1. Chapter 3: The Structures of Metals
Energy and Packing
• Non dense, random packing
Energy
typical neighbor
bond length r
typical neighbor
bond energy
• Dense, ordered packing
Energy
typical neighbor
bond length r
typical neighbor
bond energy
Dense, ordered packed structures tend to have lower energies.

2. Materials and Packing 3.3. Unit cell Crystalline materials...
• atoms pack in periodic, 3D arrays
metals, many ceramics, some polymers
Noncrystalline materials...
crystalline SiO2
• atoms have no periodic packing
-complex structures
-rapid cooling
Oxygen
"Amorphous" = Noncrystalline Si
noncrystalline SiO2
3.3. Unit cell
: smallest repetitive volume which contains the complete lattice pattern of a crystal.
a, b, and c are the lattice constants

3. 3.4 Metallic Crystal Structures
How can we stack metal atoms to minimize empty space?
2-dimensions
• 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

4. Simple Cubic Structure (SC)
• Rare due to low packing denisty (only Po has this structure)
• Close-packed directions are cube edges.
• Coordination # = 6
(# nearest neighbors)
Atomic Packing Factor (APF)
Volume of atoms in unit cell*
contains 8 x 1/8 = 1atom/unit cell
APF =
Volume of unit cell a
R=a/2
close-packed directions

5. Body Centered Cubic Structure (BCC)
Atoms touch each other along cube diagonals.
Cr, W, Fe (), Tantalum, Molybdenum
Coordination # (배위수)= 8
2 atoms/unit cell: 1 center + 8 corners x 1/8
원자의 반지름 : , 최인접 원자간 거리 : a 3 a a 2

6. Face Centered Cubic Structure (FCC)
Atoms touch each other along face diagonals
Al, Cu, Au, Pb, Ni, Pt, Ag
Coordination #(배위수) = 12
4 atoms/unit cell: 6 face x 1/2 + 8 corners x 1/8
최인접원자간 거리 :
원자 반지름 : a

7. FCC Stacking Sequence : ABCABC... Stacking Sequence
A sites B C
sites A B
sites C A C A A B C
• FCC Unit Cell

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

9. 3.5 Theoretical Density 예제)Cu는 0.128nm의 원자 반지름과 FCC결정 구조를 가지며,
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
예제)Cu는 0.128nm의 원자 반지름과 FCC결정 구조를 가지며,
원자량은 63.5g/mol이다. 밀도를 계산하고 실측치과 비교하여라.
n = number of atoms/unit cell=4
A = atomic weight = 63.5g/mol
VC = Volume of unit cell = a3 for cubic 16R3√2
NA = Avogadro’s number = x 1023 atoms/mol
Cu의 실측치는 8.94g/cm3

10. Densities of Material Classes
In general
Graphite/
Metals/
Composites/
metals
>
ceramics
>
polymers
Ceramics/
Polymers
Alloys
fibers
Semicond 30
Why?
Metals have...
• close-packing
(metallic bonding)
• often large atomic masses 2
Magnesium
Aluminum
Steels
Titanium
Cu,Ni
Tin, Zinc
Silver, Mo
Tantalum
Gold, W
Platinum 10 G
raphite
Silicon
Glass -
soda
Concrete
Si nitride
Diamond
Al oxide
Zirconia
Ceramics have...
• less dense packing
• often lighter elements 5 3 4
(g/cm ) 3
Wood
AFRE *
CFRE
GFRE*
Glass fibers
Carbon
fibers A
ramid fibers H
DPE, PS
PP, LDPE PC
PTFE
PET
PVC
Silicone
Polymers have...
• low packing density
(often amorphous)
• lighter elements (C,H,O) r 2 1
Composites have...
• intermediate values
0.5
0.4
0.3
Data from Table B1, Callister 7e.

11. 3.6  Polymorphism
Two or more distinct crystal structures for the same material (allotropy/polymorphism)
Titanium : , -Ti
Carbon : diamond, graphite
BCC
FCC
1538ºC
1394ºC
912ºC
-Fe
-Fe
-Fe
liquid
iron system

12. 3.7 결정계 입방 육방 정방 삼방 사방 단사 3사

13. 3.8 Point Coordinates z x y a b c
000
111
3.9 결정학적 방향

14. 3.9 결정학적 방향(Miller지수) (a)[100] z (b)[110] (c)[111] (d)[112] (e)[212]
(f)[231] z x y
(a)
(b)
(c) b c
(1/2, 1/2, 1)
(1/2, 1, 1/2)
(d)
(e)
(f)
(2/3, 1, 1/3)
Lecture 2 ended here

15. HCP Crystallographic Directions]
uvtw [ ' w v u - a3 a1 a2 z
[0001] ) ' v u 2 ( 3 1 - u = ) ' u v 2 ( 3 1 - v = t = ) v u ( + - w = w '
[1210]
[2110]

16. 3.10 결정학적 면

17. 3.10 결정학적 면 특정 면의 밀러 지수를 구하는 법 면이 중심을 지나면 평행 이동
면이 3축과 만나는 점을 구함(x, y, z)
역수을 취함
공통 분모를 곱하여 최소의 정수를 구함
지수는 ( )안에 콤마 없이 표시 z x y a b c z x y a b c
교차점 : 1, 1, ∞
역 수 : 1, 1, 0
지수 : (110)
교차점 : 1/2, 1, 3/4
역 수 : 2, 1, 4/3
공통분모 3을 곱하면
지수 : (634) z x y a b c
교차점 : 1/2, ∞ , ∞
역 수 : 2, 0, 0
지수 : (200)

18. 3.11 선 밀도 Hexagonal 의 경우 a1 a2 a3 c 교차점 : 1  -1 1 역 수 : 1 0 -1 1 공통분모z
교차점 : 1 
역 수 :
공통분모
지수 : (1011)
a1 a2 a3 c
3.11 선 밀도
ex: linear density of Al in [110] direction  a = nm a
[110]
Number of atoms
Linear Density of Atoms  LD =
Unit length of direction vector

19. Planar Density of (100) Iron
Solution:  At T < 912C iron has the BCC structure.
2D repeat unit R 3 4 a =
(100)
Radius of iron R = nm =
Planar Density = a 2 1
atoms
2D repeat unit
nm2
12.1 m2
= 1.2 x 1019 R 3 4
area

20. Planar Density of (111) Iron
1 atom in plane/ unit surface cell 2 a
atoms in plane
atoms above plane
atoms below plane
2D repeat unit 3 h = a 2 1 =
nm2
atoms
7.0 m2
0.70 x 1019 3 2 R 16
Planar Density =
2D repeat unit
area

21. Section 3.16 - 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.  

22. X-Rays to Determine Crystal Structure
• Incoming X-rays diffract from crystal planes.
detector
“1”
incoming
X-rays
reflections must
“2”
be in phase for
“1”
a detectable signal
outgoing X-rays
extra l
“2”
distance q q
travelled
by wave “2”
spacing d
between
planes
X-ray
intensity
(from
detector) q c
d =
n l
2 sin
Measurement of critical angle, qc, allows computation of planar spacing, d.

23. X-Ray Diffraction Patternz x y a b c z x y a b c z x y a b c
(110)
(211)
Intensity (relative)
(200)
Diffraction angle 2q
Diffraction pattern for polycrystalline a-iron (BCC)

24. 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).
• 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.

25. 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.
• 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.

26. ANNOUNCEMENTS
Reading:
Core Problems:
Self-help Problems: