Chapter 3. Crystalline Structure
•
•
•
•
•
•
•
Seven Systems and Fourteen Lattices
Metal Structures
Ceramic Structures
Polymeric Structures
Semiconductor Structures
Lattice Positions, Directions, and Planes
X-Ray Diffractions
Structure Review
• In solid state, particles are bonded together in rigid, crystalline
structure. Seven basic crystalline structures are:
–
–
–
–
–
cubic: BCC, and FCC
tetragonal: Simple T and BCT
orthorhombic: simple, Body CO, Base CO, and FCO
rhombohedral
hexagonal: hexagonal close packed, FCC closed packed, complex cubic
structure (diamond structure)
– monoclinic: simple and base-centered
– triclinic
• Each crystal system has a different axis length and angles that
separate the atoms
– Table 3.1
Structure Review
• Defining components of a general crystal system
– lengths of the axes (lattice constants) are a, b, and c
– angles between the atom planes are  (alpha),  (beta), and  (gamma)
• Components of the crystalline structure
z
Crystal
Cubic
Axis Length
a=b=c
Angles
 =  =  = 90
Rhombohedral
a=b=c
 =  =   90
Tetragonal
a=bc
 =  =  = 90
Orthorhombic
a=bc
 =  =  = 90
Hexagonal
a=bc
 =  = 90,  = 120
Monoclinic
abc
 =  = 90,   90
Triclinic
abc
      90


c
b
y
x
a

Miller Index
• Defining particular plane of the atom with Miller Index
• 3 steps to determine the Miller Index
–
–
–
–
Find the intercepts x = 2, y = 3, and z = 2 from figure
Take the reciprocal of the axis length, 1/2, 1/3, and 1/2
Find the lowest common multiplier and then multiply the reciprocal
Then, the Miler Index is (3, 2, 3)…… 6*1/2, 6*1/3, 6*1/2.. Note ( )
z
• Example,
c=2
a=2
b=3
y
x
Crystalline Structures
• Every crystal lattice structure has its own unit cell
– Smallest unit into which a lattice structure can be broken down that
sill retains the properties of the whole structure
– Table 3.2. Fourteen Crystal (Bravais) Lattices
•
•
•
•
•
•
Simple Cubic, Body-Centered Cubic, Face-Centered Cubic
Simple Tetragonal, BCT
Simple Orthorhombic, BCO, Base CO, FCO,
Rhombohedral
Hexagonal
Simple Monoclinic, BCM, Triclinic
•
Metal
Crystalline
Structures
Body centered cubic- atom centered in the cube
– Atomic packing factor (APF) is 0.68 and represents the fraction of the unit cell
occupied by the two atoms.
– Ba, Ce, Li, K, molybdenum (less ductile metals)
• Face centered cubic- atom centered on each of the faces
– Atomic packing factor (APF) is 0.74 and represents the fraction of the unit cell
occupied by the two atoms.
– Regular stackings of close-packed planes
• Fourth close pack layer lies precisely above the first one
– Al, Cu, Au, Pb, Ni, Platinum, Ag (soft metals)
• Hexagonal close packed (HCP)
– Two atoms are associated with each Bravais lattice point
• One atom centered within the unit cell and various fractional atoms at unit cells (four
1/6th atoms and four 1/12th atoms)
– Close pack is efficient packing shperes as is the fcc structure.
– Atomic packing factor (APF) is 0.74 and represents the fraction of the unit cell
occupied by the two atoms.
– Regular stackings of close-packed planes
• The third close-packed layer lies precisely above the first.
– Be, Mg, Ti, Zn, Zr
Metal Crystalline Structures
• Hexagonal Close-packed structure (continued)
– the distance between atoms in the bases are equal in the hexagonal structure.
The bases are perpendicular to the sides. The angle between the sides is 120°.
