Crystals Bravais lattice

Crystallographic nomenclature (Bravais lattices, crystal classes, space groups) The following information is generally included in a usual crystallographic description ... 

Example 16.1-1 Find the Bravais lattices, crystal systems, and crystallographic point groups that are consistent with a C3z axis normal to a planar hexagonal net. 

Crystal system Possible Bravais lattices Crystal classes or point groups Number of space groups... 

Translation Symmetry of Crystals. Point Symmetry of Bravais Lattices. Crystal Class... 

If atoms, molecules, or ions of a unit cell are treated as points, the lattice stmcture of the entire crystal can be shown to be a multiplication ia three dimensions of the unit cell. Only 14 possible lattices (called Bravais lattices) can be drawn in three dimensions. These can be classified into seven groups based on their elements of symmetry. Moreover, examination of the elements of symmetry (about a point, a line, or a plane) for a crystal shows that there are 32 different combinations (classes) that can be grouped into seven systems. The correspondence of these seven systems to the seven lattice groups is shown in Table 1. 

The crystal group or Bravais lattice of an unknown crystalline material can also be obtained using SAD. This is achieved easily with polycrystalline specimens, employing the same powder pattern indexing procedures as are used in X-ray diffraction. ... 

All crystal structures are derived from the 14 Bravais lattices. The atoms in a unit cell are counted by determining what fraction of each atom resides within the cell. The type of unit cell adopted by a metal can be determined by measuring its density. 

Bravais lattices The 14 basic patterns of unit cells from which a crystal can be built. 

These 14 Bravais Lattices are unique in themselves. If we arrange the crystal systems in terms of symmetry, the cube has the highest symmetry and the triclinic lattice, the lowest symmetry, as we showed above. The same hierarchy is maintained in 2.2.4. as in Table 2-1. The symbols used by convention in 2.2.4. to denote the type of lattice present are... 

One of the concepts in use to specify crystal structures the space lattice or Bravais lattice. There are in all fourteen possible space (or Bravais) lattices. 

Crystal family Symbol Crystal system Crystallographic point groups (crystal classes) Number of space groups Conventional coordinate system Bravais lattices... 

Figure 3.4. The crystal systems and the Bravais lattices illustrated by a unit cell of each. All the points which, within a unit cell, are equivalent to each other and to the cell origin are shown. Notice that, in the primitive lattices the unit cell edges are coincident with the smallest equivalence distances. For the rhombohedral lattice, described in terms of hexagonal axis, the symbol hR is used instead of a symbol such as rP. In the construction of the so-called Pearson symbol ( 3.6.3), oS and mS will be used instead of oC and mC.

Microdiffraction is the pertinent method to identify the crystal system, the Bravais lattices and the presence of glide planes [4] (see the chapter on symmetry determination). For the point and space group identifications, CBED and LACBED are the best methods [5]. 

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. 

With the exception of He, solid noble gases crystallize in the face-centred-cubic Bravais lattice and their cohesive energies (eV atom ) are 0.02 for Ne, 0.08 for Ar, 0.11 for Kr and 0.17 for Xe (Ashcroft Mermin, 1976 p. 401). VanderWaals-Londonforces areexpressedby theLennard-Jonespotential f/u a — CaZ> ,... 

Identify the seven crystal systems and 14 Bravais lattices. 

Our description of atomic packing leads naturally into crystal structures. While some of the simpler structures are used by metals, these structures can be employed by heteronuclear structures, as well. We have already discussed FCC and HCP, but there are 12 other types of crystal structures, for a total of 14 space lattices or Bravais lattices. These 14 space lattices belong to more general classifications called crystal systems, of which there are seven. 

If we combine the 32 crystal point groups with the 14 Bravais lattices we find 230 three-dimensional space groups that crystal structures can adopt (i.e., 230... 

It is possible to characterize the type of Bravais lattice present by the pattern of systematic absences. Although our discussion has centred on cubic crystals, these absences apply to all crystal systems, not just to cubic, and are summarized in Table 2.3 at the end of the next section. The allowed values of are listed in Table 2.2 for... 

Nickel crystallizes in a cubic crystal system. The first reflection in the powder pattern of nickel is the 111. What is the Bravais lattice ... 

A.2 The fourteen Bravais lattices and seven crystal systems Refer to Figs. A.2.1 and Table A.2.1. 

Figure 11.11. The 14 Bravais lattices arranged into the 6 crystal systems.

In Section 11.4 the fourteen 3D lattices (Bravais lattices) were derived and it was shown that they could be grouped into the six crystal systems. For each crystal system the point symmetry of the lattice was determined (there being one point symmetry for each, except the hexagonal system that can have either one of two). These seven point symmetries are the highest possible symmetries for crystals of each lattice type they are not the only ones. 

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