CHEM 548: Materials Chemistry

Lecture 7: Important Ternary and Quaternary Structures

Wed 09/16/2026

Spinel, garnet, perovskite, and Goldschmidt tolerance factors

Reading: Woodward, Karen, Evans, and Vogt, Solid State Materials Chemistry
Ch. 1 §1.5

Learning Objectives

Recognize the structural features of Spinel, Garnet, and Perovskite lattices.
Understand how cation site preferences (tetrahedral vs. octahedral) shape these structures.
Relate these structures to functional properties (magnetism, lasers, ferroelectricity, piezoelectricity).
Apply Pauling’s rule of parsimony and coordination number analysis to complex solids.
Use the tolerance factor concept to predict perovskite stability and distortions.

Spinel

The spinel structure, named after the mineral spinel (MgAl₂O₄), is a cubic oxide framework found in many important compounds.

Magnetite (Fe₃O₄): the earliest known magnetic material — Chinese compasses more than 2,000 years ago; spinels remain widely used for their magnetic properties
In MgAl₂O₄: magnesium on tetrahedral sites, aluminum on octahedral sites
Broader family AB₂O₄: tetrahedral sites can accommodate cations from +1 (lithium) up to +6 (tungsten)
Octahedral-site charge adjusts to maintain overall charge balance
This flexibility in site occupancy is central to the chemical and physical diversity of spinel compounds

Next, we build the structural description one feature at a time.

Spinel Structure

Bond graph

Crystal-chemical formula

Normal spinel

Inverse spinel

The spinel unit cell contains 56 atoms, as seen from its Wyckoff positions.

Usual description: cubic close-packed oxygens, aluminum in one half of the octahedral holes, magnesium in one eighth of the tetrahedral holes
Remove the aluminum: the Mg-centered tetrahedra are isolated, sitting at the carbon positions of diamond — same space group, Fd3̅m
Remove the magnesium: AlO₆ octahedra share edges, forming chains in multiple directions
Consistent because filling all octahedral sites would give rock salt, where every octahedron shares all its edges; here not all edges are shared, but the edge-sharing network remains
Site symmetry: magnesium sits on 4̅3m, an ideal tetrahedron

[advance: show the spinel bond graph]

We could draw a bond graph for the spinel structure — it obeys Pauling's rule of parsimony
Known already from the Wyckoff sites: all oxygens on a single Wyckoff site, so all crystallographically and chemically equivalent; same for magnesium and aluminum
Oxygen makes one bond to magnesium and three bonds to aluminum

[advance: show the crystal-chemical formula]

We could write the crystal chemical formula this way: magnesium coordination number four, aluminum six
Oxygen has four neighbors: one magnesium and three aluminum

[advance: show the normal spinel distribution]

The mineral spinel, like many others, is what we call a normal spinel structure: one magnesium and two aluminums
The element present in the smaller amount (one magnesium) goes onto the tetrahedral site; the other cation, two per formula unit, occupies the octahedral sites

[advance: show the inverse spinel distribution]

Often we find what are called inverse spinels, like TiFe₂O₄: the minority cation (titanium) prefers the octahedral site
So the majority cation is split between octahedral and tetrahedral sites
Result: a disordered or random distribution of two different cations on the octahedral sites

That extra site complexity leads naturally into garnet.

Garnet

Wyckoff-site detail

Crystal-chemical formula

The garnet structure (X₃Y₂Si₃O₁₂) is even more complex than spinel.

Larger cations such as magnesium on distorted cubic sites, aluminum on octahedral, the smallest cations such as silicon on tetrahedral
YAG (Y₃Al₅O₁₂): aluminum on both tetrahedral and octahedral positions; yttrium partially substituted with neodymium gives the most widely used laser crystal — infrared at 1064 nm that can be frequency-doubled to visible green at 532 nm
Iron for aluminum: YIG, an important magnetic material
Gemstones: chromium for aluminum gives red garnets often mistaken for rubies — the historical "Black Prince's Ruby" is actually a garnet
Family properties set by the cations on the cubic, octahedral, and tetrahedral sites

[advance: show garnet Wyckoff-site detail]

Details for magnesium aluminum silicate: space group Ia3̅d, number 230 — the very last, arguably the most symmetric, space group in the international tables
160 atoms in the unit cell, yet each atom on a single Wyckoff site
All silicons crystallographically and chemically equivalent; same for aluminum, magnesium, and oxygen — once again Pauling's rule of parsimony

[advance: show the garnet crystal-chemical formula]

The bond graph is a bit complicated with so many different atoms, but we can just write the crystal chemical formula shown here
Magnesium 8-coordinate, aluminum 6-coordinate, silicon 4-coordinate
Oxygen again has 4 bonds to the cations: 2 to magnesium, 1 to silicon, 1 to aluminum

Perovskite is structurally simpler, so it gives us a clean way to study size effects.

