CHEM 548: Materials Chemistry

Lecture 6: Networks, Silicates, and Microporous Solids

Mon 09/14/2026

Network topology, silicates, and microporous solids

Core structural motif: $\mathrm{SiO_4}$ tetrahedra

Reading: Woodward, Karen, Evans, and Vogt, Solid State Materials Chemistry
Ch. 1 §§1.4.3–1.4.4, §§1.5.4–1.5.5; Ch. 14 §14.4

Learning Objectives for Networks

Understand how coordination numbers lead to network topologies in crystal structures.
Define key terms: vertex (linking point) and linker/edge (connection between vertices).
Recognize that linkers can be bonds, atoms, or polyatomic groups (e.g., O in zeolites, CN⁻ in Prussian blue, dicarboxylates in MOFs).
Distinguish between:
- Uninodal nets (all vertices equivalent).
- Binodal nets (two different kinds of vertices).
- Regular nets (all vertices, edges, and angles equivalent).
Apply network ideas to both simple binary compounds and framework materials (zeolites, MOFs).
Relate real crystal structures (e.g., TiO₂ as a 6–3 binodal net) to their underlying network topology.

Network Vocabulary

Vertex = Linking point (in our discussion the vertex will be generally be an atom, though in some cases it will be a cluster of atoms)

Linker = Connects vertices, can take various forms

A bond, such as in diamond
Atom, such as the oxygen atoms in a zeolite/silicate
Molecule, such as a dicarboxylic acid in a MOF

Uninodal Network = All vertices are the same

Binodal Network = Two types of vertices. We use the symbolism (N,M) connected net. For example, TiO₂ is a (6,3) net.

Regular Network = All vertices, edges and angles are equivalent by symmetry

See: M. O’Keeffe, M. Eddauodi, H. Li, T. Reinke & O.M. Yaghi, J. Solid State Chem. 152, 3-20 (2000).

A few key terms:

Vertex = linking point — usually an atom, but in complex structures can be a cluster
Edge/Linker = connection between vertices — a bond in simple compounds, but may be an atom (oxygen in zeolites), a polyatomic anion (CN⁻ in Prussian blue), or a larger molecule (dicarboxylic acid in MOFs)
Uninodal = identical vertices; binodal = two kinds of vertices
Example: TiO₂ has Ti (6-coordinate) and O (3-coordinate) — a 6–3 binodal net
Regular networks: all vertices, edges, and angles equivalent
Infinitely many topologies exist, but crystal chemistry relies on a finite, well-studied set — foundational reference: O'Keeffe and Yaghi (2001); deep understanding pays off for MOFs and zeolites, here we focus on the basics

With those terms in hand, let's start with networks built from a single kind of vertex, beginning with three connections per vertex.

Two uninodal three-connected networks: graphite and SrSi2.

Uninodal four-connected networks: lonsdaleite and diamond.

Uninodal 6- and 8-Connected Nets

Two ball-and-rod net renders side by side: the six-connected primitive cubic net of polonium and the eight-connected body-centred cubic net of iron.

Regular network	Regular network
Primitive cubic (Po)	Body-centered cubic (Fe)
Space Group = Pm3̅m	Space Group = Im3̅m
Site symmetry = m3̅m (O_h)	Site symmetry = m3̅m (O_h)

Site ordering transforms uninodal nets into compound structures.

Starting from these important unionodal nets we can develop compounds in a couple of different ways.

