From Perfect Crystals to Real Ones
In Lectures 17–19 we read the perfect crystal from its diffraction pattern: peak positions gave the unit cell, intensities gave where the atoms sit. Now we let the crystal be imperfect.
Lattice + motif, with every site occupied by the right atom, forever. A useful model, but it never exists.
Thermal energy alone guarantees some vacancies at any temperature above 0 K; processing adds solutes, dislocations, grain boundaries, pores and second phases.
Atoms diffuse, ions conduct and metals deform plastically because of defects, not in spite of them.
Defects in Crystals
Defects are always present in crystals: any deviation from the ideal crystal structure may be termed a defect. They influence properties significantly, and are sometimes deliberately introduced in large numbers to obtain the properties we want.
- 0DPoint defects: vacancies, interstitials, solute atoms
- 1DLine defects: dislocations
- 2DPlanar defects: surfaces, grain boundaries, stacking faults
- 3DVolume defects: voids, pores, precipitates, inclusions
Defects at Every Length Scale
The four families differ in how many dimensions they extend in, and therefore in size: from a single missing atom to cracks and inclusions you can see under an optical microscope.
Vacancies, interstitials, solutes: an atom or two, plus a strain field around them.
Dislocations: a core about 1 nm wide, but lines µm to mm long. They carry plastic flow.
Surfaces, grain and twin boundaries, stacking faults, antiphase boundaries, phase boundaries.
Pores, cracks, inclusions, precipitates and second phases: from nm to mm.
Point Defects
Two major categories: vacancies and interstitials. Add foreign atoms and we get solid solutions; add charge, as in ionic crystals, and we need a bookkeeping notation and charge balance.
Vacancies
A vacancy is created when a site that is normally occupied in a perfect crystal is unoccupied.
Watch a Vacancy Wander
How Many Vacancies? An Equilibrium Count
Vacancies are not just processing mistakes. A crystal at any temperature above 0 K wants some of them, because of a competition between energy and disorder.
Making one vacancy costs \(Q_v \approx 1\) eV (Cu: 0.9 eV). With \(n\) vacancies, \(\Delta H = n\,Q_v\): a straight line that rises with \(n\).
There are very many ways to place \(n\) vacancies among \(N\) sites, so the configurational entropy \(\Delta S\) grows and \(-T\Delta S\) falls. \(\Delta G = \Delta H - T\Delta S\) has a minimum at \(n > 0\): vacancies are equilibrium defects.
\(n_v/N = \exp(-Q_v/k_BT)\). Cu near its melting point: \(\approx 4\times10^{-4}\), i.e. about 1 empty site in every 2000.
Find the Minimum of \(\Delta G\)
The curves use the per-site free energy \(g(x) = xQ_v + k_BT\,[x\ln x + (1-x)\ln(1-x)]\) with \(x = n/N\). Setting \(dg/dx = 0\) gives \(x/(1-x) = e^{-Q_v/k_BT}\), i.e. \(n_v/N \approx e^{-Q_v/k_BT}\). Cu melts at about 1358 K (1085 °C).
Interstitials
An interstitial defect is an atom (host or foreign) sitting at a site that is not normally occupied in the perfect crystal, e.g. when an atom leaves its regular site and moves into such a site.
An Interstitial Squeezing Through the Lattice
Foreign Atoms: Substitutional and Interstitial Solid Solutions
A foreign (solute) atom dissolved in a host (solvent) crystal is also a point defect. Where it goes depends mainly on its size.
The solute replaces a host (solvent) atom on its lattice site: Ni in Cu, Zn in Cu (brass), Cr in Fe.
Small atoms (H, B, C, N, O) squeeze into holes between host atoms: C in Fe makes steel.
For extensive substitutional solubility
- 1Atomic radii differ by less than about 15%.
- 2Same crystal structure.
- 3Similar electronegativity (otherwise compounds form).
- 4Same valence (a higher-valence solute dissolves more readily).
