Packing Efficiency Calculator

Calculator Inputs

Input Mode

Crystal Structure

Atomic Radius

Length Unit

Manual Cell Type

Atoms per Unit Cell

Cubic Edge Length (a)

Hexagonal a

Hexagonal c

Preset mode derives ideal unit-cell dimensions from atomic radius. Manual mode lets you test custom crystal cells and compare void space.

Packing Efficiency Comparison Graph

Example Data Table

Structure	Atoms per Cell	Radius (Å)	Derived Cell Parameter	Packing Efficiency (%)	Void Space (%)
Simple Cubic	1	1.20	a = 2.40 Å	52.36	47.64
Body-Centered Cubic	2	1.20	a = 2.77 Å	68.02	31.98
Face-Centered Cubic	4	1.20	a = 3.39 Å	74.05	25.95
Hexagonal Close-Packed	6	1.20	a = 2.40 Å, c = 3.92 Å	74.05	25.95

Formula Used

Packing Efficiency (%) = (Volume occupied by atoms in a unit cell / Total unit cell volume) × 100

Occupied Atomic Volume = n × (4/3)πr³

Cubic Unit Cell Volume = a³

Hexagonal Unit Cell Volume = (3√3/2)a²c

Void Percentage = 100 − Packing Efficiency

Preset Structure Relations

SC: a = 2r

BCC: a = 4r/√3

FCC: a = 2√2r

HCP: a = 2r and ideal c/a = 1.633

How to Use This Calculator

Select Preset Crystal Structure for SC, BCC, FCC, or HCP calculations.
Enter the atomic radius and choose your preferred length unit.
Use Manual Unit Cell when you already know atoms per cell and cell dimensions.
Pick cubic or hexagonal cell geometry in manual mode.
Submit the form to view packing efficiency, void fraction, and volume details.
Use the CSV or PDF buttons to save your result summary.
Review the comparison graph to understand how structures differ in packing behavior.

Frequently Asked Questions

1) What is packing efficiency in chemistry?

Packing efficiency measures how much unit-cell space is occupied by atoms, treated as hard spheres. Higher values mean less empty space and tighter crystal packing.

2) Why do FCC and HCP have similar efficiency?

FCC and HCP are both close-packed arrangements. They organize spheres with the same maximum ideal packing fraction, so each reaches about 74.05% efficiency.

3) What does void percentage represent?

Void percentage is the empty space inside the unit cell. It equals 100 minus the packing efficiency and helps compare structure compactness.

4) When should I use manual mode?

Use manual mode when experimental or textbook values for atoms per unit cell and dimensions are already known. It is useful for custom or non-ideal cases.

5) Does the calculator support hexagonal cells?

Yes. Preset HCP mode uses the ideal c/a ratio, while manual hexagonal mode accepts your own a and c values for custom calculations.

6) Are the results affected by the selected unit?

Not if you keep inputs consistent. Packing efficiency is a ratio, so unit choice cancels out, but all length values must use the same unit.

7) Why is simple cubic less efficient?

Simple cubic leaves more unoccupied space between neighboring atoms. Its geometry causes lower sphere utilization than BCC, FCC, or HCP structures.

8) Can this help compare crystal structures quickly?

Yes. It shows packing efficiency, void fraction, occupied volume, unit-cell volume, and graph-based structure comparison in one page.