Three-dimensional point lattices diffraction from

From a comparison of various spot electron diffraction patterns of a given crystal, a three-dimensional system of axis in the reeiproeal lattice may be established. The reeiproeal unit cell may be eompletely determined, if all the photographs indexed. For this it is sufficient to have two electron diffraction patterns and to know the angle between the seetions of the reeiproeal lattice represented by them, or to have three patterns which do not all have a particular row of points in common (Fig.5). Crystals of any compound usually grow with a particular face parallel to the surface of the specimen support. Various sections of the reciprocal lattice may, in this case, be obtained by the rotation method (Fig.5). 

Since the variation of any physical property in a three dimensional crystal is a periodic function of the three space coordinates, it can be expanded into a Fourier series and the determination of the structure is equivalent to the determination of the complex Fourier coefficients. The coefficients are indexed with the vectors of the reciprocal lattice (one-to-one relationship). In principle the expansion contains an infinite number of coefficients. However, the series is convergent and determination of more and more coefficients (corresponding to all reciprocal lattice points within a sphere, whose radius is given by the length of a reciprocal lattice vector) results in a determination of the stmcture with better and better spatial resolution. Both the amplitude and the phase of the complex number must be determined for any Fourier coefficient. The amplitudes are determined from diffraction... 

Diffraction patterns can be described in terms of three-dimensional arrays called lattice points.33 The simplest array of points from which a crystal can be created is called a unit cell. In two dimensions, unit cells may be compared to tiles on a floor. A unit cell will have one of seven basic shapes (the seven crystal systems), all constructed from parallelepipeds with six sides in parallel pairs. They are defined ac-... 

The most powerful use of x-ray diffraction is in conjunction with single crystals of the sample. Structures of molecules ranging in size from a few atoms to viruses with tens of thousands of atoms have been solved. In a single crystal all the molecules lie on a three-dimensional lattice with fixed relative positions and orientations. The intensities of each point in the diffraction pattern can be measured and therefore information regarding the full three-dimensional structure can be obtained. 

In the kinematical theory, we consider the diffraction of a plane wave (of wavelength X) incident upon a three-dimensional lattice array of identical scattering points, each of which consists of a group of atoms and acts as the center of a spherical scattered wave. Our problem is to find the combined effect of the scattered waves at a point outside the crystal, at a distance from the crystal that is large compared with its linear dimensions. In developing the theory, we make several important assumptions ... 

As we established earlier, a powder diffraction pattern is one-dimensional but the associated reciprocal lattice is three-dimensional. This translates into scattering from multiple reciprocal lattice vectors at identical Bragg angles. Consider two points in a reciprocal lattice, 00/ and 00/. By examining Eqs. 2.29 to 2.34 it is easy to see that in any crystal system l/c/ (00/) = l/c/ (00/). Thus, Bragg reflections from these two reciprocal lattice points will be observed at exactly the same Bragg angle. 

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. 

When we consider diffraction from a crystal, we will further find it useful and convenient to consider a point lattice in even more simplified terms. We can introduce this simplification by organizing the lattice points, distributed in a periodic manner through three-dimensional space, into easily characterized families of two-dimensional planes. This is accomplished by defining all families of planes of equal interplanar spacing that include all of the points in the lattice. Examples are shown in Figure 3.17. Every plane of the family need not contain... 

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. 

Note that the resulting function depends only on IHI - which means that the three-dimensional information, available in single-crystal diffraction, is lost, or rather, averaged over the orientations of the crystallites. Since many different Hs may have the same magnitude, peaks (maxima) of the powder spectrum may well receive contributions from more than one reciprocal lattice point (thus the multiplicity of the reflections have to be established). 

X-ray diffraction is applied to crystalline arrays of molecules, so we need to know how to describe the arrangement of molecules in a crystal. The pattern that atoms, ions, or molecules adopt in a crystal is expressed in terms of an array of points making up the lattice that identify the locations of the individual species (Fig. 11.8). A unit cell of a crystal is the small three-dimensional figure obtained by joining typically eight of these points, which may be used to construct the entire crystal lattice by purely translational displacements, much as a wall may be constructed from bricks (Fig. 11.9). An infinite number of different unit cells can describe the same structure, but it is conventional to choose the cell with sides that have the shortest lengths and are most nearly perpendicular to one another. 

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