1. Chapter 3: Structures of Metals & Ceramics

2. Structures
The properties of some materials are directly related to their crystal structures.
Significant property differences exist between crystalline and noncrystalline materials having the same composition.

3. Energy and Packing • Non dense, random packing
typical neighbor
bond length
bond energy
• Dense, ordered packing
Energy r
typical neighbor
bond length
bond energy
Dense, ordered packed structures tend to have lower energies.

4. Materials and Packing Si Oxygen Crystalline materials...
• atoms pack in periodic, 3D arrays
• typical of:
-metals
-many ceramics
-some polymers
crystalline SiO2 Si
Oxygen
Noncrystalline materials...
• atoms have no periodic packing
• occurs for:
-complex structures
-rapid cooling
"Amorphous" = Noncrystalline
noncrystalline SiO2

5. Metallic Crystal Structures
How can we stack metal atoms to minimize empty space?
2-dimensions
vs.

6. Metallic Crystal Structures
• Tend to be densely packed.
• Reasons for dense packing:
Typically, only one element is present, so all atomic radii are the same.
- Metallic bonding is not directional.
- Nearest neighbor distances tend to be small in
order to lower bond energy.
- The “electron cloud” shields cores from each other
• They have the simplest crystal structures.

7. Simple Cubic Structure (SC)
• Rare due to low packing density (only Po has this structure)
• Close-packed directions are cube edges.
• Coordination # = 6
(# nearest neighbors)

8. Atomic Packing Factor (APF)
Volume of atoms in unit cell*
APF =
Volume of unit cell
*assume hard spheres
• APF for a simple cubic structure = 0.52
close-packed directions a
R=0.5a
contains 8 x 1/8 = 1
atom/unit cell
atom
volume
atoms
unit cell 4 3 p
(0.5a) 1
APF = 3 a
unit cell
volume

9. Body Centered Cubic Structure (BCC)
• Atoms touch each other along cube diagonals.
All atoms are identical.
ex: Cr, W, Fe (), Tantalum, Molybdenum
• Coordination # = 8
2 atoms/unit cell: 1 center + 8 corners x 1/8

10. Atomic Packing Factor: BCC
• APF for a body-centered cubic structure = 0.68 a R a 3 a a 2
length = 4R =
Close-packed directions:
3 a
APF = 4 3 p (
a/4 ) 2
atoms
unit cell
atom
volume a

11. Face Centered Cubic Structure (FCC)
• Atoms touch each other along face diagonals.
--Note: All atoms are identical; the face-centered atoms are shaded
differently only for ease of viewing.
ex: Al, Cu, Au, Pb, Ni, Pt, Ag
• Coordination # = 12
4 atoms/unit cell: 6 face x 1/2 + 8 corners x 1/8

12. c03prob

13. Atomic Packing Factor: FCC
• APF for a face-centered cubic structure = 0.74 a
2 a
maximum achievable APF
Close-packed directions:
length = 4R =
2 a
Unit cell contains:
6 x 1/2 + 8 x 1/8 =
4 atoms/unit cell
APF = 4 3 p ( 2
a/4 )
atoms
unit cell
atom
volume a

14. Hexagonal Close-Packed Structure (HCP – another view)

15. Hexagonal Close-Packed Structure (HCP)
• ABAB... Stacking Sequence
• 3D Projection
• 2D Projection c a
A sites B
sites
Bottom layer
Middle layer
Top
layer
• Coordination # = 12
6 atoms/unit cell
• APF = 0.74
ex: Cd, Mg, Ti, Zn
• c/a = 1.633

16. ABAB... Stacking Sequence
c03f30

17. X-Ray Diffraction

18. X-Ray Diffractometer Bragg´s Equation d = λ/2 sinθ Detector Xray-Tube
Sample
Xray-Tube
Detector
Metal Target
(Cu or Co)
sample
d – distance between the
same atomic planes
λ – monochromatic
wavelength
θ – angle of diffracto-
meter
Bragg´s Equation
d = λ/2 sinθ