– Graphite has Close-packed hexagonal structure of Carbon
– diamond has a form of face-centered closed -pack cubic structure, or complex
cubic structure (diamond structure)
– other materials with closed-packed hexagonal structure include Be, Cadmium,
Co, Mg, Titanium, Zn, Zirconium
• Simple tetragonal and body centered tetragonal structures
– all atomic planes are still at right angles to each other ,as in cubic structure,
however, one dimension is longer than the other two
– pure tin: natural tetragonal stucture
Metal Crystalline Structures
• Simple orthorhombic
– Lengths of all three axis are different. The places of atoms are perpendicular.
– Can have body centered, face centered, base centered
• Rhombohedral
– None of the planes are perpendicular
• Monoclinic structure
– two atoms of the atomic planes are perpendicular, but the third angle is not 90°.
• Triclinic structure
– no two places are perpendicular to each other, the distances between the atoms are different,
and the angles are not equal
– elongated, thin crystal formation
Metal Crystalline Structures
• Allotropes
– some materials exist in two or more crystalline structures and depends
upon temperature and pressure changes.
– e.g., pure iron is BCC at normal temp and pressure conditions.
Change to FCC if temperature is raised to 1670 F
– allotropes are also known as polymorphs
• Freezing point of material
– As the energy in a liquid system decreases, the forces that are
grouping the atoms tend to form distinct patterns which become the
characteristic lattice structure of the material.
– Formation of lattice crystals produces heat. Lattice grows until it
meets another energy block, e.g., lattice structure or container.
– Grain Boundary- point at which two lattice structures collide.
Grains
Boundaries
Crystalline Structures
• For lattice growth to start, a nucleus (seed), must be present.
– Very pure metals, rapid cooling restricts time for nucleus growth.
– Supercooling occurs if the temperature falls below the melting
temperature. The temperature increases as the nucleus forms and
levels off as the lattice structure evolves.
– Crystal size depends upon: metal type, temperature, Cooling rate:
Rapid cooling = smaller crystals
• Lattice growth occurs more rapidly in directions perpendicular to
each other, called Dendritic nature of crystals
Ceramic Crystalline Structures
• Wide variety of chemical compositions of ceramics is reflected
in their structures
– Ionic packing factor- the fraction of the unit cell volume occupied by
various cations and anions
• Simplest chemical formula, MX, where M is the metallic
element and X is the non-metallic element
– CsCl structure Fig 3-8
– BCC- built with two ions associated with each lattice point
– Does not represent any important ceramic materials
• NaCl is shown in Figure 3-9
– Shares structure of many ceramic materials
– Two intertwining FCC structures, one of Na ions and one of Cl ions.
– Other important ceramic oxides with FCC are
• MgO, CaO, FeO, and NiO
Ceramic Crystalline Structures
• Other chemical formula, MX2, where M is the metallic element and X is the
non-metallic element
– Shares structure of importantceramic materials, Figure 3-10
• CaF2
– Built with an FCC Bravais lattice with three ions (on Ca and two F).
• 12 ions (four Ca and 8 F) per unit cell.
– Other important ceramic oxides with this structure are
• UO2, ThO2, TeO2 [Uranium, Thorium, Tellurium Oxides]
• Ref: http://www-tech.mit.edu/Chemicool/
– Silica is the most important ceramic of this form, MX2.
• Structure is not simple because it is not one structure but many depending
upon temperature and pressure (like iron phase diagram)
• Example, cristobalite (fig 3-11)
– Built upon an FCC Bravais lattice with 6 ions (two Si and 4 O ions)
– 24 ions per unit cell
– Continuously connected network of SiO4 tetrahedra (Note: the sharing
of O2 ions by adjacent tetrahedra gives the overall chemical formula,
SiO2
Ceramic Crystalline Structures
• Other chemical formula, M2X3, where M is the metallic element
and X is the non-metallic element
– Important material- Corundum (Al2O3)
– Rhombohedral Bracais lattice, Fig 3-13, but closely approximated
hexagonal lattice.
– 30 ions per lattice with 12 Al and 18 O.
