Post on 13-Apr-2018
1
CHAPTER 2
The Structure of Crystalline Solids
230.07.2007
OUTLINE� 1.0 Energy & Packing
� 2.0 Materials & Packing
� 3.0 Crystal Structures
� 4.0 Theoretical Density, r
� 5.0.Polymorphism and Allotropy
� 6.0 Close - Packed Crystal Structures
� 7.0 Structure of Compounds: NaCl
� 8.0 X-Ray Diffraction From Crystals
� 9.0 Crystallographic Points, Directions and Planes: Miller
Indices
2
330.07.2007
1.0 Energy & Packing
i. Non-Dense Random Packing
ii. Dense Regular Packing
Energy
r
typical neighbor bond length
typical neighbor bond energy
Energy
r
typical neighbor bond length
typical neighbor bond energy
�Dense, regular-packed structures tend to have lower energy.
430.07.2007
2.0 Materials & Packing
Si Oxygen •the atoms pack in periodic, 3D arrays in a
long range
•dense, regular packing
•typical of:
1. Crystalline materials...
-metals
-many ceramics
-some polymers
• atoms have no periodic packing in long
range
• occurs for:
2. Noncrystalline materials... "Amorphous"
-complex structures
-rapid cooling
crystalline SiO2
noncrystalline SiO2
Adapted from Fig. 3.18(b),Callister 6e.
3
530.07.2007
Examples
Basic Unit
Si0 4 tetrahedron4-
Si4+
O2-
• Crystalline materials: Quartz(SiO2)
630.07.2007
Examples• Noncrystalline materials(Amorphous): Glass
• Amorphous structureoccurs by adding impurities
(Na+,Mg2+,Ca2+, Al3+)
• Impurities:
interfere with formation of
crystalline structure.
Si4+
Na+
O2-
(soda glass)
4
730.07.2007
Type of Crystals:1. Single Crystals:
When the regular arrangement of atoms is perfect and extends throughout the specimen without interruption, the result is single crystal. They are difficult to grow.
GRAPHITE
830.07.2007
• Some engineering applications require single crystals:
2)turbine blades
Fig. 8.30(c), Callister 6e.
3)silicon single crystals are grown for
quantum electronics
1)diamond single crystals for abrasives
5
930.07.2007
2. Polycrystalline Minerals and Materials
� A collection of many small crystals or grains
� Random orientations
1030.07.2007
Polycrystalline Minerals and Materials
6
1130.07.2007
� Most engineering materials are polycrystalline.
� Nb-Hf-W plate with an electron beam weld.
� Each "grain" is a single crystal.
� If crystals are randomly oriented, overall component properties are not directional.
� Crystal sizes range from 1 nm to 2 cm (i.e., from a few atoms to millions of atomic layers).
Callister 6e.1 mm
1230.07.2007
Single vs Polycrystals
• Single Crystals
1. Properties vary with direction: anisotropic
Example: the modulus
of elasticity (E) in BCC iron:
• Polycrystals
1. Properties may/may not vary with
direction.
1. If grains are randomly
oriented: isotropic(Epoly iron = 210 GPa)
2. If grains are textured:anisotropic
E (diagonal) = 273 GPa
E (edge) = 125 GPa
200 mm
Data from Table 3.3,
Callister 6e.
Adapted from Fig. 4.12(b), Callister 6e.
7
1330.07.2007
3.0 Crystal Structures
� Diffraction pattern of electron beams produced for a single crystal of Ga As (Gallium Arsenide) using Transmission Electron Microscope
1430.07.2007
Definitions about Crystal Structures
� Lattice: 3D array of regularly spaced points
� (a) Hard sphere representation:atoms denoted by hard, touching spheres
� (b) Reduced sphere representation
� (c)Unit cell: basic building block unit (such as a flooring tile) that repeats in space to create the crystal structure
8
1530.07.2007
Definition of Crystal Systems
� Lattice Parameters:� The unit cell geometry is defined in terms of six parameters.
