xraytools.

Lattice explorer

https://xraytools.com/lattice?lattice=F&repeat=1&motif=none

Powder or single crystal

Start with a box that repeats. Then distinguish the lattice points from the atoms placed around them.

How to use this lesson
You supply
Choose a cubic lattice, a block size and whether to show lattice points or atoms. No crystallography background is needed.
Reading it
A finite view of an infinite repeat. Boundary points are shared between neighbouring cells; visible spheres and points per cell are different counts.

Worked examples: primitive, repeat the cell · body-centred, repeat the whole motif · face-centred, count the shared points

Next on the path: Bragg Calculator On From planes to a powder pattern, step 1 of 7

Terms here: centring

See also: Bragg Calculator · HKL Calculator

Teaching with this page
Objective
After this page a learner can distinguish lattice points from atoms, repeat a motif, and count the shared points in a conventional cubic cell.
Start from
this worked example
Ask first
A primitive cell shows a sphere at each of its eight corners. Does it contain eight lattice points?
Watch for
“Yes: every sphere belongs entirely to this cell”
Then
Bragg Calculator
Check yourself: A primitive cell shows a sphere at each of its eight corners. Does it contain eight lattice points?

No: the shared corners contribute one lattice point per cell Yes: every sphere belongs entirely to this cell

Each corner is shared by eight neighbouring cells. Eight contributions of one eighth make one point. Adding a motif changes the atoms repeated around each point, not this sharing rule.

Build the repeat

A unit cell is one box of the repeat. A motif is the atom or group repeated at each lattice point. Here the cell edges stay equal and meet at right angles. Adding cells changes how much you see, not the lattice.

Face-centred (F): the lattice points

Shown: 1 × 1 × 1 conventional cells.

lattice point 0, 0, 0lattice point 0, 1, 0blattice point ½, ½, 0 — added by the centringlattice point 1, 0, 0lattice point 0, ½, ½ — added by the centringalattice point ½, 0, ½ — added by the centringlattice point 0, 0, 1lattice point 1, 1, 0clattice point ½, 1, ½ — added by the centringlattice point 1, ½, ½ — added by the centringlattice point 0, 1, 1lattice point ½, ½, 1 — added by the centringlattice point 1, 0, 1lattice point 1, 1, 1
Drag to rotate; scroll to zoom.Look along

These spheres are lattice points, not atoms. Each marks an equivalent place in an infinite repeat. The box shows only a finite part of it.

4 lattice points per conventional cell, after sharing the boundary points.

14 points are visible, including the outer boundary. That is a drawing count, not the number belonging to one cell.

Count one conventional cell

8 corners × 1/8 + 6 face centres × 1/2 = 4.

A corner is shared by eight neighbouring cells; a face centre by two. A body centre is wholly inside one cell. A primitive cell contains one lattice point. The conventional cubic I and F cells make the cubic symmetry easy to see, but are not primitive; each lattice also has a smaller primitive cell.

Try one change

Repeat the cell

Does repeating the box create a new kind of lattice?

Try this change

Add a motif

Can a lattice point be empty while the structure still repeats?

Try this change

Centre the lattice

Does the body-centre translation repeat one atom or the whole motif?

Try this change

Reset to one primitive cell

From repeat to diffraction

A unit cell is a box whose translations tile the repeating pattern without gaps. The lattice describes its translations; the motif is the same arrangement placed at every lattice point. The conventional box is a choice, not a physical boundary in the crystal.

This explorer compares the three cubic lattice types. It does not cover all crystal systems. Next, connect plane spacing to diffraction angle in the path’s worked example, or try the diffraction playground to change a cell or its contents and compare patterns. The playground uses a different motif: atoms at the corner and centre, with the centre occupancy adjustable.

Definitions: IUCr Direct lattice, Crystal pattern and Centred lattice.