Unit Cells and Atomic Packing: Simple, BCC and FCC Crystals

What is a unit cell?

A crystal is a pattern that repeats. Pick a small block, copy it again and again by sliding it along three directions, and you fill space without gaps or overlaps. That block is the unit cell, and the grid of repeat points is the lattice.

Many metals use cells shaped like cubes, which keeps the maths friendly. Three cubic arrangements cover most of what an introduction needs:

• Simple cubic (SC)
• Body-centred cubic (BCC)
• Face-centred cubic (FCC)

The side length of the cube is written a, and the atom radius is r. Almost every calculation links those two.

Counting atoms: the sharing rule

Pictures of unit cells show whole spheres at the corners, but those spheres are shared with neighbouring cells. Only the slice that falls inside your cell counts.

• A corner atom is shared by 8 cells, so it contributes ⅛.
• A face-centre atom is shared by 2 cells, so it contributes ½.
• A body-centre atom is not shared, so it contributes 1.

Now add up. Simple cubic has eight corners: 8 × ⅛ = 1 atom per cell. BCC adds one atom in the middle: 8 × ⅛ + 1 = 2. FCC has eight corners and six faces: 8 × ⅛ + 6 × ½ = 4.

The drawings show whole spheres only so that you can see them. The counting rule is what the maths uses.

Where do atoms touch?

To link a with r, find the line along which neighbouring atoms touch, since that is where the spheres are squeezed against each other.

• Simple cubic: atoms touch along an edge, so a = 2r.
• BCC: atoms touch along the body diagonal, giving 4r = √3 a. Each atom has eight nearest neighbours.
• FCC: atoms touch along a face diagonal, giving 4r = √2 a. Each atom has twelve nearest neighbours.

The pattern is useful: the more neighbours an atom has, the more crowded the structure tends to be.

Packing fraction: how full is the cell?

The packing fraction is the share of the cell's volume taken up by atoms, treated as hard spheres:

packing = N × (4/3)πr³ / a³

where N is the number of atoms per cell. Putting in the contact condition for each structure gives well-known results:

• Simple cubic: about 52%
• BCC: about 68%
• FCC: about 74%

FCC is as densely packed as identical spheres can be in a regular arrangement. Three-quarters of the space is atom and the rest is gaps. A simple cubic structure wastes nearly half the space, which is why very few elements use it under ordinary conditions.

Remember that this is a hard-sphere model. Real atoms are fuzzy clouds, and bonding affects spacing, so the numbers describe an idealised picture.

Why packing matters

Packing links the microscopic picture to things you can measure. A more tightly packed structure generally gives a higher density for atoms of similar mass and size. The number of neighbours and the planes of atoms in a lattice also influence how a metal deforms and how X-rays scatter from it. Diffraction patterns are in fact how crystal structures are measured in real laboratories.

You do not need those details yet. The key idea is that a few geometry facts, counted carefully, predict a lot.

Mistakes to avoid

• Counting whole corner atoms. Always use ⅛ for a corner.
• Using the wrong contact line. Edge, face diagonal and body diagonal are different for each structure.
• Confusing the lattice with the atoms. The lattice is a set of points; atoms sit on or around them.
• Treating packing fractions as exact for real materials. They belong to the hard-sphere model.
• Believing smaller drawn spheres change the count. Shrinking the display leaves more visible gaps but not more lattice sites.

How to explore this in the lab

The Crystal Lab is a simplified teaching model with 3D cells you can rotate and rescale. Work through the four free lessons in this order:

1. Eight corners, one atom for the sharing rule.
2. One in the middle to add a body-centred atom.
3. Six shared faces for the FCC count.
4. Space between atoms to compare packing.

Before you open each one, predict the number of atoms per cell and the contact line. Then use the model to confirm. If bonding between atoms is the part you are missing, the Chemistry Lab covers ionic and covalent bonds.