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Instead
of stacking hexagonal closest packed planes directly above one another,
they can be stacked such that atoms in successive planes nestle in the
triangular "grooves" of the adjacent plane. (note that there are six of
these "grooves" surrounding each atom in the hexagonal plane, but only
three of them can be covered by atoms in the adjacent plane).
Let the first plane (at the bottom) be labeled
"A" and the next plane above it be labeled "B". If a third hexagonal
closest packed plane is stacked above B but in the "A" orientation, and
succeeding planes are stacked in the repeating pattern ABABA... = (AB),
a hexagonal unit cell can be chosen (using the nine atoms labeled
"h"), with Z = 2.
For the stack of hexagonally closest packed
spheres of equal radius (r) described above, the interplanar spacing
between adjacent planes is proportional to r. The proportionality
constant is an irrational number called the Closest Packed Interlayer
Spacing, CPIS, and its value is about 1.6. Thus, the interlayer
spacing is about 1.6.r (compared to 2.r
for simple hexagonal stacking).
Thus, the "c" unit cell edge (in the stacking
direction) has a length c = 2.CPIS.r, the
ratio
c:a = CPIS, and it can be shown
that this lattice has a packing efficiency which is identical to the FCC
lattice. Thus, the name Hexagonal Closest Packing (HCP) for
this array is justified.
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