---
title: "BCC unit cell: lattice constant and packing"
subject: "Electrical Engineering Material"
chapter: 1
key: "semester_3/electrical_engineering_material:bcc-lattice-constant-packing"
source: https://www.bctengineer.com/q/semester_3/electrical_engineering_material/bcc-lattice-constant-packing
site: BCT Engineering
---

# BCC unit cell: lattice constant and packing

Electrical Engineering Material · Chapter 1: Theory of Metals

- Asked in: 2076 Chaitra, 4 marks; 2070 Chaitra, 4 marks (asked 2 times)
- Video: [The Body Centered Cubic Crystal Structure](https://www.youtube.com/watch?v=cEXgiXrBliY) — Scott Ramsay
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/electrical_engineering_material/bcc-lattice-constant-packing
- Whole chapter: https://www.bctengineer.com/md/semester_3/electrical_engineering_material/chapter-1

## Question

Calculate the lattice constant, face diagonal, body diagonal and packing density of body centered cubic (BCC) crystal unit cell.

## Short answer

Atoms touch along the body diagonal: √3 a = 4R, so a = 4R/√3 ≈ 2.309R. Face diagonal = √2 a = 4√(2/3)R ≈ 3.266R, body diagonal = 4R. Packing density APF = (2 × 4πR³/3) / a³ = √3π/8 ≈ 0.68 (68%).

## Answer

In a **body centred cubic (BCC)** unit cell there is an atom at each of the 8 corners and one atom at the centre of the cube. The atoms touch each other along the **body diagonal**.

```text
        o-----------o
       /|          /|
      / |         / |      o = corner atom
     o-----------o  |      @ = body-centre atom
     |  |   @    |  |
     |  o--------|--o      atoms touch along
     | /         | /       the body diagonal
     |/          |/
     o-----------o
```

Let $R$ = atomic radius and $a$ = lattice constant (edge of the cube).

#### Lattice constant
Along the body diagonal there is one corner atom radius, one full centre atom diameter and another corner radius:

$$
\begin{aligned}
\text{Body diagonal} &= \sqrt{3}\,a = R + 2R + R = 4R \\
a &= \frac{4R}{\sqrt{3}} = 2.309\,R
\end{aligned}
$$

#### Face diagonal

$$
\text{Face diagonal} = \sqrt{2}\,a = \sqrt{2}\times\frac{4R}{\sqrt{3}} = 4\sqrt{\frac{2}{3}}\,R = 3.266\,R
$$

#### Body diagonal

$$
\text{Body diagonal} = \sqrt{3}\,a = 4R
$$

#### Packing density
Number of atoms per cell: $8\times\frac{1}{8} + 1 = 2$.

$$
\begin{aligned}
\text{APF} &= \frac{\text{volume of atoms in cell}}{\text{volume of cell}} = \frac{2\times\frac{4}{3}\pi R^3}{a^3} \\
&= \frac{\frac{8}{3}\pi R^3}{\left(\frac{4R}{\sqrt{3}}\right)^3} = \frac{\frac{8}{3}\pi R^3\cdot3\sqrt{3}}{64R^3} = \frac{\sqrt{3}\,\pi}{8} \\
&= 0.68
\end{aligned}
$$

**Answer:** $a = 4R/\sqrt{3} = 2.309R$; face diagonal $= 3.266R$; body diagonal $= 4R$; packing density $= \sqrt{3}\pi/8 = 0.68$ (68%). Coordination number is 8. Examples: Fe ($\alpha$), Cr, Na, W.

## 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 BCC, giving the lattice constant?
   A. Cube edge
   B. Face diagonal
   C. Body diagonal, sqrt(3) a = 4R
   D. Face centre
   Answer: C. Corner radius + centre diameter + corner radius = 4R = sqrt(3) a. (Key point: Touching direction)

2. What is a in terms of R?
   A. a = 4R/sqrt(3)
   B. a = 4R/sqrt(2)
   C. a = 2R
   D. a = sqrt(3) R
   Answer: A. From sqrt(3) a = 4R, a = 4R/sqrt(3) = 2.309R; face diagonal is sqrt(2) a = 3.266R. (Key point: Lattice constant)

3. BCC has 2 atoms per cell. What is its packing density?
   A. 0.52
   B. 0.74
   C. 0.68
   D. 0.34
   Answer: C. APF = 2(4/3)pi R^3 / a^3 = sqrt(3) pi / 8 = 0.68. (Key point: Packing density)

## Related in Electrical Engineering Material

- [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×
- [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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## 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
- [Density from lattice constant](https://www.bctengineer.com/question.md?key=semester_3%2Fmaterial_science%3Acopper-and-tungsten-density) — Material Science, chapter 2
- [Planar and linear density of BCC iron](https://www.bctengineer.com/question.md?key=semester_3%2Fmaterial_science%3Aplanar-and-linear-density-bcc-iron) — Material Science, chapter 2