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Chapter 3
The Structure of Crystalline Solids
EGN 3365 Materials I
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Objectives of Chapter 3
Learn classification of materials based on atomic/ionic
arrangements
Describe the arrangements in crystalline solids based onlattice, basis, and crystal structure
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Issues to Address
How do atoms assemble into solid structures?
How do the structures of metals differ from those of
other materials?
How does the density of a material depend on its
structure?
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Types of Solids
Crystalline Material: Atoms self-organize in a periodic array
Single Crystal:
Atoms are in a repeating or periodic array over the entire
extent of the material Polycrystalline Material:
Comprised of many small crystals or grains
Amorphous Material:
Lacks atomic arrangement
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(c) 2003 Brooks/Cole Publishing / Thomson
Learning
Figure: Levels ofatomic arrangementsin materials: (a) NOORDER - Inertmonoatomic gaseshave no regularordering of atoms:(b,c) SHORT RANGEORDER (SRO) - Some
materials, includingwater vapor, nitrogengas, amorphous siliconand silicate glass haveshort-range order. (d)LONG-RANGE ORDER(LRO) - Metals, alloys,many ceramics andsome polymers have
regular ordering ofatoms/ions thatextends through thematerial.
Materials and Packing
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Single Crystals vs. Polycrystalline Materials
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Classification of Materials Based on Structure
Figure 3.4 (a) Photograph of a silicon single crystal.(b) Micrograph of a polycrystalline stainless steelshowing grains and grain boundaries (Co u r t e s y D r .M . H u a , D r . I . Ga r c i a , a n d D r . A . J. D e a r d o .)
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Figure: Liquid crystal display. Thesematerials are amorphous in one stateand undergo localized crystallization inresponse to an external electric fieldand are widely used in liquid crystaldisplays. (Co u r t e s y o f N i c k
K ou d i s / Ph o t o D is c / Ge t t y I m a g es .)
Classification of Materials Based on Structure
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(c) 2003 Brooks/Cole Publishing / Thomson
Learning
Materials and Packing Summary
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Crystal Structure
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Definitions Lattice
Collection of points that divide space into smaller equally sizedsegments
Basis
Group of atoms associated with a lattice point
Unit cell
Subdivision of the lattice that still retains the overall characteristics ofthe entire lattice
Atomic radius
Apparent radius of an atom
Typically calculated from the dimensions of the unit cell, using close-packed directions (depends upon coordination number for metals,each atom has the same number of nearest-neighbor or touchingatoms)
Packing factor
The fraction of space in a unit cell occupied by atoms.
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Unit Cell
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-14- (c) 2003 Brooks/Cole Publishing / Thomson Learning
The fourteen
types of
lattices
grouped in
seven crystal
systems.
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Metallic Crystal Structures Metals are usually (poly)crystalline; although formation of
amorphous metals is possible by rapid cooling
As we learned in Chapter 2, the atomic bonding in metals isnon-directional no restriction on numbers or positions ofnearest-neighbor atoms large number of nearest neighbors
and dense atomic packing
Atomic (hard sphere) radius, R, defined by ion core radius -typically 0.1 - 0.2 nm
The most common types of unit cells are the faced-centeredcubic (FCC), the body-centered cubic (BCC) and the hexagonalclose-packed (HCP).
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Rare due to poor packing (only Po has this structure)
Close-packed directions are cube edges.
Coordination # = 6
(# nearest neighbors)
(Courtesy P.M. Anderson)
Simple Cubic Cell (SCC)
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APF for a simple cubic structure = 0.52
APF =a3
4
3 (0.5a)31
atoms
unit cellatom
volume
unit cell
volume
Atomic Packing Factor (APF)
APF =Volume of atoms in unit cell*
Volume of unit cell
*assume hard spheres
Adapted from Fig. 3.23,
Callister 7e.
close-packed directions
a
R=0.5a
contains 8 x 1/8 =1 atom/unit cell
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(Courtesy P.M. Anderson)
Close packed directions are cube diagonals.--Note: All atoms are identical; the center atom is shaded
differently only for ease of viewing.
