Crystal Mechanics
Lecture 5 – Orientation of Crystallites
Ref : Texture and Related Phenomena D N Lee 2006 Ref : Texture and Related Phenomena, D. N. Lee, 2006
Quantitative Texture Analysis, H.J. Bunge & C. Esling, 1979
Texture and Anisotropy, U.F. Kocks, C.N. Tome H.-R. Wenk, 1998
Heung Nam Han
Associate Professor
School of Materials Science & Engineering School of Materials Science & Engineering
College of Engineering Seoul National University
S 1 1 44
Seoul 151-744, Korea Tel : +82-2-880-9240 Fax : +82-2-885-9647
2010-09-27
email : [email protected]
Crystallographic Coordinates
• position: fractional multiples of the unit cell edge lengths ) P:
ex) P: q,r,s
cubic unit cell
Crystallographic Directions
• a line between two points or a vector
• [uvw] square bracket, smallest integer
• families of directions: <uvw> angle bracket
cubic cubic
Crystallographic Planes (Miller Index)
z
p m00 0n0 00p: define lattice plane
c
m00, 0n0, 00p: define lattice plane m, n, ∞ : no intercepts with axes
a
b
n yx m x
Plane (hkl)
Family of planes {hkl}
Miller indicies ; defined as the smallest integral multiples of
the reciprocals of the plane Family of planes {hkl} the reciprocals of the plane
intercepts on the axes
Crystallographic Planes
A B C D
A B C D
E F
{110} Family
{110} Family
Definite of crystal orientation Definite of crystal orientation
Specimen coordinate KA =KS and crystal coordinate KB =KC
Crystal orientation
Specimen reference frame
Crystal orientation
Specimen reference frame
NDK
sND
Normal direction
K
sTD RD
Crystal reference frame
Rolling direction
TD
Transverse direction
Crystal reference frame
Crystal orientation Euler angles
Crystal orientation g
g ={
1
2}
Crystal orientation Euler angles and
Crystal orientation E l
Euler space
Each point in this space represents a specific choice of the parameters p
11
22and hence a specific crystal orientation.
1
360
1[0 -360 ]
g ={1 2}
1[ ]
[0 -180 ]
2
2
360
180
2[0 -360 ]
Miller Indices
Crystal orientation
K {110}<001>
Crystal orientation
Ks {110}<001>
Description of the grain orientation g= (hkl)[uvw]
g={hkl}<uvw>
g={hkl}<uvw>
Definition of the Pole Figure and
Definition of the Pole Figure and
Inverse Pole Figure
Crystal orientation Matrix Representation
Crystal orientation p
u ' r ' h '
RD TD ND
v ' s ' k ' g
w ' t ' l '
R t ti f i t ti b ND TD
Representation of orientation g by ND, TD, RD parallel to [hkl], [rst], [uvw] respectively.
Crystal orientation Matrix Representation
Crystal orientation p
x u ' r ' h ' X
Z Y X
l t
k s
v
h r
u y
x
' '
'
' '
'
z w ' t ' l ' Z
X u v w x
X ' ' ' Y
u v w
r s t
x y
' ' '
Z h k l z
' ' '
[XYZ] in specimen coord.
[x,y,z] in crystal coord.
Crystal orientation Crystal orientation
i 0 1 0 0 i 0
'
2
g
Zg
X ' '1
g
Z
sin cos 0
0 sin
cos sin
cos 0
0 0
1 0
cos sin
0 sin
cos
1 1
1 1
2 2
2 2
g
0 0 1 0 sin
cos
0 0 1 i i i i i i
cos cos sin sin cos sin cos cos sin cos sin sin
cos sin sin cos cos sin sin cos cos cos cos sin
1 2 1 2 1 2 1 2 2
1 2 1 2 1 2 1 2 2
sin sin
1 cos sin
1cos
Invariant measure Invariant measure
The invariant measure is introduced because the orientation transformation is combined with a distortion orientation transformation is combined with a distortion.
For the Euler angles, the invariant measure is
I(
1
2)=sin
I(
1
2) sin
Normalization factor Normalization factor
i
2
d d d
22
8
02
0
02sin d
1d d
2 8 2
dg 1 1 d d d
8
2sin
1 2If we add the normalization factor and invariant measure
8
If we add the normalization factor and invariant measure for the orientation element in Euler space, the orientation
can be transformed without a distortion.
How can Orientation Distributions How can Orientation Distributions (Texture) be represented?
dg g
d f
) v (
v
f(g) : the orientation distribution function f(g) : the orientation distribution function
f(g)=1 for random distribution.
How Texture is represented?
(110) Pole Figure
ODF :
Orientation Distribution Function Function
