Use this chemistry calculator to reproduce a named relationship with explicit quantities and units for study, laboratory planning, or an independent numerical check.

Atomic radius (Å)

Required number input.

Atoms per unit cell

Required integer input.

Unit-cell volume (Å^3)

Required number input.

Calculate the hard-sphere packing fraction of a unit cell from atomic radius, atom count, and unit-cell volume. The tool evaluates the supplied inputs together and returns the named outputs below; it does not infer omitted operating conditions or change the units shown.^[1]

Packing fraction

The resulting packing fraction returned as a number.

• Crystallographic calculations assume the stated ideal lattice, indexing, wavelength, occupancy, and unit-cell geometry. Defects, strain, texture, thermal expansion, mixed phases, and instrumental corrections are outside the compact model.
• Use finite inputs in the displayed units and preserve more precision than the final presentation requires. Independently verify safety-critical, financial, compliance, or production decisions.

Check units and significant figures, compare the result with limiting cases, and use measured or source-specific constants where available. Laboratory, safety, and regulatory decisions require validated methods and appropriate uncertainty analysis.

• Void Fraction Calculator

  Calculate the unoccupied fraction of a unit cell as one minus its packing fraction.

• Packing Efficiency Calculator

  Compute the percent of a unit cell volume filled by hard-sphere atoms for the common cubic and close-packed packings.

• Density From Unit Cell Calculator

  Calculate crystal density from formula-unit molar mass, formula units per cell, and unit-cell volume.