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Unformatted text preview: 92 0 Chapter3 i StructumofMotalsIndCeramics REFERENCES Azaroff, L. E, Elements of X Ray Crystallography,
McGrawIIill, New York. 1968. Reprinted by
TechBooks, Marietta, OH, 1990. Buerger, M. 1., Elementary Crystallography, Wiley,
New York, 1956. Chiang, Y. M., D. P. Birnie, 1]], and W. D. Kingery,
Physical Ceramics: Principles for Ceramic Sci
ence and Engineering, Wiley, New York, 1997. Cullity, B. D., and S. R. Stock, Elements of XRay
Dzﬂraction, 3rd edition, Prentice Hall, Upper
Saddle River, NJ, 2001. Curl, R. F. and R. E. Smalley, “Fullerenes,” Scien
tiﬁc American. Vol. 265, No. 4, October 1991,
pp. 54—63. QUESTIONS AND PROBLEMS Hauth; W. 5., “Crystal Chemistry in Ceramics,”
American Ceramic Society Bulletin, Vol. 30,
1951:No. 1, pp. 5—7; No. 2, pp. 47—49; No. 3, pp.
‘76—'17; No. 4, pp. 137142; No. 5, pp. 165—167;
No. 6, pp. 203—205. A good overview of silicate
structures. Kingery, W. D., H. K. Bowen, and D. R. Uhlmann,
Introduction to Ceramics; 2nd edition, Wiley,
New York, 1976. Chapters 1—4. Richerson, D. W., The Magic ofCeramics, American
Ceramic Society, Westerville, OH, 2000. Richerson, D. W., M odem Ceramic Engineering, 3rd
edition, CRC Press, Boca Raton, FL, 2006. Additional problems and questions for this chapter may be found on both Student and
Instructor Companion Sites at WW. M'leyxom/coﬂege/callirter. Unit Cells
Metallic Crystal Structure: 3.1 If the atomic radius of lead is 0.175 nm, calcu
late the volume of its unit cell in cubic meters. 3.2 Show that the atomic packing factor for BCC
is 0.68. Density Computations—Metals 3.3 Molybdenum has a BCC crystal structure, an
atomic radius of 0.1363 nm, and an atomic
weight of 95.94 g/mol. Compute its theoretical
density and compare it with the experimental
value found inside the front cover. 3.4 Calculate the radius of a palladium atom,
given that Pd has an FCC crystal structure, a
density of 12.0 glcma, and an atomic weight of
106.4 glmoi. 3.5 Some hypothetical metal has the simple cu
bic crystal structure shown in Figure 3.42. If
its atomic weight is 14.5 glmol and the atomic
radius is 0.145 nm, compute its density. 3.6 Using atomic weight, crystal structure, and
atomic radius data tabulated inside the front
cover, compute the theoretical densities of alu
minum, nickel, magnesium, and tungsten, and
then compare these values with the measured
densities listed in this same table. The cia ratio
for magnesium is 1.624. 3.7 Below are listed the atomic weight, density,
and atomic radius for three hypothetical al
loys. For each determine whether its crystal
structure is FCC, BCC, or simple cubic and
then justify your determination. A simple cu—
bic unit cell is shown in Figure 3.42. Atomic Atomic Weight Density Radius
Alloy _ _ (lg/moi) (5mm) fun)
A 43.1 6.40 0.122
B 184.4 12.30 0.146
C 91.6 9.60 0.137 Figure 3.42 Hardsphere unit cell
representation of the simple cubic
crystal structure. 3.8 Indium has a tetragonal unit cell for which the
a and c lattice parameters are 0.459 and 0.495
urn. respectively. (a) If the atomic packing factor and atomic ra
dius are 0.693 and 0.1625 nm, respectively,
determine the number of atoms in each
unit cell. (Is) The atomic weight of indium is 114.82
g/mol; compute its theoretical density. 3.9 Magnesium has an HCP crystal structure a
cfa ratio of 1. 624 and a density of 1.74 glcmll.
