Chapter 3: The Structure of Crystalline Solids

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Chapter 3 The Structure of Crystalline Solids
ISSUES TO ADDRESS...
How do atoms assemble into solid structures?
How does the density of a material depend on
its structure?
When do material properties vary with the
sample (i.e., part) orientation?
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Attractive Energy the energy released when the
ions come close together Repulsive energy the
energy absorbed as the ions come close
together Net Energy the sum of energies
associated with the attraction and repulsion of
the ions It is minimum when the ions are at
their equilibrium separation distance r0. At the
minimum energy, the force between the ions are
zero. The ions will remain at an equilibrium
separation distance r0 ( more stable).
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Energy and Packing
Non dense, random packing
Dense, ordered packed structures tend to 1.have
lower energies and more stable energy
arrangements. 2. Atoms come closer together and
bond more tightly
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Materials and Packing
Crystalline materials...
atoms pack in periodic, 3D arrays
typical of
-metals -many ceramics -some polymers
crystalline SiO2
Si
Oxygen
Noncrystalline materials...
atoms have no periodic packing
occurs for
-complex structures -rapid cooling
noncrystalline SiO2
"Amorphous" Noncrystalline
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• Crystal is a solids in which the constituent
  atoms, molecules, or ions are packed in a
  regularly ordered, repeating pattern extending in
  all three spatial dimensions. Long range order
  exist
• Noncrystalline(amorphous) materials that dont
  crystallize, this long range atomic order is
  absent
• Crystal structure it is the manner in which
  atoms, ions, or molecules are spatially arranged.
  
• Crystal system is described in terms of the unit
  cell geometry.
• A crystal structure is described by both the
  geometry of, and atomic arrangements within the
  unit cell, whereas a
• Crystal system is described only in terms of the
  unit cell geometry.
• For example, face-centered cubic and
  body-centered cubic are crystal structures that
  belong to the cubic crystal system.

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Crystal Systems
Unit cell smallest repetitive volume which
contains the complete lattice pattern of a
crystal.
7 crystal systems 14 crystal lattices
a, b, and c are the lattice constants a, b, c, a,
ß, ? are lattice parameters
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 Section 3.4 Metallic Crystal Structures

• How can we stack metal atoms to minimize empty
  space?
• 2-dimensions

vs.
Now stack these 2-D layers to make 3-D structures
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Metallic Crystal Structures
Tend to be densely packed.
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. - Electron cloud shields cores from each
other
Have the simplest crystal structures.
We will examine three such structures...
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Simple Cubic Structure (SC)
Rare due to low packing density (only Po has
this structure) Close-packed directions are
cube edges.
Coordination 6 ( nearest neighbors)
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Atomic Packing Factor (APF)
Volume of atoms in unit cell
APF
Volume of unit cell
assume hard spheres
APF for a simple cubic structure 0.52
1
APF
3
a
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Body Centered Cubic Structure (BCC)
Atoms touch each other along cube diagonals.
--Note All atoms are identical the center atom
is shaded differently only for ease of viewing.
ex Fe (?), Tantalum, Molybdenum
Coordination 8
2 atoms/unit cell 1 center 8 corners x 1/8
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Atomic Packing Factor BCC
APF for a body-centered cubic structure 0.68
a
Adapted from Fig. 3.2(a), Callister 7e.
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Face Centered Cubic Structure (FCC)
Atoms touch each other along face diagonals.
--Note All atoms are identical the
face-centered atoms are shaded differently
only for ease of viewing.
ex Al, Cu, Pb, Ni, Ag
Coordination 12
4 atoms/unit cell 6 face x 1/2 8 corners x 1/8
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Atomic Packing Factor FCC
APF for a face-centered cubic structure 0.74
maximum achievable APF
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Hexagonal Close-Packed Structure (HCP)
ABAB... Stacking Sequence
3D Projection
2D Projection
6 atoms/unit cell
Coordination 12
ex Cd, Mg, Ti, Zn
APF 0.74
c/a 1.633
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• CLOSE-PACKED CRYSTAL STRUCTURES FOR METALS
• Both face-centered cubic and hexagonal
  close-packed crystal structures have atomic
  packing factors of 0.74, which is the most
  efficient packing of equal sized spheres or
  atoms.
• In addition to unit cell representations, these
  two crystal
• structures may be described in terms of
  close-packed planes of atoms (i.e., planes having
  a maximum atom or sphere-packing density) a
  portion of one such plane is illustrated in
  Figures bellow.
• Both crystal structures may be generated by the
  stacking of these close-packed planes on top of
  one another the difference between the two
  structures lies in the stacking sequence.

