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Cubic Cell Calculator

Ready to calculate
SC · BCC · FCC Lattices.
5 Length Units.
16 Reference Metals.
100% Free.
No Data Stored.

How it Works

01Pick Lattice Type

Simple Cubic (SC), Body-Centered Cubic (BCC), or Face-Centered Cubic (FCC) — each has a distinct geometry

02Enter Atomic Radius

Metal atomic radius in Å, pm, nm, μm, or mm — typical values 100-200 pm for metals

03Apply Geometric Relation

SC: a=2r · BCC: a=4r/√3 · FCC: a=2r√2 — derived from the close-packing geometry of each lattice

04Read All Cell Properties

Lattice parameter, cell volume, atomic packing factor, atoms per cell (Z), coordination number — plus 16 reference metals

What is a Cubic Cell Calculator?

The cubic crystal cell is the workhorse geometry of solid-state chemistry — three of the 14 Bravais lattices are cubic, and the vast majority of metallic elements crystallize in one of them: Simple Cubic (SC), Body-Centered Cubic (BCC), or Face-Centered Cubic (FCC). Each type has a precise geometric relationship between the atomic radius (r) of the constituent atom and the lattice parameter (a) — the edge length of the unit cell — derived from the close-packing geometry. Our Cubic Cell Calculator implements all three relationships: SC: a = 2r (atoms touch along edge); BCC: a = 4r/√3 (atoms touch along body diagonal); FCC: a = 2r√2 (atoms touch along face diagonal). For each, the calculator reports the lattice parameter, cell volume V = a³, atomic packing factor (APF — fraction of cell volume actually occupied by atoms), atoms per unit cell (Z), and coordination number, plus matches your input against 16 reference metals from Fe to Au.

Just pick the lattice type from the radio selector (the on-screen SVG cube illustration adapts to show corner atoms only for SC, the body-center atom for BCC, or all six face atoms for FCC), then enter the atomic radius in any of five length units (Å, pm, nm, μm, mm). The calculator instantly returns the lattice parameter and all derived properties. The atomic packing factor — the most important quantity in crystal chemistry — quantifies how efficiently spheres fill space: SC = π/6 ≈ 52.4% (lots of empty space, rare in nature); BCC = π√3/8 ≈ 68.0% (typical for high-strength metals like Fe, W, Cr); FCC = π/(3√2) ≈ 74.05% (the maximum possible for identical spheres, proven by Kepler's conjecture and Hales' 2014 proof — typical for ductile metals like Au, Cu, Al, Ag).

Designed for materials chemistry students learning crystallography for the first time, mechanical engineering students predicting metal properties from crystal structure, X-ray-diffraction users computing expected reflection positions, ceramic and metallurgy researchers analyzing close-packed arrangements, and physical chemistry students preparing for the GRE or qualifying exams, the tool runs entirely in your browser — no data is stored or transmitted.

Pro Tip: Pair this with our Lattice Constant Calculator for the inverse problem (compute r from measured a), or our Miller Indices Calculator for crystallographic plane analysis.

How to Use the Cubic Cell Calculator?

Pick the Lattice Type: Choose Simple Cubic (SC), Body-Centered Cubic (BCC), or Face-Centered Cubic (FCC). The on-screen SVG illustration updates to show the atom positions: SC has 8 corner atoms only; BCC adds 1 atom in the cube center; FCC adds 6 atoms in the face centers (one per face).
Enter Atomic Radius (r): The radius of the constituent atom in your crystal. Typical metallic radii range from ~50 pm (small atoms) to ~250 pm (large alkali metals). Common values: Fe = 124 pm, Cu = 128 pm, Au = 144 pm, Al = 143 pm, K = 227 pm.
Pick a Length Unit: Five options — ångström (Å, the crystallographic standard), picometers (pm, the modern SI form), nanometers (nm, for nanoscale work), micrometers, or millimeters. The calculator normalizes everything to meters internally.
Press Calculate: Get the lattice parameter (cell edge length) a = K_a × r, where K_a depends on lattice type: 2 for SC, 4/√3 ≈ 2.309 for BCC, 2√2 ≈ 2.828 for FCC. Cell volume V = a³ follows immediately.
Read All Cell Properties: Lattice parameter (your chosen unit + pm reference); cell volume in m³ and ų; atoms per cell (Z): SC=1, BCC=2, FCC=4; coordination number (nearest neighbors): SC=6, BCC=8, FCC=12; Atomic Packing Factor (APF) as percentage; closest-radius matches from 16 reference metals.

