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Chapter 3: Structures of Metals & Ceramics
Chapter 3 -
School of Mechanical EngineeringProfessor Choi, Hae-Jin
Materials Science - Prof. Choi, Hae-Jin 1
ISSUES TO ADDRESS...• How do atoms assemble into solid structures?
• How does the density of a material depend onits structure?
Chapter 3: Structures of Metals & Ceramics
Chapter 3 - 2
• How does the density of a material depend onits structure?
• When do material properties vary with thesample (i.e., part) orientation?
• How do the crystal structures of ceramic materials differ from those for metals?
Materials Science - Prof. Choi, Hae-Jin
• Non dense, random packing
• Dense, ordered packing
Energy and PackingEnergy
r
typical neighborbond length
typical neighborbond energy
Energy
Chapter 3 - 3
• Dense, ordered packing
Dense, ordered packed structures tend to havelower energies.
Energy
r
typical neighborbond length
typical neighborbond energy
Materials Science - Prof. Choi, Hae-Jin
• atoms pack in periodic, 3D arraysCrystalline materials...
-metals-many ceramics-some polymers crystalline SiO2
Adapted from Fig. 3.40(a),Callister & Rethwisch 3e.
Materials and Packing
Si Oxygen
• typical of:
Chapter 3 - 4
• atoms have no periodic packingNoncrystalline materials...
-complex structures-rapid cooling
noncrystalline SiO2"Amorphous" = NoncrystallineAdapted from Fig. 3.40(b),Callister & Rethwisch 3e.
Si Oxygen
• occurs for:
Materials Science - Prof. Choi, Hae-Jin
Metallic Crystal Structures • How can we stack metal atoms to minimize
empty space?2-dimensions
Chapter 3 - 5
vs.
Now stack these 2-D layers to make 3-D structures
Materials Science - Prof. Choi, Hae-Jin
• Tend to be densely packed.• Reasons for dense packing:
- Typically, only one element is present, so all atomicradii are the same.
- Metallic bonding is not directional.- Nearest neighbor distances tend to be small inorder to lower bond energy.
- Electron cloud shields cores from each other
Metallic Crystal Structures
Chapter 3 - 6
- Typically, only one element is present, so all atomicradii are the same.
- Metallic bonding is not directional.- Nearest neighbor distances tend to be small inorder to lower bond energy.
- Electron cloud shields cores from each other
• Have the simplest crystal structures.
We will examine three such structures...
Materials Science - Prof. Choi, Hae-Jin
• Coordination # = 8
• Atoms touch each other along cube diagonals.--Note: All atoms are identical; the center atom is shaded
differently only for ease of viewing.
Body Centered Cubic Structure (BCC)
ex: Cr, W, Fe (a), Tantalum, Molybdenum
Chapter 3 - 7
Adapted from Fig. 3.2,Callister & Rethwisch 3e.
2 atoms/unit cell: 1 center + 8 corners x 1/8
Materials Science - Prof. Choi, Hae-Jin
Atomic Packing Factor (APF)
APF = Volume of atoms in unit cell*
Volume of unit cell
Chapter 3 -
*assume hard spheres
Materials Science - Prof. Choi, Hae-Jin 8
Atomic Packing Factor: BCC
a
• APF for a body-centered cubic structure = 0.68
a2
a3
Chapter 3 - 9
APF =
43
p ( 3a/4)32atoms
unit cell atomvolume
a3unit cellvolume
length = 4R =Close-packed directions:
3 aaRAdapted from
Fig. 3.2(a), Callister & Rethwisch 3e.
2
Materials Science - Prof. Choi, Hae-Jin
• Coordination # = 12
• Atoms touch each other along face diagonals.--Note: All atoms are identical; the face-centered atoms are shaded
differently only for ease of viewing.
Face Centered Cubic Structure (FCC)
ex: Al, Cu, Au, Pb, Ni, Pt, Ag
Chapter 3 - 10
Adapted from Fig. 3.1, Callister & Rethwisch 3e.
