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Materials Science · Chapter 3 of 10 · Beginner

The Structure of Crystalline Solids

Most metals are crystals: atoms stacked in a pattern that repeats millions of times. The way they pack sets density, and it decides how easily the metal will later deform.

Readiness check

This chapter is geometry applied to atoms. Tick only what you can do closed-notes.

Find the volume of a cube and a sphere.
Use the Pythagorean theorem in 3D (the cube diagonal).
Recall Avogadro's number and the mole.
Work in nanometres and grams per cubic centimetre.
Picture a 3D arrangement from a 2D drawing.

0 or 1 weak itemsContinue with this chapter.

2 weak itemsRefresh solid geometry and unit conversions.

3 or more weak itemsReview Chapter 2 and basic geometry before continuing.

The core idea

A crystal is a single small arrangement of atoms, the unit cell, repeated in all directions; counting the atoms in one cell gives both the density and the packing efficiency.

APF = (atoms × volume each)/V_cellρ = nA/(V_CN_A)FCC: a = 2R√2 · BCC: a = 4R/√3

Metals adopt one of three close arrangements: face-centred cubic (FCC), body-centred cubic (BCC), or hexagonal close-packed (HCP). Each unit cell holds a definite number of atoms, has a fixed relation between the cell edge a and the atomic radius R, and packs space to a definite fraction. From that you can compute the theoretical density and predict how readily the metal will slip and deform later.

The skill works when: you count atoms per cell correctly (sharing corners and faces) and use the right a-to-R relation.

The skill breaks down when: shared atoms are over-counted, or the cube edge is confused with the close-packed direction.

The concept. Two cubic unit cells. BCC adds one atom at the body centre; FCC adds an atom at the centre of each face. The face-centred packing fills more space, which is why FCC metals tend to be more ductile.

The skills, taught in order

Crystal structure is bookkeeping with geometry. Five skills build from the unit cell to density, packing, and the planes along which metals deform.

3.1 Unit cells and lattices

A crystal is built by repeating a unit cell, the smallest box that captures the pattern, in three dimensions. Its edge length a is the lattice parameter. Atoms on corners and faces are shared with neighbouring cells, so each contributes only a fraction to one cell.

3.2 The three metallic structures

Most metals are FCC, BCC, or HCP. Each has a fixed atom count, coordination number (nearest neighbours), packing factor, and a-to-R relation.

Structure	Atoms per cell	Coordination	APF	a and R
FCC	4	12	0.74	a = 2R√2
BCC	2	8	0.68	a = 4R/√3
HCP	6	12	0.74	c/a ≈ 1.633

3.3 Atomic packing factor

The APF is the fraction of cell volume filled by atoms (treated as hard spheres): APF = (atoms per cell × ⁴⁄₃πR³)/V_cell. FCC and HCP reach 0.74, the densest possible packing of equal spheres; BCC reaches 0.68. Denser packing gives more close-packed planes, the key to ductility.

3.4 Theoretical density

Knowing the cell, the density follows: ρ = nA/(V_CN_A), where n is atoms per cell, A the atomic weight, V_C the cell volume, and N_A Avogadro's number. Computed densities match measured ones closely, confirming the structure.

3.5 Directions, planes, and polycrystals

Directions and planes in a crystal are labelled by Miller indices, such as [100] or (111). They matter because slip happens on the most densely packed planes. Real metals are polycrystalline, many small crystals (grains) in random orientations, so bulk properties average out the single-crystal anisotropy. X-ray diffraction is how the structure is measured.

Engineering connection: packing and close-packed planes set ductility (Chapter 6), grain structure sets strength (Chapter 6), and the iron polymorphism (BCC to FCC) underlies all steel heat treatment (Chapter 9).

Worked example 1: theoretical density of copper

Copper is FCC with atomic radius R = 0.128 nm and atomic weight A = 63.55 g/mol. Compute its theoretical density and compare with the measured 8.96 g/cm³.

Figure 1. The FCC cell of copper holds four atoms. With the edge fixed by a = 2R√2, the density follows directly from atom count, atomic weight, and cell volume.

ProblemFind copper's theoretical density from its FCC cell in Figure 1.
Given / findFCC (n = 4), R = 0.128 nm = 1.28×10⁻⁸ cm, A = 63.55 g/mol, N_A = 6.022×10²³. Find ρ.
AssumptionsHard-sphere atoms touching along the face diagonal, perfect crystal.
ModelGet the edge from a = 2R√2, the cell volume a³, then ρ = nA/(V_CN_A).
Equationsa = 2R√2 V_C = a³ ρ = nA/(V_CN_A)
Solvea = 2(1.28×10⁻⁸)√2 = 3.62×10⁻⁸ cm, so V_C = 4.75×10⁻²³ cm³. Then ρ = (4 × 63.55)/(4.75×10⁻²³ × 6.022×10²³) = 254.2/28.58 = 8.90 g/cm³.
CheckThe measured value is 8.96 g/cm³, so the model is within 1%. The small gap reflects the hard-sphere idealisation; the agreement confirms copper is FCC.
ConclusionFour atoms in a cell of known size reproduce the bulk density, a striking link from the atomic scale to a handbook number. The same calculation, with n = 2 and a = 4R/√3, would handle a BCC metal.

Result. Theoretical density ρ = 8.90 g/cm³, within 1% of the measured 8.96 g/cm³.

Worked example 2: packing factor, BCC versus FCC

Compute the atomic packing factor of a BCC and an FCC metal, and explain what the difference implies for how the two deform.

Figure 2. FCC reaches the close-packed limit of 0.74; BCC fills only 0.68. The denser FCC structure has more close-packed planes for slip, so FCC metals are generally more ductile.

