Structures of Solids

Crystalline solids have their constituent particles arranged in a repeating three-dimensional pattern, like quartz, while amorphous solids like fused silica glass lack an ordered internal arrangement. Crystalline solids have precise melting temperatures because atoms are held with uniform forces, whereas amorphous solids soften gradually over a temperature range as different intermolecular attractions break sequentially.

A unit cell is the smallest repeating unit of a crystalline structure that retains the symmetry of the entire solid. It is defined by three axis lengths (a, b, c) and three angles (α, β, γ) between them. The entire crystal lattice is composed of unit cells repeating in three dimensions, making it the fundamental building block for understanding crystalline structure.

There are seven lattice systems classified by their unit cell dimensions and angles: cubic, tetragonal, orthorhombic, rhombohedral, monoclinic, triclinic, and hexagonal. For example, cubic systems have equal axes (a = b = c) and all angles at 90°, while triclinic systems have unequal axes and angles all different from 90°.

Lattice points represent the locations of atoms or ions within a crystal structure, and may be positioned at corners, on faces, or at the center of a unit cell. Lattice vectors delineate the edges of the unit cell and define its three-dimensional boundaries. Together, they form the crystal lattice that describes the overall geometric arrangement of the solid.

Packing efficiency reflects the amount of a solid's volume occupied by atoms rather than empty space. A higher number of atoms in a unit cell generally corresponds to more efficient packing. Partial atoms are counted by assigning corner atoms as one-eighth and face atoms as one-half of an atom to determine the total atomic content.

The motif is the pattern of atoms in a unit cell, defined by the locations of atoms relative to a given lattice point. While lattice points mark geometric positions in the crystal structure, the motif specifies where actual atoms are positioned within those points. The positions of atoms are not necessarily identical to lattice point positions.

Atoms on unit cell corners are counted as one-eighth of an atom, while atoms on faces are counted as one-half. Alternatively, if a corner atom is assigned to the unit cell, the other seven are ignored; if a face atom is assigned, the other is ignored. This accounting method ensures accurate determination of total atoms per unit cell.