Rule 1 Check: the 15% Size Rule
Radii in nm. Presets: Cu 0.128, Ni 0.125, Zn 0.133, Fe 0.124, Cr 0.125, C 0.071.
Where Can a Small Atom Fit? Octahedral and Tetrahedral Holes
Interstitial atoms sit in the holes (voids) left between the host atoms. In cubic metals there are two kinds, named after the polyhedron formed by the host atoms around them.
Holes in FCC and BCC Unit Cells
At “Atom size” = 1 the host atoms touch as hard spheres; the coloured sphere is then the largest atom that fits in the hole without pushing its neighbours apart.
6 neighbours. FCC: at \(\tfrac12 00\) (edge centres) and \(\tfrac12\tfrac12\tfrac12\) (body centre), 4 per cell, largest sphere \(r/R = 0.414\).
4 neighbours. FCC: \(\tfrac14\tfrac14\tfrac14\), 8 per cell, \(r/R = 0.225\). BCC: \(\tfrac12\tfrac14 0\), 12 per cell, \(r/R = 0.291\).
The BCC octahedral hole has \(r/R = 0.155\) and is distorted. So C in α-Fe (BCC) dissolves only up to \(\approx 0.022\) wt% vs about 2.1 wt% in γ-Fe (FCC).
Solute Atoms Strain the Lattice Around Them
A misfitting atom pushes or pulls its neighbours. The strain field it creates is what makes solutes useful (strengthening) and harmful (lower conductivity).
Grow or Shrink the Solute
\(\Delta V = \tfrac43\pi r_a^3\cdot 3\varepsilon = 4\pi r_a^3\varepsilon\)
Why Point Defects Matter in Metals
Strength peaks at intermediate compositions while electrical resistivity rises: every solute atom both pins dislocations and scatters electrons.
Solute strain fields pin dislocations → stronger alloys, but lower electrical and thermal conductivity (more scattering).
Vacancies let atoms move: homogenisation, heat treatment, sintering and creep at high temperature.
Neutrons knock atoms off their sites (Frenkel pairs) → voids, swelling and embrittlement in reactor materials.
Single point defects are difficult to image or count, yet they change properties strongly.
Charged Point Defects
In an ionic crystal every missing or extra ion carries charge. We need a notation that tracks it (Kröger–Vink), and rules that keep the crystal neutral: Schottky and Frenkel defects, non-stoichiometry and defect reactions.
Kröger–Vink Notation for Point Defects in Ionic Crystals
A point defect is written as a symbol of the form \(X_Y^Z\), where
- Xwhat is present on the site: V for a vacancy, or the symbol of the element occupying it;
- Ythe type of site occupied: i for an interstitial site, or the symbol of the element that normally occupies it;
- Zthe charge relative to the normal charge on site Y: a dot \(^\bullet\) for each positive unit, a prime \('\) for each negative unit, and \(^\times\) for zero.
Kröger–Vink Builder
Schottky and Frenkel Defects
To keep the solid as a whole neutral, defects with charges of opposite sign must be present together. There are two ideal types.
Schottky defect
Cation vacancies (effective charge negative, e.g. \(V_{\mathrm{Na}}'\)) are balanced by anion vacancies (effective charge positive, e.g. \(V_{\mathrm{Cl}}^\bullet\)).
• In NaCl, \(n\) \(V_{\mathrm{Na}}'\) require \(n\) \(V_{\mathrm{Cl}}^\bullet\).
• In Nb\(_2\)O\(_5\), every 2 \(V_{\mathrm{Nb}}'''''\) require 5 \(V_{\mathrm O}^{\bullet\bullet}\): \(2\times(-5) + 5\times(+2) = 0\).
Frenkel defect
An ion moves from its usual site to an interstitial site, forming a vacancy and an interstitial together: a Frenkel pair. If an Na\(^+\) ion in NaCl moved to an interstitial site, \(V_{\mathrm{Na}}'\) and \(\mathrm{Na}_i^\bullet\) would form.
In practice NaCl forms mainly Schottky defects; Frenkel pairs dominate in AgCl. NaCl is used here only to show the notation.
