Materials Science and Engineering 1

Materials Science
and Engineering

Atomic Structure
& Interatomic
Bonding
CHAPTER 1
1

Fundamental Concepts
Protons
Neutrons
Electrons
Nucleons
Atom
1

e
e
e
e
e
e = 1.602 x10-19 C
-e
-e
-e
-e
-e
SHIFT 7 2 3
1

e = 1.602 x10-19 C
SHIFT 7 2 3
Atomic Number (Z)
the number of protons within
the nucleus.
equal to number of electrons in
complete atoms.
H
1
Hydrogen
U
92
Uranium
1

Mass Number (A)
is expressed as:
where N = number of neutrons
𝐴 = 𝑍 + 𝑁
Isotopes
atoms of the same element that
have different atomic masses.
𝑍
𝐴
𝐸 𝑛±
1

Sample Problem 1
An atom having a charge of +2e
has 54 electrons. How many
neutrons are there if the atom’s
mass number is 131?
75.00
1

Atomic Mass
is the mass of an atom.
𝐴 𝑚 = 𝑍𝑚 𝑝 + 𝑁𝑚 𝑛
(Am)
Atomic Mass Unit “amu”
is defined as 1
12
of the atomic
mass of Carbon-12.
1 𝑎𝑚𝑢
𝑎𝑡𝑜𝑚
= 1 𝑔
𝑚𝑜𝑙
Atomic Weight
is the weighted average of the
atomic masses of an atom’s
naturally occurring isotopes.
𝐴 𝑤 = 𝐴 𝑚 % 𝑎𝑡
(Aw)
SHIFT 7 0 1
SHIFT 7 0 2
1

Sample Problem 2
Carbon occurs in nature as a
mixture of 12C and 13C. The
isotopic mass of 13C is 13.003
amu. The atomic weight of
carbon is 12.011 amu. What is
the atom percentage of 12C in
natural carbon?
98.90%
1

Mole
the quantity of substance
corresponding to 6.022 x1023
atoms or molecules.
“mol”
Avogadro’s Number
𝑁𝐴 = 6.022 × 1023
(NA)
SHIFT 7 2 4
1

Electrons in Atoms
Bohr Atomic Model
an early atomic model, in
which electrons are assumed to
revolve around the nucleus in
discrete orbitals.
1

Electrons in Atoms
Wave-Mechanical Model
atomic model in which
electrons are treated as being
wave-like.
Orbital
Electron
Electron
Cloud
1.0
1

Electrons in Atoms
Quantum Numbers
a set of four numbers, the
values of which are used to
label possible electron states.
n l m s
Principal
is related to the distance of an
electron from the nucleus, or
its position.
(n)
K L M N O P Q
1 2 3 4 5 6 7Electron State
one of a set of discrete,
quantized energies that are
allowed for electrons.
1

Electrons in Atoms
Quantum Numbers
n l m s
Orbital
signifies the subshell and is
related to its shape.
(l)
s p d f
0 1 2 3
1

Electrons in Atoms
Quantum Numbers
n l m s
Magnetic
is related to the orientation of
orbitals within each subshells
in the presence of a magnetic
field.
(m)
s
p
d
f
0
-
1
0 1
-
2
-
1
0 1 2
-
3
-
2
-
1
0 1 2 3
1

Electrons in Atoms
Quantum Numbers
n l m s
Spin
is related to electron’s spin
moment.
(s)
−
𝟏
𝟐
𝟏
𝟐
𝑁𝑒 = 2𝑛2
1

Electrons in Atoms
Electron Configuration
the manner in which possible
electron states are filled with
electrons.
Aufbau Principle
states that electrons enter
orbitals of lowest energy first.
1s
2s
3s
4s
5s
6s
7s
2p
3p
4p
5p
6p
7p
3d
4d
5d
6d
4f
5f
Ground State
a normally filled electron
energy state from which an
electron excitation may occur.
Differentiating Electron
the last electron that enters an
orbital which makes an atom
configuration different from
other atoms.
Valence Electrons
the electrons in the outermost
occupied shell, which
participate in interatomic
bonding.
1

