Body-Centered Cubic Direct Lattice

Next consider a bcc lattice. A set of primitive bcc lattice vectors is 

We use the primitive vectors for the bcc lattice to distinguish its special symmetry from the nonprimitive bcc lattice and use the prime symbol to distinguish these primitive lattice vectors from the lattice vectors a, b, and c. The choice of the primitive vectors is arbitrary. Any set will produce similar results. Using the set of relations in Equation 6.9 to transform these vectors to reciprocal space. 

These are the Cartesian coordinates of a set of primitive translation vectors for an fee lattice. 

Primitive and nonprimitive vectors in an fee reciprocal lattice in which the primitive vectors and their translation sites are denoted by prime symbols. Trar slatioi s along these primitive vectors produce a nonprimitive fee reciprocal lattice with a basis of (000), (110), (101), (011). Notice that primitive trai slatioi s (100) are equivalent to (110), (200) to (220), etc 

Thus the primitive translation vector in reciprocal space, G = h A + fc B + C for a bcc direct lattice (fee reciprocal lattice) is 

---

Table 4.1 Points and directions of high symmetry in the first Brillouin zones, fee is the face-centered cubic crystal lattice bcc is the body-centered cubic crystal lattice hep is the hexagonal close-packed crystal lattice.

Dispersion curves of longitudinal and transverse osciQations along direction a type of that is L[lll] and T[lll] for potassium are presented in Figure 12.4. This direction is chosen because the distance between neighboring atoms is minimal along it in the body-centered cubic crystal lattice. Similar curves are typical for other alkali elements. 

A great deal of effort has been directed to determining the structures of lithium alkyls. It has been determined that in hydrocarbon solutions the dominant species is a hexamer when the alkyl groups are small. In the solid phase, the structure is body-centered cubic with the (LiCH3)4 units at each lattice site. Each unit is a tetramer in which the four lithium atoms reside at the comers of a tetrahedron and the methyl groups are located above the centers of the triangular faces. The carbon atoms of... 

Therefore lattice structures with closely packed planes allow more plastic deformation than those that are not closely packed. Also, cubic lattice structures allow slippage to occur more easily than non-cubic lattices. This is because of their symmetry which provides closely packed planes in several directions. Most metals are made of the body-centered cubic (BCC), face-centered cubic (FCC), or hexagonal close-packed (HCP) crystals, discussed in more detail in the Module 1,... 

D systems, particularly crystalline materials, favor lattice structures which can be viewed as stackings of planar layers. The unit cell is a description of the arrangement of atoms within a box unit cells may be stacked in each direction to describe an atomic lattice. Three of the most common unit cells are the body-centered cubic (bcc), face-centered cubic (fee) and hexagonal close-packed... 

For the body-centered cubic lattice (bcc), the addition of a building block is equal to a linear displacement by half a lattice constant in the three lattice directions. For building blocks that do not have a rotational symmetry, an additional rotation of the building blocks can occur one of the most prominent being a rotation by 180°. This combined displacement and rotation by 180° appears like the movement of a screw. 

Figure 4.12 A body-centered cubic lattice viewed along a [001] direction showing the atom positions in the perfect crystal and how they change with the formation of [001] twist boundaries.

TABLE 2.4 Basis Vectors of Direct and Reciprocal Space for the Three Cubic Lattices SC (Simple Cubic), BCC (Body-Centered Cubic), FCC (Face-Centered Cubic), CPH (Close-Packed Hexagonal)... 

All of these hexafluorides are dimorphic, with a high-temperature, cubic form and an orthorhombic form, stable below the transition temperature (92). The cubic form corresponds to a body-centered arrangement of the spherical units, with very high thermal disorder of the molecules in the lattice, leading to a better approximation to a sphere. Recently, the structures of the cubic forms of molybdenum (93) and tungsten (94) hexafluorides have been studied using neutron powder data, with the profile-refinement method and Kubic Harmonic analysis. In both compounds the fluorine density is nonuniformly distributed in a spherical shell of radius equal to the M—F distance. Thus, rotation is not completely free, and there is some preferential orientation of fluorine atoms along the axial directions. The M—F distances are the same as in the gas phase and in the orthorhombic form. 

---