One: a site ordering of the vertices — let me illustrate

[advance: show the lonsdaleite to wurtzite site ordering]

Lonsdaleite with the silicons replaced by alternating zinc and sulfur → a site ordered homeotype: wurtzite
We could call this a 4,4 net — the two vertices are now different atoms, but both still tetrahedra

[advance: show the diamond to sphalerite site ordering]

Same site ordering on diamond → zinc blende, or sphalerite — another 4,4 net

[advance: show the primitive cubic to rock salt site ordering]

Primitive cubic net + site ordering — we could get the rock salt structure, sodium chloride — a six net

[advance: show the bcc to CsCl site ordering]

bcc net + site ordering → the CsCl structure
Many familiar compound structures — wurtzite, sphalerite, NaCl, CsCl — can be seen as ordered versions of high-symmetry uninodal nets
Recall: sphalerite (ZnS) = ccp anions, half the tetrahedral holes filled; NaCl = ccp anions, all octahedral holes filled — so these structures can be derived from close-packing or from ordering symmetric uninodal nets
Key point, symmetry reduction: diamond → sphalerite, same cell, m3̅m drops to 4̅3m; lonsdaleite → wurtzite, same cell, 6/mmm drops to 6mm — ordering creates inequivalent sites and eliminates some symmetry operations
pc → NaCl and bcc → CsCl: point symmetry stays m3̅m, translational symmetry changes
pc → NaCl: cell doubles in all three directions, primitive cubic to fcc Bravais lattice; bcc → CsCl: cell size unchanged, body-centered lattice becomes primitive cubic

So far we changed the vertices; the next route keeps the vertices and changes the linker instead.

Another way we can go from an element in a unilateral net to a compound is that we could do something called network expansion.

Keep the same vertices, change the linker

[advance: add oxygen linkers to the lonsdaleite net]

Lonsdaleite with an oxygen atom as the linker between the two silicons → one of the polymorphs of SiO₂: the tridymite structure, hexagonal

[advance: add oxygen linkers to the diamond net]

Diamond with oxygen at the midpoint of each bond → the idealized cristobalite structure — another SiO₂ polymorph

[advance: add oxygen linkers to the primitive cubic net]

Primitive cubic net with oxygens between the vertices → the ReO₃ structure
Add a large cation at the cube center → perovskite, one of the most important ternary structure types — we'll return to this later
Na₃N (though unstable) was eventually synthesized and adopts the ReO₃-type structure — nitrogen at cube corners, each surrounded by six sodiums, highlighting 2-coordinate sodium
Na = the red atoms, N = the blue; 1 atom from all the corners, 3 atoms from all the edges

There is one more modification, where the vertices themselves become groups of atoms.

Decoration of a network using a B6 cluster.

Some Important Binodal Nets

(N,M)-Net	Coordination Figures		Net (Example)
4,2	Tetrahedron	Linear (bent)	SiO₂ (cristobalite, tridymite)
4,3	Tetrahedron	Triangle	Si₃N₄
4,4	Tetrahedron	Tetrahedron	ZnS (sphalerite, wurtzite)
4,3	Square	Triangle	Pt₃O₄
4,4	Square	Square	NbO
4,4	Square	Tetrahedron	Cooperite (PtS)
6,2	Octahedron	Linear	ReO₃
6,3	Octahedron	Triangle	TiO₂ (rutile, anatase)
6,4	Octahedron	Tetrahedron	Corundum (Al₂O₃)
6,6	Octahedron	Octahedron	Rock salt (NaCl)
6,6	Octahedron	Trigonal prism	NiAs
8,4	Cube	Tetrahedron	Fluorite (CaF₂)
8,8	Cube	Cube	CsCl

This table shows several key binodal networks.

Earlier: element nets to compounds by site ordering or expansion — the same structures can be described directly as binodal nets
Sphalerite and wurtzite: from diamond and lonsdaleite, ordered into binary compounds → 4–4 binodal nets
SiO₂ polymorphs: Si 4-coordinate tetrahedral, O 2-coordinate linear → cristobalite, tridymite; Si₃N₄: Si tetrahedral, so N must be 3-coordinate, giving triangular vertices
Important: coordination number does not uniquely define geometry — CN=4 usually tetrahedron, sometimes square planar
Squares + triangles → Pt₃O₄; squares + squares → Nb₂O; squares + tetrahedra → Cu₂O or PtS
Octahedral cations: + linear → ReO₃; + triangle → rutile/anatase; + tetrahedron → corundum; + octahedron → rock salt; + trigonal prism → NiAs; some cations act as cubic vertices
As n and m increase, the number of possible nets decreases — binodal nets systematically connect empirical formulas, coordination numbers, and real structures

Let's look more closely at two rows of that table, starting with the four-four nets built from square-planar vertices.

