CH 03 Fundamentals of Materials Science & Engineering
CH 03 Fundamentals of Materials Science & Engineering
CH 03 Fundamentals of Materials Science & Engineering
CHAPTER
3
Crystal Structures and Crystal Geometry
3-1
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Face centered
Figure 3.2
Tetragonal
a =b c = = = 900
Rhombohedral
a =b = c = = 900
Figure 3.2
Simple
3-4
Body Centered
3-5
Simple
After W.G. Moffatt, G.W. Pearsall, & J. Wulff, The Structure and Properties of Materials, vol. I: Structure, Wiley, 1964, p.47.)
After W.G. Moffatt, G.W. Pearsall, & J. Wulff, The Structure and Properties of Materials, vol. I: Structure, Wiley, 1964, p.47.)
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Monoclinic
a b c = = = 900 Simple
Triclinic
a b c = = = 900
Figure 3.2
FCC Structure
Figure 3.3
HCP Structure
3-7
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4R 3
Figure 3.5
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Vatoms =
4R 3 2. 3
= 8.373R3
3
V unit cell = a3 = 4 R 3
= 12.32 R3
Therefore APF =
3-10
3-11
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Atoms contact each other across cubic face diagonal Therefore, lattice 4 R constant a =
3-12
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Atom positions are located using unit distances along the axes.
Figure 3.10 b
3-15
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(0,0,0)
The direction indices are [212] Convert them to smallest possible integer by multiplying by an integer.
NO Are all are integers? YES Are any of the direction vectors negative?
YES Represent the indices in a square bracket without comas with a over negative index (Eg: [121])
3-17
NO
Represent the indices in a square bracket without comas (Eg: [212] )
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Miller Indices
Miller Indices are are used to refer to specific lattice planes of atoms. They are reciprocals of the fractional intercepts (with fractions cleared) that the plane makes with the crystallographic x,y and z axes of three nonparallel edges of the cubic unit cell. z Miller Indices =(111)
y x
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Figure 3.14
Enclose in parenthesis (hkl)where h,k,l are miller indicesof cubic crystal plane forx,y and z axes. Eg: (111)
3-20 3-21
Intercepts of the plane at x,y & z axes are 1, and Taking reciprocals we get (1,0,0). Miller indices are (100). ******************* Intercepts are 1/3, 2/3 & 1. taking reciprocals we get (3, 3/2, 1). Multiplying by 2 to clear fractions, we get (6,3,2). Miller indices are (632).
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Figure EP3.7 a
To show this plane a single unit cell, the origin is moved along the positive direction of y axis by 1 unit.
(110)
x
Figure EP3.7 c
3-22
3-23
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[110]
(110)
y
Figure EP3.7b
Interplanar spacing between parallel closest planes with same miller indices is given by
d
3-24
hkl
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Third layer of Atoms placed Third layer of Atoms placed in b Voids of plane B. (Identical in a voids of plane B. Resulting to plane A.) HCP crystal. In 3rd Plane C. FCC crystal.
Plane A Plane B Plane A Figure 3.19 a&b 3-28
After W.G. Moffatt, G.W. Pearsall, & J. Wulff, The Structure and Properties of Materials, vol. I: Structure, Wiley, 1964, p.51.)
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Volume Density
Volume density of metal =
Example:- Copper (FCC) has atomic mass of 63.54 g/mol and atomic radius of 0.1278 nm. a=
4R
4 0.1278nm
= 0.361 nm
Example:- In Iron (BCC, a=0.287), The (100) plane intersects center of 5 atoms (Four and 1 full atom). Equivalent number of atoms = (4 x ) + 1 = 2 atoms Area of 110 plane = 2 a a = 2a 2
=
Figure 3.22 a&b
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2 2 (0.287 )
2
17.2atoms nm 2
v
3-30
m V
= 8.98
Mg m3
= 8.98
g cm3
1.72 1013 mm 2
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Polymorphism or Allotropy
Metals exist in more than one crystalline form. This is caller polymorphism or allotropy. Temperature and pressure leads to change in crystalline forms. Example:- Iron exists in both BCC and FCC form depending on the temperature.
Liquid Iron
Number of atomic diameters intersected by selected length of line in direction of interest Selected length of line
Example:- For a FCC copper crystal (a=0.361), the [110] direction intersects 2 half diameters and 1 full diameter. Therefore, it intersects + + 1 = 2 atomic diameters. Length of line =
2 0.361nm
= 3.92 10 6 atoms mm
-2730C 9120C 13940C 15390C
3-32
2atoms 2 0.361nm
3.92atoms nm
Iron BCC
3-33
Iron FCC
Iron BCC
Figure 3.23
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35 KV
(Eg: Molybdenum)
Figure 3.26
Figure 3.25
3-34
After B.D. Cullity, Elements of X-Ray Diffraction, 2d ed., Addison-Wesley, 1978, p.23.
3-35
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X-Ray Diffraction
Crystal planes of target metal act as mirrors reflecting X-ray beam. If rays leaving a set of planes are out of phase (as in case of arbitrary angle of incidence) no reinforced beam is produced. If rays leaving are in phase, reinforced beams are produced.
Figure 3.28
3-36
After A.G. Guy and J.J. Hren, Elements of Physical Metallurgy, 3d ed., Addison-Wesley, 1974, p.201.)
= 2 dhkl.Sin
3-37
After A.G. Guy and J.J. Hren, Elements of Physical Metallurgy, 3d ed., Addison-Wesley, 1974, p.201.)
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hkl
a h2 + k 2 + l 2
2 ( h A 2 + k A 2 + l A 2 )
4a 2
Since = 2dSin
Substituting for d,
(For plane A)
2aSin h2 + k 2 + l 2
Sin 2 B =
2 ( h B 2 + k B 2 + l B 2 )
4a 2
(For plane B)
2 Therefore Sin =
2 h 2 + k 2 + l 2
4a 2
Note that the wavelength and lattice constant a are the same For both incoming and outgoing radiation.
Sin 2 A Sin 2 B
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(hA + k A + l A )
2 2 2
( hB + k B + l B )
2 2 2
3-38
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Figure 3.30
3-40
After A.G. Guy Essentials of Materials Science, McGraw-Hill, 1976.
3-41
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Unknown metal
Crystallographic Analysis
For FCC crystals the first two sets of diffracting planes are {111} and {200} planes Therefore
Sin2 A Sin B
2
= 0.75
Sin 2 A Sin 2 B
= 0 .5
Sin 2 A Sin 2 B
(12 + 12 + 12 ) (2 2 + 02 + 02 )
= 0.75
3-42
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Amorphous Materials
Glass is a ceramic made up of SiO4 4tetrahedron subunits limited mobility. Rapid cooling of metals (10 8 K/s) can give rise to amorphous structure (metallic glass). Metallic glass has superior metallic properties.