4 atoms/unit cell: 6 face x 1/2 + 8 corners x 1/8
Materials Science - Prof. Choi, Hae-Jin
• APF for a face-centered cubic structure = 0.74Atomic Packing Factor: FCC
maximum achievable APF
Close-packed directions: length = 4R = 2 a
Unit cell contains:6 x 1/2 + 8 x 1/8
2 a
Chapter 3 - 11
APF =
43
p ( 2a/4)34atoms
unit cell atomvolume
a3unit cellvolume
6 x 1/2 + 8 x 1/8 = 4 atoms/unit cella
Adapted fromFig. 3.1(a),Callister & Rethwisch 3e.
Materials Science - Prof. Choi, Hae-Jin
• ABAB... Stacking Sequence• 3D Projection • 2D Projection
Hexagonal Close-Packed Structure (HCP)
c
A sites
B sites Middle layer
Top layer
Chapter 3 - 12
• Coordination # = 12
• APF = 0.74
Adapted from Fig. 3.3(a),Callister & Rethwisch 3e.
6 atoms/unit cell
ex: Cd, Mg, Ti, Zn
• c/a = 1.633
c
a
B sites
A sites Bottom layer
Middle layer
Materials Science - Prof. Choi, Hae-Jin
Hexagonal Close-Packed Structure (HCP)
Chapter 3 - 13Materials Science - Prof. Choi, Hae-Jin
Theoretical Density, r
Density = r =
VCNA
n Ar =
CellUnitofVolumeTotalCellUnitinAtomsofMass
Chapter 3 - 14
where n = number of atoms/unit cellA = atomic weight VC = Volume of unit cell = a3 for cubicNA = Avogadro’s number
= 6.022 x 1023 atoms/mol
Materials Science - Prof. Choi, Hae-Jin
• Ex: Cr (BCC) A = 52.00 g/molR = 0.125 nmn = 2 atoms/unit cell
R
Theoretical Density, r
Chapter 3 - 15
rtheoretical
a = 4R/ 3 = 0.2887 nm
ractual
aR
r = a3
52.002atoms
unit cell molg
unit cellvolume atoms
mol
6.022 x 1023
= 7.18 g/cm3
= 7.19 g/cm3
Adapted from Fig. 3.2(a), Callister & Rethwisch 3e.
Materials Science - Prof. Choi, Hae-Jin
• 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:
Atomic Bonding in Ceramics
SiC: smallCaF2: large
Chapter 3 - 16
Adapted from Fig. 2.7, Callister & Rethwisch 3e. (Fig. 2.7 is adapted from Linus Pauling, The Nature of the Chemical Bond, 3rd edition, Copyright 1939 and 1940, 3rd edition. Copyright 1960 by Cornell University.
SiC: small
Materials Science - Prof. Choi, Hae-Jin
Ceramic Crystal Structures
Oxide structures– oxygen anions larger than metal cations– close packed oxygen in a lattice (usually FCC)– cations fit into interstitial sites among oxygen ions
Chapter 3 - 17Materials Science - Prof. Choi, Hae-Jin
Factors that Determine Crystal Structure1. Relative sizes of ions – Formation of stable structures:
--maximize the # of oppositely charged ion neighbors.
Adapted from Fig. 3.4, Callister & Rethwisch 3e.
- -
- -+
unstable
- -
- -+
stable
- -
- -+
stable
Chapter 3 - 18
unstable stable stable2. Maintenance of
Charge Neutrality :--Net charge in ceramic
should be zero.--Reflected in chemical
formula:
CaF2: Ca2+cation
F-
F-
anions+
AmXpm, p values to achieve charge neutrality
Materials Science - Prof. Choi, Hae-Jin
• Coordination # increases with
Coordination # and Ionic Radii
2
rcationranion
Coord #
< 0.155 linearAdapted from Fig. 3.7, Callister & Rethwisch 3e.
ZnS (zinc sulfide)
rcationranion
To form a stable structure, how many anions cansurround around a cation?
Chapter 3 - 19
Adapted from Table 3.3, Callister & Rethwisch 3e.
2 < 0.155
0.155 - 0.225
0.225 - 0.414
0.414 - 0.732
0.732 - 1.0
3
4
6
8
linear
triangular
tetrahedral
octahedral
cubic
Adapted from Fig. 3.5, Callister & Rethwisch 3e.
Adapted from Fig. 3.6, Callister & Rethwisch 3e.
Adapted from Fig. 3.7, Callister & Rethwisch 3e.
