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Chapter 3 -
Chapter 3: Structures of Metals & Ceramics
X-ray diffraction photograph X-ray diffraction in metals
Chapter 3 - 2
3.1 ISSUES TO ADDRESS...
• How do atoms assemble into solid structures?
• How does the density of a material depend on its structure?
• When do material properties vary with the sample (i.e., part) orientation?
Chapter 3: Structures of Metals & Ceramics
• How do the crystal structures of ceramic materials differ from those for metals?
Chapter 3 - 3
• Non dense, random packing
• Dense, ordered packing
Dense, ordered packed structures tend to have lower energies.
CRYSTAL STRUCTURES3.2 Energy and Packing
Energy
r
typical neighbor bond length
typical neighbor bond energy
Energy
r
typical neighbor bond length
typical neighbor bond energy
Chapter 3 - 4
• atoms pack in periodic, 3D arraysCrystalline materials...
-metals-many ceramics-some polymers
• atoms have no periodic packingNoncrystalline materials...
-complex structures-rapid cooling
crystalline SiO2
noncrystalline SiO2"Amorphous" = NoncrystallineAdapted from Fig. 3.40(b), Callister & Rethwisch 3e.
Adapted from Fig. 3.40(a), Callister & Rethwisch 3e.
Materials and Packing
Si Oxygen
• typical of:
• occurs for:
Chapter 3 - 5
3.4 Metallic Crystal Structures
• How can we stack metal atoms to minimize empty space?
2-dimensions
vs.
Now stack these 2-D layers to make 3-D structures
Chapter 3 - 6
• 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.- Electron cloud shields cores from each other
• Have the simplest crystal structures.
We will examine three such structures...
3.4 Metallic Crystal Structures
Chapter 3 -
FCC = face-centered cubicHCP = hexagonal close-packedBCC = body-centered cubic1 nm = 10-9 m = 10 Ǻ
Chapter 3 - 8
• Rare due to low packing density (only Po has this structure)• Close-packed directions are cube edges.
• Coordination # = 6 (# nearest neighbors)
(Courtesy P.M. Anderson)
Simple Cubic Structure (SC)
Chapter 3 - 9
• APF for a simple cubic structure = 0.52
APF = a3
4
3(0.5a) 31
atoms
unit cellatom
volume
unit cell
volume
Atomic Packing Factor (APF)
APF = Volume of atoms in unit cell*
Volume of unit cell
*assume hard spheres
Adapted from Fig. 3.42, Callister & Rethwisch 3e.
close-packed directions
a
R=0.5a
contains 8 x 1/8 = 1 atom/unit cell
Chapter 3 -
The body-centered cubic (BCC) crystal structure
Hard sphere unit cell representation
Reduced-sphere unit cell representation
Aggregate of many atoms
Chapter 3 - 11
• Coordination # = 8
Adapted from Fig. 3.2, Callister & Rethwisch 3e.
(Courtesy P.M. Anderson)
• 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 (), Tantalum, Molybdenum
QuickTime™ and aCinepak decompressor
are needed to see this picture.
2 atoms/unit cell: 1 center + 8 corners x 1/8
Chapter 3 - 12
Atomic Packing Factor: BCC
a
APF =
4
3 ( 3a/4)32
atoms
unit cell atom
volume
a3unit cell
volume
length = 4R =Close-packed directions:
3 a
• APF for a body-centered cubic structure = 0.68
aRAdapted from
Fig. 3.2(a), Callister & Rethwisch 3e.
a 2
a 3
Chapter 3 - 13
• Coordination # = 12
Adapted from Fig. 3.1, Callister & Rethwisch 3e.
(Courtesy P.M. Anderson)
• 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
QuickTime™ and aCinepak decompressor
are needed to see this picture.
