From Symmetry to Real Structures
So far we have described crystals abstractly: symmetry operations, lattices, space groups, Wyckoff positions, and how diffraction reads them. In Lectures 20–22 we saw how real crystals deviate from perfection. Now we ask the chemical question: which arrangement does a given set of atoms actually choose, and why?
Copper is FCC, iron is BCC at room temperature, NaCl is rock salt and ZnS is zinc blende. A space group tells us the symmetry of each structure, but not why nature picked it.
The type of bond decides whether atoms can be treated as hard spheres. Sphere sizes, charges and stoichiometry then decide how they pack, which holes get filled and how many neighbours each atom has.
Once the structure is known, its point group decides which properties are even possible: a centre of symmetry forbids piezoelectricity, pyroelectricity and ferroelectricity outright.
From Bonds to Packed Spheres, Their Holes, and Properties
One chain of reasoning runs through all three lectures. Every link is explained below with something you can rotate, drag or play.
- 1Bonding: types of bond, bond energy versus \(k_BT\), the interatomic potential
- 2Packing: CN and packing fraction; SC, BCC, FCC and HCP; density
- 3Voids: tetrahedral, octahedral and cubic holes, their number and size
- 4Pauling’s rules for ionic crystals
- 5Ionic structures: NaCl, CsCl, ZnS, wurtzite, spinel, perovskite
- 6Structure → properties: Neumann’s principle, piezo-, pyro- and ferroelectrics, microstructure
What Holds Atoms Together
The bond decides two things that matter for structure: how strong it is (compared with the thermal energy \(k_BT\)) and whether it has a direction. Non-directional bonds let us model atoms as hard spheres; directional bonds fix the bond angles instead.
Types of Bonds
There are three strong (primary) bonds and a family of weak (secondary) ones. Each little animation shows the mechanism; the numbers are bond energies per atom, ion or molecule from Callister, Materials Science and Engineering, Table 2.3.
Ionic
- An electron is transferred (Na → Cl), making Na+ and Cl−; the Coulomb attraction between the ions holds them.
- Non-directional: the field of an ion is spherically symmetric, so the CN is set by the ion sizes (typically 4 to 8).
- NaCl 3.3 eV, MgO 5.2 eV. The doubly charged ions of MgO bind much more strongly.
- Hard, brittle, high melting, electrically insulating.
Covalent
- Neighbours share electron pairs; the pair sits between the two nuclei.
- Strongly directional: bond angles fix the CN. C and Si use four bonds at 109.5°, so CN = 4.
- Si 4.7 eV, C (diamond) 7.4 eV
- Very hard and high melting, yet the structures are open (low packing fraction).
Metallic
- Positive ion cores sit in a “sea” of delocalised valence electrons that belong to the whole crystal.
- Non-directional, so the cores pack densely: CN 8 or 12.
- Hg 0.7, Al 3.4, Fe 4.2, W 8.8 eV: an enormous range.
- Ductile (planes can slide without breaking specific bonds) and electrically conducting.
Secondary (vdW, H-bond)
- Attraction between electric dipoles: fluctuating ones (van der Waals, even in Ar) or permanent ones (H-bonds in H2O, NH3).
- Weak: Ar 0.08, Cl2 0.32 eV; H-bond: NH3 0.36, H2O 0.52 eV.
- Hold molecules together in molecular solids and between polymer chains.
How Strong Are Bonds? Compare with \(k_BT\)
A bond energy only means something next to the energy that thermal motion supplies, \(k_BT\). At 300 K, \(k_BT = 8.617\times10^{-5}\ \text{eV/K} \times 300\ \text{K} = 0.026\ \text{eV}\). Drag the temperature and watch which substances would have boiled.
Data: Callister, Table 2.3 (per atom, ion or molecule). Bars turn pale once \(15\,k_BT\) exceeds the bond energy, the rule of thumb for boiling explained in the next section.
Why Are Cl2 and NH3 Gases at Room Temperature?
