What is the difference between cubic and hexagonal close packing?

Both pack spheres as efficiently as possible, but they differ in the stacking sequence of close-packed layers: hexagonal close packing repeats an ABAB pattern while cubic close packing repeats ABCABC, giving the face-centred cubic arrangement.

Why does the radius ratio predict coordination number?

A cation must be large enough to keep the surrounding anions from touching one another; as the cation-to-anion radius ratio increases, progressively higher coordination numbers become geometrically stable, which is the basis of the radius-ratio rules.

Close Packing and Crystal Structures

Many metals and ionic solids derive from the close packing of spheres, with cations filling octahedral and tetrahedral holes to generate the recurring structure types of inorganic chemistry.

Definition

Close packing and crystal structures is the description of how atoms and ions arrange in extended solids by efficient sphere packing, with smaller ions occupying the interstitial holes, giving rise to characteristic structure types.

Scope

This topic covers the geometric description of inorganic crystal structures: cubic and hexagonal close packing and their interstitial octahedral and tetrahedral holes; the derivation of common structure types such as rock salt, zinc blende, fluorite, rutile, and perovskite; radius-ratio rules and Pauling's rules for predicting coordination and structure; and the relationship between structure type and stoichiometry. It treats geometry and structure prediction rather than the energetics covered in the lattice-energy topic.

Core questions

What are cubic and hexagonal close packing and how many holes do they contain?
How are the common ionic structure types derived from close-packed arrays?
How do radius-ratio and Pauling's rules predict coordination and structure?
How does stoichiometry constrain which holes are filled?

Key concepts

Cubic and hexagonal close packing
Octahedral and tetrahedral holes
Rock-salt and zinc-blende structures
Fluorite and rutile structures
Perovskite structure
Radius-ratio and Pauling's rules

Key theories

Close packing and interstitial holes: Spheres pack most efficiently in cubic or hexagonal close-packed arrangements, each providing one octahedral and two tetrahedral holes per sphere into which cations can be placed to build ionic structures.
Common structure types: Filling specified fractions of the holes in a close-packed anion array generates the rock-salt, zinc-blende, fluorite, rutile, and related structure types that recur across binary and ternary inorganic solids.
Radius-ratio and Pauling's rules: The ratio of cation to anion radius predicts the preferred coordination number, and Pauling's electrostatic-valence and related rules constrain how polyhedra share corners, edges, and faces in stable structures.

Clinical relevance

Recognizing structure types underlies the design and interpretation of functional inorganic materials, including the perovskite oxides used in catalysis, ferroelectrics, and solar cells, and the spinels used in batteries and magnets.

History

Bragg's early X-ray determinations revealed that simple salts such as sodium chloride adopt close-packed structures, and Goldschmidt's compilation of ionic radii enabled radius-ratio reasoning. Pauling's 1929 rules and Wells's systematic surveys organized the vast catalogue of inorganic structure types.

Key figures

Linus Pauling
William Lawrence Bragg
Victor Goldschmidt
Alexander Wells

Seminal works

pauling1929
wells2012
west2014

Frequently asked questions

What is the difference between cubic and hexagonal close packing?: Both pack spheres as efficiently as possible, but they differ in the stacking sequence of close-packed layers: hexagonal close packing repeats an ABAB pattern while cubic close packing repeats ABCABC, giving the face-centred cubic arrangement.
Why does the radius ratio predict coordination number?: A cation must be large enough to keep the surrounding anions from touching one another; as the cation-to-anion radius ratio increases, progressively higher coordination numbers become geometrically stable, which is the basis of the radius-ratio rules.