EMSE 201 — Introduction to Materials Science & Engineering
23 September 2009
Name:
S O L U T I O N
Department of Materials Science and Engineering
1 of 6
Case Western Reserve University
Test #1 — 75 minutes; 100 points; 6 questions; 6 pages; 10% of course grade
Partial credit
will be given for correct set-ups and reasoning. Give
units
on numerical answers where
appropriate and use the
correct number of significant figures.
Please write all answers on these pages; use
the backs if needed.
Boltzmann’s constant,
k
B
= 1.381
×
10
-23
J K
-1
= 8.620
×
10
-5
eV K
-1
Avogadro’s number,
N
A
= 6.023
×
10
23
mol
-1
gas constant,
R
: 8.314 J mol
-1
K
-1
1) The sketch at right shows one unit cell of a
compound of titanium (Ti, black circles) and
oxygen (O, white circles). The coordinates of
the atoms are as follows:
Ti: 0, 0, 0
1/2, 1/2, 1/2
O:
x
1
,
x
1
, 0
1–
x
1
, 1–
x
1
, 0
1–
x
2
,
x
2
, 1/2
x
2
, 1–
x
2
, 1/2
(0 < (
x
1
,
x
2
) < 1, with specific values that
depend on the ionic radii of Ti and O.)
a)
(2 points)
Determine the
number of Ti atoms
per unit cell.
net 1 atom from the corners + 1 atom in the interior =
2
b)
(2 points)
Determine the
number of O atoms
per unit cell.
4 atoms on faces for a net of 2 atoms + 2 atoms in the interior =
4
c)
(2 points)
The coordination number of titanium in this compound is 6,
i.e.
every titanium atom is
surrounded by 6 oxygen atoms. What is the
coordination number of oxygen?
By inspection, the interior oxygens have
3
nearest Ti neighbors. Or reasoning from
the number of each atom per unit cell,
CN
O
= (2/4)
Ti
= 3
.
The values of the lattice parameters are
a
=
b
= 0.45933 nm,
c
= 0.29592 nm,
α
=
β
=
γ
= 90°.
d)
(2 points)
To what
crystal system
does this structure belong?
tetragonal
e)
(5 points)
Calculate the atomic packing fraction
in this compound. For the radius of oxygen use
0.136 nm. The radius of titanium is given on the next page.
APF =
V
atoms
/
cell
(
1 pt
)
= (4
O
+ 2
Ti
)/(
abc
)
=
4
3
π
4
r
O
3
+
2
r
Ti
3
( )
a
2
c
=
4
3
4(0.136)
3
+
2(0.0605)
3
( )
(0.45933)
2
(0.29592)
=
70.5%
(
2 pts
)
(
1 pt
)
(
1 pt
)
continued on next page
x
y
z
c
a