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Engineering Materials ME Chap 3 10th Ed Part (I) Aoh ...
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Engineering MaterialsChapter 3 (I)
The Structure of Crystalline Solids
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Prof. J.N. Aoh CCU ME 10.12.2021
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Crystalline Solids
• Periodic atom sites• Crystal directions
• Crystal planes
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SiO4 Crystal
Figun_03_p047a
Figun_03_p047b
Figun_03_p047c Figun_03_p047d
Fundamental of CrystallographyPart (I) 2021.10.12.
矽酸鹽類
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Crystal structures
• Crystal structure :the atoms are situated in a (1) repeating or periodic 3 D array over (2) large atomic distances, or “________________”.
• Crystalline amorphous• Single crystal Polycrystalline• Crystal lattice
• Unit cell
What is crystallography?
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Unit cell
The smallest portion of a crystal lattice that shows the three-dimensional pattern of the entire crystal.
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Crystallineamorphous• Amorphous: noncrystalline, “without form”• Amorphous metals“_____________”• Glass is noncrystallineamorphous.• occurs at extreme rapid cooling
Fig_03_11 SiO2
Metal Glass
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Unit cell & crystal systems
Fig_03_02
• The unit cell is the basic structural unit of the crystal structure and defines the crystal structure by its geometry and the atomic positions within.
• Lattice parameter a, b, c• Angles between axes: α, β, γ• Crystal system: axes, a, b, c and α, β, γ
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14 Bravais Crystal lattice
7 Crystal systems
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Metallic crystal structures
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Solidification and Crystallization3.14/3.15 Single vs Polycrystalline
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• Single Crystals-Properties vary with
direction: anisotropic.-Example: the modulus
of elasticity (E) in BCC iron:
• Polycrystalline-Properties may/may not
vary with direction.-If grains are randomly
oriented: isotropic.(Epoly iron = 210 GPa)
-If grains are textured,anisotropic.
200 μm
3.14/3.15 Single/Polycrystalline & AnisotropyE (diagonal) = 273 GPa
E (edge) = 125 GPa
Unit cell
Iron microstructure, magnification: x100
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Elastic modulus along different orientationCrystallographic Directions
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Growth of silicon single crystal
Single crystal growthSi single crystal Silicon wafer 晶圓
Silicon chip晶元
1420°C
Homework I – Short report
Indicate crystallographic direction of the wafer
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3.4 Metallic crystal structures• Tend to be densely packed.
• Reasons for dense packing:- For element metals, all atoms have the same
radius.- 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...
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• Coordination number CN= 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.
3.4.1.Face Centered Cubic Structure (FCC)
ex: Al, Cu, Au, Pb, Ni, Pt, Ag
atoms per unit cell: 6 face x 1/2 + 8 corners x 1/8= _____
• Close packed directions are face diagonals.
Sphere model site model
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Hard sphere model Atomic site model
•long range order”• long range repetition.
∞
∞
∞
3.4.1.Face Centered Cubic Structure (FCC)
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APF = a3
4
3π ( 2a/4)34
atoms
unit cell atomvolume
unit cell
volume
Unit cell contains: 6 x 1/2 + 8 x 1/8 = 4 atoms/unit cell
Close-packed directions: length = 4R
= 2 a
ATOMIC PACKING FACTOR (APF): FCC
a
Face diagonal 4R
• APF for a face-centered cubic structure = _______
a is lattice constant
a
APF = Volume of atoms in unit cell*
Volume of unit cell*assume hard spheres
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Examples• Determine the FCC unit cell volume
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CN = 8
(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.
ex: Cr, W, Fe (α), Ta, Mo
atoms/unit cell: 1 center + 8 corners x 1/8 = ____coordination number CN = ______
3.4.2. Body Centered Cubic Structure (BCC)
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aR
Close-packed directions: length = 4R
= 3 a
Unit cell contains: 1 + 8 x 1/8 = 2 atoms/unit cell
ATOMIC PACKING FACTOR: BCC
APF = a3
4
3π ( 3a/4)32
atoms
unit cell atomvolume
unit cell
volume
APF for a body-centered cubic structure = ______
body diagonals
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• Rare due to low packing density (only Polonium, Po 釙)• Close-packed directions are cube edges.
• Coordination number =CN= ____(No. of nearest neighbor atoms)
3.4.2. Simple Cubic Structure (SC)
Sphere model lattice model
or
Site model
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• APF for a simple cubic structure = ____
APF = a3
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π (0.5a) 31atoms
unit cellatom
volume
unit cellvolume
Atomic Packing Factor for SC
APF = Volume of atoms in unit cell*
Volume of unit cell
.
close-packed directions
a
R=0.5a
contains 8 x 1/8 = 1 atom/unit cellLattice const.
