METE 327 Physical Metallurgy Copyright 2008 Loren A. Jacobson 5/16/08
Crystal Symmetries
METE 327 Physical Metallurgy Copyright 2008 Loren A. Jacobson 5/16/08
Why should we be interested?
● Important physical properties depend on crystal structure– Conductivity– Magnetic properties– Stiffness– Strength
● These properties also often depend on crystal orientation
METE 327 Physical Metallurgy Copyright 2008 Loren A. Jacobson 5/16/08
Some Important Terms for Crystals– Crystal Structure– Bravais lattices– Symmetry operations– Basis of Close Packed
Structures– Miller Indices
● Planes● Directions
– Stereographic Projection
– Standard Projection– Principal Metal Structures
—BCC, FCC, HCP– Ionic Crystals– Diamond Structure– Twinned Crystals– Isomorphism– Polymorphism
METE 327 Physical Metallurgy Copyright 2008 Loren A. Jacobson 5/16/08
Bravais Lattices (14 Total)– Cubic—a=b=c, all angles= 90o
– Tetragonal—a=b /=c, all angles=90o – Orthorhombic—a/=b/=c, all angles=90o – Rhombohedral—a=b=c, three equal
angles, / =90o – Hexagonal—a=b/=c, two angles =90o ,
third =120o
– Monoclinic—a/=b/=c, two angles =90o , third not.
– Triclinic-- a/=b/=c, no angles =, nor =90o
METE 327 Physical Metallurgy Copyright 2008 Loren A. Jacobson 5/16/08
Cubic Bravais Lattices
METE 327 Physical Metallurgy Copyright 2008 Loren A. Jacobson 5/16/08
Tetragonal and Orthorhombic
METE 327 Physical Metallurgy Copyright 2008 Loren A. Jacobson 5/16/08
Orthorhombic, Rhombohedral, Hexagonal
METE 327 Physical Metallurgy Copyright 2008 Loren A. Jacobson 5/16/08
Monoclinic, Triclinic
METE 327 Physical Metallurgy Copyright 2008 Loren A. Jacobson 5/16/08
Miller Indices– A Convenient Way to Identify
Crystal Planes and Directions● For Planes, the index is the
reciprocal of the value of the intersection of the plane with a particular axis, converted to whole numbers.
● For Directions, the index is the axis coordinate of the end point of the vector,converted to nearest whole numbers.
METE 327 Physical Metallurgy Copyright 2008 Loren A. Jacobson 5/16/08
Miller Indices (directions)– h,k,and l refer to principal axes, x,y
and z.– Directions are indicated by square
brackets, [hkl]. Families of directions are indicated by <hkl>.
– Example of a family of directions: <100> = [100], [010], [001], [-100], [0-10] and [00-1]● The first three are principal axes,
x,y, and z.
METE 327 Physical Metallurgy Copyright 2008 Loren A. Jacobson 5/16/08
Miller Indices (planes)
– Planes are indicated by parentheses, (hkl) and families of planes by curly brackets, {hkl}.
– Example of a plane family is as follows: {100} = (100), (010), (001), (-100), (0-10), and (00-1)● These are all six faces of a cube.
METE 327 Physical Metallurgy Copyright 2008 Loren A. Jacobson 5/16/08
Calculating Miller Index for planes
xy
z
Intercepts= 0.5, 1, 1/3
Index = (213)
Intercepts= -1, 1, 0.5
Index = (-112)
METE 327 Physical Metallurgy Copyright 2008 Loren A. Jacobson 5/16/08
Calculating Miller Index, directions
End coordinates= 0.5,0.5,1
Index = [112]
End=0,1,0.5
Index=[021]
METE 327 Physical Metallurgy Copyright 2008 Loren A. Jacobson 5/16/08
Hexagonal, Miller-Bravais Indices
a1
a2
a3
c Three a directions, as shown, plus c
Indices are:
(h,k,-(h+k), l)
The plane shown is:
(11-20)
METE 327 Physical Metallurgy Copyright 2008 Loren A. Jacobson 5/16/08
Other Crystal Characterization
– It is often important to determine crystal orientation.● Single crystals.● Individual grains in a polycrystal
– If there is a preferred grain orientation, this is referred to as “texture”.
– One method is to employ the Stereographic Projection.
METE 327 Physical Metallurgy Copyright 2008 Loren A. Jacobson 5/16/08
Stereographic Projection
METE 327 Physical Metallurgy Copyright 2008 Loren A. Jacobson 5/16/08
Some Cubic Crystal Planes
001 Plane
110 Plane
111 Plane
METE 327 Physical Metallurgy Copyright 2008 Loren A. Jacobson 5/16/08
Crystal Symmetries
– Translational Symmetry—a move of one cell in each of 3 axis directions restores the structure.
