Materials Science on CDROM User Guide
Introduction to Point Defects
Version 2.1
Andrew Green, MATTER
John Humphreys, UMIST/University of Manchester
Ross Mackenzie, Open University
October 1997
Assumed Preknowledge
This module has been developed primarily as a background to the module Atomic Diffusion in Metals and Alloys for which a basic
understanding of the nature of vacancies and interstitials is essential. Before starting
this module, it is assumed that the student is familiar with:
 the basics of (metallurgical) thermodynamics;
 the terms Gibbs’ free energy, G, enthalpy, H and entropy,
S;
 the relationship G = HTS, where T is temperature;
 basic crystallography, including stacking sequences in FCC, BCC and CPH crystal
structures.
Module Structure
The module consists of two sections: Vacancies
and Interstitials. As previously mentioned, it
is not designed as a comprehensive treatment of this subject, but as a background
to the MATTER module Atomic Diffusion in Metals and Alloys.
This section begins with a brief description of some of the important types of point
defects in metals and alloys. They include:
 Vacancies
 Selfinterstitials
 (Foreign) interstitials
 Substitutional atoms
Each type is illustrated by a graphic.
The remainder of the section concentrates on vacancies, and in particular the equilibrium
vacancy concentration. Firstly, an overview is given of the variation in free energy
of a crystal, DG with vacancy concentration, X_{v}.
From this relationship, the equilibrium vacancy concentration, X_{v}^{e}
is shown to be that which results in the minimum free energy.
Having looked at the general relationship between X_{v} and DG, the module goes on to look in more detail at the
underlying theory.
The free energy term DG is first rewritten in terms
of enthalpy (DH) and entropy (DS)
of vacancy formation, using the familiar relationship:

(1) 
where T is the absolute temperature. The DH
and DS terms are each considered in turn.
Enthalpy of vacancy formation, DH
This is the change in enthalpy resulting from the addition of vacancies and arises from
the increase in internal energy caused by breaking interatomic bands (i.e. removing
atoms). By making the reasonable assumption that the X_{v} is low, such
that vacancyvacancy interactions can be ignored, it is shown that DH
is proportional to X_{v}, according to the equation:

(2) 
DH_{v} is the molar enthalpy of vacancy
formation.
Entropy of vacancy formation, DS
This arises from the increased degree of randomness introduced by the addition of
vacancies to a crystal. DS itself can be separated into
2 components:
 Thermal entropy,
 Configurational entropy.
Each of these terms is explained in further detail. It is shown that thermal entropy
can be given by:

(3) 
where DS_{v} is the molar thermal entropy.
Special attention is given to the dominant configurational entropy term, and a sidebranch
takes the user to background information and related exercises. An expression for
configurational entropy as a function of vacancy concentration is derived:

(4) 
The variation of each component, and hence DS with
vacancy concentration is illustrated by a graph.
Having seen how both DH and DS
vary with vacancy concentration, the relationship between DG
and X_{v} is revisited. An interactive graph plotting routine allows the
user to see the effect of some of the important parameters on the overall shape of the DG vs. X_{v }curve and hence on the equilibrium
vacancy concentration.
For many metallurgical processes, such as phase transformations, diffusion, creep,
etc., it is important to know how the equilibrium vacancy concentration, X_{v}^{e}
varies with temperature, T. The final page of the section shows that this
relationship can be given by:

(5) 
Often it is convenient to split DG into enthalpy and
entropy components, i.e.:

(6) 
The relatively temperatureinsensitive term exp(DS_{v
}/R) is normally taken as a constant of value ~3, to give:

(7) 
Interstitial solid solutions occur when the solute atoms occupy the interstices between
the solvent atoms. Such materials form the basis of some very important alloy systems, not
least FeC.
This section concentrates on the interstitial sites in the three most common metal
crystal structures, namely facecentred cubic (FCC), bodycentred cubic (BCC) and
closepacked hexagonal (CPH). In a series of 3 interactive exercises, the unit cell for
each structure is shown and the user asked to click the mouse at each of the points
corresponding the octahedral and tetrahedral interstices (voids). An example
of each type of interstice is available for those requiring guidance. Each individual
interstice becomes highlighted when correctly identified.
Additional questions for the FCC and BCC cases ask the student to calculate the void
radii as a fraction of the parent atom radii for both octahedral and tetrahedral
interstices.
The final page of the section compares the equilibrium concentration of
selfinterstitials with that of vacancies.
Bibliography
The student is referred to the following resources in this module:
Porter, D.A., and Easterling, K.E., Phase Transformations in Metals and Alloys,
2nd ed., Chapman & Hall, 1992
Henderson, B., Defects in Crystalline Solids, Arnold, 1972
Flynn, C.P., Point Defects and Diffusion, Clarendon Press, 1972
