צפיפות גבישית

A unit cell is the smallest repeating unit of a crystal lattice, defined by three edge lengths (a, b, c) and three angles (α, β, γ). It represents the fundamental building block of the crystal structure. Understanding the unit cell is essential for calculating crystal density because its dimensions and the number of atoms it contains directly determine the material's density.

The lattice type determines Z, the number of formula units per unit cell. Primitive cells have Z = 1, body-centered cubic cells have Z = 2, and face-centered cubic cells have Z = 4. In body-centered cubic lattices, one atom sits at the center plus corner atoms. In face-centered cubic lattices, atoms occupy corners and face centers, increasing the total count per unit cell.

Crystal density (ρ) is calculated using the formula: ρ = (Z × M) / (a × b × c × NA), where Z is the number of formula units per unit cell, M is molar mass, a, b, and c are edge lengths, and NA is Avogadro's number. If molar mass is in grams per mole and edge lengths in centimeters, density is expressed in grams per cubic centimeter.

In a face-centered cubic lattice, eight corner atoms each contribute 1/8 to the unit cell, and six face-centered atoms each contribute 1/2. The calculation is Z = (8 × 1/8) + (6 × 1/2) = 1 + 3 = 4. This higher Z value reflects the denser packing compared to primitive or body-centered cubic structures.

Crystal density depends on two primary factors: the number of particles inside the unit cell (Z) and the unit cell dimensions (edge lengths a, b, c). A larger Z value or smaller unit cell volume increases density. Materials with different lattice types or atomic arrangements will have different densities even if composed of the same element.

Each corner atom of a unit cell is shared by eight neighboring unit cells that meet at that corner. Since the atom is shared equally among all eight cells, it contributes only 1/8 of its mass to any single unit cell. This sharing principle is fundamental to accurately counting atoms when calculating Z and crystal density.

Crystal density is typically expressed in grams per milliliter (g/mL) or grams per cubic centimeter (g/cm³), since one milliliter equals one cubic centimeter. The specific units depend on the input units for molar mass and edge lengths. Using grams per mole and centimeters yields density in g/cm³.