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Atomic packing factor

The atomic packing factor (APF), also known as the packing efficiency or packing fraction, is a dimensionless quantity in crystallography that represents the fraction of the volume of a unit cell in a crystal structure occupied by the atoms, assuming the atoms are rigid, hard spheres touching their nearest neighbors.^[1] It is calculated as the ratio of the total volume of the atoms within the unit cell to the overall volume of the unit cell, providing a measure of how efficiently space is filled in the lattice without overlaps or voids beyond the atomic radii.^[2] For one-component crystals, the theoretical maximum APF is approximately 0.74, achieved in close-packed structures, while less efficient arrangements leave more interstitial space.^[3] The value of APF varies significantly with crystal structure, directly influencing the density and mechanical properties of materials. In a simple cubic (SC) lattice, with one atom per unit cell and atoms touching along the edges, the APF is 0.52, calculated as \frac{\pi}{6}.^[4] The body-centered cubic (BCC) structure, featuring two atoms per unit cell and atoms touching along the body diagonal, yields an APF of 0.68, or \frac{\pi \sqrt{3}}{8}.^[5] In contrast, the face-centered cubic (FCC) and hexagonal close-packed (HCP) structures, both with atoms touching along face diagonals or close-packed planes, achieve the highest packing efficiency of 0.74, or \frac{\pi}{3\sqrt{2}}, making them prevalent in many metals like aluminum and copper.^[4] These differences arise from the coordination number and atomic arrangement: SC has a coordination number of 6, BCC 8, and FCC/HCP 12. In materials science, APF is crucial for predicting and engineering material behavior, as higher packing densities correlate with greater theoretical density, enhanced strength, and improved ductility in metallic alloys.^[6] For instance, the denser FCC structure in metals like gold facilitates slip along close-packed planes, contributing to malleability, whereas BCC structures in iron at room temperature exhibit greater hardness but reduced ductility due to lower APF and more distorted interstitial sites.^[7] Beyond metals, APF informs the design of alloys, ceramics, and nanomaterials by quantifying void space available for dopants or defects, which affect electrical conductivity, thermal expansion, and phase stability.^[8] Experimental determination of APF often involves X-ray diffraction to measure lattice parameters and atomic radii, ensuring models align with observed densities.^[9]

Fundamentals

Definition

The atomic packing factor (APF), also known as packing efficiency or packing fraction, is defined as the fraction of the volume of a unit cell in a crystal structure that is occupied by the volumes of the constituent atoms, modeled as hard spheres that touch their nearest neighbors without overlapping.^[1]^[9] In crystallography, the unit cell is the smallest repeating parallelepiped volume that, when translated along its lattice vectors, generates the entire periodic crystal lattice, serving as the fundamental building block of the structure.^[10]^[11] The APF differs from related metrics such as the linear packing factor, which assesses the fraction of a one-dimensional line segment occupied by atomic diameters along a crystallographic direction, and planar density, which counts the number of atoms intersecting a specific two-dimensional plane per unit area; these provide directional or surface-specific insights, whereas APF evaluates overall three-dimensional space utilization. The concept of atomic packing has roots in 17th-century sphere packing theories proposed by Johannes Kepler and was advanced in the late 19th and early 20th centuries through geometrical models of crystal structures by William Barlow and William J. Pope, with experimental verification enabled by the development of X-ray diffraction techniques in 1912.^[12]

General Formula

The atomic packing factor (APF) is calculated using the general formula

\text{APF} = \frac{Z \times V_\text{atom}}{V_\text{cell}},

where Z is the number of atoms per unit cell, V_\text{atom} is the volume of a single atom, and V_\text{cell} is the volume of the unit cell.^[4] This formula quantifies the fraction of the unit cell volume occupied by the atoms, assuming a hard-sphere model for the atoms. The volume of one atom is taken as that of a perfect sphere, V_\text{atom} = \frac{4}{3} \pi r^3, where r is the atomic radius.^[4] The value of Z is determined by summing the fractional contributions of atoms based on their positions within the unit cell: for example, each corner atom is shared among eight adjacent unit cells and thus contributes \frac{1}{8}, while each face-centered atom is shared between two unit cells and contributes \frac{1}{2}.^[13] This calculation rests on key assumptions, including that atoms behave as rigid, non-overlapping hard spheres that touch along the closest-packed directions in the lattice; it neglects effects such as thermal vibrations, defects, or deviations from sphericity. To derive the APF, the process begins by identifying the lattice parameters defining the unit cell geometry—for instance, the edge length a for cubic structures or the basal plane edge a and height c for hexagonal structures. Next, the atomic radius r is related to these parameters based on the condition that nearest-neighbor atoms touch without overlap. The unit cell volume V_\text{cell} is then computed from the lattice parameters (e.g., V_\text{cell} = a^3 for cubic cells). Finally, Z is calculated as described, V_\text{atom} is substituted, and the values are plugged into the APF formula to yield the packing efficiency.^[4]