Electrons in Atoms
electrons.
Pauli Exclusion Principle
the postulate that for an
individual atom, at most two
electrons, which necessarily
have opposite spins, can
occupy the same state.
1

Electrons in Atoms
electrons.
Hund’s Rule
states that when orbitals are of
about the same energy, each
orbital is filled with parallel
spins before any of them is
filled with the opposite spin
electrons.
1

Sample Problem 3
Determine the set of quantum
numbers for the differentiating
electron of Oxygen.
2, 1, -1, -1/2
1

Electrons in Atoms
electrons.
Hund’s Rule
states that when orbitals are o
about the same energy, each
orbital is filled with parallel
spins before any of them is
filled with the opposite spin
electrons.
1

The Periodic Table
Periodic Table
the arrangement of chemical
elements with increasing
atomic number according to
the periodic variation in
electron structure.
1

Representative Elementss-blockp-block
**
*
118
Og
117
Ts
116
Lv
115
Mc
114
Fl
113
Nh
112
Cn
111
Rg
110
Ds
109
Mt
108
Hs
107
Bh
106
Sg
105
Db
104
Rf
103
Lr
102
No
101
Md
100
Fm
99
Es
98
Cf
97
Bk
96
Cm
95
Am
94
Pu
93
Np
92
U
91
Pa
90
Th
89
Ac
88
Ra
87
Fr
86
Rn
85
At
84
Po
83
Bi
82
Pb
81
Tl
80
Hg
79
Au
78
Pt
77
Ir
76
Os
75
Re
74
W
73
Ta
72
Hf
71
Lu
70
Yb
69
Tm
68
Er
67
Ho
66
Dy
65
Tb
64
Gd
63
Eu
62
Sm
61
Pm
60
Nd
59
Pr
58
Ce
57
La
56
Ba
55
Cs
54
Xe
53
I
52
Te
51
Sb
50
Sn
49
In
48
Cd
47
Ag
46
Pd
45
Rh
44
Ru
43
Tc
42
Mo
41
Nb
40
Zr
39
Y
38
Sr
37
Rb
36
Kr
35
Br
34
Se
33
As
32
Ge
31
Ga
30
Zn
29
Cu
28
Ni
27
Co
26
Fe
25
Mn
24
Cr
23
V
22
Ti
21
Sc
20
Ca
19
K
11
Na
12
Mg
4
Be
3
Li
1
H
18
Ar
10
Ne
2
He
17
Cl
9
F
16
S
15
P
14
Si
13
Al
5
B
6
C
7
N
8
O
*
**
Alkali
Alkaline Earth
Transition
Lanthanoid
Actinoid
Post-transition
Metalloid
Nonmetal
Halogen
Noble Gas
1
2
3
4
5
6
7
Period
IIA
IA
IIIA VAIVA VIIAVIA
VIIIA
IIIB IB IIBIVB VB VIB VIIB
VIIIB
1

f-blockTransition ElementsInner Transition Elements
**
*
118
Og
117
Ts
116
Lv
115
Mc
114
Fl
113
Nh
112
Cn
111
Rg
110
Ds
109
Mt
108
Hs
107
Bh
106
Sg
105
Db
104
Rf
103
Lr
102
No
101
Md
100
Fm
99
Es
98
Cf
97
Bk
96
Cm
95
Am
94
Pu
93
Np
92
U
91
Pa
90
Th
89
Ac
88
Ra
87
Fr
86
Rn
85
At
84
Po
83
Bi
82
Pb
81
Tl
80
Hg
79
Au
78
Pt
77
Ir
76
Os
75
Re
74
W
73
Ta
72
Hf
71
Lu
70
Yb
69
Tm
68
Er
67
Ho
66
Dy
65
Tb
64
Gd
63
Eu
62
Sm
61
Pm
60
Nd
59
Pr
58
Ce
57
La
56
Ba
55
Cs
54
Xe
53
I
52
Te
51
Sb
50
Sn
49
In
48
Cd
47
Ag
46
Pd
45
Rh
44
Ru
43
Tc
42
Mo
41
Nb
40
Zr
39
Y
38
Sr
37
Rb
36
Kr
35
Br
34
Se
33
As
32
Ge
31
Ga
30
Zn
29
Cu
28
Ni
27
Co
26
Fe
25
Mn
24
Cr
23
V
22
Ti
21
Sc
20
Ca
19
K
11
Na
12
Mg
4
Be
3
Li
1
H
18
Ar
10
Ne
2
He
17
Cl
9
F
16
S
15
P
14
Si
13
Al
5
B
6
C
7
N
8
O
*
**
Alkali
Alkaline Earth
Transition
Lanthanoid
Actinoid
Post-transition
Metalloid
Nonmetal
Halogen
Noble Gas
1
2
3
4
5
6
7
Period
IIA
IA
IIIA VAIVA VIIAVIA
VIIIA
VIIIB
d-block 1

