# 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. May 6, 2013 at 21:10
• Yes thanks, you are correct. I think the explanation in the link above is how crystallographers use these terms. May 6, 2013 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/… May 18, 2015 at 9:07
• 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. May 7, 2013 at 13:58