NaCl(sodium chloride)
CsCl(cesium chloride)
Materials Science - Prof. Choi, Hae-Jin
Computation of Minimum Cation-Anion Radius Ratio
• Determine minimum rcation/ranion for an octahedral site (C.N. = 6)
a = 2ranion
arr 222 cationanion =+
Chapter 3 - 20
a = 2ranion
�
2ranion + 2rcation = 2 2ranion
�
ranion + rcation = 2ranion
�
rcation = ( 2 -1)ranion
414.012anion
cation =-=rr
Materials Science - Prof. Choi, Hae-Jin
• On the basis of ionic radii, what crystal structurewould you predict for FeO?
• Answer:
550014000770
anion
cation
...
rr
=
=
Example: Predicting the Crystal Structure of FeO
Ionic radius (nm)0.0530.0770.069
CationAl3+
Fe2+
Fe3+
Chapter 3 - 21
550014000770
anion
cation
...
rr
=
=
based on this ratio,-- coord # = 6 because
0.414 < 0.550 < 0.732
-- crystal structure is NaClData from Table 3.4, Callister & Rethwisch 3e.
0.0690.100
0.1400.1810.133
Anion
Fe3+
Ca2+
O2-
Cl-
F-Materials Science - Prof. Choi, Hae-Jin
Rock Salt StructureSame concepts can be applied to ionic solids in general. Example: NaCl (rock salt) structure
rNa = 0.102 nm
rCl = 0.181 nm
Chapter 3 - 22
rNa/rCl = 0.564
\ cations (Na+) prefer octahedral sites
Adapted from Fig. 3.5, Callister & Rethwisch 3e.
Materials Science - Prof. Choi, Hae-Jin
MgO and FeO
O2- rO = 0.140 nm
Mg2+ rMg = 0.072 nm
rMg/rO = 0.514
\ cations prefer octahedral sites
MgO and FeO also have the NaCl structure
Chapter 3 - 23
rMg/rO = 0.514
\ cations prefer octahedral sites
So each Mg2+ (or Fe2+) has 6 neighbor oxygen atoms
Adapted from Fig. 3.5, Callister & Rethwisch 3e.
Materials Science - Prof. Choi, Hae-Jin
AX Crystal Structures
939.0181.0170.0
Cl
Cs ==-
+
r
r
Cesium Chloride structure:
AX–Type Crystal Structures include NaCl, CsCl, and zinc blende
Chapter 3 - 24
939.0181.0170.0
Cl
Cs ==-
+
r
r
Adapted from Fig. 3.6, Callister & Rethwisch 3e.
\ Since 0.732 < 0.939 < 1.0, cubic sites preferred
So each Cs+ has 8 neighbor Cl-
Materials Science - Prof. Choi, Hae-Jin
AX2 Crystal Structures
• Calcium Fluorite (CaF2)• Cations in cubic sites
• UO2, ThO2, ZrO2, CeO2
• Antifluorite structure –positions of cations and anions reversed
Fluorite structure
Chapter 3 - 25
• Calcium Fluorite (CaF2)• Cations in cubic sites
• UO2, ThO2, ZrO2, CeO2
• Antifluorite structure –positions of cations and anions reversed
Adapted from Fig. 3.8, Callister & Rethwisch 3e.
Materials Science - Prof. Choi, Hae-Jin
ABX3 Crystal Structures• Perovskite structure
Ex: complex oxide BaTiO3
Chapter 3 - 26
Adapted from Fig. 3.9, Callister & Rethwisch 3e.
Materials Science - Prof. Choi, Hae-Jin
Density Computations for Ceramics
A
AC )(NV
AAn
C
S+S¢=r
Number of formula units/unit cell
Avogadro’s number
Chapter 3 - 27
Volume of unit cell
= sum of atomic weights of all anions in formula unit
�
SAA
�
SAC = sum of atomic weights of all cations in formula unit
Materials Science - Prof. Choi, Hae-Jin
Densities of Material Classesrmetals > rceramics > rpolymers
Why?
(g/cm )3
Graphite/ Ceramics/ Semicond
Metals/ Alloys
Composites/ fibersPolymers
20
30Based on data in Table B1, Callister
*GFRE, CFRE, & AFRE are Glass,Carbon, & Aramid Fiber-ReinforcedEpoxy composites (values based on60% volume fraction of aligned fibers
in an epoxy matrix).10
5
Steels
Titanium
Cu,NiTin, Zinc
Silver, Mo
TantalumGold, WPlatinum
Zirconia
Metals have...• close-packing
(metallic bonding)• often large atomic masses
Ceramics have...• less dense packing• often lighter elements
In general
Chapter 3 - 28
Data from Table B.1, Callister & Rethwisch, 3e.
r(g/cm )3
1
2
345
0.30.40.5
Magnesium
Aluminum
Titanium
GraphiteSiliconGlass -sodaConcrete
Si nitrideDiamondAl oxide
HDPE, PSPP, LDPEPC
PTFE
PETPVCSilicone
Wood
AFRE*CFRE*GFRE*Glass fibers
Carbon fibersAramid fibers
Ceramics have...• less dense packing• often lighter elements
Polymers have...• low packing density
(often amorphous)• lighter elements (C,H,O)
Composites have...• intermediate values
Materials Science - Prof. Choi, Hae-Jin
Silicate CeramicsMost common elements on earth are Si & O
• SiO2 (silica) polymorphic forms are quartz, crystobalite, & tridymite
• The strong Si-O bonds lead to a high melting temperature (1710ºC) for this material
Si4+
O2-
Chapter 3 - 29
Most common elements on earth are Si & O
• SiO2 (silica) polymorphic forms are quartz, crystobalite, & tridymite
• The strong Si-O bonds lead to a high melting temperature (1710ºC) for this material
Adapted from Figs. 3.10-11, Callister & Rethwisch 3e crystobalite
Materials Science - Prof. Choi, Hae-Jin
• Basic Unit: Glass is noncrystalline (amorphous)• Fused silica is SiO2 to which no
impurities have been added• Other common glasses contain
impurity ions such as Na+, Ca2+, Al3+, and B3+
Glass Structure
Si04 tetrahedron4-
Si4+
O2-
Chapter 3 - 30
• Quartz is crystallineSiO2:
(soda glass)Adapted from Fig. 3.41, Callister & Rethwisch 3e.
Si4+Na+
O2-
Materials Science - Prof. Choi, Hae-Jin
Polymorphic Forms of CarbonDiamond– tetrahedral bonding of
carbon• hardest material known• very high thermal
conductivity– large single crystals –
gem stones– small crystals – used to
grind/cut other materials – diamond thin films
• hard surface coatings –used for cutting tools, medical devices, etc.
Chapter 3 - 31
Diamond– tetrahedral bonding of
carbon• hardest material known• very high thermal
conductivity– large single crystals –
gem stones– small crystals – used to
grind/cut other materials – diamond thin films
• hard surface coatings –used for cutting tools, medical devices, etc.
Adapted from Fig. 3.16, Callister & Rethwisch 3e.
Materials Science - Prof. Choi, Hae-Jin
Polymorphic Forms of Carbon (cont)Graphite– layered structure – parallel hexagonal arrays of
carbon atoms
– weak van der Waal’s forces between layers– planes slide easily over one another -- good
lubricant
Chapter 3 - 32
Graphite– layered structure – parallel hexagonal arrays of
carbon atoms
– weak van der Waal’s forces between layers– planes slide easily over one another -- good
lubricant
Adapted from Fig. 3.17, Callister & Rethwisch 3e.
Materials Science - Prof. Choi, Hae-Jin
Polymorphic Forms of Carbon (cont)Fullerenes and Nanotubes
• Fullerenes – spherical cluster of 60 carbon atoms, C60
– Like a soccer ball • Carbon nanotubes – sheet of graphite rolled into a tube
– Ends capped with fullerene hemispheres
Chapter 3 - 33
Adapted from Figs. 3.18 & 3.19, Callister & Rethwisch 3e.
Materials Science - Prof. Choi, Hae-Jin
Crystal Systems
7 crystal systems
14 crystal lattices
Unit cell: smallest repetitive volume which contains the complete lattice pattern of a crystal.
Chapter 3 - 34
Fig. 3.20, Callister & Rethwisch 3e.
