Close packing and packing efficiency

Result

Close packing and packing efficiency

T-089Home CU-304Threads structure · bonding

Statement

How hard spheres fill space.

Why it matters

crystal-lattices-unit-cell establishes how a repeating unit cell describes a periodic crystal in general; close packing answers the more specific geometric question of how efficiently identical spheres (a reasonable first approximation for many metallic and simple ionic solids) can actually fill three-dimensional space, and what structures result from packing them as tightly as geometry allows.

The two distinct ways of close-packing identical spheres, hexagonal close packing and cubic close packing, turn out to be the structures adopted by a large fraction of the metallic elements, and the tetrahedral and octahedral holes left between the close-packed spheres are exactly the sites that smaller ions occupy in many common ionic structures, connecting this purely geometric result directly to bragg-law's experimentally determined real structures and to born-lande-lattice-energy's electrostatic treatment of the resulting lattice.

Hypotheses

Atoms (or ions) are modelled as identical, incompressible hard spheres, each maximising the number of nearest-neighbour contacts.Real atoms are not literally hard spheres with a sharply defined boundary, but this approximation captures the dominant geometric constraint governing how metallic and simple ionic solids actually pack, and gives quantitatively useful results for many real structures despite its simplicity. The packing arrangement considered is the geometrically densest possible arrangement of identical spheres, achieved by stacking close-packed layers.There are, in fact, infinitely many distinct ways to stack close-packed layers (each layer identical, but the vertical stacking sequence can vary), all sharing the identical maximum packing efficiency; hexagonal close packing (ABAB... stacking) and cubic close packing (ABCABC... stacking) are simply the two most common, most symmetric examples among this larger family of equally efficient stacking sequences.

Proof

\text{Within one close-packed layer, each sphere touches six neighbours arranged hexagonally, the densest possible packing of a single layer.}

In two dimensions, arranging identical circles so that each touches six neighbours in a hexagonal pattern is the provably densest possible packing of circles in a plane, leaving the minimum possible unfilled area between them. A

\text{A second layer sits in the depressions of the first; a third layer can either repeat the first layer's position (ABAB, hexagonal close packing) or sit in a new, third position (ABCABC, cubic close packing).}

Once the first layer is fixed, the second close-packed layer's spheres naturally settle into the triangular depressions left between spheres of the first layer, the closest possible vertical approach; the third layer then has exactly two geometrically distinct choices of position, giving rise to the two named stacking sequences, both equally dense. A

\text{Cubic close packing (ABCABC) is equivalent to the face-centred cubic (FCC) unit cell, with 4 atoms per unit cell and } a = 2\sqrt{2}\,r

Analysing the FCC unit cell geometrically, the atoms touch along the face diagonal (length \(4r\)), which equals \(a\sqrt2\) by the Pythagorean theorem applied to the square face; solving for the cell edge \(a\) in terms of the atomic radius \(r\) gives this standard relationship. A

\text{Packing efficiency} = \frac{\text{volume of atoms in cell}}{\text{volume of unit cell}} = \frac{4\times\tfrac43\pi r^3}{(2\sqrt2\,r)^3} = \frac{\pi}{3\sqrt2} \approx 0.7405

Substituting the known number of atoms per FCC unit cell (\(4\), accounting for shared corner and face atoms) and the cell edge from Step 3 into the ratio of atomic volume to total cell volume gives the packing efficiency exactly, independent of the specific atomic radius \(r\) chosen, since \(r\) cancels entirely from the final ratio. A

Result

\text{Close packing (HCP or CCP): } 74.05\%\text{ efficiency} \quad(\text{the maximum possible for identical spheres})

Reading. Both close-packed arrangements (hexagonal and cubic) achieve the identical, maximum possible packing efficiency of about \(74\%\) for identical spheres, leaving roughly \(26\%\) of the volume as genuinely empty space between them, regardless of which specific stacking sequence is used.

Scope. Applies to identical, hard-sphere-like atoms (Hypotheses); other, less efficient packings exist and are adopted by some elements (body-centred cubic, \(68.02\%\) efficient; simple cubic, \(52.36\%\) efficient), typically because of directional bonding character not captured by the pure hard-sphere model.

Corollaries & converses

Close packing generates two distinct types of interstitial holes between the packed spheres, tetrahedral holes (surrounded by four spheres) and octahedral holes (surrounded by six); in ionic close-packed structures, smaller cations commonly occupy these holes between larger, close-packed anions, with the specific hole type occupied predicted by radius-ratio considerations.
bragg-law's experimentally determined interplanar spacings for real metallic elements can be checked directly against the geometric spacing predicted from close packing (Step 3), a standard cross-validation of a proposed structural model.
Converse: a measured density substantially below the value predicted for ideal close packing at a given atomic radius is itself evidence that a solid adopts a less efficient packing arrangement (body-centred cubic or simple cubic) rather than true close packing, without needing a full crystallographic structure determination to establish this.