Rutile and anatase as six-three networks.

Corundum structure shown in two representations.

Finally, let's take a closer look at corundum (Al₂O₃).

Structure: hexagonal close-packed O²⁻ with Al³⁺ filling two-thirds of the octahedral sites; vacancies ordered so the Al cations shift apart
Coordination: Al octahedral, O tetrahedral — among the two most common coordination environments in inorganic chemistry
But the network is not perfectly symmetric: corundum is rhombohedral, not cubic, so neither polyhedron is ideal
Al octahedra noticeably distorted; O tetrahedra highly distorted
The distortion reflects the lower symmetry of the rhombohedral space group compared to cubic structures, where perfect octahedral or tetrahedral coordination would occur

That completes the network story, so let me summarize it before you practice.

Summary for Networks

Networks describe how atoms connect, abstracted into vertices and linkers.
Uninodal nets: all vertices identical (e.g., diamond, graphite, cubic/bcc nets).
Binodal nets: two vertex types with different coordination (e.g., Ti 6–O 3 in TiO₂).
Regular nets: rare, highly symmetric cases where all vertices, edges, and angles are equivalent.
Transformations of nets:
- Site ordering → compounds like NaCl, CsCl, sphalerite, wurtzite.
- Network expansion → insert atoms as linkers (e.g., ReO₃, perovskite).
- Decoration → replace simple vertices with polyatomic clusters (e.g., CaB₆).
- Interpenetration → overlapping nets (e.g., Cu₂O).
Binodal nets provide a systematic way to connect empirical formulas, coordination numbers, and real crystal structures.

To summarize networks: networks describe how atoms connect, abstracted into vertices and linkers.

Uninodal: all vertices identical — diamond, graphite, the cubic and bcc nets
Binodal: two vertex types with different coordination — six-coordinate Ti and three-coordinate O in TiO₂
Regular: the rare, highly symmetric cases — all vertices, edges, and angles equivalent
Four transformations: site ordering (NaCl, CsCl, sphalerite, wurtzite); network expansion inserts atoms as linkers (ReO₃, perovskite)
Decoration replaces simple vertices with polyatomic clusters (CaB₆); interpenetration gives overlapping nets (Cu₂O)
Binodal nets: a systematic way to connect empirical formulas, coordination numbers, and real crystal structures

Let's put that connectivity bookkeeping to work on the ReO₃ framework in problem 1.27.

Practice problem 1.27 on the ReO3 network.

Answer: ReO3 is a six-two connected binodal net.

Silicates and Microporous Solids

Learning Objectives

By the end of this lecture, you should be able to:

Explain how silicate structures are built from corner-sharing SiO₄ tetrahedra.
Use the Niggli formula to relate oxygen: silicon ratios to structural motifs.
Identify structural motifs (isolated tetrahedra, chains, rings, sheets, frameworks) from O:Si ratios.
Describe how zeolites form from Al/Si tetrahedra and why cations create porosity.
Recognize key zeolite building blocks (sodalite cage, zeolite A, faujasite) and their pore sizes.
Distinguish microporous, mesoporous, and macroporous solids.
Understand the basics of MOFs and how MOF-5 is constructed from nets.

Silicates

Silicates are the most common minerals in the earth’s crust
Silicates represent about one third of known inorganic crystal structure types
Silicates generally share a number of common features
- Tetrahedrally coordinated silicon
- If connected tetrahedra are connected through corner sharing
The O:Si ratio is an important parameter in dictating the extent to which the tetrahedra are connected

Oxygen to silicon ratio and Niggli formula table.

Knowing the Nikle formula, we can examine oxygen-to-silicon ratios in silicates.

