3.7 Iron has a BCC crystal structure, an atomic radius of 0.124 nm, and an atomic weight of 55.85
g/mol. Compute and compare its theoretical density with the experimental value found inside the
front cover.
Solution
This problem calls for a computation of the density of iron. According to Equation 3.5
ρ =
nAFe
VC N A
For BCC, n = 2 atoms/unit cell, and
⎛ 4 R ⎞3
VC = ⎜
⎟
⎝ 3⎠
Thus,
ρ =
=
nAFe
⎛ 4 R ⎞3
⎜
⎟ NA
⎝ 3⎠
(2 atoms/unit cell)(55.85 g/mol)
[(4) (0.124 × 10-7 cm) / 3] /(unit cell) (6.022 × 1023 atoms/mol)
3
= 7.90 g/cm3
The value given inside the front cover is 7.87 g/cm3.
3.14 Below are listed the atomic weight, density, and atomic radius for three hypothetical alloys.
For each determine whether its crystal structure is FCC, BCC, or simple cubic and then justify your
determination. A simple cubic unit cell is shown in Figure 3.24.
Alloy
Atomic Weight
Density
Atomic Radius
3
(g/mol)
(g/cm )
(nm)
A
77.4
8.22
0.125
B
107.6
13.42
0.133
C
127.3
9.23
0.142
Solution
For each of these three alloys we need, by trial and error, to calculate the density using Equation
3.5, and compare it to the value cited in the problem.
For SC, BCC, and FCC crystal structures, the
respective values of n are 1, 2, and 4, whereas the expressions for a (since VC = a3) are 2R, 2 R 2 ,
and
4R
.
3
For alloy A, let us calculate ρ assuming a simple cubic crystal structure.
ρ =
=
=
nAA
VC N A
nAA
(2R)3 N A
(1 atom/unit cell)(77.4 g/mol)
3
⎧
⎫
⎨ (2)(1.25 × 10−8 ) /(unit cell)⎬ (6.022 × 10 23 atoms/mol)
⎩
⎭
[
]
= 8.22 g/cm3
Therefore, its crystal structure is simple cubic.
For alloy B, let us calculate ρ assuming an FCC crystal structure.
ρ =
=
nAB
(2 R 2) 3 N A
(4 atoms/unit cell)(107.6 g/mol)
3
⎧
⎫
⎨ 2 2 (1.33 × 10 -8 cm) /(unit cell)⎬ (6.022 × 10 23 atoms/mol)
⎩
⎭
[( )
]
= 13.42 g/cm3
Therefore, its crystal structure is FCC.
For alloy C, let us calculate ρ assuming a simple cubic crystal structure.
=
=
nAC
(2R)3 N A
(1 atom/unit cell)(127.3 g/mol)
3
⎧
⎫
⎨ (2)(1.42 × 10 -8 cm) /(unit cell)⎬ (6.022 × 10 23 atoms/mol)
⎩
⎭
[
]
= 9.23 g/cm3
Therefore, its crystal structure is simple cubic.
3.23 List the point coordinates of the titanium, barium, and oxygen ions for a unit cell of the
perovskite crystal structure (Figure 12.6).
Solution
In Figure 12.6, the barium ions are situated at all corner positions. The point coordinates for
these ions are as follows: 000, 100, 110, 010, 001, 101, 111, and 011.
The oxygen ions are located at all face-centered positions; therefore, their coordinates are
11
22
1, 1
11
22
, 0
11
22
,
1
2
1
1 1
2
2 2
0 , and
11
22
0,
1 .
And, finally, the titanium ion resides at the center of the cubic unit cell, with coordinates
1 11
2 22
.
3.30 Within a cubic unit cell, sketch the following directions:
(a) [1 10] ,
(b) [1 2 1] ,
(c) [01 2] .
3.31 Determine the indices for the directions shown in the following cubic unit cell:
Solution
Direction A is a [01 1 ]direction, which determination is summarized as follows. We first of all
position the origin of the coordinate system at the tail of the direction vector; then in terms of this
new coordinate system
x
y
z
Projections
0a
–b
–c
Projections in terms of a, b, and c
0
–1
–1
Reduction to integers
not necessary
Enclosure
[01 1 ]
Direction B is a [ 210] direction, which determination is summarized as follows. We first of all
position the origin of the coordinate system at the tail of the direction vector; then in terms of this
new coordinate system
x
y
z
Projections
–a
b
2
0c
Projections in terms of a, b, and c
–1
Reduction to integers
–2
Enclosure
1
2
1
[ 2 10]
0
0
Direction C is a [112] direction, which determination is summarized as follows. We first of all
position the origin of the coordinate system at the tail of the direction vector; then in terms of this
new coordinate system
Projections
Projections in terms of a, b, and c
Reduction to integers
x
y
z
a
2
b
2
c
1
1
2
2
1
1
Enclosure
1
2
[112]
Direction D is a [112 ] direction, which determination is summarized as follows. We first of all
position the origin of the coordinate system at the tail of the direction vector; then in terms of this
new coordinate system
Projections
Projections in terms of a, b, and c
Reduction to integers
Enclosure
x
y
z
a
2
b
2
–c
1
1
2
2
1
1
[112 ]
–1
–2
3.40 Sketch within a cubic unit cell the following planes:
(a)
(01 1 )
(b)
(112 )
3.42 Determine the Miller indices for the planes shown in the following unit cell:
Solution
For plane A we will move the origin of the coordinate system one unit cell distance to the upward
along the z axis; thus, this is a (322) plane, as summarized below.
Intercepts
Intercepts in terms of a, b, and c
Reciprocals of intercepts
x
y
a
b
3
2
1
1
3
2
3
2
Reduction
(not necessary)
Enclosure
(322)
z
c
–
2
1
–
2
–2
For plane B we will move the original of the coordinate system on unit cell distance along the x
axis; thus, this is a (1 01) plane, as summarized below.
x
Intercepts
–
Intercepts in terms of a, b, and c
–
a
2
1
2
y
z
∞b
c
∞
1
2
2
Reciprocals of intercepts
–2
0
2
Reduction
–1
0
1
Enclosure
(1 01)
3.57 Explain why the properties of polycrystalline materials are most often isotropic.
Solution
Although each individual grain in a polycrystalline material may be anisotropic, if the grains have
random orientations, then the solid aggregate of the many anisotropic grains will behave isotropically.
11.A
12.C