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I am trying to get a firm understanding of crystal structures in solid state physics but having some issues with the terminology.

If I understand correctly, the lattice are points in a 3-D space so that each point has an identical surrounding. Does this mean the points on a lattice form the unit cell and the basis form the location of the atom within each unit cell? I am getting confused because these points could also be atoms surrounding other atoms. I'm trying to understand how lattice and basis vectors are applied to form a crystal. Any assistance is greatly appreciated.

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Let me try to explain with an example. Let's consider for instance the honeycomb lattice shown below. What type of lattice is this? Now since this is a two dimensional structure, we might expect that we are able to identify all the points with a linear combination of two vectors $\bf{a}_1$ and $\bf{a}_2$: $n\bf{a}_1 + m\bf{a}_2$, where the coefficients $m,n$ are integers, right? Now if you consider the two red vectors in the figure as candidates, you immediately realize that there is no way to combine them with integer coefficients to get the whole lattice, you can only get the red vertices!

This means that, despite being a regular structure, the honeycomb lattice is not a simple Bravais lattice, because you can't chose a pair of vectors whose linear combinations with integer coefficients reach all the vertices. So how can we solve this problem? The idea is to introduce a basis. Consider the green vectors $\bf{d}_1$, $\bf{d}_2$: now what if we take as the building block of the lattice not only the red site in the origin, but the red site plus the two blue sites described by $\bf{d}_1$, $\bf{d}_2$?

Well, now you can think to the honeycomb lattice as a Bravais lattice described by (combinations of) the Bravais vectors $\bf{a}_1$, $\bf{a}_2$, but where the repeating structure is not a single vertex, but a more complicated structure described by the three vertices in $\bf{0}$, $\bf{d}_1$, $\bf{d}_2$, which is called basis.

To answer more precisely the question, you see that blue points and red points don't have the same surroundings!, and that's why the honeycomb lattice is not a simple Bravais lattice. On the other hand, each building block has exactly the same surroundings of all the others.

Finally let me stress that this is a redundant description: you have clearly only two different types of vertices, so why taking three vertices to form a basis? You can actually retain only the vertical green vector and forget about the other, and now you still have a basis (pairs of blue-red vertices), which is no longer redundant. So we say that the honeycomb lattice has 2 vertices per unit cell (not 3!). Sorry but I couldn't find a better image.

enter image description here

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