Dislocation Density

Dislocation density
The number of dislocations in a specimen is clearly an important quantity. We define the dislocation density Nd as the total length of dislocation line in unit volume. It is also equal to the number of dislocations which cut through a unit area which is randomly oriented in the crystal.
In annealed crystals Nd is of the order of 10^10 m-2; in work hardened materials it can be as high as 10^16 m-2.



The Burger’s vector
A dislocation can either extend right through a crystal or it can form a closed loop. Therefore it cannot be described just in terms of its orientation since this can vary with position. The entire dislocation can be characterized however by a vector which represents the amount and direction of slip which is produced when that dislocation has passed right through the crystal (in a certain direction to be established by convention). This vector is called the Burgers’ vector b. its magnitude (sometimes called the strength of the dislocation) is a repeat vector of the lattice and it is usually the smallest one, so that it is of the order of the interatomic spacing. From the definition of a dislocation it should be clear that its Burgers’ vector is constant throughout its length even though the orientation of the dislocation itself may change. The Burgers’ vector lies in the slip plane and it will be perpendicular to an edge dislocation and parallel to a screw dislocation.




The conservation of the Burger’s vector
Dislocations may join up with one another to form a series of networks within the crystal. If a dislocation splits into a number of other dislocations then the sum of the Burgers’ vectors of the components must equal the Burgers’ vector of the original dislocation. This is of course an analogue of Kirchhoff’s law for electric currents at a node.




Impurities and dislocations
At an edge dislocation the atoms are closer together just above the slip plane where the extra half plane stops, and therefore any impurity atom A which is smaller than the host atoms can be accommodated in this part of the crystal with a reduction in the elastic energy. On the other hand, a large atom B will be attracted to the other lower side of the slip plane where the lattice is extended. Thus edge dislocations act as a place where impurity atoms will tend to be concentrated. Reference is often made in the literature to the impurity atmosphere around a dislocation. For a similar reason dislocations on parallel planes will tend to line up under one another because the elastic energy around them is thereby reduced.





Conservative and non-conservative motion
An edge dislocation can move easily in either horizontal direction along the slip plane. This is called conservative motion because when the dislocation passes through a region the number of lattice sites remains constant. If however the dislocation was made to move vertically upwards some of the extra half plane would have to be dismantled, and so a trail of interstitial atoms would be left behind. Similarly if the motion was downwards then the extra atoms which would be needed to extend the extra half plane would result in vacancies being produced. Both these types of motion are non-conservative and are referred to as dislocation climb. Because extra energy is required to produce the point defects, dislocation climb only occurs at high temperatures. Pure screw dislocations do not have an extra half plane and they can move conservatively on any slip plane.




Stress and strains around a dislocation
Whenever a lattice is distorted from its regular structure extra elastic energy is associated with the distortion. This is so because the original structure must have been the one with the lowest (free) energy; if it were not, it would not have been the original structure! We need to know the extra energy which is associated with the presence of a dislocation and to calculate this we require a detailed knowledge of the way the lattice is strained in that region. This is fairly easy to do for a screw dislocation.

The magnitude of the Burgers’ vector
In all our discussion we have assumed that the slip associated with a dislocation has been equal to one lattice spacing. Since the energy of a dislocation is proportional to b2, it is energetically more favourable for a dislocation with a large value of b to separate into several dislocations each of which has as small a value of b as possible. In many cases this means that b is equal in magnitude to the lattice spacing a, although in some structures it is possible to have partial dislocations in which b is less than a. the tendency to minimize the value of b implies that there will be a repulsion between like dislocations which are on the same slip plane.

The line tension of a dislocation
Since there is an additional elastic energy E associated with unit length of dislocation there will always be a tendency for a dislocation to make itself as short as possible so as to minimize this energy. The dislocation behaves like a piece of elastic; closed loops will therefore tend to contract and disappear and other dislocations will become as straight as possible.

If the length of a dislocation is increased by ∂x then work equal to E∂x must be done. This can be thought of as being produced by a tensile force T moving through the distance ∂x, and hence T = E. The dislocation, therefore, can be considered to have a line tension T equal to E~ Gb^2. This is of course completely analogous to the surface tension in a liquid surface which is equal to the surface energy per unit area. A straightforward calculation shows that in order to bend a dislocation into an arc with a radius of curvature R requires a normal force F per unit length given by

F = T/R ~ Gb^2




The force on a dislocation
Although we deform a crystal by applying an external shear stress to it, we have described the mechanism of deformation in terms of the motion of dislocations. These must therefore experience a force when a stress is applied to the crystal otherwise they would not move. Simple considerations enable us to derive an expression for this force.
When a dislocation of Burgers’ vector b, sweeps over the entire area A, of a slip plane, slip equal to b is produced. If part of the dislocation of length l moves a distance ∂x over the slip plane thereby seeping out an area l ∂x, a proportionate amount of slip (l ∂x/A)b will result. But this slip has actually been produced by an external shear stress τ acting on area A.

