Space Lattice Geometry

A crystal is a physical object - it can be touched. However, an abstract construction in Euclidean space may be envisioned, known as a direct space lattice (also referred to as the real space lattice, space lattice, or just lattice for short), which is comprised of equidistant lattice points representing the geometric centers of the structural motifs. Any two of these lattice points are connected by a primitive translation vector, r, given by  

It is often convenient to choose a unit cell larger than the primitive unit cell. Nonprimitive unit cells contain extra lattice points, not at the vertices. For example, in three dimensions, nonprimitive unit cells may be of three kinds  

Face-centered, where a lattice point resides at the center of each face of the unit cell 

Side-centered, where an extra lattice point resides at each of two opposing faces of the unit cell. 

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Auguste Bravais (1811-1863) first proposed the Miller-Bravais system for indices. Also, as a result of his analyses of the external forms of crystals, he proposed the 14 possible space lattices in 1848. His Etudes Cristallographiques, published in 1866, after his death, treated the geometry of molecular polyhedra. 

First, the chemical activity of the metal surface is dependent on the atomic spacing and geometry of the surface atoms. Second, the mechanical properties, such as permeability and cohesion, of the very thin oxide films formed in the initial stages of attack are influenced by the short-range forces of the underlying metal and by the mechanism by which the surface metal lattice is converted into an ordered oxide lattice. 

FIGURE 16-3 The grid of wave numbers allowed by periodic boundary conditions, as in Fig. 15-2. Also shown are several lattice wave numbers (neavier dots). The relative spacings and geometry correspond to a simple cubic lattice of 6859 ions (see Problem 16-1). A sample wave number k is shown. 

The diffraction pattern (Fignre 2.1) then simply results from the interference of the reflections from sets of parallel planes within the crystal. The spacing of the lattice planes is determined by the lattice geometry, that is, it is a function of the unit cell parameters. The orientation of the plane with respect to the axes of the unit cell is defined by three integers, h, k, and / (Miller indices) that denoted the points where the plane intersects the three unit cell axes. Miller indices are defined as h = a X, k = blY, and I = dZ, where X, Y, and Z are the points where the plane intersects the a, b, and c nnit cell axes. Thus, the plane intersecting the unit cell at all, bll, and c/2 would have indices 222. 

S. Square vs. Hexagonal Rod Spacing in Pile. Since there is no preference in lattice geometry from the standpoint of physics, the square lattice spacing of rods will be adopted for its easier adaption to construction of the tank shields, piping, valve arrangements, etc. 

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