CHAPTER 2
Q1- How many grams are there in a one amu of a material?
A1- In order to determine the number of grams in one amu of material, appropriate manipulation of the
amu/atom, g/mol, and atom/mol relationships is all that is necessary, as

 1 g / mol 
1 mol
# g/amu = 


23
6.023 x 10
atoms 1 amu / atom 
= 1.66 x 10
-24
g/amu
Q2- Show the electron configuration for Fe3+, Cu+, and S23+
2 2 6 2 6 5
A2Fe - 1s 2s 2p 3s 3p 3d
+
2 2 6 2 6 10
Cu - 1s 2s 2p 3s 3p 3d
22 2 6 2 6
S - 1s 2s 2p 3s 3p
Q3. Why water expands on freezing?
A3. The geometry of the H O molecules, which are hydrogen bonded to one another, is more
2
restricted in the solid phase than for the liquid. This results in a more open molecular structure in
the solid, and a less dense solid phase.
CHAPTER 3
Q1- What is the difference between a crystal structure and a crystal system?
A1- A crystal structure is described by both the geometry of, and atomic arrangements within, the unit
cell, whereas a crystal system is described only in terms of the unit cell geometry. For example,
face-centered cubic and body-centered cubic are crystal structures that belong to the cubic
crystal system.
Q2- Show that the ideal c/a ratio for HCP is 1.633.
A2- A sketch of one-third of an HCP unit cell is shown below.
Consider the tetrahedron labeled as JKLM, which is reconstructed as
The atom at point M is midway between the top and bottom faces of the unit cell--that is MH =
c/2. And, since atoms at points J, K, and M, all touch one another,
JM = JK = 2R = a
where R is the atomic radius. Furthermore, from triangle JHM,
(JM) 2 = (JH)2  (MH)2, or
2
c 
a 2 = (JH)2 +  
2 
Now, we can determine the JH length by consideration of triangle JKL, which is an equilateral
triangle,
a/ 2
3
=
, and
JH
2
a
JH =
3
Substituting this value for JH in the above expression yields
cos 30 =
2
2
 a 2 c 2
a
c
a =   +   =
+
2 
3
4
 3 
2
and, solving for c/a
c
=
a
8
= 1.633
3
Q3- Calculate the unit cell volume for Co which has an HCP crystal structure.
A3- In order to do this, it is necessary to use a result of Problem 3.7, that is
VC = 6R2c 3
The problem states that c = 1.623a, and a = 2R. Therefore


= (1.623)(12 3) 1.253 x 10 -8 cm
3
VC = (1.623)(12 3 ) R 3
= 6.64 x 10 -23 cm3 = 6.64 x 10 -2 nm3
Q4- List the point coordinates for all of the atoms that are associated with the FCC unit cell.
A4- From Figure 3.1b, the atom located of the origin of the unit cell has the coordinates 000.
Coordinates for other atoms in the bottom face are 100, 110, 010, and
for all these points is zero.)
11
2 2
0 . (The z coordinate
For the top unit cell face, the coordinates are 001, 101, 111, 011, and
11
2 2
1 . (These
coordinates are the same as bottom-face coordinates except that the “0” z coordinate has been
replaced by a “1”.)
Coordinates for only those atoms that are positioned at the centers of both side faces,
and centers of both front and back faces need to be specified. For the front and back-center
face atoms, the coordinates are 1
11
22
and 0
1 1
22
center-face atoms, the respective coordinates are
, respectively. While for the left and right side
1
2
0
1
2
and
1 1
1 .
2 2
Q&A5- Some crystallographic directions in the cubic cell.
Q&A6- Some crystallographic planes in the cubic unit cells are shown below:
for -iron.
111
A7- -iron has a BCC crystal structure and an atomic radius of 0.1241 nm. Using Equation (3.3) the
lattice parameter, a, may be computed as
4R (4)(0.1241 nm)
a=
=
= 0.2866 nm
3
3
Now, the d
interplanar spacing may be determined using Equation (3.3W) as
111
a
0.2866 nm
d111 =
=
= 0.1655 nm
2
2
2
3
(1) + (1) + (1)
Q7- Calculate d
CHAPTER 4
Q1- Cite which of the elements listed form with Cu the three possible solid solution types.
A1- For complete substitutional solubility the following criteria must be met: 1) the difference in atomic
radii between Cu and the other element (R%) must be less than ±15%, 2) the crystal structures
must be the same, 3) the electronegativities must be similar, and 4) the valences should be the
same, or nearly the same. Below are tabulated, for the various elements, these criteria.
Element
R%
Cu
C
H
O
Ag
Al
Co
Cr
Fe
Ni
Pd
Pt
Zn
-44
-64
-53
+13
+12
-2
-2
-3
-3
+8
+9
+4
Crystal
Structure
Electronegativity
FCC
FCC
FCC
HCP
BCC
BCC
FCC
FCC
FCC
HCP
Valence
2+
0
-0.4
-0.1
-0.3
-0.1
-0.1
+0.3
+0.3
-0.3
1+
3+
2+
3+
2+
2+
2+
2+
2+
(a) Ni, Pd, and Pt meet all of the criteria and thus form substitutional solid solutions having
complete solubility.
(b) Ag, Al, Co, Cr, Fe, and Zn form substitutional solid solutions of incomplete solubility. All
these metals have either BCC or HCP crystal structures, and/or the difference between their
atomic radii and that for Cu are greater than ±15%, and/or have a valence different than 2+.
(c) C, H, and O form interstitial solid solutions. These elements have atomic radii that are
significantly smaller than the atomic radius of Cu.
Q2- Determine the weight percent of Ge that must be added to Si such that the resultant alloy will
21
contain 2.43 x10 Ge atoms per cubic centimeter.
A2- To solve this problem, employment of Equation (4.18) is necessary, using the following values:
21
3
N =N
= 2.43 x 10 atoms/cm
1
Ge
3
 =
= 5.32 g/cm
1
Ge
3
 =  = 2.33 g/cm
2
Si
A =A
= 72.59 g/mol
1
Ge
A = A = 28.09 g/mol
2
Si
Thus
100
C Ge =
N A Si
Si
1