Diatomic Elements
**
*
112
Cn
111
Rg
110
Ds
109
Mt
108
Hs
107
Bh
106
Sg
105
Db
104
Rf
103
Lr
102
No
101
Md
100
Fm
99
Es
98
Cf
97
Bk
96
Cm
95
Am
94
Pu
93
Np
92
U
91
Pa
90
Th
89
Ac
88
Ra
87
Fr
86
Rn
85
At
84
Po
83
Bi
82
Pb
81
Tl
80
Hg
79
Au
78
Pt
77
Ir
76
Os
75
Re
74
W
73
Ta
72
Hf
71
Lu
70
Yb
69
Tm
68
Er
67
Ho
66
Dy
65
Tb
64
Gd
63
Eu
62
Sm
61
Pm
60
Nd
59
Pr
58
Ce
57
La
56
Ba
55
Cs
54
Xe
53
I
52
Te
51
Sb
50
Sn
49
In
48
Cd
47
Ag
46
Pd
45
Rh
44
Ru
43
Tc
42
Mo
41
Nb
40
Zr
39
Y
38
Sr
37
Rb
36
Kr
35
Br
34
Se
33
As
32
Ge
31
Ga
30
Zn
29
Cu
28
Ni
27
Co
26
Fe
25
Mn
24
Cr
23
V
22
Ti
21
Sc
20
Ca
19
K
11
Na
12
Mg
4
Be
3
Li
1
H
18
Ar
10
Ne
2
He
17
Cl
9
F
16
S
15
P
14
Si
13
Al
5
B
6
C
7
N
8
O
*
**
Alkali
Alkaline Earth
Transition
Lanthanoid
Actinoid
Post-transition
Metalloid
Nonmetal
Halogen
Noble Gas
1
2
3
4
5
6
7
Period
IIA
IA
IIIA VAIVA VIIAVIA
VIIIA
VIIIB
118
Og
117
Ts
116
Lv
115
Mc
114
Fl
113
Nh
Noble MetalsRefractory Metals 1

The Periodic Table
Periodic Table
the arrangement of chemical
elements with increasing
atomic number according to
the periodic variation in
electron structure.
Electropositive
the tendency of an atom to
release valence electrons.
Electronegative
the tendency of an atom to
accept valence electrons.
1

Bonding Forces and Energies
FA
FR
r
𝐹 𝑁 = 𝐹𝐴 + 𝐹𝑅
ro
ro corresponds to the
separation distance at the
minimum of the potential
energy curve.
1

ER
EA
r
ro corresponds to the
separation distance at the
minimum of the potential
energy curve.
Eo
Bonding Energy
the energy required to
separate two atoms to an
infinite separation.𝐸 𝑁 = 𝐸𝐴 + 𝐸 𝑅
(Eo)
𝐸 𝑁 = −
𝐴
𝑟
+
𝐵
𝑟 𝑛
1

Sample Problem 4
For a given ion pair the following
relations are given:
-5.32 eV
𝐸𝐴 = −
1.436
𝑟
𝐸 𝑅 = −
7.32 × 10−6
𝑟8
where EA and ER in eV and r in
nm.
Determine the magnitude of the
bonding energy between the
two ions
1

Primary Bonds
interatomic bonds that are
relatively strong and for which
bonding energies are relatively
large.
Secondary Bonds
interatomic and intermolecular
bonds that are relatively weak
and for which bonding
energies are relatively small.
Chemical Physical
1

