Material Science I
Exercise 1
24.11.2008
Structure of Ceramics
Question 1:
Show that the minimum cation to anion radius ratio (rc/ra) for a coordination number of 8 is 0.732.
Coordination Number = 8
a = 2ra
Room diagonal = 3a = 2ra + 2rc
⇒ 3 ⋅ 2ra = 2(ra + rc ) ⇒
rc
= 3 − 1 = 0.732
ra
Question 2:
On the basis of ionic charge and ionic radii, predict the crystal structures for MgO and KCl. Justify your selections.
For MgO:
rMg 2+
rO 2−
=
86 pm
= 0.683
126 pm
The cordination number for the cation is 6. MgO with ions Mg+2 and O-2 will have a AX type stochiometry exhibiting the
crystal structure of sodium chloride.
For KCl:
rK +
rCl −
=
152 pm
= 0.910
167 pm
The cordination number for the cation is 8. KCl with ions K+ and Cl- will have a AX type stochiometry exhibiting the
crystal structure of cesium chloride.
Question 3:
Calculate the theoretical density of MgO and SiC using following atomic radii:
The density of a solid can be calculated from its crystal structure:
ρ=
weight of ions per unit cell
na:
nc :
MA,c:
Vuc:
NA:
Number of anions/unit cell
Number of cations/ unit cell
Atomic weight of anion/cation
Volume of unit cell
Avogadro number
unit cell volume
=
n a ⋅ M a + nc ⋅ M c
N A ⋅ VUC
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Material Science I
Exercise 1
24.11.2008
Structure of Ceramics
MgO: NaCl type
The ions touch along the edge with a lattice constant of 2rMg2t + 2rO2- = 2(86 pm + 126 pm) = 424 pm (values for
octahedral coordination of the ions).
Atomic weight of Mg = 24.31 g/mol
Atomic weight of O = 16 g/mol
The unit cell contains 4 Mg2+ ions and 4 O2- ions:
⇒ρ=
4(16 g / mol + 24.31g / mol )
= 3.51g / cm3
(6.022 ⋅10 23 mol −1 )(424 ⋅10 −10 cm) 3
SiC: ZnS („Zinkblende“) type
The ions touch on the diagonal of the cube with a length of 3a . Therefore the Si-C-distance is 3a / 4 .
Using the radii 118pm for Si and 71pm for C:
3a
= 118 pm + 71 pm ⇒ a = 436 pm
4
Atomic weight of Si = 28.09 g/mol
Atomic weight of C = 12 g/mol
The unit cell contains 4 Si atoms and 4 C atoms:
⇒ρ=
4(12 g / mol + 28.09 g / mol )
= 3.214 g / cm3
23
3
−1
−10
(6.022 ⋅ 10 mol )(436 ⋅ 10 cm)
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Material Science I
Exercise 1
24.11.2008
-3-
Structure of Ceramics