# Crystalline Structures

Topics: Crystal system, Atomic packing factor, Crystal Pages: 24 (1388 words) Published: February 19, 2014
CHAPTER 3: CRYSTAL
STRUCTURES & PROPERTIES
• How do atoms assemble into solid structures?
(for now, focus on metals)
• How does the density of a material depend on
its structure?
• When do material properties vary with the
sample (i.e., part) orientation?

Chapter 3- 1

ENERGY AND PACKING
• Non dense, random packing

Energy
typical neighbor
bond length

typical neighbor
bond energy

• Dense, regular packing

r

Energy
typical neighbor
bond length

typical neighbor
bond energy

r

Dense, regular-packed structures tend to have
lower energy.
Chapter 3- 2

MATERIALS AND PACKING
Crystalline materials...
• atoms pack in periodic, 3D arrays
• typical of: -metals
-many ceramics
-some polymers

crystalline SiO2
Adapted from Fig. 3.22(a),
Callister 7e.

Si

Oxygen

Noncrystalline materials...
• atoms have no periodic packing
• occurs for: -complex structures
-rapid cooling
"Amorphous" = Noncrystalline

noncrystalline SiO2
Adapted from Fig. 3.22(b),
Callister 7e.
Chapter 3- 3

METALLIC CRYSTALS
• tend to be densely packed.
• have several 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.

• have the simplest crystal structures.
We will look at three such structures...

Chapter 3- 4

Chapter 3-

α=β==90o

Cubic

a=b=c

Hexagonal

a=b≠c α=β=90o,=120o

Tetragonal

a=b≠c

Rhombohedral a=b=c

α=β==90o
α=β=≠ 90o

Orthorrhombic a≠b≠c α=β==90o
Monoclinic
Triclinic

a≠b≠c α==90o,β ≠ 90o
a≠b≠c α≠β≠≠90o

Chapter 3-

SIMPLE CUBIC STRUCTURE (SC)
• Rare due to poor packing (only Po has this structure)
• Close-packed directions are cube edges.
• Coordination # = 6
(# nearest neighbors)

(Courtesy P.M. Anderson)
Chapter 3- 5

ATOMIC PACKING FACTOR
APF =

Volume of atoms in unit cell*
Volume of unit cell

*assume hard spheres
• APF for a simple cubic structure = 0.52

atoms
unit cell

a
R=0.5a

APF =

volume
atom
4
(0.5a) 3
1
3
a3

close-packed directions
contains 8 x 1/8 =
1 atom/unit cell
Adapted from Fig. 3.23,
Callister 7e.

volume
unit cell
Chapter 3- 6

BODY CENTERED CUBIC
STRUCTURE (BCC)
• Close packed directions are cube diagonals.
--Note: All atoms are identical; the center atom is shaded
differently only for ease of viewing.

• Coordination # = 8

Adapted from Fig. 3.2,
Callister 7e.
(Courtesy P.M. Anderson)

Chapter 3- 7

Chapter 3-

ATOMIC PACKING FACTOR: BCC
• APF for a body-centered cubic structure = 0.68

Close-packed directions:
length = 4R
= 3a

R
Fig. 3.2,
Callister 7e.

Unit cell c ontains:
1 + 8 x 1/8
= 2 atoms/unit cell

a

atoms
unit cell
APF =

2

4
3

( 3a/4 ) 3

a3

volume
atom

volume
unit cell
Chapter 3- 8

FACE CENTERED CUBIC
STRUCTURE (FCC)
• Close packed directions are face diagonals.
--Note: All atoms are identical; the face-centered atoms are shaded differently only for ease of viewing.

• Coordination # = 12

Adapted from Fig. 3.1(a),
Callister 7e.
(Courtesy P.M. Anderson)
Chapter 3- 9

Chapter 3-

ATOMIC PACKING FACTOR: FCC
• APF for a body-centered cubic structure = 0.74

Close-packed directions:
length = 4R
= 2a
Unit cell c ontains:
6 x 1/2 + 8 x 1/8
= 4 atoms/unit cell

a
Fig. 3.1(a),
Callister 7e.

atoms
unit cell
APF =

4

4
3

( 2a/4 ) 3

a3

volume
atom

volume
unit cell
Chapter 3- 10

FCC STACKING SEQUENCE
• ABCABC... Stacking Sequence
• 2D Projection
A
B
B
C
A
B
B
B
A sites
C
C
B sites
B
B
C sites

• FCC Unit Cell

A
B
C

Chapter 3- 11

HEXAGONAL CLOSE-PACKED
STRUCTURE (HCP)
• ABAB... Stacking Sequence
• 3D Projection

• 2D Projection
A sites

Top layer

B sites

Middle layer

A...

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