Fundamental Concepts

• Atoms in crystalline solids are positioned in orderly and repeated patterns that are in contrast to the random and disordered atomic distribution found in noncrystalline or amorphous materials

Unit cells

• Crystal structures are specified in terms of parallelepiped units cells, which are characterized by geometry and atom positions within

Metallic crystal structures

• Most common metals exist in at least one of three relatively simple crystal structures:

  • Face-centered cubic (FCC), which has a cubic unit cell

    

  • Body-centered cubic (BCC), which also has cubic unit cell

    

  • Hexagonal close-packed, which has a unit cell of hexagonal symmetry

    

• Two features of a crystal structure are

  • Coordination number - the number of nearest-neighbor atoms

  • Atomic packing factor - the fraction of solid sphere volume in the unit cell

Density computations

• The theoretical density of a metal is a function of the number of equivalent atoms per unit cell, the atomic weight, unit cell volume, and Avogadro’s number

  

Polymorphism and allotropy

• polymorphism is when a specific material can have more than one crystal structure. Allotropy is polymorphism for elemental solids

Crystal systems

• used to classify crystal structures on the basis of unit cell geometry

• 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 defined by the unit cell for the particular crystal structure

Crystallographic directions

• the location of a point within a unit cell is specified using coordinated that are fractional multiples of the cell edge lengths

  

• directional indices are computed in terms of differences between vector head and tail coordinates

  

Crystallographic planes

• for hexagonal unit cells, a four-index scheme for both directions and planes is found to be more convenient. Directions may be determined using

  

Linear and planar densities

• crystallographic directional and planar equivalencies are related to atomic linear and planar densities, respectively

• for a given crystal structure, planes having identical atomic packing yet different Miller indices belong to the same family

Close-packed crystal structures

• both FCC and HCP crystal structures may be generated by the stacking of close-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…

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 faces and regular geometric shapes

Polycrystalline materials

• the vast majority of crystalline solids

• 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

Anisotropy

• anisotropy is the directionality dependence of properties

• for isotropic materials, properties are independent of the direction of measurement

X-ray diffraction: determination of crystal structures

• X-ray diffractometry is used for crystal structure and interplanar spacing determinations. A beam of x-rays directed on a crystalline material may experience diffraction (constructive interference) as a result of its interaction with a series of parallel atomic planes.

• Bragg's law specifies the condition for diffraction of x-rays

  

Noncrystalline solids

• Noncrystalline solid materials lack a systematic and regular arrangement of atoms or ions over relatively large distances (on an atomic scale). Sometimes the term amorphous is also used to describe these materials.

List of symbols