Page 120 - Materials Science and Engineering An Introduction

92 • Chapter 3 / The Structure of Crystalline Solids

EXAMPLE PROBLEM 3.15

Interplanar Spacing and Lattice Parameter Computations for Lead
Figure 3.24 shows an x-ray diffraction pattern for lead taken using a diffractometer and mono-
chromatic x-radiation having a wavelength of 0.1542 nm; each diffraction peak on the pattern
has been indexed. Compute the interplanar spacing for each set of planes indexed; also, de-
termine the lattice parameter of Pb for each of the peaks. For all peaks, assume the order of
diffraction is 1.

Solution
For each peak, in order to compute the interplanar spacing and the lattice parameter we must
employ Equations 3.21 and 3.22, respectively. The first peak of Figure 3.24, which results from
diffraction by the (111) set of planes, occurs at 2u 31.3 ; the corresponding interplanar spacing
for this set of planes, using Equation 3.21, is equal to

nl (1)(0.1542 nm)
d 111 = = = 0.2858 nm
2 sin u 31.3
(2)c sin a b d
2
And, from Equation 3.22, the lattice parameter a is determined as
2
2
a = d hkl 2h + k + l 2
2
2
= d 111 2(1) + (1) + (1) 2
= (0.2858 nm)13 = 0.4950 nm
Similar computations are made for the next four peaks; the results are tabulated below:

Peak Index 2U d hkl (nm) a(nm)
200 36.6 0.2455 0.4910
220 52.6 0.1740 0.4921
311 62.5 0.1486 0.4929
222 65.5 0.1425 0.4936

3.17 NONCRYSTALLINE SOLIDS

noncrystalline It has been mentioned that noncrystalline solids lack a systematic and regular arrange-
ment of atoms over relatively large atomic distances. Sometimes such materials are also
amorphous called amorphous (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 may ex-
ist in both states. Figures 3.25a and 3.25b present two-dimensional schematic diagrams
for both structures of SiO 2 . Even though each silicon ion bonds to three oxygen ions
for both states, beyond this, the structure is much more disordered and irregular for the
noncrystalline structure.
Whether a crystalline or an amorphous solid forms depends on the ease with which
a random atomic structure in the liquid can transform to an ordered state during so-
lidification. Amorphous materials, therefore, are characterized by atomic or molecular
structures that are relatively complex and become ordered only with some difficulty.