4 atoms/unit cell: 6 face x 1/2 + 8 corners x 1/8
Chapter 3 - 14
• APF for a face-centered cubic structure = 0.74Atomic Packing Factor: FCC
maximum achievable APF
APF =
4
3( 2a/4)34
atoms
unit cell atom
volume
a3unit cell
volume
Close-packed directions: length = 4R = 2 a
Unit cell contains: 6 x 1/2 + 8 x 1/8 = 4 atoms/unit cell
a
2 a
Adapted fromFig. 3.1(a),Callister & Rethwisch 3e.
Chapter 3 - 15
A sites
B B
B
BB
B B
C sites
C C
CA
B
B sites
• ABCABC... Stacking Sequence• 2D Projection
• FCC Unit Cell
FCC Stacking Sequence
B B
B
BB
B B
B sitesC C
CA
C C
CA
AB
C
Chapter 3 -
16
• Coordination # = 12
• ABAB... Stacking Sequence
• APF = 0.74
• 3D Projection • 2D Projection
Adapted from Fig. 3.3(a), Callister & Rethwisch 3e.
6 atoms/unit cell
ex: Cd, Mg, Ti, Zn
• c/a = 1.633
c
a
A sites
B sites
A sites Bottom layer
Middle layer
Top layer
The Hexagonal Closed-Packed Crystal Structure (HCP)
Chapter 3 - 17
3.5 Theoretical Density,
where n = number of atoms/unit cell A = atomic weight VC = Volume of unit cell = a3 for cubic NA = Avogadro’s number = 6.022 x 1023 atoms/mol
Density = =
VC NA
n A =
Cell Unit of VolumeTotal
Cell Unit in Atomsof Mass
Chapter 3 - 18
• Ex: Cr (BCC)
A = 52.00 g/mol
R = 0.125 nm
n = 2 atoms/unit cell
theoretical
a = 4R/ 3 = 0.2887 nm
actual
aR
= a3
52.002
atoms
unit cellmol
g
unit cell
volume atoms
mol
6.022 x 1023
Theoretical Density,
= 7.18 g/cm3
= 7.19 g/cm3
Adapted from Fig. 3.2(a), Callister & Rethwisch 3e.
Chapter 3 - 19
3.6 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 -
c03tf02
Chapter 3 - 21
• Bonding: -- Can be ionic and/or covalent in character. -- % ionic character increases with difference in electronegativity of atoms.
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 byCornell University.
• Degree of ionic character may be large or small:
Atomic Bonding in Ceramics
SiC: small
CaF2: large
Chapter 3 - 22
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
2. Maintenance of Charge Neutrality : --Net charge in ceramic should be zero. --Reflected in chemical formula:
CaF2: Ca2+cation
F-
F-
anions+
AmXp
m, p values to achieve charge neutrality
Chapter 3 -
c03tf03
Chapter 3 -
c03prob
Example 3.4: Computation of minimum cation-to-anion radius ration for a coordination number of 3
Chapter 3 - 25
• Coordination # increases with
Coordination # and Ionic Radii
Adapted from Table 3.3, Callister & Rethwisch 3e.
2
rcationranion
Coord #
< 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.
ZnS (zinc blende)
NaCl(sodium chloride)
CsCl(cesium chloride)
rcationranion
To form a stable structure, how many anions can surround around a cation?
Chapter 3 -
c03tf04
Chapter 3 - 27
Computation of Minimum Cation-Anion Radius Ratio
• Determine minimum rcation/ranion for an octahedral site (C.N. = 6)
a 2ranion
2ranion 2rcation 2 2ranion
ranion rcation 2ranion
rcation ( 2 1)ranion
arr 222 cationanion
414.012anion
cation r
r
Chapter 3 - 28
Bond Hybridization
Bond Hybridization is possible when there is significant covalent bonding– hybrid electron orbitals form– For example for SiC
• XSi = 1.8 and XC = 2.5
%.)XXionic% 511]}exp[-0.25(-{1 100 character 2CSi
• ~ 89% covalent bonding• Both Si and C prefer sp3 hybridization• Therefore, for SiC, Si atoms occupy tetrahedral sites
Chapter 3 - 29
• On the basis of ionic radii, what crystal structure would you predict for FeO?
