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Crystallographic Directions And Planes
Lattice Directions Individual directions: [uvw] Symmetry-related directions:
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1 Chem 253, UC, Berkeley
Crystallographic Directions And Planes
Miller Indices:
1. Find the intercepts on the axes in terms of the lattice constant a, b, c 2. Take the reciprocals of these numbers, reduce to the three integers having the same ratio (hkl)
Set of symmetry-related planes: {hkl}
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(100) (111)
(200) (110)
2 Chem 253, UC, Berkeley
Chem 253, UC, Berkeley
Crystallographic Directions And Planes
Miller-Bravais indices [uvtw], (hkil) t=-(u+v) i=-(h+k)
3 Chem 253, UC, Berkeley
In cubic system,
[hkl] direction perpendicular to (hkl) plane
Chem 253, UC, Berkeley Lattice spacing
1 h2 k 2 l 2 2 2 For cubic system dhkl a
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Chem 253, UC, Berkeley
Crystal Structure Analysis
X-ray diffraction
Electron Diffraction
Neutron Diffraction
Essence of diffraction: Bragg Diffraction
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Interference fringes Light
Constructive
Destructive
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Bragg’s Law
n SQ QT
dhkl sin dhkl sin
2dhkl sin
For cubic system: a But not all planes have the dhkl diffraction !!! h2 k 2 l 2
6 Chem 253, UC, Berkeley X-Ray Diffraction
E h hc /
Mo: 35KeV ~ 0.1-1.4A Cu K 1.54 A
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X-Ray Diffraction
7 Chem 253, UC, Berkeley Powder diffraction
X-Ray
(200)
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Powder diffraction
8 Chem 253, UC, Berkeley Laue Diffraction
Single crystal
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Electron Diffraction
9 Chem 253, UC, Berkeley Layered Cuprates Thin film, growth oriented along c axis Example: La2CuO2
Cu K 1.54 A 2d sin n
2*theta d (hkl)
7.2 12.1 (001)
14.4 6.1 (002)
(00l) 22 4.0 (003) c=12.2 A
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c
12.2 A CuO2 LaO
LaO CuO2
10 Chem 253, UC, Berkeley
Layered Cuprates Example: Ca0.5Sr0.5CuO2 Thin film, growth oriented along c axis 2d sin n
2*theta d (hkl)
12.7 6.96 (001)
26 3.42 (002) (00l) 42.2 2.15 (003) c=6.96 Å
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c
Sr 6.96 Å CuO2 Ca
11 Chem 253, UC, Berkeley What we will see in XRD of simple cubic, BCC, FCC?
2 sin A 2 0.5 (BCC); 0.75 (FCC) sin B
a dhkl h2 k 2 l 2
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Reciprocal Lattice
n ' d n R R n1 a n2 b n3 c Path Difference: R n R n' R(n n' ) m
2 R (n n' ) 2m
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Reciprocal Lattice
' 2 k k n d R R n a n b n c 1 2 3 2 R (n n' ) 2m R (k k ' ) 2m Correspond to plane wave: ' ei R( k k ) ei2m 1
Chem 253, UC, Berkeley Reciprocal Lattice
' k k n ' d n R R n1 a n2 b n3 c ' ei R( k k ) 1 Laue Condition
' K k k Reciprocal lattice vector iKR e 1 For all R in the Bravais Lattice
13 Chem 253, UC, Berkeley Reciprocal Lattice
k ' d K Reciprocal lattice vector R k
i R( k k ' ) e 1 Laue Condition
' K k k Reciprocal lattice vector iKR e 1 For all R in the Bravais Lattice
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Reciprocal Lattice iKR e 1 For all R in the Bravais Lattice A reciprocal lattice is defined with reference to a particular Bravias Lattice.
a b c Primitive vectors bc a 2 a (bc) c a b 2 a (bc) ab c 2 a (bc)
14 Chem 253, UC, Berkeley Reciprocal Lattice
iKR e 1 For all R in the Bravais Lattice
bc a 2 a (bc) a a 2 c a Verify: b 2 a (bc) a b 0 ab c 2 a c 0 a (bc)
For any K k1a k2b k3c R n1 a n2 b n3 c
K R 2 (k1n1 k2n2 k3n3 )
Chem 253, UC, Berkeley Reciprocal Lattice Reciprocal lattice is always one of 14 Bravais Lattice.
a a x b a y Simple cubic c a z
bc 2 a 2 x a (bc) a c a Simple cubic b 2 2 a (bc) y ab a c 2 2 a (bc) z a
15 Chem 253, UC, Berkeley
Primitive Cell of FCC
1 a a(x y) 1 2 1 a a(z y) 2 2
1 a a(z x) 3 2
o •Angle between a1, a2, a3: 60
Chem 253, UC, Berkeley Primitive Cell of BCC •Rhombohedron primitive cell
0.53a
1 a a (x y z109) o28’ •Primitive Translation Vectors: 1 2 1 1 a a ( x y z) a a (x y z) 2 2 3 2
16 Chem 253, UC, Berkeley Reciprocal Lattice Reciprocal lattice is always one of 14 Bravais Lattice. a a (y z) 2 a b (x z) 2 FCC a c (x y) 2
bc 4 1 a 2 (y z x) a (bc) a 2 c a b 2 4 1 a (bc) (x z y) BCC a 2 ab 4 1 c 2 (x y z) a (bc) a 2
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Reciprocal Lattice
BCC FCC
Simple hexagonal Simple hexagonal
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bc a 2 a (b c)
c a b 2 a (b c)
ab c 2 a (bc)
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bc a 2 a (b c)
c a b 2 a (b c)
ab c 2 a (bc)
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Chem 253, UC, Berkeley
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Wigner-Seitz cells of reciprocal lattice
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L
X (a 0 0)
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Theorem: For any family of lattice planes separated by distance d, there are reciprocal lattice vectors perpendicular to the planes, the shortest being 2/d.
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Orientation of plane is determined by a normal vector The miller indices of a lattice plane are the coordination at the reciprocal lattice vector normal to the plane.
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1 1 1 Plane (hkl) (hkl) ( , , ) x1 x2 x3
x2b B K ha kb lc 2 2 2 a b c x1 x2 x3 x3c A C x a 1 bc a 2 K AB AC (x1a x2b)(x1a x3c) a (bc) 1 1 1 c a a b c b 2 x1 x2 x3 a (bc) ab c 2 K ha kb lc a (bc)
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