Enter values to see results
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ų
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nm³
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×10⁻²⁴ cm³
Enter values to see results
—
ų
—
nm³
—
×10⁻²⁴ cm³
The Unit Cell Volume Calculator computes the volume of crystallographic unit cells for cubic, tetragonal, and orthorhombic crystal systems. The unit cell is the smallest repeating unit that, when translated in three dimensions, generates the entire crystal lattice. Understanding unit cell geometry is fundamental to crystallography, materials science, and solid-state chemistry.
Unit cell volume is essential for calculating crystal density, determining atomic packing efficiency, interpreting X-ray diffraction data, and computing many thermodynamic properties of crystalline materials. The seven crystal systems (cubic, tetragonal, orthorhombic, hexagonal, trigonal, monoclinic, triclinic) each have distinct relationships between lattice parameters and angles.
For orthogonal crystal systems (all angles = 90°), the unit cell volume is simply the product of the lattice parameters:
$$V = a \times b \times c$$
For the specific systems:
$$\text{Cubic: } V = a^3 \quad (a = b = c)$$
$$\text{Tetragonal: } V = a^2 c \quad (a = b \neq c)$$
$$\text{Orthorhombic: } V = abc \quad (a \neq b \neq c)$$
For the general triclinic case with angles α, β, γ:
$$V = abc\sqrt{1 - \cos^2\alpha - \cos^2\beta - \cos^2\gamma + 2\cos\alpha\cos\beta\cos\gamma}$$
The lattice parameter $$a$$ is typically measured in Ångströms (1 Å = 10⁻¹⁰ m = 10⁻⁸ cm). The volume in ų is converted to cm³ by multiplying by 10⁻²⁴. The number of atoms per unit cell (Z) depends on the lattice type: Z = 1 for simple, Z = 2 for body-centered, Z = 4 for face-centered.
Larger unit cell volumes indicate either larger atoms or more complex crystal structures. Silicon has a diamond cubic structure with a = 5.43 Å and Z = 8, giving V = 160.2 ų. Simple metals like copper (FCC, a = 3.61 Å, Z = 4) have V = 47.1 ų. The unit cell volume directly feeds into density calculations via ρ = ZM/(NAV), making it one of the most important crystallographic parameters.
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Results
V = 5.43³ = 160.1 ų, containing 8 silicon atoms per unit cell
Inputs
Results
V = 4.594² × 2.959 = 62.43 ų, with 2 formula units TiO₂ per cell
A unit cell is the smallest repeating structural unit of a crystal. When translated along its lattice vectors in all three dimensions, it generates the complete crystal structure. Unit cells are characterized by six parameters: three edge lengths (a, b, c) and three angles (α, β, γ).
The seven crystal systems are: cubic (a=b=c, α=β=γ=90°), tetragonal (a=b≠c, α=β=γ=90°), orthorhombic (a≠b≠c, α=β=γ=90°), hexagonal (a=b≠c, α=β=90°, γ=120°), trigonal/rhombohedral (a=b=c, α=β=γ≠90°), monoclinic (a≠b≠c, α=γ=90°, β≠90°), and triclinic (a≠b≠c, no angle restrictions).
It depends on the lattice type and basis. Simple cubic: 1, BCC: 2, FCC: 4, HCP: 2, diamond cubic: 8. Corner atoms contribute 1/8, edge atoms 1/4, face atoms 1/2, and body-centered atoms 1. For compounds, Z counts formula units, not individual atoms.
Unit cell volume is commonly expressed in cubic Ångströms (ų), where 1 Å = 10⁻¹⁰ m. For density calculations, convert to cm³ by multiplying by 10⁻²⁴. Some crystallographic software uses nm³ (1 nm³ = 1000 ų) or pm³.
Lattice parameters are determined by X-ray diffraction using Bragg's law. The positions of diffraction peaks give precise d-spacings, which are related to lattice parameters through the crystal system equations. Modern powder diffractometers can determine lattice parameters to ±0.001 Å accuracy.
Silicon has the diamond cubic structure, which consists of two interpenetrating FCC lattices offset by (a/4, a/4, a/4). Each FCC lattice contributes 4 atoms (8 corners × 1/8 + 6 faces × 1/2 = 4), so the total is 4 + 4 = 8 atoms per unit cell. Each Si atom is tetrahedrally coordinated to 4 neighbors.
Thermal expansion increases lattice parameters and therefore unit cell volume as temperature rises. The coefficient of thermal expansion α relates the change: ΔV/V = 3α×ΔT for cubic systems. Typical values are 10⁻⁵ to 10⁻⁶ per Kelvin for metals and ceramics, respectively. Precise crystallographic measurements must specify temperature.
There are 14 Bravais lattices, which are the distinct lattice types obtained by combining the 7 crystal systems with 4 centering types (primitive P, body-centered I, face-centered F, base-centered C). Not all combinations are unique: for example, face-centered tetragonal is equivalent to body-centered tetragonal with a smaller cell.
Crystal density is calculated from: ρ = ZM/(NA×V), where Z is atoms or formula units per cell, M is molar mass, NA is Avogadro's number, and V is cell volume. This allows density prediction from crystallographic data or, conversely, determination of Z from measured density and lattice parameters.
This calculator handles cubic, tetragonal, and orthorhombic systems (all angles 90°). For hexagonal systems, the volume formula is V = a²c×sin(60°) = a²c×(√3/2). For monoclinic: V = abc×sin(β). For triclinic, the full formula with all three angles is needed.
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