CHAPTER 3
THE STRUCTURE
OF
CRYSTALLINE SOLIDS
Crystal Structures
►
Why study the structure of crystalline structure?
►
The properties of some materials are directly related to their crystal
structure.
ƒ For example, pure and undeformed magnesium and beryllium,
having one crystal structure, are much more brittle than pure and
undeformed metals such as gold and silver that have yet another
crystal structure.
►
Significant property differences exist between crystalline and
noncrystalline materials having same composition.
ƒ Noncrystalline ceramics and polymers normally are optically
transparent; the same material in crystalline form tend to be
opaque or, at best, translucent.
3.2 Fundamental Concepts
►
Chapter 2 was concerned primarily with the various types of atomic
bonding, which are determined by the electron structure of the
individual atoms.
►
Next level of the structure of the materials deals with some of the
arrangement that may be assumed by the atoms in the solid state.
►
Within this framework, concepts of crystallinity and noncrystallinity are
introduced.
►
Solid materials may be classified according to the regularity with which
atoms or ions are arranged with respect to one another.
Crystalline material Î atoms are situated
in a repeating or periodic array over large
atomic distances
►
►
►
►
Upon solidification, the atoms will position themselves in a
repetitive three-dimensional pattern, in which each
atom is bonded to its nearest-neighbor atoms.
All metals, many ceramic materials, and certain polymers
form crystalline structure under normal solidification
conditions.
Noncrystalline or amorphous materials Î do not
crystallize.
3.2 FUNDAMENTAL CONCEPTS
SOLIDS
AMORPHOUS
CRYSTALLINE
Atoms in a crystalline solid
are arranged in a repetitive
three dimensional pattern
Long Range Order
All metals are crystalline solids
Atoms in an amorphous
solid are arranged
randomly- No Order
Many ceramics are crystalline solids
Some polymers are crystalline solids
3.2 Fundamental Concepts (Contd.)
►
When describing crystalline
structures, atoms ( or ions ) are
thought of as being solid spheres
having well-defined diameters.
►
This is termed the atomic hard
sphere model in which spheres
representing nearest-neighbor
atoms touch one another.
►
An example of the hard sphere
model for the atomic arrangement
found in some common elemental
metals is displayed in Figure 3.1c
►
Lattice Î a 3-D array of points
coinciding with atom positions or
sphere centers.
LATTICE
Lattice -- points arranged in a pattern that
repeats itself in three dimensions.
The points in a crystal lattice coincides
with atom centers
3-D view of a lattice
3.3 Unit Cells
The atomic order in crystalline solids indicates that small groups
of atoms form a repetitive pattern.
Î in describing crystal structures, it is often convenient to
subdivide the structure into small repeat entities called unit
cells.
► Unit cell:
The basic structural unit or building block of the
crystal structure and defines the crystal structure by virtue of its
geometry and the atom positions within.
►
►
For most crystal structures are parallelepipeds or prisms having
three sets of parallel faces ( In case of Figure 3.1c, it is cube )
►
Parallelepiped corners coincides with centers of the hard sphere
atoms.
►
Generally use the unit cell having the highest level of
geometrical symmetry. More than single unit cell may be
chosen for a particular crystal structure.
Unit cell & Lattice
Lattice
Unit Cell
3.4 Metallic Crystal Structures
►
The atomic bonding in this group of materials is metallic, and
thus nondirectional.
Î there are no restrictions as to the number and position of
nearest-neighbor atoms; this leads to relatively large numbers
of nearest neighbors and dense atomic packings for most
metallic crystal structures.
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 have lower bonding energy.
►
Using the hard sphere model for the crystal structure of metals, each sphere
represents an ion core.
Table 3.1 presents the atomic radii and crystal structure type.
4
► Three
►
►
►
relatively simple structures found are:
Face-Centered Cubic (FCC);
Body-Centered Cubic (BCC);
Hexagonal Close-Packed (HCP).
FACE CENTERED CUBIC STRUCTURE (FCC)
3.4 Metallic Crystal Structures (Contd.)
The Face-Centered Cubic Crystal Structure
►
Unit cells of cubic geometry, with atoms located at each of the corners
and the centers of all the cube faces.
►
FCC Î Face-Centered Cubic
►
Found in Copper, Aluminum, Silver, and Gold.
