Crystalline Solids

 

Books

 

There are many good texts on crystalline solids which cover the defects we will be covering here, many of which will already be familiar to you from Solid State.

 

e.g

Kittel

Ashcroft and Mermin

Rosenberg

 

 

 

More specialist texts:

 

Hull and Bacon – Introduction to Dislocations

 

Ashby and Jones – Engineering Materials

Defects

 

Three basic (geometric) types:

·       Point – vacancy, interstitial, substitutional

·       Line – dislocations

·       Plane – grain boundaries

 

 

 

Point Defects

 

1.    Vacancy or Schottky Defect

 


 


Perfect Crystal                              Defect Crystal

Free energy Go                              Free energy G

More complicated in ionic crystals, where still need to maintain charge neutrality in the bulk.

Positive and negative ions both move to  surface, leaving a pair of vacancies.

 

Defects will affect both optical and electronic properties.

 

 

In general, the energy of formation Ev depends on site to which atom moved.

 

Ev lower if transferred to kink site

(crystal ledge) than perfect surface.

 

On average Ev corresponds to net breaking of ~1/2 neighbouring bonds

~1/2 latent heat of sublimation/atom

 

Ev~1eV

2)  Interstitial vacancy – Frenkel defect


 

 

 

 


Ionic crystal – 2 types


 


More likely since cations tend to be smaller than anions  Þ lower associated strain energy.

Energy  due to strain (non-ionic case)

 

Strain energy = 1/2 elastic constant x strain2 /vol

 

Define shear modulus G =

 

Strain energy = 1/2 G g2 (or equivalently 1/2 tg)

 

If b = lattice parameter

Volume ~ b3

Strain ~1

 

ÞStrain energy = 1/2 Gb3

 

hence EFrenkel~ 5-6ev

 

Much larger than ESchottky and also EFrenkel> kBT

 

In general not thermodynamically stable, and won't be discussed further.

Equilibrium number of vacancies in monatomic crystal

 

(For complete discussion see Waldram, Theory of Thermodynamics)

 

Compute F for crystal with N atoms, n vacanices on N+n sites.

 

3 contributions to toal entropy

·       Sc determined by density of states etc for given configuration of atoms.

·       Sbµ number of bulk configurations

·       Ssµ number of surface arrangements

 

And         Sc= kBln gc(E)

                Sb= kBln Wb

                Ss= kBln Ws

 

At equilibrium

        = 0

where      dFc = dE-TdSc-TdSs, the change in free energy when we move an atom from a particular bulk site to a particular surface site, without allowing lattice rearrangements to occur.

 

DFc ~ 6 nearest neighbour bond energies (since break on average 1/2 the bonds in the surface)

 

Now 

 

If 1 vacancy added Wb multiplied by

 

       

 

\  

 

For large crystals dSs<<dSb

 

\

\n ~ N exp –dFc/kBT

 

This is generally quite small, but can become appreciable towards the melting point.

 

We will see later how vacancies are important for creep and diffusion

Dislocations – Line Defects

 

Dislocations were originally invoked to explain the discrepancy between theoretical shear stress and that experimentally determined, long before a dislocation was directly seen.

 

 

Theoretical Shear Stress

 

As two atom planes move past one another, the stress must increase and then decrease.

 

 

           

 

Assume a sinusoidal    form for the variation of shear stress t with displacement x.

Shear stress          with k = const

 

Near origin, slope is measure of elastic shear modulus G.

Hence, within this linear regime       and

 

 

Þ              and therefore 

 

\

 


Maximum shear stress t0 is given by

Better models give to ~ G/30

 

Experiment shows this is far too high

e.g Copper     G=4.6 GN m-2  Þto = 0.72 GN m-2

 

Experimentally a good single crystal gives to 100 kN m-2
Dislocations

 

The origin of the discrepancy between theory and experiment lies in the existence of dislocations.

 

Dislocations are characterised by their Burger's vectors.  These represent the 'failure closure' in a Burger's circuit in imperfect (top) and perfect (bottom) crystal.

 

Edge                                                       Screw

Vectors describing dislocation line and Burger's vector are

Perpendicular                                       Parallel

Dislocation Motion

 

Dislocations make a material softer because they permit crystals to deform without moving one entire crystal plane over the one below.

 

e.g. movement of edge dislocations

 

 

The slip (also known as glide) plane is the plane on which the dislocation moves.

 

The glide plane is defined by the vectors b and l.

 

This means edge dislocations have a unique glide plane, but screw dislocations do not and can move on a whole family of planes.