3.6204
Å
47.4531
ų
4
12
0.74048
35.1381
ų
12.315
ų
3.6204
Å
47.4531
ų
4
12
0.74048
35.1381
ų
12.315
ų
The Cubic Unit Cell Calculator determines the lattice parameter, cell volume, atoms per unit cell, and coordination number for the three cubic Bravais lattices: simple cubic (SC), body-centered cubic (BCC), and face-centered cubic (FCC). These are the most important crystal structures in metallurgy and materials science, describing the arrangement of atoms in most metals and many ionic and covalent compounds.
Given the atomic radius, the calculator computes the lattice parameter using the geometric relationship specific to each structure type. The touching direction (where atoms are in contact) differs: along the cube edge for SC, along the body diagonal for BCC, and along the face diagonal for FCC.
The relationship between atomic radius $$r$$ and lattice parameter $$a$$ depends on the touching direction:
$$\text{Simple Cubic: } a = 2r \quad \text{(atoms touch along edge)}$$
$$\text{BCC: } a = \frac{4r}{\sqrt{3}} \quad \text{(atoms touch along body diagonal)}$$
$$\text{FCC: } a = 2\sqrt{2}r \quad \text{(atoms touch along face diagonal)}$$
The unit cell volume is:
$$V = a^3$$
Atoms per unit cell: SC has 8 corner atoms × 1/8 = 1. BCC has 8 corners × 1/8 + 1 body center = 2. FCC has 8 corners × 1/8 + 6 faces × 1/2 = 4. The coordination number (nearest neighbors) is 6 for SC, 8 for BCC, and 12 for FCC.
These relationships are fundamental for computing packing fraction, crystal density, and understanding the physical properties that arise from different atomic arrangements.
FCC structures have the highest packing efficiency (74.0%) and coordination number (12), making them common among ductile metals like copper, aluminum, gold, and silver. BCC structures pack less efficiently (68.0%, CN = 8) but are found in strong metals like iron (α-Fe), tungsten, and chromium. SC structures have the lowest packing (52.4%, CN = 6) and are rare in nature, with polonium being the only elemental SC structure.
Inputs
Results
a = 2√2 × 1.28 = 3.62 Å, V = 3.62³ ≈ 47.5 ų, 4 atoms/cell with 12 nearest neighbors
Inputs
Results
a = 4 × 1.26 / √3 = 2.91 Å, V = 24.6 ų, 2 atoms/cell with 8 nearest neighbors
Simple cubic (SC) has atoms only at the 8 corners of the cube (1 atom/cell). BCC adds an atom at the body center (2 atoms/cell). FCC adds atoms at the center of each of the 6 faces (4 atoms/cell). Each structure has a different packing efficiency, coordination number, and set of available slip systems that determine mechanical behavior.
FCC achieves 74% packing efficiency because atoms are arranged in the closest-packed configuration with stacking sequence ABCABC. BCC has 68% efficiency because its 8-coordinate arrangement leaves more void space. The difference arises from the geometry of the touching directions: face diagonal (FCC) versus body diagonal (BCC).
Common BCC metals include iron (α-Fe below 912°C), tungsten, chromium, molybdenum, vanadium, niobium, tantalum, and alkali metals (Li, Na, K, Rb, Cs). BCC metals tend to be strong but less ductile than FCC metals due to fewer active slip systems at room temperature.
Common FCC metals include copper, aluminum, gold, silver, nickel, platinum, lead, and γ-iron (912–1394°C). FCC metals are generally more ductile than BCC metals because they have 12 slip systems ({111}<110>) that are active at all temperatures.
SC has only 52.4% packing efficiency, the lowest of any cubic structure. Most atoms prefer denser packing to maximize bonding and minimize energy. Only polonium crystallizes in the SC structure among elements, and it is radioactive and rare. Most materials prefer BCC, FCC, or HCP structures.
The relationship depends on which direction atoms touch. In SC, atoms touch along the cube edge: a = 2r. In BCC, atoms touch along the body diagonal (length a√3): 4r = a√3. In FCC, atoms touch along the face diagonal (length a√2): 4r = a√2. These geometric relationships are derived from the hard-sphere model.
Coordination number is the number of nearest neighbors surrounding each atom. SC = 6 (along ±x, ±y, ±z), BCC = 8 (at body center positions), FCC = 12 (along face diagonal directions). Higher coordination generally means stronger metallic bonding and higher density.
Yes. NaCl has the rock salt structure (two interpenetrating FCC lattices). CsCl has a BCC-like structure (but is actually two interpenetrating SC lattices). ZnS has the zinc blende structure (two interpenetrating FCC lattices offset by a/4). Diamond and silicon have the diamond cubic structure (FCC with tetrahedral interstitial sites filled).
Many materials undergo polymorphic phase transitions with temperature. Iron transforms from BCC (α-Fe) to FCC (γ-Fe) at 912°C and back to BCC (δ-Fe) at 1394°C. These allotropic transitions change density, magnetic properties, and solubility of interstitial atoms like carbon, which is the basis of steel heat treatment.
Use the metallic radius, which is half the nearest-neighbor distance in the elemental crystal. This differs from ionic radius, covalent radius, and van der Waals radius. Common values: Cu 1.28 Å, Al 1.43 Å, Fe 1.26 Å, Au 1.44 Å, Ag 1.44 Å, Ni 1.25 Å.
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