Packing Efficiency: How Tightly Can We Pack Spheres?
Imagine you're filling a suitcase with tennis balls. No matter how carefully you arrange them, there will always be some empty space between the balls — you can't fill every corner. The packing efficiency is simply a measure of how much of the total space is actually occupied by the balls (the atoms), and how much is wasted as empty space.
In a crystal, atoms are modelled as hard spheres of equal size. They touch each other along specific directions, but they can't overlap. The unit cell is the box that contains the repeating pattern. Packing efficiency tells us: out of the total volume of that box, what fraction is actually filled by the atoms?
Packing efficiency=Total volume of the unit cellVolume occupied by atoms in one unit cell×100%
The key is to count how many whole atoms belong to a single unit cell, and then use the geometry of the cell to relate the atom's radius to the cell's edge length.
Simple Cubic (SCC) — 52.4%
In a simple cubic cell, atoms sit only at the eight corners. Each corner atom is shared by 8 neighbouring cells, so each contributes 81 of an atom. Total atoms per cell: 8×81=1.
The atoms touch along the edge of the cube. If the edge length is a and the atomic radius is r, then a=2r.
Volume of one atom: 34πr3. Volume of the cube: a3=(2r)3=8r3.
Efficiency=8r334πr3×100%=6π×100%≈52.4%
That's barely more than half the space. The rest is empty — a very loose packing.
Body-Centred Cubic (BCC) — 68%
Now we add one atom right at the centre of the cube. The corner atoms still contribute 1 atom total, and the body-centred atom is entirely inside the cell — it's not shared. So total atoms per cell: 1+1=2.
The atoms do not touch along the edge. They touch along the body diagonal — from one corner, through the centre atom, to the opposite corner. The body diagonal length is a3, and it contains two full radii from the corner atoms plus two radii from the centre atom: 4r=a3. So a=34r.
Volume of two atoms: 2×34πr3=38πr3. Volume of cube: a3=(34r)3=3364r3.
Efficiency=3364r338πr3×100%=8π3×100%≈68%
A significant improvement — the central atom fills a lot of the void.
Face-Centred Cubic (FCC) and Hexagonal Close-Packed (HCP) — 74%
In FCC, atoms sit at all eight corners and at the centre of each of the six faces. A face-centred atom is shared by 2 cells, so each contributes 21. Total atoms: 8×81+6×21=1+3=4.
Atoms touch along the face diagonal. The face diagonal length is a2, and it contains 4 atomic radii: 4r=a2, so a=24r=22r.
Volume of four atoms: 4×34πr3=316πr3. Volume of cube: a3=(22r)3=162r3. …
Packing efficiency measures what fraction of a unit cell's volume is actually occupied by the spheres, and the two true close-packed arrangements both reach the same maximum value, higher than the looser bcc arrangement. …
ccp and hcp are both close-packed arrangements with 74% packing efficiency — the maximum possible for spheres of one size — while bcc reaches only 68% and simple cubic only 52.4%.
Packing efficiency is the percentage of a unit cell's volume actually occupied by the constituent spheres (atoms/ions).
Simple cubic: 1 atom/cell, spheres touch along the edge (a = 2r). Packing efficiency = 52.4%.
Body-centred cubic (bcc): 2 atoms/cell, spheres touch along the body diagonal (√3·a = 4r). Packing efficiency = 68%.
Cubic close packed (ccp, = face-centred cubic) and hexagonal close packed (hcp): both are true 'close packed' arrangements built by stacking layers of spheres so each sphere touches 12 neighbours (coordination number 12). Both achieve the theoretical maximum packing efficiency for identical spheres, 74%.
…
For a body-centred cubic lattice, atoms touch along the body diagonal, giving the relation 4r = √3 a (r = radius, a = edge). The unit cell has 2 atoms.
Q.What percentage of a body centred cubic structure is vacant?
(a) 68%
(b) 52.4%
(c) 74%
(d) 32%
›Reveal solutionSolution
A bcc unit cell has 2 atoms per cell and packing efficiency 68% (=√3π/8), so the remaining 32% of the cell's volume is empty space.
In a body-centred cubic (bcc) arrangement, atoms touch along the body diagonal, giving edge length a related to atomic radius r by a = 4r/√3. There are 2 atoms per unit cell (1 corner-contribution + 1 body-centre atom). Packing efficiency = (volume occupi …
Q.Efficiency of packing in Simple Cubic Lattice is:
(a) 52.4%
(b) 62.4%
(c) 68%
(d) 74%
›Reveal solutionSolution
In simple cubic (SC) packing the atoms touch along the cell edge and occupy only about half the available volume, giving 52.4% packing efficiency - the least efficient of the common cubic packings.
In a simple cubic lattice, atoms are present only at the corners of the cube, and each corner atom touches the atoms in the adjacent cells along the edge. So the edge length a = 2r, where r is the radius of the atom.
Each unit cell contains 8 corner atoms x 1/8 = 1 atom.
Q.Which of the following lattices has the highest packing efficiency (assuming that atoms are touching each other)?
(a) Simple cubic
(b) Body centred cubic
(c) Face centred cubic
›Reveal solutionSolution
Face centred cubic (fcc) has the highest packing efficiency (74%) of the three cubic lattices.
Packing efficiency is the fraction of total volume of a unit cell occupied by the constituent spheres (atoms), assuming they touch each other along the relevant direction.
Simple cubic (sc): atoms touch along the cell edge; packing efficiency = 52.4%
Body centred cubic (bcc): atoms touch along the body diagonal; packing efficiency = 68% …
In a bcc unit cell, atoms occupy 68% of the total volume.
In a body-centred cubic (bcc) arrangement, atoms touch each other along the body diagonal of the cube (not along the edge).
Step 1 — Relation between radius and edge length.
For a cube of edge a, the body diagonal length is 3a. Along this diagonal, one corner atom, the body-centre atom, and the opposite corner atom touch, so the diagonal spans 4r (radius r of two half-atoms at the ends plus the full diameter of the centre atom):