Are face-centred cubic and hexagonal close packing of equal spheres really two different arrangements, or...

Peter John Revelent <prevelen@extro.ucc.su.OZ.AU> wrote:
>Anyone with a materials science background is familiar with the
>densest possible arrangement of spheres of uniform size.  For metal
>crystals in particular there are three very common packing structures.
>The density is described by an 'atomic packing factor' or PF = volume
>of atoms divided by the total volume.  They are:
>BCC - Body Centred Cubic - PF=0.68
>FCC - Face Centred Cubic - PF=0.74
>HCP - Hexagonal Close Packed - PF=0.74
>
>Any introductory text on materials science will have diagrams of these
>near the start.  The BCC has a coordination number of 8, meaning it
>touches eight neighbouring atoms.  FCC and HCP have CN=12 - each atom
>touches 12 neighbours.  FCC and HCP are the most dense packings
>possible for uniform spheres.

This has always annoyed me.  Face Centred Cubic and Hexagonal Closest
Packed are always presented as two different packing arrangements when
in fact they are one and the same!  One is simply a rotation of the
other.  This was a source of great confusion for me in my early
chemistry classes.  There are really only two packing structures here:

Cubic     - PF=0.68 (Rectangular layers of atoms)
Hexagonal - PF=0.74 (Hexagonal layers of atoms)

This seems to me a much clearer way to present it.

-- Tom Maciukenas <tom.maciukenas@amd.com>

If hexagonal layers of atoms are shifted by a constant offset wrt. the
previous layers, other layers (both rectangular and hexagonal) are
formed, tilted at various angles.  This is the "Face Centered Cubic"
packing and has 4 hexagonal and 3 rectangular planes.

However, if each hexagonal layer is placed to a random position the
patterns on all other layers are distorted.

I don't know if these "random" hexagonal layers occur in nature or if
crystals with this structure have different properties than the
"regular" hexagonal packing.  If they do, they might deserve a specific
name.

-- Tapani Lindgren <nispa@vipunen.hut.fi>

Although FCC and HCP have the same packing density, they are not "one
and the same."  Imagine a single layer of spheres, packed in a
hexagonal array like so:

o     o     o     o     o


   o     o     o     o


o     o     o     o     o


   o     o     o     o

(The spheres would actually be touching each other; I'm just showing
where the centers are.)

Now we place another layer atop this one, like so:

o     o     o     o     o
   +     +     +     +

   o     o     o     o
+     +     +     +     +

o     o     o     o     o
   +     +     +     +

   o     o     o     o
+     +     +     +     +

When we get to the third layer, we have two choices.  The first is, we
can place each new sphere directly above an "o" from the first layer.
Then a fourth layer could be placed above each "+", a fifth above each
"o", and so on.  The lattice thus obtained is completely equivalent to
a face-centered cubic lattice, although some rotation is necessary to
see the cubes.

However, the second option is to place the third layer at the
positions labelled "*" below.

o     o     o     o     o
   +     +     +     +
*     *     *     *     *
   o     o     o     o
+     +     +     +     +
   *     *     *     *
o     o     o     o     o
   +     +     +     +
*     *     *     *     *
   o     o     o     o
+     +     +     +     +
   *     *     *     *

Then the fourth layer is placed above each "o", a fifth above each "+",
a sixth above each "*", and so on.  *This* is what is meant by
hexagonal close-packed, and it is *not* equivalent to the face-centered
cubic shown above.  Both schemes do have the same packing density, and
they're similar in other ways, but they are nonetheless two different
geometries.

-- Jim Yingst <yingst@gas.uug.arizona.edu>

[ Editor: the two names are swapped in this posting: the stacking
o, +, o, +, ... is hexagonal close packing, and o, +, *, o, +, *, ...
(each layer shifted by the same offset, as Lindgren says above) is
face-centred cubic. ]