� the three edge lengths a, b, c, and
� the three interaxial angels α, β, γ
� Seven possible combinations of a, b, c& α, β, γ, resulting in seven crystal systems
1630.07.2007
Crystal Systems
9
1730.07.2007
•Atomic Packing Factor
APF = Volume of atoms in unit cell*
Volume of unit cell
*assume hard spheres
•Coordination Number
The number of closest atoms to a selected atom in a unit cell
1830.07.2007
Cubic Crystal Structures
1. Simple Cubic (SC)
2. Face Centered Cubic ( FCC )
3. Body Centered Cubic ( BCC )
10
1930.07.2007
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)
Click on image to animate
2030.07.2007
Volume of Atoms In Unit Cell(SC)
Fig. 3.19,Callister 6e.
a3Volume of unit cell=
atoms
unit cell= 8 x 1/8 = 1 atom/unit cell
close-packed directions
a
R=0.5a
R=0.5aStep1:
4
3π (0.5a) 3Volume of atom =
Step2:
Step3:
Step4:
11
2130.07.2007
Atomic Packing Factor: SC
APF = Volume of atoms in unit cell*
Volume of unit cell
*assume hard spheres
APF for a simple cubic structure = 0.52
APF =
a3
4
3π (0.5a)31
atoms
unit cellatom
volume
unit cell
volume
a
R=0.5a
2230.07.2007
Face Centered Cubic Structure (FCC)
• Coordination # = 12
• Close packed directions are face diagonals.
Note: All atoms are identical; the face-
centered atoms are shaded
differently only for ease of viewing.
Click on image to animate
This structure is exhibited by metals Ag, Al,
Au, Cu, Ir, Ni, Pb, Pd, Pt, and Rh, as well
as the noble gases Ar, Kr, Ne, and Xe.
This arrangement (but without actual contact
by atoms of the same element) is also
assumed by the anions (or, less often, the
cations) in numerous ionic compounds.
12
2330.07.2007
Atomic Packing Factor: FCC
APF =
a3
4
3π ( 2a/4)34
atoms
unit cell atom
volume
unit cell
volume
Unit cell c ontains: 6 x 1/2 + 8 x 1/8 = 4 atoms/unit cell
• APF for a body-centered cubic structure = 0.74
Close-packed directions: length = 4R = 2 a
Adapted from
Fig. 3.1(a),
Callister 6e.
2430.07.2007
Body Centered Cubic Structure (BCC)
Coordination # = 8
The metals Ba, Cr, Cs, α-Fe, K, Li, Mo, Na, Rb,
Ta, V, and W adopt this structure. If the atom
in the center of the cube is different than
those at the corners, the cesium chloride
structure results.
• Close packed directions are cube diagonals.
Click on image to animate
13
2530.07.2007• APF for a body-centered cubic structure = 0.68
Close-packed directions: length = 4R = 3 a
APF =
a3
4
3π ( 3a/4)32
atoms
unit cell atom
volume
unit cell
volume
Unit cell contains: 1 + 8 x 1/8 = 2 atoms/unit cell
Atomic Packing Factor: BCC
2630.07.2007
4.0 Theoretical Density, ρρρρDensity = mass/volume
mass = (number of atoms per unit cell) x (mass of each atom)
mass of each atom = Molar Mass /Avogadro’s number
ρ = nA
VcNA
# atoms/unit cell Atomic weight (g/mol)
Volume/unit cell
(cm3/unit cell)
Avogadro's number
(6.023 x 1023 atoms/mol)
Molar Mass
14
2730.07.2007
Theoretical Density, ρρρρ
Example: Copper
• crystal structure = FCC: 4 atoms/unit cell
• molar mass = 63.55 g/mol (1 amu = 1 g/mol)
• atomic radius R = 0.128 nm (1 nm = 10 cm)-7
Vc = a3 ; For FCC, a = 4R/ 2 ; Vc = 4.75 x 10
-23cm3
Compare to actual: ρCu = 8.94 g/cm3
Result: theoretical ρCu = 8.89 g/cm3
ρ = nA
VcNA
# atoms/unit cell Atomic weight (g/mol)
Volume/unit cell
(cm3/unit cell)
Avogadro's number
(6.023 x 1023 atoms/mol)
Molar Mass
2830.07.2007
Symbol Al Ar Ba Be B Br Cd Ca C Cs Cl Cr Co Cu F Ga Ge Au He H
Atomic radius (nm) 0.143 ------ 0.217 0.114 ------ ------ 0.149 0.197 0.071 0.265 ------ 0.125 0.125 0.128 ------ 0.122 0.122 0.144 ------ ------
Crystal Structure FCC ------ BCC HCP Rhomb ------ HCP FCC Hex BCC ------ BCC HCP FCC ------ Ortho. Dia. cubic FCC ------ ------
Callister 6e.
