Dense atomic packing for crystal structures
of metals
High degree of shielding (of ion cores) provided
Crystal structures for metals simpler than
by free electron cloud
structures for ceramics and polymers.
Centers of atoms located at the eight corners of a cube
number of nearest-neighbor
or touching atoms
a
π
a
a
a

a
a
a
a
a
π
a
a
Atoms located at 8 cube corners and at the centers of the 6 faces
a
a
a
π
a
a
a
a
A sketch of one-third of an HCP unit cell is shown below.
JH =
JM = JK = 2R = a
(JM) 2 = ( JH) 2  ( MH) 2
 a 2 c 2
a2
c2
a =   +   =
+
 3 
2 
3
4

2

c 2
2
2
a = ( JH ) +  

a
3
2 


a /2
cos 30 =
=
JH
3
2
c
=
a


8
= 1.633
3
a
4 R 3 
3

VS = 6 
=
8
R
 3 




2R 3
BC = 2R cos (30) =
2
VC = (AREA)(c) = 6 R2c 3
= (6 R2 3) (2)(1.633 )R = 12 3 (1.633) R3


2 R 3 
2
AREA = (3)(CD)(BC) = (3)(2 R) 
= 6R 3
2


and since c = 1.633a = 2R(1.633)


APF =
VS
VC
=
8 R3
12 3 (1.633 ) R3
= 0.74
Stacking Sequence
Referenced to an
FCC Unit Cell.
Close-Packed
Plane

