Physical Chemistry Third Edition

1158 28 The Structure of Solids, Liquids, and Polymers

cobalt crystallizes in the hexagonal lattice. All of the inert gases crystallize in the

face-centered cubic lattice except for helium, which crystallizes in the hexagonal

lattice.

EXAMPLE28.1

Show that the packing fraction for the fcc close-packed lattice is equal to 0.74.

Solution

For spheres of radiusrtouching each other, the diagonal of the cube face is equal to 4r, and

is also equal to

√

2 a, whereais the dimension of the unit cell.

r

1

4

√

2 a

1

23 /^2

a

There are 6 atoms in the faces of the unit cell, and half of each is in the cell:

(6)

(

1

2

)

4

3

πr^3  4 πr^3  4 π 2 −^9 /^2 a^3  0. 55536 a^3

There are 8 atoms in the corners of the unit cell, and 1/8 of each is in the cell:

(8)

(

1

8

)

4

3

πr^3 

4

3

πr^3 

4

3

π 2 −^9 /^2 a^3  0. 18512 a^3

The volume of the unit cell isa^3 and the total volume of the spheres in the unit cell is

V(spheres)(0. 5536 + 0 .18512) 0. 74048 a^3

Packing fraction

0. 74048 a^3

a^3

 0. 74048

Exercise 28.2

Iron crystallizes in the body-centered cubic lattice. Give the coordination number for this lattice

and calculate the packing fraction for this lattice, assuming spheres that touch.

X-Ray Diffraction and Miller Indices

Since X-rays are electromagnetic radiation with wavelengths roughly equal to crystal

lattice spacings, crystals can act as diffraction gratings for X-rays. Study of the angles

and intensities of diffracted X-ray beams can allow unit cell dimensions and positions

of atoms within the unit cells to be calculated. To describe the diffraction of X-rays,

it is necessary to specify the directions of planes that fit into a crystal lattice in such a

way that the plane contains a repeating regular pattern of lattice points. These planes

intersect with one or more of the axes within a unit cell. We define a plane that intersects

with theaaxis at a distancea/hfrom the origin, with thebaxis at a distanceb/kfrom

the origin, and with thecaxis at a distancec/lfrom the origin, whereh,k, andlare