– Rotational Symmetry—rotation of specific angle (90o, 120o, 180o) restores the structure.
– Mirror Symmetry—reflection across a plane restores the structure.
METE 327 Physical Metallurgy Copyright 2008 Loren A. Jacobson 5/16/08
A simple Cubic Structure(illustrating translational symmetry)
The cubic unit cell
Eight unit cells, the start of a crystal structure.
METE 327 Physical Metallurgy Copyright 2008 Loren A. Jacobson 5/16/08
Rotational Symmetry, Cubic
Four-foldRotation
Two-FoldRotation
Three-FoldRotation
METE 327 Physical Metallurgy Copyright 2008 Loren A. Jacobson 5/16/08
{100} Poles of a Cubic Crystal
METE 327 Physical Metallurgy Copyright 2008 Loren A. Jacobson 5/16/08
Interplanar Angles
Points on the sphere are intersections of plane normals.
METE 327 Physical Metallurgy Copyright 2008 Loren A. Jacobson 5/16/08
Calculating Interplanar Angles(Cubic Crystals)
Cos φ = h1h2 + k1k2 + l1l2
-------------------------- SQRT((h1
2 + k12 + l1
2 )(h22
+ k22 + l2
2 ))
METE 327 Physical Metallurgy Copyright 2008 Loren A. Jacobson 5/16/08
Calculating Interplanar Spacing(Cubic Crystals)
1/d2 = (h2 + k2 + l2)/ a2
METE 327 Physical Metallurgy Copyright 2008 Loren A. Jacobson 5/16/08
Standard (001) Cubic Projection
METE 327 Physical Metallurgy Copyright 2008 Loren A. Jacobson 5/16/08
The Standard Stereographic Triangle
Useful for showing crystal axis orientations.
METE 327 Physical Metallurgy Copyright 2008 Loren A. Jacobson 5/16/08
Metallic Bonding
– Valence, or outer electrons of metallic atoms are distributed throughout the structure. Positively charged metal ions are distributed within this “sea” of electrons.
– This allows metals to be electrical conductors.– There are second nearest neighbor interactions that influence the
crystal structure
METE 327 Physical Metallurgy Copyright 2008 Loren A. Jacobson 5/16/08
Body Centered Cubic (BCC) Metals
α-Fe, Cr, Mo, V, β-Ti, β-Zr
METE 327 Physical Metallurgy Copyright 2008 Loren A. Jacobson 5/16/08
Face-Centered Cubic (FCC) Metals
Cu, Al, Ni, Pb,
γ-Fe
METE 327 Physical Metallurgy Copyright 2008 Loren A. Jacobson 5/16/08
Hexagonal Close-Packed (HCP) Metals
Be, Mg, Zn, Cd, α-Ti, α-Zr
METE 327 Physical Metallurgy Copyright 2008 Loren A. Jacobson 5/16/08
Relation Between FCC and HCPABCABC...
ABABAB...
METE 327 Physical Metallurgy Copyright 2008 Loren A. Jacobson 5/16/08
BCC to HCP Transformation
Close Packed BCC plane {110} becomes Close Packed HCP plane (0002).
METE 327 Physical Metallurgy Copyright 2008 Loren A. Jacobson 5/16/08
BCC to HCP Transformation (2)1.633 a
BCC {110}
1.155 a
1.732 a
a
HCP (0002)In both cases the diagonal, where atoms touch, is of length = 2a, where a is the atom diameter. A small distortion is needed for BCC to transform to HCP. (Note that there can be 6 orientation variants of HCP.)
METE 327 Physical Metallurgy Copyright 2008 Loren A. Jacobson 5/16/08
Types of Transformations
– Displacive, which means that atoms do not have to move very far, and often a shear displacement can lead from one crystal structure to another.
– Replacive, which means that atoms will move some distance, to their new locations and so longer range diffusion is required.
– These topics will be treated in more detail later
METE 327 Physical Metallurgy Copyright 2008 Loren A. Jacobson 5/16/08
a
b
α
a
b
α
c = a x b
a 1
b 3b 3 = a1 x a2 a3 ( a1 x a2)
a 3a 2
c = ab sin a
c = a x b a b = ab cos a
Vector Multiplication Examples
METE 327 Physical Metallurgy Copyright 2008 Loren A. Jacobson 5/16/08
METE 327 Physical Metallurgy Copyright 2008 Loren A. Jacobson 5/16/08
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