Metal Structure: ENT 145 Materials Engineering
Metal Structure: ENT 145 Materials Engineering
Metal Structure: ENT 145 Materials Engineering
Chapter 3 - 1 Chapter 3 - 2
Chapter 3 - 3 Chapter 3 - 4
1
Atomic Packing Factor (APF) Body Centered Cubic Structure (BCC)
Volume of atoms in unit cell* • Atoms touch each other along cube diagonals.
APF = --Note: All atoms are identical; the center atom is shaded
Volume of unit cell differently only for ease of viewing.
*assume hard spheres ex: Cr, W, Fe (), Tantalum, Molybdenum
• APF for a simple cubic structure = 0.52 • Coordination # = 8
volume
atoms atom
a 4
unit cell 1 p (0.5a) 3
3
R=0.5a APF =
a3 volume
close-packed directions
unit cell Click once on image to start animation
Adapted from Fig. 3.2,
Callister & Rethwisch 8e.
contains 8 x 1/8 =
(Courtesy P.M. Anderson)
1 atom/unit cell 2 atoms/unit cell: 1 center + 8 corners x 1/8
Adapted from Fig. 3.24,
Callister & Rethwisch 8e. Chapter 3 - 7 Chapter 3 - 8
2
Theoretical Density, r Theoretical Density, r
Exercise 2.
Iron has a BCC crystal structure, an atomic radius of 0.124 nm,
and an atomic weight of 55.85 g/mol. Compute and compare its
theoretical density with the experimental value found in the
front section of the book.
Chapter 3 - 17 Chapter 3 - 18
Exercise 3.
Rhodium has an atomic radius of 0.1345 nm and a density of
12.41 g/cm3. Determine whether it has an FCC or BCC crystal
structure.
Chapter 3 - 19 Chapter 3 - 20
3
Densities of Material Classes Polycrystalline Anisotropic
In general Graphite/ • Most engineering materials are polycrystalline.
rmetals > rceramics > rpolymers
Metals/ Composites/
Ceramics/ Polymers
Alloys fibers
Semicond
30
Why? Platinum
B ased on data in Table B1, Callister
*GFRE, CFRE, & AFRE are Glass,
20 Gold, W Adapted from Fig. K,
Metals have... Tantalum Carbon, & Aramid Fiber-Reinforced
color inset pages of
Epoxy composites (values based on
• close-packing 60% volume fraction of aligned fibers
Callister 5e.
(Fig. K is courtesy of
10 Silver, Mo in an epoxy matrix).
(metallic bonding) Cu,Ni
Steels
Paul E. Danielson,
Teledyne Wah Chang
• often large atomic masses Tin, Zinc
Zirconia Albany)
r (g/cm3 )
5
Ceramics have... 4
Titanium
Al oxide
1 mm
• less dense packing 3
Diamond
Si nitride
Aluminum
• often lighter elements Glass -soda
Concrete
Silicon PTFE
Glass fibers
GFRE*
2
Polymers have... Magnesium G raphite
Silicone
Carbon fibers
CFRE*
• Nb-Hf-W plate with an electron beam weld. Isotropic
PVC A ramid fibers
• low packing density AFRE *
• Each "grain" is a single crystal.
PET
1 PC
(often amorphous) HDPE, PS
PP, LDPE
• lighter elements (C,H,O) • If grains are randomly oriented,
Composites have...
0.5 Wood overall component properties are not directional.
0.4
• intermediate values 0.3
• Grain sizes typically range from 1 nm to 2 cm
Data from Table B.1, Callister & Rethwisch, 8e.
(i.e., from a few to millions of atomic layers).
Chapter 3 - 21 Chapter 3 - 22
4
Point Coordinates
z
111 Point coordinates for unit cell
c center are
a/2, b/2, c/2 ½½½
000
y
a b
Point coordinates for unit cell
x corner are 111
z 2c
Translation: integer multiple of
lattice constants identical
b y position in another unit cell
b
Chapter 3 - Chapter 3 - 28
Point Coordinates
Point Coordinates
tableun_03_p57
Crystallographic Directions
z Algorithm
1. Vector repositioned (if necessary) to pass
through origin.
2. Read off projections in terms of
unit cell dimensions a, b, and c
y 3. Adjust to smallest integer values
4. Enclose in square brackets, no commas
x [uvw]
5
Crystallographic Planes 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. If the plane passes through selected origin, either
another parallel plane must be constructed within
the unit cell or new origin must be construct.
2. Read off intercepts of plane with axes in
terms of a, b, c
3. Take reciprocals of intercepts
4. Reduce to smallest integer values
5. Enclose in parentheses, no commas i.e., (hkl) Adapted from Fig. 3.10,
Callister & Rethwisch 8e.
Chapter 3 - 33 Chapter 3 - 34
reflections must
be in phase for
a detectable signal
extra Adapted from Fig. 3.20,
q q
distance
Callister & Rethwisch 8e.
travelled
by wave “2” spacing
dhkl between
planes
Braggs Law
n
dhkl = = wavelength
• Diffraction gratings must have spacings comparable to 2 sin q
the wavelength of diffracted radiation. dhkl = Interplanar spacing
a
• Can’t resolve spacings dhkl =
a= lattice parameter
√h2 + k2 + l2
• Spacing is the distance between parallel planes of n = order of reflection
atoms.
Chapter 3 - 37 Chapter 3 - 38
6
Exercise 3.
X-Ray Diffraction Pattern Using the data for molybdenum in Table 3.1, compute
z z z
c c c the interplanar spacing for the (111) set of planes
y (110) y y
a b a b a b
Intensity (relative)
x x x (211)
(200)
Diffraction angle 2q
Chapter 3 - 39 Chapter 3 - 40
Exercise 4.
Determine the expected diffraction angle for the first-
order reflection from the (113) set of planes for FCC
platinum when monochromatic radiation of wavelength
0.154 nm is used.
Chapter 3 - 41 Chapter 3 - 42
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.
Chapter 3 - 43 Chapter 3 - 44
7
SUMMARY ANNOUNCEMENTS
Reading:
• 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 Core Problems:
with randomly oriented grains.
• Some materials can have more than one crystal
structure. This is referred to as polymorphism (or Self-help Problems:
allotropy).
• X-ray diffraction is used for crystal structure and
interplanar spacing determinations.
Chapter 3 - 45 Chapter 3 - 46