Lattices, planes and directions

The development of the idea of a lattice was among the earliest mathematical explorations in crystallography. Crystal structures and crystal lattices are different, although these terms are frequently (and incorrectly) used as synonyms. A crystal structure is built of atoms. A crystal lattice is an infinite pattern of points, each of which must have the same surroundings in the same orientation. A lattice is a mathematical concept. 

All crystal structures can be built up from a lattice by placing an atom or a group of atoms at each lattice point. The crystal structure of a simple metal and that of a complex protein may both be described in terms of the same lattice, but whereas the number of atoms allocated to each lattice point is often just one for a simple metallic crystal, it may easily be thousands for a protein crystal. 

---

Figure 4.4. X-rays scattered by atoms in an ordered lattice interfere constructively in directions given by Bragg s law. The angles of maximum intensity enable one to calculate the spacings between the lattice planes and allow furthermore for phase identification. Diffractograms are measured as a function of the angle 26. When the sample is a...

The geometrical aspect concerns the position of the diffracted beams on a pattern it only depends on the direct lattice of the crystal through the Bragg law =2dhkisin9B - dhu being the interplanar distance of the diffracted (hkl) lattice planes and 0b the Bragg angle. In other words, it only depends on the lattice parameters of the crystal a, b, c, a, P and y. 

Crystal Locations, Planes, and Directions. In order to calculate such important quantities as cell volumes and densities, we need to be able to specify locations and directions within the crystal. Cell coordinates specify a position in the lattice and are indicated by the variables u, v, w, separated by commas with no brackets ... 

In some structures, several planes and directions may be equivalent by symmetry. For example, this is the case for the (100), (010), (001), (100), (010), and (OOl) planes in the diamond cubic structure. Equivalent directions are denoted concisely as a group by using angular brackets. Thus, the (100) directions in a diamond cubic lattice include all of the directions that are perpendicular to the six planes noted above. The Miller index notation thus provides a concise designation for describing the surfaces of semiconductor crystals. 

X-ray diffraction analysis of the samples sintered from powder A has shown that theoretical density decreases with the growth of the sintering temperature, while the experimental values increase. At the same time lattice parameter a increases and parameter c decreases under the sintering temperature higher than 1200°C, i.e. the anisotropy of the TiB2 lattice decreases. Apparently this is caused by different value of compressibility of the lattice in basal plane and direction of hexagonal axis. 

Figure 2.9 Relationship between a reciprocal lattice plane and real space directions. The reciprocal lattice plane is composed of diffraction spots from crystal planes (hkO). The zone axis [001 ] is perpendicular to the reciprocal lattice plane.

Directions in a hexagonal lattice are best expressed in terms of the three basic vectors a, a2, and c. Figure 2-11(b) shows several examples of both plane and direction indices. Another system, involving four indices, is sometimes used to designate directions. The required direction is broken up into four component vectors, parallel to a, a2, as, and c and so chosen that the third index is the... 

Planes, Directions and Plastic Deformation. When dealing with the modern interpretation of the mechanism of plastic deformation of metals and alloys and with the associated problems of hardness and strength—resistance to plastic deformation—we find it essential to be familiar with the more important crystallographic planes and crystallographic directions. We shall, therefore, with this in mind, devote our attention now to a brief discussion of the planes and directions of the atoms in the throe main types of metal crystal lattices... 

NiO grows on MgO with both grains aligned so that corresponding planes and directions in the two crystals are nearly parallel. The difference in lattice parameter (Aa) will be accommodated by a square array of misfit dislocations. 

X-Rays have been used to study the interior of crystals since 1912. Exactly as a diffraction grating produces interference effects with light, crystals diffract the incident X-ray beam. Indeed, as already stated, the lattice of the crystal can be divided into an indefinite number of parallel and equidistant planes called the lattice planes, and each of these lattice planes is to the X-ray beam what a line of the diffraction grating is to the light beam. When an X-ray source passes through the crystal, the beam is reflected in different directions, and interference effects are obtained. The direction of the outgoing beams is determined by the direction of the incident rays, their wavelength, and the distance between two lattice planes. In the crystal. 

The base vectors of an elementary cell in a translation lattice together with one lattice point define a crystollogrophic coordinote system, which is suitable for description of crystallographic planes and directions by means of vectors with integer components. But for description of crystal pltysical properties (e.g. elastic, dielectric or pyroelectric properties, etc.) rising material constants another coordinate system, so called the Cortesion system, is preferred. 

The nanostructure with no defects or artificial defects are not suitable to simulate the material properties of practical nanostructure, in this part, we introduce an integrated MD simulation method, in order to study the mechanical properties of nanostructure with practical processing defects. We consider the nanostructure scratched by diamond tip under the machining conditions of different scratching depth of different scratching lattice plane or direction, just like we have described above, as the practical processing defects, then, apply tension or shearing force on such nanostructure to observe their physical and mechanical behavior. 

---