0.7405
74.05
%
25.95
%
0.7405
74.05
%
25.95
%
The Packing Fraction Calculator determines the atomic packing fraction (APF) for common crystal structures and custom unit cell configurations. The packing fraction represents the volume fraction of a unit cell that is occupied by atoms, assuming a hard-sphere model. This fundamental parameter governs crystal density, void volume available for interstitial atoms, and many physical properties of crystalline materials.
The calculator offers two modes: selecting from standard crystal types with known analytical packing fractions, or entering custom parameters for any crystal structure. Standard values include the classic results: FCC and HCP achieve 74.05% (the theoretical maximum for equal spheres), BCC achieves 68.02%, SC achieves 52.36%, and diamond cubic achieves 34.01%.
The atomic packing fraction is calculated as:
$$\text{APF} = \frac{n \times \frac{4}{3}\pi r^3}{V_{\text{cell}}}$$
Where $$n$$ is the number of atoms per unit cell, $$r$$ is the atomic radius, and $$V_{\text{cell}}$$ is the unit cell volume.
For standard cubic structures using the hard-sphere model:
$$\text{SC: APF} = \frac{\pi}{6} = 0.5236 \quad (1 \text{ atom}, a = 2r)$$
$$\text{BCC: APF} = \frac{\pi\sqrt{3}}{8} = 0.6802 \quad (2 \text{ atoms}, a = 4r/\sqrt{3})$$
$$\text{FCC: APF} = \frac{\pi}{3\sqrt{2}} = 0.7405 \quad (4 \text{ atoms}, a = 2\sqrt{2}r)$$
$$\text{Diamond: APF} = \frac{\pi\sqrt{3}}{16} = 0.3401 \quad (8 \text{ atoms}, a = 8r/\sqrt{3})$$
The Kepler conjecture (proven in 2005) confirms that FCC/HCP packing at 74.05% is the densest possible arrangement of equal spheres.
Higher packing fractions indicate denser structures with less void space. FCC metals like copper and aluminum are dense and ductile. BCC metals like iron have more void space, allowing greater interstitial solubility for small atoms like carbon and nitrogen. Diamond cubic structures (Si, Ge, diamond) have very low packing because tetrahedral bonding forces atoms far apart. The void fraction determines how much space is available for interstitial atoms, defects, and diffusion pathways.
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FCC is a close-packed structure with 74.05% of space occupied by atoms and 25.95% void
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Custom calculation: APF = 2 × (4/3)π(1.241)³ / (2.866)³ = 16.03 / 23.56 ≈ 0.680
Atomic packing fraction (APF) is the ratio of the volume of atoms in a unit cell to the total volume of the unit cell, assuming atoms are hard spheres. APF = (number of atoms × volume per atom) / cell volume. It ranges from 0 (no atoms) to the theoretical maximum of about 0.74 for equal spheres.
The Kepler conjecture, proven by Thomas Hales in 2005, states that no arrangement of equal spheres can exceed 74.05% packing density. FCC and HCP both achieve this limit because they are built from close-packed layers (each atom touches 6 neighbors) stacked in the densest possible way (ABCABC for FCC, ABAB for HCP).
Diamond cubic has only 34% packing because each atom forms four tetrahedral bonds at 109.5° angles, forcing large separations between atoms. The lattice parameter is a = 8r/√3 (much larger than 2r for SC), but with only 8 atoms per cell. The directed covalent bonding sacrifices packing efficiency for bond strength and directionality.
Higher packing fraction means more mass per unit volume, so denser crystals. However, density also depends on atomic mass and lattice parameter. Osmium (HCP, APF = 0.74) is the densest element because it has both high packing efficiency and heavy atoms. Lithium (BCC, APF = 0.68) is very light despite reasonable packing because lithium atoms are light.
Void space (interstitial sites) accommodates small interstitial atoms. Carbon in iron occupies octahedral voids in FCC (γ-Fe) and tetrahedral/octahedral voids in BCC (α-Fe). Interstitial sites are also important for hydrogen storage, ionic conductivity in solid electrolytes, and diffusion mechanisms in alloys.
Interstitial sites are the voids between atoms in a crystal structure. FCC has 4 octahedral sites (radius ratio 0.414r) and 8 tetrahedral sites (radius ratio 0.225r) per unit cell. BCC has 6 octahedral and 12 tetrahedral sites. These sites can accommodate small atoms like H, C, N, O, and B.
FCC metals (APF = 0.74) are generally more ductile because they have 12 independent slip systems on close-packed {111} planes. BCC metals (APF = 0.68) have more slip systems (48) but fewer are active at low temperatures, making them brittle below a transition temperature. HCP metals have limited slip systems and are less ductile.
For identical hard spheres, no. But real atoms are not hard spheres, and compounds with different-sized atoms can achieve effective packing above 74%. For example, NaCl has APF ≈ 0.67 considering both ion sizes, but optimized sphere mixtures can pack above 74% when radii differ appropriately.
Random close packing (RCP) of equal spheres achieves approximately 64% density, significantly below the crystalline maximum of 74%. Random loose packing is about 60%. These values are relevant to granular materials, powders, and amorphous metals. The gap between random and ordered packing drives crystallization.
The hard-sphere model is an approximation where atoms are rigid, non-overlapping spheres. Real atoms have electron clouds that compress under pressure and thermal vibration that effectively increases their average size. Despite this simplification, the model accurately predicts packing fractions, coordination numbers, and structural stability for metallic systems.
Roboculator Team
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