Chemistry - A Molecular Science

Chapter 8 Solid Materials

8.3

UNIT CELL STOICHIOMETRY

(b) edge

(c) corner

(a) face

Figure 8.5 Atoms in more than one unit cell (a) Atoms on the faces are shared by two unit cells. (b) Atoms on an edge are shared by four unit cells.

The stoichiometry of a compound can be determined from the composition of its unit cell. However, recall that unit cells pack to completely fill space, so they must share faces, corners, and edges with other unit cells. C

onsequently, atoms sitting on these sites are

shared by more than one unit

cell. For example, consider the two-dimensional lattice of

A’s and

B’s in Figure 8.1. The

A’s on the corners of a cell are each shared by four unit

cells, so each

A on a corner contributes only

1 /^4

A

to any one unit cell. As a result, the four

corner

A’s combine to contribute only one

A to the overall stoichiometry (4 corners

1 x

/^4

A^

per corner = 1

A). The

B in the center of the cell is completely within the cell, so it

contributes a full

B to the overall stoichiometry of the cell. Consequently, the

stoichiometry of the unit cell is

AB

. The same stoichiometry is achieved using the cell

with

B’s on the corner and an

A in the center. Figure 8.5 shows that an atom (the yellow

sphere) is shared by two unit cells when it occupies a face center (Figure 8.5a), by four unit cells (Figure 8.5b) when it sits on an edge,

and by eight unit cells when it is located on

a corner (Figure 8.5c).

(c) Atoms on a corner are shared by eight unit cells.

The contribution of each atom position in the unit cell (

uc

) to the overall unit cell

stoichiometry can be summarized as follows:

(^) •
Each atom located totally within the
cell contributes 1 atom to the unit cell.
(^) •
Atoms on the faces contribute a total of
3 atoms to the unit cell. A cube has
six faces
, but
only
1 /^2
of each atom is in any one unit cell (Figure 8.5a); (
1 /^2
)(6) = 3 atoms.
(^) •
Atoms on the edges contribute a total of 3 atoms to the unit cell. A cube has twelve edges, but only
1 /^4
of each atom on an edge is in any one cell (Figure 8.5b); (
1 /^4
)(12) = 3 atoms.
(^) •
Atoms on the corners contribute at total of 1 atom to the unit cell. A cube has eight corners, but only
1 /^8
of each atom is in any on
e unit cell (Figure 8.5c); (
1 /^8
)(8) = 1 atom.
Example 8.2
How many atoms are in each of the three cubic unit cells?
a) Simple cubic (sc)
Atoms are on the eight corners onl
y, so there is only one atom
in a simple cubic unit cell.
b) Body-centered cubic (bcc)
Atoms on the eight corners contribute one
atom and the atom in the body center
contributes another atom, so there are tw
o atoms in a body-centered cubic unit cell
.
c) Face-centered cubic (fcc)
Atoms are on the eight corners contribute one atom and those in the six face centers contribute three more atoms, so there are fo
ur atoms in a face-centered cubic unit cell
.
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