Divide each bond energy by the thermal energy at the boiling point, \(k_BT_b\). Something remarkable happens: the ratio is about the same for every bond type, from argon to tungsten. Click any row to see it on the bar.
| Substance | Eb (eV) | Tb (K) | kBTb (meV) | Eb/kBTb | At 300 K |
|---|
Bond energies: Callister, Table 2.3. Boiling points at 1 atm: CRC Handbook. \(k_B = 8.617\times10^{-5}\) eV/K.
Boiling is decided by the Gibbs free energy, \(G = H - TS\), not by energy alone. The vapour costs enthalpy (the bonds must be broken) but gains an enormous entropy, because each molecule can now be anywhere in a much larger volume. Trouton’s rule says this entropy gain is roughly the same for most liquids: \(\Delta S_\text{vap} \approx 10.5R\). So the vapour wins at \(T_b = \Delta H_\text{vap}/\Delta S_\text{vap}\), long before \(k_BT\) reaches \(E_b\).
\(15\,k_BT \approx 15 \times 0.026 \approx 0.4\) eV. Cl2 (0.32 eV) and NH3 (0.36 eV) fall just below the cutoff and are gases. H2O (0.52 eV) is just above, so it stays liquid. The extra strength of the hydrogen bond is the only reason oceans exist.
Energy vs Separation: The Interatomic Potential
Bring two atoms together. Far apart they attract; pushed too close, their electron clouds overlap and they repel violently. The sum of the two gives a potential well \(U(r)\), and everything about the bond can be read from its shape.
More energy is needed to pull the atoms apart, so the solid melts at a higher temperature. W (8.8 eV) melts at 3695 K; Hg (0.7 eV) is liquid at room temperature.
For small displacements \(U \approx U(r_0) + \tfrac12 U''(r_0)(r-r_0)^2\): a spring with constant \(U''(r_0)\). A narrow, sharply curved well means a stiff bond and a high elastic modulus.
The repulsive wall is steeper than the attractive tail. A vibrating atom spends more of its time on the far side, so the mean spacing grows with temperature: thermal expansion. A perfectly parabolic well would show none.
Atoms as Hard Spheres
If the bond has no preferred direction, the only thing that matters is fitting as many neighbours as possible around each atom. Crystal structures then become a geometry problem: how do you pack spheres?
From Bonds to Hard Spheres
Look again at the potential: the repulsive wall is so steep that atoms almost never get closer than \(r_0\). That is the justification for modelling them as incompressible spheres of fixed radius.
Dense or Directional?
The steep repulsive wall in \(U(r)\) acts like a hard surface, so each atom or ion is modelled as an incompressible sphere of fixed radius. The structure is then the densest arrangement that the sphere sizes, charges and stoichiometry allow, giving a high CN (typically 6 to 12). The left view shows one atom with its 12 touching neighbours in FCC.
Bond angles, not packing, decide the arrangement. In diamond each C bonds to exactly 4 neighbours at 109.5°. CN = 4 and the packing fraction is only \(\pi\sqrt3/16 = 0.34\): an open structure, despite very strong bonds.
Coordination Number and Packing Fraction: Start in 2D
Two numbers describe any packing. Slide the rows of a square array sideways and let them settle into the hollows; watch both numbers change.
Simple Cubic and Body-Centred Cubic
Now stack in 3D. For each cell we need three things: the direction along which the spheres touch (this links \(a\) to \(r\)), the number of atoms per cell (corners count ⅛, body centre counts 1), and the resulting packing fraction.
Stacking Close-Packed Layers: ABAB or ABCABC
Build your own stack. Start with layer A, then place the next layer into one set of hollows. The third layer is where the choice is made. A layer can never sit directly on an identical layer (no AA), because its spheres would land on top of each other instead of in the hollows.
FCC Is Cubic Close Packing: Finding the ABC Layers
The ABC layers are hidden inside the FCC cube, tilted. Every atom whose fractional coordinates add up to the same integer, \(x + y + z = n\), lies on the same (111) plane. Colour the atoms by plane and pull the planes apart.
Packing Fraction of FCC (ccp)
In FCC the corner atoms do not touch each other; the face-centre atom pushes them apart. Contact runs along the face diagonal. Step through the derivation, then rotate the cell to count the 12 neighbours.
Count the 12 Neighbours
Pick a {100} plane through the face-centred atom: each one holds 4 of its neighbours.