Primary Atomic Bonds
Ionic Bonds
a coulombic interatomic bond
that exist between two
adjacent and oppositely
charged ions.
11
Na
17
Cl
18
Ar
10
Ne
Electrovalent
1

Ionic Bonds
charged ions.
11
Na
17
Cl
18
Ar
10
Ne
Coulombic Force
a force between charged
particles like ion; attractive if
opposite charged
NaCl (salt)
Non-directional
1

Ionic Bonds
charged ions.
11
Na
17
Cl
18
Ar
10
Ne
Attractive Energy
𝐸𝐴 = −
𝑉1 𝑉2 𝑒2
4𝜋𝜖 𝑜
1
𝑟
SHIFT 7 2 3
IA VIIA
SHIFT 7 3 2
1

**
*
112
Cn
111
Rg
110
Ds
109
Mt
108
Hs
107
Bh
106
Sg
105
Db
104
Rf
103
Lr
102
No
101
Md
100
Fm
99
Es
98
Cf
97
Bk
96
Cm
95
Am
94
Pu
93
Np
92
U
91
Pa
90
Th
89
Ac
88
Ra
87
Fr
86
Rn
85
At
84
Po
83
Bi
82
Pb
81
Tl
80
Hg
79
Au
78
Pt
77
Ir
76
Os
75
Re
74
W
73
Ta
72
Hf
71
Lu
70
Yb
69
Tm
68
Er
67
Ho
66
Dy
65
Tb
64
Gd
63
Eu
62
Sm
61
Pm
60
Nd
59
Pr
58
Ce
57
La
56
Ba
55
Cs
54
Xe
53
I
52
Te
51
Sb
50
Sn
49
In
48
Cd
47
Ag
46
Pd
45
Rh
44
Ru
43
Tc
42
Mo
41
Nb
40
Zr
39
Y
38
Sr
37
Rb
36
Kr
35
Br
34
Se
33
As
32
Ge
31
Ga
30
Zn
29
Cu
28
Ni
27
Co
26
Fe
25
Mn
24
Cr
23
V
22
Ti
21
Sc
20
Ca
19
K
11
Na
12
Mg
4
Be
3
Li
1
H
18
Ar
10
Ne
2
He
17
Cl
9
F
16
S
15
P
14
Si
13
Al
5
B
6
C
7
N
8
O
*
**
Alkali
Alkaline Earth
Transition
Lanthanoid
Actinoid
Post-transition
Metalloid
Nonmetal
Halogen
Noble Gas
1
2
3
4
5
6
7
Period
IIA
IA
IIIA VAIVA VIIAVIA
VIIIA
VIIIB
118
Og
117
Ts
116
Lv
115
Mc
114
Fl
113
Nh
Ionic Pairs 1

Sample Problem 5
Calculate the attractive energy
between a Ca+2 and an O-2 ion,
the centers of which are
separated by a distance of 12.5 Å
(units in eV)
13.83 eV
1 J = 1.602 x10-19 eV
1 nm = 10 Å
1

Ionic Bonds
ionic materials are
characteristically hard and
brittle, furthermore, electrically
and thermally insulative.
the predominant bonding in
ceramic materials is ionic.
1

Covalent Bonds
an interatomic bond that is
formed by the sharing of
electrons between neighboring
atoms.
6
C
1
H
10
Ne
Molecular
1

Covalent Bonds
an interatomic bond that is
formed by the sharing of
electrons between neighboring
atoms.
6
C
1
H
2
He
10
Ne
CH4 (methane)
Directional
Number of Covalent Bonds
8 − 𝑉
Percent Ionic Character
%𝑖𝑐 = 1 − 𝑒−
𝑋1−𝑋2
2
4 × 100%
1

