# Lattice points and lattice basis

• Kitten
In summary, the conversation discusses the concept of lattice points and atom basis in 2D and 3D structures. It is explained that in a 2D square net, each corner only holds one fourth of the total motif (atoms), while in a cubic structure, each lattice point only contains one eighth of the motif. This can be confusing when trying to define a basis with each lattice point.
Kitten
Hi!
I'm struggling in identifying the lattice points and atom basis.

As I understand in a cube, there are 8 lattice points, on on each corner of a cube. But in 2d it is any square between 4 points which are the lattice points. Is this correct?

So if the points on the corners are the lattice points. What confuses me is that a primitive cell can only have 1 lattice point and so if it's a square I thought it would have four lattice points not one.

Also I don't understand how to define a basis with each lattice point?

In a square net, or any net for that matter, each corner holds only one fourth of the total motif. So if your motif is atoms, then each corner of that square net will only have a single atom even though there are four total lattice points. Same thing with a cube. Though there are a total of eight sites where a motif can reside, each of those sites will only contain one eighth of the total motif (in this case atoms). So for a primitive cubic structure, there are eight lattice points but only a single atom!

Kitten

## 1. What are lattice points?

Lattice points are the set of points in a regular, repeating pattern in a mathematical space. They are the points where two or more grid lines intersect.

## 2. How are lattice points related to crystal structures?

In crystal structures, the atoms are arranged in a regular, repeating pattern. These atoms can be thought of as lattice points, with the crystal lattice acting as a framework connecting them.

## 3. What is a lattice basis?

A lattice basis is a set of vectors that can be used to define any lattice point in a mathematical space. These vectors are typically chosen to be the shortest possible and are often referred to as primitive vectors.

## 4. How is the lattice basis related to the lattice points?

The lattice basis is used to define the coordinates of the lattice points. Each lattice point can be expressed as a linear combination of the lattice basis vectors.

## 5. Can different lattice points have the same lattice basis?

Yes, different lattice points can have the same lattice basis. This is because the lattice basis only defines the coordinates of the lattice points and does not determine their arrangement in space.

• Atomic and Condensed Matter
Replies
4
Views
2K
• Atomic and Condensed Matter
Replies
2
Views
1K
• Atomic and Condensed Matter
Replies
1
Views
1K
• Atomic and Condensed Matter
Replies
1
Views
1K
• Atomic and Condensed Matter
Replies
5
Views
2K
• Atomic and Condensed Matter
Replies
2
Views
6K
• Linear and Abstract Algebra
Replies
7
Views
1K
• Atomic and Condensed Matter
Replies
5
Views
3K
• Atomic and Condensed Matter
Replies
20
Views
6K
• Atomic and Condensed Matter
Replies
1
Views
2K