Last updated: July 3, 2026
Cubic Cell Calculator
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Quick Answer
The cubic cell calculator finds the lattice parameter a for simple cubic, body-centered cubic, or face-centered cubic crystals from atomic radius r. It uses a = 2r for SC, a = 4r/√3 for BCC, and a = 2√2r for FCC, then estimates density with ρ = ZM/(NAa³).
For a cubic crystal, simple cubic has edge length two times the atomic radius, body-centered cubic has four radius divided by square root of three, and face-centered cubic has two square root two times the radius.
Key Takeaways
- Simple cubic cells use a = 2r and contain Z = 1 atom per cell.
- Body-centered cubic cells use a = 4r/√3 and contain Z = 2 atoms per cell.
- Face-centered cubic cells use a = 2√2r and contain Z = 4 atoms per cell.
- Theoretical density is ρ = ZM/(NAa³) after converting a from Å to cm.
- Choose the radius type and crystal structure carefully because density depends on both.
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Formula
SC: a = 2r, Z = 1; BCC: a = 4r/√3, Z = 2; FCC: a = 2√2r, Z = 4; ρ = ZM/(NAa³)
Where:
- a=Cubic unit-cell edge length(Å (or cm for density))
- r=Atomic radius(Å)
- Z=Atoms per unit cell(dimensionless)
- M=Molar mass of the element(g/mol)
- NA=Avogadro constant(mol⁻¹)
- ρ=Theoretical density(g/cm³)
Worked Examples
BCC iron-like cell with r = 1.24 Å
Body-centered cubic geometry uses the body diagonal, so four radii span √3 times the edge length.
- 1Use the BCC relation a = 4r/√3 and Z = 2.
- 2Substitute r = 1.24 Å: a = 4 × 1.24 / √3 = 4.96 / 1.7320508.
- 3The edge length is a ≈ 2.864 Å; density uses a = 2.864 × 10⁻⁸ cm.
FCC gold-like cell with r = 1.43 Å
Face-centered cubic atoms touch along a face diagonal, giving a = 2√2r and Z = 4.
- 1Use the FCC relation a = 4r/√2 = 2√2r and Z = 4.
- 2Substitute r = 1.43 Å: a = 2.8284271 × 1.43.
- 3The edge length is a ≈ 4.045 Å, before density is evaluated from ZM/(NAa³).
Simple cubic cell with r = 1.50 Å
Simple cubic atoms touch along the cube edge, so the edge is exactly two radii.
- 1Use the simple cubic relation a = 2r and Z = 1.
- 2Substitute r = 1.50 Å: a = 2 × 1.50 Å.
- 3The edge length is a = 3.000 Å.
Introduction
A cubic unit cell is the repeating cube used to describe many crystalline metals and ionic solids. This cubic cell calculator converts an atomic radius into the unit-cell edge length for simple cubic (SC), body-centered cubic (BCC), or face-centered cubic (FCC) packing, then estimates ideal density from molar mass. It complements structure-focused tools such as the lattice energy calculator and composition tools such as the molar mass calculator. The geometry follows the hard-sphere crystal models summarized in LibreTexts solid-state chemistry and standard materials-science texts.
What is a cubic unit cell?
A unit cell is the smallest repeating block that reproduces a crystal by translation in three dimensions. In cubic systems, the three edge lengths are equal and all angles are 90°. The simple cubic, body-centered cubic, and face-centered cubic arrangements differ in where atoms sit inside that cube, which changes both the edge-length formula and the number of atoms counted per cell.
Simple cubic (SC) has atoms only at corners; eight corner eighths add to Z = 1.
Body-centered cubic (BCC) adds one atom at the cube center; Z = 2.
Face-centered cubic (FCC) adds atoms on the six faces; Z = 4.
These ideal models assume atoms touch along a specific line in the cube.
Edge length formulas for SC, BCC, and FCC
The geometry comes from the line along which hard-sphere atoms touch. In SC, contact is along an edge, so a = 2r. In BCC, contact is along the body diagonal, whose length is √3a, and four radii fit on that diagonal, so a = 4r/√3. In FCC, contact is along a face diagonal, whose length is √2a, and four radii fit there, so a = 4r/√2 = 2√2r.
Use the same length unit for a and r. This calculator accepts r in Å and returns a in Å.