Compute the atomic radius for Mg. Ceramic Crystal Structures 3.10 Show that the minimum cationto—anion ra
dius ratio for a coordination number of 4 is 0.225. 3.11 Demonstrate that the minimum cation—to—
anioni radius ratio for a coordination number
of 8 is 0.732. 3.12 On the basis of ionic charge and ionic radii,
predict crystal structures for the following ma
terials: (u) C30
(13) KBr.
Justify your selections. Density Computations—Ceramics 3.13 Compute the atomic packing factor for the
rock salt crystal structure in which refer:
0.414. 3.14 Compute the atomic packing factor for cesium
chloride using the ionic radii in Table 3.4 and
assuming that the ions touch along the cube
diagonals. 3.15 Iron oxide (FeO) has the rock salt crystal struc
ture and a density of 5.?0 g/cm3. (a) Determine the unit cell edge length. (b) How does this result compare with the
edge length as determined from the radii
in Table 3 .4, assuming that the Fe2+ and
02 ions just touch each other along the
edges? 3.16 One crystalline form of silica (SiOz) has a
cubic unit cell, and from xray diffraction data it is known that the cell edge length is
0.700 nm. If the measured density is 2.32 glcrn3, Questions and Problems 0 93 how many Si“ and 02‘ ions are there per unit
cell? 3.17 A hypothetical AX type of ceramic material
is known to have a density of 2.10 glcm3 and
a unit cell of cubic symmetry with a cell edge
length of 0.57 nm. The atomic weights of the
A and X' elements are 28.5 and 30.0 g/mol, re
spectively. On the basis of this information,
which of the following crystal structures is
(are) possible for this material: sodium chlo
ride, cesium chloride, or zinc blende? Justify
your choice(s). Silicate Ceramics 3.18 Determine the angle between covalent bonds
in an no? tetrahedron. Carbon 3.19 Compute the theoretical density of ZnS given
that the Zn—S distance and bond angle
are 0.234 nm and 1095", respectively. How
does this value compare with the measured
density? 3.20 Compute the atomic packing factor for the di
amond cubic crystal structure (Figure 3.16).
Assume that bonding atoms touch one an
other. that the angle between adjacent bonds
is 109.5“, and that each atom internal to the
unit cell is positioned r114 of the distance away
from the two nearest cell faces (a is the unit
cell edge length). Crystal Systems 3.21 Sketch a unit cell for the facecentered or
thorhombic crystal structure. Point Coordinate: 3.22 List the point coordinates of both the sodium
and chlorine ions for a unit cell of the sodium
chloride crystal structure (Figure 3.5). 3.23 Sketch a tetragonal unit cell, and] within that
cell indicate locations of the 1 1— Iand 1 1 %
point coordinates. 3.24 Using the Molecule Deﬁnition Utility found in ._ lography“ and “Ceramic Crystal Structures”
' modules of VMSE, located on the book's web
site [www.wileycomlcollegefcallister (Student @I both “Metallic Crystal Structures and Crystal 4 ' Chnphr3 I mummcmnm Companion Site)], generate (and print out)
a threedimensional unit cell for lead oxide.
PbO, given the following: (1) The unit cell is
tetragonal with a = 0.397 nm and c = 0.502 nm,
(2) oxygen atoms are located at the following point coordinates: 000 001
100 101
010 011
110 111
11 11
ii” 551 and (3) Pb atoms are located at the following
point coordinates: ‘ 1 0 0 763 0 1 o 237
2 . 2 .
1 1
— . 3 — 0.