How is the cubic cell parameter calculated?

The cubic-cell relationships fall out of pure geometry — the constraint that nearest-neighbor atoms touch along a specific direction in each lattice. Once you know the geometry, every property of the unit cell follows from atom-counting and Pythagoras. Here's the complete derivation:

Auguste Bravais classified the 14 possible 3D space lattices in 1848. The three cubic ones — SC, BCC, FCC — together account for the vast majority of metallic crystal structures observed in nature, plus many ionic and molecular crystals (NaCl, diamond, silicon, etc.).

1. Simple Cubic (SC) — a = 2r

Atoms only at the 8 cube corners. Adjacent corner atoms touch along the cube edge (length a). Since each atom has radius r, two atoms touching span 2r along the edge:

a = 2r

Atoms per cell: each corner shared by 8 adjacent cells → 8 × (1/8) = Z = 1 atom/cell. Coordination = 6 (each atom touches its 6 ortho-axial neighbors).

APF = (Z × volume per atom) / (cell volume) = (1 × 4πr³/3) / (2r)³ = (4π/3) / 8 = π/6 ≈ 0.5236 = 52.36%. About half the cell is empty space — too inefficient to be common in nature.

2. Body-Centered Cubic (BCC) — a = 4r/√3

Atoms at the 8 corners + 1 atom in the cube center. Now atoms touch along the body diagonal (the line from one corner through the center to the opposite corner), not along the edge. The body diagonal of a cube of side a has length a√3 (Pythagoras: √(a² + a² + a²) = a√3). Three atoms touch along this body diagonal — corner + body-center + opposite corner — spanning 4r:

a√3 = 4r → a = 4r/√3 ≈ 2.309·r

Atoms per cell: 8 corners × 1/8 + 1 body atom = Z = 2 atoms/cell. Coordination = 8 (each atom touches 8 nearest neighbors).

APF = (2 × 4πr³/3) / (4r/√3)³ = (8πr³/3) / (64r³/3√3) = π√3/8 = 0.6802 = 68.02%. Better than SC but still not maximal. Typical of high-strength metals: Fe (α), W, Cr, V, Mo, alkali metals.

3. Face-Centered Cubic (FCC) — a = 2r√2

Atoms at the 8 corners + 1 atom in each of the 6 cube faces. Atoms touch along the face diagonal (the diagonal across a single face), which has length a√2 (Pythagoras: √(a² + a²) = a√2). Three atoms touch along this face diagonal — corner + face-center + opposite corner — spanning 4r:

a√2 = 4r → a = 4r/√2 = 2r√2 ≈ 2.828·r

Atoms per cell: 8 corners × 1/8 + 6 faces × 1/2 = 1 + 3 = Z = 4 atoms/cell. Coordination = 12 (each atom touches 12 nearest neighbors — the absolute maximum possible for spheres).

APF = (4 × 4πr³/3) / (2r√2)³ = (16πr³/3) / (16√2·r³) = π/(3√2) = 0.7405 = 74.05%. The MAXIMUM possible packing of identical spheres — proven by Kepler's conjecture (1611), finalized by Thomas Hales in 2014 after a 16-year computer-assisted proof. Hexagonal Close Packed (HCP) achieves the same 74.05% but isn't cubic.

Density from Lattice Data

For any cubic crystal:

ρ = (Z × M) / (NA × V) = (Z × M) / (NA × a³)

where ρ is density (g/cm³), Z is atoms per cell, M is molar mass (g/mol), N_A = 6.022 × 10²³ mol⁻¹, and V = a³ in cm³. For Cu (M = 63.55 g/mol, FCC, Z = 4, a = 361.5 pm = 3.615 × 10⁻⁸ cm): ρ = (4 × 63.55) / (6.022 × 10²³ × 4.725 × 10⁻²³) = 254.2 / 28.46 = 8.93 g/cm³ — matching the textbook density of copper exactly.