Characteristics of Selected Elements at 200C
Element Aluminum Argon Barium Beryllium Boron Bromine Cadmium Calcium Carbon Cesium Chlorine Chromium Cobalt Copper Flourine Gallium Germanium Gold Helium Hydrogen
Molarmass
(g/mol) 26.98 39.95 137.33 9.012 10.81 79.90 112.41 40.08 12.011 132.91 35.45 52.00 58.9363.5519.00 69.72 72.59 196.97 4.003 1.008
19.32 ------
Density
(g/cm3)
2.71 ------3.5 1.85 2.34 ------8.65 1.55 2.25 1.87 ------7.19 8.9 8.94------5.90 5.32
------
15
2930.07.2007
ρ (g/cm3)
Graphite/ Ceramics/ Semicond
Metals/ Alloys
Composites/ fibers
Polymers
1
2
20
30Based on data in Table B1, Callister *GFRE, CFRE, & AFRE are Glass, Carbon, & Aramid Fiber-Reinforced Epoxy composites (values based on 60% volume fraction of aligned fibers
in an epoxy matrix). 10
3
4
5
0.3
0.4 0.5
Magnesium
Aluminum
Steels
Titanium
Cu,Ni
Tin, Zinc
Silver, Mo
Tantalum Gold, W Platinum
Graphite
Silicon
Glass-soda Concrete
Si nitride Diamond Al oxide
Zirconia
HDPE, PS PP, LDPE
PC
PTFE
PET PVC Silicone
Wood
AFRE*
CFRE*
GFRE*
Glass fibers
Carbon fibers
Aramid fibers
Why?Metals have...• close-packing
(metallic bonding)
• large atomic mass
Ceramics have...• less dense packing
(covalent bonding)
• often lighter elements
Polymers have...• poor packing
(often amorphous)
• lighter elements (C,H,O)
Composites have...• intermediate values Data from Table B1, Callister 6e.
Densities of Material Classes
ρmetals> ρceramics> ρpolymers
3030.07.2007
5.0.Polymorphism and Allotropy
� HEATING AND COOLING OF AN IRON WIRE
Demonstrates "polymorphism"
The same atoms can
have more than one
crystal structure.Temperature, C
BCC Stable
FCC Stable
914
1391
1536
shorter
longer!shorter!
longer
Tc 768 magnet falls off
BCC Stable
Liquid
heat up
cool down
Polymorphism
Same compound occurring in more than one crystal structure.
Crystal structure depends on Temperature and Pressure.
Allotropy
Polymorphism in elemental solids (e.g., sulphur,carbon, oxygen)
16
3130.07.2007
Shape memory alloys
� materials that "remember" their original shape and return to it when heated,
� even if apparent residual deformation was introduced below a certain temperature.
� The original shape can be set easily by heat treatment.
� These alloys are used in orthopedic implants and other medical applications, as well as in air conditioners and other home appliances, automobiles, etc.
3230.07.2007
6.0 Close - Packed Crystal StructuresClose packed planes of atoms:Planes
having a maximum atom density.
17
3330.07.2007
Close - Packed Crystal Structures
Close packed planes of atoms:Planes
having a maximum atom density.
Compare with open pack
3430.07.2007
7.0 Structure of Compounds: NaCl
• Compounds: Often have similar close-packed structures.
• Structure of NaCl
Click on image to animate
Click on image to animate
18
3530.07.2007
8.0 X-ray Diffraction From Crystals
� Diffraction based techniques.� X-ray :X-rays interact with electrons in matter.
X-rays are scattered by the electron clouds of atoms.
� Neutron
� Electron
� Microscopy.
� Thermal Analysis.
� Spectroscopy.
There are many experimental techniques that can be
used to study the properties and structure of solids.