**
*
112
Cn
111
Rg
110
Ds
109
Mt
108
Hs
107
Bh
106
Sg
105
Db
104
Rf
103
Lr
102
No
101
Md
100
Fm
99
Es
98
Cf
97
Bk
96
Cm
95
Am
94
Pu
93
Np
92
U
91
Pa
90
Th
89
Ac
88
Ra
87
Fr
86
Rn
85
At
84
Po
83
Bi
82
Pb
81
Tl
80
Hg
79
Au
78
Pt
77
Ir
76
Os
75
Re
74
W
73
Ta
72
Hf
71
Lu
70
Yb
69
Tm
68
Er
67
Ho
66
Dy
65
Tb
64
Gd
63
Eu
62
Sm
61
Pm
60
Nd
59
Pr
58
Ce
57
La
56
Ba
55
Cs
54
Xe
53
I
52
Te
51
Sb
50
Sn
49
In
48
Cd
47
Ag
46
Pd
45
Rh
44
Ru
43
Tc
42
Mo
41
Nb
40
Zr
39
Y
38
Sr
37
Rb
36
Kr
35
Br
34
Se
33
As
32
Ge
31
Ga
30
Zn
29
Cu
28
Ni
27
Co
26
Fe
25
Mn
24
Cr
23
V
22
Ti
21
Sc
20
Ca
19
K
11
Na
12
Mg
4
Be
3
Li
1
H
18
Ar
10
Ne
2
He
17
Cl
9
F
16
S
15
P
14
Si
13
Al
5
B
6
C
7
N
*
**
Alkali
Alkaline Earth
Transition
Lanthanoid
Actinoid
Post-transition
Metalloid
Nonmetal
Halogen
Noble Gas
1
2
3
4
5
6
7
Period
IIA
IA
IIIA VAIVA VIIAVIA
VIIIA
VIIIB
118
Og
117
Ts
116
Lv
115
Mc
114
Fl
113
Nh
Covalent Pairs
8
O
1

Sample Problem 6
Determine the percent ionic
character of the ceramic Fe3O4
given that 1.8 and 3.5 are the
electronegativities of iron and
oxygen respectively.
51.45%
1

Covalent Bonds
covalent materials can be hard
as in diamond or may be very
weak as with bismuth.
the predominant bonding in
polymeric materials is covalent.
1

Metallic Bonds
an interatomic bond involving
the sharing of nonlocalized
valence electrons.
18
Ar
26
Fe
36
Kr
Ion Cores
Sea of
Electrons
Non-directional
1

Metallic Bonds
these bonds may be weak
(mercury) or strong (tungsten)
metallic materials are good
conductors of both electricity
and heat.
1

Secondary Atomic Bonds
Fluctuating Induced Dipole
Bonds
bonds that exist between
atoms having short-lived
distortions of its electrical
symmetry caused by constant
vibrational motion.
Van Der Waals Bonding
exists when there is some
separation of positive and
negative portions of an atom
or molecule.
18
Ar
1

Fluctuating Induced Dipole
Bonds
bonds that exist between
atoms having short-lived
distortions of its electrical
symmetry caused by constant
vibrational motion.
London Forces
1

a molecule in which there exist
a permanent electric dipole
moment by virtue of the
asymmetrical distribution of
positively and negatively
charged regions.
Polar Molecule – Induced Dipole
Bonds
bonds that exist between polar
molecules and the affected
nonpolar ones.
1
H
17
Cl
1

Polar Molecule – Induced Dipole
Bonds
molecules and the affected
nonpolar ones.
1

Permanent Dipole Bonds
molecules.
Dipole ForcesHydrogen Bonding
the strongest van der waals
bond.
1
H
9
F
7
N
8
O
1

The Structure of
Crystalline Solids
CHAPTER 2
2

Atomic Hard Sphere Model
a model in which spheres
representing nearest-neighbor
atoms touch one another.
Solid Materials:
Crystalline Solids
Amorphous Solids
2

Crystal StructureUnit Cell
Crystalline Solid
is one in which the atoms are
situated in a repeating or
periodic array over large
atomic distances.
Amorphous Solid
long-range atomic order is
absent.
the manner in which atoms,
ions, or molecules are spatially
arranged.
the basic structural unit of a
crystal structure.
2

means a three-dimensional
array of points coinciding with
atom positions
Unit Cell
the basic structural unit of a
crystal structure.
Crystalline Solid
is one in which the atoms are
situated in a repeating or
periodic array over large
atomic distances.
Lattice
2

Crystal Systems
Lattice Parameters
the combination of unit cell
edge lengths and interaxial
angles that defines the unit cell
geometry.
y
z
x
ac
b
α
β
γCrystal System
a scheme in which crystal
structures are classified by unit
cell geometry.
2

