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Cubic Cell Calculator

Chemistry

Calculate unit cell parameters for simple cubic (SC), body-centred cubic (BCC), and face-centred cubic (FCC) structures — lattice constant and density.

Reviewed by the thecalcu.com team · Last updated June 27, 2026

4.05 Å
Å
26.98 g/mol
g/mol

Atomic Radius (r)

1.432
Atoms per Unit Cell
4
Packing Efficiency
74.05
Theoretical Density
2.698

This calculator computes your Atomic Radius (r), Atoms per Unit Cell, Packing Efficiency, Theoretical Density from the values you enter.

Inputs
Crystal StructureKnown ParameterLattice Constant (a)Molar Mass
Outputs
Atomic Radius (r)Atoms per Unit CellPacking EfficiencyTheoretical Density

What is a Cubic Cell?

The Cubic Cell Calculator computes atomic radius, atoms per unit cell, atomic packing factor, and theoretical density for Simple Cubic (SC), Body-Centred Cubic (BCC), and Face-Centred Cubic (FCC) crystal structures from the lattice constant. Enter the lattice constant (Å) and molar mass (g/mol).

Cubic unit cells are the three simplest crystal structures in materials science and solid-state chemistry. The geometric relationships between lattice constant and atomic radius (a = 2r for SC; a√3 = 4r for BCC; a√2 = 4r for FCC) determine packing efficiency and theoretical density. The theoretical density ρ = zM/(a³Nₐ) can be compared to measured density, agreement confirms the crystal structure type.

Crystal structure is characterised by X-ray diffraction using Bragg's law, the Miller Indices Calculator computes d-spacing and 2θ peak positions from (hkl) indices and the lattice constant computed here. For the molar mass M used in the density formula, the Molar Mass Calculator computes M from chemical formula.

Why Use a Cubic Cell Calculator?

The geometric relationships for SC, BCC, and FCC are different, using the BCC formula for FCC (or vice versa) gives wrong results. The density formula requires converting Å to cm (1 Å = 10⁻⁸ cm) and using Avogadro's number, straightforward but error-prone manual calculations, especially in examination conditions. This calculator implements all three geometries correctly.

Who Should Use This Calculator?

Chemistry and materials science students at Indian IITs and NITs solving solid-state chemistry problems for JEE Advanced (Unit Cell and Crystal Structure is a standard topic) and GATE Metallurgy.

Materials scientists at BARC Mumbai, DRDO, and CSIR-IMTECH characterising metal alloy and ceramic crystal structures.

Physics students learning about crystal structure in condensed matter physics, standard in BSc Physics curriculum at Indian universities.

Pharmaceutical scientists characterising active pharmaceutical ingredient (API) polymorphs, different crystal forms have different densities, solubility, and bioavailability.

What Insights Does the Cubic Cell Calculator Give You?

Atomic Radius (Å) is the primary output, the hard-sphere radius inferred from the lattice constant and cell geometry.

Atoms per Unit Cell, z = 1 (SC), 2 (BCC), or 4 (FCC), used in density and packing calculations.

Packing Efficiency (APF %), how much of the unit cell volume is occupied by atoms: SC 52.4%, BCC 68.0%, FCC 74.1%.

Theoretical Density (g/cm³), compare to experimentally measured density to confirm crystal structure.

How to use this Cubic Cell calculator

  1. Select the Crystal Structure (SC, BCC, or FCC) from the dropdown.
  2. Select Known Parameter, Lattice Constant (a) or Atomic Radius (r).
  3. Enter the Lattice Constant (Å), from XRD measurement or literature. Common values: Al FCC: 4.05 Å; Fe BCC: 2.87 Å; Cu FCC: 3.61 Å; W BCC: 3.16 Å.
  4. Enter Molar Mass (g/mol) for the theoretical density calculation.
  5. Read Atomic Radius, APF, and Density, compare density to measured value to verify structure assignment.
Show formula & methodology ↓Show less ↑

Formula & Methodology

Cubic cell geometry (hard-sphere model):

SC:  a = 2r         → r = a/2           z = 1  APF = π/6 ≈ 52.36% BCC: a√3 = 4r      → r = a√3/4         z = 2  APF = π√3/8 ≈ 68.02% FCC: a√2 = 4r      → r = a√2/4         z = 4  APF = π/(3√2) ≈ 74.05%  Theoretical density:   ρ = (z × M) / (a³ × Nₐ)   [a in cm = a_Å × 10⁻⁸; M in g/mol; Nₐ = 6.022 × 10²³ mol⁻¹]