A slightly more complex example
A Bi\(_2\)O\(_3\) molecule is added to a CaTiO\(_3\) crystal such that Bi substitutes for Ca, there is no substitution at the Ti site and no vacancies at the Ti site, and no interstitials form. What happens?
The extra CaO is either removed from the crystal or placed on its surface, so that the rest of the crystal remains unchanged.
Check: charge \(4(+1)+2(-2) = 0\); Ca sites \(4+2 = 6\) = Ti sites; O: \(6+18 = 18+6\).
Schottky and Frenkel, Step by Step
Stoichiometric and Non-Stoichiometric Point Defects
Stoichiometry = the ratio of the numbers of atoms of each element. Point defects can keep it (Schottky, Frenkel) or shift it (oxygen loss or gain, mixed valence).
Stoichiometric defects
- Schottky in A\(_2\)X\(_3\): for every \(2n\) \(V_A'''\) there are \(3n\) \(V_X^{\bullet\bullet}\). Cations and anions go missing in the formula ratio.
- Frenkel: the displaced ions stay inside the crystal, on interstitial sites, so the composition is unchanged.
Non-stoichiometric defects
- MgO\(_{1-\delta}\): oxygen vacancies, \(0 < \delta \ll 1\).
- Fe\(_{1-x}\)O (wüstite): Fe\(^{2+}\) vacancies, each compensated by two Fe\(^{3+}\): \(V_{\mathrm{Fe}}'' + 2\mathrm{Fe}_{\mathrm{Fe}}^{\bullet}\).
- The extra charge sits on electrons or holes → electronic conductivity.
Build Fe1−xO One Vacancy at a Time
32 Fe sites and 32 O sites. Each Fe vacancy removes +2; two Fe²⁺ → Fe³⁺ add +1 each, so the crystal stays neutral while the Fe : O ratio drops below 1.
Losing Oxygen: MgO Becomes MgO1−δ
Heated in a low-oxygen atmosphere, oxygen ions leave as O\(_2\) gas. Each one leaves behind a doubly charged oxygen vacancy and two electrons.
Oxygen Leaves as Gas
Left: oxygen-rich, low T. Right: lower oxygen partial pressure or higher T (qualitative).
Writing Defect Reactions: Three Balances
Every atom on the left appears on the right. Vacancies, electrons e′ and holes h\(^\bullet\) carry no mass.
The host ratio of cation : anion sites is fixed: 1:1 in NaCl and MgO, 1:2 in ZrO\(_2\), 2:3 in Al\(_2\)O\(_3\). Vacancies occupy sites; interstitials do not.
The sum of effective charges is the same on both sides (\(^\bullet = +1\), \('=-1\), \(^\times = 0\)); for a dopant reaction it is zero.
Defect Reaction Checker
Dissolving CaCl2 in NaCl, Step by Step
Every ion must sit on a site of the host, and the crystal must stay neutral. Follow the sites, then the charges.
Effective charge \(+2-(+1)=+1\) → \(\mathrm{Ca}_{\mathrm{Na}}^{\bullet}\).
\(2\mathrm{Cl}_{\mathrm{Cl}}^{\times}\). NaCl needs Na sites : Cl sites = 1 : 1.
2 Cl sites need 2 Na sites: one holds Ca, one stays empty → \(V_{\mathrm{Na}}'\).
\(\mathrm{CaCl_2}\xrightarrow{\mathrm{NaCl}}\mathrm{Ca}_{\mathrm{Na}}^{\bullet}+V_{\mathrm{Na}}'+2\mathrm{Cl}_{\mathrm{Cl}}^{\times}\). Mass: 1 Ca + 2 Cl both sides. Sites: 2 Na : 2 Cl. Charge: \(+1-1+0=0\).
Your Turn: Kröger–Vink Defect Reactions
Try each one before opening the answer. Then put it into the checker above to see the three balances.
