Three-dimensional diffraction

A crystal therefore acts as a three-dimensional diffraction grating for these x-rays, and three equations (the Laue equations) must be satisfied if there is to be constructive interference of these monochromatic x-rays. 

Fig. 1 Comparison of single-crystal and powder XRD measurements. In powder XRD, the diffraction phenomenon for each individual crystallite in the powder is the same as the diffraction phenomenon in single-crystal XRD. However, the powder comprises a large collection of crystallites with (in principle) a random distribution of crystallite orientations, and as a consequence, the three-dimensional diffraction data are effectively compressed into one dimension (intensity vs diffraction angle 26) in the powder XRD measurement...

This is a simple example of the synthesis of an image from a diffraction pattern by calculation. The synthesis of an image of a crystal structure from its X-ray diffraction pattern is more complex (because a three-dimensional diffraction grating is involved), but similar in principle, because the X-ray diffraction spots produced by an atomic pattern are absolutely analogous to the diffracted light beams formed by a pattern whose repeat distance is comparable with the wavelength of light. 

Opal. Opal is the only commercial synthetic gemstone material that is not a single crystal. It consists of a three-dimensional diffraction grating of geometrically aligned spheres of S1O2 . rH 0, where x is usually < 10 . 

Unlike the two-dimensional arrays in these examples, a crystal is a three-dimensional array of objects. If we rotate the crystal in the X-ray beam, a different cross section of objects will lie perpendicular to the beam, and we will see a different diffraction pattern. In fact, just as the two-dimensional arrays of objects we have discussed are cross sections of objects in the three-dimensional crystal, each two-dimensional array of reflections (each diffraction pattern recorded on film) is a cross section of a three-dimensional lattice of reflections. Figure 2.11 shows a hypothetical three-dimensional diffraction pattern, with the reflections that would be produced by all possible orientations of a crystal in the X-ray beam. 

Notice that only one plane of the three-dimensional diffraction pattern is superimposed on the film. With the crystal in the orientation shown, reflections shown in the plane of the film (solid spots) are the only reflections that produce spots on the film. In order to measure the directions and intensities of... 

The number of low-temperature, high-pressure (LT + HP) structure determinations of organic conductors is not yet very large, but a number of other crystallographic studies under constraint have also been reported. To limit the field of the discussion below, we consider only the case of techniques allowing for the measurement of full crystal structures, since collecting a three-dimensional diffraction data set is more demanding than simple observation of a few main or satellite reflections. 

Measure three-dimensional diffraction data with appropriate diffractometer settings for scanning each reflection and recording its direction and intensity. Select 3-5 standard reflections and measure them at regular intervals. If possible, the data are measured at low temperatures. 

The more detailed description of the non-conventional symmetry goes beyond the scope of this book as it has little use in powder diffraction, because even the three-dimensional diffraction from aperiodic crystals is quite complex. When the diffraction picture is projected along one dimension, its treatment becomes too complicated and the crystal structure of aperiodic crystals is rarely, if ever, completely studied by means of powder diffraction techniques beyond simple phase identification. Nevertheless, this section has been included here for completeness, and to give the reader a flavor of recent developments in crystallography. ... 

Laue equations once again indicate that a periodic lattice produces diffraction maxima at specific angles, which are defined by both the lattice repeat distances (a, b, c) and the wavelength (A,). Laue equations give the most general representation of a three-dimensional diffraction pattern and they may be used in the form of Eq. 2.20 to describe the geometry of diffraction from a single crystal. 

The Ewald s sphere and the reciprocal lattice are essential tools in the visualization of the three-dimensional diffraction patterns from single crystals, as will be illustrated in the next few paragraphs. They are also invaluable in the understanding of the geometry of diffraction from polycrystalline (powder) specimens, which will be explained in the next section. 

