= Crystal lattice
{title2=$L$}
A <crystal lattice> records the translation symmetry of a periodic <crystal>. With a primitive basis of three linearly independent <lattice vectors> $a_1,a_2,a_3$, its translation vectors form
$$
L=\{n_1a_1+n_2a_2+n_3a_3:n_i\in\mathbb Z\}.
$$
The <crystal structure> additionally specifies the arrangement of atoms associated with each lattice point. A lattice point is therefore a symmetry marker, not necessarily one atom.<Matrices, Mappings and Crystallographic Symmetry by Hans Wondratschek>[Section 1.5, Definition D 1.5.2 and primitive bases]{ref}
For example, a one-dimensional chain with alternating species $A,B$ at spacing $a$ has translation period $2a$, not $a$: shifting by $a$ exchanges distinct species. It can be described by lattice $2a\mathbb Z$ with a two-atom basis at offsets zero and $a$. In a diffraction measurement, the translation lattice selects possible reciprocal-space peak positions, while the atomic basis affects their intensities and systematic absences. This distinction lets diffraction reveal internal structure rather than just the visible shape of the specimen.
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