Cubic Unit Cell
SC, BCC, FCC structures. a = 2r (SC), 4r/√3 (BCC), 2√2r (FCC). Packing efficiency: SC 52%, BCC 68%, FCC 74%.
Why This Chemistry Calculation Matters
Why: Unit cell geometry determines density, coordination, and materials properties. Essential for X-ray diffraction.
How: SC: 1 atom/cell, r=a/2. BCC: 2 atoms, r=a√3/4. FCC: 4 atoms, r=a√2/4. ρ = (n×M)/(a³×N_A).
- ●FCC has highest packing (74%); SC lowest (52%).
- ●Copper, gold, aluminum are FCC; iron α is BCC.
- ●Polonium is rare SC example.
Crystal Examples
🔷 Copper (FCC)
Face-centered cubic structure - common metal
⚙️ Iron α (BCC)
Body-centered cubic structure - ferromagnetic
✈️ Aluminum (FCC)
Lightweight FCC metal - aerospace applications
☢️ Polonium (SC)
Simple cubic structure - rare example
💎 Gold (FCC)
Precious metal with FCC structure
💡 Tungsten (BCC)
High melting point BCC metal
🥈 Silver (FCC)
High conductivity FCC metal
🔩 Chromium (BCC)
Hard BCC transition metal
🪙 Nickel (FCC)
Magnetic FCC metal
🧂 Sodium (BCC)
Alkali metal with BCC structure
💍 Platinum (FCC)
Dense FCC precious metal
🔧 Molybdenum (BCC)
High strength BCC metal
Calculate Cubic Cell Properties
⚠️For educational and informational purposes only. Verify with a qualified professional.
🔬 Chemistry Facts
SC: 1 atom/cell, CN=6, APF=52.4%.
— IUPAC
BCC: 2 atoms/cell, CN=8, APF=68%.
— IUCr
FCC: 4 atoms/cell, CN=12, APF=74%.
— Crystallography
ρ = (n×M)/(a³×N_A). Density from structure.
— Materials
Cubic Crystal Structures
Cubic crystal structures are among the most common and important crystal systems in materials science. They include Simple Cubic (SC), Body-Centered Cubic (BCC), and Face-Centered Cubic (FCC) structures, each with distinct geometric properties and packing efficiencies.
Simple Cubic (SC)
Atoms at cube corners only
1 atom/unit cell
Coordination: 6
APF: 52.4%
Body-Centered Cubic (BCC)
Atoms at corners + center
2 atoms/unit cell
Coordination: 8
APF: 68.0%
Face-Centered Cubic (FCC)
Atoms at corners + face centers
4 atoms/unit cell
Coordination: 12
APF: 74.0%
Key Formulas
Lattice Constant from Atomic Radius
Simple Cubic
Body-Centered Cubic
Face-Centered Cubic
Unit Cell Volume
where a is the lattice constant
Density Calculation
where n = atoms per unit cell, M = atomic mass, NA = Avogadro's number
Atomic Packing Factor
APF = n × (4πr³/3) / a³
How Do Cubic Crystal Structures Work?
Cubic crystal structures are defined by their unit cell geometry and atomic arrangement. The unit cell is the smallest repeating unit that, when stacked in three dimensions, creates the entire crystal lattice.
🔬 Simple Cubic (SC) Structure
Structure
- • Atoms located only at cube corners
- • Each corner atom shared by 8 unit cells
- • 8 corners × 1/8 = 1 atom per unit cell
- • Coordination number: 6 (nearest neighbors)
- • Atomic packing factor: π/6 ≈ 52.4%
Examples
- • Polonium (α-Po) - rare example
- • Some ionic crystals at high pressure
- • Not common for metals
⚙️ Body-Centered Cubic (BCC) Structure
Structure
- • Atoms at cube corners + one at body center
- • Corner atoms: 8 × 1/8 = 1 atom
- • Center atom: 1 atom
- • Total: 2 atoms per unit cell
- • Coordination number: 8
- • Atomic packing factor: √3π/8 ≈ 68.0%
Examples
- • Iron (α-Fe) - ferromagnetic
- • Chromium, Tungsten, Molybdenum
- • Vanadium, Sodium, Potassium
- • Many transition metals
💎 Face-Centered Cubic (FCC) Structure
Structure
- • Atoms at cube corners + centers of all faces
- • Corner atoms: 8 × 1/8 = 1 atom
- • Face atoms: 6 × 1/2 = 3 atoms
- • Total: 4 atoms per unit cell
- • Coordination number: 12 (highest for cubic)
- • Atomic packing factor: √2π/6 ≈ 74.0%
Examples
- • Copper, Aluminum, Gold, Silver
- • Nickel, Platinum, Lead
- • Many noble metals
- • Most common for close-packed metals
When to Use This Calculator
Understanding cubic crystal structures is essential for materials scientists, chemists, physicists, and engineers working with metals, ceramics, and other crystalline materials.
Materials Science
Analyze crystal structures, predict properties, and design new materials.
- Metal alloy design
- X-ray diffraction analysis
- Phase transformations
Metallurgy
Understand metal properties, heat treatment effects, and crystal defects.
- Steel microstructure
- Grain boundary analysis
- Mechanical properties
Education
Learn crystal structures, solid-state chemistry, and materials properties.