It is clear from looking at diffraction patterns obtained from real crystals, such as those in Figures 1.13 and 1.14, that all of the reflections are not equal. They span a broad range of intensity values from very strong to completely absent. It will be shown in Chapter 5 (and was already demonstrated by Figure 1.8) that the variation in intensity from reflection to reflection is a direct function of the atomic structure of the macromolecules that comprise the crystal and occupy its lattice points. That is, the relative intensities of the reflections that make up the three-dimensional diffraction pattern, or Fourier transform, of a crystal are directly related to the relative xj, yj, Zj coordinates of all of the atoms j that define an individual molecule, and to the relative strength, Zj, with which the different atoms scatter X rays. Zj is the electron complement of each atom and is, therefore, its atomic number. 

The complete diffraction pattern from a protein crystal is not limited to a single planar array of intensities like those seen in Figures 1.13 and 1.14. These images represent, in each case only a small part of the complete diffraction pattern. Each photo corresponds to only a limited set of orientations of the crystal with respect to the X-ray beam. In order to record the entire three-dimensional X-ray diffraction pattern, a crystal must be aligned with respect to the X-ray beam in all orientations, and the resultant patterns recorded for each. From many two-dimensional arrays of reflections, corresponding to cross sections through diffraction space, the entire three-dimensional diffraction pattern composed of ten to hundreds of thousands of reflections is compiled. 

Remember further that each reciprocal lattice point represents a vector, which is normal to the particular family of planes hkl (and of length 1 /d u) drawn from the origin of reciprocal space. If we can identify the position in diffraction space of a reciprocal lattice point with respect to our laboratory coordinate system, then we have a defined relationship to its family of planes, and the reciprocal lattice point tells us the orientation of that family. In practice, we usually ignore families of planes during data collection and use the reciprocal lattice to orient, impart motion to, and record the three-dimensional diffraction pattern from a crystal. Note also that if we identify the positions of only three reciprocal lattice points, that is, we can assign hkl indexes to three reflections in diffraction space, then we have defined exactly the orientation of both the reciprocal lattice, and the real space crystal lattice. 

Similarly, if all of the electron density in the unit cell were projected onto a single plane, let us say the ab plane in real space, then the diffraction pattern of this two-dimensional crystal would be the hk0 zone of reflections in the three-dimensional diffraction pattern. As with line projections, this theorem holds true for any and every plane passing through the origin. 

The next question, with the objective of determining the space group of the crystal, is What is the symmetry of the entire, three-dimensional diffraction pattern This is the most demanding aspect of the analysis and deserves some care. It is greatly simplified for... 

In Figures 6.17, 6.18, and 6.19 are presented pairs of diffraction intensity planes for three different protein crystals. From two photographs such as these, which are usually (e.g., Figure 6.19) orthogonal to one another, the symmetry of the entire three-dimensional diffraction pattern may be deduced. In some cases, however, additional diffraction images... 

In identifying symmetry elements present in reciprocal space, we are seeking to establish symmetry relationships between intensities in various parts of the three-dimensional diffraction pattern. In doing so, it is necessary to remember that a symmetry relationship observed for a single plane of the diffraction pattern, because of Freidel s law, may not pertain to the entire pattern, and this can only be ascertained by examining additional planes through reciprocal space. 

The precession camera is still the most useful instrument in the preliminary analysis of a crystal. It enables an undistorted photograph of a plane of the three-dimensional diffraction pattern to be recorded and is invaluable in the estimation of unit cell parameters, the detection of symmetry and for screening heavy atom derivatives (see, for example, Ch. 9 of Ref. 25 for a fuller description). 

Diffraction of light in two-dimensional lattices and their calculations were known in Laue s time an additional information. They have been the basis for Laue s calculation of three-dimensional diffraction lattices from X-ray experiments as a result he formulated a model of a spatial symmetrical structure of ions in a salt crystal abstract mental model. Laue proposed the use of realistic models in order to better visualize the concepts - but needed irrelevant items like balls, sticks and glue, in order to construct closest packings of spheres or spatial lattice models concrete models. 

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