– Stucture similar to HCP
• Close-packed O– sheets with two-thirds of the small interstices
between sheets filled with Al+++.
• Cr2O3 and Fe2O3 have corundum structure
Ceramic Crystalline Structures
• Other chemical formula, M’M”X3, where M’ and M” are the metallic
elements and X is the non-metallic element
– Important material- Perovskite (CaTiO3)
– Simple cubic Bracais lattice, Fig 3-14,
• Different atoms occupy the corner (Ca++), body centered (Ti4+), and face
centered (O--) positions
– 5 ions per lattice point with one Ca, one Ti, and three O per unit cell.
– Properties
• Have important ferroelectric and piezoelectric properties
– Related to the relative locations of cations and anions as a function of T
• Other chemical formula, M’M”2X4, involves magnetic ceramics based on
the spinel structure (MgAl2O4)- Figure 3-15
– 56 ions per unit cell with 8 Mg, 16 Al, and 32 O
– Important materials- NiAl2O4, ZnAl2O4, and ZnFe2O4
– Mg are in tetrahedral positions that are coordinated by 4 oxygens with the Al in
octrahedral positions
• Other chemical formula, M’’(M’M”)X4, include Fe(MgFe)O4, FeFe2O4
Fe(NiFe)O4, and many other commercially important ferrites or ceramics
Silicate Structures
• Chemical reaction of SiO2 with other ceramic oxides
– Nature of silicate structures is the traditional oxides tend to break up the
continuity of the SiO4 tetrahedra connections.
• The remaining connectedness of tetrahedra may be in the form of silicate
chains or sheets.
• Example, Fig 3-16.
– Kaolinite structure, [2(OH)4Al2Si2O5] is a hydrated aluminosilicate and
a good example of clay mineral
– Structure is typical of sheet silicates
– Built on triclinic Bravais lattice and two laolinite molecules per unit cell
– Many clay minerals have a platelike or flaky structure (fig 3-17) due to
the crystal structure.
• Graphite- Exceptions to general description of ceramics as compounds.
– Layered crystal structure of carbon at room temperature- Fig 3-18
– Graphite is monoatomic it is more ceramic than metallic
– Hexagonal rings of C are bonded strongly with covalent bonds.
» Bonds between layers are of van der Waals type accounting for
graphite’s friable nature and use as dry lubricant.
• Diamond is cubic structure of C (Fig 3-23
Carbon/Graphite Fibers
• Need for reinforcement fibers with strength and modulii
higher than those of glass fibers has led to development of
carbon
• Thomas Edison used carbon fibers as a filament for electric
light bulb
• High modulus carbon fibers first used in the 1950s
• Carbon and graphite are based on layered structures of
hexagonal rings of carbon
• Graphite fibers are carbon fibers that
– Have been heat treated to above 3000°F that causes 3 dimensional
ordering of the atoms and
– Have carbon contents GREATER than 99%
– Have tensile modulus of 344 Gpa (50Mpsi)
Carbon/Graphite Fibers
• Manufacturing Process
– Current preferred methods of producing carbon fibers are from
polyacrylonitrile (PAN), rayon (regenerated cellulose), and pitch.
• PAN
– Have good properties with a low cost for the standard modulus
carbon
– High modulus carbon is higher in cost because high temperatures
required
• PITCH
– Lower in cost than PAN fibers but can not reach properties of PAN
– Some Pitch based fibers have ultra high modulus (725 GPa versus
350GPa) but low strength and high cost (Table 3-2)
Carbon/Graphite Fibers
• PAN Manufacturing Process Figures 3-3 and 3-4
– Polyacrylonitrile (PAN) is commercially available textile fiber and is a ready
made starting material for PAN-based carbon fibers
– Stabilized by thermosetting (crosslinking) so that the polymers do not melt in
subsequent processing steps. PAN fibers are stretched as well
– Carbonize: Fibers are pyrolyzed until transformed into all-carbon
• Heated fibers 1800°F yields PAN fibers at 94% carbon and 6% nitrogen
• Heated to 2300°F to remove nitrogen yields carbon at 99.7% Carbon
– Graphitize: Carried out at temperatures greater than 3200° F to
• Improve tensile modulus by improving crystalline structure and three dimensional
nature of the structure.