• Answer:
5500
1400
0770
anion
cation
.
.
.
r
r
based on this ratio,-- coord # = 6 because
0.414 < 0.550 < 0.732
-- crystal structure is NaCl
Data from Table 3.4, Callister & Rethwisch 3e.
Example Problem: Predicting the Crystal Structure of FeO
Ionic radius (nm)
0.053
0.077
0.069
0.100
0.140
0.181
0.133
Cation
Anion
Al3+
Fe2+
Fe3+
Ca2+
O2-
Cl-
F-
Chapter 3 - 30
Rock Salt StructureSame concepts can be applied to ionic solids in general. Example: NaCl (rock salt) structure
rNa = 0.102 nm
rNa/rCl = 0.564
cations (Na+) prefer octahedral sites
Adapted from Fig. 3.5, Callister & Rethwisch 3e.
rCl = 0.181 nm
Chapter 3 - 31
MgO and FeO
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
Adapted from Fig. 3.5, Callister & Rethwisch 3e.
MgO and FeO also have the NaCl structure
Chapter 3 - 32
AX Crystal Structures
939.0181.0
170.0
Cl
Cs
r
r
Adapted from Fig. 3.6, Callister & Rethwisch 3e.
Cesium Chloride structure:
Since 0.732 < 0.939 < 1.0, cubic sites preferred
So each Cs+ has 8 neighbor Cl-
AX–Type Crystal Structures include NaCl, CsCl, and zinc blende
Chapter 3 -
c03f07
Zinc blende or sphalerite or zinc sulfide; highly covalent
AX typeCN = 4All ions tetrahedrally coordinated
Zn ↔ S
Chapter 3 - 34
AX2 Crystal Structures
• 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.
Fluorite structure
Chapter 3 - 35
ABX3 Crystal Structures
Adapted from Fig. 3.9, Callister & Rethwisch 3e.
• Perovskite structure
Ex: complex oxide
BaTiO3
Chapter 3 -
c03tf05
Chapter 3 - 37
3.7 Density Computations for Ceramics
A
AC )(
NV
AAn
C
Number of formula units/unit cell
Volume of unit cell
Avogadro’s number
= sum of atomic weights of all anions in formula unit
AA
AC = sum of atomic weights of all cations in formula unit
Chapter 3 -
c03pro
b
Theoretical Density Calculations for Sodium Chloride
Chapter 3 - 39
Densities of Material Classesmetals > ceramics > polymers
Why?
Data from Table B.1, Callister & Rethwisch, 3e.
(g
/cm
)3
Graphite/ Ceramics/ Semicond
Metals/ Alloys
Composites/ fibers
Polymers
1
2
20
30Based on data in Table B1, Callister
*GFRE, CFRE, & AFRE are Glass, Carbon, & Aramid Fiber-Reinforced Epoxy composites (values based on 60% volume fraction of aligned fibers
in an epoxy matrix). 10
3
4 5
0.3
0.4 0.5
Magnesium
Aluminum
Steels
Titanium
Cu,Ni
Tin, Zinc
Silver, Mo
Tantalum Gold, W Platinum
Graphite
Silicon
Glass -soda Concrete
Si nitride Diamond Al oxide
Zirconia
HDPE, PS PP, LDPE
PC
PTFE
PET PVC Silicone
Wood
AFRE*
CFRE*
GFRE*
Glass fibers
Carbon fibers
Aramid fibers
Metals have... • close-packing (metallic bonding) • often large atomic masses Ceramics have... • less dense packing • often lighter elements Polymers have... • low packing density (often amorphous) • lighter elements (C,H,O)
Composites have... • intermediate values
In general
Chapter 3 -
c03f10
3.8 Silicate CeramicsSiO4-
4
Chapter 3 - 41
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-
Adapted from Figs. 3.10-11, Callister & Rethwisch 3e crystobalite
Chapter 3 -
c03f12
Silicate ion structures formed from Si tetrahedra
Chapter 3 - 43
Bonding of adjacent SiO44- accomplished by the sharing
of common corners, edges, or faces
Silicates
Mg2SiO4 Ca2MgSi2O7
Adapted from Fig. 3.12, Callister & Rethwisch 3e.