7 crystal systems
14 crystal lattices
a, b, and c are the lattice constants
Materials Science - Prof. Choi, Hae-Jin
Point CoordinatesPoint coordinates for unit cell
center area/2, b/2, c/2 ½ ½ ½
Point coordinates for unit cell corner are 111
Translation: integer multiple of lattice constants à identical position in another unit cell
z
x
ya b
c
000
111
Chapter 3 - 35
Point coordinates for unit cell center area/2, b/2, c/2 ½ ½ ½
Point coordinates for unit cell corner are 111
Translation: integer multiple of lattice constants à identical position in another unit cell
x
y
z
·
2c
·
·
·
b
b
Materials Science - Prof. Choi, Hae-Jin
Crystallographic Directions
1. Vector repositioned (if necessary) to pass through origin.
2. Read off projections in terms of unit cell dimensions a, b, and c
3. Adjust to smallest integer values4. Enclose in square brackets, no commas
[uvw]
z Algorithm
y
Chapter 3 - 36
1. Vector repositioned (if necessary) to pass through origin.
2. Read off projections in terms of unit cell dimensions a, b, and c
3. Adjust to smallest integer values4. Enclose in square brackets, no commas
[uvw]
ex: 1, 0, ½ => 2, 0, 1 => [ 201 ]
-1, 1, 1
families of directions <uvw>
x
where overbar represents a negative index
[111 ]=>
Materials Science - Prof. Choi, Hae-Jin
ex: linear density of Al in [110] direction
a = 0.405 nm
Linear Density• Linear Density of Atoms º LD =
[110]
Unit length of direction vectorNumber of atoms
Chapter 3 - 37
ex: linear density of Al in [110] direction
a = 0.405 nm
a# atoms
length
13.5 nma2
2LD -==
Materials Science - Prof. Choi, Hae-Jin
HCP Crystallographic Directions
1. Vector repositioned (if necessary) to pass through origin.
2. Read off projections in terms of unitcell dimensions a1, a2, a3, or c
3. Adjust to smallest integer values4. Enclose in square brackets, no commas
[uvtw]-
a2
zAlgorithm
Chapter 3 - 38
1. Vector repositioned (if necessary) to pass through origin.
2. Read off projections in terms of unitcell dimensions a1, a2, a3, or c
3. Adjust to smallest integer values4. Enclose in square brackets, no commas
[uvtw]
[ 1120 ]ex: ½, ½, -1, 0 =>
Adapted from Fig. 3.24(a), Callister & Rethwisch 3e.
dashed red lines indicate projections onto a1 and a2 axes a1
a2
a3
-a32
a2
2a1
-a3
a1
Materials Science - Prof. Choi, Hae-Jin
HCP Crystallographic Directions• Hexagonal Crystals
– 4 parameter Miller-Bravais lattice coordinates are related to the direction indices (i.e., u'v'w') as follows.
1]uvtw[]'w'v'u[ ®
z
Chapter 3 - 39
=
=
=
'wwt
v
u
)vu( +-
)'u'v2(31 -
)'v'u2(31 -=
Fig. 3.24(a), Callister & Rethwisch 3e.
-a3
a1
a2
Materials Science - Prof. Choi, Hae-Jin
Crystallographic Planes
Chapter 3 - 40
Adapted from Fig. 3.25, Callister & Rethwisch 3e.
Materials Science - Prof. Choi, Hae-Jin
Crystallographic Planes• Miller Indices: Reciprocals of the (three) axial
intercepts for a plane, cleared of fractions & common multiples. All parallel planes have same Miller indices.
• Algorithm1. Read off intercepts of plane with axes in
terms of a, b, c2. Take reciprocals of intercepts3. Reduce to smallest integer values4. Enclose in parentheses, no
commas i.e., (hkl)
Chapter 3 - 41
• Miller Indices: Reciprocals of the (three) axial intercepts for a plane, cleared of fractions & common multiples. All parallel planes have same Miller indices.