The spheres or ion cores in FCC touch one another across a face
diagonal; the cube edge length a and the the atomic radius R are
related through
a = 2R√2
► Each corner atom is shared among eight unit cells, whereas a facecentered atom belong to only two.
Î 1/8 of each of the eight corner atoms and ½ of each of the six face
atoms, or a total of four whole atoms, may be assigned to a given unit
cell.
►
FACE CENTERED CUBIC STRUCTURE (FCC)
Al, Cu, Ni, Ag, Au, Pb, Pt
3.4 Metallic Crystal Structures (Contd.)
The Face-Centered Cubic Crystal Structure
►
Two other important characteristics of a crystal structure are:
Coordination number and the atomic packing factor (APF).
►
Coordination number
ƒ The number of nearest-neighbor or touching atoms for an atom.
ƒ For metals, the number is same. For FCC, the coordination number
is 12.
►
Atomic Packing Factor (APF)
ƒ APF is the fraction of solid sphere volume in a unit cell, assuming
the atomic hard sphere model, or
APH = (Volume of atoms in a unit cell ) / (Total unit cell volume )
ƒ For FCC, APF=0.74
BODY CENTERED CUBIC STRUCTURE (BCC)
3.4 Metallic Crystal Structures (Contd.)
The Body-Centered Cubic Crystal Structure
►
BCC Î Body-Centered Cubic
►
This metallic crystal structure has a cubic unit cell with atoms located
at all eight corners and a single atom at the cube center.
►
Center and corner atoms tough one another along cube diagonals.
►
Unit cell length (a) and atomic radius (R) are related through
a = (4R) / √3
►
Examples: Chromium, iron, tungsten, and others exhibit BCC structure.
BODY CENTERED CUBIC STRUCTURE (BCC)
Cr, Fe, W, Nb, Ba, V
3.4 Metallic Crystal Structures
The Body-Centered Cubic Crystal Structure (Contd.)
► Two
atoms are associated with each BCC unit cell.
► The
coordination number for BCC is 8.
► Since
the coordination number is less for BCC than
FCC, so also is the atomic packing factor for BCC
lower _____ 0.68 versus 0.74.
3.4 Metallic Crystal Structures
The Hexagonal Close-Packed Crystal Structure
►
HCP Î Hexagonal Close Packed crystal structure.
►
Not all metals have unit cells with cubic symmetry; in HCP it is
hexagonal.
►
The top and bottom faces of the unit cell consists of six atoms that
form regular hexagons and surround a single atom in the center.
►
Another plane that provides three additional atoms is situated between
the top and the bottom planes.
►
The equivalent of six atoms is contained in each unit cell; one-sixth of
each of the 12 top and bottom face corner atoms, one-half of each of
the 2 center face atoms, and all the 3 midplane interior atoms.
22 September 2003
ME215: Chapter 3
23
HEXAGONAL CLOSE-PACKED STRUCTURE HCP
Mg, Zn, Cd, Zr, Ti, Be
The Hexagonal Close-Packed Crystal Structure
(Contd.)
► If
a and c represent, respectively, the short and
long unit cell dimensions, c/a ratio should be
1.633.
► The
coordination number and APF for HCP crystal
structure are same as for FCC: 12 and 0.74,
respectively.
► HCP
metal includes: cadmium, magnesium,
titanium, and zinc.
22 September 2003
ME215: Chapter 3
25
SIMPLE CUBIC STRUCTURE (SC)
• Rare due to low packing density (only Pd has this structure)
• Close-packed directions are cube edges.
