Key Concepts of Coordination Numbers

Why This Matters

Coordination numbers are one of the most fundamental concepts you'll encounter in crystallography because they directly determine crystal geometry, stability, and physical properties. When you're analyzing any crystal structure—whether it's a simple ionic salt, a complex transition metal compound, or a pure metal—the coordination number tells you how atoms arrange themselves in space and why certain structures are more stable than others. You'll be tested on connecting coordination numbers to concepts like radius ratios, packing efficiency, bond angles, and structure types.

Don't just memorize that sodium chloride has a coordination number of 6. Instead, understand why the relative sizes of sodium and chloride ions favor octahedral coordination, and how this connects to the rock salt structure type. The real exam questions ask you to predict coordination based on ionic radii, explain why metals prefer close-packed arrangements, or compare the stability of different coordination geometries. Master the underlying principles, and the specific examples become intuitive.

---

Fundamentals: What Coordination Numbers Tell Us

Before diving into specific geometries, you need to understand what coordination numbers actually represent and how they're determined. The coordination number emerges from the balance between attractive forces pulling atoms together and repulsive forces keeping them apart.

Definition and Significance

• Coordination number (CN) refers to the count of nearest neighbor atoms directly surrounding a central atom—not second-nearest neighbors or beyond
• Determines geometry by dictating the spatial arrangement; CN = 4 must be tetrahedral or square planar, CN = 6 must be octahedral
• Predicts stability because the "right" CN for a given system minimizes energy and maximizes favorable bonding interactions

Radius Ratio Principle

• Radius ratio ($r_{cation}/r_{anion}$) predicts which coordination number will be most stable for ionic compounds
• Critical thresholds exist: ratios below 0.155 favor CN = 2, 0.155–0.225 favor CN = 3, 0.225–0.414 favor CN = 4, 0.414–0.732 favor CN = 6, and above 0.732 favor CN = 8
• Practical application means you can predict structure type from ionic radii alone—a key exam skill for unknown compounds

---

Low Coordination Numbers (CN = 2–4)

Lower coordination numbers occur when the central atom is relatively small compared to its neighbors, or when directional covalent bonding dominates. These geometries maximize orbital overlap in covalent systems or accommodate small cations in ionic systems.

Linear Coordination (CN = 2)

• Two neighbors arranged at 180° to each other, creating a straight-line geometry through the central atom
• Common in molecular compounds with $sp$ hybridization, such as $CO_2$ and certain silver complexes like $[Ag(NH_3)_2]^+$
• Rare in extended crystals because most atoms can accommodate more neighbors; typically seen in chain structures or molecular crystals

Trigonal Planar Coordination (CN = 3)

• Three neighbors at 120° in a flat triangular arrangement around the central atom
• Found in borate compounds and some small, highly charged cations where only three anions can fit
• $sp^2$ hybridization in covalent cases produces this geometry, as seen in $BF_3$ and carbonate ions ($CO_3^{2-}$)

Tetrahedral Coordination (CN = 4)

• Four neighbors at 109.5° arranged at the corners of a tetrahedron—the most common low-CN geometry in crystals
• Zinc blende structure ($ZnS$) exemplifies this, as do silicates ($SiO_4^{4-}$ tetrahedra) that form the backbone of most minerals
• $sp^3$ hybridization drives this geometry in covalent compounds; radius ratios of 0.225–0.414 favor it in ionic systems

---

High Coordination Numbers (CN = 6–8)

Higher coordination numbers appear when the central atom is large enough to accommodate more neighbors without excessive repulsion. Ionic compounds with larger cations and metallic systems commonly exhibit these geometries.

