Geometry and material structure
A lattice abstracts repeated geometric relationships into points related by translations. A full crystal description also specifies the atoms, ions or molecules associated with those points. It is not a photograph showing identical frozen spheres, and a regular external shape alone does not determine internal structure.
References: [1]
Using a unit cell
A unit cell is a spatial unit whose repetition builds the structure; its choice is not always unique. Counting particles requires accounting for shared corners, faces and edges. Different arrangements change coordination and density, allowing identical elemental or chemical compositions to exhibit different structures and properties.
References: [1]
Structural evidence from diffraction
Scattering from periodic structures produces characteristic distributions. X-ray diffraction measurements constrain spacings and arrangements when compared with structural models and instrument parameters. They do not display every aspect of atomic motion directly. Lattice descriptions link geometric regularity with experimentally testable properties of materials.
References: [1]
Ideal periodicity and real materials
Real materials contain thermal motion, impurities and defects and may comprise grains with different orientations. Controlled semiconductor doping shows that departures from ideal repetition can be useful. An ideal lattice is a starting model; actual performance also depends on electronic states, real structure and processing history.