Coordination # = 6
(# nearest neighbors)
Number of atoms per unit cell
BCC
1/8 corner atom x 8 corners + 1 body center atom
=2 atoms/uc
FCC 1/8 corner atom x 8 corners + ½ face atom x 6
faces
=4 atoms/uc
HCP 3 inside atoms + ½ basal atoms x 2 bases + 1/ 6
corner atoms x 12 corners
=6 atoms/uc
Relationship between atomic radius and
edge lengths
For FCC:
For BCC:
For HCP
a = 2R√2
a = 4R /√3
a = 2R
c/a = 1.633 (for ideal case)
Note: c/a ratio could be less or more than the ideal value of
1.633
Face Centered Cubic (FCC)
2 a 0 = 4r
r
2r
r
a0
a0
Body Centered Cubic (BCC)
3a 0 = 4r
2a0
a0
3a0
Coordination Number
► The
number of touching or nearest
neighbor atoms
► SC is 6
► BCC is 8
► FCC is 12
► HCP is 12
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
a
R=0.5a
close-packed directions
contains 8 x 1/8 =
1 atom/unit cell
atoms
unit cell
APF =
volume
atom
4
π (0.5a)3
1
3
a3
volume
unit cell
6
ATOMIC PACKING FACTOR: BCC
• APF for a body-centered cubic structure = 0.68
a = 4R /√3
R
a
Unit cell contains:
1 + 8 x 1/8
= 2 atoms/unit cell
atoms
volume
4
3
π ( 3a/4)
2
unit cell
atom
3
APF =
volume
a3
unit cell
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
ATOMIC PACKING FACTOR: FCC
• APF for a face-centered cubic structure = 0.74
a = 2R√2
a
Unit cell contains:
6 x 1/2 + 8 x 1/8
= 4 atoms/unit cell
atoms
volume
4
3
π ( 2a/4)
4
unit cell
atom
3
APF =
volume
3
a
unit cell
3.5 Density Computations
►
► Density
of a material can be determined theoretically
from the knowledge of its crystal structure (from its
Unit cell information)
► Density= mass/Volume
► Mass is the mass of the unit cell and volume is the
unit cell volume.
► mass = ( number of atoms/unit cell) “n” x mass/atom
► mass/atom = atomic weight “A”/Avogadro’s Number
“NA”
► Volume = Volume of the unit cell “Vc”
THEORETICAL DENSITY
# atoms/unit cell
ρ = nA
Volume/unit cell VcNA
(cm3/unit cell)
Atomic weight (g/mol)
Avogadro's number
(6.023 x 1023 atoms/mol)
Example problem on Density Computation
Problem: Compute the density of Copper
Given: Atomic radius of Cu = 0.128 nm (1.28 x 10-8 cm)
Atomic Weight of Cu = 63.5 g/mol
Crystal structure of Cu is FCC
Solution:
ρ = n A / Vc NA
n= 4
Vc= a3 = (2R√2)3 = 16 R3 √2
NA = 6.023 x 1023 atoms/mol
ρ
= 4 x 63.5 g/mol / 16 √2(1.28 x 10-8 cm)3 x 6.023 x
1023 atoms/mol
Ans = 8.98 g/cm3
Experimentally determined value of density of Cu = 8.94 g/cm3
Densities of Material Classes
In general
ρmetals > ρceramics > ρpolymers
30
Why?
Metals have...
Ceramics have...
• less dense packing
• often lighter elements
Polymers have...
ρ (g/cm3 )
• close-packing
(metallic bonding)
• often large atomic masses
• low packing density
(often amorphous)
• lighter elements (C,H,O)
Composites have...
• intermediate values
Metals/
Alloys
20
Platinum
Gold, W
Tantalum
10
Silver, Mo
Cu,Ni
Steels
Tin, Zinc
5
4
3
2
1
0.5
0.4
0.3
Titanium
Aluminum
Magnesium
Graphite/
Ceramics/
Semicond
Polymers
Composites/
fibers
Based 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).
Zirconia
Al oxide
Diamond
Si nitride
Glass -soda
Concrete
Silicon
Ggraphite
PTFE
Silicone
PVC
PET
PC
HDPE, PS
PP, LDPE
Glass fibers
GFRE*
Carbon fibers
CFRE*
Aramid fibers
AFRE*
Wood
Data from Table B1, Callister 7e.
3.6 Polymorphism and Allotropy
►
Polymorphism Î The phenomenon in some metals, as
well as nonmetals, having more than one crystal
structures.
►
When found in elemental solids, the condition is often
called allotropy.
►
Examples:
ƒ Graphite is the stable polymorph at ambient conditions,
whereas diamond is formed at extremely high
pressures.
ƒ Pure iron is BCC crystal structure at room temperature,
which changes to FCC iron at 912oC.
►
Two or more distinct crystal structures for the same
material (allotropy/polymorphism)
iron system
titanium
liquid
α, β-Ti
1538ºC
δ-Fe
BCC
carbon
1394ºC
diamond, graphite
FCC
γ-Fe
912ºC
BCC
α-Fe
POLYMORPHISM AND ALLOTROPY
BCC (From room temperature to 912 oC)
Fe
FCC (at Temperature above 912 oC)
912 oC
Fe (BCC)
Fe (FCC)
3.7 Crystal Systems
►
Since there are many different possible crystal structures,
it is sometimes convenient to divide them into groups
according to unit cell configurations and/or atomic
arrangements.