Worked example, Iron (Fe, BCC at room temperature):

XRD measurement: a = 2.87 Å (α-Fe, BCC). Molar mass: 55.845 g/mol.

r = 2.87 × √3 / 4 = 2.87 × 1.732 / 4 = 1.241 Å z = 2 APF = π√3/8 = 68.02% a_cm = 2.87 × 10⁻⁸ cm  ρ = (2 × 55.845) / ((2.87 × 10⁻⁸)³ × 6.022 × 10²³)   = 111.69 / (2.365 × 10⁻²³ × 6.022 × 10²³)   = 111.69 / 14.24   = 7.84 g/cm³

Measured density of iron: 7.87 g/cm³, excellent agreement (< 0.5% error), confirming BCC structure. India produces ~120 million tonnes of steel annually, SAIL, RINL, and JSW control iron's BCC↔FCC phase transformation by temperature and carbon content to produce steels ranging from soft low-carbon structural steel (BCC ferrite dominant) to hard high-carbon tool steel (FCC austenite quenched to BCT martensite).

Frequently Asked Questions

What are the three types of cubic crystal structures?
Cubic crystal systems have three Bravais lattice types: Simple Cubic (SC / Primitive Cubic, P): atoms at cube corners only. Lattice points per unit cell: 8 × (1/8) = 1. Examples: Polonium (the only metallic element with SC structure at room temperature). Body-Centred Cubic (BCC): atoms at corners + 1 atom at body centre. Atoms per unit cell: 2. Examples: Fe (iron, α-phase below 912°C), Cr, Mo, W, Na, K, Ba. Face-Centred Cubic (FCC / Cubic Close-Packed, CCP): atoms at corners + 1 atom at centre of each face. Atoms per unit cell: 4. Examples: Cu, Ag, Au, Al, Ni, Pb. FCC and HCP (hexagonal close-packed) are the two close-packed structures with the highest packing efficiency.
How do I use the Cubic Cell Calculator?
Select the Crystal Structure type (SC, BCC, or FCC). Select the Known Parameter (Lattice Constant or Atomic Radius) and enter its value in Angstroms (Å). Enter Molar Mass (g/mol) for theoretical density calculation. The calculator returns Atomic Radius (Å), Atoms per Unit Cell, Atomic Packing Factor (%), and Theoretical Density (g/cm³). Default: FCC with lattice constant a = 4.05 Å (aluminium), molar mass = 26.98 g/mol → r = 1.432 Å, density = 2.70 g/cm³.
What is atomic packing factor (APF) and how does it compare between cubic structures?
Atomic Packing Factor (APF) = (volume of atoms in unit cell) / (volume of unit cell) = (z × 4πr³/3) / a³. SC: r = a/2 → APF = π/6 ≈ 52.4%. BCC: r = a√3/4 → APF = π√3/8 ≈ 68.0%. FCC: r = a√2/4 → APF = π/(3√2) ≈ 74.1%. HCP also has APF = 74.1%, the maximum possible for identical spheres (Kepler conjecture, proved by Hales 1998). Higher APF → denser structure → higher theoretical density. Industrial implication: FCC metals (Al, Cu, Ag) are more ductile than BCC metals (Fe, Mo), FCC has more slip systems (12 vs 48 for BCC, but BCC has higher Peierls stress at low T).
How is theoretical density calculated from unit cell parameters?
Theoretical density: ρ = (z × M) / (a³ × Nₐ), where z = atoms per unit cell, M = molar mass (g/mol), a = lattice constant (cm), Nₐ = Avogadro's number = 6.022 × 10²³ mol⁻¹. The lattice constant in Angstroms is converted: 1 Å = 10⁻⁸ cm. For Al FCC (z=4, M=26.98, a=4.05 Å=4.05×10⁻⁸ cm): ρ = (4 × 26.98) / ((4.05×10⁻⁸)³ × 6.022×10²³) = 107.92 / (6.644×10⁻²³ × 6.022×10²³) = 107.92/40.0 = 2.70 g/cm³, matching the measured density (2.70 g/cm³), confirming FCC structure.
What is the relationship between atomic radius and lattice constant for each cubic type?
The relationship is derived from touching atoms along the close-packed direction: SC: atoms touch along cube edge → a = 2r, so r = a/2. BCC: atoms touch along body diagonal (length = a√3) → a√3 = 4r → r = a√3/4. FCC: atoms touch along face diagonal (length = a√2) → a√2 = 4r → r = a√2/4. These are the hard-sphere model relationships, real ionic crystal lattice constants may differ due to ionic radii differences and electrostatic repulsion. For NaCl (FCC: Na⁺ and Cl⁻ occupy alternating positions): the calculation is modified using effective ionic radii (Na⁺ = 1.02 Å, Cl⁻ = 1.81 Å) → a = 2(r_Na + r_Cl) = 5.66 Å.
