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the free encyclopedia
Close-packing of
spheres is the arranging of an
infinite lattice of spheres so
that they take up the greatest
possible fraction of an infinite
3-dimensional space.
Carl Friedrich Gauss proved
that the highest average density
that can be achieved by a regular
lattice arrangement is
.
The
Kepler conjecture states that
this is the highest density that
can be achieved by any arrangement
of spheres, either regular or
irregular.
There are two regular lattices
that achieve this highest average
density. They are called face-centred
cubic (FCC) and
hexagonal close-packings (HCP),
based on their
symmetry. Both are based upon
sheets of spheres arranged at the
vertices of a triangular tiling;
they differ in how the sheets are
stacked upon one another. In both
arrangements each sphere has
twelve neighbors. For every sphere
there is one gap surrounded by six
neighbors (octahedral) and two
smaller gaps surrounded by four
neighbors (tetrahedral).
Relative to a reference layer
with positioning A, two more
positionings B and C are possible.
Every sequence of A, B, and C
without immediate repetition of
the same one is possible and gives
an equally dense packing for
spheres of a given radius.
The most regular ones are:
- HCP = ABABABA
- FCC = ABCABCA
Many
crystal structures are based
on a close-packing of atoms, or of
large ions with smaller ions
filling the spaces between them.
The cubic and hexagonal
arrangements are very close to one
another in energy, and it may be
difficult to predict which form
will be preferred from first
principles.
The
coordination number of HCP and
FCC is 12 and its
atomic packing factor (APF)
is the number mentioned above,
0.74.