►
One such scheme is based on the unit cell geometry, i.e.
the shape of the appropriate unit cell parallelepiped
without regard to the atomic positions in the cell.
►
Within this framework, an x, y, and z coordinate system is
established with its origin at one of the unit cell corners;
each x, y, and z-axes coincides with one of the three
parallelepiped edges that extend from this corner, as
illustrated in Figure.
The Lattice Parameters
Lattice parameters
a, b, c, α, β, γ are called the lattice
Parameters.
►
Seven different possible
combinations of edge
lengths and angles give
seven crystal systems.
►
Shown in Table 3.2
►
Cubic system has the
greatest degree of
symmetry.
►
Triclinic system has the
least symmetry.
3.7 CRYSTAL SYSTEMS
3.8 Point Coordinates in an Orthogonal
Coordinate System Simple Cubic
3.9 Crystallographic Directions in Cubic System
Determination of the directional indices in cubic
system:
Four Step Procedure (Text Book Method)
►
Draw a vector representing the direction within the unit
cell such that it passes through the origin of the xyz
coordinate axes.
2. Determine the projections of the vector on xyz axes.
3. Multiply or divide by common factor to obtain the three
smallest integer values.
4. Enclose the three integers in square brackets [ ].
1.
e.g.
[uvw]
u, v, and w are the integers
Crystallographic Directions in Cubic System
[111]
[120]
[110]
Crystallographic Directions in Cubic System
Head and Tail Procedure for determining
Miller Indices for Crystallographic Directions
1.
2.
3.
4.
Find the coordinate points of head and tail
points.
Subtract the coordinate points of the tail
from the coordinate points of the head.
Remove fractions.
Enclose in [ ]
Indecies of Crystallographic Directions in Cubic System
Direction A
Head point – tail point
(1, 1, 1/3) – (0,0,2/3)
1, 1, -1/3
Multiply by 3 to get smallest
integers
3, 3, -1
A = [33Ī]
Direction B
Head point – tail point
(0, 1, 1/2) – (2/3,1,1)
-2/3, 0, -1/2
Multiply by 6 to get smallest
integers
_ _
B = [403]
C = [???]
D = [???]
Indices of Crystallographic Directions in Cubic System
Direction C
Head Point – Tail Point
(1, 0, 0) – (1, ½, 1)
0, -1/2, -1
Multiply by 2 to get the smallest integers
__
C = [0I2]
Direction D
Head Point – Tail Point
(1, 0, 1/2) – (1/2, 1, 0)
1/2, -1, 1/2
Multiply by 2 to get the smallest
integers
_
D = [I2I]
B= [???]
A = [???]
Crystallographic Directions in Cubic System
[210]
Crystallographic Directions in Cubic System
Crystallographic Directions in Cubic System
Indices of a Family or Form
< 100 > ≡ [100], [010], [001], [010], [001], [100]
< 111 > ≡ [111], [11 1 ], [1 1 1], [ 1 11],
[ 1 1 1 ], [ 1 1 1], [ 1 1 1 ], [1 1 1 ]
< 110 > ≡ [110], [011], [101], [110], [011], [101]
[110], [011], [101], [110], [011], [101]
3.10 MILLER INDICES FOR
CRYSTALLOGRAPHIC PLANES
►
►
1.
2.
3.
4.
Miller Indices for crystallographic planes are the
reciprocals of the fractional intercepts (with
fractions cleared) which the plane makes with
the crystallographic x,y,z axes of the three
nonparallel edges of the cubic unit cell.
4-Step Procedure:
Find the intercepts that the plane makes with the three
axes x,y,z. If the plane passes through origin change
the origin or draw a parallel plane elsewhere (e.g. in
adjacent unit cell)
Take the reciprocal of the intercepts
Remove fractions
Enclose in ( )
Miller Indecies of Planes in Crystallogarphic
Planes in Cubic System
Drawing Plane of known Miller Indices in a cubic unit
cell
Draw (011) plane
Miller Indecies of Planes in Crystallogarphic Planes in
Cubic System
Origin for A
Origin for B
Origin
for A
A = (IĪ0)
B = (I22)
A = (2IĪ)
B = (02Ī)
CRYSTALLOGRAPHIC PLANES AND
DIRECTIONS IN HEXAGONAL UNIT CELLS
Miller-Bravais indices -- same as Miller
indices for cubic crystals except that there
are 3 basal plane axes and 1 vertical axis.