These include:
3630.07.2007
X-ray Diffraction From Crystals
W. H. Bragg (father) and William Lawrence Bragg (son)
developed a simple relation for scattering angles,
nowadays known as Bragg’s law.
X Rays
λ :0.01-10 nm
ν:1018 Hz
(visible light 1015 Hz)
19
3730.07.2007
X-ray Diffraction From Crystals
� nλ = SQ + QT
� nλ = dhkl sinθ + dhkl sinθ=2 dhkl sinθ
� nλ = 2 dhkl sinθ
3830.07.2007
X-ray Diffraction From Crystals
20
3930.07.2007
A Modern Diffractometer
4030.07.2007
Powder Diffraction
� Powder diffraction experiment requires only as small quantity of a mineral:
� 10-500mg
� Sample preparation is very simple and fast
� Reliable accurate results are obtained in a relatively short time:10 minutes to 2 hours.
� The ideal powder size is 5-10 microns.
21
4130.07.2007
Applications of XRD
� Identification of unknowns.
� Phase purity.
� Determination of lattice parameters.
� Determination of crystal size.
� Variable temperature studies.
� Structure determination and refinement.
4230.07.2007
9.0 Crystallographic Points,
Directions And Planes: Miller Indices
22
4330.07.2007
Crystallographic Directions
� It is a vector between two points.
1. The length of the vector projection on the
three axis is determined as a, b, c.
2. These three numbers are multiplied or by
divided by a common number to reduce to
the smallest integers.
3. Three indices are written as [u, v, w]
4. Negative indices are represented by a bar
over the index.
4430.07.2007
Crystallographic Directions And Planes: Miller Indices
23
4530.07.2007
Crystallographic Directions And Planes: Miller Indices
1. If the plane passes through the selected
origin, either another parallel plane or a
new origin must be established.
2. Find intersects of the plane with the three
axis
3. Take the reciprocals. A plane that
parallels and axis has an infinite intercept
4. If necessary find the smallest integers
5. Write the indices as (h k l)
4630.07.2007
Miller Indices of Planes:Examples
24
4730.07.2007
Miller Indices of Planes:Examples
4830.07.2007
FCC Unit Cell With (110) Plane
25
4930.07.2007
5030.07.2007
Problem 2.10
� (a) The tetragonal crystal system since a = b= 0.35
nm, c = 0.45 nm, and α = β = γ = 90°.
� (b) The crystal structure can be called body-centered
tetragonal.
� (c) BCC, n = 2 atoms/unit cell. A=141g/mol
VC= (3.5 x 10−8 cm)2(4.5 x 10−8
cm)
=5.51 x 10−23cm3/unit cell
ρ = nA / VCNA
= (2 atoms/unit cell)(141 g/mol) /(5.51 x 10−23cm3/unit
cell )(6.023 x 1023 atoms/mol)
ρ = 8.49 g/cm3
Below is a unit cell for a
hypothetical metal.
a) To which
crystal system
does this unit
cell belong?
b) What would this
crystal structure
be called?
c) Calculate the
density of the
material, given
that its molar
mass is 141
g/mol.
26
5130.07.2007
Problem 2.11 What are the indices for the direction indicated by the two
vectors in the sketch?
� For direction 1,
� the projection on the x-axis is zero (since it lies in
the y-z plane),
� while projections on the y- and z-axes are b/2
and c,
For direction 2
For direction 1
5230.07.2007
Problem about direction(Callister 3.52,p.88) Determine the indices for the directions shown.
Direction A : We position the origin of
the coordinate system at the tail of
the direction vector; then in terms of
this new coordinate system
[ -110] direction
27
5330.07.2007
Problem 2.13 What are the indices for the two planes drawn in the
sketch below?
� Plane 1 :(020)
� Plane 2 :
5430.07.2007
Problem2.12 Sketch within a cubic unit cell the following planes:
28
5530.07.2007
Problem 2.15 Determine the Miller indices for the planes shown in the
following unit cell.
Plane A
Plane B
5630.07.2007
Problem 2.16 Determine the Miller indices for the planes shown in the
following unit cell.
29
5730.07.2007
Problem 2.17 Determine the Miller indices for the planes shown in the
following unit cell.