Crystal lattice ($L$)

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 formThe 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.^[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.