What common metals have BCC and FCC structures?
BCC metals: α-Fe (iron, stable below 912°C), δ-Fe (above 1394°C), Cr (chromium, a=2.88 Å), Mo (molybdenum, a=3.15 Å), W (tungsten, a=3.16 Å, highest melting point: 3422°C), Na (a=4.29 Å), K (a=5.33 Å), V, Nb, Ta. FCC metals: Cu (a=3.61 Å, r=1.28 Å), Ag (a=4.09 Å), Au (a=4.08 Å), Al (a=4.05 Å), Ni (a=3.52 Å), Pb (a=4.95 Å), Pt (a=3.92 Å), γ-Fe (austenite, 912–1394°C). India's steel industry (SAIL, JSW, Tata Steel, world's top 10 producers) treats steel (Fe-C alloy) at different temperatures to exploit BCC (α-Fe)↔FCC (γ-Fe) phase transitions for strength and ductility control.
How are cubic cell calculations used in materials science and semiconductor industry?
Semiconductor industry (India: Intel fab in Magadh economic zone planned; existing assembly at Foxconn Tamil Nadu): Silicon has diamond cubic structure (FCC with 2-atom basis, z=8, a=5.43 Å, r=1.18 Å, APF=34%, much lower than close-packed). Using cubic cell calculations: (1) Dopant density: atoms/cm³ = z/a³; Si: 5.00×10²² atoms/cm³. (2) Wafer thickness: crystal planes for cleavage. (3) X-ray diffraction: d-spacing for XRD calibration (see [Miller Indices Calculator](/miller-indices-calculator/)). (4) Epitaxial growth: lattice mismatch between Si and Ge (Si: 5.43 Å, Ge: 5.66 Å → 4% mismatch) determines strain in Si-Ge heterojunctions.
What is the coordination number in cubic structures?
Coordination number (CN) = number of nearest neighbours each atom has: SC: CN = 6 (along ±x, ±y, ±z directions). BCC: CN = 8 (body diagonal neighbours). FCC: CN = 12 (face-touching neighbours, same as HCP). Higher CN → more stable, more dense structure. Ionic crystals: CN depends on radius ratio r_cation/r_anion. NaCl structure (FCC): CN = 6 for both Na⁺ and Cl⁻. CsCl structure (similar to BCC): CN = 8 for both Cs⁺ and Cl⁻. ZnS (zinc blende, FCC-based): CN = 4 (tetrahedral). These coordination numbers determine crystal field splitting of d-orbitals in transition metal complexes, important for understanding colour and magnetic properties of Indian mineralogical specimens and industrial catalysts.
How do I use the cubic cell calculator for ionic crystal NaCl?
NaCl has an FCC structure where Na⁺ and Cl⁻ alternate: effective lattice constant a = 5.64 Å. Using FCC calculation (z = 4 formula units per cell): M_NaCl = 58.44 g/mol, a = 5.64 Å = 5.64 × 10⁻⁸ cm. ρ = (4 × 58.44) / ((5.64×10⁻⁸)³ × 6.022×10²³) = 233.76 / (1.795×10⁻²² × 6.022×10²³) = 233.76/108.1 = 2.16 g/cm³. Measured: 2.165 g/cm³. Indian rock salt (sendha namak) from Sambhar Lake (Rajasthan, India's largest inland salt lake) has the NaCl structure, XRD confirms the FCC cubic pattern used in industrial salt quality control.
What is the difference between FCC and HCP (hexagonal close-packed) structures?
Both FCC and HCP have APF = 74.1% and CN = 12. The difference is in layer stacking: HCP: ABABAB... stacking (layers repeat every 2). FCC (CCP): ABCABC... stacking (layers repeat every 3). The structures have identical APF but different symmetry and slip systems, leading to different mechanical properties. Many metals can form either structure depending on temperature: Co transforms HCP (below 417°C) → FCC (above 417°C). Ti: HCP (α, below 882°C) → BCC (β, above 882°C). This HCP→BCC transformation in titanium is critical for Indian aerospace applications (ISRO, HAL use Ti alloys in rocket and aircraft components).