Significance

Relation to Density

The theoretical density \rho of a crystalline material is calculated using the formula

\rho = \frac{n \times A}{V_C \times N_A},

where n is the number of atoms per unit cell, A is the atomic mass in g/mol, V_C is the volume of the unit cell in cm³, and N_A is Avogadro's number ($6.023 \times 10^{23} atoms/mol).^[14] This expression provides the mass per unit volume assuming a perfect crystal lattice with no defects.^[4] The atomic packing factor (APF) directly influences theoretical density through its effect on V_C. For a given atomic radius r and atomic mass A, a higher APF corresponds to a smaller unit cell volume, as more space is efficiently occupied by atoms, leading to greater mass concentration and thus higher density.^[14] This proportionality highlights APF's role as a geometric efficiency metric that scales density independently of material composition.^[4] To link APF explicitly to density, substitute the relation V_C = \frac{n \times \frac{4}{3}\pi r^3}{\text{APF}} into the density formula, yielding

\rho = \frac{\text{APF} \times A}{N_A \times \frac{4}{3}\pi r^3}.

Here, APF acts as a dimensionless geometric factor that modulates density for fixed atomic size and mass, with the number of atoms per unit cell n canceling out in the derivation.^[4]^[14] An example calculation framework begins by determining n and V_C from the crystal structure, then applying the density formula; APF can be incorporated by first computing the atomic volume n \times \frac{4}{3}\pi r^3 and dividing by V_C to verify packing efficiency before scaling to \rho. This approach establishes the ideal density limit without requiring structure-specific values beyond general parameters.^[4] In practice, actual densities may deviate slightly from theoretical values due to factors such as isotopic variations in atomic mass or the presence of impurities and defects, which alter the effective mass or volume. However, APF provides the ideal geometric upper bound for packing density in a defect-free crystal.^[14]

Impact on Material Properties

The atomic packing factor (APF) significantly influences the mechanical properties of crystalline materials, particularly in metals, by determining the density and arrangement of atoms, which in turn affects stiffness, strength, and ductility. Higher APF values correlate with greater atomic density, leading to enhanced stiffness and overall strength due to reduced void space that limits atomic mobility under stress. For instance, close-packed structures like face-centered cubic (FCC) exhibit high ductility because their efficient packing facilitates multiple slip systems, allowing plastic deformation without fracture.^[6]^[15] The void fraction, given by 1 minus the APF, plays a critical role in processes such as atomic diffusion, dislocation motion, and fracture toughness. In structures with lower APF, such as simple cubic (APF ≈ 0.52), larger interstitial voids promote easier diffusion of atoms or impurities but also result in brittleness, as dislocations are less constrained and fracture occurs more readily under load. Conversely, high-APF structures like FCC and hexagonal close-packed (HCP, APF ≈ 0.74) restrict void-mediated mechanisms, enhancing fracture toughness while enabling controlled dislocation glide for improved toughness. In body-centered cubic (BCC) structures (APF ≈ 0.68), the intermediate void space provides more interstitial sites for alloying elements, which can strengthen materials through solid solution hardening but may reduce ductility compared to FCC.^[16]^[17] APF also impacts thermal properties by influencing lattice vibrations. High-APF structures can exhibit higher thermal conductivity through more efficient phonon propagation. For example, FCC metals often display more isotropic thermal behavior due to their symmetric packing, which minimizes directional variations in expansion. In modern applications, deviations from ideal APF are particularly relevant in nanomaterials and quasicrystals, where reduced packing efficiency alters property landscapes. Nanomaterials with engineered low-APF regions exhibit enhanced diffusion for applications in catalysis, while quasicrystals, featuring aperiodic structures with APF typically below 0.70, demonstrate unique combinations of high strength and low thermal conductivity due to disordered packing that scatters phonons effectively. These considerations guide the design of advanced alloys and coatings for high-performance environments.^[18]^[19]