Presence of cations such as Ca2+, Mg2+, & Al3+ 1. maintain charge neutrality, and 2. ionically bond SiO4
4- to one another
Chapter 3 - 44
• Quartz is crystalline SiO2:
• 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+
(soda glass)
Adapted from Fig. 3.41, Callister & Rethwisch 3e.
Glass Structure
Si04 tetrahedron4-
Si4+
O2-
Si4+
Na+
O2-
Chapter 3 - 45
Layered Silicates• Layered silicates (e.g., clays, mica,
talc)– SiO4 tetrahedra connected
together to form 2-D plane
• A net negative charge is associated with each (Si2O5)2- unit
• Negative charge balanced by adjacent plane rich in positively charged cations
Adapted from Fig. 3.13, Callister & Rethwisch 3e.
Silicate sheet structure of (Si2O5)2-
Chapter 3 - 46
• Kaolinite clay alternates (Si2O5)2- layer with Al2(OH)42+
layer
Layered Silicates (cont)
Note: Adjacent sheets of this type are loosely bound to one another by van der Waal’s forces.
Adapted from Fig. 3.14, Callister & Rethwisch 3e.
Kaolinite ClayTwo layer silicate sheet structure: Al2(Si2O5)(OH)4
15,000x
Electron micrograph of kaolinate crystals
Chapter 3 - 47
3.9 Polymorphic Forms of Carbon
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.
Chapter 3 - 48
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
Adapted from Fig. 3.17, Callister & Rethwisch 3e.
Chapter 3 - 49
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
Adapted from Figs. 3.18 & 3.19, Callister & Rethwisch 3e.
Chapter 3 -
c03impf01
STM
Chapter 3 - 51
• Some engineering applications require single crystals:
• Properties of crystalline materials often related to crystal structure.
(Courtesy P.M. Anderson)
-- Ex: Quartz fractures more easily
along some crystal planes than others.
-- diamond single crystals for abrasives
-- turbine blades
Fig. 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.)
3.11 Crystals as Building Blocks
Chapter 3 -
c03tf06
Chapter 3 - 53
• Most engineering materials are polycrystals.
• 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).
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
Isotropic
Anisotropic
Chapter 3 - 54
• Single Crystals-Properties vary with direction: anisotropic.
-Example: the modulus of 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
-Properties may/may not vary with direction.-If grains are randomly oriented: isotropic. (Epoly iron = 210 GPa)-If grains are textured, anisotropic.
200 m 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].)
Single vs PolycrystalsE (diagonal) = 273 GPa
E (edge) = 125 GPa
Chapter 3 - 55
Polymorphism
• Two or more distinct crystal structures for the same material (allotropy/polymorphism) titanium
, -Ti
carbon
diamond, graphite
BCC
FCC
BCC
1538ºC
1394ºC
912ºC
-Fe
-Fe
-Fe
liquid
iron system
Chapter 3 -
c03impf02a
Allotropic transformation of tin
Chapter 3 -
c03impf02b
white tin
gray tin
Cooled below 13.2 °C for an extended period of time
Chapter 3 - 58
Fig. 3.20, Callister & Rethwisch 3e.
3.11 Crystal Systems
7 crystal systems
14 crystal lattices
Unit cell: smallest repetitive volume which contains the complete lattice pattern of a crystal.
