About the Tetragonal Atomic Packing Factor Calculator
This tool estimates the atomic packing factor for a tetragonal metal unit cell using lattice dimensions, density, atomic weight, and atomic radius. It is useful for materials science students, researchers, and engineers who want to compare how much of a tetragonal cell volume is occupied by idealized atomic spheres.
How to use this calculator
- Enter the lattice parameter a and choose its length unit.
- Enter the lattice parameter c and choose its length unit.
- Enter the density in g/cm³ and atomic weight in g/mol.
- Enter the atomic radius and choose its length unit.
- Click Calculate APF to view the packing factor, inferred atoms per cell, and related cell details.
- Click Reset to restore the default example values.
How it works
For a tetragonal unit cell, the cell volume is calculated as Vc = a²c. The calculator converts the entered lattice parameters and atomic radius to centimeters so they are consistent with density in g/cm³.
The number of atoms per unit cell is inferred from the density relation n = ρVcNA / A, where ρ is density, Vc is unit-cell volume, NA is Avogadro’s number, and A is atomic weight. This inferred value is also compared with the nearest whole atom count because crystallographic unit cells normally contain an integer number of atoms.
The atomic packing factor is then APF = n(4πr³/3) / Vc. This treats each atom as a hard sphere of radius r and assumes the density, lattice parameters, and radius all describe the same material state and unit-cell convention.
Example calculation
Using the default values a = 0.2866 nm, c = 0.3220 nm, density = 7.87 g/cm³, atomic weight = 55.845 g/mol, and radius = 0.124 nm: the unit-cell volume is about 2.645 × 10^-23 cm³. The inferred atoms per cell are about 2.244, the sphere volume per atom is about 7.986 × 10^-24 cm³, so APF = 2.244 × 7.986 × 10^-24 / 2.645 × 10^-23 ≈ 0.677, or 67.7%.
Frequently asked questions
What does atomic packing factor mean?
Atomic packing factor is the fraction of a unit cell’s volume occupied by atoms modeled as hard spheres.
Why does the calculator infer atoms per cell from density?
Density, unit-cell volume, and atomic weight are related through the number of atoms in the cell, so the calculator uses n = ρVcNA / A to estimate that count.
What if the inferred atoms per cell is not close to an integer?
That usually indicates inconsistent inputs, such as lattice parameters from a different cell convention, mismatched temperature data, incorrect units, or an unsuitable atomic radius.
Can APF be greater than 1?
Physically, APF should not exceed 1 for hard spheres. A result above 1 means the inputs are inconsistent or the radius is too large for the given cell and density.
Does this apply to all tetragonal crystals?
It applies as an ideal estimate when the tetragonal cell volume is a²c and atoms can be approximated as spheres. Real crystals may differ because of bonding, defects, alloys, and thermal expansion.