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Theoretical Density, r
A knowledge of crystal structure of a metallic
solid permits computation density
Density ?
where n number of atoms/unit cell
A atomic weight g/mole VC Volume of
unit cell a3 for cubic NA Avogadros
number 6.023 x 1023 atoms/mol
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Theoretical Density, r

• Ex Cr (BCC)
• A 52.00 g/mol
• R 0.125 nm
• n 2

a 4R/ 3 0.2887 nm
?theoretical
7.18 g/cm3
ractual
7.19 g/cm3
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Densities of Material Classes
In general
Graphite/
Metals/
Composites/
Ceramics/
Polymers
gt
gt
Alloys
fibers
Semicond
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Why?
2
0

GFRE, CFRE, AFRE are Glass,
Metals have... close-packing
(metallic bonding) often large atomic
masses
Carbon, Aramid Fiber-Reinforced

Epoxy composites (values based on

60 volume fraction of aligned fibers

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in an epoxy matrix).

Ceramics have... less dense packing
often lighter elements
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3
4

(g/cm )
3

r
2
Polymers have... low packing density
(often amorphous) lighter elements
(C,H,O)
1
0.5

Composites have... intermediate values
0.4

0.3

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Single crystal For a crystalline solid, when the
periodic and repeated arrangement of atoms is
perfect or extends throughout the entirety of the
specimen without interruption. All unit cells
interlock in the same way and have the same
orientation. Single crystals exist in nature,
but they may also be produced artificially. They
are ordinarily difficult to grow, because the
environment must be carefully controlled.
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POLYCRYSTALLINE MATERIALS

• Most crystalline solids are composed of a
  collection of many small crystals or grains such
  materials are termed polycrystalline
• The Various stages in the solidification of a
  polycrystalline specimen are
• Initially, small crystals or nuclei form at
  various positions.
• The small grains grow by the successive addition
  from the surrounding liquid of atoms to the
  structure of each.
• The extremities of adjacent grains impinge on
  one another as the solidification process
  approaches completion
• exists some atomic mismatch within the region
  where two grains meet this area, called a grain
  boundary

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Crystals as Building Blocks
Some engineering applications require single
crystals
--turbine blades
--diamond single crystals for abrasives
Properties of crystalline materials
often related to crystal structure.
--Ex Quartz fractures more easily along some
crystal planes than others.
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Polycrystals
Anisotropic
Most engineering materials are polycrystals.
1 mm
Isotropic
Each "grain" is a single crystal. If grains
are randomly oriented, overall component
properties are not directional. Grain sizes
typ. range from 1 nm to 2 cm (i.e., from a
few to millions of atomic layers).
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Single vs Polycrystals
Single Crystals
-Properties vary with direction anisotropic.
-Example the modulus of elasticity (E) in BCC
iron
Polycrystals
200 mm
-Properties may/may not vary with
direction. -If grains are randomly oriented
isotropic. (Epoly iron 210 GPa) -If grains
are textured, anisotropic.
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Polymorphism

• Two or more distinct crystal structures for the
  same material (allotropy/polymorphism)  
  ex. titanium
•   ?, ?-Ti
• ex. carbon
• (cubic)diamond,
• (hexagonal)graphite

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CRYSTALLOGRAPHIC POINTS, DIRECT IONS AND PLANES

• Crystallographic planes and directions are
  specified in terms of an indexing scheme.
• Crystallographic directional and planar
  equivalencies are related to atomic linear and
  planar densities, respectively.
• The atomic packing (i.e., planar density) of
  spheres in a crystallographic plane depends on
  the indices of the plane as well as the crystal
  structure.

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Point Coordinates
To specify the position of any point located
within unit cell.

