Lattice explorer
https://xraytools.com/lattice?lattice=I&repeat=2&motif=pair
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
Body-centred (I): a motif repeated on the lattice
Shown: 2 × 2 × 2 conventional cells.
These spheres are atoms. The lattice points are hidden. The same silicon-and-oxygen pair is placed with the same offset at every lattice point.
2 atoms per motif × 2 lattice points per conventional cell = 4 atoms per conventional cell.
32 atoms are drawn in this block. Si is tan and O is red. The chosen offsets put all atoms inside the cells; none sits on a lattice point. This is an illustrative arrangement, not a model of a measured compound. A motif can lower the symmetry of the full structure while retaining these lattice translations.
Count one conventional cell
8 corners × 1/8 + 1 body centre = 2.
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 changeAdd a motif
Can a lattice point be empty while the structure still repeats?
Try this changeCentre the lattice
Does the body-centre translation repeat one atom or the whole motif?
Try this changeFrom 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.