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CHAPTER 12
STRUCTURES AND PROPERTIES OF CERAMICS
PROBLEM SOLUTIONS
12.1
The two characteristics of component ions that determine the crystal structure are:
1) the
magnitude of the electrical charge on each ion; and 2) the relative sizes of the cations and anions.
12.2
In this problem we are asked to show that the minimum cationtoanion radius ratio for a
coordination number of four is 0.225. If lines are drawn from the centers of the anions, then a
tetrahedron is formed.
The tetrahedron may be inscribed within a cube as shown below.
The spheres at the apexes of the tetrahedron are drawn at the corners of the cube, and designated
as positions
A
,
B
,
C
, and
D
.
(These are reduced in size for the sake of clarity.)
The cation resides
at the center of the cube, which is designated as point
E
.
Let us now express the cation and anion
radii in terms of the cube edge length, designated as
a
.
The spheres located at positions
A
and
B
touch each other along the bottom face diagonal.
Thus,
AB
= 2r
A
But
(AB
)
2
=
a
2
+
a
2
=
2a
2
1
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AB
= a 2
= 2r
A
And
a =
2r
A
2
There will also be an anion located at the corner, point
F
(not drawn), and the cube diagonal
AEF
will be related to the ionic radii as
AEF
A
+ r
C
(
)
(The line
AEF
has not been drawn to avoid confusion.)
From the triangle
ABF
(AB
)
2
+ (FB
)
2
= (AEF
)
2
But,
FB
= a
=
A
2
and
AB
A
from above.
Thus,
A
()
2
+
A
2
⎛
⎝
⎜
⎞
⎠
⎟
2
= 2 r
A
C
[]
2
Solving for the
r
C
/
r
A
ratio leads to
r
C
r
A
6
−
2
2
= 0.225
12.3
This problem asks us to show, using the rock salt crystal structure, that the minimum cationtoanion
radius ratio is 0.414 for a coordination number of six.
Below is shown one of the faces of the rock
salt crystal structure in which anions and cations just touch along the edges, and also the face
diagonals.
2
From triangle
FGH
,
GF
= 2r
A
and
FH
= GH
= r
A
+ r
C
Since
FGH
is a right triangle
(GH
)
2
+ (FH
)
2
= (FG
)
2
or
r
A
C
()
2
A
C
2
A
(
)
2
which leads to
r
A
C
=
2r
A
2
Or, solving for
r
C
/
r
A
r
C
r
A
=
2
2
−
1
⎛
⎝
⎜
⎞
⎠
⎟
=
0.414
12.4
This problem asks us to show that the minimum cationtoanion radius ratio for a coordination
number of 8 is 0.732.
From the cubic unit cell shown below
3
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r
A
, and from the base of the unit cell
x
2
= 2r
A
(
)
2
+ 2r
A
(
)
2
= 8r
A
2
Or
x = 2r
A
2
Now from the triangle that involves
x
,
y
, and the unit cell edge
x
2
A
(
)
2
= y
2
A
+2r
C
(
)
2
2r
A
2
()
2
+ 4r
A
2
A
C
(
)
2
Which reduces to
A
3
−
1
(
)
C
Or
r
C
r
A
= 3
−
1 = 0.732
12.5
This problem calls for us to predict crystal structures for several ceramic materials on the basis of
ionic charge and ionic radii.
(a)
For CsI, from Table 12.3
4
r
Cs
+
r
I
−
=
0.170 nm
0.220 nm
= 0.773
Now, from Table 12.2, the coordination number for each cation (Cs
+
) is eight, and, using Table 12.4,
the predicted crystal structure is cesium chloride.
(b)
For NiO, from Table 12.3
r
Ni
2
+
r
O
2
−
0.069 nm
0.140 nm
= 0.493
The coordination number is six (Table 12.2), and the predicted crystal structure is sodium chloride
(Table 12.4).
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This homework help was uploaded on 02/07/2008 for the course MATENG MAT201 taught by Professor Na during the Spring '08 term at Wisconsin Milwaukee.
 Spring '08
 na

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