Calculations for Common Structures

Simple Cubic

The simple cubic lattice features atoms located at the eight corners of a cubic unit cell, contributing a total of one atom per unit cell as each corner atom is shared equally among eight adjacent cells. In this arrangement, nearest-neighbor atoms contact each other directly along the cube edges, establishing the geometric relation a = 2r, where a is the lattice parameter (edge length) and r is the atomic radius. Each atom in the simple cubic structure has a coordination number of 6, corresponding to its six nearest neighbors positioned along the positive and negative x, y, and z directions. The atomic packing factor (APF) quantifies the fraction of the unit cell volume occupied by atoms, providing a measure of packing efficiency. For the simple cubic lattice, the derivation begins by determining the volumes involved. The volume of a single spherical atom is \frac{4}{3} \pi r^3, and with one atom per unit cell, the total atomic volume is \frac{4}{3} \pi r^3. The unit cell volume is a^3. Substituting the edge length relation a = 2r yields a^3 = (2r)^3 = 8r^3. Thus, the APF is calculated as:

\text{APF} = \frac{\frac{4}{3} \pi r^3}{8r^3} = \frac{\pi}{6} \approx 0.52.

This value indicates that approximately 52% of the unit cell space is filled by atoms, leaving significant interstitial void space. The simple cubic structure demonstrates the lowest packing density among common cubic lattices, rendering it inefficient and thus rare in metallic elements. Only polonium adopts this configuration as its ground-state crystal structure at ambient conditions, attributed to relativistic effects stabilizing the otherwise unstable lattice against denser alternatives. In contrast, the structure appears in select ionic compounds like cesium chloride (CsCl), where it accommodates a 1:1 ratio of similarly sized cations and anions, with Cs⁺ at cube corners and Cl⁻ at the body center (or vice versa in equivalent descriptions). A representative diagram of the simple cubic unit cell illustrates hard spheres centered at the corners, touching pairwise along each edge while leaving octahedral voids at the body center and edge centers unoccupied.

Body-Centered Cubic

The body-centered cubic (BCC) lattice features atoms positioned at each of the eight corners of a cubic unit cell, with an additional atom at the exact center of the cube. Each corner atom is shared among eight adjacent unit cells, contributing an effective 1/8 atom per cell from the corners (8 × 1/8 = 1 atom total), combined with the fully enclosed central atom, yielding 2 atoms per unit cell overall. In the BCC structure, nearest-neighbor atoms contact one another along the body diagonal of the unit cell rather than along the edges or faces. The body diagonal spans a distance of a\sqrt{3}, where a is the lattice parameter (edge length of the cube); this diagonal accommodates four atomic radii—from one corner atom, through the center atom, to the opposite corner—establishing the geometric relation $4r = a\sqrt{3}, or equivalently, a = \frac{4r}{\sqrt{3}}.^[14] The atomic packing factor (APF) for BCC quantifies the fraction of unit cell volume occupied by atoms and is derived as follows. The total volume of the 2 atoms is $2 \times \frac{4}{3}\pi r^3 = \frac{8}{3}\pi r^3. The unit cell volume is a^3 = \left( \frac{4r}{\sqrt{3}} \right)^3 = \frac{64 r^3}{3\sqrt{3}}. Thus,

\text{APF} = \frac{\frac{8}{3}\pi r^3}{\frac{64 r^3}{3\sqrt{3}}} = \frac{8}{3}\pi \cdot \frac{3\sqrt{3}}{64} = \frac{\pi \sqrt{3}}{8} \approx 0.68.

This value reflects a moderate packing efficiency, balancing atomic density with available space for defects.^[14] The BCC structure occurs in several metals, including alpha-iron (Fe) at room temperature and tungsten (W). It exhibits a coordination number of 8, where each atom bonds to eight nearest neighbors along the body diagonals. Notably, the lattice provides interstitial sites—such as octahedral voids—that accommodate smaller solute atoms, as seen in iron where carbon occupancy distorts the lattice and promotes hardening via solid solution strengthening.^[6]^[20] This structure was first elucidated through pioneering X-ray diffraction studies of metals in the early 1910s, enabling direct observation of atomic arrangements in crystalline solids like iron.