The Hexagonal Close-Packed (HCP) Cell
The hexagonal prism is drawn because it shows the 6-fold symmetry, but it is three primitive cells. The true unit cell is the 120° rhombic prism, holding 2 atoms.
HCP: Deriving c/a and the Packing Fraction
The B atom rests in the hollow of three touching A atoms. Find how high it sits and the whole cell follows.
How Ideal Are They?
Click a bar. Most HCP metals lie close to the ideal 1.633; Zn and Cd are stretched along c, so their atoms are not ideal hard spheres.
Four Metallic Structures at a Glance
Click a row to load the structure. The Wyckoff column is a reminder of the ITC route: put one atom on the listed site, apply the centring translations and site operations, and the whole cell follows.
| Structure | Atoms / cell | a in terms of r | CN | PF | Space group, Wyckoff | Examples |
|---|---|---|---|---|---|---|
| Simple cubic | 1 | a = 2r | 6 | 0.524 | Pm3̅m (221), 1a | α-Po |
| BCC | 2 | a = 4r/√3 | 8 | 0.680 | Im3̅m (229), 2a | α-Fe, Cr, Mo, W, Na, K |
| FCC (ccp) | 4 | a = 2√2 r | 12 | 0.740 | Fm3̅m (225), 4a | Cu, Al, Ni, Ag, Au, γ-Fe |
| HCP | 6 per prism (2 per primitive cell) | a = 2r, c = 1.633a | 12 | 0.740 | P63/mmc (194), 2c | Mg, Ti, Zn, Co, Zr |
Does the Hard-Sphere Model Give the Right Density?
Copper is FCC with atomic radius \(r = 0.128\) nm and molar mass \(A = 63.55\) g mol−1. Predict its density and compare it with the measured 8.96 g cm−3. Then try other metals.
\(\Rightarrow\; V_c = a^3 = 4.74\times10^{-23}\ \text{cm}^3\)
Density from Structure
For HCP the calculator assumes ideal \(c/a = 1.633\), so \(V = 24\sqrt2\,r^3\) per 6-atom prism.
The Holes Between the Spheres
Even the densest packing leaves 26% of space empty. That space is not random: it comes as a small number of well-defined holes (interstitial voids) with fixed shapes, sizes and positions. Small atoms dissolve in them, and ionic crystals are built by filling them.
Holes Between Close-Packed Layers
Put layer B on layer A and look between them. Only two kinds of hole appear, named after the polyhedron formed by the spheres around them.
Where exactly does layer B sit? Two fixed views
The same two layers seen from directly above and from 35° above the layer plane. Layer B is drawn see-through so the A atoms underneath stay visible. Step through to find each kind of void.
Voids in the FCC Cell: Where Each One Is Formed
The same two holes, now located in the cubic cell. Click any void marker to see the polyhedron of atoms around it.
Voids in the HCP Cell: Where Each One Is Formed
In HCP the voids line up in columns along c. Toggle each kind and click a marker to see its polyhedron.
Heights measured from the bottom A layer (A at 0 and c, B at c/2).
How Big Are the Holes? The Largest Sphere That Fits
Grow a guest sphere in each hole and watch for the moment it touches its neighbours. Host atoms have radius \(R\); the guest has radius \(r\).
Voids in the BCC Cell: More of Them, but Smaller and Distorted
BCC is less densely packed than FCC, so you might expect bigger holes. The opposite is true: the empty space is split into many small, irregular holes. Click a void to measure its neighbours.
Why Void Size Matters: Carbon in Iron
Steel is iron with a little carbon in the voids. Carbon is too big for every hole in iron, so how much dissolves depends on which structure offers the least-bad hole.
Hole Sizes in Iron (R = 0.124 nm)
From BCC to Tetragonal Martensite
All the C atoms sit in octahedral sites on edges parallel to one axis (vertical here). Each pushes its two Fe neighbours apart along that axis only. This one-axis strain is why martensite is tetragonal (body-centred tetragonal) rather than cubic.
Voids in the Simple Cubic Cell: Only One Kind
Simple cubic is so open that its single hole is almost as large as an atom.