Fails without

Model an atom with strongly directional (non-hard-sphere) bonding using the close-packing result: covalently bonded solids with directional bonding, such as diamond, adopt structures far less dense than close packing predicts, since the isotropic-contact, hard-sphere assumption (Hypotheses, first point) does not hold for them.
Assume every metallic element adopts a close-packed structure: several important metals (e.g. iron at room temperature) instead adopt the less efficient body-centred cubic arrangement, so the maximum packing efficiency of Step 4 is an upper bound achieved by many, but not all, metals.

Common errors

Assuming hexagonal close packing and cubic close packing differ in packing efficiency; both achieve the identical \(74.05\%\), differing only in stacking sequence and resulting symmetry, not in density.
Confusing the number of nearest-neighbour contacts (the coordination number, \(12\) for both close-packed structures) with the packing efficiency itself, two related but numerically distinct quantities.
Forgetting that the atomic radius \(r\) cancels entirely from the final packing-efficiency ratio (Step 4); packing efficiency is a purely geometric property of the arrangement, independent of the specific atom's size.
Assuming every metallic element adopts a close-packed structure; several important metals (including iron at room temperature) instead adopt the less efficient body-centred cubic arrangement.

Discussion

The question of how densely identical spheres can be packed has a long mathematical history, culminating in the Kepler conjecture (proposed by Johannes Kepler in 1611, that no packing of identical spheres in three dimensions can exceed the close-packed density) which was not given a rigorous, generally accepted mathematical proof until the late twentieth and early twenty-first centuries, well after chemists had long established close packing's relevance to real crystal structures experimentally via X-ray diffraction (bragg-law).

Common misconception: that the roughly \(26\%\) of "empty" volume in a close-packed structure is somehow available for extra atoms of the same size to be inserted without disrupting the structure. In fact this empty volume is fragmented into the tetrahedral and octahedral interstitial holes (Corollaries), each individually far too small to accommodate another sphere of the identical size as the close-packed spheres themselves; only substantially smaller ions or atoms can occupy these specific interstitial sites.

Worked examples

\text{Body-centred cubic: 2 atoms/cell}, \quad a = \frac{4r}{\sqrt3} \quad(\text{atoms touch along the body diagonal})

In BCC, atoms touch along the cube's body diagonal (length \(4r\)), which equals \(a\sqrt3\) by the three-dimensional Pythagorean relationship; this gives the BCC edge length in terms of atomic radius, directly analogous to the FCC derivation of Step 3 but for a different, less efficient structure. A

\text{Packing efficiency (BCC)} = \frac{2\times\tfrac43\pi r^3}{\left(\tfrac{4r}{\sqrt3}\right)^3} = \frac{\pi\sqrt3}{8} \approx 0.6802

Following the identical procedure used for FCC (Step 4 of the Proof) but with BCC's different atom count and cell-edge relationship gives a packing efficiency of about \(68\%\), noticeably lower than close packing's \(74\%\), confirming BCC is a genuinely less space-efficient arrangement. A

\text{BCC: } 68.02\% \quad<\quad \text{Close packing (FCC/HCP): } 74.05\%

Reading. Comparing the two structures' packing efficiencies directly and quantitatively confirms that close packing is genuinely denser than the body-centred cubic alternative, not merely qualitatively "more compact."

Scope. The same style of derivation, repeated for the simple cubic structure (\(1\) atom per cell, atoms touching along the cell edge), gives the lowest common packing efficiency, \(52.36\%\), for comparison.

Problems

Confirm that the coordination number in a close-packed structure (each sphere touching \(12\) nearest neighbours: \(6\) in its own layer, \(3\) in the layer above, \(3\) in the layer below) is consistent with the high packing efficiency found in the Result.
Solution
A coordination number of \(12\) is, in fact, the maximum possible number of identical spheres that can simultaneously touch one central sphere without overlapping (a geometric fact closely related to the Kepler conjecture); the very high packing efficiency of \(74.05\%\) found in the Result is a direct, quantitative consequence of this maximal nearest-neighbour contact, since more contacts per sphere generally correlates with less wasted interstitial space.
A metal is found experimentally to have a packing efficiency close to \(52\%\), far lower than close packing. What structural arrangement does this suggest, and what does it imply about the metal's coordination number?
Solution
A packing efficiency near \(52\%\) matches the simple cubic structure (Common errors, fourth point references BCC and FCC; simple cubic is the third, least efficient common structure), which has the lowest coordination number of the common metallic structures, only \(6\) nearest neighbours per atom, directly consistent with its comparatively low packing efficiency.
Explain why the tetrahedral and octahedral holes in a close-packed structure cannot each accommodate another sphere identical in size to the close-packed spheres themselves, using the packing-efficiency result.
Solution
Since close packing already achieves the maximum possible packing efficiency for identical spheres (Result, and ultimately the Kepler conjecture), inserting an additional identical sphere anywhere within the structure without first expanding it would necessarily require exceeding \(100\%\) packing efficiency, a geometric impossibility; the interstitial holes are consequently sized to accommodate only substantially smaller spheres, which is exactly why they are occupied by smaller cations, not by additional same-sized atoms, in real ionic close-packed structures.