– Fibers are surface treated
• Sizing agent is applied
• Finish is applied
• Coupling agent is applied
– Fibers are wound up for shipment
Carbon/Graphite Fibers
• PITCH Manufacturing Process Figure 3-3
– Pitch must be converted into a suitable fiber from petroleum tar
• Pitch is converted to a fiber by going through a meso-phase where the polymer
chains are somewhat oriented though is a liquid state (liquid crystal phase)
• Orientation is responsible for the ease of consolidation of pitch into carbon
– Stabilized by thermosetting (crosslinking) so that the polymers do not melt in
subsequent processing steps
– Carbonize: Fibers are pyrolyzed until transformed into all-carbon
• Heated fibers 1800°F
• Heated to 2300°F
– Graphitize: Carried out at temperatures greater than 3200° F to
• Improve tensile modulus by improving crystalline structure and three dimensional
nature of the structure.
– Fibers are surface treated
• Sizing agent is applied
• Finish is applied
• Coupling agent is applied
– Fibers are wound up for shipment
Carbon Fiber Mechanical Properties
• Table 3-2
Carbon Fiber Mechanical Properties
PAN Based
Tensile Modulus (Mpsi)
33 - 56
Tensile Strength (Msi)
0.48 - 0.35
Elongation (%)
1.4 - 0.6
Density (g/cc)
1.8 - 1.9
Carbon Assay (%)
92 - 100
PITCH Based Rayon Based
23 -55
5.9
0.2 - 0.25
0.15
0.9 - 0.4
25
1.9 - 2.0
1.6
97 - 99
99
Ceramic Crystalline Structures
• Diamond
States of Thermoplastic Polymers
• Amorphous- Molecular structure is incapable of forming regular order
(crystallizing) with molecules or portions of molecules regularly
stacked in crystal-like fashion.
• A - morphous (with-out shape)
• Molecular arrangement is randomly twisted, kinked, and coiled
Amorphous Materials
•
•
•
•
•
•
•
•
•
PVC
Amorphous
PS
Amorphous
Acrylics
Amorphous
ABS
Amorphous
Polycarbonate Amorphous
Phenoxy
Amorphous
PPO
Amorphous
SAN
Amorphous
Polyacrylates
Amorphous
States of Thermoplastic Polymers
• Crystalline- Molecular structure forms regular order (crystals) with
molecules or portions of molecules regularly stacked in crystal-like
fashion.
• Very high crystallinity is rarely achieved in bulk polymers
• Most crystalline polymers are semi-crystalline because regions are
crystalline and regions are amorphous
• Molecular arrangement is arranged in a ordered state
Crystalline Materials
•
•
•
•
•
•
•
•
•
•
•
LDPE
HDPE
PP
PET
PBT
Polyamides
PMO
PEEK
PPS
PTFE
LCP (Kevlar)
Crystalline
Crystalline
Crystalline
Crystalline
Crystalline
Crystalline
Crystalline
Crystalline
Crystalline
Crystalline
Crystalline
Factors Affecting Crystallinity
•
•
•
•
Cooling Rate from mold temperatures
Barrel temperatures
Injection Pressures
Drawing rate and fiber spinning: Manufacturing of
thermoplastic fibers causes Crystallinity
• Application of tensile stress for crystallization of
rubber
• Processing can produce an amorphous structure
from a semi-crystalline material
– Clear PET bottles for soda
– Clear Polyethylene plastic bags
X-Ray Diffraction
• X-ray diffraction is used to determine the crystalline structure of
materials.
• Diffraction is the result of radiation’s being scattered by a regular
array of scattering centers whose spacing is about the same as the
wavelength of the radiation.