Every corner shared, as in SiO₂ polymorphs: formula SiO₄/2, O:Si = 2:1 → a 3D network of corner-connected tetrahedra
Other extreme, O:Si = 4:1: no oxygens shared → isolated tetrahedra, as in zircon (ZrSiO₄)

[advance: add the 2.5 ratio sheet row]

Ratio 2.5: 1 terminal + 3 bridging oxygens → most common motif: a 2-dimensional sheet of corner connected tetrahedra

[advance: add the 3 and 3.5 ratio rows]

Ratio 3: 2 terminal + 2 bridging — a variety of motifs, most notably an infinite one-d chain
Ratio 3.5: 3 terminal + 1 bridging → the dimers of corner connected tetrahedra we saw on the last slide

Let's see what these motifs look like in real silicate structures.

Porous Materials

Zeolites (Boiling stone) = Framework materials built from corner connected SiO₄/₂ and AlO₄/₂ tetrahedra

Zeotypes = Similar to zeolites but with other elements on the tetrahedral sites

Metal Organic Frameworks (MOFs) = Porous coordination polymers made up of clusters of metal ions or clusters connected by multidentate polyatomic ligands.

Focusing on 3D networks of corner-connected tetrahedra, a key class is zeolites.

Frameworks of Si and Al tetrahedra
"Zeolite" ("boiling stone"): heating released trapped solvents like water, causing them to fizz — coined 1756 by Swedish mineralogist Axel Fredrik Cronstedt, heating stilbite
Synthetic zeolites from the mid-20th century (Barrer's work in the 1940s) for industrial uses
Crucial applications: ion exchange, catalysis (petroleum cracking to gasoline), detergents, molecular sieves, gas separation
Zeotypes: same framework structures, other elements substituted for Si and Al
MOFs: coordination polymers with open pores — metal ions or clusters linked by multidentate organic ligands

[advance: add the pore-size classification]

We can classify porous materials as microporous, mesoporous, or macroporous by the size of the pores by which molecules can come in or out
Zeolites and MOFs: by and large microporous

With those classes defined, let's compare the most important zeolites.

A few important zeolites illustrate the key concepts.

Sodalite, zeolite A, faujasite: all share the same basic building block, the sodalite cage; sodalite itself is not porous, the others are
Sodalite and zeolite A: Si:Al ratio 1:1, alternating Si and Al tetrahedra
Replacing Si⁴⁺ with Al³⁺ → negative framework charges, balanced by cations (often Na⁺) and their waters of solvation inside the pores
These counterbalancing ions are crucial — without them, the framework would likely collapse into a dense silicate rather than a porous one
Key features: ring sizes, accessible pore volume (about 25% in zeolite A and faujasite for a 2.8 Å probe sphere), and the pore-opening dimensions that control molecular access
Next: the sodalite cage as the fundamental building block

To see where the sodalite cage comes from, let's go back to our four-connected nets.

Uninodal four-connected nets including the sodalite net.

The sodalite network takes its name from the mineral sodalite (Na–Al–Si–O–Cl).

Framework contains cages shaped like truncated octahedra — blue soccer balls
Faces: hexagonal six-rings and square/diamond four-rings — matching the sodalite net topology
In sodalite itself, Cl⁻ ions occupy the cage centers; Na and O are not shown here
The cage is the fundamental building unit of many zeolites, but here the cages are fused — the structure is non-porous, preventing molecular access

[advance: apply site ordering and oxygen bridges to the cage]

Single cage + site ordering → a homeotype: alternating Al–Si–Al–Si around the vertices
Then oxygen bridges: Al and Si do not bond directly but connect through O atoms at the tetrahedral vertices
O is not at the edge midpoint (that would give a linear 180° Al–O–Si bond); the angle is ~145°, maintaining a continuous tetrahedral network of Al and Si

Let's examine that single cage more closely.

Several representations of the sodalite cage.

Sodalite cage connectivity to zeolite A and faujasite.

Hierarchical faujasite catalyst used for petroleum cracking.

Fluid catalytic cracking (FCC) started in the 1940s with simple silica–alumina catalysts, but their activity was limited.