Coordination Number Geometry

The number of nearest-neighbor atoms each atom touches:

  • SC (CN=6): 6 atoms in ±x, ±y, ±z directions at distance a = 2r.
  • BCC (CN=8): 8 atoms — body-center has 8 corners around it; corner has 8 body-centers in the 8 surrounding cubes. All at distance a√3/2 = 2r.
  • FCC (CN=12): 12 atoms in the 12 close-packed positions. Face-center has 4 corners + 4 same-layer face-centers + 4 face-centers of adjacent cells. All at distance a√2/2 = 2r.

Why Some Metals Are Hard, Others Ductile

FCC metals (Cu, Au, Ag, Al, Pt, Ni) are typically ductile — many slip systems (12 active per atom) allow plastic deformation without fracture. BCC metals (Fe, W, Cr, V) are typically stronger but less ductile — fewer active slip systems at room T, harder to deform. The crystal structure is the foundation of macroscopic mechanical properties; understanding cubic-cell geometry is the first step in metallurgy.

Real-World Example

Cubic Cell Calculator – Worked Examples

Example 1 — Iron (BCC at room temperature). Atomic radius r = 124 pm. Compute lattice parameter and APF.
  • Lattice type: BCC (α-Fe at room temperature).
  • a = 4r/√3 = 4 × 124 / √3 = 496 / 1.732 = 286.4 pm. Textbook a = 286.6 pm — matches within rounding.
  • Cell volume V = a³ = (286.4 pm)³ = 2.349 × 10⁷ pm³ = 2.349 × 10⁻²³ cm³.
  • Atoms per cell Z = 2.
  • APF = π√3/8 = 0.6802 = 68.02%. About 32% empty space — interstitial sites for C diffusion (steel hardening!).
  • Density: ρ = 2 × 55.85 / (6.022×10²³ × 2.349×10⁻²³) = 111.7 / 14.15 = 7.89 g/cm³ — matches textbook density of iron (7.87).

Example 2 — Copper (FCC, the most common close-packed metal). r = 128 pm.

  • Lattice: FCC.
  • a = 2r√2 = 2 × 128 × 1.414 = 362.0 pm. Textbook 361.5 pm — excellent agreement.
  • V = a³ = 4.745 × 10⁷ pm³ = 4.745 × 10⁻²³ cm³.
  • Z = 4 atoms.
  • APF = π/(3√2) = 74.05%. Maximum possible for identical spheres.
  • Density: ρ = 4 × 63.55 / (6.022×10²³ × 4.745×10⁻²³) = 254.2 / 28.57 = 8.90 g/cm³. Textbook 8.96. ✓
  • Each Cu atom has 12 nearest neighbors at distance a√2/2 = 256 pm. The high coordination + many slip systems → Cu is highly ductile (drawn into wire).

Example 3 — Gold (FCC, identical structure to Cu, just larger atom). r = 144 pm.

  • a = 2 × 144 × √2 = 407.3 pm. Textbook 407.8 pm. ✓
  • V = 6.755 × 10⁻²³ cm³; Z = 4; APF = 74.05%.
  • Density: ρ = 4 × 196.97 / (6.022×10²³ × 6.755×10⁻²³) = 787.9 / 40.67 = 19.37 g/cm³. Textbook 19.32. ✓
  • The dramatic density vs Cu comes from gold's heavier atomic mass (197 vs 63.5) more than from differing cell volumes.

Example 4 — Tungsten (BCC, highest melting point of all metals). r = 137 pm.

  • a = 4r/√3 = 4 × 137 / 1.732 = 316.4 pm. Textbook 316.5 pm. ✓
  • V = 3.166 × 10⁻²³ cm³; Z = 2; APF = 68.02%.
  • Density: ρ = 2 × 183.84 / (6.022×10²³ × 3.166×10⁻²³) = 367.7 / 19.07 = 19.28 g/cm³. Textbook 19.30. ✓
  • Tungsten's very high density combined with very strong metallic bonds (W-W bond order ~3 from sextuple-bond theory) → melting point 3422 °C, the highest of any metal.