Crystal Systems
Crystal System: Cubic
a b c α β γ
x x x 90o 90o 90o
x
x
x
θ θ θ
x
x
x
Rhombohedral
θ
θ
θ
2

Crystal Systems
Crystal System: Tetragonal
a b c α β γ
x x x 90o 90o 90o
x
x
x
y
y
2

120o
y
y
Crystal Systems
Crystal System: Hexagonal
a b c α β γ
x x 90o 90o 90o
x
x120o
Orthorhombic
2

Orthorhombic
90o
y
y
Crystal Systems
Crystal System:
a b c α β γ
x x 90o 90o
x
x
z
z
2

z 90o
y
y
Crystal Systems
Crystal System:
a b c α β γ
x 90o 90o
z
x
θ
Monoclinic
θ
2

z 90o
y
y
Crystal Systems
Crystal System:
a b c α β γ
x 90o
z
x
θ
Triclinic
θ
φ
φ
2

z 90o
y
y
Crystal Systems
Crystal System:
a b c α β γ
x
z
x
θ
Triclinic
φ
θ
φ
ψ
ψ
2

Metallic Crystal Structures
Pointers:
- the hard spheres represents
the ion cores of metals.
- cubic and hexagonal crystal
systems are the predominant
structures in common metals
Coordination Number
the number of atomic or ionic
nearest neighbors.
Atomic Packing Factor
the fraction of the unit cell
occupied by atoms.
(APF)(Nc)
2

the fraction of the unit cell
occupied by atoms.
Pointers:
- the hard spheres represents
the ion cores of metals.
- cubic and hexagonal crystal
systems are the predominant
structures in common metals
𝐴𝑃𝐹 =
𝑁 4
3
𝜋𝑟3
𝑉𝑢
(APF)
2

Face-Centered Cubic
a crystal structure found in
some of the common
elemental metals; where within
the cubic unit cell, atoms are
located at all corner and face-
centered positions.
(FCC)
2

Face-Centered Cubic (FCC)
Face-Centered AtomsCorner Atoms
Parameter Value
Coordination Number
Edge Length “f(r)”
12
2

Sample Problem 7
Determine the APF of FCC
structure.
0.74
2

Face-Centered Cubic (FCC)
Parameter Value
Coordination Number
12
0.74
4𝑟 2
FCC Metals at Room Temperature
Aluminum (Al) Nickel (Ni)
Copper (Cu) Platinum (Pt)
Gold (Au) Silver (Ag)
Lead (Pb)
2

Body-Centered Cubic
a common crystal structure
found in some elemental
metals; where within the cubic
unit cell, atoms are located at
corner and cell center
positions.
(BCC)
2

Body-Centered Cubic (BCC)
Body-Centered AtomCorner Atoms
Parameter Value
Coordination Number
8
2

Sample Problem 8
Determine the APF of BCC
structure.
0.68
2

Body-Centered Cubic (BCC)
Parameter Value
Coordination Number
8
0.68
4𝑟 3
BCC Metals at Room Temperature
Chromium (Cr) Tungsten (W)
Iron (Fe)
Molybdenum (Mo)
Tantalum (Ta)
2

Hexagonal Close-Packed
a crystal structure found for
some metals. It is of hexagonal
geometry and is generated by
the stacking of close-packed
planes of atoms.
(HCP)
2

Midplane Interior Atoms
Center Face AtomsCorner Atoms
Parameter Value
Coordination Number
Height/Edge Length
12
Hexagonal Close-Packed (HCP)
0.74
1.6330
HCP Metals at Room Temperature
Cadmium (Cd)
Cobalt (Co)
Titanium (Ti)
Zinc (Zn)
2

Simple Cubic
a crystal structure, which
Polonium (Po) only have,
consists of only corner atoms.
2

Simple Cubic
Parameter Value
Coordination Number
8
0.52
4𝑟 4
Simple Cubic Metal
Polonium (Po)
2r
2