Perovskite

Crystal-chemical formula

Coordination environments

The perovskite structure (ABO₃) is simpler than spinel or garnet, with only five atoms per unit cell.

Small B-site cation (e.g., Ti⁴⁺) at the cube corners; oxygen at the midpoints of the edges; larger A-site cation (e.g., Sr²⁺) at the body center
Remove the A cation: TiO₆ octahedra connected at the corners, an infinite 3D network equivalent to the ReO₃ structure
Inserting the A cation "stuffs" this framework — it fits into the cavity formed by the corner-sharing octahedra
The A cation is 12-coordinate in a cubic environment, corresponding to the number of cube edges

[advance: show the perovskite crystal-chemical formula]

Crystal chemical formula: strontium 12-coordinate, titanium 6-coordinate
Oxygen has four strontiums around it and two titanium

[advance: show the coordination environments]

Another view of the coordination environments of the three ions in the perovskite structure

Those coordination environments make ionic size the next question.

Tolerance Factor

Unit-cell geometry

Two expressions for the cell edge

Goldschmidt tolerance factor

Tolerance-factor interpretation

$t = 1$ means a perfect fit.

Per Solid State Materials Chemistry, Ch. 1 §1.5.

Goldschmidt developed a simple yet accurate geometric description of the perovskite structure, which allows us to predict the likelihood of an octahedral tilting distortion. He referred to this parameter as the tolerance factor, t.

[advance: show the unit-cell geometry]

An important factor in perovskite stability: the relative size of the ions
One parameter — the cubic cell edge — must satisfy two constraints: the optimal B–O distance (e.g., Ti–O) and the optimal A–O distance (e.g., Sr–O)
How well both are met is the tolerance factor, introduced by Victor Goldschmidt about a century ago, who also synthesized some of the first perovskites
The mineral perovskite itself: calcium titanate (CaTiO₃)

[advance: show the cell-edge equations]

Take a cut right out of the middle of the unit cell: there the strontium and the oxygen touch along the face diagonal.

The length of the edge of the cubic unit cell can be defined either in terms of the titanium and oxygen radius or in terms of the strontium and the oxygen radius.

[advance: show the tolerance-factor result]

Defined by the smaller octahedral cation: one radius of the small cation, two radii of the anion, one more radius of the small cation. Call that small cation B: the unit cell edge length is twice the radius of B plus the radius of the anion X.

In the lower drawing the ions touch along the face diagonal, so that distance is twice the radius of A plus the radius of the anion X.

This is a 45 degree right triangle: if the hypotenuse is two times the radius of A plus the radius of X, the edge is that value divided by square root of two. Simplify to get this.

Now we have two expressions for what the unit cell edge should be, and we can use those to calculate the tolerance factor.

[advance: show the tolerance-factor interpretation]

Goldschmidt's reading: how well the lattice accommodates both the large A-site cation and the smaller B-site cation
t = the ratio between the unit cell size that best fits the A cation and the size that best fits the B cation
Values match: t = 1, a perfect fit
t > 1: the A cation is too large and stretches the framework of corner-sharing octahedra
t < 1: the A cation is too small and sits loosely in the cavity, leaving extra space inside the structure

We can now test that factor against real compounds.

Size Mismatch and ABO₃ Compounds

Table of cell-edge estimates and Goldschmidt tolerance factors for six ABO3 compounds.

ATiO₃ Structures: t > 1

SrTiO3, BaTiO3, and BaMnO3 structures as the tolerance factor rises above one.

Hexagonal and Rhombohedral Perovskites

Hexagonal and rhombohedral perovskites form when cubic close-packed AO₃ layers are partially replaced by hexagonal close packing.