The work done by this stress is force × distance which is equal to τ A (l ∂ x/A)b. however, we could ignore the external stress and say that the deformation was produced as if the dislocation was being driven by a force F per unit length through the distance ∂x. the work done must be the same in both cases and so we can equate:

Fl ∂x = τ A (l ∂x/A)b




The force of one dislocation on another

If two dislocations are close together the stress field of one Gb / 2πr will act on the other just as if it were an external stress.
The radial force F between two screw dislocations:

F = Gb^2 / 2π d per unit length

Where d is the distance between the dislocations.

The multiplication of dislocations
Many crystals can double their length during tensile deformation. If we have a specimen 1 cm long with a 1 mm square cross section and each dislocation can produce a deformation of b~ 2.5 x 10 ^-10 m, the number of dislocations which are necessary to produce an extension of 1 cm would be about 4 x 10^7. The area of the side of the specimen is 10 ^-5 m^2 and hence we would require a dislocation density of 4 x 10 ^12 m^-2 to produce this deformation. Now annealed crystals only contain about 10^10 dislocations per square metre, and thus they could produce a maximum deformation of less than 1 percent. We are led to the conclusion therefore that extra dislocations must be produced during deformation.

Clarendon publishing, p 59


Thermal energy and lattice vibrations
Our general ideas about temperature and thermal equilibrium are based on the fact that he individual particles in a system are endowed with some type of vibrational motion which increases as the temperature is raised. In a solid the energy associated with this vibration and perhaps also with the rotation of the atoms or molecules is called the thermal energy. In a gas the translational motion of the atoms and molecules will also contribute to this energy.
A full appreciation of the thermal energy is fundamental to an understanding of many of the basic properties of solids. For example, we would like to know the value of this energy and how much is available to scatter a conduction electron in a metal, since this scattering gives rise to an electrical resistance. Or the energy might be used to activate a crystallographic or a magnetic transition, or a dislocation interaction. We are also interested to know how the vibrational energy changes with temperature because this gives a measure of the heat energy which is necessary in order to raise the temperature of the material. (It will be recalled that the specific heat is the thermal energy which is required in order to raise the temperature of unit mass, or of 1 g mole, by one Kelvin).

For simplicity we shall only discuss the thermal energy which is due to the vibration of atoms in a solid. It is usual to discuss this in terms of the classical theory of atomic vibrations and then to show that his is completely unsatisfactory at low temperatures.

Quantum mechanics is then brought in with a round of applause to save the day. Whilst this approach was intriguing during the first half of this century it would now appear to be a little dated. We therefore intend to give a unified discussion.



StatisticsThe system of vibrating atoms in a crystal is of course very complicated and a calculation of the total thermal energy from a knowledge of the energy of each individual atom is clearly impossible. However, if the system is in thermal equilibrium then we do have rules which give us the relative probability that a particle in that system will have an energy, say E1, rather than E2. This probability function f (E) is usually referred to as the statistics of the system. Depending on the type of system and its constituent particles, there are three possible functions which may be used, and we quote these without proof. *
In all the systems which we shall consider it is assumed that there are no interactions between the particles, ie. The energy state of one particle is not influenced by that of its neighbour. Systems may be divided into two main types depending on whether they are comprised of particles which are distinguishable or indistinguishable.

Distinguishability in this context is a rather formal concept and it is used to describe whether in principle the particles can be distinguished from one another. For our purposes the only system of distinguishable particles is the assembly of atoms that forms a solid. In such a system each atom can be designated by a unique set of position coordinates. Indistinguishable particles, e.g. an atom in a gas or an electron in a metal, are best thought of as being described by a wave packet which is sufficiently extended in space that it overlaps the wave packets of the other particles. If this occurs it is impossible even in principle to distinguish one particle form another.
Systems of distinguishable particles are described by the Maxwell-Boltzmann statistics
Fmb = A exp (- E/kT)

Where k is Boltzmann’s constant.
Indistinguishable particles must be divided into two types. If they have half-integral spin (e.g. electrons) and hence are subject to the Pauli exclusion principle they obey the Fermi-Dirac statistics

F fd = [exp { (E- Ef)/kT + 1 }] ^ -1

Where Ef is a parameter called the Fermi energy.

If the particles have zero or integral spin (photons, phonons, or 4He nuclei) they are controlled by the Bose-Einstein statistics

F be = [exp {(E –a)/kT – 1}] ^ -1

It should be noted that in the limit of high energies both Ffd and Fbe approach Fmb and this approximation is often made to simplify calculations.

Rosenberg; Clarendon P 79-81

* Statistical Mechanics:
- MacDonald D K C (1963), introductory statistical mechanics for physicists, Wiley, NY
- Dugdale J S (1966), Entropy and low temperature physics, Hutchinson, London