Sphere packing

A close packing is the geometric arrangement of infinitely many spheres of the same size in 3 -dimensional space in such a way that they only touch each other and do not overlap and thereby allow the remaining space minimal. Such an arrangement results when many balls are stacked in layers. Within each layer coming into contact with each ball six neighboring spheres. The packing density of a densest packing of spheres is:

Historically, the problem goes back to Sir Walter Raleigh. This raised the question would be how to stack as in a ship's cannon balls in the densest way. In 1611 it turned Johannes Kepler his famous conjecture on: The maximum packing density of any arrangement of infinitely many balls in three -dimensional space is the above number. This Kepler conjecture was proved in 1831 by Carl Friedrich Gauss for arrangements in which the spheres lie on a grid. Only in 1998 succeeded the American mathematician Thomas Hales to show for the general case of the conjecture using a computer proof. However, it is not yet adopted by parts of the mathematical experts of this evidence.

The observation of a finite number of balls leads to the theory of finite sphere packings, a no less complex problem in mathematics.

Closest packing occur in crystal structures and are therefore in crystallography and materials science and crystal chemistry of importance.

In two dimensions, the densest circle packing is the equivalent of the densest sphere packing in three dimensions.


A close-packed hexagonal sphere consists of layers. In one of these layers ( labeled A ) there are two types of triangular spaces, one with the top down (labeled B) and a (labeled C) with the tip upwards. In this layer can now make another hexagonal close- packed spherical layer are positioned so that all the balls sitting in the B- or C- gaps either. A structure formed by an appropriate stacking of these layers of spheres, using the same layer are not allowed to follow itself twice, called densest sphere packing. It is usually described by the sequence of the stack. Basically, there are endless possibilities for the formation of a close packing of spheres. Regardless of the order of the layers coming into contact with each ball always 12 neighbors ( Kiss: 12 ), six in its own layer, and three each in the overlying and underlying.

Stacking the layers to each other in any order, the crystal has at least a three-fold axis in the stacking direction. He has thus at least the space group or. With appropriate stacking but can also occur higher symmetric structures. Altogether, the following nine space groups are possible:.


The arrangement of atoms in a close packing corresponds to an important basic principle in the formation of crystals: the principle of minimizing the volume. In this case, one speaks also of a close packing of spheres, if the atoms do not lie exactly on the positions theoretically prescribed.

Monatomic systems

The structure of many metals corresponds to a close packing of spheres. Of particular importance are the hexagonal close -packed ( hcp ) and cubic close packing ( ccp ). In particular, in the rare earths, there is another type of structure ( dhcp ).

The hexagonal close packing of spheres has the layer sequence ABABAB .... This leads to the space group. This type of structure has the number A3 in the structure reports and is also called magnesium type. Among other things crystallize beryllium, scandium and titanium in this type of structure.

The cubic closest packing of spheres has the layer sequence ABCABC .... This leads to the space group. This type of structure has the number A1 in the structure reports and is also called copper type. In addition to copper crystallize among other things, silver and gold in this type of structure.

In particular, the lighter lanthanides and actinides heavier take at standard conditions yet another densest sphere packing with the layer sequence ... ABACABAC to. This has the same space group as the hcp structure, but with four atoms in the unit cell and indeed to (0,0,0 ) / ( 0,0,1 / 2) ( Wyckoff position 2a ) and (1 /3, 2/3, 3/4) / (2 /3, 1 /3, 1 /4) ( Wyckoff positions 2d). It will therefore also double hexagonal closest packed ( dhcp ) - called structure. Praseodymium, or curium are elements which are examples of this type of structure.

In real crystals, there are often variations in the order of the layers from the ideal structure. This structural defects is called stacking faults.

Polyatomic systems

Many crystal structures with predominantly ionic bond type based on a close packing of a portion of the ions and the incorporation of other ions in the gaps. Are these storage ions too large for the gap that sphere packing is deformed accordingly. The nature and extent of this deformation depend on the size ratio of the framework ions from the storage ions. For some stoichiometries there are relationships to so-called ionic radii of the tolerance factors to calculate. Based on these tolerance factors, one can derive predictions about the structure and behavior of these systems. A well-known example is the perovskite structure.


As polytype crystals are referred to that have a stacking sequence with a long repetition unit. Examples are zinc sulfide (ZnS) with over 150 polytypes, and forms silicon carbide (SiC). This polytype some have extremely large lattice constants. Thus, the polytype of SiC has called 393R, the lattice constants a = 3.079 Å and c = 989.6 Å.

More sphere packings

Not every type of structure occurring as a sphere packing is a densest packing of spheres. A well-known example of this is the structure type A2 ( tungsten - type), which is often called body-centered cubic lattice (bcc ). This sphere packing has a packing density of. Among other things crystallize all alkali metals in this type of structure. In contrast, the structure type Ah ( α - polonium ), also known as primitive cubic lattice (sc ), rather rare. It occurs as a structural type of high-pressure modifications of some elements. The packing density is: