2-1 Basic concept Crystal structure = lattice structure + basis Lattice points: positions (points)...
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Transcript of 2-1 Basic concept Crystal structure = lattice structure + basis Lattice points: positions (points)...
2-1 Basic concept Crystal structure = lattice structure + basis Lattice points: positions (points) in the structure which are identical.
II Crystal Structure
X X X X
X X X X
X X X X
X X X X
Not a lattice pointlattice point
Lattice translation vector Lattice plane Unit cell Primitive unit cell【 1 lattice point/unit cell】 Examples : CsCl
CS.
Fe (ferrite) b.c.c (not primitive)Al f.c.c (not primitive)Mg h.c.p simple hexagonal latticeSi diamond f.c.c (not primitive)
T
1T
2T
21 12 TTT
'21 03.143.2 TTT
'
2T
'21 TT
Cartesian coordinate
Rational direction
integer
Use lattice net to describe is much easier!
Rational direction & Rational plane are chosen to describethe crystal structure!
2-2 Miller Indices in a crystal (direction, plane) 2-2-1 direction
The direction <u v w>are all the [u v w] types of direction, which are crystallographic equivalent.
The direction [u v w] is expressed as a vector cwbvaur ˆˆˆ
2-2-2 plane The plane (h k l) is the Miller index of the plane in the figure below.
x
ha /
c
y
z)(hkl
a
ha /
lc /
kb /
1///
lc
z
kb
y
ha
x
{h k l} are the (h k l) types of planes which are crystllographic equivalent.
http://www.materials.ac.uk/elearning/matter/Crystallography/IndexingDirectionsAndPlanes/index.html
You can practice indexing directions and planesin the following website
(hkl) Individual plane
Symmetry related set{hkl}
DifferentSymmetry related set
)100(
)010(
)001(
)001(
)010(
)100(
)100( )001(
)001( )100(
xy
z
x y
z
{100}
{100}
Crystallographic equivalent?
Example:
2-2-3 meaning of miller indices
)200(
)100(
100200 2
1dd
>Low index planes are widely spaced.
x
(110)
[120]
>Low index directions and planes areimportant for slip, cross slip, and electronmobility.
>Low index directions correspond to short lattice translation vectors.
x
y
[110]
[120]
2-3 Miller Indices and Miller - Bravais Indices(h k l) (h k i l)
2-3-1 in cubic system(1)Direction [h k l] is perpendicular to (h k l) plane in the cubic system, but not true for other crystal systems.
x y
|x| = |y|
[110] (110) [110]
|x| |y|
(110)yx
2-3-2 In hexagonal system using Miller - Bravais indexing system: (hkil) and [hkil]
[100]
[010]x
y
[110]
Reason (i): Type [110] does not equal to [010] , but these directions are crystallographic equivalent.Reason (ii): z axis is [001], crystallographically distinct from [100] and [010]. (This is not a reason!)
Miller indces
If crystal planes in hexagonal systems are indexed using Miller indices, then crystallographically equivalent planes have indices which appear dissimilar. Þ the Miller-Bravais indexing system (specific for hexagonal system)
http://www.materials.ac.uk/elearning/matter/Crystallography/IndexingDirectionsAndPlanes/indexing-of-hexagonal-systems.html
2-3-3 Miller-Bravais indices
The direction [h k i l] is expressed as a vector zluiykxhr ˆˆˆˆ
x
y
u
[211 0]
[1120 ]
[121 0]
𝑎3
[2 110 ]Note : is the shortest translation vector on the basal plane.
(a) direction
u
x
y
𝑎3
[2 110 ] 𝑎3
[1 210 ]
You can check
[100] [010]
(b) planes (h k i l) ; h + k + i = 0
Plane (h k l) (h k i l)
(100)
(100) x
y
u
x
y
u
(010)
planeplane
x
y
u
? plane(2110)(10)
(h k l) (h k i l)
Proof for general case:
x
y
u
For plane (h k l), the intersection with the basal plane (001) is a line that is expressed as
𝑥1h
+𝑦1𝑘
=1line equation :
Where we set the lattice constant a = b = 1 in the hexagonal lattice for simplicity.
line equation :
Check back to page 5
The line along the axis can be expressed as
𝑥=𝑦line equation : or 𝑥− 𝑦=0
h𝑥+𝑘𝑦=1 𝑥− 𝑦=0Intersection points of these two lines
and
()is at
The vector from origin to the point can be expressed along the axis as
i = - (h + k) x
y
u
x
y
u()
1h
1𝑘
1h+𝑘
11/h
:1
1/𝑘:
− 11/(h+𝑘)
=h :𝑘:−(h+𝑘)
xy
1h+𝑘
1h+𝑘
?
