Simple Cubic Packing: The Simplest Crystal Structure

What Is Simple Cubic Packing?

Simple cubic (SC) packing is the most basic three-dimensional crystal structure. Each atom sits at the corners of a cube, with atoms touching along the cube edges. It is also called primitive cubic.

Despite its simplicity, SC packing is rare in nature because it has the lowest packing efficiency of all cubic structures. Only polonium (Po) crystallises in a simple cubic structure at room temperature.

Learning Goals: By the end of this guide, you should be able to:

Describe the arrangement of atoms in a simple cubic unit cell.

Calculate the number of atoms per unit cell, coordination number, and packing efficiency.

Relate the edge length to the atomic radius.

Compare SC with BCC and FCC/CCP structures.

Anatomy of the SC Unit Cell

Atom Positions

Each of the 8 corners of the cube has one atom. Each corner atom is shared among 8 adjacent unit cells, so each corner contributes $\frac{1}{8}$ of an atom.

\text{Atoms per unit cell} = 8 \times \frac{1}{8} = 1

Edge Length and Atomic Radius

Atoms touch along the edge of the cube:

a = 2r

where $a$ is the edge length and $r$ is the atomic radius.

Coordination Number

Each atom in SC is in direct contact with 6 neighbours — one on each face of the cube (top, bottom, left, right, front, back).

\text{Coordination number} = 6

Packing Efficiency

Packing efficiency measures what fraction of the unit cell volume is occupied by atoms.

\text{Packing efficiency} = \frac{V_{\text{atoms}}}{V_{\text{cell}}} \times 100\%

= \frac{1 \times \frac{4}{3}\pi r^3}{(2r)^3} \times 100\% = \frac{\frac{4}{3}\pi r^3}{8r^3} \times 100\% = \frac{\pi}{6} \times 100\% \approx 52.4\%

Structure	Atoms/cell	Coordination	Packing Efficiency
Simple Cubic (SC)	1	6	52.4%
Body-Centered Cubic (BCC)	2	8	68.0%
Face-Centered Cubic (FCC/CCP)	4	12	74.0%

SC has the lowest efficiency — nearly half the volume is empty space. This explains why so few elements adopt this structure.

Worked Examples

Example 1: Calculate the Density of Polonium

Given: $M = 209\ g/mol$ , $r = 167\ pm$ , SC structure.

a = 2r = 2(167) = 334\ pm = 3.34 \times 10^{-8}\ cm

\rho = \frac{Z \times M}{N_A \times a^3} = \frac{1 \times 209}{6.022 \times 10^{23} \times (3.34 \times 10^{-8})^3}

= \frac{209}{6.022 \times 10^{23} \times 3.726 \times 10^{-23}} = \frac{209}{22.44} = 9.31\ g/cm^3

(Experimental value: 9.32 g/cm³ ✅)

Example 2: Finding Atomic Radius from Density

Given: A metal with SC structure, $\rho = 7.87\ g/cm^3$ , $M = 55.85\ g/mol$ .

a^3 = \frac{Z \times M}{N_A \times \rho} = \frac{1 \times 55.85}{6.022 \times 10^{23} \times 7.87} = 1.179 \times 10^{-23}\ cm^3

a = 2.275 \times 10^{-8}\ cm = 227.5\ pm, \quad r = \frac{a}{2} = 113.7\ pm

Common Mistakes

Forgetting atom sharing at corners — Each corner atom is shared by 8 unit cells, not 1. Always divide by 8.
Confusing edge-touching with face or body diagonal touching — In SC, atoms touch along the edge ( $a = 2r$ ). In FCC they touch along the face diagonal; in BCC along the body diagonal.
Using the wrong Z value — SC has $Z = 1$ atom per unit cell, not 8. The 8 corner atoms each contribute $\frac{1}{8}$ .
Unit conversion errors in density — Be careful: $1\ pm = 10^{-10}\ cm$ . Many students lose marks here.

Exam Tips (A-Level / AP / IB)

Memorise the three packing efficiencies: SC ≈ 52%, BCC ≈ 68%, FCC ≈ 74%.
Be able to derive $a = 2r$ for SC from a diagram — draw the unit cell face showing atoms touching along the edge.
Know that SC is extremely rare — polonium is the only element with SC at standard conditions.
For density calculations, clearly state $Z$ , $a$ , and units at each step.

Frequently Asked Questions

Why is simple cubic packing so rare?

It has the lowest packing efficiency (52.4%), meaning atoms leave a lot of empty space. More tightly packed arrangements (BCC, FCC) are energetically more favourable because they maximise attractive interactions between atoms.

Does temperature affect which structure a metal adopts?

Yes! Some metals undergo allotropic transitions. For example, iron is BCC at room temperature (α-Fe), converts to FCC at 912°C (γ-Fe), and back to BCC at 1394°C (δ-Fe).

What is the void space in a SC structure?

The void space is $100\% - 52.4\% = 47.6\%$ . The largest void is at the body centre of the cube, equidistant from all 8 corner atoms.

Body-Centered Cubic Packing — The next step up in packing efficiency with an atom at the cube centre.
Cubic Close Packing (CCP) — The most efficient cubic structure (FCC) with 74% packing.
Octahedral Voids — Void spaces in close-packed structures and their role in crystal chemistry.