19. c03tf01

20. Theoretical Density, r Density =  = n A  = VC NA
Cell
Unit of
Volume
Total in
Atoms
Mass
Density =  =
VC NA
n A
 =
where n = number of atoms/unit cell
A = atomic weight
VC = Volume of unit cell = a3 for cubic
NA = Avogadro’s number
= x 1023 atoms/mol

21. Theoretical Density, r  = a R Ex: Cr (BCC)
A (atomic weight) = g/mol
n = 2 atoms/unit cell
R = nm
a = 4R/ 3 = nm
 = a 3
52.00 2
atoms
unit cell
mol g
volume
6.022 x 1023
theoretical
= 7.18 g/cm3
ractual
= 7.19 g/cm3

22. Factors that Determine Crystal Structure
1. Relative sizes of ions – Formation of stable structures:
--maximize the # of oppositely charged ion neighbors. - +
unstable - + - +
stable
stable
2. Maintenance of
Charge Neutrality :
--Net charge in ceramic
should be zero.
--Reflected in chemical
formula:
CaF 2 : Ca 2+
cation F -
anions + A m X p
m, p values to achieve charge neutrality

23. Atomic Bonding in Ceramics
-- 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

24. Coordination # and Ionic Radii
cation
anion
• Coordination # increases with
To form a stable structure, how many anions can
surround a cation?
ZnS
(zinc blende)
NaCl
(sodium
chloride)
CsCl
(cesium r
cation
anion
Coord #
< 0.155 2
linear 3
triangular 4
tetrahedral 6
octahedral 8
cubic

25. c03tf03

26. Computation of Minimum Cation-Anion Radius Ratio
Determine minimum rcation/ranion for an octahedral site (C.N. = 6) a
Measure the radii (blue and yellow spheres)
a = 2ranion
Substitute for “a” in the above equation

27. Example Problem: Predicting the Crystal Structure of FeO
• On the basis of ionic radii, what crystal structure would you predict for FeO?
Cation
Anion Al 3+ Fe 2 + Ca 2+ O 2- Cl - F
Ionic radius (nm)
0.053
0.077
0.069
0.100
0.140
0.181
0.133
• Answer:
based on this ratio,
-- coord # = 6 because
0.414 < < 0.732
-- crystal structure is similar to NaCl

28. Rock Salt Structure
Same concepts can be applied to ionic solids in general.
Example: NaCl (rock salt) structure
rNa = nm
rCl = nm
rNa/rCl = 0.564
cations (Na+) prefer octahedral sites

29. MgO and FeO MgO and FeO also have the NaCl structure
O2- rO = nm
Mg2+ rMg = nm
rMg/rO = 0.514
cations prefer octahedral sites
So each Mg2+ (or Fe2+) has 6 neighbor oxygen atoms

30. AX Crystal Structures
AX–Type Crystal Structures include NaCl, CsCl, and zinc blende
Cesium Chloride structure:
 Since < < 1.0, cubic sites preferred
So each Cs+ has 8 neighbor Cl-

31. AX2 Crystal Structures Fluorite structure Calcium Fluorite (CaF2)
Cations in cubic sites
UO2, ThO2, ZrO2, CeO2
Antifluorite structure –
positions of cations and anions reversed

32. ABX3 Crystal Structures
Perovskite structure
Ex: complex oxide
BaTiO3

33. SUMMARY • Atoms may assemble into crystalline or amorphous structures.
• Common metallic crystal structures are FCC, BCC and HCP Coordination number and atomic packing factor are the same for both FCC and HCP crystal structures.
• We can predict the density of a material, provided we know the atomic weight, atomic radius, and crystal geometry (e.g., FCC, BCC, HCP).
• Interatomic bonding in ceramics is ionic and/or covalent.
• Ceramic crystal structures are based on:
-- maintaining charge neutrality
-- cation-anion radii ratios.