(nA / N A )
VC


a
a
a
a

VC
ρ
Atomic Bonding in Ceramics
• Bonding:
-- Can be ionic and/or covalent in character.
-- % ionic character increases with difference in
electronegativity of atoms.
• Degree of ionic character may be large or small:
CaF2: large
SiC: small
Factors that Determine Crystal Structure
1. Relative sizes of ions – Formation of stable structures:
--maximize the # of oppositely charged ion neighbors.
-
+
-
-
+
-
unstable
2. Maintenance of
Charge Neutrality :
-
stable
--Net charge in ceramic
should be zero.
--Reflected in chemical
formula:
CaF 2 :
-
+
-
stable
Ca 2+ +
cation
A m Xp
m, p values to achieve
charge neutrality
Fanions
F-
Coordination Number and Ionic Radii
r cation
• Coordination Number increases with r
anion
To form a stable structure, how many anions can
surround around a cation?
r cation
r anion
< 0.155
Coord.
Number
linear
2
triangular
0.155 - 0.225
3
0.225 - 0.414
4 tetrahedral
0.414 - 0.732
6 octahedral
0.732 - 1.0
8
cubic
ZnS
(zinc blende)
NaCl
(sodium
chloride)
CsCl
(cesium
chloride)
Computation of Minimum Cation-Anion
Radius Ratio
• Determine minimum rcation/ranion for an octahedral site
(C.N. = 6)
2ranion + 2rcation = 2a
a = 2ranion
2ranion + 2rcation = 2 2ranion
ranion + rcation = 2ranion
rcation = ( 2 - 1)ranion
rcation
= 2 - 1 = 0.414
ranion
Ionic Radii for Several Cations and Anions
for a Coordination Number of 6
Example Problem: Predicting the Crystal
Structure of FeO
• On the basis of ionic radii, what crystal structure
would you predict for FeO?
Cation Ionic radius (nm)
Al 3+
0.053
Fe 2+
0.077
Fe 3+
0.069
Ca 2+
0.100
Anion
O2Cl F-
0.140
0.181
0.133
• Answer:
rcation 0.077
=
ranion 0.140
= 0.550
based on this ratio,
-- coord # = 6 because
0.414 < 0.550 < 0.732
-- crystal structure is NaCl
based on Octahedral structure
FeO and MgO
FeO and MgO also have the NaCl structure
O2-
rO = 0.140 nm
Mg2+
rMg = 0.072 nm
rMg/rO = 0.514
 cations prefer octahedral sites
So each Mg2+ (or Fe2+) has 6 neighbor oxygen atoms
AX Crystal Structures
AX–Type Crystal Structures include NaCl, CsCl, and zinc blende
Cesium Chloride structure:
rCs +
rCl -
=
0.170
= 0.939
0.181
 Since 0.732 < 0.939 < 1.0,
cubic sites preferred
So each Cs+ has 8 neighbor Cl-
AX2 Crystal Structures
Fluorite structure
• Calcium Fluorite (CaF2)
• Cations in cubic sites
• UO2, ThO2, ZrO2, CeO2
• Antifluorite structure –
positions of cations and
anions reversed
AX2 Crystal Structures
Fluorite structure
Question: the ionic radii for K+ and O2- as 0.138 and 0.140 nm,
respectively.
(a) What is the coordination number for each O2- ion?
(b) Briefly describe the resulting crystal structure for K2O.
(c) Explain why this is called the antifluorite structure.
r +
K = 0.138 nm = 0.986
r 2- 0.140 nm
O
(a) The coordination number for oxygen is eight.
(b) For a coordination number of eight for both cations and anions,
the crystal structure should be cesium chloride. However, there are
twice as many K+ as O2- ions. Therefore, the centers of the K+ ions
are positioned at the corners of cubic unit cells, while half of the cube
centers are occupied by O2- ions.
(c) This structure is called the antifluorite crystal structure because
anions and cations are interchanged with one another from the
fluorite structure.
ABX3 Crystal Structures
• Perovskite structure
Ex: complex oxide
BaTiO3
Density Computations for Ceramics
Number of formula units/unit cell
n¢(SAC + SAA )
r=
VC N A
Avogadro’s number
Volume of unit cell
SAC = sum of atomic weights of all cations in formula unit
SAA= sum of atomic weights of all anions in formula unit
Atomic Bonding in Ceramics
ρ
ρ
ρ
ρ
’
single crystal for
turbine blade
are composed of many small,
single crystals (i.e., are polycrystalline).
Temperature
β
T
°
δ
°
γ
°
—
E(edge) ≠ E(diagonal)
Unit cell of BCC iron
μ
a, b, and c
unit cell edge lengths
a
a
a, b, and c) and
remove commas
a
Example Problem I
a
a
1. Point coordinates of tail and head
a
a
2 & 3. Subtract and normalize
4 & 5. Multiply by 2 to eliminate the fraction, then place in
square brackets (no commas)
Family of directions – all directions that are crystallographically equivalent
(have the same atomic spacing) – indicated by indices in angle brackets
z
Algorithm
a2
-
a3
a1
1. Vector repositioned (if necessary) to pass
through origin.
2. Read off projections in terms of unit
cell dimensions a1, a2, a3, or c
3. Adjust to smallest integer values
4. Enclose in square brackets, no commas
[uvtw]
a
2
-a3
a2
2
ex:
½, ½, -1, 0
=>
[ 1120 ]
a3
dashed red lines indicate
projections onto a1 and a2 axes
a1
2
a1
z
a2
-
a3
a1
a
a
a
a
aa
a
a
a
aa
a
a
·
a
aa
a
·
·
Family of planes – all planes that are crystallographically equivalent (have
the same atomic packing) – indicated by indices in braces
a
a
l
a
a
a
1a
aa
a
a
a
a
a
l
a
a
There are 2 half atoms and 1 full atom
= 2 atoms centered on vector
a
a
=
=
a
=
-
There are 4 quarter atoms
= 1 atom centered on plane
a=
=
a=
=
=
a
=
a
a
a
• Ceramic crystal structures are based on:
-- maintaining charge neutrality
-- cation-anion radii ratios.
• Crystallographic points, directions and planes are specified in
terms of indexing schemes. Crystallographic directions and
planes are related to atomic linear densities and planar densities.
For most single crystals, properties vary with
crystallographic
orientation (i.e., are anisotropic)
For polycrystalline materials having randomly oriented
grains, properties are independent of crystallographic
orientation (i.e., they are isotropic)