Voids in SC, BCC, FCC and HCP at a Glance
| Structure | Atoms per cell | Octahedral voids per cell (per atom) | r/R | Wyckoff | Tetrahedral voids per cell (per atom) | r/R | Wyckoff |
|---|---|---|---|---|---|---|---|
| SC | 1 | none; the only hole is the cubic void: 1 per cell (1 per atom), r/R = 0.732, Wyckoff 1b | none | ||||
| BCC | 2 | 6 (3), distorted | 0.155 | 6b | 12 (6), distorted | 0.291 | 12d |
| FCC | 4 | 4 (1) | 0.414 | 4b | 8 (2) | 0.225 | 8c |
| HCP | 6 per prism | 6 (1) | 0.414 | 2a | 12 (2) | 0.225 | 4f |
Your Turn: Aluminium and Magnesium
Type your answers and press Check. Answers within about 2% are accepted. The worked solutions are below.
Aluminium is FCC with \(a = 0.405\) nm (\(A = 26.98\) g mol−1). Find (a) the atomic radius \(R\), (b) the density, (c) the radius of the largest atom that fits in an octahedral void, and in a tetrahedral void.
?Show the worked answer
(a) Contact along the face diagonal: \(R = \sqrt2\,a/4 = 0.143\) nm.
(b) \(\rho = \dfrac{4\times26.98}{(0.405\times10^{-7}\ \text{cm})^3\times6.022\times10^{23}} = 2.70\) g cm−3 (measured 2.70).
(c) Octahedral: \(r = 0.414R = 0.059\) nm; tetrahedral: \(r = 0.225R = 0.032\) nm.
Magnesium is HCP with \(c/a = 1.624\). Compared with ideal hard spheres (1.633), is the structure slightly stretched or slightly squashed along c?
?Show the worked answer
\(c/a = 1.624 < 1.633\): slightly squashed along c.
The atoms are not perfect hard spheres; real bonding makes the in-plane and out-of-plane spacings differ a little. Zn and Cd (\(c/a \approx 1.86\)–\(1.89\)) are strongly stretched.
Take-Aways: Bonds, Packing and Voids
Bonds
Ionic and metallic bonding is non-directional, so atoms behave as hard spheres. Covalent bonds are directional and give open structures.
Packing
SC 0.52 · BCC 0.68 · FCC = HCP 0.74. FCC and HCP differ only in stacking: ABC vs AB.
Close-packed voids
FCC and HCP: per atom 1 octahedral (\(r = 0.414R\)) + 2 tetrahedral (\(r = 0.225R\)) voids.
BCC and SC voids
BCC: 3 octahedral (0.155R) + 6 tetrahedral (0.291R) per atom, both distorted. SC: 1 cubic void (0.732R).
Why it matters
Hole sizes control which atoms dissolve interstitially (C in γ-Fe vs α-Fe) and which ions fit where (next).
Ionic Crystals: Pauling’s Approach
Linus Pauling (1929) condensed the structures of ionic crystals into five rules. They are not laws, but they predict the coordination of most ionic solids from nothing more than ionic radii and charges.
Assumptions and the Five Rules
- Ions are hard spheres with fixed ionic radii and formal charges (Na+, Cl−, Ti4+ …).
- Anions are usually larger: they form the framework, and each cation sits in a coordination polyhedron of anions.
- Electrical neutrality must hold locally, around every ion, not only for the crystal as a whole.
A cation (+) inside its polyhedron of anions (−)
\(r_\text{cation}/R_\text{anion}\) fixes the coordination polyhedron.
The bond strengths reaching each anion add up to its charge.
Sharing brings cations closer → repulsion.
e.g. SiO4 tetrahedra share only corners.
Few kinds of essentially different sites in a structure.
Rule 1: The Radius Ratio Sets the Cation Coordination Number
A cation wants as many anion neighbours as possible (more attraction), but it must touch them. Shrink the cation and the anions close in until they touch each other; shrink it further and it rattles inside its cage, and the structure is unstable.
