What Is the Coordination Number in 3D Close Packing?

The coordination number of a particle in three-dimensional close packing is 12. It counts the nearest neighbors surrounding one particle—not the number of particles in the whole unit cell.

The result is the same for the two common close-packed structures: face-centered cubic (FCC), with ABC layer stacking, and hexagonal close-packed (HCP), with ABAB stacking. Their stacking sequences differ, but each particle has the same number of nearest neighbors.

Count neighbors by layer

Imagine a central sphere in one layer of equal-sized spheres. Six other spheres touch it in that same layer, arranged around it in a hexagonal pattern. That gives six neighbors.

The next layer sits in the triangular hollows between spheres in the first layer. Three spheres in that upper layer touch the central sphere. A corresponding arrangement below the central layer contributes three more. Therefore,

$$6+3+3=12.$$

These are all at the same nearest-neighbor distance in an ideal close-packed structure. The central sphere has 12 nearest neighbors in total.

Why the stacking pattern does not change the count

In close packing, each layer is a two-dimensional hexagonal arrangement. The next layer occupies one set of triangular hollows, called the B positions. A third layer can either return to the original A positions or occupy a distinct set of C positions.

• In HCP, the layers repeat ABAB.
• In FCC, the layers repeat ABCABC.

This choice changes how layers line up beyond the central layer, and it changes the crystal structure. It does not change the immediate environment of a sphere: six touching neighbors in its own layer, three above, and three below. Thus both structures have coordination number 12.

A geometric check

The count can also be checked using the spacing between layers. Let each sphere have radius $R$, so touching centers are separated by $2R$. In one layer, the centers form equilateral triangles with side length $2R$. The horizontal distance from the center of such a triangle to one of its vertices is

$$\frac{2R}{\sqrt{3}}.$$

A sphere in the next layer sits above the triangle’s hollow and touches all three spheres at its vertices. If the vertical layer spacing is $h$, the touching condition gives

$$\begin{aligned} h^2+\left(\frac{2R}{\sqrt{3}}\right)^2&=(2R)^2,\\ h&=2R\sqrt{\frac{2}{3}}. \end{aligned}$$

For a sphere in the upper layer, its horizontal offset from the central sphere below is also $2R/\sqrt{3}$. Its center-to-center distance from that central sphere is therefore

$$\sqrt{\left(\frac{2R}{\sqrt{3}}\right)^2+h^2}=2R.$$

So the upper-layer spheres are indeed at the same nearest-neighbor distance as the six in-plane spheres. The same geometry applies to the layer below.

Worked example

Suppose the spheres have radius $R=0.15\,\text{nm}$. Their diameter, and hence the nearest-neighbor center distance, is

$$2R=0.30\,\text{nm}.$$

The distance between adjacent layers is

$$h=2(0.15)\sqrt{\frac{2}{3}}\approx 0.245\,\text{nm}.$$

The horizontal offset between a sphere in one layer and a touching sphere in the next is

$$\frac{0.30}{\sqrt{3}}\approx 0.173\,\text{nm}.$$

Combining these perpendicular components gives a center-to-center distance of approximately $0.30\,\text{nm}$, the sphere diameter. Thus the three spheres above touch the central sphere; the three below do as well. Adding those six to the six in the central layer gives 12 nearest neighbors.

Common mistakes

• Counting only the central layer: Six is the coordination number for a two-dimensional hexagonal layer considered by itself. In a three-dimensional close-packed solid, the layers above and below add six more neighbors.
• Confusing coordination number with packing efficiency: Coordination number counts nearest neighbors. Packing efficiency is the fraction of space occupied by spheres; it is about 74% for ideal FCC and HCP packing. These are different properties.
• Assuming FCC and HCP have different coordination numbers: Their stacking sequences differ, but both have 12 nearest neighbors per particle.
• Mixing up geometric and chemical coordination: For an elemental hard-sphere model, every touching nearest neighbor is counted. In a compound, coordination number is usually defined for a particular atom or ion by counting its nearest surrounding atoms or ions; the relevant neighbors and conventions depend on the structure and chemistry.