Chemistry - A Molecular Science

8.2

CUBIC UNIT CELLS AND METALLIC RADII

(a) (b) (c)

Simple cubic (sc)Body-centered cubic (bcc)Face-centered cubic (fcc)

Figure 8.3 Cubic unit cells Different colors are used to distinguish between the different lattice sites, not different atom types. (a)

sc unit cell2r=a

(c)

a

fd = 2 a

bcc unit cellbd=4r= a +fd4r = a + 2a4r = 3 a

22

22

a

a

fcc unit cellfd=4r= a +a4r = 2 a

22

(b)

fd

bd

For simplicity, the following discussion is

based on metallic solids because all of the

atoms in a metal are identical. There are three

types of cubic unit cells that differ in the

locations of the sites that are occupied by the atoms. Simple-cubic (sc)

unit cell: The atoms reside only on

the eight corners (Figure 8.3a).

Body-centered cubic (bcc)

unit cell: the atoms occupy the eigh

t corners and the center of the cell

body (Figure 8.3b).

Face-centered cubic (

fcc

) unit cell: The atoms occupy the eigh

t corners and the six face centers

(Figure 8.3c). The relative positions of the atoms in a unit

cell are determined by scattering x-rays

from the atoms with a technique called x-ray di

ffraction. One application of this method is

the determination of atomic radii, which are

inferred from the distances between the atoms

by assuming that the atoms touch along one line in the unit cell. In this method, the radius of an atom is related to the unit cell edge length (

a). As shown in Figure 8.4, the line of

contact in metallic solids is along the edge of a

sc

unit cell, along the face diagonal (

fd

) of

a fcc

unit cell, and along the body diagonal (

bd

) of a

bcc

unit cell. The edge of the

sc

unit

cell equals two atomic radii, while the lengths of the

fd

and

bd

of the

fcc

and

bcc

unit cells

are each equal to four atomic radii. Setti

ng the edge length equal to the 2r in the

sc

unit

cell, and applying the Pythagorean Theorem to the triangles shown in Figures 8.4b and 8.4c, we obtain the relationships between the atomic radius (

r) and the edge length (

a).

a

a) sc unit cells: r =

2 2a

b) fcc unit cells: r =

4 3a

c) bcc unit cells: r =

4

Eq. 8.1

Atomic radii obtained from the structures of metallic solids are also called

metallic radii

.

Example 8.1

Figure 8.4 Atom contact in cubic unit cells

-iron adopts a fcc crystal structure with an edge length of 3.56 Å. What is the γ metallic radius of iron inferred from this structure?

Atom contact is along the edge of

a sc cell (a), along the face

diagonal (fd) of a fcc cell (b), and

along the body diagonal (bd) of a

bcc cell (c). The arrows along the diagonals represent the atomic radii of which they are composed.

Use Equation 8.1b for fcc unit cells and the known edge length.

o

2

r =

a = (0.354)(3.56) = 1.26 A

4

Chapter 8 Solid Materials

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