Unit Cells and Packing Patterns
The
choice of unit cell axes for a lattice pattern is subject to some restrictions
but is otherwise quite arbitrary. For example, in the cubic closest
packed pattern (ABC) shown at the right, the axes a and b
have been chosen and the resulting unit cell is outlined in blue.
The host atoms are shown as small black circles and are labeled according
to the layer in which they reside.
This is an elevation diagram, with the c stacking direction rising
from the page toward the viewer so that the coordinate system defined
by the three unit cell edges is right-handed (on your right hand,
if lattice vector a extends from the center of the palm toward the
tips of the fingers, and lattice vector b extends from the center
of the palm toward the wrist, then the vector resultant "a cross
b"
is lattice vector c along your thumb).
The letter designations of the hexagonal closest packed planes are also
arbitrary. To start a stacking pattern, any of the three orientations
A, B or C can be chosen, so why not start with A? The A layer has
elevation 0. The second layer has been designated B, and the elevation
of the B layer along the c unit cell axis is c/3, or in units of
c/3,
the elevation is 1. The third layer is C and has elevation 2c/3
or 2, and the next layer is again A with elevation c = 3c/3
or 3. For the purposes of this discussion, let us arbitrarily designate
the B layer as the layer which has a host atom covering the equilateral
triangle bounded by the a lattice vector; the C layer then has a
host atom covering the equilateral triangle bounded by the b lattice
vector.
Now
consider what happens if the lattice vectors are reshosen as shown in the
diagram at the left. The convention for naming layers dicatates that
the "B" and "C" designations be switched, so that the B layer now has elevation
2 and the C layer has elevation 1. Thus, in order of stacking (bottom
to top), this pattern would be called (ACB). However, disregarding
the labels, it is clear that the actual lattice pattern has not changed
at all, so the two stacking patterns (ABC) and (ACB) are identical.
In the same way, the diagrams below illustrate the fact that the two
four-layer stacking patterns (ABCB) and (ACBC) are identical, "differing"
only in the choice of unit cell axes chosen to describe the pattern.
Thus, the elevation diagram is a useful graphic device for determining
equivalency or non-equivalency of two literal stacking pattern designations.