Example 5 — α-Polonium (the only Simple Cubic element at STP). r = 168 pm.

  • Lattice: SC (unique among elements at room T).
  • a = 2r = 2 × 168 = 336 pm. Textbook 336.6 pm. ✓
  • V = a³ = 3.795 × 10⁻²³ cm³; Z = 1.
  • APF = π/6 = 52.36% — the lowest packing of any cubic structure. Why does Po do this? Theorized to be due to relativistic effects in the heavy 6p electrons.
  • Density: ρ = 1 × 209 / (6.022×10²³ × 3.795×10⁻²³) = 209 / 22.85 = 9.15 g/cm³. Textbook 9.20. ✓
  • SC is so rare in nature that polonium remains a textbook curiosity. All other elements settle into BCC, FCC, HCP, or more complex structures.

Who Should Use the Cubic Cell Calculator?

1
Materials Chemistry Students: Solve textbook crystal-structure problems; understand the geometric basis of metallic, ionic, and molecular crystals; predict density from lattice data.
2
Mechanical Engineering Students: Connect crystal structure to mechanical behavior — FCC ductile (12 slip systems), BCC strong (high yield strength), SC brittle (rare in nature).
3
X-Ray Crystallographers: Compute expected lattice parameters from atomic radii to validate XRD measurements; predict reflection positions from a in Bragg's law.
4
Metallurgists: Design steel grades (BCC ferrite vs FCC austenite) by understanding lattice transformations and interstitial-carbon solubility (FCC dissolves more C than BCC).
5
Solid-State Physicists: Compute Brillouin zones, density of states, and band structures starting from the unit-cell geometry.
6
Ceramic / Mineralogy Researchers: Analyze rock-salt (NaCl), zincblende (ZnS), fluorite (CaF₂), and perovskite (CaTiO₃) structures all built on cubic frameworks.

Technical Reference

Bravais Lattices. Auguste Bravais classified the 14 possible 3D space lattices in his 1848 paper. The cubic system has 3 of these (SC, BCC, FCC); other systems include hexagonal (1), trigonal (1), tetragonal (2), orthorhombic (4), monoclinic (2), triclinic (1). For metals specifically, the most common structures are FCC, BCC, and HCP (hexagonal close packed) — together accounting for ~90% of all elemental metals.

Key Cubic-Lattice Relationships:

  • Simple Cubic (SC): a = 2r; Z = 1; APF = π/6 = 52.36%; CN = 6; nearest-neighbor distance = a = 2r.
  • Body-Centered Cubic (BCC): a = 4r/√3 = 2.309r; Z = 2; APF = π√3/8 = 68.02%; CN = 8; NN distance = a√3/2 = 2r.
  • Face-Centered Cubic (FCC): a = 2r√2 = 2.828r; Z = 4; APF = π/(3√2) = 74.05%; CN = 12; NN distance = a√2/2 = 2r.

Reference Metal Lattice Data (CRC Handbook):

  • FCC metals: Aluminum (a = 404.6 pm), Copper (361.5), Gold (407.8), Silver (408.6), Platinum (392.4), Nickel (352.4), Lead (495.0), γ-Iron austenite (359.2), Calcium (558.4)
  • BCC metals: α-Iron ferrite (286.6 pm), Tungsten (316.5), Chromium (288.4), Vanadium (303.0), Molybdenum (314.7), Sodium (429.1), Potassium (532.8), Lithium (350.9), Niobium (330.0)
  • SC metals: α-Polonium (336.6 pm) — the only one at standard conditions
  • HCP metals: Magnesium, Zinc, Cadmium, Titanium (α), Cobalt — same 74.05% APF as FCC but hexagonal symmetry

Density from Lattice Parameter. For any cubic crystal: ρ (g/cm³) = (Z × M) / (N_A × a³), where M is molar mass (g/mol), N_A = 6.022 × 10²³ mol⁻¹, and a is in cm. Convert pm → cm: divide by 10¹⁰. This formula bridges atomic-scale geometry and macroscopic density and is one of the most important calculations in solid-state chemistry.