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3.4.3. HEXAGONAL close-packed structure (HCP)
a and c are lattice constants in HCP
ex: Cd, Mg, Ti, Zn
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• =
HEXAGONAL CLOSE-PACKED STRUCTURE (HCP)
CN= 126 atoms/unit cell
APF= 6Vsphere/3Vparall = 0.74
Example 3.3. Determine unit cell volume HCP
36
22
33
)2
3(3A
2
2
cRV
Ra
caAcV
aa
parall
=
=
==
=
Figun_04_p080b
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c /a =1.633R
HCP c/a ratio
c
a
h=1/2 C
a
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Example: Copper
ρ = n AVcNA
# atoms/unit cell Atomic weight (g/mol)
Volume/unit cell
(cm3/unit cell)Avogadro's number
(6.023 x 1023 atoms/mol)
Data from Table inside front cover of Callister:• crystal structure = FCC: 4 atoms/unit cell• atomic weight = 63.55 g/mol (1 amu = 1 g/mol) • atomic radius R = 0.128 nm (1 nm = 10 cm)-7
Vc = a3 ; For FCC, a = 4R/ 2 ; Vc = 4.75 x 10-23cm3
Compare to actual: ρCu = 8.94 g/cm3Result: theoretical ρCu = 8.89 g/cm3
3.5 THEORETICAL DENSITY, ρ
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• Ex: Cr (BCC) A = 52.00 g/molR = 0.125 nmn = 2
ρtheoretical
a = 4R/ 3 = 0.2887 nm
ρactual
aR
ρ = a3
52.002atoms
unit cell molg
unit cellvolume atoms
mol
6.023 x 1023
= 7.18 g/cm3
= 7.19 g/cm3
3.5 THEORETICAL DENSITY, ρ
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3.12. Close-packed crystal structure--plane stacking sequence for HCP and FCC
A site
B site
C site
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• 3D Projection • 2D Projection
Hexagonal Close-Packed Structure (HCP)
c
a
A sites
B sites
A sitesBottom layer
Middle layer
Top layer
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Stacking Sequence ABAB... observed from z-direction
Close-packed stacking sequence for HCP
A sites
B B
B
BB
B BC C
CA BB sites B B
B
BB
B BB sites
A
A A
AA
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Stacking Sequence ABCABC... Along body diagonal
• FCC Unit Cell
A sites
B B
B
BB
B B
C sites
C C
CA BB sites B B
B
BB
B BB sites
C C
CAC C
CA
AB
C
http://www.dawgsdk.org/crystal/index.en
Close-packed stacking sequence for FCC
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Close-packed stacking sequence for FCC
AB
C
Body diagonal
Body diagonal
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Coordination Numbers for various radius ratiosusually refers to ceramic structures metallic ionscations, rc , nonmetallic ionsanions, rA
12.2 Ceramic crystal structure
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Table 12. 2 Coordination Numbers andGeometries for Various Cation–AnionRadius Ratios (rC/rA)
(rC/rA)
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frRR
−+=
3
30cos
RrR f =+ − 30cos30cos 3
30cos30cos 3 frRR −=−30cos)30cos1( 3 frR −=−
155.030cos
30cos13 =−=−
Rr f
Ex.12.1 Coordination Numbers (CN)
r3-f R
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• r4-f = 0.225R
Coordination Numbers (CN)
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12.2 Ceramic crystal structureClose-packed anion
Spheres with different radius, R and r
CN=4 CN=6
Tetrahedral and octahedral positions
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12.2 Rock salt structure
• Anion (green)• Cation (red)-octahedral positions
Fig_04_27
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• Polymorphism - metals and nonmetals having more than one crystal structure. Water, steel, etc.
• Allotropy - polymorphism in elemental solid, e.g. C, S,
3.6 Polymorphism and Allotropy
BCC
FCC
BCC
1538ºC
1394ºC
912ºC
δ-Fe
γ-Fe
α-Fe
liquidiron system
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Tin and its allotropic transformation(Tin desease)
Increase in volume 27% and decrease in density from 7.3 to 5.77 g/ccm
Body Centered Tetragonal
Diamond cubic
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Exercise Crystal structure of Tin• The unit cell for tin has tetragonal symmetry, with a and b
lattice parameters of 0.583nm and 0.318nm, respectively. If its density=7.30 g/cm3, atomic weight=118.69 g/mol, and atomic radius=0.152 nm, compute the atomic packing factor.
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ρVC NA
A=n =
APF = = 0.544