Calculate all key parameters for SC, BCC, and FCC cubic crystal structures. Enter the lattice constant or select a metal preset to instantly get atomic radius, packing efficiency, coordination number, atoms per cell, and theoretical density.
✓ Verified: IUCr crystallographic data & NIST Elemental Properties — April 2026
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Packing Efficiency
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Sources & Methodology
🛡️Lattice constants from NIST Elemental Properties database. Formulas follow International Union of Crystallography (IUCr) standards for cubic Bravais lattices.
🔬
NIST Elemental Properties — Lattice Parameters
Standard reference lattice constants for metallic elements at room temperature. nist.gov
📖
International Tables for Crystallography (IUCr)
Definitive reference for crystallographic symmetry, lattice types, and unit cell calculations. it.iucr.org
🏫
Callister: Materials Science and Engineering, 10th Ed.
Standard textbook for unit cell calculations, atomic packing factors, and crystal density formula (Chapter 3).
Key formulas:
SC: Z=1, r=a/2, APF=52.36% (π/6)
BCC: Z=2, r=a√3/4, APF=68.02% (π√3/8)
FCC: Z=4, r=a√2/4, APF=74.05% (π√2/6)
Density = (Z × M) / (Nₐ × a³) | a in cm, M in g/mol
ρ = (Z × M) / (NA × a³)
Iron (BCC): Z=2, M=55.845, a=2.87×10⁻⁸ cm, NA=6.022×10²³
ρ = (2×55.845) / (6.022e23 × (2.87e-8)³) = 7.87 g/cm³ (matches measured 7.87)
Cubic Crystal Structures: SC, BCC, and FCC Explained
The three cubic Bravais lattices — simple cubic (SC), body-centered cubic (BCC), and face-centered cubic (FCC) — are the most important crystal structures in materials science and metallurgy. Understanding how atoms pack into these structures determines a metal's density, ductility, strength, melting point, and thermal properties.
The key parameter describing any cubic crystal is the lattice constant (a) — the edge length of the unit cell cube. From this single value, all other structural parameters can be derived: atomic radius, packing efficiency, coordination number, and theoretical density.
Comparison of the Three Cubic Structures
Property
SC
BCC
FCC
Atoms per cell (Z)
1
2
4
Atomic radius r
a/2
a√3/4
a√2/4
Coordination number
6
8
12
Packing efficiency
52.36%
68.02%
74.05%
Touch direction
Edge (a)
Body diagonal (a√3)
Face diagonal (a√2)
Example metals
Polonium only
Fe, W, Cr, Mo
Cu, Al, Au, Ag, Ni
How to Calculate Atomic Radius from Lattice Constant
The atomic radius relationship depends on which direction the atoms touch in the unit cell. In SC, atoms at adjacent corners touch along the edge: 2r = a, so r = a/2. In BCC, the body-center atom touches all 8 corner atoms along the body diagonal (length a√3): 4r = a√3, so r = a√3/4 ≈ 0.433a. In FCC, atoms touch along the face diagonal (length a√2): 4r = a√2, so r = a√2/4 ≈ 0.354a.
Theoretical vs Measured Crystal Density
The theoretical crystal density calculated from unit cell parameters almost exactly matches measured values for pure metals. Discrepancies arise from lattice defects (vacancies, dislocations), impurities, and temperature effects. The formula ρ = ZM/(Nₐa³) is used to:
Verify crystal structure assignments from X-ray diffraction
Calculate lattice constant from measured density (reverse calculation)
Determine the molar mass of unknown crystalline compounds
Design ceramic and alloy compositions with target densities
💡 Why FCC is More Ductile Than BCC: FCC has 12 slip systems (4 {111} planes × 3 <110> directions each), compared to BCC which also has 12 slip systems but requires higher critical resolved shear stress. The close-packed {111} planes in FCC slip past each other more easily, making FCC metals like copper, gold, and aluminum exceptionally ductile and formable. This is why copper wire, gold leaf, and aluminum foil are all FCC.
Common Lattice Constants of Metals
Metal
Structure
a (Å)
r (Å)
Density (g/cm³)
Iron (Fe)
BCC
2.87
1.24
7.87
Copper (Cu)
FCC
3.615
1.28
8.96
Aluminum (Al)
FCC
4.046
1.43
2.70
Gold (Au)
FCC
4.078
1.44
19.32
Tungsten (W)
BCC
3.165
1.37
19.25
Nickel (Ni)
FCC
3.524
1.25
8.91
Chromium (Cr)
BCC
2.884
1.25
7.19
Frequently Asked Questions
A unit cell is the smallest repeating structural unit of a crystal that, repeated in 3D, produces the complete lattice. For cubic systems, all edge lengths are equal (a = b = c) and all angles are 90°. The three types are simple cubic (SC), body-centered cubic (BCC), and face-centered cubic (FCC).
BCC has 2 atoms per unit cell: 8 corner atoms each shared by 8 cells (8 × 1/8 = 1) plus 1 atom fully inside the body center. Total = 2. Common BCC metals: iron (at room temperature), chromium, tungsten, molybdenum, vanadium.
FCC has 4 atoms per unit cell: 8 corners × 1/8 = 1, plus 6 face atoms each shared by 2 cells: 6 × 1/2 = 3. Total = 4. Common FCC metals: copper, gold, silver, aluminum, nickel, platinum.
FCC packing efficiency = 74.05% (π√2/6). This is the maximum possible for equal spheres and was proven by Hales in 1998 (Kepler conjecture). BCC = 68.02% and SC = 52.36%. Higher packing efficiency generally means denser, harder material.
SC: r = a/2 (atoms touch along edge). BCC: r = a√3/4 ≈ 0.433a (touch along body diagonal). FCC: r = a√2/4 ≈ 0.354a (touch along face diagonal). These come from the geometry of how atoms contact each other in the unit cell.
Density ρ = (Z × M) / (NA × a³), where Z = atoms per cell, M = molar mass (g/mol), NA = 6.022×10²³ mol⁻¹, and a = lattice constant in cm. For iron (BCC): ρ = (2 × 55.845) / (6.022e23 × (2.87e-8)³) = 7.87 g/cm³.
Common BCC metals: iron (alpha), chromium, tungsten, molybdenum, vanadium, niobium, tantalum, and barium. BCC metals tend to have higher melting points but lower ductility than FCC metals. Iron changes from BCC to FCC (austenite) above 912°C.
Common FCC metals: copper, aluminum, gold, silver, nickel, lead, platinum, palladium, calcium. FCC metals are generally more ductile because their 12 close-packed slip systems allow plastic deformation more easily. This makes copper wire, gold leaf, and aluminum foil possible.
Coordination number = number of nearest-neighbor atoms touching each atom. SC = 6 (front, back, left, right, up, down). BCC = 8 (all 8 body-diagonal neighbors). FCC = 12 (6 face neighbors + 6 from adjacent cells). Higher coordination = more efficient packing.
Polonium (Po) is the only pure element known to form a simple cubic crystal at room temperature. SC has only 52.36% packing efficiency, making it energetically unfavorable. Most elements prefer BCC or FCC for their greater stability and denser atomic packing.