Reciprocal lattice specimens

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). 

Polycrystal-type (rings) electron diffraction patterns (Fig.6) are especially valuable for precision studies - checking on the scattering law, identification of the nature of chemical bonding, and refinement of the chemical composition of the specimen - because these patterns allow the precision measurements of reflection intensities. The reciprocal lattice of a polycrystal is obtained by spherical rotation of the reciprocal lattice of a single crystal around a fixed 000 point it forms a system of spheres placed one inside the other and has the symmetry co oo.m. It is also important for structure... 

The reciprocal lattice of single crystal is a system of points. In the case of a plate texture, the axis of the reciprocal lattice is perpendicular to the specimen support. When a plate texture specimen is perpendicular to the electron beam, the diffraction pattern becomes a system of concentric rings (equivalent to the rotation of single crystal about the texture axis). 

Figure 9. Formation of circular scattering regions (rings) in the reciprocal lattice of a texture, and relationship between their shape and the structure of the specimen. Transition from a point to a ring (a) for an ideal texture without disorder (c), having a distribution function (e) (d), (f) - corresponding diagrams for a real texture with some disorder.

Figure 4.7 The accessible, allowed reflections in a high resolution experiment, shown in reciprocal space, for a silicon specimen with an (001) surface, using CuK radiation, (a) incident beam in (100) plane, reciprocal lattice section perpendicular to [100], (b)...

The deviation vector q, with origin at the end of the reciprocal lattice vector h, has two components, q y horizontal, positive rightwards going, and q vertical, positive upwards going. For the symmetric reflection, these components are related to the deviations of specimen ( ) and analyser ( ) from their zero... 

Figure 7.8 A scattering map in reciprocal space. Equal intensity contours are shown schematically, and the Ewald sphere is represented as a plane near reciprocal lattice points 0 and h. The dynamical diffraction from the specimen is displaced slightly from the relp and from the centre of the diffuse scatter by the refractive index effect...

As will become apparent, it is important to place the photographic plate as close to the specimen as possible. With rotating anode generators, care should be taken not to allow the full power of the beam to fall on the plate when stationary as this leads to an unsightly overexposed vertical line on the topograph. The presence of a horizontal stripe on the recorded topograph is often due to the presence of a second reciprocal lattice point lying on the Ewald sphere. It can be removed by a small rotation of the crystal about the diffraction vector as if to take a stereo pair. 

This intensity distribution means that each reciprocal lattice point is effectively extended into a rod that is normal to the thin crystal specimen. Due to this extension of the reciprocal lattice points and to the fact that the radius of the Ewald sphere is large compared with the spacing between reciprocal lattice points, the Ewald sphere usually intersects many reciprocal lattice rods, as shown in Figure 3.12(a). Thus, many diffracted... 

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. 

The powder diffraction experiment is the cornerstone of a truly basic materials characterization technique - diffraction analysis - and it has been used for many decades with exceptional success to provide accurate information about the structure of materials. Although powder data usually lack the three-dimensionality of a diffraction image, the fundamental nature of the method is easily appreciated from the fact that each powder diffraction pattern represents a one-dimensional snapshot of the three-dimensional reciprocal lattice of a crystal. The quality of the powder diffraction pattern is usually limited by the nature and the energy of the available radiation, by the resolution of the instrument, and by the physical and chemical conditions of the specimen. Since many materials can only be prepared in a polycrystalline form, the powder diffraction experiment becomes the only realistic option for a reliable determination of the crystal structure of such materials. 

In the first three chapters of this book, we considered the fundamentals of crystallographic symmetry, the phenomenon of diffraction from a crystal lattice, and the basics of a powder diffraction experiment. Familiarity with these broad subjects is essential in understanding how waves are scattered by crystalline matter, how structural information is encoded into a three-dimensional distribution of discrete intensity maxima, and how it is convoluted with numerous instrumental and specimen-dependent functions when projected along one direction and measured as the scattered intensity V versus the Bragg angle 20. We already learned that this knowledge can be applied to the structural characterization of materials as it gives us the ability to decode a one-dimensional snapshot of a reciprocal lattice and therefore, to reconstruct a three-dimensional distribution of atoms in an infinite crystal lattice by means of a forward Fourier transformation. 

Figure 2.12 The Ewald sphere method illustrates the ring pattern of diffraction from a powder specimen. The Debye ring recorded by the Elull-Debye-Scherrer method results from randomly oriented crystals in the powder specimen, in which reciprocal lattice points of (hkl) touch the Ewald sphere surface in various directions to form individual rings. It is equivalent to rotating a reciprocal lattice along an incident beam axis. (Reproduced with permission from R. Jenkins and R.L. Snyder, Introduction to X-ray Powder Diffractometry, John Wiley Sons Inc., New York. 1996 John Wiley Sons Inc.)...

Electron diffraction in the TEM often does not satisfy the exact conditions of Bragg s Law. This relaxation of Bragg s Law in a TEM is illustrated in Figure 3.29, in which the reciprocal lattice points are elongated in the transmitted beam direction due to the small thickness of a TEM specimen. The deviation from the exact Bragg conditions can be represented by a deviation... 

The random orientations of the individual crystals in a powder specimen are equivalent to the rotation of a single crystal about all possible axes during the x-ray exposure. The reciprocal lattice therefore takes on all possible orientations relative to the incident beam, but its origin remains fixed at the end of the Sq/A vector. 

It should be noted that any defects and imperfection in structure and orientation would give rise to rather coarse points and doughnut-shaped zones in the reciprocal lattice rather than clearly defined points and circles. This is the case for oriented liquid crystals and polymer fibers. Diffused diffraction patterns are therefore obtained for these specimens. 

Figure 9 shows the uniaxial orientation distribution functions qjiCp 0), determined from X-ray diffraction measurement, for a high-density polyethylene specimen stretched to an extension ratio of 1.4. Twelve different 7th reciprocal lattice vectors were observed. With qjiCp 0) for7 = 1 to 12, the coefficients Qj were calculated from equation (9) for any higher orders of I, and then the coefficients Wion were calculated from the simultaneous equations of equation (11) with respect to 7 by the weighted least-square method up to /= 18. 

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