Fundamentals of Materials Science and Engineering: An Integrated Approach, 3e

GTBL042-03 GTBL042-Callister-v2 September 6, 2007 15:33

3.21 Noncrystalline Solids • 87

Intensity

0.0 10.0 20.0 30.0 40.0

(111)

(200)

(220)

(311)(222)

(400) (331)(420) (422)

50.0

Diffraction angle 2

60.0 70.0 80.0 90.0 100.0

Figure 3.39

Diffraction pattern

for powdered lead.

(Courtesy of Wesley

L. Holman.)

by a specific set of crystallographic planes. Other uses of x-rays include qualitative

and quantitative chemical identifications and the determination of residual stresses

and crystal size.

EXAMPLE PROBLEM 3.15

Interplanar Spacing and Diffraction Angle Computations

For BCC iron, compute (a) the interplanar spacing and (b) the diffraction angle

for the (220) set of planes. The lattice parameter for Fe is 0.2866 nm. Also,

assume that monochromatic radiation having a wavelength of 0.1790 nm is

used, and the order of reflection is 1.

Solution

(a)The value of the interplanar spacingdhklis determined using Equation 3.15,

witha=0.2866 nm, andh= 2 ,k=2, andl=0, since we are considering

the (220) planes. Therefore,

dhkl=

a

√

h^2 +k^2 +l^2

=

0 .2866 nm

√

(2)^2 +(2)^2 +(0)^2

= 0 .1013 nm

(b)The value ofθmay now be computed using Equation 3.14, withn=1, since

this is a first-order reflection:

sinθ=

nλ

2 dhkl

=

(1)(0.1790 nm)

(2)(0.1013 nm)

= 0. 884

θ=sin−^1 (0.884)= 62. 13 ◦

The diffraction angle is 2θ,or

2 θ=(2)(62. 13 ◦)= 124. 26 ◦

3.21 NONCRYSTALLINE SOLIDS

noncrystalline It has been mentioned thatnoncrystallinesolids lack a systematic and regular ar-

rangement of atoms over relatively large atomic distances. Sometimes such materials

amorphous are also calledamorphous(meaning literally without form) or supercooled liquids,

inasmuch as their atomic structure resembles that of a liquid.

An amorphous condition may be illustrated by comparison of the crystalline

and noncrystalline structures of the ceramic compound silicon dioxide (SiO 2 ), which