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Crystallographic Directions and Planes Chem 253, UC, Berkeley Crystallographic Directions And Planes Lattice Directions Individual directions: [uvw] Symmetry-related directions: <uvw> Chem 253, UC, Berkeley 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} Chem 253, UC, Berkeley (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 4 Chem 253, UC, Berkeley Chem 253, UC, Berkeley Crystal Structure Analysis X-ray diffraction Electron Diffraction Neutron Diffraction Essence of diffraction: Bragg Diffraction 5 Chem 253, UC, Berkeley Interference fringes Light Constructive Destructive Chem 253, UC, Berkeley 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 Chem 253, UC, Berkeley X-Ray Diffraction 7 Chem 253, UC, Berkeley Powder diffraction X-Ray (200) Chem 253, UC, Berkeley Powder diffraction 8 Chem 253, UC, Berkeley Laue Diffraction Single crystal Chem 253, UC, Berkeley 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 Chem 253, UC, Berkeley 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 Chem 253, UC, Berkeley 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 Chem 253, UC, Berkeley 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 12 Chem 253, UC, Berkeley 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 Chem 253, UC, Berkeley 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 Chem 253, UC, Berkeley Reciprocal Lattice BCC FCC Simple hexagonal Simple hexagonal 17 Chem 253, UC, Berkeley bc a 2 a (b c) c a b 2 a (b c) ab c 2 a (bc) Chem 253, UC, Berkeley bc a 2 a (b c) c a b 2 a (b c) ab c 2 a (bc) 18 Chem 253, UC, Berkeley Chem 253, UC, Berkeley 19 Chem 253, UC, Berkeley Wigner-Seitz cells of reciprocal lattice Chem 253, UC, Berkeley 20 Chem 253, UC, Berkeley L X (a 0 0) Chem 253, UC, Berkeley 21 Chem 253, UC, Berkeley 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. Chem 253, UC, Berkeley 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. 22 Chem 253, UC, Berkeley 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) 23.
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