= Atomic basis of a crystal
{title2=$\{(s_j,r_j)\}_{j=1}^{N}$}
= Atomic basis
{synonym}
An <atomic basis of a crystal> is the motif of <chemical species> $s_j$ and relative positions $r_j$ attached identically to every <lattice point> $R$ of a <Bravais lattice> $\Lambda$. The ideal periodic structure is the labeled set
$$
\{(s_j,R+r_j):R\in\Lambda,\quad j=1,\ldots,N\}.
$$
The <primitive lattice translation vectors> generate $R$ by <integer> combinations; the <atomic basis> lists what is repeated at $R$. These are different meanings of the word <basis>. In particular, the offsets $r_j$ need not be <linearly independent vectors>, and one of them may be zero. A <primitive unit cell> represents $N$ motif sites, even though it represents only one <lattice point>.
Changing the origin, choosing another <primitive unit cell>, or moving a motif site by a <Bravais lattice> translation changes the coordinates used to describe the same infinite <crystal structure>. The grouping into a motif does not, by itself, identify bonded <molecules>. In solid <sodium chloride>, grouping one <sodium> <cation> and one <chlorine> <anion> into an <atomic basis> does not create a collection of discrete NaCl <molecules>; the <ionic crystal> has an extended network of equivalent neighboring <ions>.
Back to article page
