Coordination Number Identifier
Identify the coordination number, atoms per unit cell, and packing context for the five common crystal structure types.
🔗 What is a Coordination Number Identifier?
A coordination number is the number of nearest-neighbor atoms that directly touch a given atom in a crystal lattice, the atoms in the closest surrounding shell. This calculator identifies the coordination number, the number of atoms per conventional unit cell, and the idealized atomic packing factor for the five crystal structures most commonly taught in materials science and solid-state chemistry courses: simple cubic (SC), body-centered cubic (BCC), face-centered cubic (FCC), hexagonal close-packed (HCP), and diamond cubic.
Materials scientists use coordination number as a quick fingerprint for a structure's packing efficiency and bonding character. A metallurgist comparing iron's two allotropes notes that alpha-iron (BCC, CN=8) is less densely packed than gamma-iron (FCC, CN=12), which partly explains why the FCC phase is more ductile. A semiconductor engineer recognizes that silicon's diamond cubic structure (CN=4) reflects its four covalent bonds per atom, fundamentally different from the metallic bonding in close-packed structures. A crystallography student uses coordination number as a first checkpoint before working through more detailed unit cell geometry problems.
A common point of confusion is treating coordination number and atomic packing factor as the same idea. They are related but distinct: coordination number is a simple integer count of touching neighbors, while packing factor is the fraction of unit cell volume actually filled by atomic spheres. FCC and HCP share the identical coordination number (12) and packing factor (0.7405) despite being visibly different structures, because both are close-packed, differing only in how their identical layers stack.
This calculator is useful for materials science and condensed matter physics coursework, for quickly checking a structure's basic packing statistics, and as a companion to the Atomic Packing Factor Calculator and Unit Cell Volume Calculator for deeper unit cell analysis.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — Copper (FCC)
Copper adopts the face-centered cubic structure
Example 2 — Alpha-Iron (BCC)
Alpha-iron (α-Fe) adopts the body-centered cubic structure at room temperature
Example 3 — Silicon (Diamond Cubic)
Silicon adopts the diamond cubic structure, four covalent bonds per atom
❓ Frequently Asked Questions
🔗 Related Calculators
What is coordination number in crystallography?
Coordination number (CN) is the number of nearest-neighbor atoms directly touching a given atom in a crystal structure, the atoms in the closest surrounding shell. It is a key descriptor of how efficiently a structure packs atoms in space.
What is the coordination number of FCC?
Face-centered cubic (FCC) has a coordination number of 12. Each atom touches 4 neighbors in its own close-packed layer, 4 in the layer above, and 4 in the layer below, giving the densest possible packing of identical spheres alongside HCP.
What is the coordination number of BCC?
Body-centered cubic (BCC) has a coordination number of 8. The central atom of each cube touches all 8 corner atoms, which are its true nearest neighbors, at a distance of a√3/2.
What is the coordination number of simple cubic?
Simple cubic (SC) has a coordination number of 6, the lowest of the common metallic structures. Each atom touches its 6 face-adjacent neighbors along the ±x, ±y, and ±z directions, one cube edge length away.
Why do FCC and HCP have the same coordination number?
Both FCC and HCP are close-packed structures built from stacking identical close-packed atomic layers, each atom touches 12 neighbors regardless of stacking order. They differ only in whether the third layer repeats the first (ABAB, giving HCP) or introduces a new position (ABCABC, giving FCC).
What is the coordination number of diamond cubic?
Diamond cubic has a coordination number of 4, each atom forms four covalent bonds to its nearest neighbors in a tetrahedral arrangement. This is far lower than the close-packed metallic structures because covalent bonding, not sphere-packing efficiency, dictates the geometry.
How many atoms are in each unit cell type?
Simple cubic has 1 atom per unit cell, BCC has 2, FCC has 4, the conventional hexagonal HCP cell has 6, and diamond cubic has 8, counted from the fractional contributions of corner, face, edge, and interior atoms shared between neighboring cells.
Does a higher coordination number always mean higher density?
For a fixed atomic radius, yes, a higher coordination number generally comes with a higher packing factor and therefore higher density. Simple cubic (CN=6, 52.4% packed) is the least dense of the five, while FCC and HCP (CN=12, 74.05% packed) are the densest possible sphere packings.
Which elements have a body-centered cubic structure?
Common BCC elements include alpha-iron (α-Fe, the room-temperature form of iron), chromium, tungsten, sodium, and vanadium. BCC metals are generally less ductile than FCC metals because they have fewer close-packed slip planes.
How is coordination number different from atomic packing factor?
Coordination number (CN) counts nearest-neighbor atoms, a simple integer. Atomic packing factor (APF) measures the fraction of unit cell volume filled by those atoms, modeled as touching spheres, a number between 0 and 1. The two are related but distinct: higher CN usually correlates with higher APF, but APF also depends on how the atoms are arranged, not just how many touch each one.
Why is diamond cubic's packing factor so much lower than FCC?
Diamond cubic has an idealized packing factor of only about 0.340 (34.0%), far below FCC's 0.7405 (74.05%), because its open tetrahedral bonding geometry leaves large gaps between atoms. This open structure, not close-packing efficiency, is what gives diamond and silicon their characteristic hardness and semiconductor properties.