Cubic Cell Calculator
Calculate atomic radius, atomic packing factor (APF), coordination number, and theoretical density for simple cubic (SC), body-centred cubic (BCC), face-centred cubic (FCC), and diamond cubic unit cells from the lattice parameter a. Furthermore, enter molar mass to get theoretical density.
How to use the Cubic Cell Calculator
SC: 1 atom/cell, APF=52.4%. BCC: 2 atoms/cell, APF=68%. FCC: 4 atoms/cell, APF=74%. Diamond: 8 atoms/cell, APF=34%. Furthermore, most metals adopt BCC or FCC; noble gases often BCC at low T.
Type a in Ångströms. Furthermore, common FCC metals: Cu 3.615 Å, Ni 3.524 Å, Al 4.046 Å, Au 4.078 Å. BCC metals: Fe 2.866 Å, W 3.165 Å, Mo 3.147 Å.
For theoretical density: type M in g/mol. Furthermore, density = n×M/(Nₐ×a³) — the "hard sphere" theoretical density based on perfect crystal packing.
Atomic radius, APF, coordination number, cell volume, and density appear. Moreover, the atomic radius formulas: SC: r=a/2; BCC: r=a√3/4; FCC: r=a√2/4; Diamond: r=a√3/8.
Theoretical density should closely match measured density for pure metals. Furthermore, significant differences indicate impurities, vacancies, or non-metallic bonding character.
Variants, options and when to use each
| Structure | Atoms/cell | APF | CN | Radius formula |
|---|---|---|---|---|
| SC | 1 | 52.4% | 6 | a/2 |
| BCC | 2 | 68.0% | 8 | a√3/4 |
| FCC | 4 | 74.0% | 12 | a√2/4 |
| Diamond | 8 | 34.0% | 4 | a√3/8 |
The formula explained
a = lattice parameter (Å)
n = atoms per unit cell (SC=1, BCC=2, FCC=4, diamond=8)
M = molar mass (g/mol)
Nₐ = 6.022×10²³ mol⁻¹
APF = 4nπr³/3 / a³ (packing efficiency)
In hard-sphere models, atoms touch along specific directions. Furthermore, in SC, atoms touch along the cell edge (face diagonal); in BCC, along the body diagonal; in FCC, along the face diagonal. Moreover, these contact conditions give the radius-lattice parameter relationships. The atomic packing factor APF = (volume of atoms)/(volume of cell) measures how efficiently space is filled.
Worked example — copper (FCC, a = 3.615 Å, M = 63.55 g/mol)
| Property | Calculation | Result |
|---|---|---|
| r = a√2/4 | 3.615×1.4142/4 | 1.278 Å |
| APF | 4×(4/3)π×1.278³/3.615³ | 74.05% |
| ρ = nM/(Nₐa³) | 4×63.55/(6.022×10²³×(3.615×10⁻⁸)³) | 8.933 g/cm³ |
| Experimental density | — | 8.960 g/cm³ |
What is a cubic unit cell in crystallography?
A unit cell is the smallest repeating unit of a crystal lattice. Furthermore, cubic unit cells have a = b = c and all angles 90°. The simple cubic (SC), body-centred cubic (BCC), and face-centred cubic (FCC) arrangements are the three cubic Bravais lattices. Diamond cubic is an FCC lattice with a two-atom basis (each "lattice point" has two atoms).Atomic packing factor (APF) measures efficiency of space usage in the unit cell. Moreover, FCC and HCP achieve 74.05% APF — the maximum theoretical packing for equal spheres (Kepler conjecture). BCC achieves 68.0% and SC only 52.4%. Most metals adopt FCC or BCC — FCC is preferred for ductile metals (Cu, Al, Ni, Au) due to its 12 close-packed slip systems.
The coordination number is the number of nearest neighbours. Additionally, FCC: 12 nearest neighbours (highest for equal spheres); BCC: 8; SC: 6; diamond: 4. Higher CN generally correlates with higher ductility and conductivity in metals.
Who uses this calculator?
Materials scientists and metallurgists calculate unit cell properties to determine atomic radii, density, and packing for crystal structure determination. Furthermore, semiconductor engineers work with diamond cubic silicon and FCC compound semiconductors (GaAs, InP). Ceramic scientists analyse oxide crystal structures for catalysts and electronic ceramics. Moreover, students use cubic cell calculations in introductory solid-state chemistry and materials science courses.
Historical context and related concepts
The cubic crystal system was identified by Haüy in the late 18th century from the geometric regularity of crystals. Furthermore, the X-ray determination of crystal structures began with Bragg's analysis of NaCl (1913) — confirming the FCC arrangement of Na⁺ and Cl⁻. The systematic classification of crystal structures and packing calculations developed throughout the early 20th century. Moreover, the Kepler conjecture — that FCC/HCP packing at 74% is optimal — was finally proved by Thomas Hales in 1998.
Why unit cell calculations are fundamental to materials design
Density prediction from crystal structure is used to verify phase purity and identify phase transformations. Furthermore, the BCC-to-FCC transformation in iron at 912°C (allotropic transformation) involves a density change that drives the characteristic volume contraction — important for steel heat treatment. Moreover, thin film deposition requires lattice parameter matching between substrate and film — mismatches > 1–2% cause defects and stress in semiconductor heterostructures.Unit cell calculations in battery electrode materials
Lithium-ion battery cathode materials (LiCoO₂, LiFePO₄, NMC) have layered or olivine crystal structures. Furthermore, the unit cell volume changes during lithium insertion/extraction — volume changes of > 5% cause mechanical cracking and capacity fade. Materials scientists design new cathode structures with minimal volume change upon cycling (0–2%). Moreover, the theoretical specific capacity = n_Li × F/M_cathode (mAh/g) where n_Li is lithium ions per formula unit from the unit cell — directly linking crystal chemistry to battery performance.
Frequently asked questions
Related tools
Miller Indices Calculator
d-spacing from cubic lattice parameter. Furthermore, unit cell parameter a is the key input for Miller indices d-spacing.
→Significant Figures Calculator
Round density to 4 significant figures. Furthermore, lattice parameters are reported to 4–5 decimal places in Å.
→Avogadro's Number Calculator
Nₐ in density formula. Furthermore, Avogadro's number connects molar to per-atom quantities.
→Lattice Energy Calculator
Lattice energy uses ionic radii. Moreover, atomic radius from cubic cell feeds into ionic radius estimation.
→Scientific Notation Converter
Express very small unit cell volumes in cm³. Moreover, a = 3.615 Å = 3.615×10⁻⁸ cm → a³ = 4.727×10⁻²³ cm³.
→