M
a
t
e
r
i
a
l
s
C
r
i
b
S
h
e
e
t
Mike Bebjak
Chapter 3 – Crystal Structures
Constants & Conversions
giga
G
10
9
milli
m
-3
mega
M
6
micro
µ
-6
kilo
k
3
nano
n
-9
centi
c
-2
pico
p
-12
k = 1.38× 10
-23
J/ atom
⋅
K = 8.62× 10
-5
eV/ atom
⋅
K
R = 8.314 J/ mol
⋅
K
N
A
= 6.023× 10
23
atoms/ mol
e = 1.602× 10
-19
C
m
e
= 9.11× 10
-31
kg
h = 6.63× 10
-34
J
⋅
s
Atomic Structures
Atomic packing factor (APF) =
atoms in unit cell
unit cell
/
VV
CN = number of atoms that each given atom touches in a unit cell
FCC
– close packed
ABCABC sequence
22
aR
=
APF = 0.74
CN = 12
4 atoms/ cell
BCC
– not close packed
4
3
=
APF = 0.68
CN = 8
2 atoms/ cell
HCP
ABABAB
sequence
=
Braggs’ Law:
2s
i
n
nd
λ
θ
=
, Density:
ii
cA
nA
nA
VN
ρ
==
∑
, Planar spacing:
222
2sin
na
d
hkl
+
+
. Miller Indices
: Steps – lines: 1. Start at any cell corner; 2.
Find coordinates of vectors, subtract; 3. Multiply by common factor, planes: 1. Find intercepts in a, b, c; 2. Find reciprocals 1/ a, 1/ b, 1/ c; 3. Multiply to make
integer.
Notations –
lines: [ u v w] , planes: (h k l), families of planes: { 1 1 1} family includes (1 1 1),
(1 1 1)
, etc.
uu
=−
Chapter 4 - I m perfections
Point Defects
- Vacancies:
Q
V
RT
V
NN
e
−
=
(N is the total # of atomic sites, Q the energy required to form a vacancy) Thermodynamics: perfect crystal
⇒
S= 0, add
n vacancies to N atoms
⇒
n
↑
⇒
S
↑
⇒
G
↓
(G =
∆
H - T
∆
S). Will create vacancies until G is a minimum. Impurities
– Substitutional: replace an atom; Interstitial:
lie between atoms;
Q
sol
kT
sol
Xe
−
=
. Linear Defects
– Dislocations: edge (extra plane of atoms), screw (shear distortion of lattice)
↑
dislocations =
↓
S
⇒
thermodynamically unstable. Planar Defects
– Grain boundaries:
1
2
n
N
−
=
(N – avg. # of grains/ sq. inch, n – grain size # ). Solidification Process
: 1. Nucleation;
2. Growth; 3. Impingement
⇒
grain boundaries form. Metals have more vacancies than self-interstitials.
Conditions:
⇒
32
0 J/m
0 J/m
VS
GG
∆<
G
V
RT
r
Pe
∆
−
=
(probability of getting a critical radius r
*
),
⇒
1/
tp
∼
4
4
3
Ts
o
l
s
Gr
π
πγ
∆=
+
,
*
2
V
r
G
γ
−
=
∆
'
11
1
''
1
12
2 2
mC
A
C
mm
CA CA
+
+
'
2
1
21
m
n
CA
C
nn
C
AC
A
++
'
1
1
C
C
CC
=
+
100
avg
ρρ
+
+
+
100
100
avg
A
AA
+
+
(C-wt% , C’-at% , C’’-conc.)
Chapter 5 - Diffusion
Types
–
Interdiffusion
: atoms of one metal diffuse into another,
Self-diffusion
: atoms of metal exchange positions
Mechanisms
Vacancy Diffusion
: substitutional atoms diffuse into vacancies (used by interdiffusion & self-diffusion).
Interstitial Diffusion
: atoms move into
interstitials locations (gases, much faster than vacancy). Steady-state Diffusion
– diffusion flux doesn’t change with time:
1
M
dM
J
At
A
dt
⎛⎞
==⎜
⎝⎠
⎟
, Fick’s 1
st
Law:
dC
JD
dx
(D – diffusion coefficient),
(q – amount diffusing),
qJ
A
=
0
Q
RT
D
De
−
=
Chapter 6 – Mechanical Properties of Metals
Types of stresses
: Tensile, Compressive, Shear, Biaxial Tension Compression, Hydrostatic
Stress:
0
F
A
σ
=
, Strain:
0
l
l
ε
∆
=
, Shear stress:
0
F
A
τ
=
, Shear strain:
0
tan
l
l
∆
E
=
G
=
, Physics of elasticity:
2
2
FU
E
r
r
∂∂
=
∂
∂
∼∼
, Energy stored:
2
0
0
1
V
WV
E
σε
, Poisson’s ratio:
x
z
υ
(
)
EG
=+
. Work hardening
: after they yield, metals get harder to deform
Hardness tests
:
indent