a, b, and c are the lattice constants
Chapter 3 - 59
Point CoordinatesPoint coordinates for unit cell
center are
a/2, b/2, c/2 ½ ½ ½
Point coordinates for unit cell corner are 111
Translation: integer multiple of lattice constants identical position in another unit cell
z
x
ya b
c
000
111
y
z
2c
b
b
Chapter 3 -
c03f21
Chapter 3 - 61
3.13 Crystallographic Directions
1. Vector repositioned to pass through origin2. Read off projections in terms of unit cell
dimensions a, b, and c3. 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>
z
x
Algorithm
where overbar represents a negative index
[ 111 ]=>
y
Chapter 3 -
c03f22
Chapter 3 -
c03f23
Chapter 3 - 64
HCP Crystallographic Directions
1. Vector repositioned (if necessary) to pass through origin.2. Read off projections in terms of unit cell dimensions a1, a2, a3, or c3. 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
-a3
2
a2
2
a1
-a3
a1
a2
z
Algorithm
Chapter 3 -
c03f24
Chapter 3 - 66
HCP Crystallographic Directions• Hexagonal Crystals
– 4 parameter Miller-Bravais lattice coordinates are related to the direction indices (i.e., u'v'w') as follows.
'ww
t
v
u
)vu( +-
)'u'v2(3
1-
)'v'u2(3
1-
]uvtw[]'w'v'u[
Fig. 3.24(a), Callister & Rethwisch 3e.
-a3
a1
a2
z
Chapter 3 - 67
3.14 Crystallographic Planes
Adapted from Fig. 3.25, Callister & Rethwisch 3e.
Chapter 3 - 68
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.
• Algorithm 1. 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 - 69
Crystallographic Planesz
x
ya b
c
4. Miller Indices (110)
example a b cz
x
ya b
c
4. Miller Indices (100)
1. Intercepts 1 1 2. Reciprocals 1/1 1/1 1/
1 1 03. Reduction 1 1 0
1. Intercepts 1/2 2. Reciprocals 1/½ 1/ 1/
2 0 03. Reduction 2 0 0
example a b c
Chapter 3 - 70
Crystallographic Planes
z
x
ya b
c
4. Miller Indices (634)
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
(001)(010),
Family of Planes {hkl}
(100), (010),(001),Ex: {100} = (100),
Chapter 3 - 71
Crystallographic Planes (HCP)
• In hexagonal unit cells the same idea is used
example a1 a2 a3 c
4. Miller-Bravais Indices (1011)
1. Intercepts 1 -1 12. Reciprocals 1 1/
1 0 -1-1
11
3. Reduction 1 0 -1 1
a2
a3
a1
z
Adapted from Fig. 3.24(b), Callister & Rethwisch 3e.
Chapter 3 - 72
Crystallographic Planes
• Atomic packing of crystallographic planes• The atomic packing of the exposed planes
is important in catalysis
• Preparing for the exam:
a) Draw (100) and (111) crystallographic planes
for Fe.
b) Calculate the planar density for each of these planes.
Chapter 3 -
c03f26
Atomic Arrangements in the (100) plane of a FCC
Chapter 3 -
c03f27
Atomic Arrangements in the (100) plane of a BCC
Chapter 3 - 75
ex: linear density of Al in [110] direction
a = 0.405 nm
Linear Density
• Linear Density of Atoms LD =
a
[110]
Unit length of direction vector
Number of atoms
# atoms
length
13.5 nma2
2LD
Chapter 3 - 76
Planar Density of (100) IronSolution: At T < 912C iron has the BCC structure.
(100)
Radius of iron R = 0.1241 nm
R3
34a
Adapted from Fig. 3.2(c), Callister & Rethwisch 3e.
2D repeat unit
= Planar Density = a2
1
atoms
2D repeat unit
= nm2
atoms12.1
m2
atoms= 1.2 x 1019
12
R3
34area
2D repeat unit
Chapter 3 -
c03f28
Chapter 3 - 78
Planar Density of (111) IronSolution (cont): (111) plane 1 atom in plane/ unit surface cell
333 2
2
R3
16R
34
2a3ah2area
atoms in plane
atoms above plane
atoms below plane
ah2
3
a 2
2D re
peat
uni
t
1
= = nm2
atoms7.0m2
atoms0.70 x 1019
3 2R3
16Planar Density =
atoms
2D repeat unit
area
2D repeat unit
Chapter 3 -
c03f29
3.16 Closed-packed crystal structures
3.16 Closed-Packed Crystal Structures
Chapter 3 -
c03f32
Ceramics
Chapter 3 -
c03f33
(111) PLANES OF ROCK SALT
Chapter 3 -c03f34
Crystalline and non crystalline materials3.17 Single Crystals
A garnet single crystalTongbei, Fujian, China
Chapter 3 -
c03f35
3.18 Solidification of a polycrystalline material
Chapter 3 -
3.19 Anisotropy
Chapter 3 -
c03f40
Noncrystalline solids
Chapter 3 - 88
3.20 X-Ray Diffraction
• Diffraction gratings must have spacings comparable to the wavelength of diffracted radiation.