Phase Transformations in Iron. Iron is the canonical example of polymorphism: α-Fe (BCC, ferrite) stable below 912 °C, density 7.87 g/cm³; γ-Fe (FCC, austenite) stable 912-1394 °C, density 7.95 g/cm³ (more close-packed); δ-Fe (BCC again) stable 1394-1538 °C. The α↔γ transformation at 912 °C is the foundation of steel heat treatment — austenite dissolves up to 2.0% carbon (interstitial in larger FCC sites), then on quenching forms martensite (distorted BCC) trapping the carbon, producing hardened steel.

Kepler's Conjecture (Sphere-Packing Maximum). Johannes Kepler conjectured in 1611 that no arrangement of identical spheres has higher density than FCC and HCP (both at π/(3√2) = 74.05%). Thomas Hales proved this in 1998 with a computer-assisted argument; the formal proof was published in Annals of Mathematics in 2005, and the verification was completed in 2014 (Project Flyspeck). FCC and HCP are not the only optimal packings — there are infinitely many close-packed sequences with the same density, all built from layered hexagonal sheets stacked in different orders.

Coordination Number and Mechanical Properties.

  • FCC (CN=12): 12 close-packed slip systems → highly ductile (Cu, Al, Au, Ag, Pb). Drawn into wires, beaten into foils.
  • BCC (CN=8): 48 slip systems but harder to activate at room T → strong but brittle below DBTT (Fe, W, Cr). Below the ductile-to-brittle transition temperature, BCC metals fail catastrophically (RMS Titanic hull steel failed this way at low ocean temperatures).
  • HCP (CN=12 but only ~3 active slip systems at room T): intermediate (Mg, Zn, Ti). Anisotropic — much weaker in some directions than others.

Beyond Pure Metals. Cubic structures also describe many ionic and molecular crystals: NaCl (rock salt): two interpenetrating FCC lattices (Na⁺ + Cl⁻), CN = 6. CsCl: two interpenetrating SC lattices, CN = 8. ZnS (zincblende): two interpenetrating FCC lattices, CN = 4 (tetrahedral). CaF₂ (fluorite): Ca FCC + F in tetrahedral holes. Perovskite (CaTiO₃): ABX₃ cubic with Ca at corners, Ti at body center, O at face centers. The cubic-cell geometry calculations apply with appropriate adaptations.

Key Takeaways

Cubic cell geometry distills crystallography down to three relationships: SC: a = 2r (52.4% APF, Z=1, CN=6 — rare in nature); BCC: a = 4r/√3 (68.0% APF, Z=2, CN=8 — Fe, W, Cr, alkali metals); FCC: a = 2r√2 (74.05% APF, Z=4, CN=12 — Cu, Au, Al, Ag, Pt — the absolute maximum possible packing of identical spheres). The 74.05% maximum was conjectured by Kepler in 1611 and proven by Hales in 2014 after a 16-year computer-assisted proof. Density follows from ρ = ZM/(N_A a³) — combining lattice parameter with atomic mass gives the bulk density of the metal. Use the ToolsACE Cubic Cell Calculator with the three lattice types (with adaptive SVG illustrations), 5 length units, all derived properties, and 16-metal reference matching. Bookmark it for materials chemistry, metallurgy, X-ray crystallography, and any time you need to translate atomic radius into a complete crystallographic description.

Frequently Asked Questions

What is the Cubic Cell Calculator?
It computes the lattice parameter (a) and all derived properties (cell volume V, atomic packing factor APF, atoms per cell Z, coordination number CN) for the three Bravais cubic lattices — Simple Cubic (SC), Body-Centered Cubic (BCC), and Face-Centered Cubic (FCC) — given the atomic radius r. Inputs: lattice type (radio selector with adaptive SVG illustration) and atomic radius (5 length units). Output: lattice parameter, cell volume in m³ and ų, APF as percentage, atoms per cell, coordination number, a/r ratio, and closest-radius matches against 16 reference metals.

