How to Calculate Planar Atomic Density
Calculate planar atomic density by constructing a repeating planar cell, counting atom centers with sharing factors, and dividing by area.
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Definition
Planar atomic density is the number of atom centers lying in a crystallographic plane per unit area of a repeating two-dimensional cell.
Use the Planar Density Calculator after constructing the correct in-plane cell.
Step 1: Identify the Plane
Use the crystal structure, basis, lattice parameter, and Miller indices to locate the plane. MIT OpenCourseWare's crystallographic-notation material covers Miller planes and planar density.
Step 2: Build a Repeating Planar Cell
Choose two nonparallel translations that lie in the plane. If their lengths are a and b and their included angle is theta, the area is:
`A = a b sin(theta)`
Step 3: Count Atom Centers
Count only atom centers lying in the plane:
- corner center: 1/4 in a standard repeating 2D cell,
- edge center: 1/2,
- interior center: 1.
The effective count is divided by area.
Unit Conversion
If dimensions are in nanometers, the calculator returns atoms/nm². Since 1 m² equals 10¹⁸ nm², multiply atoms/nm² by 10¹⁸ to obtain atoms/m².
Common Error
Do not count every atomic sphere visually intersected by the plane. The standard planar-density definition counts lattice or basis atom centers in that plane.
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Frequently Asked Questions
- Does planar density depend on crystal structure?
- Yes. The lattice and basis determine which atom centers lie in a selected plane.
- Why can equivalent-looking cells give the same answer?
- A larger repeat cell contains proportionally more effective atoms and area, so the density remains unchanged.
- What if the planar cell is not rectangular?
- Use the included-angle area formula a × b × sin(theta).
Last updated 7/20/2026