Three-dimensional point lattices

Figure 14 A three-dimensional point lattice and some examples of set of parallel planes of points...

This is the familiar formulation of Bragg s law for a three-dimensional point lattice. It says that the Fourier transform of a point lattice is absolutely discrete and periodic in diffraction space, and that we can predict when a nonzero diffraction intensity will appear for any family of planes hkl, and what the angle of incidence and reflection 0 must be in order for an intensity to appear. Bragg s law, notice, is completely independent of atoms, or molecules, or unit cell contents. The law is imposed by the periodicity of the crystal lattice, and it strictly governs where we may observe any nonzero intensity in diffraction space. It tells us when the resultant waves produced by the scattering of all of the atoms in the many individual unit cells, each represented by a single lattice point, are exactly in phase. 

When such features exist, they are penetrated by the electron beam so the material is represented by a three-dimensional point lattice and diffraction only occurs when the Ewald sphere intersects a point. This produces a transmission-type spot pattern. For smooth surfaces, the diffraction pattern appears as a set of streaks normal to the shadow edge on the fluorescent screen, due to the interaction of the Ewald sphere with the rods projecting orthogonally to the plane of the two-dimensional reciprocal lattice of the surface. The reciprocal lattice points are drawn out into rods because of the very small beam penetration into the crystal (2—5 atomic layers). We would emphasize, however, that despite contrary statements in the literature, the appearance of a streaked pattern is a necessary but not sufficient condition by which to define an atomically flat surface. Several other factors, such as the size of the crystal surface region over which the primary wave field is coherent and thermal diffuse scattering effects (electron—phonon interactions) can influence the intensity modulation along the streaks. 

A lattice is not a structure per se. A lattice is defined as a set of three-dimensional points, having a certain symmetry. These points may, or may not, be totally occupied by the atoms eomposing the structure. Consider a cubic structure such as that given in the following dicigram ... 

In general, the lattice points forming a three-dimensional space lattice should be visualized as occupying various sets of parallel planes. With reference to the axes of the unit cell (Fig. 16.2), each set of planes has a particular orientation. To specify the orientation, it is customary to use the Miller indices. Those are defined in the following manner Assume that a particular plane of a given set has intercepts p, q, and r... 

Lattice Points These are the points which indicates the position of atoms, ions or molecules in a crystal. These are equivalent points and are arranged in some regular pattern in three-dimensional space lattice. Each one of these lattice points have the same environment. 

In direct analogy with two dimensions, we can define a primitive unit cell that when repeated by translations in space, generates a 3D space lattice. There are only 14 unique ways of connecting lattice points in three dimensions, which define unit cells (Bravais, 1850). These are the 14 three-dimensional Bravais lattices. The unit cells of the Bravais lattices may be described by six parameters three translation vectors (a, b, c) and three interaxial angle (a, (3, y). These six parameters differentiate the seven crystal systems triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic. 

Bravais lattice — used to describe atomic structure of crystalline -> solid materials [i,ii], is an infinite array of points generated by a set of discrete translation operations, providing the same arrangement and orientation when viewed from any lattice point. A three-dimensional Bravais lattice consists of all points with position vectors R ... 

Bravais in 1849 showed that there are only 14 ways that identical points can be arranged in space subject to the condition that each point has the same number of neighbors at the same distances and in the same directions.Moritz Ludwig Frankenheim, in an extension of this study, showed that this number, 14, could also be used to describe the total number of distinct three-dimensional crystal lattices.These are referred to as the 14 Bravais lattices (Figure 4.9), and they represent combinations of the seven crystal systems and the four lattice centering types (P, C, F, I). Rhombohedral and hexagonal lattices are primitive, but the letter R is used for the former. 

FIGURE 5.5 A wavefront along k0 encounters a two-dimensional point lattice and is scattered in the general direction k. All the points in the planar lattice can be organized into lines of periodically spaced points as in Figure 5.4. Thus all points on any two-dimensional plane through a three-dimensional lattice can be considered to reflect waves according to conventional laws of optics, just as did the one-dimensional line of points. 

Mathematics (Hassel, 1830) has shown that there are only 32 combinations of symmetry operations (rotation, inversion, and reflection) that are consistent with a three-dimensional crystal lattice. These 32 point groups, or crystal classes, can be grouped into one of the seven crystal systems given in Table 2.1. There are four types of crystal lattices primitive (P), end-centered (C, B, and A), face-centered (/O, and body-centered (/). The primitive lattice contains a lattice point at each comer of the unit cell, the end-centered lattice has an additional lattice point on one of the lattice faces, the face-centered lattice has an extra lattice on each of the lattice faces, and the body-centered lattice has an extra lattice point at the center of the crystal lattice. By combining the seven crystal systems with the four lattice types (P, C, I, F), 14 unique crystal lattices, also known as Bravais lattices (Bravais, 1849), are produced. 

A solid can belong to one of an infinite number of general three-dimensional point groups. However, if the rotation axes are restricted to those that are compatible with the translation properties of a lattice, a smaller number, the crystallographic point groups, are found. The operators allowed within the crystallographic... 

We shall now apply the general formalism described in Section II to the study of an electron gas in a positive medium consisting of point charges fixed on a three-dimensional periodic lattice. For the moment the exact structure of the lattice will not be specified later on this model will be specialized to the cases of a pure metal and binary alloys. 

In order for the molecular lattice to accurately represent the molecule, it is necessary to add information to each lattice point reflecting the atom within which it is found. This process is illustrated in Figure 1 (steps D-E) wherein a portion of the molecule is shaded to represent different types of atoms (e.g. electron-rich, electron-poor). The correspondingly shaded lattice points reflect the atom type and thus can be considered a fourth dimensional extension of the three dimensional molecular lattice. 

Understand why there are a limited number of lattices. Be able to recognize the four two-dimensional and the seven three-dimensional primitive lattices. Know the locations of lattice points for body-centered and face-centered lattices. [Section 12.2]... 

Perhaps the most effective method of immobilization for stabilizing a protein is through multipoint covalent bindings between the protein surface and a polymer support. In this way, the protein is covalently attached to a three-dimensional support lattice through multiple points leading to a significant reduction in protein mobility. Enzymes immobilized using this technique far exceed the stability... 

---