For the purpose of this question, I'm making the assumption that atoms are fixed in space, and as such possess no vibrational degrees of freedom.
A crystalline solid is a solid which exhibits periodicity in the arrangement of its constituent atoms. That is, there exists a translation vector, $T=n_1\textbf{a}+n_2\textbf{b}+n_3\textbf{c},$ where $\textbf{a},\,\textbf{b}$, and $\textbf{c}$, are non-coplanar vectors, such that $r'=r+T$ is equivalent to $r$, such that $r'\equiv r$. Due to this equivalence, the Hamiltonian, $H$ is invariant under translation by $T$, such that $H(r)\equiv H(r+T)$.
The following is a passage from my textbook, Atoms, Molecules, and Solids, by Morrison:
The lattice is the set of all points in space that have identical environments. (Thus the lattice does not "exist" in a physical sense.)
A physical crystal is obtained by associating a basis with each lattice point; a basis is one or more atoms "attached" to each point of the lattice. Thus we may consider a crystal to be a periodic array of atoms with a structure described by the lattice of locations in space to which each atom (or group of atoms) in the basis is attached.
Now, the first part of this is easy enough to understand. It is not necessary that the lattice points correspond to the locations of atoms within the crystal. Consider a 2D square lattice. The lattice points may just as well be the points corresponding to the centers between four atoms (as the electronic environments of those points are still equivalent), rather than the locations of where the atoms are fixed in space.
However, I find the definition of the basis to be much more abstract. As I understand it, the basis seems simply to be a way of "fixing" the lattice to a set of points (i.e. giving the lattice a well defined origin). Still, I'm not entirely clear on this. Any help would be appreciated.
