This notes sheet looks at crystalline defects & deformation, based on slip systems and close-packed planes & directions, as well as how to calculate and avoid deformation by strengthening.

There are two types of deformation of crystalline materials: elastic and plastic. The former is fully reversible, whereas in the latter, atomic planes slip over one another to form a new, different formation. This kind of deformation is irreversible.

Elastic & plastic deformation, elastic vs plastic deformation
  • Note that the elastic deformation graph could be non-linear (e.g. rubber)

Slip & Slip Systems

The slipping of atomic planes (slip planes) happens in whatever direction requires the least energy. It is brought about by the application of an external force, resulting in shear stresses within the crystal structure.

In a perfect sample, the slip would occur over the whole plane at once, however imperfections in the crystal structure prevent this: slip is a gradual process.

Atomic slip, slip systems, atoms sliding, crystal deformation

Close-packed planes are the most susceptible to slip. This is because more energy is required for a non-close-packed plane to slip, as there is a greater distance for each displaced atom to move.

The number of slip systems a crystal lattice has reflects how likely slip in the structure is.

The more slip systems present, the more susceptible to slip.

This is determined as the product of close-packed directions and non-parallel close-packed planes:

Slip  systems = close-packed  directions \times close-packed  planes

Lattices with few slip systems are said to show brittle deformation, while those that have many undergo ductile deformation.

HCP Slip Systems

There are three close-packed directions per close-packed plane, however all close-packed planes are parallel. Therefore, the number of slip systems is 3, so the deformation is brittle:

3 HCP slip systems, hexagonal-close packed slip systems

FCC Slip Systems

The close-packed planes in an FCC lattice are along the diagonal-corner, and there are four:

12 FCC slip systems, 12 face-centred cubic slip systems

Each plane has three close-packed directions, so the number of slip systems is 12, so the deformation is ductile.

BCC Slip Systems

There are no close-packed planes, but there are 6 nearly-close-packed planes, each with two close-packed directions:

12 BCC Slip systems, 12 body-centred cubic slip systems

Therefore, the number of slip systems is 12, so the deformation is ductile.

Calculating Deformation

The applied stress must be resolved in terms of the slip plane, \phi, and direction, \lambda:

\tau_R = \sigma cos\phi cos\lambda

  • \sigma is the normal stress, given by \sigma = \frac{F}{A}

Under the application of a shear stress, slip will occur in the most favourably orientated slip system. This is the system with the largest value of resolved shear stress:

\tau_{R, max} = \sigma (cos\phi cos\lambda)_{max}

The value of shear stress required for this slip to occur in the most favourable direction is the critical resolved shear stress:

\sigma_y = \frac{\tau_{crss}}{(cos\phi cos\lambda)_{max}}

For single crystals, slip occurs when φ = λ = 45°.

Defects in Crystal Structures

Crystals are never perfect – they are littered with defects that allow slip to occur at significantly lower values of applied stress than you would calculate from the equations above. There are three types of defects:

  1. Point Defects – defects with atomic dimensions in all three directions, e.g. a missing atom (a vacancy)
  2. Line Defects – defects in two atomic dimensions but are normal in the third direction.
  3. Area Defects – defects in only one dimension, e.g. a grain boundary

All defects distort the lattice, increasing the energy stored inside it.

Line Defects

Line defects, also known as dislocations, is when a row of atoms is added or removed. These allow lattices to slip in steps, instead of all at once. Two types of dislocations are edge and screw dislocations:

An edge dislocation is when a row of atoms is removed:

edge dislocation, line defects, defects in crystal structures

A screw dislocation is when a set of rows slips a certain amount to one side:

screw dislocation, line defect, defects in crystal structures

In both cases, the Burgers vector, b, measures the slip.

edge dislocations, edge dislocation dteps, how edge dislocations occur, defects in crystal structures

As you can see from this schematic, edge dislocations happen in steps, until the edge of the crystal has been reached and no more slip can occur. This is because far less energy is required to move only a small amount every time.

Area Defects & Grain Boundaries

Grain boundaries occur between uniform regions of crystal structures. These could be crystals of the same atom, or different. Either way, they provide a blockage to slip, as slip cannot occur across such a boundary.

A material with multiple crystal grains is known as a polycrystal.

area defects, grain boundaries, defects in crystal structures

The smaller the grains, the higher the stress required to cause slip. This is expressed by the Hall-Petch equation:

\sigma_y = \sigma_i + \frac{f}{\sqrt{d}}

  • \sigma(y) is the yield strength
  • \sigma(i) is the intrinsic lattice strength
  • k is a material constant
  • d is the average diameter of grains


It may seem intuitive that more dislocations = less strong material, but this is not the case. Yes, a perfect material with no defects at all would be very strong, but this is impossible to achieve.

A small number of dislocations reduces the strength, but as you increase the number of dislocations, the strength actually increases. This is because the defects then interact and obstruct each other, reducing the likeliness of slip.

Therefore, the four main ways of strengthening a material are:

  1. Reduce the grain size
  2. Generate more dislocations
  3. Adding smaller atoms into the lattice
  4. Adding larger atoms into the lattice

The easiest methods of generating more dislocations to a structure are work hardening or melting and solidifying.

It is important to note, however, that you cannot add infinite dislocations to a structure. This is because two opposite dislocations will meet and annihilate:

work hardening, dislocations, annihilation, dislocation annihilation, defects in crystal structures

This leads to a resolved shear stress-strain curve like this:

resolved shear stress-strain graph, stress-strain graph for dislocations, defects in crystal structures
  • Deformation can be elastic (reversible), or plastic (irreversible).
  • Slip will happen in the most energy-favourable direction.
    • Close-packed planes are the most susceptible to slip
  • The number of slip systems is the product of close-packed planes and directions
    • HCP has 3 slip systems
    • FCC & BCC both have 12 slip systems
  • There are three types of defects in crystal structures:
    • Point Defects – defects with atomic dimensions in all three directions, e.g. a missing atom (a vacancy)
    • Line Defects – defects in two atomic dimensions but are normal in the third direction
    • Area Defects – defects in only one dimension, e.g. a grain boundary
  • Up to a certain point, increasing the number of dislocations present in a crystal lattice increases the overall strength of the lattice