Close-Packed Layers

Introduction

Equal spheres can form a dense triangular layer, and the next layer nests in some of its hollows. Repeating this construction in two different ways produces cubic close packing, equivalent to FCC, and hexagonal close packing, HCP. Both give twelve nearest neighbors and the same ideal sphere packing fraction, but their stacking symmetries differ.

Core explanation

Within a close-packed layer A, each sphere touches six neighbors. The triangular hollows between spheres provide positions for a second layer B, where each B sphere contacts three A spheres below. A third layer can sit directly above A positions, producing ABAB... stacking. This is the hexagonal close-packed sequence. Alternatively, the third layer can occupy a new set of hollows C, producing ABCABC... stacking. This is cubic close packing, which has an FCC lattice description.

Both structures give each sphere six neighbors in its own layer, three above and three below, totaling twelve. They also fill the same ideal fraction of space, π/(3√2)≈74.0%. Therefore coordination number and packing fraction alone cannot distinguish HCP from FCC. The layer order and diffraction pattern provide the distinction.

The layers are described by letters only as relative lateral positions. A, B and C do not denote different chemical elements. In a monatomic metal all sphere positions can contain the same atom. A chemical compound may place different ions into the close-packed array and its voids, requiring separate occupancy analysis.

Real close-packed crystals can have stacking faults: a local ABC sequence may briefly switch toward AB or another order. A fault is a defect, not automatically a different bulk phase. Such defects affect mechanical and electronic properties. The ideal infinite sequences are reference structures against which real material is compared.

Several metals have HCP structures under ordinary conditions, such as magnesium, while aluminium and copper are FCC. It is unsafe to rank their strength simply from close-packing fraction because they share that fraction. Their slip systems, bonding, temperature and microstructures differ. The structure-property connection must include more than void percentage.

The same close-packed geometry creates interstitial spaces called tetrahedral and octahedral voids. Counting these voids helps analyze ionic structures and alloying. For N close-packed host spheres, there are N octahedral and 2N tetrahedral sites in the ideal array, though not all must be occupied. The next pages derive and use this counting.

Step-by-step reasoning

1. Draw a triangular A layer with six same-layer neighbors. 2. Nest B spheres in one set of triangular hollows. 3. Place the third layer over A for ABAB or at new C positions for ABCABC. 4. Count six same-layer and three each above and below. 5. Distinguish stacking by sequence, not packing fraction.

Visual explanation

Draw three triangular layers from above, with A dots black, B dots blue and C dots red. For HCP show the third layer overlapping A positions; for FCC show C in the remaining hollows. Label ABAB and ABCABC repeats and twelve-neighbor count.

Real-world analogy

Oranges stacked in a market crate can put the third tier over the first or over a different set of gaps. Both arrangements are dense, yet a side view reveals a different repeating pattern.

Real-world example

Magnesium and aluminium can have different close-packed stacking at ordinary conditions. Comparing them teaches that equal nearest-neighbor count does not make crystal symmetry or mechanical behavior identical.

Why?

Why do FCC and HCP have the same ideal packing fraction? Each layer is equally dense and each new layer nests in equivalent hollows; changing the third-layer position changes sequence but not local sphere contacts.

Common misconception

“ABC means layers contain three different elements.” The letters mark positional offsets of otherwise identical close-packed layers, not chemical identities.

Worked example

An unknown monatomic sample has close-packed layers A, B, C, A, B, C. It is cubic close packed, with FCC geometry. Each atom has six neighbors in its own layer and three in each neighboring layer, so CN=12. Its ideal hard-sphere fraction is about 74%, the same as AB-stacked HCP; the sequence, not fraction, identifies it.

Quick check

1. Which close-packed sequence corresponds to HCP? Answer: ABAB... stacking.

Exam focus

Draw or state AB versus ABC, give CN=12 and equal 74% packing, and explain why layer labels do not represent chemical species.

Advanced insight

Stacking faults can be described as local deviations in the layer sequence. They influence dislocation motion and may change diffraction peak shapes, connecting an ideal geometry exercise to real material behavior.

Summary

Close-packed triangular layers stack ABAB for HCP or ABCABC for cubic close packing/FCC. Both give twelve neighbors and about 74% ideal packing. Their layer sequence and symmetry differ.

Practice questions

1. What is the stacking sequence of cubic close packing? Answer: ABCABC... . 2. How many nearest neighbors surround one sphere in either ideal close-packed structure? Answer: Twelve, six in its layer and three each above and below. 3. Can packing fraction distinguish HCP from FCC? Answer: No. Both have the same ideal equal-sphere packing fraction.