Post on 05-Jul-2020
Chapter 3 -
Crystal System Review
Property SC FCC BCC HCP# atoms/cell 1 4 2 6
Lattice Parameter
a,c
Coordination # 6 12 8 12
APF 0.52 0.74 0.68 0.74
2
Chapter 3 -
VOLUME DENSITY
3
Chapter 3 - 4
Theoretical Density, ρ
where n = number of atoms/unit cellA = atomic weight (g/mol) VC = Volume of unit cell = a3 for cubicNA = Avogadro’s number
= 6.022 x 1023 atoms/mol
Density = ρ =
VC
n Aρ =
CellUnitofVolumeTotalCellUnitinAtomsofMass
Book Section 3.5
How much stuffHow much spaceNA
Chapter 3 - 5
• Ex: CrA = 52.00 g/molR = 0.125 nmn = 2 atoms/unit cell
ρtheoretical
a = 4R/ 3 = 0.2887 nm
ρactual
aR
ρ = a3
52.002atoms
unit cell molg
unit cellvolume atoms
mol
6.022 x 1023
Theoretical Density, ρ
= 7.18 g/cm3
= 7.19 g/cm3
Adapted from Fig. 3.2(a), Callister & Rethwisch 8e.
(BCC)
Book Section 3.5
Table 3.1Back of book
Chapter 3 -
So what
• When you sinter a ceramic, you have pores, which make it less dense
• You can compare manufactured density to theoretical density for process quality
6http://www.additivalab.com/en/blog/laser-sintering-vs-laser-melting/ DOI: 10.1016/j.matlet.2015.12.042
Chapter 3 - 7
Densities of Material Classesρmetals > ρceramics > ρpolymers
Why?
Data from Table B.1, Callister & Rethwisch, 8e.
ρ(g
/cm
)3
Graphite/ Ceramics/ Semicond
Metals/ Alloys
Composites/ fibersPolymers
1
2
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
345
0.30.40.5
Magnesium
Aluminum
Steels
Titanium
Cu,NiTin, Zinc
Silver, Mo
TantalumGold, WPlatinum
GraphiteSiliconGlass -sodaConcrete
Si nitrideDiamondAl oxide
Zirconia
HDPE, PSPP, LDPEPC
PTFE
PETPVCSilicone
Wood
AFRE*CFRE*GFRE*Glass fibers
Carbon fibersAramid 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
Book Section 3.5
Chapter 3 -
Crystallographic planes are typically identified with which of the following:
A. Direction indicesB. Planar DirectionsC. Miller Indices
8
Chapter 3 -
The direction of the red arrow is:
A. [1 0 0]B. <1 0 1>C. <1 1 1>D. [1 1 0]E. [1 1 1]
9
Chapter 3 -
Which of the following has the highest atomic packing factor (APF)?(Give me your best answer)
A. Simple CubicB. FCCC. BCCD. HCP
10
Chapter 3 -
Select the correct set of indices for this plane
A. (211)B. (213)C. (332)D. (201)E. (110)
11
Chapter 3 -
Crystal Coordinates, Directions, Planes
• How can we identify crystal coordinates, directions, and planes?– Or how we talk a common language about
crystals
12
Chapter 3 -
POINTS
13
Chapter 3 - 14
Point CoordinatesPoint coordinates for unit cell
center area/2, b/2, c/2 ½ ½ ½
Point coordinates for unit cell corner are 111
z
x
ya
b
c
000
111
Book Section 3.8
Chapter 3 -
DIRECTIONS
15
Chapter 3 - 16
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]
families of directions <uvw>
ex: 1, 0, ½ 2, 0, 1 [201 ]
-1, 1, 1
z
x
Algorithm
where overbar represents a negative index
[111 ]
y
Book Section 3.9
× 2
Chapter 3 -
PLANES(MILLER INDICES)
17
Chapter 3 - 18
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)
Book Section 3.10
(100) (110) (111)Why?(FCC)
z
x
ya b
c
Chapter 3 - 19
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/∞
3. Reduction 1 0 0
example a b c
Book Section 3.10
2 0 0
Chapter 3 - 20
Crystallographic Planesz
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÷¾
3. Reduction 6 3 4
(001)(010),
Family of Planes {hkl}
(100), (010),(001),Ex: {100} = (100),
Book Section 3.10
2 1 4/3
Chapter 3 - 21
Crystallographic Planes
Adapted from Fig. 3.10, Callister & Rethwisch 8e.
Book Section 3.10
Chapter 3 -
Summary
• Points– 111
• Directions– [111]– Families <111>
• Will be used in Ch 7
• Planes– (111)– Families {111}
• Will be used in Ch 722
z
x
ya b
c
z
x
ya b
c
Chapter 3 - 23
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 1a2
a3
a1
z
Adapted from Fig. 3.8(b), Callister & Rethwisch 8e.
Book Section 3.10
Chapter 3 -
Bigger Picture
• Classifying crystals– Metrics ✔– Planes/directions ✔– Densities
• Real Materials– Single/Polycrystal– Anisotropy– Characterization
24
(100)(111)