Ions in a ceramic: ionic radius and coordination number
Most ceramics are made of ions: tiny charged particles. Metals give up electrons and become small positive cations. Oxygen and similar atoms take electrons and become big negative anions. Opposite charges pull, so ions pack together like balls.
The ionic radius is the size of the ion. The coordination number (CN) is how many ions of the opposite charge touch one ion. A cation wants as many anion neighbours as possible, but they must all touch it and must not be squeezed against each other.
That is why one number decides it all: the radius ratio = r(cation) ÷ r(anion).
| Radius ratio | CN | Shape of neighbours |
|---|---|---|
| 0.155 to 0.225 | 3 | Triangle |
| 0.225 to 0.414 | 4 | Tetrahedron |
| 0.414 to 0.732 | 6 | Octahedron |
| 0.732 to 1.0 | 8 | Cube corners |
Examples: in NaCl the ratio is about 0.56, so CN = 6. In CsCl the ratio is about 0.9, so CN = 8. In MgO the ratio is about 0.51, so CN = 6. This is a good rule of thumb, not a law: bonds with a lot of sharing (covalent) can break it.
Crystal structure and its properties
In a crystal the ions sit in a pattern that repeats in every direction. The smallest repeating block is the unit cell. Because every ion has its own fixed place, a crystal has a sharp melting point and breaks along flat planes (cleavage).
Crystals can be anisotropic: a property can be different in different directions, because the ions are packed differently along each direction.
Why are ceramics hard and heat-proof? The ionic and covalent bonds are very strong, so a lot of energy is needed to move or pull ions apart. Why are they brittle? If a layer slips by one place, plus meets plus and minus meets minus. They push apart and a crack runs. Metals do not crack like this, because their electrons keep the layers together.
Why are ceramics good insulators? All the electrons are stuck inside the ions or bonds. No free electrons, no current.
Glass structure and its properties
A glass has the same building blocks as a crystal but no repeating pattern. Each ion still has its usual neighbours (short-range order), but the pattern is random after a few ions (no long-range order). Glass forms when a melt is cooled so fast, or is so thick and sticky, that the ions are stuck before they can find their places in a crystal.
In silica glass every silicon sits inside a small tetrahedron of 4 oxygen, and these tetrahedra share corners in a random net.
Properties of glass: no sharp melting point (it softens slowly through the glass transition), the same properties in all directions (isotropic), it breaks with curved shiny surfaces, and it is clear because there are no crystal edges to scatter light.
| Crystal | Glass | |
|---|---|---|
| Pattern | Repeats | Random |
| Melting | Sharp point | Softens slowly |
| Direction | Can differ | Same everywhere |
| Break | Flat planes | Curved |
Key formulas and definitions
- Radius ratio = r(cation) ÷ r(anion)
- 0.225 to 0.414: CN 4; 0.414 to 0.732: CN 6; 0.732 to 1: CN 8
- 1 nm = 10⁻⁹ m = 10 Å; 1 Å = 100 pm
- Key terms: cation, anion, coordination number, unit cell, long-range order, glass transition
Worked examples
1. Na⁺ has radius 0.102 nm and Cl⁻ has 0.181 nm. Find the radius ratio and CN.
Ratio = 0.102 ÷ 0.181 = 0.56. This lies between 0.414 and 0.732, so CN = 6 (octahedral).
2. A cation of radius 0.06 nm sits with anions of radius 0.14 nm. How many anions can touch it?
Ratio = 0.06 ÷ 0.14 = 0.43. This is just above 0.414, so CN = 6.
3. Why is a crystal of salt flat-faced when it breaks but window glass breaks with curves?
Salt ions are in layers that give flat planes to split along. Glass ions are random, so there are no planes to follow and the crack curves.
4. Cs⁺ is 0.167 nm and Cl⁻ is 0.181 nm. Which CN does the rule give?
Ratio = 0.167 ÷ 0.181 = 0.92, which is above 0.732, so CN = 8.
Common mistakes
- Dividing the other way round. Always take the small cation over the big anion, so the ratio is below 1.
- Thinking a glass has no order at all. It still has the same neighbours around each ion; only the long pattern is missing.
- Saying glass is a very slow liquid that flows in old windows. Old panes are uneven because of how they were made; glass is a solid.
- Saying ceramics are brittle because the bonds are weak. The bonds are strong; the ions just cannot slip without like charges meeting.