• Can’t resolve spacings • Spacing is the distance between parallel planes of
atoms.
Chapter 3 -
c03f36
Chapter 3 -
c03f37
Chapter 3 - 91
X-Rays to Determine Crystal Structure
X-ray intensity (from detector)
c
d n
2 sin c
Measurement of critical angle, c, allows computation of planar spacing, d.
• 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 planes
d
incoming
X-rays
outg
oing
X-ra
ys
detector
extra distance travelled by wave “2”
“1”
“2”
“1”
“2”
Chapter 3 -
c03f38
Chapter 3 - 93
X-Ray Diffraction Pattern
Adapted from Fig. 3.20, Callister 5e.
(110)
(200)
(211)
z
x
ya b
c
Diffraction angle 2
Diffraction pattern for polycrystalline -iron (BCC)
Inte
nsity
(re
lativ
e)
z
x
ya b
cz
x
ya b
c
Chapter 3 -
Some hypothetical metal has this simple cubic crystal structure. If its atomic weight is 74.5 g/mol and the atomic radius is 0.145 nm, compute its density.
For the simple cubic crystal structure, the value of n in Equation 3.5 is unity since there is only a single atom associated with each unit cell.
Furthermore, for the unit cell edge length, a = 2R (Figure 3.42).
Therefore, employment of Equation 3.5 yields
= nA
VC N A=
nA
(2 R)3 N A
= (1 atom/unit cell)(74.5 g/mol)
(2)(1.45 10-8
cm)
3
/(unit cell)
(6.02 10
23 atoms/mol)
= 5.07 g/cm3
Problem
Chapter 3 -
problemFirst five peaks, x-ray diffraction, tungsten, BCC monochromatic radiation; wavelength = 0.1542 nm(a) Index (h, k, and l indices) for each of these peaks.(b) Interplanar spacing for each of the peaks.(c) For each peak, determine the atomic radius for W and compare these with the value presented in Table 3.1.
(a) Since W has a BCC crystal structure, only those peaks for which h + k + l are even will appear. Therefore, these five peaks result by diffraction from the following planes: (110), (200), (211), (220), and (310).
Chapter 3 -
First five peaks, x-ray diffraction, tungsten, BCC, monochromatic radiation; wavelength = 0.1542 nm
(b) Interplanar spacing for each of the peaks.
d110 =n
2 sin =
(1)(0.1542 nm)
(2) sin40.2
2
= 0.2244 nm
(b) For each peak, in order to calculate the interplanar spacing we must employ Equation 3.14. For the first peak, which occurs at 40.2°
Chapter 3 -
R110 =(0.2244 nm)( 3) (1)2 + (1)2 + (0)2
4 R110 = 0.1374 nm
(c) Employment of Equations 3.15 and 3.3 is necessary for the computation of R as {a=4R/sqrt(3) and dhkl = a/sqrt(h2+k2+l2)}
R =a 3
4=
(dhkl)( 3) (h)2 + (k)2 + (l)2
4
Similar computations are made for the other peaks:Peak Index 2 dhkl(nm) R (nm)
200 58.4 0.1580 0.1369211 73.3 0.1292 0.1370220 87.0 0.1120 0.1371310 100.7 0.1001 0.1371
Experimental R = 0.1371 nm
Chapter 3 - 98
• 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.
• 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.
Chapter 3 - 99
• 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.
• X-ray diffraction is used for crystal structure and interplanar spacing determinations.
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