Body Centered Cubic Cell (BCC)
2 atoms/unit cell: 1 center + 8 corners x 1/8
Coordination # = 8
ex: Cr, W, Fe (), Tantalum, Molybdenum
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Atomic Packing Factor: BCC
a
APF =
4
3 ( 3a/4)32
atoms
unit cell atom
volume
a3 unit cell
volume
length = 4R =Close-packed directions:
3 a
APF for a body-centered cubic structure = 0.68
aR
Adapted fromFig. 3.2(a), Callister 7e.
a2
a3
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(Courtesy P.M. Anderson)
Close packed directions are face diagonals.--Note: All atoms are identical; the face-centered atoms are shaded
differently only for ease of viewing.
Face Centered Cubic Cell (FCC)
4 atoms/unit cell: 6 face x 1/2 + 8 corners x 1/8
Adapted from Fig. 3.1, Callister 7e.
ex: Al, Cu, Au, Pb, Ni, Pt, Ag
Coordination # = 12
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APF for a face-centered cubic structure = 0.74
Atomic Packing Factor: FCC
maximum achievable APF
APF =
4
3 ( 2a/4)34
atoms
unit cell atom
volume
a3
unit cell
volume
Close-packed directions:length = 4R = 2 a
Unit cell contains:6 x1/2 + 8 x1/8
=4 atoms/unit cella
2 a
Adapted from
Fig. 3.1(a),Callister 7e.
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Determine the relationship between the atomic radius and
the lattice parameter in SC, BCC, and FCC structures when
one atom is located at each lattice point.
Determining the Relationship between
Atomic Radius and Lattice Parameters
(c) 2003 Brooks/Cole Publishing / Thomson
Learning
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We find that atoms touch along the edge of the cube in SC structures.
3
40
ra =
In FCC structures, atoms touch along the face diagonal of the cube.
There are four atomic radii along this lengthtwo radii from the face-
centered atom and one radius from each corner, so:
2
40
ra =
ra 20 =
In BCC structures, atoms touch along the body diagonal. There are two
atomic radii from the center atom and one atomic radius from each of
the corner atoms on the body diagonal, so
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A sites
B B
B
BB
B B
C sites
C C
CA
B
B sites
ABCABC... Stacking Sequence
2D Projection
FCC Unit Cell
FCC Stacking Sequence
B B
B
BB
B B
B sites
C C
CA
C C
CA
A
BC
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Example: Copper
crystal structure = FCC: 4 atoms/unit cell
atomic weight = 63.55 g/mol (1 amu = 1 g/mol)
atomic radius R = 0.128 nm (1 nm = 10 cm)-7
Compare to actual: Cu = 8.94 g/cm3
Result: theoretical Cu = 8.89 g/cm3
Theoretical Density
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Ex: Cr (BCC)A = 52.00 g/mol
R = 0.125 nm
n = 2
theoretical
a = 4R/ 3 = 0.2887 nm
actual
a
R
=a3
52.002
atoms
unit cell mol
g
unit cell
volume atoms
mol
6.023x1023
Theoretical Density,
= 7.18 g/cm3
= 7.19 g/cm3
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Determining the Density of BCC Iron
Determine the density of BCC iron, which has a lattice parameter of 0.2866
nm.
SOLUTION
Atoms/cell = 2, a0 = 0.2866 nm = 2.866 10-8 cmAtomic mass = 55.847 g/mol
Volume of unit cell = = (2.866 10-8 cm)3 = 23.54 10-24cm3/cell
Avogadros number NA
= 6.02 1023 atoms/mol
3
0a
3
2324/882.7
)1002.6)(1054.23(
)847.55)(2(
number)sadro'cell)(Avogunitof(volume
iron)ofmass)(atomicatoms/cellof(number
Density
cmg=
=
=
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metals ceramics polymers
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
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