Calculator Input
Use preset geometry for common crystal types, or switch to manual mode for custom cells and experimental values.
Plotly Graph
The chart compares standard packing factors for SC, BCC, FCC, and HCP. Your current result is added as a marker.
Example Data Table
| Structure | Atoms per Cell | Typical Radius (Å) | Cell Relation | APF | Void % | Coordination Number |
|---|---|---|---|---|---|---|
| SC | 1 | 1.25 | a = 2r | 0.5236 | 47.64 | 6 |
| BCC | 2 | 1.25 | a = 4r / √3 | 0.6802 | 31.98 | 8 |
| FCC | 4 | 1.25 | a = 2√2 r | 0.7405 | 25.95 | 12 |
| HCP | 6 | 1.25 | a = 2r, c = 1.633a | 0.7405 | 25.95 | 12 |
Formula Used
APF = (N × (4/3) × π × r³) / Vcell
N = effective atoms per unit cell, r = atomic radius, Vcell = unit cell volume.
SC: a = 2r, V = a³, N = 1
BCC: a = 4r / √3, V = a³, N = 2
FCC: a = 2√2 r, V = a³, N = 4
HCP: a = 2r, c = (c/a) × a, V = (3√3/2)a²c, N = 6
The calculator first determines unit cell geometry, then computes the occupied atomic volume, then divides by total cell volume. Packing efficiency is APF × 100, while void fraction is 1 − APF.
How to Use This Calculator
- Select a crystal structure such as SC, BCC, FCC, HCP, or Custom.
- Choose Preset Geometry for standard crystal relations or Manual Unit Cell Data for custom research values.
- Enter the atomic radius in ångströms.
- For HCP preset mode, confirm or edit the c/a ratio.
- For manual mode, enter effective atoms per cell and total unit cell volume.
- Press Calculate APF to show the result above the form.
- Review APF, packing efficiency, void fraction, and geometry outputs.
- Use the CSV or PDF buttons to export the computed result.
Frequently Asked Questions
1) What does atomic packing factor measure?
Atomic packing factor measures the fraction of a unit cell actually occupied by atoms, modeled as hard spheres. It shows how efficiently a crystal structure packs space.
2) Why do FCC and HCP have the highest common APF?
FCC and ideal HCP are closest-packed arrangements. Their spheres fill space more efficiently than SC or BCC, so both reach an APF of about 0.7405.
3) Can APF be greater than 1?
No. A physically valid packing factor cannot exceed 1. If manual mode gives a value above 1, the radius, atom count, or cell volume inputs are inconsistent.
4) Which units should I use?
Use any consistent length unit for radius and the matching cubic unit for cell volume. This file labels radius in Šand cell volume in ų for convenience.
5) Why does HCP use a c/a ratio?
HCP cells are not cubic. The c/a ratio controls cell height relative to the basal edge. The ideal ratio is about 1.633 for close packing.
6) Is APF the same as density?
No. APF measures geometric packing only. Density also depends on atomic mass and actual cell dimensions, so two materials can share APF but have different densities.
7) Can I use this for alloys or nonideal crystals?
Yes, manual mode is suitable for approximate studies when you know an effective radius, atoms per cell, and unit cell volume. Interpret mixed-radius systems carefully.
8) When should I choose manual mode?
Choose manual mode when experimental lattice data, simulation outputs, or nonstandard unit cells are available and preset textbook geometry is not appropriate.