Critical Radius Ratios: Where They Come From
At the critical ratio, the distance from the cation centre to an anion centre, \(R + r\), equals the circumradius of a polyhedron whose edge is \(2R\) (anions touching). Click a row to load it into the 3D model above at exactly the critical ratio.
| Cation CN | Polyhedron | Contact condition (anions just touching) | Stable range of r/R | Example |
|---|---|---|---|---|
| 2 | Linear | — | < 0.155 | — |
| 3 | Triangle | \(R + r = 2R/\sqrt3\) | 0.155 – 0.225 | — |
| 4 | Tetrahedron | \(R + r = \sqrt{3/2}\,R\) | 0.225 – 0.414 | ZnS (zinc blende) |
| 6 | Octahedron | \(R + r = \sqrt2\,R\) | 0.414 – 0.732 | NaCl, MgO |
| 8 | Cube | \(R + r = \sqrt3\,R\) | 0.732 – 1.0 | CsCl, CaF2 |
| 12 | Cuboctahedron (close packing) | \(R + r = 2R\) | ≥ 1.0 | Ba2+ in BaTiO3 |
Rule 2: Electrostatic Valence Fixes the Anion CN
A cation of charge \(z_c\) with \(\text{CN}_c\) neighbours sends a fraction of its charge along each bond: the electrostatic bond strength. For local neutrality, the bonds arriving at every anion must exactly cancel its charge.
Drawn flat; in 3D the cations around the anion form a polyhedron (an octahedron for NaCl).
| Crystal | Cation CN (rule 1) | s | Anion CN |
|---|---|---|---|
| NaCl | 6 | 1/6 | 1 ÷ 1/6 = 6 |
| CsCl | 8 | 1/8 | 1 ÷ 1/8 = 8 |
| ZnS | 4 | 2/4 = 1/2 | 2 ÷ 1/2 = 4 |
| MgO | 6 | 2/6 = 1/3 | 2 ÷ 1/3 = 6 |
| CaF2 | 8 | 2/8 = 1/4 | 1 ÷ 1/4 = 4 |
Rule 3: Corners Are Shared More Readily than Edges or Faces
Neighbouring polyhedra must share something (every anion belongs to several of them). Watch how the two cations at the centres approach each other as the polyhedra share a corner, then an edge, then a face.
Rules 4 and 5
High-charge, low-CN cations avoid sharing
In a crystal with different cations, those with high valence and small CN tend not to share polyhedron elements with each other. Example: in silicates, SiO4 tetrahedra share only corners, never edges or faces.
Parsimony
The number of essentially different kinds of constituents (sites and polyhedra) in a crystal tends to be small. Example: perovskite BaTiO3 has just two cation environments, TiO6 octahedra and BaO12 cuboctahedra. Try the explorer below.
Rule 1 → the cation polyhedron (CN of the cation)
Rule 2 → how many polyhedra meet at each anion (CN of the anion)
Rules 3–5 → how the polyhedra link: corners preferred, few kinds of sites
Then look up the space group: the ITC page gives every ion position from a few Wyckoff sites
Rule 5 in Practice: Count the Kinds of Site
Parsimony means economy: a crystal tends to use as few kinds of local environment as it can. What is counted is not the number of elements, but the number of essentially different coordination polyhedra each ion sits in. Pick a structure, predict, then reveal.
Sites are not kinds
Kyanite has 16 distinct sites and sillimanite only 7, yet kyanite is the more parsimonious: all four of its Al sites are AlO6 octahedra. “Essentially different” means a different polyhedron, not a slightly distorted copy of the same one.
Why it usually holds
Rules 1–4 pick one best polyhedron for each ion and one best way of linking it. Repeating that best choice everywhere gives the lowest energy; every extra kind of site puts some ions in a worse one.
When it gives way
Something outside the hard-sphere picture tips the balance: crystal-field energy of d electrons in Fe3O4, or temperature and pressure among the Al2SiO5 polymorphs. Rule 5 is a tendency, the weakest of the five.
Close-Packed Anions + Filled Holes
Most simple ionic crystals can be read as a close-packed array of the big ions with the small ions sitting in some fraction of the octahedral or tetrahedral holes. Pauling’s rules tell us which holes, and the space group gives every position.
