# Solutions PS1 ```
MECH 6501
Assignment 1 (Solutions)
1. Calculate the minimum radius ratio for an AB ceramic compound with a
coordination number of four assuming a hard sphere model for both the cations
and anions.
2. Give the expected coordination number of both the anions and cations, taking into
account their charge balance as well possible geometric arrangement, for the
following ceramics: a) ZrO2 b) MgO and c) BeO. Explain any differences
between the calculated and actual observed CNs.
3. For the above compounds, calculate the percent ionic/covalent nature of these
ceramics using Pauling's scale and briefly outline the implication of these values
on the general properties of these ceramics.
4. Li2O forms with the antifluorite structure. Sketch this structure labelling the ions.
Sketch the following crystallographic planes of this structure indicating those ions
that appear on the particular plane: a) (100), b) (110) and (004). Provide the
crystallographic coordinates of the ions where appropriate.
5. Calculate the crystallographic density of CaO, where the atomic masses of Ca and
O are 40 and 16, respectively (Avogadro's number is 0.602 x 1024 atoms mol-1).
What assumptions are made? The measured density of CaO is 3.25 g cm-3.
Explain the difference between your calculated value and this value.
Solution to Assignment #1
1.(a)
B
E
AB= 2R = 2 d
d = 2R/ 2 = 2 R
DE = 3 d
X
D
C
XE = r + R =
3 d/2
3
2R
2
r/R =  3 2  / 2 - 1 = 0.225

 
r+R=
A
d
(b)
B
A
2R
A
+
2r
B
D
C
C
D
2R
2R + 2r = 2R 2
2 R  2r
 2
2R
2 R 2r

 2
2R 2R
minimum
r
 2 -1 = 0.414
R
2. (a) ZrO2
r
Zr 4 
0.72
 0.514
1.40
=
R
O 2
CN = 6
CN (Zr4+) = 8
CN (O2-) = 4
Actual
Vc
Va

(CN )c (CN )a
4
2

(8)c ( 4)a
i.e. Fluorite structure
(b) MgO
r
Mg 2 
R
=
O 2
0.72
 0.51
1.40
CN = 6
Actual CN = 6
(c) BeO
rBe 2 
RO 2 
=
0.35
 0.25
1.40
Actual CN = 4
Zinc blende or Wurtzite structure
CN = 4
3. Using Pauling’s Electronegativity sale:
X  X A  X B
ZrO2
X  1.6  3.4 = 1.8
55% ionic/45% covalent
MgO
X  1.2  3.4 = 2.2
70% Ionic/30% covalent
BeO
X  1.4  3.4 = 2.0
60% ionic/40% covalent
4. Li2O
Structure is same as CaF2 except Ca2+ sites are occupied by O, and F- sites are
occupied by Li+
a) (100)
b) (110)
c) (004)
&frac14;, &frac14;, &frac14;
&frac14;, &frac14;, &frac14;
&frac12;, &frac12;, 0
O2-
Li+
O2-
5. Crystallographic density CaO
CaO
NaCl structure
4Ca2+ + 4O2- / cell
RO = 1.40 &Aring;
rCa = 1.00 &Aring;
Ca2+
a = 2 + 2.8 = 4.8 &Aring;
O2Vcell = a3 = (4.8 x 10-8) cm3
=
a = 2R + 2r
4 x40  4 x16 / 0.602 x10 24  3.353 g.cm-3
4.8x10 
8 3
 CaOXRD = 3.25 g.cm-3
-porosity
-impurities
```