Q1KCl: \(\text{null} \rightarrow V_{\mathrm K}' + V_{\mathrm{Cl}}^{\bullet}\). Which defect is this?
Q2Schottky disorder in Al\(_2\)O\(_3\): \(\text{null} \rightarrow \_\_V_{\mathrm{Al}}''' + \_\_V_{\mathrm O}^{\bullet\bullet}\)
Q3CaO dissolved in ZrO\(_2\) (Ca on Zr sites)
Q4Al\(_2\)O\(_3\) dissolved in MgO (Al on Mg sites)
Importance of Point Defects in Ionic Solids (and Beyond)
Line Defects: Dislocations
The puzzle: pure metals start to deform permanently at stresses thousands of times below what it takes to slide perfect atomic planes over each other. Why? The answer is the dislocation.
Why Crystals Do Not Slip All at Once
Compare shearing the top half rigidly with letting a dislocation carry the slip across, one bond row at a time.
Rigid Slip vs Dislocation Glide
Edge Dislocations and the Burgers Vector
Line defects run along a curve through an otherwise ordered material and involve a translation of one part of the crystal relative to the other. There are two basic types: edge and screw.
Edge dislocation
- • An extra half-plane of atoms is inserted into the crystal, causing distortion around its lower edge, the dislocation line (marked \(\perp\); line AA′ in Borchardt-Ott’s figure).
- • It is measured by the Burgers vector \(\mathbf b\), which for an edge dislocation is normal to the dislocation line.
- On a projection normal to the dislocation line, walk a closed loop of lattice-point hops around the distorted region, with as many hops in each direction as in the opposite direction: 4 right, 3 down, 4 left, 3 up.
- In a perfect crystal this loop would close. Around a dislocation it does not.
- The extra step needed from the final point back to the starting point gives the magnitude and direction of \(\mathbf b\).
After Fig. 13.2, Crystallography by W. Borchardt-Ott (Springer, 2nd ed.).
An Edge Dislocation Gliding Under Shear
A Dislocation Is the Boundary of the Slipped Area
Inside the shaded area the crystal has slipped by \(\mathbf b\); outside it has not. The line marks where slip stops, so it cannot just end inside a crystal.
Grow the Slipped Area
Same \(\mathbf b\), same slip plane: only the direction in which the slipped area has grown differs. When the slipped area reaches the far face, the line leaves the crystal and a surface step of size \(b\) remains.
Making a Screw Dislocation
Cut part-way into the crystal, shift one side by \(\mathbf b\) parallel to the edge of the cut, and re-join. The atomic planes become a single spiral ramp.
Cut, Shift, Re-join
The cut runs from the front face to the red line. One side is shifted by \(\mathbf b\), parallel to the edge of the cut.
The planes crossing the line join into one helical ramp: a spiral staircase of atoms. Atom shades alternate from one horizontal layer to the next.
The closure F→S = \(\mathbf b\) lies along the line: \(\mathbf b \parallel \mathbf l\) is the signature of a screw (edge: \(\mathbf b \perp \mathbf l\)).
One Line, One Burgers Vector: the Mixed Dislocation
A dislocation line can curve. Here it enters the cube as an edge dislocation on the front face, curls through the crystal as a mixed dislocation, and leaves through the adjacent right face as a screw dislocation. The line direction changes all the way; the Burgers vector never does.
Edge → Mixed → Screw in a Cubic Crystal
Showing the whole crystal.
Model: a simple cubic lattice with b = one lattice spacing along x. Each atom is displaced along b by \(u = b\,\varphi/2\pi\), where \(\varphi\) is its angle around the nearest point of the line. That is the exact screw field and a simplified edge field (the small displacements perpendicular to b near the edge part are left out). Atoms are drawn shrunk so that the lattice can be seen through.
Volterra’s Construction: Cut, Displace, Re-join
Cut a hollow cylinder along a radial plane, displace one cut face relative to the other by \(\mathbf b\), and glue. The axis of the cylinder becomes a dislocation line.
