GTBL042-03 GTBL042-Callister-v2 September 6, 2007 15:33
96 • Chapter 3 / Structures of Metals and CeramicsLinear and Planar Densities
3.43 (a)Derive linear density expressions for FCC
[100] and [111] directions in terms of the
atomic radiusR.
(b)Compute and compare linear density val-
ues for these same two directions for cop-
per.
3.44 (a)Derive planar density expressions for
BCC (100) and (110) planes in terms of
the atomic radiusR.
(b)Compute and compare planar density val-
ues for these same two planes for molyb-
denum.
Close-Packed Crystal Structures
3.45The corundum crystal structure, found for
Al 2 O 3 , consists of an HCP arrangement of
O^2 −ions; the Al^3 +ions occupy octahedral po-
sitions.
(a)What fraction of the available octahedral
positions are filled with Al^3 +ions?
(b)Sketch two close-packed O^2 − planes
stacked in anABsequence, and note octa-
hedral positions that will be filled with the
Al^3 +ions.
3.46Iron titanate, FeTiO 3 , forms in the ilmenite
crystal structure that consists of an HCP ar-
rangement of O^2 −ions.
(a)Which type of interstitial site will the Fe^2 +
ions occupy? Why?
(b)Which type of interstitial site will the Ti^4 +
ions occupy? Why?
(c)What fraction of the total tetrahedral sites
will be occupied?
(d)What fraction of the total octahedral sites
will be occupied?0.0 20.0 40.0 60.0 80.0 100.0
Diffraction angle 2IntensityFigure 3.43 Diffraction pattern for
powdered tungsten. (Courtesy of Wesley
L. Holman.)X-Ray Diffraction: Determination of
Crystal Structures
3.47Determine the expected diffraction angle for
the first-order reflection from the (310) set of
planes for BCC chromium when monochro-
matic radiation of wavelength 0.0711 nm is
used.
3.48Using the data forα-iron in Table 3.1, compute
the interplanar spacings for the (111) and (211)
sets of planes.
3.49The metal niobium has a BCC crystal struc-
ture. If the angle of diffraction for the (211) set
of planes occurs at 75.99◦(first-order reflec-
tion) when monochromatic x-radiation hav-
ing a wavelength of 0.1659 nm is used, com-
pute(a)the interplanar spacing for this set of
planes, and(b)the atomic radius for the nio-
bium atom.
3.50Figure 3.39 shows an x-ray diffraction pat-
tern for lead taken using a diffractometer and
monochromatic x-radiation having a wave-
length of 0.1542 nm; each diffraction peak on
the pattern has been indexed. Compute the
interplanar spacing for each set of planes in-
dexed; also determine the lattice parameter of
Pb for each of the peaks.
3.51Figure 3.43 shows the first five peaks of the
x-ray diffraction pattern for tungsten, which
has a BCC crystal structure; monochromatic
x-radiation having a wavelength of 0.1542 nm
was used.
(a)Index (i.e., giveh,k, andlindices) for each
of these peaks.
(b)Determine the interplanar spacing for
each of the peaks.
(c)For each peak, determine the atomic ra-
dius for W and compare these with the
value presented in Table 3.1.