32 3·Mechanisms that is,repetitive,pattern which forms a three-dimensional grid called a lattice.The smallest unit of such a lattice that still pos- sesses the characteristic symmetry of the entire lattice is called a conventional unit cell.(Occasionally smaller or larger unit cells are used to better demonstrate the particular symmetry of a unit.) The entire lattice can be generated by translating the unit cell into three-dimensional space. Bravais Bravais!has identified 14 fundamental unit cells,often referred Lattice to as Bravais lattices or translation lattices,as depicted in Figure 3.8.They vary in the lengths of their sides (called lattice constants, a,b,and c)and the angles between the axes (a,B,y).The char- acteristic lengths and angles of a unit cell are termed lattice pa- rameters.The arrangement of atoms into a regular,repeatable lat- tice is called a crystal structure. The most important crystal structures for metals are the face- centered cubic(FCC)structure,which is typically found in the case of soft (ductile)materials,the body-centered cubic (BCC) structure,which is common for strong materials,and the hexag- onal close-packed (HCP)structure,which often is found in brit- tle materials.It should be emphasized at this point that the HCP structure is not identical with the simple hexagonal structure shown in Figure 3.8 and is not one of the 14 Bravais lattices since HCP has three extra atoms inside the hexagon.The unit cell for HCP is the shaded portion of the conventional cell shown in Fig- ure 3.9.It contains another "base"atom within the cell in con- trast to the hexagonal cell shown in Figure 3.8. The lattice points shown as filled circles in Figure 3.8 are not necessarily occupied by only one atom.Indeed,in some materi- als,several atoms may be associated with a given lattice point; this is particularly true in the case of ceramics,polymers,and chemical compounds.Each lattice point is equivalent.For ex- ample,the center atom in a BCC structure may serve as the cor- ner of another cube. We now need to define a few parameters that are linked to the mechanical properties of solids. cla Ratio The separation between the basal planes,co,divided by the length of the lattice parameter,ao,in HCP metals(Figure 3.9),is theoret- ically V8/3 =1.633,assuming that the atoms are completely spherical in shape.(See Problem 3.6.)Deviations from this ideal ratio result from mixed bondings and from nonspherical atom shapes.The c/a ratio influences the hardness and ductility of ma- terials;see Section 3.4. See Section 3.1.that is, repetitive, pattern which forms a three-dimensional grid called a lattice. The smallest unit of such a lattice that still pos￾sesses the characteristic symmetry of the entire lattice is called a conventional unit cell. (Occasionally smaller or larger unit cells are used to better demonstrate the particular symmetry of a unit.) The entire lattice can be generated by translating the unit cell into three-dimensional space. Bravais1 has identified 14 fundamental unit cells, often referred to as Bravais lattices or translation lattices, as depicted in Figure 3.8. They vary in the lengths of their sides (called lattice constants, a, b, and c) and the angles between the axes (, , ). The char￾acteristic lengths and angles of a unit cell are termed lattice pa￾rameters. The arrangement of atoms into a regular, repeatable lat￾tice is called a crystal structure. The most important crystal structures for metals are the face￾centered cubic (FCC) structure, which is typically found in the case of soft (ductile) materials, the body-centered cubic (BCC) structure, which is common for strong materials, and the hexag￾onal close-packed (HCP) structure, which often is found in brit￾tle materials. It should be emphasized at this point that the HCP structure is not identical with the simple hexagonal structure shown in Figure 3.8 and is not one of the 14 Bravais lattices since HCP has three extra atoms inside the hexagon. The unit cell for HCP is the shaded portion of the conventional cell shown in Fig￾ure 3.9. It contains another “base” atom within the cell in con￾trast to the hexagonal cell shown in Figure 3.8. The lattice points shown as filled circles in Figure 3.8 are not necessarily occupied by only one atom. Indeed, in some materi￾als, several atoms may be associated with a given lattice point; this is particularly true in the case of ceramics, polymers, and chemical compounds. Each lattice point is equivalent. For ex￾ample, the center atom in a BCC structure may serve as the cor￾ner of another cube. We now need to define a few parameters that are linked to the mechanical properties of solids. The separation between the basal planes, c0, divided by the length of the lattice parameter, a0, in HCP metals (Figure 3.9), is theoret￾ically 8/3
1.633, assuming that the atoms are completely spherical in shape. (See Problem 3.6.) Deviations from this ideal ratio result from mixed bondings and from nonspherical atom shapes. The c/a ratio influences the hardness and ductility of ma￾terials; see Section 3.4. Bravais Lattice 32 3 • Mechanisms 1See Section 3.1. c/a Ratio