Volterra’s Recipe
Elastic Energy per Unit Length
\(G\) = shear modulus, \(b = |\mathbf b|\), \(\nu\) = Poisson’s ratio, \(R\) = outer cut-off radius, \(r_0\) = core radius. The strain field is perfect everywhere except near the red axis.
Defaults: G = 48 GPa for Cu; b = a/√2 with a = 0.3615 nm. Both energies scale as \(Gb^2\) (about \(\tfrac12 Gb^2\) per unit length), so dislocations with the shortest \(\mathbf b\) are the cheapest.
Burgers Vectors in Common Lattices
A perfect dislocation has a lattice translation as its Burgers vector. Because \(E \propto Gb^2\), the shortest one is the usual choice. Click a row.
| Lattice | Burgers vector b | No. of equivalent b | b² | relative b² |
|---|
After Kelly & Knowles, Crystallography and Crystal Defects. Bars compare b² for each lattice taking c = 1.633a for HCP.
- Shaded rows: the shortest lattice translation, the usual \(\mathbf b\), because \(E \propto Gb^2\).
- Counts include both senses (FCC: 6 directions × 2 = 12).
- Longer \(\mathbf b\) (e.g. <100> in BCC) are rarer and often unstable.
Frank’s Rule: Will a Dislocation Reaction Happen?
\(\mathbf b_1 = \mathbf b_2 + \mathbf b_3\): the Burgers vector is conserved at a node.
Node: a point where three or more dislocation lines meet. Like currents at a junction, the total \(\mathbf b\) flowing in = the total \(\mathbf b\) flowing out.
Since \(E \propto Gb^2\): \(\mathbf b_1 \rightarrow \mathbf b_2 + \mathbf b_3\) is favourable if \(b_1^2 > b_2^2 + b_3^2\).
Shockley partial: in FCC, \(\mathbf b = \tfrac16\langle 112\rangle\). Not a lattice vector, so it leaves a stacking fault behind it.
Frank’s Rule Checker (FCC)
Type each b as a fraction and three indices, e.g. 1/6 and -2 1 1. Energies are in units of a². A dislocation glides only on a plane that contains both its b and its line: see (c), where the product’s line is not in any of these planes.
Dislocation Reactions in FCC
For each: (i) do the vectors add up? (ii) does Frank’s rule allow it? (iii) on which plane does each dislocation glide (\(\mathbf b\cdot\mathbf n = 0\))?
a\(\tfrac12[\bar110] \rightarrow \tfrac16[\bar12\bar1] + \tfrac16[\bar211]\)
b\(\tfrac16[\bar12\bar1] + \tfrac16[\bar211] \rightarrow \tfrac12[\bar110]\)
c\(\tfrac12[\bar110] + \tfrac12[101] \rightarrow \tfrac12[011]\)
Seeing Dislocations and Slip
Click any micrograph to enlarge it.
nanometre scale
micrometre scale
specimen scalePlanar Defects
Interfaces across which the crystal changes orientation, stacking, ordering or phase — and the most obvious one: the free surface of the crystal.
Five Kinds of Internal Interface
Two grains of different orientation (low- or high-angle).
One wrong layer in the ABC stacking.
Mirror-image lattice on either side.
Ordered domains out of step.
Interface with another phase (e.g. a precipitate).
Describing a Grain Boundary: Tilt and Twist
A Low-Angle Tilt Boundary Is a Wall of Edge Dislocations
Tilt two grains by \(\theta\) about an axis lying in the boundary: one extra half-plane is needed every \(D = b/\theta\).
Grain Boundaries Block Slip: Pile-Ups and Hall–Petch
Dislocations from a source glide until the boundary stops them; the pile-up concentrates stress at its head.
Pile-Up at a Grain Boundary
Stacking Faults and Twins: Errors in ABC Stacking
FCC stacks close-packed {111} layers ABCABC…; shifting layers by a partial vector \(\mathbf b_p\) changes the sequence.