ProblemFind the APF for BCC and FCC and interpret the difference, as in Figure 2.
Given / findBCC: n = 2, a = 4R/√3. FCC: n = 4, a = 2R√2. Find APF for each.
AssumptionsHard spheres of radius R touching along the close-packed direction of each cell.
ModelDivide the total atom volume in a cell by the cell volume.
EquationsAPF = n(⁴⁄₃πR³)/a³
SolveBCC: a³ = (4R/√3)³ = 12.3R³, so APF = 2(4.19R³)/12.3R³ = 0.68. FCC: a³ = (2R√2)³ = 22.6R³, so APF = 4(4.19R³)/22.6R³ = 0.74.
CheckBoth are below 1 (atoms cannot fill all space), and FCC exceeds BCC, matching the known close-packed value of 0.74 for FCC and HCP. The radius R cancels, so the APF is a pure geometric property of the structure.
ConclusionFCC packs denser and has more close-packed slip planes, so FCC metals (copper, aluminium, gold) tend to be ductile, while BCC metals can be stronger but less forgiving. Packing geometry foreshadows mechanical behaviour.

Result. APF = 0.68 (BCC) and 0.74 (FCC); the denser FCC favours ductility.

Misconceptions and diagnostics

Mistake	Symptom	Diagnostic question	Correction
Over-counting shared atoms	FCC counted as 14 atoms, not 4	"How much of each atom is inside this cell?"	Corner atoms count ⅛, face atoms ½; FCC totals 4.
Wrong a-to-R relation	Edge used as the touching direction	"Along which line do atoms touch?"	FCC touch along the face diagonal (a = 2R√2); BCC along the body diagonal (a = 4R/√3).
Unit slips in density	Density off by powers of ten	"Is R in cm and A in g/mol?"	Keep consistent units; nm to cm is 10⁻⁷.
Single crystal assumed	Anisotropy expected in a bulk metal	"Is it one crystal or many grains?"	Most metals are polycrystalline, so bulk properties are averaged.

Practice ladder

Level 1 · Direct skill

Iron is BCC with R = 0.124 nm. Find its lattice parameter a.

Show answer

a = 4R/√3 = 4(0.124)/1.732 = 0.286 nm. This is the edge of the iron unit cell at room temperature, the well-known 0.287 nm of ferrite.

Level 2 · Mixed concept

How many atoms belong to one FCC unit cell, counting the sharing of corner and face atoms?

Show answer

8 corners × ⅛ + 6 faces × ½ = 1 + 3 = 4 atoms. Correct counting of shared atoms is what makes the density formula work.

Level 3 · Independent problem

Compute the theoretical density of BCC iron (R = 0.124 nm, A = 55.85 g/mol, n = 2) and compare with the measured 7.87 g/cm³.

Show answer

a = 0.286 nm = 2.86×10⁻⁸ cm, V_C = 2.34×10⁻²³ cm³. ρ = (2 × 55.85)/(2.34×10⁻²³ × 6.022×10²³) = 111.7/14.1 = 7.92 g/cm³, within 1% of 7.87. The BCC count n = 2 is the only change from the copper example.

Level 4 · Transfer to real engineering

Iron changes from BCC to FCC when heated above 912 °C. Predict what happens to its density at that transformation, and why it matters for heat treatment.

What good work looks like

Recognising FCC packs denser (0.74 vs 0.68), so the metal contracts slightly on transforming to FCC; noting this volume change and the denser packing underlie carbon solubility and steel heat treatment in Chapter 9.

Working with AI, and proving it yourself

Use AI as an examiner, not a solver

"Check that I counted shared corner and face atoms correctly."

"Give me five metals; I will state the structure and the a-to-R relation."

"Compute the density." Setting up n, a, and V_C yourself is the skill.

"What is the packing factor?" Deriving it from the geometry is the point.

Portfolio task

Compute the theoretical density of two metals of different structure, compare with handbook values, and explain any difference in terms of packing.

Must include: correct atom counts, the right a-to-R relation, consistent units, and a comparison with measured density.

Retrieval and spaced review

Closed notes. Answer out loud, then reveal.

1. How many atoms are in an FCC and a BCC unit cell?

FCC has 4; BCC has 2.

2. Give the APF of FCC, BCC, and HCP.

FCC and HCP are 0.74; BCC is 0.68.

3. Write the a-to-R relations for FCC and BCC.

FCC: a = 2R√2; BCC: a = 4R/√3.

4. State the theoretical density formula.

ρ = nA/(V_CN_A).

5. Why are most metals not anisotropic in bulk?

They are polycrystalline, with many randomly oriented grains that average out the single-crystal directionality.

TodayFinish this quiz and Levels 1 and 2 of the ladder.

+1 dayRe-derive copper's density from a blank page.

+3 daysCompute one BCC and one FCC density.

+7 daysCarry the close-packed planes into defects, Chapter 4.

+30 daysRecall iron's BCC-to-FCC change for heat treatment, Chapter 9.

Textbook mapping

Item	Mapping
Primary source	Callister and Rethwisch, Materials Science and Engineering: An Introduction, Chapter 3 (The Structure of Crystalline Solids)
Cross-reference	Askeland, Ch. 3 · Shackelford, Ch. 3
Core topics	3.1 Unit cells · 3.2 FCC, BCC, HCP · 3.3 Packing factor · 3.4 Theoretical density · 3.5 Directions, planes, polycrystals
Engineering connection	Packing sets ductility; iron polymorphism underlies steel heat treatment.
Read next	Chapter 4: Imperfections and Diffusion.