Coordination number, voids (tetrahedral, octahedral): Foundations

This page builds every word, letter, and picture the parent note Coordination number, voids uses — starting from "what is a sphere touching another sphere". If a symbol appears in the parent, it is earned here first.

0. The starting picture: hard spheres that touch

Before any letter, fix the mental model. An atom or ion is drawn as a hard sphere — a solid ball that cannot squash or overlap. Two spheres touch when the distance between their centres equals the sum of their radii, and not one bit less.

Figure — Coordination number, voids (tetrahedral, octahedral)

1. Radius — the symbols $R$ and $r$

A radius is the distance from the centre of a ball to its surface. We need two different radius letters because the topic always mixes big balls and small balls.

r/R means... ::: how large the small sphere is compared to the large sphere (a pure number, no units)

2. The letter $N$ — a count of packed atoms

$N$ is simply how many big spheres live in the piece of crystal we are looking at (usually one unit cell).

3. "Coordination number" (CN) — counting neighbours

Now combine touch (Section 0) with counting (Section 2).

The parent note writes lines like $8 \times \frac{1}{8}$ and $12 \times \frac{1}{4}$ . These fractions come from sharing atoms between neighbouring boxes, not from cutting atoms.

Formula Sharing fractions

Position	Boxes sharing it	Fraction owned
Corner	8	$\frac{1}{8}$
Edge centre	4	$\frac{1}{4}$
Face centre	2	$\frac{1}{2}$
Body centre	1	$1$ (all ours)

Why the topic needs it: counting atoms ( $N = 8 \cdot \frac{1}{8} + 6 \cdot \frac{1}{2} = 4$ ) and counting voids at edge centres ( $12 \cdot \frac{1}{4} = 3$ ) both live or die on these fractions.

A corner atom is shared by how many cells? ::: 8, so it contributes 1/8 An edge-centre void is shared by how many cells? ::: 4, so it contributes 1/4

5. The cube and its three special lines

Every void-size derivation in the parent note is really Pythagoras on a cube. So we must name the cube's parts.

Length of a cube's face diagonal (edge a) ::: a√2 Length of a cube's body diagonal (edge a) ::: a√3

6. Tetrahedron and octahedron — the two hole shapes

The voids are named after the shape made by the centres of the surrounding balls.

7. The square-root numbers you must trust

The parent uses $2 \approx 1.414$ , $3 \approx 1.732$ , and $3/2 \approx 1.225$ . These are just decimal values of the diagonal lengths.

√2 − 1 equals about ::: 0.414 (octahedral radius ratio) √(3/2) − 1 equals about ::: 0.225 (tetrahedral radius ratio)

8. How the foundations feed the topic

This foundation feeds directly into Close packing in solids (hcp, ccp, bcc), Packing efficiency and unit cell dimensions, and Radius ratio rule and ionic structures; the void counts then decide formulas in NaCl, ZnS, CaF2 crystal structures and the counting recurs in Density of a unit cell. All of it sits inside States of Matter (Quantitative).

Equipment checklist

I can say what it means for two hard spheres to "touch" ::: their surfaces meet at one point; centre-to-centre distance = sum of radii I know the difference between R and r ::: R is the big packed sphere, r is the small guest sphere in the hole I know why we use the ratio r/R not r alone ::: fit depends on guest size compared to host size, a pure unitless number I know what N counts ::: the number of large packed atoms in the region (N = 4 for fcc) I can define coordination number ::: number of neighbours that directly touch a given sphere I know the sharing fractions 1/8, 1/4, 1/2, 1 ::: corner, edge centre, face centre, body centre I can compute a cube's face and body diagonals ::: a√2 and a√3, from Pythagoras I know why Pythagoras appears at all ::: spheres touch along slanted diagonal lines, so we need the diagonal length I can picture a tetrahedron (4 corners) and octahedron (6 corners) ::: triangular pyramid vs two square pyramids joined I know octahedral holes are bigger than tetrahedral ::: 0.414 > 0.225