Density Computations
Theoretical Density
the density of a given unit cell
which intrinsically represents
the whole solid with no regard
to imperfections. 𝜌𝑡 = 𝑚 𝑎
𝑉𝑢
= 𝐴 𝑤 𝑛
𝑉𝑢 =
𝐴 𝑤
𝑁
𝑁 𝐴
𝑉𝑢
𝜌𝑡 = 𝑁𝐴 𝑤
𝑉𝑢 𝑁 𝐴
= 𝐴𝑃𝐹 𝐴 𝑤
4
3
𝜋𝑟3 𝑁 𝐴
(ρt)
2

Sample Problem 9
The unit cell for a uranium has
orthorhombic symmetry, with a,
b, and c lattice parameters of
0.286, 0.587, and 0.495 nm,
respectively. If its density, atomic
weight, and atomic radius are
19.05 g/cm3, 238.03 g/mol, and
0.1385 nm, respectively,
compute the atomic packing
factor.
0.54
2

Density Computations
Theoretical Density
the density of a given unit cell
which intrinsically represents
the whole solid with no regard
to imperfections. 𝜌𝑡 = 𝑁𝐴 𝑤
𝑉𝑢 𝑁 𝐴
= 𝐴𝑃𝐹 𝐴 𝑤
4
3
𝜋𝑟3 𝑁 𝐴
(ρt)
2

Polymorphism and Allotropy
Polymorphism
the ability of a solid material to
exist in more than one form or
crystal structure.
Allotropy
the term used for
polymorphism of elemental
solids.
Graphite Diamond
2

Crystallographic
Points, Directions, and Planes
Point Coordinate
set of 3 numbers that specifies
the location of a point inside a
unit cell as fractional multiples
of the edge lengths. y
z
x
ac
b
qa
rb
sc
q r s
2

Sample Problem 10
Locate the body-centered atom
of Niobium BCC having an
atomic radius of 0.143 nm.
𝟏
𝟐
𝟏
𝟐
𝟏
𝟐
2

qa
rb
sc
Crystallographic
Crystallographic Direction
is a vector defined by a set of 3
numbers that specifies its
direction as reduced lowest set
of integers.
y
z
x
[u v w]
y
z
x
ac
b
ua
vb
wc
2

Sample Problem 11
Determine the indices for the
direction shown below.
𝟑 𝟔 𝟏
y
z
x
1
3
1
2
,
1
2
2

Crystallographic
is a vector defined by a set of 3
numbers that specifies its
of integers.
[u v w]
Antiparallel Direction
y
z
x
y
z
x
ac
b
ua
vb
wc
2

<u v
w>
Crystallographic
is a vector defined by a set of
3 numbers that specifies its
of integers.
[u v w]
Family
x y
z
[0 1 0]
[0 0 1]
[-1 0 0]
<1 0 0>
2

[u’v’w’]
Crystallographic
is a vector defined by a set of
3 numbers that specifies its
of integers.
Miller–Bravais Coordinate
System
z
a1
a2
a3
Basal Plane
[u v t w]
𝑢 = 1
3
2𝑢′−𝑣′
𝑣 = 1
3
2𝑣′−𝑢′
𝑡 = − 𝑢+𝑣
𝑤 = 𝑤′
2

Sample Problem 12
Determine the indices for the
direction shown below.
𝟏 𝟏 𝟐 𝟑
z
a1
a2
a3
2

Crystallographic
Crystallographic Plane
is a plane defined by a set of 3
numbers called Miller Indices
that specifies its position inside
the cell as reciprocals of its
intersections with the axes.
(h k l)
y
z
x
ac
b
ha
kb
lc
2

Sample Problem 13
Determine the Miller indices for
the plane shown below.
𝟎 𝟏 𝟐
y
z
x
2

(h k l){h k l}
Crystallographic
Family
x y
z
(0 1 2)
(2 0 -1)
{0 1 2}
(1 -2 0)
2

(h’k’l’)
Crystallographic
Hexagonal Crystals
z
a1
a2
a3
(h k i l)
𝑖 = − ℎ+𝑘
𝑙 = 𝑙′
ℎ = ℎ′
𝑘 = 𝑘′
2

Sample Problem 14
Determine the Miller indices for
the plane shown below.
𝟏 𝟏 𝟐 𝟐
z
a1
a2
a3
2