Many Ionic Structures = Close-Packed Anions + Filled Holes
Click a row to build that structure in the viewer. Colours: anions are the large spheres; cations are the small ones.
| Structure | Close-packed array | Holes filled | CN (cation : anion) | Examples |
|---|---|---|---|---|
| Rock salt (NaCl) | Cl− fcc | all octahedral | 6 : 6 | NaCl, MgO, NiO, FeO |
| Zinc blende (ZnS) | S2− fcc | ½ tetrahedral | 4 : 4 | ZnS, GaAs, CdTe, SiC (3C) |
| Wurtzite (ZnS) | S2− hcp | ½ tetrahedral | 4 : 4 | ZnO, GaN, AlN |
| Spinel (MgAl2O4) | O2− fcc | ⅛ tetrahedral (Mg) + ½ octahedral (Al) | Mg 4, Al 6 : O 4 | MgAl2O4, Fe3O4, NiFe2O4 |
| Perovskite (BaTiO3) | Ba2+ + O2− ccp | ¼ octahedral (by Ti4+) | Ti 6, Ba 12 : O 6 | BaTiO3, SrTiO3, PbTiO3 |
| Fluorite (CaF2) | Ca2+ fcc | all tetrahedral (by F−) | 8 : 4 | CaF2, ZrO2 (cubic), UO2 |
| Corundum (Al2O3) | O2− hcp | ⅔ octahedral | 6 : 4 | Al2O3, Cr2O3, Fe2O3 |
Rock Salt (NaCl): Apply Rules 1 and 2
Before building anything, predict. The SVG is a (100) layer of NaCl drawn to scale; drag the ratio to see what changes when the cation is too small.
A (100) layer drawn to scale: ions touch along the rows, Cl− do not touch each other. 4 neighbours in the plane + 1 above + 1 below = 6.
Rock Salt: Cl− FCC with Na+ in Every Octahedral Void
NaCl from Its ITC Page: Fm3̅m (No. 225)
The space group symbol already tells us most of the story. F: face-centred, so every position comes with the translations (0,0,0)+ (0,½,½)+ (½,0,½)+ (½,½,0)+. m3̅m: full cubic point symmetry (Oh). Apply the translations one at a time.
| Ion | Wyckoff | Site sym. | Representative | All 4 positions in the cell |
|---|---|---|---|---|
| Cl− | 4a | m3̅m | 0, 0, 0 | |
| Na+ | 4b | m3̅m | ½, ½, ½ |
Caesium Chloride: CN 8 : 8, and Not BCC
Zinc Blende (ZnS): Zn2+ in Half the Tetrahedral Voids
| Wyckoff | Site sym. | Representative | What sits there in ZnS |
|---|---|---|---|
| 4a | 4̅3m | 0, 0, 0 | S2− (the fcc array) |
| 4b | 4̅3m | ½, ½, ½ | empty (octahedral voids) |
| 4c | 4̅3m | ¼, ¼, ¼ | Zn2+ (half of the tetrahedral voids) |
| 4d | 4̅3m | ¾, ¾, ¾ | empty (the other half) |
Wurtzite: the HCP Cousin of Zinc Blende
Spinel AB2O4: Two Kinds of Holes Filled at Once
The spinel cell is big: 32 O2− + 8 Mg2+ + 16 Al3+ = 56 ions. Build it up one kind of ion at a time.
Perovskite BaTiO3: Apply Rules 1 and 2
Ba2+ at the corners, O2− at the edge midpoints, Ti4+ at the centre (one more O2− directly above and below it).
Perovskite: a Network of Corner-Sharing TiO6 Octahedra
Six Ionic Structures at a Glance
Click a row to load it. Keep an eye on the point-group column: it decides which properties a crystal can have, which is where we go next.