Full hexagonal packing: the 2H structure — face-sharing octahedra form infinite chains that are not interconnected, leaving space for large cations such as barium
Mixed cubic and hexagonal stacking: more complex connectivity
4H: four layers repeat — dimers of face-sharing octahedra linked by corners
9R: nine layers repeat — trimers of face-sharing octahedra connected by corners to other trimers
6H, barium titanate under high pressure: corner-sharing octahedra plus dimers of face-sharing ones
These hexagonal perovskites appear when the tolerance factor is greater than one

We now turn the factor in the other direction.

Crystal Structure: CaTiO₃

CaTiO3 structure showing an octahedral tilting distortion and reduced calcium coordination.

ATiO₃ Structures: t < 1

SrTiO3, CaTiO3, and MgTiO3 structures as the tolerance factor falls below one.

If the tolerance factor decreases far enough, even distorted perovskites become unstable.

The structure often transforms into ilmenite, as in magnesium titanate (MgTiO₃)
Ilmenite is closely related to corundum (Al₂O₃), which contains pairs of face-sharing octahedra
In ilmenite these octahedra are ordered — titanium in one, magnesium in the other — producing alternating magnesium and titanium layers
Alkaline earth coordination number reduced to six, more appropriate for a small ion such as magnesium
Overall: some perovskites remain cubic, but a much larger fraction are distorted like calcium titanate or barium titanate
The family is vast — thousands of members — and will appear repeatedly in discussions of functional properties

The summary gathers the structural patterns we have used.

Summary

Spinel (AB₂O₄):
- Cubic close-packed O²⁻ lattice.
- Normal vs. inverse site distributions.
- Important for magnetism (e.g., Fe₃O₄).
Garnet (X₃Y₂Si₃O₁₂):
- Multiple cation sites: cubic (8), octahedral (6), tetrahedral (4).
- Technological uses: lasers (Nd:YAG), magnetic materials (YIG), gemstones.
Perovskite (ABO₃):
- Corner-sharing octahedra with A in 12-coordination.
- Stability governed by Goldschmidt tolerance factor (t).
- Distortions → ferroelectric (BaTiO₃), piezoelectric, or hexagonal variants.
- Ubiquitous in functional oxides (superconductors, dielectrics, photovoltaics).

CaF₂: Fluorite

Fluorite CaF2 space group, lattice parameter, Wyckoff positions, and mineral photograph, then the unit cell generated from those Wyckoff sites.

Fluorite unit cell

CaF2, fluorite, has space group Fm3m, number 225, and lattice parameter a equals 5.46 angstroms.

Calcium on the 4a Wyckoff site at zero, zero, zero
Fluorine on the 8c Wyckoff site at one quarter, one quarter, one quarter
We now use the unit cell to practice placing all of the atoms generated by symmetry

[advance: show the fluorite unit cell]

Drawn with all atoms fully or partially inside the cell: a calcium atom sits at the origin, and the generators place others at (½,½,0), (½,0,½), and (0,½,½).

The remaining calcium atoms come from applying translational symmetry — adding integer multiples of the unit cell vectors to those four positions.

We can always add or subtract integers to bring atoms back inside (or partly inside) the unit cell.

That same crystallographic bookkeeping describes the sodalite network.

Sodalite Network

Sodalite cages sharing hexagonal faces in a body-centered cubic arrangement.

Analyzing Sphere Packing in Hexagonal Structures

Sphere-packing analysis in a 4H hexagonal structure with cubic and hexagonal stacking labels.

Generalized 8 minus N Rule

Practice questions

Work the Lecture 7 practice questions before the next class. They cover the spinel and garnet structures, cation site preferences, the perovskite framework, and the Goldschmidt tolerance factor.

Open the Lecture 7 practice questions

Every question carries a worked explanation, so you can check your reasoning as you go.

CHEM 548: Materials Chemistry

Lecture 7: Important Ternary and Quaternary Structures

Learning Objectives

Spinel

Spinel Structure

Garnet

Perovskite

Tolerance Factor

Size Mismatch and ABO3 Compounds

ATiO3 Structures: t > 1

Hexagonal and Rhombohedral Perovskites

Crystal Structure: CaTiO3

ATiO3 Structures: t < 1

Summary

CaF2: Fluorite

Sodalite Network

Analyzing Sphere Packing in Hexagonal Structures

Generalized 8 minus N Rule

Practice questions

Size Mismatch and ABO₃ Compounds

ATiO₃ Structures: t > 1

Crystal Structure: CaTiO₃

ATiO₃ Structures: t < 1

CaF₂: Fluorite