1962: Union Carbide introduced Zeolite Y — strong acid sites, large pores that let bulky oil molecules enter and react
Result: doubled gasoline yields, boosted octane, longer catalyst life; later USY, rare-earth exchange, and ZSM-5 additives made FCC the backbone of refining
Framework: faujasite — AlO₄ and SiO₄ tetrahedra forming spacious ~12 Å cages connected by 12-membered oxygen rings
Al charges balanced by cations (Na⁺), exchangeable for H⁺ → Brønsted acid sites that protonate molecules to form carbocations
At ~500 °C in the FCC riser: β-scission, isomerization, hydride transfer, aromatization → gasoline-range alkanes and alkenes plus lighter gases; coke blocks sites but is burned off in the regenerator
Why it works: pores ~7.4 Å just wide enough for heavy hydrocarbons, surface area ~900 m²/g, shape selectivity limits over-cracking and steers toward high-octane products
Payoff: gasoline yield ~45% to ~70%, higher octane from branched alkanes and olefins, tunable gasoline/propylene/diesel; today FCC with zeolite Y processes about 40% of global crude oil

From these inorganic frameworks we now step to metal-organic frameworks, built from the same network ideas.

MOFs (metal–organic frameworks) are a whole field, alongside COFs (covalent organic frameworks).

To illustrate: MOF-5, introduced by Omar Yaghi in a Nature paper in 1999
Demonstrated exceptionally high permanent porosity (free volume, very large internal cavity), very high surface area — established many of the design principles for the field, and helped move it from proof of concept to practical relevance, especially gas storage
Built from a simple cubic net: at each vertex an OZn₄ cluster — one oxygen surrounded by four Zn²⁺ cations
Clusters connected by benzene dicarboxylate linkers — deprotonated carboxylates at both ends coordinate to Zn; the Zn atoms end up tetrahedrally coordinated
Decorating the cubic vertices and linking with extended organic ligands yields MOF-5; a whole series can be made by enlarging the linker, which increases pore size
When the void space grows too large, interpenetration can occur — avoiding it is a central design challenge; the reference at the bottom relates MOF frameworks to inorganic crystal structures

Let me say more about why that 1999 paper mattered.

MOF-5 publication and permanent porosity.

The Mof field really go going when Omar Yagi published this paper in 1999.

The 1999 Nature paper on MOF-5 was pivotal because it showed, for the first time, that you could deliberately build a crystalline solid with permanent porosity, very high surface area (~2900 m²/g), and modular designability.

Unlike earlier coordination polymers that collapsed on solvent removal, MOF-5 remained stable
Proof that metal clusters and organic linkers could be stitched into robust, tunable frameworks
Launched reticular chemistry — designing solids from molecular building blocks
Made clear MOFs could rival zeolites and carbons with vastly greater flexibility
Transformed MOFs from a curiosity into a general design platform for gas storage, separations, and catalysis, igniting the field now central to porous materials research

After all that promise, the natural question is whether MOFs have reached any real commercial application.

ION-X: the status quo problem

1. The status quo problem

Ion-implant tools in chip fabs need toxic dopant gases: arsine (AsH₃), phosphine (PH₃), BF₃, etc.
Traditionally stored in high-pressure steel cylinders (pressures often >100 psig).
Hazards:
- Catastrophic leak risk → a single bottle contains enough toxic gas to endanger workers and trigger fab-wide shutdowns.
- Regulatory & insurance burden → fabs need redundant ventilation, scrubbing, and emergency protocols, raising costs.
- Cylinder heel (unused residual gas) → waste that still poses disposal risks.

2. What the MOF cylinder changes

MOFs adsorb and store the gas inside their pores, allowing the cylinder to hold the same usable inventory at sub-atmospheric pressure (near vacuum to ~1 atm).
On use, the gas desorbs in a controlled way to feed the ion-implant tool.