Basal plane -- close packed plane similar
to the (1 1 1) FCC plane.
contains 3 axes 120o apart.
Direction Indices in HCP Unit Cells –
[uvtw] where t=-(u+v)
Conversion from 3-index system to 4-index system:
[u v w ] → [uvtw]
' '
'
n
u = (2u ' − v ' )
3
n
'
'
v = ( 2v − u )
3
t = −(u + v)
w = nw
Miller Bravais indices are h,k,i,l
with i = -(h+k).
Basal plane indices
(0 0 0 1)
'
HCP Crystallographic Directions
z
Algorithm
a2
-
a3
1. Vector repositioned (if necessary) to pass
through origin.
2. Read off projections in terms of unit
cell dimensions a1, a2, a3, or c
3. Adjust to smallest integer values
4. Enclose in square brackets, no commas
[uvtw]
a2
a1
ex:
½, ½, -1, 0
-a3
a2
2
Adapted from Fig. 3.8(a), Callister 7e.
=>
[ 1120 ]
a3
dashed red lines indicate
projections onto a1 and a2 axes
a1
2
a1
HCP Crystallographic Directions
► Hexagonal
Crystals
ƒ 4 parameter Miller-Bravais lattice coordinates are
related to the direction indices (i.e., u'v'w') as
follows.
z
[ u 'v 'w ' ] → [ uvtw ]
a2
-
a3
a1
Fig. 3.8(a), Callister 7e.
1
u = (2 u ' - v ')
3
1
v = (2 v ' - u ')
3
t = - (u +v )
w = w'
Crystallographic Planes (HCP)
► In
hexagonal unit cells the same idea is used
z
example
1. Intercepts
2. Reciprocals
a1
1
1
1
1
3.
Reduction
4.
Miller-Bravais Indices
a2
∞
1/∞
0
0
a3
-1
-1
-1
-1
(1011)
c
1
1
1
1
a2
a3
a1
Adapted from Fig. 3.8(a), Callister 7e.
Miller-Bravais Indices for crystallographic planes
in HCP
_
(1211)
Miller-Bravais Indices for crystallographic
directions and planes in HCP
Atomic Arrangement on (110) plane in FCC
Atomic Arrangement on (110) plane in BCC
Atomic arrangement on [110] direction
in FCC
3.11 Linear and Planar Atomic Densities
► Linear
Density “LD”
is defined as the number of atoms per unit
length whose centers lie on the direction
vector of a given crystallographic direction.
Linear Density
►
Number of atoms
Linear Density of Atoms ≡ LD =
Unit length of direction vector
[110]
a
ex: linear density of Al in [110]
direction
a = 0.405 nm
# atoms
LD =
length
2
2a
= 3.5 nm −1
Linear Density
LD for [110] in BCC.
# of atom centered on the direction
vector [110]
= 1/2 +1/2 = 1
Length of direction vector [110] = √2 a
a = 4R/ √ 3
1
1
3
LD =
=
=
2a
2 (4 R / 3 )
2 4R
[110]
√ 2a
Linear Density
►
LD of [110] in FCC
# of atom centered on the direction
vector [110] = 2 atoms
Length of direction vector [110] = 4R
LD = 2 /4R
LD = 1/2R
Linear density can be defined as
reciprocal of the repeat distance
‘r’
LD = 1/r
Planar Density
► Planar
Density “PD”
is defined as the number of atoms per unit
area that are centered on a given
crystallographic plane.
No of atoms centered on the plane
PD = —————————————
Area of the plane
Planar Density of (110) plane in FCC
# of atoms centered on the
plane (110)
= 4(1/4) + 2(1/2) = 2
atoms
Area of the plane
= (4R)(2R √ 2) = 8R22
(111) Plane in FCC
PD110
2 atoms
1
=
=
2
2
8R 2 4 R 2
a = 2R √ 2
4R
Planar Density of (111) Iron
Solution (cont): (111) plane
1 atom in plane/ unit surface cell
2a
atoms in plane
un
it
atoms above plane
rep
ea
t
atoms below plane
2D
h=
3
a
2
2
atoms
2D repeat unit
Planar Density =
area
2D repeat unit
⎛ 4 3 ⎞ 16 3 2
2
area = 2 ah = 3 a = 3 ⎜⎜
R ⎟⎟ =
R
3
⎝ 3
⎠
1
16 3
3
atoms =
= 7.0
2
R
2
nm
0.70 x 1019
atoms
m2
Planar Density of (100) Iron
Solution: At T < 912°C iron has the BCC structure.
