Chemistry Reference
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indices. Another shortcoming is that in this example the [
hkl
] direction is not
normal to the (
hkl
) plane.
A better solution is to use a four-axis, four-index system for hcp solids. A
few examples are depicted in Fig. 4.10. The Miller indices are found as before
by taking the reciprocals of the intercepts of the plane with the four axes. Using
this system, the equivalent planes discussed above become (1100) and (1010).
Now the six equivalent planes resulting from the sixfold axis of symmetry can
be identified as the {1100} family of planes. Using four axes, the [
hkil
] direc-
tion is normal to the (
hkil
) plane, in the same way it was for cubic solids using
the three-axis system.
§
4.5 SURFACE RELAXATION
In the example above, we placed atoms in our slab model in order to create a
five-layer slab. The positions of the atoms were the ideal, bulk positions for the
fcc material. In a bulk fcc metal, the distance between any two adjacent layers
must be identical. But there is no reason that layers of the material near a sur-
face must retain the same spacings. On the contrary, since the coordination of
atoms in the surface is reduced compared with those in the bulk, it is natural to
expect that the spacings between layers near the surface might be somewhat
different from those in the bulk. This phenomenon is called surface relaxation,
and a reasonable goal of our initial calculations with a surface is to characterize
this relaxation.
In Fig. 4.11, the positions of atoms in a DFT calculation of a five-layer slab
model of a surface before and after relaxation are indicated schematically. On
the left is the original slab model with the atoms placed at bulk positions and
on the right is the slab model after relaxation of the top three layers. Surface
relaxation implies that the relaxed surface has a lower energy than the original,
ideal surface. We can find the geometry of the relaxed surface by performing
an energy minimization as a function of the positions of the atoms in the super-
cell. We imagine the bottom of the slab as representing the bulk part of a
material and constrain the atoms in the bottom layers in their ideal, bulk
positions. The calculation then involves a minimization of the total energy
of the supercell as a function of the positions of the atoms, with only the
atoms in the top layers allowed to move, as described in Section 3.5.2. This
results in a structure such as the one shown on the right in Fig. 4.11. In reality,
the movements of the atoms in surfaces are on the order of 0.1
˚
. In this figure,
the magnitude of the relaxation of the top layers is exaggerated.
§
Because the four axes are not independent, the first three indices in the four index system will
always sum to zero. (This serves as a convenient check.)