How theoretical density is calculated
Theoretical density divides the mass of atoms in one cell by the cell volume. The mass per cell is Z × M / NA, where M is molar mass and NA is Avogadro's constant. The volume is a³. Because the density output is in g/cm³, the calculator converts a from angstroms to centimetres using 1 Å = 10⁻⁸ cm before cubing. The constants align with the NIST value for the Avogadro constant.
If the density seems too high or low, check that the radius is metallic/atomic radius rather than ionic radius for a different coordination environment.
Common cubic structures and reference values
Many metals adopt BCC or FCC structures near room temperature. The table gives representative hard-sphere examples; real lattice parameters vary with temperature, alloying, defects, and measurement method.
| Structure | Touching direction | Atoms per cell | Example materials |
|---|---|---|---|
| SC | Cube edge | 1 | α-polonium (classic example) |
| BCC | Body diagonal | 2 | Fe (α), Cr, W, Mo |
| FCC | Face diagonal | 4 | Al, Cu, Ni, Ag, Au |
| Diamond cubic | Tetrahedral network | 8 | C, Si, Ge; not covered by this SC/BCC/FCC calculator |
How to use this cubic cell calculator
Enter the atomic radius in Å, choose SC, BCC, or FCC, and enter the molar mass if you want density. The primary answer is edge length a. Secondary outputs show Z and ideal density. For mass-based chemistry calculations, pair the density with the atomic mass calculator or the molar mass calculator to verify M.
Use SC when atoms touch along the edge.
Use BCC when atoms touch along the body diagonal.
Use FCC when atoms touch along the face diagonal.
Use molar mass in g/mol and leave radius in Å.
Limitations and best practices
The model is an ideal hard-sphere calculation. It does not include thermal expansion, vacancies, interstitial atoms, alloy composition, non-cubic distortions, or polymorphic phase changes. For precise crystallography, compare against measured lattice parameters from X-ray or neutron diffraction databases such as the Crystallography Open Database and use the calculator as a transparent first check.
Report temperature when comparing density or lattice constants.
Do not use this SC/BCC/FCC tool for hexagonal close packed or diamond cubic structures.
Use consistent radius definitions: metallic, covalent, and ionic radii are not interchangeable.
Round final results to match the precision of the radius and molar mass inputs.
Quick Reference Card
Cubic Unit Cell — Quick Reference
Quick reference • Cubic Cell Calculator
SC: a = 2r; BCC: a = 4r/√3; FCC: a = 2√2r; ρ = ZM/(NAa³)Valid range: Use positive atomic radii and molar masses; typical metallic radii are about 1–2 Å.
Common Values
⚠ Watch Out
- •Do not mix atomic radius definitions; metallic, ionic, and covalent radii can differ.
- •Convert edge length to centimetres before computing density in g/cm³.
- •Do not apply SC/BCC/FCC formulas to HCP, diamond cubic, or tetragonal crystals.
- •Real samples can deviate from theoretical density because of vacancies, impurities, and porosity.
Pro Tips
- →For BCC examples, check that 4r is approximately √3a.
- →For FCC examples, check that 4r is approximately √2a.
- →Use measured lattice constants when available, then compare the implied radius.
- →Keep at least three significant figures in radius before cubing a for density.
FAQs
What does the cubic cell calculator compute?
It computes the cubic unit-cell edge length a from atomic radius r for SC, BCC, or FCC structures. It also reports atoms per cell Z and theoretical density when molar mass is supplied.
Why is BCC edge length a = 4r/√3?
In a BCC cell, atoms touch along the body diagonal. The body diagonal is √3a long and contains four atomic radii from one corner atom through the body-center atom to the opposite corner, so √3a = 4r.
Why is FCC edge length a = 2√2r?
In an FCC cell, atoms touch along a face diagonal. The face diagonal is √2a and contains four radii, so √2a = 4r and a = 4r/√2 = 2√2r.
How many atoms are in SC, BCC, and FCC unit cells?
Simple cubic has Z = 1, body-centered cubic has Z = 2, and face-centered cubic has Z = 4. These are effective counts after sharing corner and face atoms with neighbouring cells.
Why must edge length be converted to centimetres for density?
Density is reported in g/cm³. Since the edge length is computed in Å, the calculator converts a to cm using 1 Å = 10⁻⁸ cm before calculating the volume a³.
Can I use this calculator for HCP or diamond cubic crystals?
No. HCP and diamond cubic structures use different geometry and atoms-per-cell relationships. This tool is specifically for simple cubic, BCC, and FCC hard-sphere models.