Octahedral Coordination (CN = 6)

• Six neighbors at 90° forming an octahedron—the most important geometry for transition metal chemistry and many ionic compounds
• Rock salt structure ($NaCl$, $MgO$) features mutual CN = 6 for both cation and anion; transition metal complexes like $[Fe(H_2O)_6]^{2+}$ also adopt this geometry
• Radius ratio range of 0.414–0.732 predicts octahedral coordination; efficient packing with moderate-sized cations

Cubic Coordination (CN = 8)

• Eight neighbors at the corners of a cube surrounding the central atom, with neighbor distances along cube diagonals
• Cesium chloride structure ($CsCl$) is the classic example, where the large $Cs^+$ cation requires eight chloride neighbors
• Requires large cations because fitting eight anions around a center demands significant space; radius ratios above 0.732 favor this geometry

---

Close-Packed Structures (CN = 12)

The highest common coordination number occurs in metallic systems where non-directional bonding allows atoms to pack as efficiently as possible. Metals maximize the number of nearest neighbors to strengthen metallic bonding and achieve the densest possible arrangement.

Face-Centered Cubic (FCC)

• Twelve nearest neighbors arranged with atoms at face centers of a cubic unit cell—the ABCABC stacking sequence
• 74% packing efficiency—the theoretical maximum for identical spheres, shared with HCP
• Common metals include copper, aluminum, gold, and silver; this structure contributes to their malleability and ductility

Hexagonal Close-Packed (HCP)

• Twelve nearest neighbors in an ABAB stacking sequence, creating hexagonal symmetry rather than cubic
• Same 74% packing efficiency as FCC, but different symmetry affects properties like slip systems
• Common metals include zinc, magnesium, and titanium; HCP metals often show more anisotropic properties than FCC metals

---

Coordination Numbers in Different Bonding Types

The type of bonding—ionic, metallic, or covalent—fundamentally influences which coordination numbers are possible and preferred. Understanding these differences helps you predict structures across different material classes.

Ionic Compounds

• Radius ratio determines CN because ions behave approximately as charged spheres seeking electrostatic stability
• Charge balance constraints mean that CN depends on stoichiometry; in $CaF_2$ (fluorite), $Ca^{2+}$ has CN = 8 while $F^-$ has CN = 4
• Common examples include $NaCl$ (CN = 6/6), $CsCl$ (CN = 8/8), and $ZnS$ (CN = 4/4)

Metallic Structures

• Non-directional bonding allows high coordination numbers because metallic bonds don't require specific orbital orientations
• CN typically 8–12 in metals, with close-packed structures (CN = 12) maximizing metallic bond strength
• Body-centered cubic (BCC) metals like iron have CN = 8, trading some packing efficiency for other stability factors

Covalent Compounds

• Hybridization dictates CN because covalent bonds require orbital overlap in specific directions
• Lower coordination numbers (CN = 2–4) dominate because atoms typically form limited numbers of covalent bonds
• Network solids like diamond (CN = 4) show how covalent bonding constrains geometry despite having space for more neighbors

---

Effects on Crystal Properties

Coordination number doesn't just describe geometry—it directly influences the physical and chemical properties of crystalline materials. Higher CN generally correlates with greater stability, but the relationship with other properties is more nuanced.

Stability and Energy

• Higher CN often means lower energy because more neighbors provide more stabilizing interactions (ionic attractions or metallic bonding)
• Madelung constant increases with CN in ionic compounds, reflecting stronger electrostatic stabilization
• Phase transitions can change CN; pressure often increases coordination number by forcing atoms closer together

Physical Properties

• Density correlates with CN because higher coordination typically means more efficient packing
• Hardness and melting point relate to CN through bond strength and the number of bonds that must break
• Optical and electronic properties depend on the geometry that CN creates; octahedral vs. tetrahedral crystal fields split d-orbitals differently

---

Quick Reference Table

---

Self-Check Questions

1. Both $NaCl$ and $CsCl$ are 1:1 ionic compounds, yet they have different coordination numbers. Explain why, using the radius ratio principle.

2. FCC and HCP structures both have CN = 12 and identical packing efficiency. What distinguishes them, and how might this affect material properties?

3. Why do covalent network solids like diamond have lower coordination numbers than metals, even though both are composed of single element types?

4. Compare the coordination environments in the fluorite structure ($CaF_2$): what is the CN of $Ca^{2+}$ versus $F^-$, and why do they differ?

5. If you were given an unknown ionic compound and told the cation-to-anion radius ratio is 0.52, what coordination number would you predict, and what structure type might you expect?