# What is the difference between lattice vectors and basis vectors?

Google has not been very useful in this regard. It seems no one has clearly defined terms and Kittel has too little on this.

There could be more to it. I have learned a quite different meaning of "basis" when it comes to crystallography:

Of course, lattice vectors are the vectors that span the lattice. Now, at each lattice site, the crystal can have one or more "basis atoms". That's when we speak of a one-atomic, two-atomic basis etc... The positions of the basis atoms are usually described by vectors with lengths relative to the size of the unit cell (in units of the lattice parameter a). The first atom is usually at (0, 0, 0), the other for example at (1/2, 1/2, 1/2) (for a bcc-lattice) or at any other position within the unit cell. A specific basis vector of all the unit cells together is then forming a certain sublattice.

Basis vectors and lattice vectors are alternative ways to represent vectors in a vector space.

In mathematics (linear algebra,) basis vectors are mutually orthogonal and form a set of linearly independent vectors that, in a linear combination, can represent every vector in a given vector space. A set of basis vectors define what we usually think of as a conventional "coordinate system."

Lattice vectors represent the edges of a unit cell of a lattice. They are not necessarily mutually orthogonal. A linear combination of lattice vectors, with integral parameters, can represent every vector that belongs to the lattice.

The definitions for basis and lattice vectors, are much better described by LATTICE GEOMETRY, LATTICE VECTORS, AND RECIPROCAL VECTORS .

• The OP is talking about crystallography. In this context, basis vectors don't have to be orthogonal. – Peter Shor May 6 '13 at 21:10
• Yes thanks, you are correct. I think the explanation in the link above is how crystallographers use these terms. – Mark Rovetta May 6 '13 at 21:15
• I'm writing this in case anybody else is curious about basics of lattices: the provided link is very useful, it may be supplemented by the very first figure of this slideshow, to make everything perfectly clear: navrotsky.engr.ucdavis.edu/pages/classes/2006ClassArchive/… – user3237992 May 18 '15 at 9:07
• @user3237992 link is dead. – Pranav Feb 13 '19 at 6:23

Basis vectors are 3 shortest independent lattice vectors

• That is not correct. In linear algebra, basis vectors are any set of linearly independent vectors that span the entire space. In crystallography, basis vectors have a quite different meaning. – Lagerbaer May 7 '13 at 4:55
• yes as I've said they are "shortest". – richard May 7 '13 at 5:43
• That's still not correct. Heck, basis vectors in crystallography aren't even linearly independent. You can have a 3D crystal structure where the unit cell contains a large number of atoms. Then you have way more than 3 basis vectors, who can't possibly be linearly independent. Basis vector in crystallography $\neq$ basis vector in linear algebra. – Lagerbaer May 7 '13 at 13:58
• maybe what I described are "primitive vectors" – richard May 7 '13 at 14:12
• Yes. I know that it's confusing, because the primitive vectors are basis vectors in the linear-algebra sense... – Lagerbaer May 7 '13 at 14:14