N GeA Ge
 Ge
100
=
6.023 x10 atoms/ mol(2.33 g / cm3 )  2.33
2.43 x10 21 atoms / cm3 (72.59 g / mol) 5.32
23
1
g/ cm 3 


g/ cm 3 
= 11.7 wt%
Q3- Compare the energies of a) a surface to a grain boundary and b) a small angle boundary to high
angle boundary.
A3- (a) The surface energy will be greater than the grain boundary energy since some atoms on one
side of the boundary will bond to atoms on the other side--i.e., there will be fewer unsatisfied
bonds along a grain boundary.
(b) The small angle grain boundary energy is lower than for a high angle one because more
atoms bond across the boundary for the small angle, and, thus, there are fewer unsatisfied
bonds.
CHAPTER 5
Q1- Compare self-diffusion and interdiffusion.
A1- Self-diffusion is atomic migration in pure metals--i.e., when all atoms exchanging positions are of
the same type. Interdiffusion is diffusion of atoms of one metal into another metal.
Q2- How do you monitor self diffusion?
A2- Self-diffusion may be monitored by using radioactive isotopes of the metal being studied. The
motion of these isotopic atoms may be monitored by measurement of radioactivity level.
Q3- A sheet of 1.5 mm thick steel has nitrogen atmosphere on both sides at 1200 oC and is permitted
to achieve a steady-state diffusion condition. D = 6x10-11 m2/s and diffusion flux is found to be
1.2x10-7 kg/m2s. Also, it is known that the concentration of nitrogen in the steel at the high
pressure surface is 4 kg/m 3. How far into the sheet from this high pressure side will the
concentration be 2.0 kg/m 3? Assume a linear concentration profile.
3
A3- We are asked to determine the position at which the nitrogen concentration is 2 kg/m . This
problem is solved by using Equation (5.3) in the form
C  C
B
J =  D A
xA  xB
3
to be the point at which the concentration of nitrogen is 4 kg/m , then it becomes
A
necessary to solve for x , as
B
C  CB 
xB = xA + D  A

J


Assume x is zero at the surface, in which case
A
 4 kg / m3  2 kg / m3 
-11 2

xB = 0 + 6 x 10
m /s 
 1.2 x 10 7 kg / m2 - s 




-3
= 1 x 10 m = 1 mm
If we take C



CHAPTER 6
Q1- Calculate the elastic strain that results for an aluminum specimen stressed in tension. The cross2
2
-4 2
sectional area is (10 mm) x (12.7 mm) = 127 mm (= 1.27 x 10 m = 0.20 in. ); also, the elastic
9
2
modulus for Al is 69 GPa (or 69 x 10 N/m ).
A1- By using Hooke’s law:

F
35,500 N
 = =
=
= 4.1 x 10-3
4
E A oE
1.27 x 10 m 2 69 x 10 9 N/ m 2



CHAPTER 7
Q1- consider a single crystal of silver oriented such that a tensile stress is applied along a [001]
direction. If slip occurs on a (111) plane and in a [101 ] direction, and is initiated at an applied
tensile stress of 1.1 MPa, compute the critical resolved shear stress.
A1- This problem asks that we compute the critical resolved shear stress for silver. In order to do this,
we must employ Equation (7.3), but first it is necessary to solve for the angles  and  from the
sketch below.
If the unit cell edge length is a, then
 = tan
-1 a 
  = 45
a 
For the angle , we must examine the triangle OAB. The length of line OA is just a, whereas,
the length of AB is a 2 . Thus,
a 2 
 = tan -1 
 = 54.7 
 a 
And, finally
 crs s =  y (cos  cos )
= (1.1 MPa)cos(54.7) cos(45) = 0.45 MPa