Point lattices diffraction patterns from

FIGURE 1.14 Seen here is the hk0 zone diffraction pattern from a crystal of M4 dogfish lactate dehydrogenase obtained using a precession camera. It is based on a tetragonal crystal system and, therefore, exhibits a fourfold axis of symmetry. The hole at center represents the point where the primary X-ray beam would strike the film (but is blocked by a circular beamstop). Note the very predictable positions of the diffraction intensities. All the intensities, or reflections, fall at regular intervals on an orthogonal net, or lattice. This lattice in diffraction space is called the reciprocal lattice. 

Because the observed diffraction pattern is a product of the diffraction patterns from the two distributions, what is observed at each nonzero point in the combined transform, or diffraction pattern determined by the periodic point lattice in Figure 5.8c, is the value of the Fourier transform at that point from the continuous distribution in Figure 5.8a. That is, the combined diffraction pattern samples the continuous Fourier transform of the object making up the array, that of Figure 5.8b, but only at those discrete points permitted by the array s periodic, discrete transform seen in Figure 5.8d. 

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. 

FIGURE 4.1.3 (a) Schematic diagram of a GIXD experiment (b) Rods formed in reciprocal space by a 2-D lattice of points, and the reciprocal disk of an extended molecule, which combines to give the diffraction pattern from a 2-D array of extended molecules (c) real-space and reciprocal-space views and characteristic diffraction patterns of the monolayer in untilted phase (d) AW-titled phase (e) AWA -titled phase (f) intermediate titled phase. (From Kaganer, V.M. et ah. Rev. Mod. Phys. 71, 779, 1999. With permission.)... 

The point scatters are equally spaced with distances a, b and c between adjacent pairs along X-, y- and z-axes. The angles, oto, po and Yo and ot, p and y are the angles that the incident and diffracted beams make with the three axes respectively. The integers, h, k and 1 are Miller indices. The diffraction pattern from a crystal will thus consist of spots of scattered intensity whose positions depend on the crystal lattice. 

Olesen and Szabo obtained crystals from ethanol and acetone53. They found the crystals to have different solubility, melting point and x-ray diffraction patterns. Since acetone is retained in the crystalline lattice, it was indicated that the forms are pseudopolymorphs. 

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. 

Electron Diffraction (CBED) and Large-Angle Convergent-Beam Electron Diffraction (LACBED) allow the identification of the crystal system, the Bravais lattice and the point and space groups. These crystallographic features are obtained at microscopic and nanoscopic scales from the observation of symmetry elements present on electron diffraction patterns. 

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). 

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