Cubic Unit Cell & Crystallography Calculator

Solid State Physics, Crystallography & Lattice Geometry Metrology

Cubic crystal systems are defined by orthogonal axes of equal length:

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• Lattice Parameter (a) & Geometry Formulas**:

• Simple Cubic (SC): a = 2r, N = 1 atom/cell, CN = 6, APF = (π) / (6) = 52.36\%

• Body-Centered Cubic (BCC): a = (4r) / (√(3)), N = 2 atoms/cell, CN = 8, APF = π√(3)8 = 68.02\%

• Face-Centered Cubic (FCC): a = 2√(2r), N = 4 atoms/cell, CN = 12, APF = π√(2)6 = 74.05\%

• Diamond Cubic: a = (8r) / (√(3)), N = 8 atoms/cell, CN = 4, APF = 34.01\%

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• Theoretical Density Formula**:

ρ = N_atoms × M_wN_A × V_cell = N_atoms × M_wN_A × (a × 10^-8 cm)^3

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• Standard Elemental Metal Benchmarks**:

• Aluminum: a = 405.03 pm ρ = 2.70 g/cm^3

• Copper: a = 361.47 pm ρ = 8.94 g/cm^3

• Alpha-Iron: a = 286.60 pm ρ = 7.88 g/cm^3

• Silicon: a = 543.17 pm ρ = 2.33 g/cm^3

How to Use This Calculator

Enter your Cubic Crystal Lattice Geometry — choose from Simple Cubic (SC: 1 atom/cell, a = 2r, CN = 6, APF = 52.4%), Body-Centered Cubic (BCC: 2 atoms/cell, a = 4r/√3, CN = 8, APF = 68.0%), Face-Centered Cubic (FCC: 4 atoms/cell, a = 2√2 r, CN = 12, APF = 74.0%), Atomic Radius (r in Picometers - pm) (e.g. 143.2 in pm), Element Molar Mass (g/mol) (e.g. 26.98 in g/mol), and Elemental Crystal Benchmark — choose from Custom Crystal Lattice, Aluminum Al (FCC, r=143.2 pm, Mw=26.98) → a=405.03 pm | APF: 74.05% | Density: 2.70 g/cm³, Copper Cu (FCC, r=127.8 pm, Mw=63.55) → a=361.47 pm | Density: 8.94 g/cm³. Results update instantly as you type — adjust any value to compare scenarios.

The Formula

Theoretical Density: rho = (N Mw) / (N_A V_cell_cm3)

a in cm = aEdgePm * 1e-10 cm

Worked Example

A materials engineer calculating the crystal density of Aluminum determines its lattice parameter is 405.03 pm (4.05 Å) and its theoretical density is 2.70 g/cm³—matching empirical lab measurements.

Understanding Your Result

Unit conversions are exact by definition within a given measurement system (e.g., 1 km = exactly 1,000 m). Cross-system conversions (metric to imperial) use defined exchange factors: 1 inch = exactly 2.54 cm; 1 pound = 0.45359237 kg. Round your converted result to match the precision of your original measurement — expressing more decimal places than your input justifies implies false accuracy. For regulatory, legal, or safety-critical applications, verify which measurement system is required by the applicable standard.