Materials Crib Sheet
Mike Bebjak
Chapter 3 – Crystal Structures
Constants & Conversions
giga
G
10
9
milli
m
10
3
mega
M
10
6
micro
µ
10
6
kilo
k
10
3
nano
n
10
9
centi
c
10
2
pico
p
10
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
/
V
V
CN = number of atoms that each given atom touches in a unit cell
FCC
– close packed
ABCABC sequence
2
2
a
R
=
APF = 0.74
CN = 12
4 atoms/ cell
BCC
– not close packed
4
3
a
R
=
APF = 0.68
CN = 8
2 atoms/ cell
HCP
– close packed
ABABAB
sequence
2
2
a
R
=
APF = 0.74
CN = 12
Braggs’ Law:
2
sin
n
d
λ
θ
=
, Density:
i
i
c
A
c
A
n A
nA
V N
V N
ρ
=
=
∑
, Planar spacing:
2
2
2
2sin
n
a
d
h
k
l
λ
θ
=
=
+
+
. 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.
u
u
= −
Chapter 4  I m perfections
Point Defects
 Vacancies:
Q
V
RT
V
N
Ne
−
=
(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
X
e
−
=
. 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 selfinterstitials.
Conditions:
⇒
3
2
0 J/m
0 J/m
V
S
G
G
∆
<
∆
<
G
V
RT
r
P
e
∆
−
=
(probability of getting a critical radius r
*
),
⇒
1 /
t
p
∼
3
2
4
4
3
T
sol
s
G
r G
r
π
π
γ
∆
=
+
,
*
2
V
r
G
γ
−
=
∆
'
1
1
1
'
'
1
1
2
1
1
2
2
m
C A
C
m
m
C A
C A
=
=
+
+
,
'
1
1
2
1
1
2
2
1
1
2
m
m
m
n
C A
C
n
n
C A
C A
=
=
+
+
,
''
1
1
1
2
1
2
C
C
C
C
ρ
ρ
=
+
,
'
'
1
1
2
2
'
'
1
2
1
1
2
2
1
2
1
2
100
avg
C A
C A
C
C
C A
C A
ρ
ρ
ρ
ρ
ρ
+
=
=
+
+
,
'
'
1
1
2
2
1
2
1
2
100
100
avg
C A
C A
A
C
C
A
A
+
=
=
+
(Cwt% , C’at% , C’’conc.)
Chapter 5  Diffusion
Types
–
Interdiffusion
: atoms of one metal diffuse into another,
Selfdiffusion
: atoms of metal exchange positions
Mechanisms
–
Vacancy Diffusion
: substitutional atoms diffuse into vacancies (used by interdiffusion & selfdiffusion).
Interstitial Diffusion
: atoms move into
interstitials locations (gases, much faster than vacancy). Steadystate Diffusion
– diffusion flux doesn’t change with time:
1
M
dM
J
At
A
dt
⎛
⎞
=
=
⎜
⎝
⎠
⎟
, Fick’s 1
st
Law:
dC
J
D
dx
= −
(D – diffusion coefficient),
(q – amount diffusing),
q
JA
=
0
Q
RT
D
D e
−
=
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
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 Spring '11
 Ghosh
 Force, Interstitial defect

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