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Lattice Constant Calculator

Ready to calculate
SC · BCC · FCC.
5 Length Units (Å → mm).
APF · CN · Volume.
100% Free.
No Data Stored.

How it Works

01Pick the Lattice

SC (atoms touch on the edge), BCC (touch on body diagonal), FCC (touch on face diagonal)

02Enter Atomic Radius

5 length units supported — Å (default), pm, nm, μm, mm

03Apply the Geometry

SC: a = 2r · BCC: a = 4r/√3 · FCC: a = 4r/√2 · derived from the close-packed touching condition

04Get Properties

Lattice constant + cell volume + APF + coordination number + atoms-per-cell

What is a Lattice Constant Calculator?

The lattice constant — symbol a — is the edge length of the cubic unit cell that repeats throughout a crystal. Knowing it lets you predict density, X-ray diffraction patterns, mechanical properties, and how impurity atoms slot into the structure. Our Lattice Constant Calculator computes a instantly from the atomic radius for the three cubic Bravais lattices: simple cubic (SC), body-centered cubic (BCC), and face-centered cubic (FCC). It's an essential tool for materials science students, solid-state chemists, and anyone working with crystalline solids.

The geometry comes from the close-packing condition — atoms modeled as hard spheres touching their nearest neighbors. In SC, atoms touch along the cell edge, so a = 2r. In BCC, atoms touch along the body diagonal (length a√3, holding 4 atomic radii), giving a = 4r/√3. In FCC, atoms touch along the face diagonal (length a√2, holding 4 radii), giving a = 4r/√2 = 2r√2. The calculator handles atomic radius input in five units (Å, pm, nm, μm, mm) and outputs the lattice constant, unit-cell volume, atomic packing factor (APF), coordination number, and atoms-per-cell — everything you need for a routine homework or research calculation.

Designed for first-year materials science through graduate solid-state coursework, the tool shows every calculation step transparently and includes example metals for each lattice (iron is BCC, copper is FCC, polonium is the rare SC). It's free, fast, and runs entirely in your browser.

Pro Tip: For more relevant tools in the chemistry category, try our Molarity Calculator or the Ideal Gas Temperature calculator.

How to Use the Lattice Constant Calculator?

Pick the Lattice Type: Simple cubic (SC), body-centered cubic (BCC), or face-centered cubic (FCC). The visual schematic in the input panel updates with your choice so you can see the geometry.
Enter the Atomic Radius: The radius of the atom (or ion) you're modeling — choose Å (default for crystallography), pm, nm, μm, or mm from the unit dropdown. Iron has r ≈ 1.24 Å, copper r ≈ 1.28 Å, sodium r ≈ 1.86 Å.
Press Calculate: The tool applies the close-packing geometry — SC: a = 2r, BCC: a = 4r/√3, FCC: a = 4r/√2 = 2r√2 — and returns the lattice constant in the most readable unit (typically Å or pm for atomic-scale calculations).
Read the Properties: Lattice constant in all five units, unit-cell volume (a³), atomic packing factor (APF — fraction of cell volume occupied by atoms), coordination number (nearest neighbors), and atoms-per-unit-cell. A "real-world examples" panel shows which elements adopt the chosen lattice.

How do I calculate the lattice constant?

The lattice constant comes from the close-packing geometry — modeling atoms as hard spheres that touch their nearest neighbors. Each cubic lattice has its own touching condition, which gives a different formula:


Think of it as packing tennis balls in a box: in different arrangements, the box's edge length is a different multiple of the ball's radius. Crystallographers formalized this for the three cubic Bravais lattices.

Simple Cubic (SC)


a = 2r


Atoms sit only at the cube's 8 corners (each shared with 8 cells, so 8/8 = 1 atom per unit cell). Adjacent corners touch along the edge — so the edge length is just two atomic radii. Coordination number is 6 (each atom has 6 nearest neighbors). APF = π/6 ≈ 0.5236 — the lowest packing efficiency of the three.

Body-Centered Cubic (BCC)


a = 4r/√3 ≈ 2.309 r


Atoms at the 8 corners plus 1 in the center of the cube (1 corner-equivalent + 1 center = 2 atoms per unit cell). Atoms touch along the body diagonal of length a√3, which holds 4 atomic radii (corner + center + corner). Solving 4r = a√3 gives a = 4r/√3. Coordination number is 8. APF = π√3/8 ≈ 0.6802.

Face-Centered Cubic (FCC)


a = 4r/√2 = 2r√2 ≈ 2.828 r


Atoms at the 8 corners plus 1 in the center of each of the 6 faces (1 corner-equivalent + 6×½ face-equivalent = 4 atoms per unit cell). Atoms touch along the face diagonal of length a√2, holding 4 atomic radii. Solving 4r = a√2 gives a = 4r/√2 = 2r√2. Coordination number is 12 — the maximum for any sphere packing in 3D. APF = π/(3√2) ≈ 0.7405 — tied with HCP for the densest possible packing of identical spheres.

Atomic Packing Factor (APF)


APF = (atoms × volume per atom) ÷ unit cell volume = (n × ⁴⁄₃πr³) ÷ a³


APF tells you what fraction of the unit cell is actually occupied by atomic spheres. The remainder is empty space. SC's low APF (0.524) explains why no normal metal adopts it — the structure is just inefficient. FCC and HCP at 0.740 are mathematically tied for the densest packing of identical spheres.

Real-World Example

Lattice Constant Calculator – Cubic Crystal Structures In Practice

Consider face-centered cubic (FCC) copper with atomic radius r = 1.28 Å:
  • Step 1: Identify the lattice. Copper is FCC. Apply a = 4r/√2 = 2r√2.
  • Step 2: Substitute. a = 2 × 1.28 × √2 = 2 × 1.28 × 1.4142 ≈ 3.62 Å.
  • Step 3: Check against experiment. Reported lattice constant for copper at 25°C: 3.6149 Å. Our calculation gives 3.62 Å — within 0.2%, excellent for the hard-sphere model.
  • Step 4: Compute unit cell volume. V = a³ = 3.62³ ≈ 47.4 ų.
  • Step 5: Read other properties. FCC: 4 atoms per cell · CN = 12 · APF = 0.7405.
  • Step 6: (Optional) Compute density. ρ = (n × M) ÷ (V × N_A) = (4 × 63.55 g/mol) ÷ (47.4 × 10⁻²⁴ cm³ × 6.022 × 10²³) = 8.91 g/cm³ — matches measured copper density (8.96 g/cm³).

Now consider body-centered cubic (BCC) iron with r = 1.24 Å: a = 4 × 1.24 ÷ √3 = 4.96 ÷ 1.732 ≈ 2.86 Å. Reported value: 2.866 Å — within 0.1%. The hard-sphere model is remarkably accurate for close-packed metals.

Who Should Use the Lattice Constant Calculator?

1
Materials Science Students: Solve homework on crystal structures, derive lattice constants from atomic radii, verify worked examples in textbooks (Callister, Smith, Shackelford).
2
Solid-State Chemistry: Quick reference for crystal-structure problems. Compute density from lattice constant, predict X-ray diffraction patterns, work with Bragg's law.
3
Crystallographers: Validate refined lattice parameters from XRD against close-packed expectations. Identify when atoms deviate from rigid-sphere behavior.
4
Metallurgy & Engineering: Predict density of pure metals and alloys (with Vegard's law correction). Useful for FCC/BCC phase identification in steel and brass research.
5
Physics Students: Compute Brillouin zone dimensions, verify reciprocal lattice geometry, and check atomic-density assumptions in solid-state physics problems.

Technical Reference

Cubic Bravais Lattices. Three of the 14 Bravais lattices have cubic symmetry:

  • Simple cubic (P, primitive): a = 2r, n = 1, CN = 6, APF = π/6 ≈ 0.5236
  • Body-centered cubic (I, innenzentriert): a = 4r/√3, n = 2, CN = 8, APF = π√3/8 ≈ 0.6802
  • Face-centered cubic (F): a = 4r/√2, n = 4, CN = 12, APF = π/(3√2) ≈ 0.7405

Maximum sphere packing. The Kepler conjecture (proven 2014) establishes that no arrangement of equal-sized spheres can exceed APF ≈ 0.7405. This bound is achieved by both FCC and HCP (hexagonal close-packed). SC and BCC are necessarily less dense.