ION-X: why fabs pay for it

3. Why fabs pay for this

Massively reduced safety risk: Even if the cylinder is breached, no pressurized toxic gas rushes out. That alone cuts the probability and severity of a catastrophic event.
Regulatory relief: Lower safety classification than high-pressure bottles → lower facility cost, easier permitting, fewer insurance penalties.
Operational continuity: Prevents fab shutdowns due to leaks or alarms, which are extremely expensive (>$1M/day lost production).
Waste reduction: Lower “heel” volumes mean less toxic waste to manage and dispose of.
Compatibility: ION-X cylinders plug directly into existing ion-implant tools (no redesign needed).

4. The value proposition in one line

They convert a toxic-gas supply chain from a high-pressure hazard into a sub-atmospheric, “fail-safe” consumable, saving fabs money on risk, compliance, and downtime.

Why do fabs pay for this?

First, massively reduced safety risk: even if the cylinder is breached, no pressurized toxic gas rushes out — that alone cuts the probability and severity of a catastrophic event
Second, regulatory relief: a lower safety classification than high-pressure bottles — lower facility cost, easier permitting, fewer insurance penalties
Third, operational continuity: prevents fab shutdowns due to leaks or alarms — more than a million dollars a day in lost production
Fourth, waste reduction: lower heel volumes, less toxic waste to manage and dispose of
Fifth, compatibility: ION-X cylinders plug directly into existing ion-implant tools, no redesign needed
Value in one line: a toxic-gas supply chain converted from a high-pressure hazard into a sub-atmospheric, fail-safe consumable — saving fabs money on risk, compliance, and downtime

Here is the framework chemistry that makes that work.

MOF channel structure with open Cu sites after solvent removal.

ION-X dopant gas delivery performance at Axcelis.

ION-X (MOF) vs SDS (Carbon) Cylinders

Metric	ION-X (MOF)	SDS (Carbon)
Gas delivered	Higher usable capacity (20–50% more for some gases)	Lower deliverable, larger residual heel
Flow stability	Stable delivery down to very low pressures (few Torr)	Delivery falls off sooner; larger heel
Residual gas (heel)	Very low; most gas is recoverable	Higher heel, more waste
Purity / contamination	Low metals, stable across cycles	Acceptable, but batch variability possible
Reproducibility	Highly consistent (crystalline pores)	More variability (heterogeneous carbon)
Safety	Sub-atmospheric storage	Also sub-atmospheric storage (same safety principle)

This table compares ION-X MOF cylinders with the incumbent SDS carbon cylinders, metric by metric.

Gas delivered: MOF gives higher usable capacity, 20 to 50 percent more for some gases; carbon delivers less, larger residual heel
Flow stability: MOF holds stable delivery down to a few torr; carbon's delivery falls off sooner
Residual gas, the heel: very low for the MOF, most gas recoverable; carbon leaves a higher heel and more waste
Purity and contamination: MOF shows low metals, stable across cycles; carbon acceptable but can vary batch to batch
Reproducibility: highly consistent for the MOF's crystalline pores; more variable for heterogeneous carbon
Safety: both are sub-atmospheric storage — the same safety principle

That brings us to the end of the new material, so let me summarize.

Summary

Silicates dominate Earth’s crust; their tetrahedral networks are defined by O:Si ratios.
Structural motifs range from isolated tetrahedra to 1D chains, 2D sheets, and 3D frameworks.
Zeolites are Al/Si tetrahedral frameworks; charge-balancing cations generate accessible pores.
Key zeolites (sodalite, zeolite A, faujasite) differ in connectivity and pore size (4–7.4 Å).
Porous materials are classified as microporous (<2 nm), mesoporous (2–50 nm), macroporous (>50 nm).
MOFs (e.g., MOF-5) extend these ideas: metal–oxide clusters linked by organic ligands create highly porous, tunable frameworks.

Homework

1.22 Write the Niggli formula and the simple crystal-chemical formula for the CrO₃ structure that contains chains of corner-sharing chromium-centered tetrahedra.

