---
title: "Lattice, basis and FCC packing density"
subject: "Electrical Engineering Material"
chapter: 1
key: "semester_3/electrical_engineering_material:lattice-basis-fcc"
source: https://www.bctengineer.com/q/semester_3/electrical_engineering_material/lattice-basis-fcc
site: BCT Engineering
---

# Lattice, basis and FCC packing density

Electrical Engineering Material · Chapter 1: Theory of Metals

- Asked in: 2078 Bhadra, 2+6 marks
- Video: [Unit Cell Chemistry   Simple Cubic, Body Centered Cubic, Face Centered Cubic Crystal Lattice Structu](https://www.youtube.com/watch?v=HCWwRh5CXYU) — The Organic Chemistry Tutor
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/electrical_engineering_material/lattice-basis-fcc
- Whole chapter: https://www.bctengineer.com/md/semester_3/electrical_engineering_material/chapter-1

## Question

Define lattice and basis of a crystal structure. Draw the face centered cubic (FCC) unit cell and find the body diagonal and packing density.

## Short answer

A lattice is a periodic array of points; a basis is the atom(s) attached to each point. For FCC (atoms at corners and face centres): body diagonal = 2√6 R ≈ 4.90R, packing density = π/(3√2) ≈ 0.74 (74%).

## Answer

#### Lattice and basis
- **Lattice:** a regular, periodic array of imaginary points in space such that every point has exactly the same surroundings. It shows only the geometry of repetition.
- **Basis:** the atom or group of atoms attached to each lattice point. Repeating the basis at every lattice point builds the real crystal.

$$
\text{Crystal structure} = \text{Lattice} + \text{Basis}
$$

Example: copper = FCC lattice + basis of one Cu atom; NaCl = FCC lattice + basis of one Na⁺ and one Cl⁻ ion.

#### FCC unit cell
An atom sits at each of the 8 corners and at the centre of each of the 6 faces. The atoms touch along the **face diagonal**.

```text
         o------------o
        /|    x      /|
       / |          / |     o = corner atom
      o------------o  |     x = face-centre atom
      |  |  x      |x |
      |x |    x    |  |
      |  o---------|--o
      | /    x     | /
      |/           |/
      o------------o

  One face:   o-------o
              | \   / |    face diagonal
              |   x   |    = R + 2R + R = 4R
              | /   \ |
              o-------o
```

Let $R$ = atomic radius, $a$ = lattice constant.

Along a face diagonal: $\sqrt{2}\,a = 4R \Rightarrow a = 2\sqrt{2}\,R = 2.828R$.

#### Body diagonal

$$
\text{Body diagonal} = \sqrt{3}\,a = \sqrt{3}\times2\sqrt{2}\,R = 2\sqrt{6}\,R = 4.899R
$$

#### Packing density
Atoms per cell: $N = 8\times\frac{1}{8} + 6\times\frac{1}{2} = 4$.

$$
\begin{aligned}
\text{APF} &= \frac{N\cdot\frac{4}{3}\pi R^3}{a^3} = \frac{4\times\frac{4}{3}\pi R^3}{16\sqrt{2}\,R^3} \\
&= \frac{\pi}{3\sqrt{2}} = 0.74
\end{aligned}
$$

**Answer:** body diagonal $= 2\sqrt{6}\,R \approx 4.90R$; packing density $= 0.74$ (74 % of the cell volume is filled). FCC is close-packed with coordination number 12; examples are Cu, Al, Ag, Au.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Crystal structure equals:
   A. Lattice x Basis
   B. Lattice + Basis
   C. Basis - Lattice
   D. Lattice only
   Answer: B. The basis (atoms) attached to each lattice point builds the real crystal. (Key point: Lattice vs basis)

2. In FCC the atoms touch along the:
   A. Edge
   B. Body diagonal
   C. Face diagonal
   D. Space diagonal of half cell
   Answer: C. Face diagonal = 4R, so sqrt(2) a = 4R. (Key point: FCC contact)

3. The body diagonal of the FCC cell in terms of R is:
   A. 4.899R
   B. 4R
   C. 2.828R
   D. 5.657R
   Answer: A. a = 2 sqrt(2) R, body diagonal = sqrt(3) a = 2 sqrt(6) R = 4.899R. (Key point: Body diagonal)

4. Packing density of FCC is:
   A. 0.68
   B. 0.52
   C. 0.74
   D. 0.91
   Answer: C. 4 atoms per cell: pi/(3 sqrt(2)) = 0.74. (Key point: Packing density)

## Close variants set in other exams

- [FCC unit cell: body diagonal and packing density](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Afcc-body-diagonal-packing) — Electrical Engineering Material, chapter 1, asked 3×

## Related in Electrical Engineering Material

- [BCC unit cell: lattice constant and packing](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Abcc-lattice-constant-packing) — Electrical Engineering Material, chapter 1, asked 2×
- [Copper FCC: atoms, packing, concentration and density](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Acopper-fcc-atoms-density) — Electrical Engineering Material, chapter 1
- [Copper FCC: packing density and atomic concentration](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Acopper-fcc-packing-concentration) — Electrical Engineering Material, chapter 1
- [Copper FCC: packing, density and Fermi energy](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Acopper-fcc-fermi-energy) — Electrical Engineering Material, chapter 1
- [Lattice constant of a simple cubic structure](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Asimple-cubic-lattice-constant) — Electrical Engineering Material, chapter 1
- [BCC chromium: packing density and coordination number](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Abcc-chromium-packing) — Electrical Engineering Material, chapter 1

## Related in other subjects

- [BCC, FCC and HCP crystal structures](https://www.bctengineer.com/question.md?key=semester_3%2Fmaterial_science%3Abcc-fcc-hcp-structures) — Material Science, chapter 2