Page 28 - Modern physical chemistry

1.13 Surface Structure 17

The best results from X-ray determinations have been compared with those found in
other ways. Critical evaluations have led to the value

N A = 6.0221 x 10 23 mori. [1.10]

Example 1.6

At room temperature, the density of a good lithium fluoride crystal is 2.640 g cm- 3 .
X-ray studies show that the LiF has a face-cantered cubic lattice with a unit-cell edge
4.016 A long. From these data, calculate Avogadro's number.
Each P- at a comer of a unit cube is shared with seven other unit cells, while each
P- at the middle of a face is shared with one other unit cell. Since there are eight comers
to the cube and six faces, the number of F- ions effectively in the unit cube is
8 6
n =-+-=4.
8 2

Because the cube is electrically neutral, the unit cell contains a like number of cations.
Substituting into formula (1.8),

V (2.640 g cm .3X4.016 x 10-8 cm)3
m = L = = 4.275 x 10- 23 g (formula units/ I
n 4 formula units

and then into (1.9) gives us
M
N A = - = 25.939 g mol- I = 6.068 x 10 23 formula units mol-I.
m 4.275 x 10- 23 g (formula units)"!

1. 13 Surface Structure

At the surface of a condensed phase, the coordination number of a unit (atom, ion,
or molecule) is lower than it is in a homogeneous region of the phase. As a result, the
units there may adopt a geometric arrangement different from that in the interior.
When the condensed phase is a solid conductor, the surface structure may be deter-
mined directly by moving a relatively charged sharp metal tip over the surface, keeping
the resulting current low and constant. How the tip moves may be analyzed and screened
by a computer. The current is said to tunnel between the tip and the closest point on the
surface. So the apparatus is called a scanning tunneling microscope. See figure 1.16. The
technique is referred to as scanning tunneling microscopy (STM).
To move the tip over the solid with atomic-sized resolution, one employs orthogonal
piezoelectric positioners. In figure 1.16, these are labeled x, y, and z. A bias voltage is
applied between the tip and the sample to be studied. The resulting current is held con-
stant by controlling the vertical position of the tip, using the z position~r, as the tip is
moved over the surface, using the x and y positioners. The computer is programmed so
that an image of the surface appears on the screen.
As an example, consider semiconductor silicon. Within the silicon crystal, each atom
is bound tetrahedrally to its nearest neighbors. But when such a crystal is cleaved, bonds
are broken along the cleavage surfaces. If no rearrangements occurred, the surface atoms
would sport dangling bonds, valence electrons not involved in bonding. To reduce the