• Algorithm1. Read off intercepts of plane with axes in
terms of a, b, c2. Take reciprocals of intercepts3. Reduce to smallest integer values4. Enclose in parentheses, no
commas i.e., (hkl)
Materials Science - Prof. Choi, Hae-Jin
Crystallographic Planesz
x
ya b
c
4. Miller Indices (110)
z
1. Intercepts 1 1 ¥2. Reciprocals 1/1 1/1 1/¥
1 1 03. Reduction 1 1 0
example a b c
Chapter 3 - 42
xexample a b c
z
x
ya b
c
4. Miller Indices (100)
1. Intercepts 1/2 ¥ ¥2. Reciprocals 1/½ 1/¥ 1/¥
2 0 03. Reduction 2 0 0
Materials Science - Prof. Choi, Hae-Jin
Crystallographic Planesz
ya b
c·
··
example1. Intercepts 1/2 1 3/4
a b c
2. Reciprocals 1/½ 1/1 1/¾2 1 4/3
3. Reduction 6 3 4
Chapter 3 - 43
xa b
4. Miller Indices (634)
3. Reduction 6 3 4
(001)(010),
Family of Planes {hkl}
(100), (010),(001),Ex: {100} = (100),
Materials Science - Prof. Choi, Hae-Jin
Crystallographic Planes (HCP)• In hexagonal unit cells the same idea is used
example a1 a2 a3 c1. Intercepts 1 ¥ -1 12. Reciprocals 1 1/¥
1 0 -1-1
11 a2
z
Chapter 3 - 44
4. Miller-Bravais Indices (1011)
1 0 -1 13. Reduction 1 0 -1 1
a2
a3
a1
Adapted from Fig. 3.24(b), Callister & Rethwisch 3e.
Materials Science - Prof. Choi, Hae-Jin
Crystallographic Planes• We want to examine the atomic packing of
crystallographic planes• Iron foil can be used as a catalyst. The
atomic packing of the exposed planes is important.
a) Draw (100) and (111) crystallographic planes for Fe.
b) Calculate the planar density for each of these planes.
Chapter 3 - 45
• We want to examine the atomic packing of crystallographic planes
• Iron foil can be used as a catalyst. The atomic packing of the exposed planes is important.
a) Draw (100) and (111) crystallographic planes for Fe.
b) Calculate the planar density for each of these planes.
Materials Science - Prof. Choi, Hae-Jin
Planar Density of (100) IronSolution: At T < 912°C iron has the BCC structure.
(100) R3
34a =
2D repeat unit
Chapter 3 - 46
Radius of iron R = 0.1241 nmAdapted from Fig. 3.2(c), Callister & Rethwisch 3e.
= Planar Density =a2
1atoms
2D repeat unit
= nm2
atoms12.1m2
atoms= 1.2 x 10191
2
R3
34area2D repeat unit
Materials Science - Prof. Choi, Hae-Jin
Planar Density of (111) IronSolution (cont): (111) plane 1 atom in plane/ unit surface cell
atoms in plane
atoms above plane
atoms below plane
ah23=
a2
Chapter 3 - 47
333 2
2
R3
16R3
42a3ah2area =÷÷ø
öççè
æ===
ah2
=
1= =
nm2atoms7.0
m2atoms0.70 x 1019
3 2R3
16Planar Density =
atoms2D repeat unit
area2D repeat unit
Materials Science - Prof. Choi, Hae-Jin
A sites
B B
B
BB
B B
C sites
C C
CA
B
B sites
• ABCABC... Stacking Sequence• 2D Projection
FCC Stacking Sequence
B B
B
BB
B B
B sitesC C
CA
C C
CA
Chapter 3 - 48
BC sites
• FCC Unit Cell
B
AB
C
Materials Science - Prof. Choi, Hae-Jin
• Some engineering applications require single crystals:
• Properties of crystalline materials often related to crystal structure.
-- diamond singlecrystals for abrasives
-- turbine bladesFig. 9.40(c), Callister & Rethwisch 3e. (Fig. 9.40(c) courtesy of Pratt and Whitney).
(Courtesy Martin Deakins,GE Superabrasives, Worthington, OH. Used with permission.)
Crystals as Building Blocks
Chapter 3 - 49
• Properties of crystalline materials often related to crystal structure.
(Courtesy P.M. Anderson)
-- Ex: Quartz fractures more easily along some crystal planes than others.
Materials Science - Prof. Choi, Hae-Jin
Polycrystalline Materials
Small crystallite nuclei Growth of crystallites
Chapter 3 -Materials Science - Prof. Choi, Hae-Jin 50
Small crystallite nuclei Growth of crystallites
Completion of solidification Grain structure under microscope
Grain boundary
• Most engineering materials are polycrystals.