Designed for materials chemistry students, mechanical engineers studying metal mechanical properties, X-ray crystallographers, metallurgists, solid-state physicists, and ceramic/mineralogy researchers. Runs entirely in your browser — no data stored.

Pro Tip: Use our Lattice Constant Calculator for the inverse problem.

What's the formula for each cubic lattice?
Simple Cubic (SC): a = 2r (atoms touch along edge). Body-Centered Cubic (BCC): a = 4r/√3 ≈ 2.309r (atoms touch along body diagonal a√3 = 4r). Face-Centered Cubic (FCC): a = 2r√2 ≈ 2.828r (atoms touch along face diagonal a√2 = 4r). Each formula comes from the geometric constraint that nearest-neighbor atoms touch along a specific direction in each lattice.
What is the Atomic Packing Factor (APF)?
The fraction of unit-cell volume occupied by atoms (treated as hard spheres). APF = (Z × volume per atom) / (cell volume) = (Z × 4πr³/3) / a³. Values: SC = π/6 = 52.36% (lots of empty space, rare in nature); BCC = π√3/8 = 68.02% (typical for high-strength metals); FCC = π/(3√2) = 74.05% (maximum possible for identical spheres, proven by Kepler's conjecture). The remaining 26-48% is empty space — interstitial sites that determine defect chemistry, diffusion rates, and dopant solubility.
How many atoms are in each unit cell?
SC: 8 corners × 1/8 (each corner shared by 8 cells) = 1 atom/cell. BCC: 8 corners × 1/8 + 1 body atom = 2 atoms/cell. FCC: 8 corners × 1/8 + 6 faces × 1/2 (each face shared by 2 cells) = 1 + 3 = 4 atoms/cell. The atom-counting rule: corner atoms count 1/8, edge atoms 1/4, face atoms 1/2, body atoms 1.
Why is FCC the most efficient packing?
Because it's the close-packed structure — atoms in each layer are arranged hexagonally (each touches 6 others in the same layer), and successive layers stack in the close-packed sequence ABCABC... that maximizes 3D density. Each atom touches 12 nearest neighbors (the maximum possible for spheres in 3D), and 74.05% of space is filled. Kepler's conjecture (1611): no arrangement of identical spheres can exceed 74.05% packing density. Thomas Hales proved this in 1998 with a 250-page computer-assisted argument; formal verification completed 2014 (Project Flyspeck). HCP achieves the same 74.05% but with hexagonal (not cubic) symmetry.
What's the coordination number in each lattice?
Coordination number = number of nearest-neighbor atoms each atom touches. SC: CN = 6 (atoms in ±x, ±y, ±z directions). BCC: CN = 8 (the body-center has 8 corners around it; symmetric for corner atoms). FCC: CN = 12 (6 in same close-packed layer + 3 above + 3 below — the maximum possible for identical spheres). Higher CN generally correlates with greater ductility — FCC metals (Cu, Al, Au) are easy to deform; SC structures are rare and brittle.
How do I compute density from lattice parameter?
ρ = (Z × M) / (N_A × a³), where ρ is density (g/cm³), Z is atoms per cell, M is molar mass (g/mol), N_A = 6.022 × 10²³ mol⁻¹, and a is in cm (convert from pm by dividing by 10¹⁰). Example for Cu (FCC, Z=4, M=63.55, a=361.5 pm = 3.615×10⁻⁸ cm): ρ = (4 × 63.55) / (6.022×10²³ × 4.725×10⁻²³) = 8.93 g/cm³ — matching the textbook density of copper to 0.5%.
Which metals have which structure?
FCC (most ductile metals): Aluminum, Copper, Gold, Silver, Platinum, Nickel, Lead, γ-Iron (austenite), Calcium. BCC (high-strength metals): α-Iron (ferrite), Tungsten, Chromium, Vanadium, Molybdenum, Sodium, Potassium, Lithium, Niobium. SC: α-Polonium only (a quirky relativistic-effects case). HCP (hexagonal, not cubic but same 74% APF): Magnesium, Zinc, Cadmium, Titanium (α), Cobalt. Iron famously transitions from BCC (α) to FCC (γ) at 912 °C — the foundation of steel heat treatment.
Why is α-polonium the only SC element?
Polonium's anomalous SC structure (rather than the typical close-packed BCC/FCC/HCP) is attributed to relativistic effects in the heavy 6p electrons. The 6p orbitals undergo relativistic spin-orbit splitting that destabilizes close-packed arrangements relative to SC. This is one of the few cases where relativistic chemistry dominates the crystal structure of an element. Above 36 °C, polonium converts to a slightly less symmetric β-form (rhombohedral). All other elements at standard conditions adopt FCC, BCC, HCP, or more complex (often layered) structures because those give better packing energies.
How does crystal structure affect mechanical properties?
FCC metals (Al, Cu, Au, Ag) are typically ductile because they have 12 active slip systems at room temperature — easy to deform plastically without fracture. Drawn into wires, beaten into foils. BCC metals (Fe, W, Cr) are typically strong but less ductile at room T because slip is harder to activate; they're the workhorses of structural engineering (steel, tungsten lamp filaments). Below the ductile-to-brittle transition temperature (DBTT), BCC metals fail catastrophically — this is what doomed the RMS Titanic's BCC steel hull in icy water. HCP metals (Mg, Ti, Zn) are intermediate — anisotropic mechanical properties because slip occurs preferentially in basal planes.
How is this related to ionic crystals like NaCl?
Many ionic and molecular crystals are built on cubic frameworks. NaCl (rock salt): two interpenetrating FCC lattices — Na⁺ at FCC positions, Cl⁻ at FCC positions offset by a/2. CN = 6 for each ion. CsCl: two interpenetrating SC lattices, CN = 8 for each ion. ZnS (zincblende, also diamond and silicon): two interpenetrating FCC lattices but offset by (1/4, 1/4, 1/4), giving tetrahedral coordination CN = 4. Fluorite (CaF₂): Ca²⁺ FCC + F⁻ in tetrahedral holes. The cubic-cell calculator works for these structures with appropriate atom-counting and effective radius interpretations.