Density formula. Once you know the lattice constant, density follows from:

ρ = (n × M) ÷ (V_cell × N_A)

where n = atoms per cell, M = molar mass (g/mol), V_cell = a³, and N_A = 6.022 × 10²³. The calculator's unit-cell volume output feeds directly into this formula.

Bragg's law. Diffraction angles 2θ are linked to lattice spacing d_hkl, which for cubic systems is:

d_hkl = a / √(h² + k² + l²)

A correctly computed lattice constant lets you predict every line in an XRD pattern.

Limitations of the hard-sphere model.

  • Soft cores: Real atoms have electron-density tails; "touching" is approximate. Errors typically 1–3%.
  • Thermal expansion: Lattice constants increase with temperature (typical α ≈ 10⁻⁵ K⁻¹). The calculator's output is at 0 K equivalent / room temperature with the radius you supply.
  • Ionic compounds: Use Shannon ionic radii (1976), which depend on coordination number — not metallic radii.
  • Alloys: For random solid solutions, apply Vegard's law: a_alloy = x_A × a_A + x_B × a_B (works well for similar atomic sizes; deviations grow with size mismatch).

Key Takeaways

The cubic lattice constant is the most fundamental geometric parameter of a crystalline solid — it sets the unit-cell volume, the atomic density, and the diffraction pattern. The hard-sphere model captures the relationship between atomic radius and lattice constant remarkably well for close-packed metals, with errors typically below 2% compared to experiment. Use the ToolsACE Lattice Constant Calculator to compute a from atomic radius for SC, BCC, and FCC structures, with full transparency on the geometry and additional outputs (cell volume, APF, coordination number) you'll need for downstream calculations like density, diffraction, and elastic properties. Bookmark it as your everyday crystallography utility.

Frequently Asked Questions

What is the Lattice Constant Calculator?
The lattice constant a is the edge length of a cubic unit cell — the repeating geometric building block of a crystal. Our Lattice Constant Calculator computes a from the atomic radius for the three cubic Bravais lattices: SC (a = 2r), BCC (a = 4r/√3), and FCC (a = 4r/√2). Beyond the lattice constant itself, the tool returns the unit cell volume, atomic packing factor (APF), coordination number, and atoms-per-unit-cell — everything you need for solid-state and materials science calculations.

The calculator handles atomic radius in five units (Å, pm, nm, μm, mm) and shows results in all five output units automatically. A live SVG schematic of the chosen lattice updates with your selection, and a real-world-examples panel shows which elements adopt each structure (iron is BCC, copper is FCC, polonium is the unusual SC).

Designed for materials science, solid-state chemistry, and crystallography coursework, the tool runs entirely in your browser — no data is stored or transmitted.

Pro Tip: For more chemistry tools, try our Molarity Calculator.

