---
title: "FCC unit cell: body diagonal and packing density"
subject: "Electrical Engineering Material"
chapter: 1
key: "semester_3/electrical_engineering_material:fcc-body-diagonal-packing"
source: https://www.bctengineer.com/q/semester_3/electrical_engineering_material/fcc-body-diagonal-packing
site: BCT Engineering
---

# FCC unit cell: body diagonal and packing density

Electrical Engineering Material · Chapter 1: Theory of Metals

- Asked in: 2074 Chaitra, 6 marks; 2070 Asar, 4 marks (asked 2 times)
- 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/fcc-body-diagonal-packing
- Whole chapter: https://www.bctengineer.com/md/semester_3/electrical_engineering_material/chapter-1

## Question

Draw face centered cubic (FCC) unit cell and find body diagonal and packing density.

## Short answer

In FCC, atoms touch along face diagonal: √2 a = 4R, so a = 2√2 R. Body diagonal = √3 a = 2√6 R ≈ 4.90 R. Packing density = (4 × 4/3 π R³) / a³ = π/(3√2) ≈ 0.74 (74%).

## Answer

In a **face centred cubic (FCC)** unit cell there is an atom at each of the 8 corners and one 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    |  |     atoms touch along
      |  o---------|--o     each face diagonal
      | /    x     | /
      |/           |/
      o------------o

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

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

#### Lattice constant
Along a face diagonal: corner radius + face atom diameter + corner radius.

$$
\sqrt{2}\,a = 4R \quad\Rightarrow\quad a = \frac{4R}{\sqrt{2}} = 2\sqrt{2}\,R = 2.828\,R
$$

#### Body diagonal

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

#### Number of atoms per cell

$$
N = 8\times\frac{1}{8} + 6\times\frac{1}{2} = 1 + 3 = 4
$$

#### Packing density

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

**Answer:** body diagonal $= 2\sqrt{6}\,R = 4.90R$ (with $a = 2\sqrt{2}R$), packing density $= \pi/(3\sqrt{2}) = 0.74$ (74%). FCC is a close-packed structure with coordination number 12. Examples: Cu, Al, Ag, Au, Ni.

## Check yourself

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

1. Along which direction do atoms touch in FCC?
   A. Body diagonal
   B. Cube edge
   C. Face diagonal, sqrt(2) a = 4R
   D. Half the body diagonal
   Answer: C. Corner radius, face-atom diameter and corner radius along the face diagonal give 4R. (Key point: Touching direction)

2. What is a in terms of R for FCC?
   A. 4R/sqrt(3)
   B. 2 sqrt(2) R
   C. 2R
   D. sqrt(2) R
   Answer: B. sqrt(2) a = 4R gives a = 2 sqrt(2) R = 2.828R. (Key point: Lattice constant)

3. What is the body diagonal of the FCC cell?
   A. 4R
   B. 2.828R
   C. 2 sqrt(6) R = 4.90R
   D. sqrt(6) R
   Answer: C. Body diagonal = sqrt(3) a = sqrt(3) x 2 sqrt(2) R = 2 sqrt(6) R. (Key point: Body diagonal)

4. How many atoms per FCC cell and what is the packing density?
   A. 2 atoms, 0.68
   B. 4 atoms, 0.74
   C. 4 atoms, 0.52
   D. 6 atoms, 0.74
   Answer: B. N = 8(1/8) + 6(1/2) = 4, and APF = pi / (3 sqrt(2)) = 0.74. (Key point: Atoms per cell and APF)

## Close variants set in other exams

- [Lattice, basis and FCC packing density](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Alattice-basis-fcc) — Electrical Engineering Material, chapter 1, asked 3×

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