| Structure | Cation CN | Anion CN | Space group | Wyckoff sites | Point group | Also adopted by |
|---|---|---|---|---|---|---|
| NaCl (rock salt) | 6 | 6 | Fm3̅m (225) | Cl 4a, Na 4b | m3̅m | MgO, NiO, LiF, TiC |
| CsCl | 8 | 8 | Pm3̅m (221) | Cl 1a, Cs 1b | m3̅m | CsBr, CsI, β-brass (CuZn) |
| ZnS (zinc blende) | 4 | 4 | F4̅3m (216) | S 4a, Zn 4c | 4̅3m | GaAs, InP, CdTe, 3C-SiC |
| ZnS (wurtzite) | 4 | 4 | P63mc (186) | S 2b, Zn 2b (u ≈ 3/8) | 6mm | ZnO, GaN, AlN, CdS |
| MgAl2O4 (spinel) | Mg 4, Al 6 | 4 | Fd3̅m (227) | Mg 8a, Al 16d, O 32e | m3̅m | Fe3O4, NiFe2O4, ZnFe2O4 |
| BaTiO3 (cubic perovskite) | Ti 6, Ba 12 | 6 | Pm3̅m (221) | Ba 1a, Ti 1b, O 3c | m3̅m | SrTiO3, PbTiO3, KNbO3 |
Your Turn: Predict the Structure
Use the predictor to run rules 1 and 2 on any binary ionic compound, then check the two exercise answers.
KCl is a good borderline case: \(r/R = 1.38/1.81 = 0.76\) predicts CN 8, yet KCl adopts rock salt. Radius ratios are a guide, not a law.
1Magnesium oxide, MgO: \(r(\text{Mg}^{2+}) = 0.72\) Å, \(r(\text{O}^{2-}) = 1.40\) Å. Predict the CN of each ion and the structure type.
\(r/R = 0.72/1.40 = 0.51\) → between 0.414 and 0.732 → Mg2+ CN 6.
\(s = 2/6 = \tfrac13\); O2− needs 2 → O2− CN = 6.
6 : 6 with octahedra → rock salt structure, Fm3̅m (as observed; \(a = 4.21\) Å).
2Calcium fluoride, CaF2: \(r(\text{Ca}^{2+}, \text{CN 8}) = 1.12\) Å, \(r(\text{F}^-, \text{CN 4}) = 1.31\) Å. Predict the CN of Ca2+ and of F−.
\(r/R = 1.12/1.31 = 0.85\) → above 0.732 → Ca2+ CN 8 (a cube of F−).
\(s = 2/8 = \tfrac14\); F− needs 1 → F− CN = 4 (a tetrahedron of Ca2+).
Ca2+ fcc with F− in all the tetrahedral voids: the fluorite structure. Load “Fluorite” in the first viewer of this part to see it.
What the Symmetry Allows
Everything in this course so far pays off here. The point group of a crystal sets hard limits on its physical properties: how many numbers you need to describe its conductivity, and whether it can be piezoelectric, pyroelectric or ferroelectric at all.
Neumann’s Principle: Properties Inherit the Crystal’s Symmetry
A second-rank property such as electrical conductivity links two vectors, \(\mathbf J = \sigma\,\mathbf E\), and can be drawn as an ellipsoid whose semi-axes are its principal values. Pick a crystal system: its symmetry decides how much freedom the ellipsoid has.
A 4th-Rank Property: Young’s Modulus
Cubic crystals are isotropic for 2nd-rank properties only. Elasticity is 4th rank: cubic crystals need 3 constants (\(C_{11}, C_{12}, C_{44}\)) and are elastically anisotropic. The surface shows Young’s modulus in every direction, \(1/E(\mathbf n) = S_{11} - 2\left(S_{11} - S_{12} - \tfrac12 S_{44}\right)(n_1^2n_2^2 + n_2^2n_3^2 + n_3^2n_1^2)\).
\(A = 2C_{44}/(C_{11}-C_{12})\); \(A = 1\) means isotropic. Elastic constants at room temperature from standard tables.
Why a Centre of Symmetry Switches Off Piezo- and Pyroelectricity
Inversion takes every vector \(\mathbf r\) to \(-\mathbf r\). Watch what it does to a polarization arrow, and to the coordination polyhedra of NaCl and ZnS.
Invert a Polyhedron
The 32 Point Groups, Sorted by Property
Each split below removes the groups that cannot show the next property. Click a box in the tree to light up its point groups.
