases cannot normally be determined experimentally (see Chapter 8). The determination of the relative phases of the Bragg reflections is crucial to the calculation of the correct electron density map. 

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

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). 

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