Linear and Planar Densities
Linear Density
defined as the number of
atoms per unit length whose
centers lie on the direction
vector for a specific
crystallographic direction.
(LD)
𝐿𝐷 =
𝑁
𝐿
2

Sample Problem 15
Determine the linear density of
Molybdenum BCC [101] (for Mo:
R=0.1363 nm)
2.25 nm-1
2

Linear and Planar Densities
Linear Density
atoms per unit length whose
centers lie on the direction
vector for a specific
crystallographic direction.
(LD)
𝐿𝐷 =
𝑁
𝐿
Planar Density
atoms per unit area that are
centered on a particular
crystallographic plane.
𝑃𝐷 =
𝑁
𝐴
2

Sample Problem 16
Determine the planar density of
Copper FCC (110) (for Cu:
R=0.1280 nm)
10.79 nm-2
2

Close-Packed Crystal Structures
Close-Packed
Planes of Atoms
planes having a maximum
atom or sphere packing
density.
0.74 is the most efficient
packing of equal-sized spheres
or atoms.
2
FCC HCP

Close-Packed
Planes of Atoms
planes having a maximum
atom or sphere packing
density.
0.74 is the most efficient
packing of equal-sized spheres
or atoms.
2
B
B B B B
B
B
B B B
BB B B
B B B B
C C C
C C C C
C
C
A A A A
A A A A A
A A A A A A

2
A
A
B
C
A
{1 1 1}
B
C
A
FCC
A
B
A
{0 0 0 1}
B
HCP

Crystalline Materials
Single Crystal
a crystalline solid for which the
periodic and repeated atomic
pattern extends throughout its
entirety without interruption.
2
Anisotropy
the directionality of properties.

Crystalline Materials
Polycrystalline
is a collection of multiple
crystals with different
orientations.
2
Isotropy
having identical properties in
all crystallographic directions.
Grain
Grain
Boundary

X-ray Diffraction
Diffraction
constructive interference of
waves that are scattered by
atoms of a crystal.
2
Constructive
Interference
Destructive
Interference
X-ray
a form EM radiation that have
high energy and short
wavelength (order of atomic
spacings)

X-ray Diffraction
Diffraction
constructive interference of
waves that are scattered by
atoms of a crystal.
2
X-ray
a form EM radiation that have
high energy and short
wavelength (order of atomic
spacings)
Bragg’s Law
a relationship that stipulates
the condition for diffraction by
a set of crystallographic planes.
𝑛𝜆 = 2 𝑑ℎ𝑘𝑙 sin 𝜃

X-ray Diffraction
2
Bragg’s Law
a relationship that stipulates
the condition for diffraction by
a set of crystallographic planes.
𝑛𝜆 = 2 𝑑ℎ𝑘𝑙 sin 𝜃
𝑑ℎ𝑘𝑙 = 𝑎
ℎ2+𝑘2+𝑙2
(hkl)
(hkl)
θ θ
dhkl
λ
θθ

Sample Problem 17
For BCC iron, compute the lattice
parameter (a) in angstrom. It has
a diffraction angle of 124.26o for
the (220) set of planes. Also,
assume that monochromatic
radiation having λ = 0.179 nm is
used, and the order of reflection
is 1.
2.86 Å
2

X-ray Diffraction
Diffractometer
is an apparatus used to
determine the angles at which
diffraction occurs for powdered
specimens
2
TS
C
θ
2θ

0.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0
Diffraction Angle
X-ray Diffraction
Diffractometer
is an apparatus used to
determine the angles at which
diffraction occurs for powdered
specimens
2
TS
θ
2θ
C
(111)
(200)
(220)
(311)
(222)
(400)
Intensity

Non Crystalline Solids
Non Crystalline Solid
their atomic structure
resembles that of a liquid.
2
rapidly cooling through the
freezing temperature favors its
formation.

Materials Science and Engineering
– An Introduction (7th Edition)
by William D. Callister, Jr.
Materials Science
and Engineering
Source

Materials Science and Engineering 1

Recommended

Recommended

More Related Content

What's hot

What's hot (20)

Similar to Materials Science and Engineering 1

Similar to Materials Science and Engineering 1 (20)

Recently uploaded

Recently uploaded (20)

Materials Science and Engineering 1