Concise Physical Chemistry

c11 JWBS043-Rogers September 13, 2010 11:26 Printer Name: Yet to Come

CRYSTALS 175

FIGURE 11.11 A two-dimensional unit cell for packing of discs.

the three-dimensional structure of atomic, molecular, and ionic solids in the regular

arrangements we find in crystals. The concept of the unit cell is central in X-ray

crystallography.

A crystal with the simple square atomic or molecular packing arrangement in all

three directions is completely described as a repeating three-dimensional pattern of

cubic unit cells. Once we know the length of the edge of the cubic unit cell, we know

the geometry of the entire crystal (aside from impurities and structural imperfections).

The three-dimensional analog of Fig. 11.11 has a simple cubic unit cell

(Fig. 11.12).

To simplify, let us go back to the unit cell of discs in Fig. 11.11. The total areaAof

discs within the cell is 4 timesA/4, whereA=πr^2 because, although there are four

discs, one on each corner of the cell, only one-fourth of each discis actually inside

the cell. The dimension of the cell itself is 2ron an edge so the area of the unit cell is

( 2 r)^2 = 4 π^2. The area of the discs within the cell isπr^2 , so the packing fraction is

P=

Adisc

Acell

=

πr^2

4 r^2

=

π

4

= 0. 785

When this structure is taken to a three-dimensional simple cubic structure, the inter-

stitial space at the center of the cube takes a larger proportion of the whole and the

packing fraction is a rather inefficient 0.52. Simple cubic packing is not favored by

many crystals. An exception is the metal polonium,^84 Po.

FIGURE 11.12 A simple cubic cell.