Face-Centered Cubic

The face-centered cubic (FCC) lattice consists of atoms positioned at the eight corners of a cubic unit cell, contributing a total of 1 atom (each corner atom shared among 8 cells), and one atom at the center of each of the six faces, contributing 3 atoms (each face-centered atom shared between 2 cells), for a total of 4 atoms per unit cell.^[4]^[5] In this structure, nearest-neighbor atoms touch along the face diagonal of the cube, leading to the geometric relation $4r = a\sqrt{2}, where r is the atomic radius and a is the lattice parameter (edge length of the unit cell).^[21]^[4] To compute the atomic packing factor (APF) for FCC, first determine the volume occupied by the atoms and the volume of the unit cell. The total volume of the 4 atoms, modeled as hard spheres, is $4 \times \frac{4}{3}\pi r^3 = \frac{16}{3}\pi r^3.^[21] The unit cell volume is a^3, and substituting the face-diagonal relation gives a = \frac{4r}{\sqrt{2}}, so a^3 = \left(\frac{4r}{\sqrt{2}}\right)^3 = \frac{64 r^3}{2\sqrt{2}} = 16\sqrt{2} r^3.^[21]^[4] The APF is then the ratio:

\text{APF} = \frac{\frac{16}{3}\pi r^3}{16\sqrt{2} r^3} = \frac{\pi}{3\sqrt{2}} \approx 0.74.

This derivation simplifies by canceling r^3 terms and rationalizing the denominator if needed, confirming that 74% of the unit cell volume is occupied by atoms.^[21]^[5]^[22] The FCC structure represents the highest packing efficiency among cubic lattices, with a coordination number of 12, meaning each atom is surrounded by 12 nearest neighbors.^[5]^[22] It is commonly observed in noble metals such as copper (Cu) and aluminum (Al).^[4]^[5] The FCC lattice is equivalent to the cubic close-packed structure, featuring an ABCABC... stacking sequence of close-packed planes.^[22]^[4]

Hexagonal Close-Packed

The hexagonal close-packed (HCP) structure features an alternating ABAB stacking sequence of close-packed atomic planes, where each plane consists of atoms arranged in a hexagonal array. This arrangement results in 6 atoms per conventional hexagonal unit cell, equivalent to 2 atoms per primitive cell, with each atom surrounded by 12 nearest neighbors for a coordination number of 12. Within each plane, atoms touch their six in-plane neighbors, yielding the relation a = 2r, where a is the basal lattice parameter and r is the atomic radius. The ideal axial ratio c/a = \sqrt{8/3} \approx 1.633 arises from the geometry of atoms in adjacent planes touching through the centers of triangular voids, forming tetrahedral coordination between layers.^[23] This structure is commonly observed in metals such as beryllium, cobalt, magnesium, and zinc. Compared to the face-centered cubic (FCC) structure, HCP exhibits hexagonal symmetry rather than cubic, yet both share identical ideal packing efficiency due to their close-packed nature. HCP proves more stable than FCC in certain elements owing to the lower stacking fault energy favoring the ABAB sequence over ABC stacking.^[24] The atomic packing factor (APF) for HCP is derived by comparing the volume occupied by atoms to the unit cell volume. In the basal plane, atoms form a close-packed hexagonal layer with a packing density of \pi / (2 \sqrt{3}) \approx 0.907, but the three-dimensional efficiency accounts for interlayer spacing. The height between consecutive planes equals the distance across a regular tetrahedron of edge length $2r, given by h = (2 \sqrt{6}/3) r; since the unit cell height c spans two such intervals, c = 2h = (4 \sqrt{6}/3) r = \sqrt{8/3} \, a.^[23] The total volume of the 6 atoms is

$6 \times \frac{4}{3} \pi r^3 = 6 \times \frac{4}{3} \pi \left( \frac{a}{2} \right)^3 = \pi a^3,

as r = a/2. The unit cell volume is the hexagonal base area times c, where the base area is (3 \sqrt{3}/2) a^2, yielding

V = \frac{3 \sqrt{3}}{2} a^2 c = \frac{3 \sqrt{3}}{2} a^2 \left( \sqrt{\frac{8}{3}} a \right) = 3 \sqrt{2} \, a^3.

Thus, the APF is

\text{APF} = \frac{\pi a^3}{3 \sqrt{2} \, a^3} = \frac{\pi}{3 \sqrt{2}} \approx 0.74,

confirming HCP achieves the maximum theoretical packing density for equal spheres, equivalent to FCC.^[23]

References

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