(c) Transformation from Miller [x y z] to Miller-Bravais index [h k i l]
rule
h=2𝑥− 𝑦
3
𝑘=2 𝑦−𝑥
3
𝑖=−(𝑥+𝑦 )
3
𝑧=𝑙
x
y
u
The same vector is expressed as [x y z] inmiller indices and as [h k I l] inMiller-Bravais indices!
Proof:
[ 𝑥 𝑦 𝑧 ]=𝑥 ��+𝑦 ��+ 𝑧 ��[ h𝑘 𝑖𝑙 ]=h ��+𝑘 ��+𝑖 ��+𝑙 ��
¿h ��+𝑘 ��+𝑖 (− ��− �� )+𝑙 ��¿ (h− 𝑖 ) ��+(𝑘− 𝑖 ) ��+𝑙 ��
𝑥=h −𝑖=h+h+𝑘=2h+𝑘𝑦=𝑘−𝑖=𝑘+h+𝑘=h+2𝑘𝑧=𝑙
Moreover,
h=2𝑥− 𝑦
3
𝑘=2 𝑦−𝑥
3 𝑖=
−(𝑥+𝑦 )3
𝑧=𝑙
2-4 Stereographic projections
N
S
Horizontalplane
[h𝑘𝑙 ]
[𝑢𝑣𝑤 ]
[𝑢𝑣𝑤 ]
[h𝑘𝑙 ]
[𝑢𝑣𝑤 ]
[𝑢𝑣𝑤 ]
2-4-1 direction
Representation of relationship of planes and directions in 3D on a 2D plane. Useful for theorientation problems.
A line (direction) a point.
(100)
2-4-2 planeGreat circle: the plane passing through the center of the sphere.
S
N
A plane (Great Circle) trace
http://courses.eas.ualberta.ca/eas233/0809winter/EAS233Lab03notes.pdf
Small circle: the plane not passing through the center of the sphere.
S
N
B.D. Cullity
Example: [001] stereographic projection; cubic
B.D. Cullity
Zoneaxis
2-4-2 Stereographic projection of different Bravais systems
Cubic
How about a standard (011) stereographicprojection of a cubic crystal?
��
�� ��
Start with what you know!
What does (011) look like?
��
�� ��
(011)
[100]
[01]
[]
[]
(011)
[][11]109.47o
70.53o
��
�� ��
[100]
[01]
[] (011)
[011][][001]45o
[011] [001] [01]
��
�� ��
[01]
[] (011)
[011] []
35.26o
[011] [111] [100]
[100][111]
1 10
111
HexagonalTrigonal 3a
3a
c[0001]
[110]
[111]
tan−1( 3𝑎𝑐 )
Orthorhombic Monoclinic
2-5 Two convections used in stereographic projection
(1) plot directions as poles and planes as great circles
(2) plot planes as poles and directions as great circles
2-5-1 find angle between two directions
(a) find a great circle going through them(b) measure angle by Wulff net
Meridians: great circle
Parallels except theequator are small circles
Equal anglewith respectto N or S pole
Measure the anglebetween two points:
Bring these two points onthe same great circle; counting the latitude angle.
(i) If two poles up
(ii) If one pole up, one pole down
angle
angle
2-5-2 measuring the angle between planes This is equivalent to measuring angle between poles
http://en.wikipedia.org/wiki/Pole_figure
Pole and trace
Angle between the planes of two zone circles is the anglebetween the poles of the corresponding
use of stereographic projections(i) plot directions as poles
---- used to measure angle between directions
---- use to establish if direction lie in a particular plane
(ii) plot planes as poles---- used to measure angles between
planes---- used to find if planes lies in the
same zone