Author Spotlight

The ToolsACE Team - ToolsACE.io Team

The ToolsACE Team

Our chemistry tools team implements the geometric relationships of the three Bravais cubic lattices that underpin all of crystallography and solid-state chemistry. <strong>Simple Cubic (SC):</strong> a = 2r, Z = 1 atom/cell, APF = π/6 ≈ 52.4%. <strong>Body-Centered Cubic (BCC):</strong> a = 4r/√3, Z = 2, APF = π√3/8 ≈ 68.0%. <strong>Face-Centered Cubic (FCC):</strong> a = 2r√2, Z = 4, APF = π/(3√2) ≈ 74.05% (the maximum possible for identical spheres — proven by Kepler's conjecture, finalized by Hales 2014). The calculator handles five length units (Å, pm, nm, μm, mm) for atomic radius input, displays a custom SVG lattice illustration that adapts to the selected type (corners only for SC, body atom for BCC, six face atoms for FCC), and reports lattice parameter, cell volume in m³ and ų, atomic packing factor, atoms per cell (Z), coordination number, and the a/r ratio. Closest-radius matching against 16 reference metals (Fe, Cu, Au, Al, W, Cr, Pb, Pt, Ni, Pb, Na, K, Mo, V, Ag, α-Po) shows you the real materials that match your input.

CrystallographySolid-State ChemistrySoftware Engineering Team

Disclaimer

Calculations assume ideal hard-sphere atom packing — real metals deviate from this slightly due to ionic-vs-metallic bonding character, thermal vibrations (Debye-Waller factor), and electronic structure. Atomic radii vary 5-10% between sources (CRC, Slater, Pauling, Shannon ionic radii). For X-ray-diffraction lattice-parameter measurements, use the experimental value rather than computed value. Hexagonal Close Packed (HCP) achieves the same 74.05% APF as FCC but isn't a cubic lattice and requires separate analysis. The calculator gives bulk-crystal properties; nanoparticles and surfaces show modified geometries.