Adapted from Fig. K, color inset pages of Callister 5e.(Fig. K is courtesy of Paul E. Danielson, Teledyne Wah Chang Albany)
1 mm
Polycrystals Anisotropic
Chapter 3 - 51
• Nb-Hf-W plate with an electron beam weld.• Each "grain" is a single crystal.• If grains are randomly oriented,
overall component properties are not directional.• Grain sizes typ. range from 1 nm to 2 cm
(i.e., from a few to millions of atomic layers).
1 mm
Isotropic
Materials Science - Prof. Choi, Hae-Jin
• Single Crystals-Properties vary withdirection: anisotropic.
-Example: the modulusof elasticity (E) in BCC iron:
Data from Table 3.7, Callister & Rethwisch 3e. (Source of data is R.W. Hertzberg, Deformation and Fracture Mechanics of Engineering Materials, 3rd ed., John Wiley and Sons, 1989.)
• Polycrystals
Single vs PolycrystalsE (diagonal) = 273 GPa
E (edge) = 125 GPa
Chapter 3 - 52
• Polycrystals-Properties may/may notvary with direction.
-If grains are randomlyoriented: isotropic.(Epoly iron = 210 GPa)
-If grains are textured,anisotropic.
200 mm Adapted from Fig. 5.19(b), Callister & Rethwisch 3e.(Fig. 5.19(b) is courtesy of L.C. Smith and C. Brady, the National Bureau of Standards, Washington, DC [now the National Institute of Standards and Technology, Gaithersburg, MD].)
Materials Science - Prof. Choi, Hae-Jin
Polymorphism • Two or more distinct crystal structures for the same
material (allotropy/polymorphism)
titaniuma, b-Ti
carbondiamond, graphite
1538ºCd-Fe
liquid
iron system
Chapter 3 - 53
• Two or more distinct crystal structures for the same material (allotropy/polymorphism)
titaniuma, b-Ti
carbondiamond, graphite
BCC
FCC
BCC
1538ºC
1394ºC
912ºC
d-Fe
g-Fe
a-Fe
Materials Science - Prof. Choi, Hae-Jin
X-Ray Diffraction
Chapter 3 - 54
• Diffraction gratings must have spacings comparable to the wavelength of diffracted radiation.
• Can’t resolve spacings < l• Spacing is the distance between parallel planes of
atoms. Materials Science - Prof. Choi, Hae-Jin
X-Rays to Determine Crystal Structure• Incoming X-rays diffract from crystal planes.
Adapted from Fig. 3.37, Callister & Rethwisch 3e.
reflections must be in phase for a detectable signal
spacing between d
ql
qextra distance travelled by wave “2”
Chapter 3 - 55
X-ray intensity (from detector)
q
qc
d =nl
2 sin qc
Measurement of critical angle, qc, allows computation of planar spacing, d.
spacing between planes
d
Materials Science - Prof. Choi, Hae-Jin
X-Ray Diffraction Pattern
(110)
(211)
z
x
ya b
c
Intensity (relative)
z
x
ya b
cz
x
ya b
c
Chapter 3 - 56
Adapted from Fig. 3.20, Callister 5e.
(200)
Diffraction angle 2q
Diffraction pattern for polycrystalline a-iron (BCC)
Intensity (relative)
Materials Science - Prof. Choi, Hae-Jin
• Atoms may assemble into crystalline or amorphous 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).
SUMMARY
• 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.
Chapter 3 - 57
• We can predict the density of a material, provided we know the atomic weight, atomic radius, and crystal geometry (e.g., FCC, BCC, HCP).
• 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.
• Ceramic crystal structures are based on:-- maintaining charge neutrality-- cation-anion radii ratios.
• Interatomic bonding in ceramics is ionic and/or covalent.
Materials Science - Prof. Choi, Hae-Jin
• Some materials can have more than one crystal structure. This is referred to as polymorphism (or allotropy).
SUMMARY
• Materials can be single crystals or polycrystalline. Material properties generally vary with single crystal orientation (i.e., they are anisotropic), but are generally non-directional (i.e., they are isotropic) in polycrystals with randomly oriented grains.
Chapter 3 - 58
• Some materials can have more than one crystal structure. This is referred to as polymorphism (or allotropy).
• X-ray diffraction is used for crystal structure and interplanar spacing determinations.
Materials Science - Prof. Choi, Hae-Jin