Physical Chemistry , 1st ed.

standing of Miller indices, as defined and described in Chapter 21. Suppose we

cut off one side of the crystal so that the atoms in the defined plane are now

the surface of the solid. The Miller indices of that plane of atoms are sufficient

to describe the surface of that solid. In this case, we can say that this surface is

the (XYZ) plane of the crystal.

Miller indices are commonly used to describe surfaces of crystalline solids.

Figure 22.15 shows some examples.

Example 22.5

Refer to Figure 22.16, which illustrates a surface of a body-centered cubic

crystalline solid. What are the Miller indices of the surface indicated?

Solution

Since the crystal is a cubic solid, the unit cell parameters a,b, and care equal.

The surface plane indicated intercepts the aand baxes at one unit each, and

does not intercept the c-axis at all. Another way of saying this is that the plane

intercepts the c-axis at . Miller indices are reciprocals of intercepts, so the

Miller indices of this surface plane are (^11 ^11 ^1 ) or (110). This surface is the

(110) surface plane. Other surfaces of crystals can be labeled similarly.

As with liquid surfacces, the surface of a solid is associated with an energy.

However, the term “surface tension” is not usually associated with solid sur-

faces. Instead, the term surface energyis used (although the concept is the

same: it takes energy to increase the surface area of a solid). Table 22.2 lists

some surface energies of solids. Generally, metals have relatively high surface

energies, and ionic compounds have lower surface energies. Generally speak-

ing, solids have higher surface energies than liquids, although a comparison of

Tables 22.1 and 22.2 shows that at least one liquid (mercury) has a higher sur-

face energy (“surface tension”) than solids.

Surface energies for solids also vary depending on the arrangement of atoms

making the surface plane; that is, surface planes having different Miller indices

will have different surface energies.

The idea of different surface energies and an understanding of thermo-

dynamics suggest that surfaces might tend to adopt the surface that has the

lowest surface energy. Indeed they would tendto, but two factors work

against this.

1. It is impossible for any solid except a cubic crystal to have the same
   Miller indices for all of its surfaces. This is because for any general solid,

22.5 Solid Surfaces 779

(a) fcc (111)

(b) hcp (0001)

(c) bcc (110)

(d) fcc (100)
Figure 22.15 Atomic arrangements for some
crystals and the different Miller indices of the
surface. In the diagrams, the darker atoms are
the top layer, which has a different arrangement
depending on the crystal lattice and (hk).

Figure 22.16 See Example 22.5. What are the
Miller indices of the indicated surface of this
body-centered cubic crystal?

Table 22.2 Surface energies of solids

Solid Temperature (°C) Surface energy (dyn/cm or erg/cm^2 )a

gold 1027 1410

iron 1400 2150

LiF 195 340

NaCl 25 227

KCl 25 110

MgO 25 1200

CaF 2 195 450

BaF 2 195 280

aTo convert to units of J/m (^2) , multiply by 1  10  (^3).