Atomic Packing Factor Calculator
Find the atomic packing factor, the fraction of unit cell volume occupied by atoms, for simple cubic, BCC, and FCC crystal structures.
🧊 What is Atomic Packing Factor?
Atomic packing factor (APF) is the fraction of a crystal's unit cell volume that is actually occupied by atoms, modeled as touching hard spheres, with the remainder counted as empty space. It is calculated as APF = (n × volume of one atom) / (volume of the unit cell), where n is the number of atoms per unit cell, giving a dimensionless value always between 0 and 1.
Materials scientists use packing factor to compare how efficiently different crystal structures fill space, which correlates with density, and to a lesser extent with mechanical properties like ductility, since densely packed structures like face-centered cubic (FCC) offer more slip planes for plastic deformation. It is a standard early topic in materials science and solid-state chemistry courses, usually introduced right alongside the three common cubic structures: simple cubic, body-centered cubic (BCC), and face-centered cubic (FCC).
A common point of confusion is assuming APF is a fixed constant for a given structure type regardless of the actual atomic radius and lattice parameter. In the idealized touching-spheres model it is (0.5236 for simple cubic, 0.6802 for BCC, 0.7405 for FCC), but this calculator computes APF directly from whatever radius and lattice parameter you enter, which also lets it handle real, experimentally measured lattice parameters that deviate slightly from the idealized geometry.
This calculator computes APF = n(4/3)πr³/a³ for simple cubic, BCC, and FCC structures from your entered atomic radius and lattice parameter, showing the result as both a decimal fraction and a percentage.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — Copper (FCC)
Copper: FCC (n=4), r = 127.8 pm, a = 361.5 pm
Example 2 — Iron (BCC)
Alpha-iron: BCC (n=2), r = 124.1 pm, a = 286.65 pm
Example 3 — Polonium (Simple Cubic)
Polonium: Simple cubic (n=1), r = 167 pm, a = 336 pm
❓ Frequently Asked Questions
🔗 Related Calculators
What is atomic packing factor?
Atomic packing factor (APF) is the fraction of a unit cell's volume that is actually occupied by atoms, modeled as touching hard spheres. It is calculated as APF = (volume of atoms in the cell) / (volume of the unit cell), always between 0 and 1.
What is the atomic packing factor for FCC?
Face-centered cubic (FCC) has an APF of exactly pi/(3*sqrt(2)) ≈ 0.7405 (74.05%) when the lattice parameter matches the ideal touching-spheres geometry, tied with hexagonal close-packed (HCP) for the densest possible packing of identical spheres.
What is the atomic packing factor for BCC?
Body-centered cubic (BCC) has an APF of exactly pi*sqrt(3)/8 ≈ 0.6802 (68.02%) at the ideal touching-spheres lattice parameter, less densely packed than FCC but more than simple cubic.
What is the atomic packing factor for simple cubic?
Simple cubic has an APF of exactly pi/6 ≈ 0.5236 (52.36%) at the ideal touching-spheres lattice parameter, the lowest packing efficiency of the three common cubic structures. Very few elements adopt this structure; polonium is the notable exception.
How many atoms are in each unit cell type?
Simple cubic has n=1 atom per unit cell, body-centered cubic (BCC) has n=2, and face-centered cubic (FCC) has n=4, counted from the fractional contributions of corner, face-centered, and body-centered atoms shared between neighboring cells.
Why does this calculator ask for both atomic radius and lattice parameter?
In the idealized touching-spheres model, the lattice parameter a is fully determined by the atomic radius r (for example a=2r for simple cubic). This calculator accepts both independently so it also works with real measured lattice parameters, which can deviate slightly from the idealized geometry.
What is the maximum possible packing factor for any sphere packing?
The Kepler conjecture, proven in 1998, establishes that no arrangement of equal spheres can exceed an APF of pi/(3*sqrt(2)) ≈ 0.7405, achieved by both the FCC and hexagonal close-packed (HCP) structures. This is the densest possible packing of identical spheres in three dimensions.
Does a higher packing factor mean a denser material?
For a given element (same atomic mass and radius), yes, a higher APF generally correlates with higher density, since more of the unit cell's volume is filled with mass rather than empty space. However, density also depends on atomic mass, so different elements with the same crystal structure can still have very different densities.
How is packing factor related to a material's mechanical properties?
Densely packed structures like FCC tend to have more available slip systems (closest-packed planes and directions), which generally correlates with greater ductility, this is part of why FCC metals such as copper, aluminum, and gold are known for good formability compared to BCC metals like iron and chromium.
Can atomic packing factor exceed 100%?
No, an APF above 1.0 (100%) would mean the modeled spheres physically overlap, which is not geometrically possible for hard spheres. If a calculation using real (non-ideal) atomic radius and lattice parameter values produces a result above 1.0, it signals that the radius or lattice parameter used does not correspond to a physically consistent touching-spheres model for that structure.