2 10 76 1 2 237
Crystallographic Directions
3.25 Sketch} monoclinic unit cell. and within that
cell a [101] direction. 3.26 What are the indices for the direction in
dicated by the vector in the sketch below? (1:) [212],
(a) [301]. 3.28 Determine the indices for the directions
_.‘_ shown in the following cubic unit cell: +2 . T H
.5! _.....+y top ”In / +1 3.29 For tetragonal crystals, cite the inclines of di
rections that are equivalent to each of the fol
lowing directions: (a) [011]
m)uam 3.30 Convert the [001] direction into the four
index Miller—Bravais scheme for hexagonal
unit cells. 3.31 Determine the indiccs for the two directions
shown in the following hexagonal unit cell: 3.32 Using Equations 3.7a, 3.7b. 3.?c, and 3.7d, de
rive expressions for each of the three primed
indices set (14’ , v’, and w’) in terms of the four
unprimed indices (u, v, t, and w). Crystallographic Planes 3.33 Draw an orthorhombic unit cell, and within
that cell a (021) plane. Questions and Problems ' 95 3.34 What are the indices for the plane drawn in 3.38 For each of the following crystal structures,
the sketch below? represent the indicated plane in the manner of Figures 3.26 and 3.27, showing both anions and cations: (in) (111) plane for the diamond cubic crystal
structure, and (b) (110) plane for the ﬂuorite crystal struc
ture. 3.39 Below are shown three different crystallo
graphic planes for a unit cell of some hypo
thetical metal. The circles represent atoms: 3.35 Sketch within a cubic unit cell the following T T T
5.: ' '2 planes: 5 g E
2:! (a) (012) . (c) (191), g a a
99 (b) (313) (a) (211). 1 j: 0‘
3.36 Determine the Miller indices for the planes H0 30 MRI
' ' _'shown tn the following unit cell. m ’ o. 25 m
I (110) {101) {011} (a) To what crystal system does the unit cell
belong? (b) What would this crystal structure be
called? (c) If the density of this metal is 18.91 glen13,
determine its atomic weight. 3.40 Convert the (0T2) plane into the fourindex
Miller—Bravais scheme for hexagonal unit cells.
/ H: 3.41 Determine the indices for the planes shown in
the hexagonal unit cells below: 3.37 Determine the Miller indices for the planes
"I".3 shown in the following unit cell: I
'1‘. .
a: 1"“ +3" 3.42 Sketch the (2110) plane in a hexagonal unit
cell. 96 I Chephr3 I'Shuehrmanctalsand Ceramics Linearandﬂamrbemﬂies 3.43 (a) Derive linear density expressions for FCC
[100] and [111] directions in terms of the
atomic radius R. ' (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 radius R.
(1)) Compute and compare planar density val
ues for these same two planes for molyb—
denum. I ClosePacked Crystal Structures 3.45 The comndum crystal structure, found for
A1203, consists of an HCP arrangement of
02‘ ions; the Al“ ions Occupy octahedral po
sitions. (a) 'What fraction of the available octahedral
positions are ﬁlled with A1“ ions? (Ir) Sketch two closepacked 02* planes
stacked in an AB sequence, and note octa
hedral positions that will be ﬁlled with the
Al3+ ions. 3.46 Iron titanate. FeTiOg, forms in the ilmenite
crystal structure that consists of an llCP ar
rangement of Oz'ions. (In) Which type of interstitial site will the Fe“ ions occupy? Why? (h) Which type of interstitial site will the 'Ii4+
ions occupy? Why? (c) What fraction of the total tetrahedral sites
will be occupied? (d) What traction of the total octahedral sites
will be occupied? g E 0.0 20.0 «0.0 50.0 50.0
Diﬁraction angle 29 100.0 X—Ray Dalmatian: Detemu'nalz'an of
Cranial 5mm 3.47 Determine the expected diffraction angle for
the ﬁrstorder reﬂection from the (310) set of
planes for BCC chromium when monochro
matic radiation of wavelength 0.0711 nm is
used. 3.48 Using the data for ctiron in Table 3.1, compute
the interplanar spacings for the (11 1) and (21 1)
sets of planes 3.49 The metal niobium has a BCC crystal struc
ture. If the angle of diffraction for the (211) set
of planes occurs at 75.95” (ﬁrstorder reﬂec
tion) when monochromatic xradiation 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. 3.50 Figure 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 run; each diffraction peak on
the pattern has been indexed. Compute the
interplanar spacing for each set of planes m
dexed; also determine the lattice parameter of
Pb for each of the peaks. 3.51 Figure 3.43 shows the ﬁrst ﬁve peaks of the
x—ray diffraction pattern for tungsten, which
has a BCC crystal structure; monochromatic
xradiation having a wavelength of 0.1542 nm
was used. (a) Index (i.e.. give h, k, and l indices) 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. Figure 3.43 Diffraction pattern for
powdered tungsten. (Courtesy of Wesley
L. Holman.) ...
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