What is the meaning of a basis?

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SUMMARY

A basis in the context of crystallography refers to a set of points within a unit cell that, when translated by integer multiples of lattice vectors, generates the entire lattice structure. In the discussed honeycomb structure, the basis can consist of either two points (0, a/2(x+y+z)) or four points (0, a/2(x+y), a/2(x+z), a/2(y+z)). The inclusion of the point 0 in the basis signifies the origin of the lattice, which is essential for defining the spatial arrangement of the lattice points. Understanding this concept is crucial for grasping the relationship between lattice structures and their corresponding basis points.

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noamriemer
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Hi!
There is a concept I don't understand and would love to have is cleared...
What is the meaning of a lattice with a basis?
What do I need it for?
Say I have a honeycomb structure. (fig 1) and a basis as mentioned there (did I understand it right? is it the basis? )
why does it become a simple lattice spanned by ax,ay,az
with a two point basis

0, a/2(x+y+z)

or a four -point basis

0, a/2(x+y), a/2(x+z), a/2(y+z) ?

I don't understand the whole thing. For a start, how can in be (mathematically) that 0 is a part of a basis? What is this 0?
Second thing, how can something be spread by either 2 or 4 dimension basis?
It does not fit what I know about a basis...
I can't find answers to these questions... and I guess I miss the whole concept...
Thank you in advance for your help :)
 

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A basis in the crystallographic sense not a basis in the sense of a vector space. The two are completely disjunct concepts, which just happen to have the same name by accident.

The unit-cell basis you are describing is effectively just a bunch of points in the unit-cell. The entire lattice is then spanned by translating these points (i.e., all the points separately) by integer-amounts of the lattice vectors.
 
Thank you!
 

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