Planar Density Calculator
Calculate planar density for common cubic crystal planes, convert between atoms per square nanometer and square meter, and review the atom count and planar repeat area used.
Planar Density Calculator inputs
Select an SC, BCC or FCC low-index plane, or enter a custom plane-cell atom count and area factor. Lattice units and Miller indices are handled explicitly.
Planar Density Calculator results
How to use the planar density calculator
The planar density calculator divides the effective lattice points centered on a two-dimensional repeat area by that area. A separate cubic-spacing calculation reports dₕₖₗ from the entered Miller indices.
Definition of planar atomic density
This planar density calculator divides the effective number of atom centers lying in a crystallographic planar repeat unit by that repeat unit’s area. Corner, edge and face atoms are counted fractionally according to how many adjacent planar cells share their centers. The result is an areal density, not the three-dimensional atomic packing factor.
Common cubic structure presets
The SC, BCC and FCC presets include the standard low-index (100), (110) and (111) formulas used in introductory materials science. FCC (111) is a close-packed plane and therefore has a high planar density. BCC (110) is the densest of the common BCC planes. The calculator shows the coefficient multiplying one over the lattice parameter squared.
Interplanar spacing
For a cubic lattice, the spacing of the (hkl) plane family is the lattice parameter divided by the square root of h squared plus k squared plus l squared. The h, k and l inputs are therefore used for spacing even when a preset density formula is selected. A zero-zero-zero Miller index is invalid.
Custom structures and basis atoms
More complicated crystals can contain a basis, non-cubic lattice vectors, reconstructed surfaces or multiple species. Use custom mode only when the correct planar atom count and repeat area have already been derived. The visual is an educational representation, not an atomistic surface simulation.
Planar Density Calculator questions
What units does planar density use?
The calculator reports atoms per square nanometer and atoms per square meter. Any consistent length unit can be converted through the lattice-parameter selector.
Is planar density the same as linear density?
No. Linear density counts atom centers per unit length along a crystallographic direction; planar density counts atom centers per unit area on a plane.
Why are boundary atoms fractional?
An atom center on a planar-cell boundary belongs to more than one neighboring repeat unit, so only the appropriate fraction is assigned to one unit.