What's the difference between SC, BCC, and FCC?
Simple Cubic (SC): 8 atoms at the cube corners only. 1 effective atom per cell. Lowest packing density (52%) — only polonium adopts it among elements at standard conditions. Body-Centered Cubic (BCC): 8 corner atoms + 1 center atom = 2 effective per cell. 68% packing. Iron (α-Fe at room temperature), chromium, tungsten, sodium, and potassium are BCC. Face-Centered Cubic (FCC): 8 corner atoms + 6 face-center atoms = 4 effective per cell. 74% packing — tied with HCP for the densest possible. Copper, silver, gold, aluminum, nickel, lead, and γ-iron are FCC.
Why is FCC so common among metals?
Because it's the densest packing — 74% atomic packing factor, the maximum possible for identical spheres (Kepler conjecture, proven 2014). Dense packing means lower energy per atom, which makes FCC thermodynamically favored for many metals. FCC also has 12 nearest neighbors (the maximum), giving rise to ductility and good metallic bonding. Copper, gold, aluminum, lead, and nickel are FCC for these reasons. BCC (68%) is favored when bonding is more directional (iron, chromium, tungsten) or when entropy stabilizes a less-dense structure at high temperature.
Where does the formula a = 4r/√3 come from for BCC?
From the touching condition along the body diagonal. In BCC, the corner atoms don't touch each other — but they do touch the central atom along the body diagonal. The body diagonal has length a√3 (Pythagoras in 3D). Going along this diagonal you pass through one corner atom (radius r), the central atom (diameter 2r = 2r), and the opposite corner atom (radius r) — total 4 radii of atomic material. Setting a√3 = 4r and solving gives a = 4r/√3. Same logic for FCC but along the face diagonal (length a√2): a√2 = 4r, so a = 4r/√2 = 2r√2.
What atomic radius value should I use?
Use the metallic radius for elemental metals (typical values from Slater 1964 or Pauling 1960 tables, expressed in Å). For ionic compounds (NaCl, CsCl), use Shannon ionic radii (1976), which depend on coordination number — Na⁺ has different radii in 4-, 6-, and 8-coordinated environments. The choice depends on context: hard-sphere calculations need the radius that corresponds to the touching condition in the actual structure.
Why does my calculated lattice constant differ from the experimental value?
Two main reasons. First, the hard-sphere model is an approximation — real atoms have soft electron-density tails, so the touching distance isn't exactly 2r. Second, atomic radius is itself an empirically derived value with ~1% uncertainty. Together, these typically yield 1–3% deviation from XRD-measured lattice constants. For copper, calculated 3.62 Å vs measured 3.6149 Å — within 0.2%. For more exotic metals, deviations can be larger.
What is the atomic packing factor (APF)?
APF is the fraction of unit-cell volume actually occupied by atomic spheres: APF = (atoms × ⁴⁄₃πr³) ÷ a³. The remaining space is geometric void. APF is a structure-only property — it doesn't depend on what element you put at the lattice points. SC = 0.524, BCC = 0.680, FCC = HCP = 0.740 (the maximum for identical spheres). High APF correlates with mechanical and thermal properties: dense packing typically means stiffer, denser metals.
How do I get density from the lattice constant?
Use ρ = (n × M) ÷ (V × N_A), where n = atoms per unit cell (1 for SC, 2 for BCC, 4 for FCC), M = atomic molar mass in g/mol, V = a³ (unit cell volume), and N_A = 6.022 × 10²³ /mol. The calculator gives you V directly. For copper (FCC, a = 3.62 Å, M = 63.55 g/mol): ρ = (4 × 63.55) ÷ (47.4 × 10⁻²⁴ × 6.022 × 10²³) = 8.91 g/cm³, matching the measured 8.96 g/cm³.
Does the calculator handle non-cubic crystals (HCP, tetragonal)?
No — only the three cubic lattices (SC, BCC, FCC). Hexagonal close-packed (HCP) requires two parameters (a and c, with ideal c/a ≈ 1.633), and tetragonal/orthorhombic systems have three independent edge lengths. Those need separate calculators. About 60% of metallic elements crystallize in cubic systems, so this calculator covers the most common cases — including iron (BCC), copper (FCC), aluminum (FCC), and tungsten (BCC).
What's the relationship between lattice constant and X-ray diffraction?
Bragg's law (nλ = 2d sin θ) connects the diffraction angle to the inter-planar spacing d. For cubic crystals, d_hkl = a / √(h² + k² + l²). So once you know a, you can predict every diffraction line for any allowed (hkl) reflection. The calculator gives you a from atomic radius; XRD gives you a from diffraction angles. Comparing the two is the standard validation in materials characterization.

Author Spotlight

The ToolsACE Team - ToolsACE.io Team

The ToolsACE Team

Our chemistry tools team implements the hard-sphere close-packing geometry for the three cubic Bravais lattices. SC: atoms touch along the cell edge, giving a = 2r. BCC: atoms touch along the body diagonal, giving a = 4r/√3. FCC: atoms touch along the face diagonal, giving a = 4r/√2 = 2r√2. The calculator also returns the atomic packing factor (APF), coordination number, atoms-per-unit-cell, and unit cell volume for each lattice.

Solid State ChemistryCrystallography (Bravais Lattices)Software Engineering Team

Disclaimer

The calculator uses the hard-sphere model — atoms treated as rigid touching spheres. Real metals show 1–3% deviation due to electron-cloud softness and thermal vibrations. For ionic compounds use Shannon ionic radii. For alloys apply Vegard's law to interpolate between pure-element constants.