Which of Our Crystals Qualify?
| Crystal | Point group | Centre of symmetry? | Piezoelectric | Pyroelectric (polar) | Ferroelectric |
|---|---|---|---|---|---|
| NaCl, CsCl, MgAl2O4 | m3̅m | yes | — | — | — |
| ZnS (zinc blende), GaAs | 4̅3m | no | ✓ | — | — |
| α-quartz (SiO2) | 32 | no | ✓ | — | — |
| ZnO, GaN, AlN (wurtzite) | 6mm | no | ✓ | ✓ | — (P cannot be switched) |
| BaTiO3 at room temperature | 4mm | no | ✓ | ✓ | ✓ |
BaTiO3: a Ferroelectric You Can Switch
Cool the crystal through its Curie temperature and the Ti4+ ion settles off-centre, giving each cell a dipole. Then push it back and forth with an electric field and trace the hysteresis loop.
The well and the loop come from a simple Landau model, \(F = \alpha(T - T_c)P^2 + \beta P^4 - EP\), in arbitrary units. Real BaTiO3 has a first-order transition, but the picture is the same.
Same Crystal, Different Behaviour
The crystal structure sets what is possible; the microstructure (Lectures 20–22) tunes how much of it you get.
| Feature (Lectures 20–22) | How it changes properties | Example |
|---|---|---|
| Grain size | Grain boundaries block slip: \(\sigma_y = \sigma_0 + k_y d^{-1/2}\) (Hall–Petch). They also scatter phonons. | Fine-grained steels; nanostructured thermoelectrics with low thermal conductivity |
| Precipitates | Dislocations must cut through them or loop around them: precipitation hardening | Age-hardened Al–Cu aircraft alloys |
| Solutes and point defects | Solid-solution strengthening; dopants and vacancies set ionic and electronic conduction | Y3+ in ZrO2 makes O2− vacancies: an oxygen-ion conductor |
| Texture (preferred orientation) | Grains aligned → the polycrystal keeps the single crystal’s anisotropy | Grain-oriented Fe–Si sheet for transformer cores |
| Pores and cracks | Lower strength, stiffness and thermal conductivity | Sintered ceramics; thermal-barrier coatings |
| Ferroelectric domains | Poling aligns domains; domain-wall motion adds to the piezoelectric response | Poled PZT ceramics |
Poling a Ferroelectric Ceramic
Each grain’s polarization can only point along its own allowed directions. A strong field switches each grain to the allowed direction closest to the field; most of that alignment remains when the field is removed.
The Course in One Picture
Structure (and how we characterise it) ↔ properties ↔ processing ↔ performance: the materials science tetrahedron from Lecture 0. Every block of this course fills in one part of it.
Operations, groups, 32 point groups, stereograms, H-M symbols
2D plane groups, 14 Bravais lattices, space groups, ITC pages, Wyckoff sites
Miller indices, reciprocal lattice, structure factor, absences, indexing
Point, line, planar and volume defects
Packing, voids, Pauling’s rules, metals and ionic crystals
Neumann’s principle, tensors, piezo/pyro/ferroelectrics, microstructure
Take-Aways: Ionic Crystals and Properties
Pauling’s rules
Radius ratio → cation CN; bond strengths → anion CN; corners shared rather than edges or faces; few kinds of site.
Ionic structures
Close-packed anions + filled holes: NaCl (all oct.), zinc blende and wurtzite (½ tet., ABC vs AB), spinel (⅛ tet. + ½ oct.), perovskite.
Neumann’s principle
A property has at least the point-group symmetry: cubic crystals are isotropic for 2nd-rank properties.
Inversion
Centrosymmetric crystals cannot be piezo-, pyro- or ferroelectric. 20 piezoelectric classes, 10 polar, ferroelectrics among the polar.
Microstructure
Grain size, precipitates, solutes, texture, pores and domains tune properties on top of the crystal structure.
Revision Problem (XRD): Which Reflections Does Tungsten Show?
Tungsten is BCC, space group Im3̅m (No. 229), with W on Wyckoff 2a. Using the reflection conditions on its ITC page, list the reflections present in its powder pattern. Click each family to mark it present or absent, then check.
Peak positions for Cu Kα1 (λ = 1.5406 Å), a = 3.165 Å. Heights are not to scale.
Did It Sink In? - 23-25
60 questions on bonds and kBT, packing and density, interstitial voids, Pauling’s rules, the common ionic structures, and how symmetry decides piezo-, pyro- and ferroelectricity.