2D repeat unit
(100)
Planar Density =
area
2D repeat unit
1
a2
=
4 3
R
3
Radius of iron R = 0.1241 nm
Adapted from Fig. 3.2(c), Callister 7e.
atoms
2D repeat unit
a=
1
4 3
3
2 = 12.1
R
atoms
19 atoms
=
1.2
x
10
nm2
m2
Closed Packed Crystal Structures
► FCC
and HCP both have:
CN = 12
and
APF = 0.74
APF= 0.74 is the most efficient packing.
Both FCC and HCP have Closed Packed Planes
FCC ----(111) plane is the Closed Packed Plane
HCP ----(0001) plane is the Closed Packed Plane
The atomic staking sequence in the above two
structures is different from each other
Closed Packed Structures
Closed Packed Plane Stacking in HCP
Closed Packed Plane Stacking in FCC
Crystalline and Noncrystalline Materials
3.13 Single Crystals
►
For a crystalline solid, when the periodic and repeated
arrangement of atoms is perfect or extends throughout
the entirety of the specimen without interruption, the
result is a single crystal.
►
All unit cells interlock in the same way and have the
same orientation.
►
Single crystals exist in nature, but may also be produced
artificially.
►
They are ordinarily difficult to grow, because the
environment must be carefully controlled.
►
Example: Electronic microcircuits, which employ single
crystals of silicon and other semiconductors.
Polycrystalline Materials
3.13 Polycrytalline Materials
Polycrystalline Î crystalline solids
composed of many small
crystals or grains.
Various stages in the solidification :
a) Small crystallite nuclei Growth
of the crystallites.
b) Obstruction of some grains that
are adjacent to one another is
also shown.
c) Upon completion of
solidification, grains that are
adjacent to one another is also
shown.
d) Grain structure as it would
appear under the microscope.
Crystals as Building Blocks
• Some engineering applications require single crystals:
--diamond single
crystals for abrasives
(Courtesy Martin Deakins,
GE Superabrasives,
Worthington, OH. Used with
permission.)
--turbine blades
Fig. 8.33(c), Callister 7e.
(Fig. 8.33(c) courtesy
of Pratt and Whitney).
• Properties of crystalline materials
often related to crystal structure.
--Ex: Quartz fractures more easily
along some crystal planes than
others.
(Courtesy P.M. Anderson)
Polycrystals
• Most engineering materials are polycrystals.
1 mm
• Nb-Hf-W plate with an electron beam weld.
• 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).
Anisotropic
Adapted from Fig. K,
color inset pages of
Callister 5e.
(Fig. K is courtesy of
Paul E. Danielson,
Teledyne Wah Chang
Albany)
Isotropic
Single vs Polycrystals
• Single Crystals
E (diagonal) = 273 GPa
Data from Table 3.3,
Callister 7e.
(Source of data is R.W.
Hertzberg, Deformation
and Fracture Mechanics
of Engineering
Materials, 3rd ed., John
Wiley and Sons, 1989.)
-Properties vary with
direction: anisotropic.
-Example: the modulus
of elasticity (E) in BCC iron:
• Polycrystals
-Properties may/may not
vary with direction.
-If grains are randomly
oriented: isotropic.
(Epoly iron = 210 GPa)
-If grains are textured,
anisotropic.
E (edge) = 125 GPa
200 µm
Adapted from Fig.
4.14(b), Callister 7e.
(Fig. 4.14(b) is courtesy
of L.C. Smith and C.
Brady, the National
Bureau of Standards,
Washington, DC [now
the National Institute of
Standards and
Technology,
Gaithersburg, MD].)
3.15 Anisotropy
►
The physical properties of single crystals of some
substances depend on the crystallographic direction in
which the measurements are taken.
►
For example, modulus of elasticity, electrical
conductivity, and the index of refraction may have
different values in the [100] and [111] directions.
►
This directionality of properties is termed anisotropy.
►
Substances in which measured properties are
independent of the direction of measurement are
isotropic.
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.
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.