Page 122 - Materials Science and Engineering An Introduction

94 • Chapter 3 / The Structure of Crystalline Solids

• Two features of a crystal structure are
Coordination number—the number of nearest-neighbor atoms, and
Atomic packing factor—the fraction of solid sphere volume in the unit cell.

Density • The theoretical density of a metal (r) is a function of the number of equivalent atoms per
Computations unit cell, the atomic weight, unit cell volume, and Avogadro’s number (Equation 3.8).
Polymorphism and • Polymorphism is when a specific material can have more than one crystal structure.
Allotropy Allotropy is polymorphism for elemental solids.

Crystal Systems • The concept of a crystal system is used to classify crystal structures on the basis of unit
cell geometry—that is, unit cell edge lengths and interaxial angles. There are seven
crystal systems: cubic, tetragonal, hexagonal, orthorhombic, rhombohedral (trigonal),
monoclinic, and triclinic.

Point Coordinates • Crystallographic points, directions, and planes are specified in terms of indexing
schemes. The basis for the determination of each index is a coordinate axis system
Crystallographic
Directions defined by the unit cell for the particular crystal structure.
The location of a point within a unit cell is specified using coordinates that are
Crystallographic fractional multiples of the cell edge lengths (Equations 3.9a–3.9c).
Planes Directional indices are computed in terms of differences between vector head
and tail coordinates (Equations 3.10a–3.10c).
Planar (or Miller) indices are determined from the reciprocals of axial intercepts
(Equations 3.14a–3.14c).
• For hexagonal unit cells, a four-index scheme for both directions and planes is found
to be more convenient. Directions may be determined using Equations 3.11a–3.11d
and 3.12a–3.12d.

Linear and Planar • Crystallographic directional and planar equivalencies are related to atomic linear and
Densities planar densities, respectively.
Linear density (for a specific crystallographic direction) is defined as the number
of atoms per unit length whose centers lie on the vector for this direction
(Equation 3.16).
Planar density (for a specific crystallographic plane) is taken as the number of
atoms per unit area that are centered on the particular plane (Equation 3.18).
• For a given crystal structure, planes having identical atomic packing yet different
Miller indices belong to the same family.
Close-Packed • Both FCC and HCP crystal structures may be generated by the stacking of close-
Crystal Structures packed planes of atoms on top of one another. With this scheme A, B, and C denote
possible atom positions on a close-packed plane.
The stacking sequence for HCP is ABABAB. . . .
The stacking sequence for FCC is ABCABCABC. . . .
• Close-packed planes for FCC and HCP are 111 and 0001 , respectively.

Single Crystals • Single crystals are materials in which the atomic order extends uninterrupted over
the entirety of the specimen; under some circumstances, single crystals may have flat
Polycrystalline
Materials faces and regular geometric shapes.
• The vast majority of crystalline solids, however, are polycrystalline, being composed
of many small crystals or grains having different crystallographic orientations.
• A grain boundary is the boundary region separating two grains where there is some
atomic mismatch.