• Point coordinates for unit cell center are
• a/2, b/2, c/2 ½ ½ ½
• Point coordinates for unit cell corner are 111
• Translation integer multiple of lattice
  constants ? identical position in another unit
  cell
• Lattice constants a, b, c

z
2c
y
b
b
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Crystallographic Directions
Is defined as a line between two points, or a
vector.
Algorithm
z
1. Vector repositioned (if necessary) to pass
through origin.( parallelism
maintained)2. Read off projections in terms of
unit cell dimensions a, b, and c3. Adjust to
smallest integer values( multiplied or
divided)4. Enclose in square brackets, no
commas uvw
y
x
ex 1, 0, ½
gt 2, 0, 1
gt 201
-1, 1, 1
families of directions ltuvwgt
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• For each of the three axes, there will exist both
  positive and negative coordinates. Thus negative
  indices are also possible, which are represented
  by a bar over the appropriate index.
• For example, the direction would have a
  component
• in the - y direction. Also, changing the signs of
  all indices produces an antiparallel direction
  that is, is directly opposite to

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Ex. The 100, 110, and 111 directions are
common ones they are drawn in the unit cell
shown in Figure
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Example
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Example
Draw a direction within a cubic unit
cell. SOLUTION First construct an appropriate
unit cell and coordinate axes system. In the
accompanying figure the unit cell is cubic, and
the origin of the coordinate system, point O, is
located at one of the cube corners.
For this direction, the projections
along the x, y, z axes are 1a, -1a, and 0a,
respectively. This direction is defined by a
vector passing from the origin to point P, which
is located by first moving along the x axis a
units, and from this position, parallel to the y
axis a units, as indicated in the figure. There
is no z component to the vector, since the z
projection is zero.
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Linear Density
 

• Linear Density ? LD  
• j

ex linear density of Al in 110
direction  a 0.405 nm
½1 ½2 atoms
LD is important relative to the process of
slip-the mechanism by which metals plastically
deform
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Crystallographic Planes
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Crystallographic Planes

• Miller Indices Reciprocals of the (three) axial
  intercepts for a plane, cleared of fractions
  common multiples. All parallel planes have same
  Miller indices.
• Algorithm 
• 1.  Read off intercepts of plane with axes in
• terms of a, b, c
• 2. Take reciprocals of intercepts
• 3. Reduce to smallest integer values
• 4. Enclose in parentheses, no
• commas i.e., (hkl)

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Crystallographic Planes
4. Miller Indices (110)
4. Miller Indices (100)
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Crystallographic Planes
example
a b c
4. Miller Indices (634)
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Crystallographic Planes

• We want to examine the atomic packing of
  crystallographic planes
• Iron foil can be used as a catalyst. The atomic
  packing of the exposed planes is important.
• Draw (100) and (111) crystallographic planes
• for Fe.
• b) Calculate the planar density for each of
  these planes.

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Planar Density of (100) Iron

• Solution  At T lt 912?C iron has the BCC
  structure.

2D repeat unit
(100)
Radius of iron R 0.1241 nm
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Planar Density of (111) Iron

• Solution (cont)  (111) plane

1 atom in plane/ unit surface cell
a

2
2D repeat unit
3

a
h
2
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X-Ray Diffraction

• Diffraction gratings must have spacings
  comparable to the wavelength of diffracted
  radiation.
• Cant resolve spacings ? ?
• Spacing is the distance between parallel planes
  of atoms.  

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X-Rays to Determine Crystal Structure
Incoming X-rays diffract from crystal planes.
Measurement of critical angle, qc, allows
computation of planar spacing, d.
n is the order of reflection Which may be any
integer(1,2,3,)
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X-Ray Diffraction Pattern
(110)
(211)
Intensity (relative)
(200)
Diffraction angle 2q
Diffraction pattern for polycrystalline a-iron
(BCC)
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SUMMARY
Atoms may assemble into crystalline or
amorphous structures.
Common metallic crystal structures are FCC,
BCC, and HCP. Coordination number and
atomic packing factor are the same for both
FCC and HCP crystal structures.
We can predict the density of a material,
provided we know the atomic weight, atomic
radius, and crystal geometry (e.g., FCC,
BCC, HCP).
Crystallographic points, directions and planes
are specified in terms of indexing schemes.
Crystallographic directions and planes are
related to atomic linear densities and
planar densities.
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SUMMARY
Materials can be single crystals or
polycrystalline. Material properties
generally vary with single crystal
orientation (i.e., they are anisotropic), but are
generally non-directional (i.e., they are
isotropic) in polycrystals with randomly
oriented grains.
Some materials can have more than one crystal
structure. This is referred to as
polymorphism (or allotropy).
X-ray diffraction is used for crystal
structure and interplanar spacing
determinations.