- Solid-state chemistry
- Materials engineering
- Physical chemistry
Real-World Crystal Examples
| Element | Structure | Lattice Constant (Å) | Atomic Mass (g/mol) | Density (g/cm³) | Atomic Radius (Å) |
|---|---|---|---|---|---|
| Polonium (Po) | SC | 3.35 | 209 | 9.32 | 1.680 |
| Iron (α) (Fe) | BCC | 2.87 | 55.85 | 7.87 | 1.240 |
| Chromium (Cr) | BCC | 2.88 | 52 | 7.19 | 1.250 |
| Tungsten (W) | BCC | 3.16 | 183.84 | 19.25 | 1.370 |
| Molybdenum (Mo) | BCC | 3.15 | 95.94 | 10.22 | 1.360 |
| Vanadium (V) | BCC | 3.02 | 50.94 | 6.11 | 1.310 |
| Sodium (Na) | BCC | 4.29 | 22.99 | 0.97 | 1.860 |
| Copper (Cu) | FCC | 3.61 | 63.55 | 8.96 | 1.280 |
| Aluminum (Al) | FCC | 4.05 | 26.98 | 2.70 | 1.430 |
| Gold (Au) | FCC | 4.08 | 196.97 | 19.32 | 1.440 |
| Silver (Ag) | FCC | 4.09 | 107.87 | 10.49 | 1.440 |
| Nickel (Ni) | FCC | 3.52 | 58.69 | 8.90 | 1.240 |
| Platinum (Pt) | FCC | 3.92 | 195.08 | 21.45 | 1.390 |
| Lead (Pb) | FCC | 4.95 | 207.2 | 11.34 | 1.750 |
Example 1: Copper (FCC) - From Lattice Constant
Given:
- Crystal type: FCC
- Lattice constant (a) = 3.61 Å
- Atomic mass (M) = 63.55 g/mol
Calculations:
r = a√2/4 = 3.61 × √2/4 = 1.28 Å
V = a³ = (3.61)³ = 47.0 ų
APF = √2π/6 ≈ 0.740 (74.0%)
ρ = (4 × 63.55) / (47.0 × 10⁻²⁴ × 6.022×10²³)
ρ = 8.96 g/cm³
Example 2: Iron α (BCC) - From Lattice Constant
Given:
- Crystal type: BCC
- Lattice constant (a) = 2.87 Å
- Atomic mass (M) = 55.85 g/mol
Calculations:
r = a√3/4 = 2.87 × √3/4 = 1.24 Å
V = a³ = (2.87)³ = 23.6 ų
APF = √3π/8 ≈ 0.680 (68.0%)
ρ = (2 × 55.85) / (23.6 × 10⁻²⁴ × 6.022×10²³)
ρ = 7.87 g/cm³
Example 3: Polonium (SC) - From Lattice Constant
Given:
- Crystal type: SC (rare!)
- Lattice constant (a) = 3.35 Å
- Atomic mass (M) = 209.0 g/mol
Calculations:
r = a/2 = 3.35/2 = 1.68 Å
V = a³ = (3.35)³ = 37.6 ų
APF = π/6 ≈ 0.524 (52.4%)
ρ = (1 × 209.0) / (37.6 × 10⁻²⁴ × 6.022×10²³)
ρ = 9.32 g/cm³
Example 4: Aluminum (FCC) - From Atomic Radius
Given:
- Crystal type: FCC
- Atomic radius (r) = 1.43 Å
- Atomic mass (M) = 26.98 g/mol
Calculations:
a = 2√2·r = 2√2 × 1.43 = 4.05 Å
V = a³ = (4.05)³ = 66.4 ų
ρ = (4 × 26.98) / (66.4 × 10⁻²⁴ × 6.022×10²³)
ρ = 2.70 g/cm³
Example 5: Tungsten (BCC) - From Density
Given:
- Crystal type: BCC
- Density (ρ) = 19.25 g/cm³
- Atomic mass (M) = 183.84 g/mol
Calculations:
a³ = (2 × 183.84) / (19.25 × 6.022×10²³)
a³ = 31.6 ų
a = ∛31.6 = 3.16 Å
r = a√3/4 = 3.16 × √3/4 = 1.37 Å
Lattice constant = 3.16 Å
Important Considerations
⚠️ Limitations
- • Assumes perfect crystal structure (no defects)
- • Temperature effects not included (thermal expansion)
- • Only valid for pure cubic structures
- • Does not account for impurities or alloys
- • Assumes hard-sphere model (no electron overlap)
✓ Assumptions Made
- • Atoms are perfect spheres
- • No crystal defects or vacancies
- • Room temperature conditions
- • Pure element (no alloying)
- • Ideal cubic symmetry
Complete Formula Reference
Lattice Constant Formulas
Simple Cubic (SC)
r = a/2
Body-Centered Cubic (BCC)
r = a√3/4
Face-Centered Cubic (FCC)
r = a√2/4
Density Formula
where:
ρ = density (g/cm³)
n = atoms per unit cell (1 for SC, 2 for BCC, 4 for FCC)
M = atomic mass (g/mol)
a = lattice constant (Å)
NA = Avogadro's number (6.022 × 10²³ mol⁻¹)
Atomic Packing Factor (APF)
APF = n × (4πr³/3) / a³
SC
APF = π/6 ≈ 0.524
52.4%
BCC
APF = √3π/8 ≈ 0.680
68.0%
FCC
APF = √2π/6 ≈ 0.740
74.0%
Unit Cell Volume
where a is the lattice constant in Angstroms (Å)
Volume in ų = (lattice constant)³
Nearest Neighbor Distance
Simple Cubic
Body-Centered Cubic
Face-Centered Cubic
📚 Official Data Sources
⚠️ Disclaimer: This calculator uses IUPAC-recommended crystallography conventions. For precise work, consult the latest IUCr and NIST crystal structure data.