1.23 Construct a bond graph or Niggli formula to determine if it is possible for all anions to be equivalent in a structure of tetrahedrally coordinated cations and stoichiometry of C₂A₃? Which alternative Niggli formula complies best with the rule of parsimony?

1.24 Using the Niggli formula and the rule of parsimony, determine the stoichiometry that results from sharing (a) all corners, (b) all edges, and (c) all faces of a cation-centered cube of anions. Note the structure prototype where you recognize it.

1.25 Write the Niggli formula for C₃N₄ made of identical CN₄ tetrahedra. How many different types of nitrogen vertices are there? What is the coordination number of each?

1.26 In β-Li₃N, nitrogen is 11-coordinated. Write down the Niggli formula of the NLi₃ polyhedron.

For homework, work problems 1.22 through 1.26.

1.22: the Niggli formula and the simple crystal-chemical formula for the CrO₃ structure with chains of corner-sharing chromium-centered tetrahedra
1.23: a bond graph or Niggli formula — can all anions be equivalent for tetrahedrally coordinated cations and stoichiometry C₂A₃; which alternative Niggli formula complies best with the rule of parsimony
1.24: with the Niggli formula and the rule of parsimony, the stoichiometry from sharing all corners, all edges, and all faces of a cation-centered cube of anions; note the structure prototype where you recognize it
1.25: the Niggli formula for C₃N₄ made of identical CN₄ tetrahedra — how many types of nitrogen vertices, and the coordination number of each
1.26: in β-Li₃N, nitrogen is 11-coordinated — the Niggli formula of the NLi₃ polyhedron

To start you off on 1.22, let me work the CrO₃ chain's Niggli formula with you right now.

1.22

Corner sharing tetrahedra in 1‑D chains -> CrO₄ tetrahedra

What the Niggli formula describes

It’s a connectivity-based formula for one polyhedron (e.g., a tetrahedron or octahedron around a cation).
It tells you how many anion vertices there are, and how many other polyhedra each vertex connects to.
You always write it per one cation-centered polyhedron.
C A(numerator/denominator + numerator/denominator …)

Niggli formula: CrO₂/₂+₂/₁

The SiO4 4- silicate tetrahedron with its oxygen and silicon spheres labelled, above the 1. Single Chain drawing of corner-sharing tetrahedra.

Silicate tetrahedron SiO₄⁴⁻: oxygen and silicon labelled, above “1. Single Chain”

Crystal-Chemical Formula Algorithm

1. List atoms: Write down each unique cation and anion site.

2. Tag cations: Add coordination number (how many neighbors) + geometry (t = tetra, o = octa, etc.).

3. Tag anions: Mark how many cations each anion connects to: O[1] = terminal, O[2] = bridging, O[3] = 3-connected, etc.

4. Check balances:

Bonds from cations = bonds accepted by anions.
Count anions = formula stoichiometry.
Charges must cancel.

5. Keep it simple: Only include distinct sites/connectivities needed to show bonding.

Worked crystal-chemical formula for the CrO3 chain.

Here is the worked solution.

Given: chromium is tetrahedral, and the tetrahedra link by corners to form 1D chains.

The sites are chromium, bridging oxygen, and terminal oxygen.

Chromium is Cr[4t]; the bridging oxygen is two-coordinate with a bent Cr–O–Cr bond; the terminal oxygen is one-coordinate.

Balances per one chromium: stoichiometry requires b plus t equals three, and connectivity requires four equals two b plus t, since chromium has a coordination number of four.

Subtracting gives b equals one, then t equals two.

So the crystal-chemical formula describes one crystallographically distinct infinite chain, in which each chromium is tetrahedral, with one bridging oxygen per formula, nonlinear, and two terminal oxygens.

That is the pattern to follow on the rest of the homework. That is all for today; next time we take up the important ternary and quaternary structures.

Practice questions

Work the Lecture 6 practice questions before the next class. They cover network vocabulary, uninodal and binodal nets, silicates and the Niggli formula, zeolites and pore size, and metal-organic frameworks.

Open the Lecture 6 practice questions

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