Stacking Sequence Builder
A layer is outlined in red when the layers directly above and below it are the same (…BAB…): locally HCP-like stacking. In perfect FCC every layer has different neighbours above and below. Two identical layers can never sit directly on top of each other.
Twin Boundaries
A twin boundary is a special grain boundary: one side is the mirror image of the other. It is coherent and of very low energy.
Atoms on one side are mirror images of the other across the twin plane (dashed).
EBSD
TEM
Annealing twins form during recrystallisation (common in FCC metals with low stacking-fault energy \(\gamma_{SF}\): Cu, brass, austenitic steel).
Deformation twins form under shear (high strain rate, low temperature, BCC and HCP metals).
Antiphase Boundaries in Ordered Alloys
Ordering domains start independently. Where a domain with A on the “wrong” sublattice meets another, an antiphase boundary (APB) forms.
Order on Cooling, Then Look at the Seam
Volume Defects
Precipitates, pores, cracks, inclusions and second phases: from nanometres to millimetres. Some are designed in, some are fought against.
Precipitates and Phase Boundaries
How well the precipitate lattice (orange) matches the matrix (brown) decides the interface.
Coherent, Semi-Coherent, Incoherent
Small misfit → coherent · larger misfit / bigger particles → semi-coherent · different structure → incoherent.
Dislocations Meet Precipitates: Cut or Loop
Top view of a slip plane: a dislocation line (red) sweeps upward past a row of particles (gold).
Cut Through or Loop Around
Pores, Cracks and Second Phases
SEM · TEM
optical
SEMPores: gas, shrinkage, poor sintering.
Cracks: solidification and thermal stresses.
Second phases: designed (dual-phase steel) or harmful.
Inclusions: Unwanted Foreign Particles
Oxides, sulphides and silicates trapped during steelmaking. What they do during hot working depends on how they deform.
Hot Rolling Three Kinds of Inclusion
- • Hard ones do not deform: voids open at their ends.
- • Brittle ones break into stringers.
- • Plastic ones (MnS) elongate → direction-dependent toughness.
- • Inclusions are crack starters in fatigue and in wire drawing.
SEM
fractureSecond Phases on Purpose: Cast Irons and Composites
The shape and distribution of a second phase matter as much as its amount.
Same Phase, Very Different Toughness
flakes
nodules
rosettesCallister, Ch. 11. Sharp-tipped flakes act like internal cracks; rounded nodules do not, which is why ductile iron is so much tougher than grey iron.
Particles, Fibres or Layers Added on Purpose
Composites add particles, fibres or layers on purpose to carry load or block dislocations.
fibres
laminateThe Defect Family and What Each One Does
0D · point
- What
- Vacancies, interstitials, solutes; Schottky, Frenkel
- Ideas
- \(n_v/N = e^{-Q_v/kT}\); \(\varepsilon = (r-r_a)/r_a\); Kröger–Vink with mass, site and charge balance
- Controls
- Diffusion, solid-solution strength, ionic and electronic conduction
1D · line
- What
- Edge, screw and mixed dislocations; partials
- Ideas
- \(\mathbf b\), \(\mathbf l\); Burgers circuit; Frank’s rule; \(E \approx \tfrac12 Gb^2\)
- Controls
- Plastic flow, work hardening, ductility
2D · planar
- What
- Grain and twin boundaries, stacking faults, APBs, phase boundaries
- Ideas
- \(D = b/\theta\); \(\gamma_{SF}\); Hall–Petch
- Controls
- Strength, toughness, corrosion, transport
3D · volume
- What
- Pores, cracks, inclusions, precipitates, second phases
- Ideas
- Coherent → incoherent; cutting vs Orowan (\(\tau \approx Gb/L\))
- Controls
- Fracture, fatigue, precipitation strengthening
Sort the Defects by Dimension
Pick 0D, 1D, 2D or 3D for each, then check.
Did It Sink In? - 20-22
75 questions on point, line, planar and volume defects: vacancies and solid solutions, Kröger–Vink defect reactions, dislocations and Frank’s rule, grain boundaries and twins, and precipitates and second phases.