Planar Density Calculator
Enter the dimensions of the repeating planar cell and count atoms whose centers lie in that plane. The calculator applies boundary sharing factors and returns atoms per square nanometer and per square meter.
- Planar Cell Area
- 0.16 nm²
- Effective Atom Count
- 1 atoms
- Density in Atoms per m²
- 6,249,999,999,999,999,000 atoms/m²
- Mean Area per Atom
- 0.16 nm²/atom
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The result shows the effective number of atoms assigned to one planar cell, its area, density in atoms/nm², density in atoms/m², and mean area per atom.
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How to Use This Calculator
Choose the smallest repeating two-dimensional cell in the crystallographic plane. Enter both in-plane side lengths in nanometers and their included angle in degrees. Count corner, edge-centered, and fully interior atom centers in that planar cell. Do not count atoms whose centers are above or below the plane.
Formula & Methodology
A corner atom in a two-dimensional repeating cell contributes one quarter, an edge-centered atom contributes one half, and an interior atom contributes one. The plane area is a × b × sin(theta). Planar density is the effective atom count divided by that area.
Example: square planar cell with four corner atoms
For a 0.4 nm by 0.4 nm square cell containing four corner atoms and no other atom centers, the effective count is 1 atom and the planar density is 6.25 atoms/nm².
The calculator does not infer which atoms intersect a Miller plane. The user must construct the correct two-dimensional repeat cell and count only atom centers in the plane. Sharing factors differ if a nonstandard or nonperiodic region is used.
Educational engineering tool. Verify crystallographic geometry, basis atoms, Miller indices, and the selected planar repeat cell independently before using the result in research or design.
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Frequently Asked Questions
- What is planar atomic density?
- It is the number of atom centers lying in a crystallographic plane divided by the area of a repeating two-dimensional cell in that plane.
- Why does a corner atom count as one quarter?
- In a repeating two-dimensional tiling, a corner is shared by four neighboring planar cells.
- Why does an edge atom count as one half?
- An atom centered on a planar-cell edge is shared by two neighboring cells.
- Do I count atoms that the plane only touches?
- Planar atomic density normally counts atom centers lying in the plane, not every atomic sphere intersected by a drawing.
- Can I use angstroms instead of nanometers?
- Convert lengths first. One angstrom equals 0.1 nanometer.
Last updated 7/20/2026