Chapter 4. Imperfections

Home ] Up ] Chapter 1. Introduction ] Chapter 2. Atomic Structure and Bonding ] Chapter 3. Structure of Crystals ] [ Chapter 4.  Imperfections ] Chapter 5. Diffusion ] Chapter 6. Mechanical Properties of Metals ] Chapter 7. Dislocations and Strengthening Mechanisms ] Chapter 8. Failure ] Chapter 9. Phase Diagrams ] Chapter 10: Phase Transformations in Metals ] Chapter 11. Thermal Processing of Metal Alloys ] Chapter 13. Ceramics - Structures and Properties ] Chapter 14. Ceramics - Applications and Processing ] Chapter 15. Polymer Structures ] Chapter 16. Polymers. Characteristics, Applications and Processing ] Chapter 17. Composites ] Chapter 19. Electrical Properties ]


Imperfections in Solids

4.1 Introduction

Materials are often stronger when they have defects.  The study of defects is divided according to their dimension:

0D (zero dimension) – point defects: vacancies and interstitials. Impurities.

1D – linear defects: dislocations (edge, screw, mixed)

2D – grain boundaries, surfaces.

3D – extended defects: pores, cracks.

Point Defects

4.2 Vacancies and Self-Interstitials

A vacancy is a lattice position that is vacant because the atom is missing. It is created when the solid is formed. There are other ways of making a vacancy, but they also occur naturally as a result of thermal vibrations.

An interstitial is an atom that occupies a place outside the normal lattice position. It may be the same type of atom as the others (self interstitial) or an impurity atom.

In the case of vacancies and interstitials, there is a change in the coordination of atoms around the defect. This means that the forces are not balanced in the same way as for other atoms in the solid, which results in lattice distortion around the defect.

The number of vacancies formed by thermal agitation follows the law:

NV = NA exp(-QV/kT)

where NA is the total number of atoms in the solid, QV is the energy required to form a vacancy, k is Boltzmann constant, and T the temperature in Kelvin (note, not in oC or oF).

When QV is given in joules, k = 1.38 10-23 J/atom-K. When using eV as the unit of energy, k = 8.62 10-5 eV/atom-K.

Note that kT(300 K) = 0.025 eV (room temperature) is much smaller than typical vacancy formation energies. For instance, QV(Cu) = 0.9 eV/atom. This means that NV/NA at room temperature is exp(-36) = 2.3 10-16, an insignificant number. Thus, a high temperature is needed to have a high thermal concentration of vacancies. Even so, NV/NA is typically only about 0.0001 at the melting point.

4.3 Impurities in Solids

All real solids are impure. A very high purity material, say 99.9999% pure (called 6N – six nines) contains ~ 6 1016 impurities per cm3.

Impurities are often added to materials to improve the properties. For instance, carbon added in small amounts to iron makes steel, which is stronger than iron. Boron impurities added to silicon drastically change its electrical properties.

Solid solutions are made of a host, the solvent or matrix) which dissolves the solute (minor component). The ability to dissolve is called solubility. Solid solutions are:

  • homogeneous
  • maintain crystal structure
  • contain randomly dispersed impurities (substitutional or interstitial)

Factors for high solubility

  • Similar atomic size (to within 15%)
  • Similar crystal structure
  • Similar electronegativity (otherwise a compound is formed)
  • Similar valence

Composition can be expressed in weight percent, useful when making the solution, and in atomic percent, useful when trying to understand the material at the atomic level.

Miscellaneous Imperfections

4.4 Dislocations—Linear Defects

Dislocations are abrupt changes in the regular ordering of atoms, along a line (dislocation line) in the solid. They occur in high density and are very important in mechanical properties of material. They are characterized by the Burgers vector, found by doing a loop around the dislocation line and noticing the extra interatomic spacing needed to close the loop. The Burgers vector in metals points in a close packed direction.

Edge dislocations occur when an extra plane is inserted. The dislocation line is at the end of the plane. In an edge dislocation, the Burgers vector is perpendicular to the dislocation line.

Screw dislocations result when displacing planes relative to each other through shear. In this case, the Burgers vector is parallel to the dislocation line.

4.5 Interfacial Defects

The environment of an atom at a surface differs from that of an atom in the bulk, in that the number of neighbors (coordination) decreases. This introduces unbalanced forces which result in relaxation (the lattice spacing is decreased) or reconstruction (the crystal structure changes).

The density of atoms in the region including the grain boundary is smaller than the bulk value, since void space occurs in the interface.

Surfaces and interfaces are very reactive and it is usual that impurities segregate there. Since energy is required to form a surface, grains tend to grow in size at the expense of smaller grains to minimize energy. This occurs by diffusion, which is accelerated at high temperatures.

Twin boundaries: not covered

4.6 Bulk or Volume Defects

A typical volume defect is porosity, often introduced in the solid during processing. A common example is snow, which is highly porous ice.

4.7 Atomic Vibrations

Atomic vibrations occur, even at zero temperature (a quantum mechanical effect) and increase in amplitude with temperature. Vibrations displace transiently atoms from their regular lattice site, which destroys the perfect periodicity we discussed in Chapter 3.

Macroscopic Examination

Sections 4.8 to 4-10 were not covered.



  • Alloy
  • Atom percent
  • Atomic vibration
  • Boltzmann’s constant
  • Burgers vector
  • Composition
  • Dislocation line
  • Edge dislocation
  • Grain size
  • Imperfection
  • Interstitial solid solution
  • Microstructure
  • Point defect
  • Screw dislocation
  • Self-Interstitial
  • Solid solution
  • Solute
  • Solvent
  • Substitutional solid solution
  • Vacancy
  • Weight percent