– Example, parallel scratch lines spaced repeatedly about 1 m apart causes
diffraction of the visible light (electromagnetic radiation with a wavelength just
under 1 m.
– The diffraction grating causes the light to be scattered with a strong intensity in a
few directions (Fig. 3-33)
• Phenomenon in which the atoms of a crystal, by virtue of their
uniform spacing, cause an interference pattern of the waves in an
incident beam of X rays.
– The crystal's atomic planes act on the X rays in the same way a uniformly ruled
grating acts on a beam of light (see polarization).
– The interference pattern is specific to each substance and gives information on
the structure of the atoms or molecules in the crystal.
X-Ray Diffraction
• Appendix 2 shows that atoms and ions are on the order of
0.1 nm in size
– Crystal structures as being diffraction gratings on a sub nm scale.
– X-ray diffraction characterizes crystalline structures, because
• Fig 3-34. Portion of electromagnetic spectrum with a wavelength in this
range is x-radiation (compared to 1000-nm range for visible light.
– In x-rays, atoms are the scattering centers.
• Mechanism is the interaction of photon of electromagnetic radiation with an
orbital electron in the atom.
• A crystal acts as a 3-dimensional diffraction grating, repeated stacking of
crystal planes serves the same function serves the same purpose as the
parallel scratch lines in Fig 3-33.
– For diffraction to occur, x-ray beams scattered off adjacent crystal
planes must be in phase. Otherwise, destructive interference of
waves occurs which blocks the scattering pattern. (no intensity)
Miller Index
• Defining particular plane of the atom with Miller Index
• 3 steps to determine the Miller Index
–
–
–
–
Find the intercepts x = 2, y = 3, and z = 2 from figure
Take the reciprocal of the axis length, 1/2, 1/3, and 1/2
Find the lowest common multiplier and then multiply the reciprocal
Then, the Miler Index is (3, 2, 3)…… 6*1/2, 6*1/3, 6*1/2.. Note ( )
z
• Example,
c=2
a=2
b=3
y
x
X-Ray Diffraction
• Bragg equation relates the spacing between adjacent crystal
planes and the  angle of diffraction (Bragg angle).
– n = 2d sin , Bragg’s Law,
• Where d is the spacing between adjacent crystal planes and  is the angle of
scattering.  is the wavelength of the x-ray beam. 2 is the diffraction angle.
• Note: William Bragg (1862-1942) and son were first to demonstrate the
dhkl 
power of x-ray diffraction by identifying the chemical structure of NaCl.
Today over 70,000 materials have been identified.
– Magnitude of interplanar spacing, d, is a direct function of the
Miller indices for the plane For a cubic system, the relationship is
simple.
• The spacing between adjacent hkl planes is
dhkl =
a…
(h2+k2+l2)
– Where a is the lattice parameter )edge of the unit cell
X-Ray Diffraction
• Crystal structures with nonprimitive unit cells have atoms at
additional lattice sites located along a unit cell edge, within a unit cell
face, or in the interior of the unit cell.
– Extra scattering can cause out-of phase scattering which will eliminate the
diffraction.
• Table 3.4
Diffraction does not
crystall structure occur when
BCC
FCC
3-39HCP
h+k+l = odd number
h,k,l mixed (enev and
odd numbers)
(h+2k)=3n, l odd
Diffraction occurs
when
h+k+l = even
number
h,k,l unmixed (enev
and odd numbers)
All other cases
• Example, Fig
– Diffraction pattern for a specimen of aluminum powder.
» Each peak represents a solution to Bragg’s law.
» Powder consists of many small crystal grains of oriented randomly, single
wavelength of radiation is used.
» The pattern is compared with a database of known diffraction patterns.
X-Ray Diffraction
• Corrosion and scale analysis by XRD (X-ray diffraction)
– This involves qualitative identification of corrosion products.
• The peaks in the diffraction pattern are checked against a
database of previously identified phases using a search match
program.
• Any chemical data you have available helps to narrow the search scope
http://www.ktgeo.com/tCS.html