Solid State Physics 460- Lecture 2a Structure of Crystals (Kittel Ch. 1)
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1 Solid State Physics 460- Lecture 2a Structure of Crystals (Kittel Ch. 1) See many great sites like ob s rock shop with pictures and crystallography info: Physics 460 F 2006 Lect 2
2 From Last Time Crystals A crystal is a repeated array of atoms Crystal Lattice + asis Lattice of points Crystal (ravais Lattice) asis of atoms Crystals can be classified into a small number of types See text for more details Physics 460 F 2006 Lect 2
3 From Last Time Two Dimensional Crystals φ asis Lattice Infinite number of possible lattices and crystals Finite number of possible lattice types and crystal types Physics 460 F 2006 Lect 2a 3
4 From Last Time Possible Two Dimensional Lattices φ General oblique Hexagonal Φ = 60, = 6-fold rotation, reflections Square 4-fold rot., reflect. Rectangular 2-fold rot., reflect. Centered Rectangular 2-fold rot., reflect. These are the only possible special crystal types in two dimensions Physics 460 F 2006 Lect 2a 4
5 Three Dimensional Lattices z a 3 y Every point on the ravais lattice is a multiple of 3 primitive lattice vectors T(n 1,n 2,n 3 ) = n 1 + n 2 + n 3 a 3 where the n s are integers X Physics 460 F 2006 Lect 2a 5
6 Three Dimensional Lattices Simplest examples a 3 a 3 Simple Orthorhombic ravais Lattice Hexagonal ravais Lattice Orthorhombic: angles 90 degrees, 3 lengths different Tetragonal: 2 lengths same Cubic: 3 lengths same Hexagonal: a 3 different from, by symmetry Physics 460 F 2006 Lect 2a 6
7 Length of each side - a Cubic Lattices a 3 a a a Simple Cubic Primitive lattice vectors = (1,0,0) a = (0,1,0) a a 3 = (0,0,1) a One atom per cell at position (0,0,0) ody Centered Cubic (CC) Conventional Cell with 2 atoms at positions (000), (1/2,1/2,1/2) a Physics 460 F 2006 Lect 2a 7
8 Length of each side - a Cubic Lattices a 3 a 3 Simple Cubic Primitive lattice vectors = (1,0,0) a = (0,1,0) a a 3 = (0,0,1) a One atom per cell at position (0,0,0) Face Centered Cubic (FCC) Conventional Cell with 4 atoms at positions (000 ), (0,1/2,1/2), (1/2,0,1/2), (1/2,1/2,0)a Physics 460 F 2006 Lect 2a 8
9 Face Centered Cubic Two views - Conventional Cubic Cell z z y y X X Conventional Cell of Face Centered Cubic Lattice 4 times the volume of a primitive cell Physics 460 F 2006 Lect 2a 9
10 Face Centered Cubic (fcc) Also called cubic closed packed (ccp) z z y y X X Each atom has 12 equal neighbors We will see later that this is a close packed lattice Physics 460 F 2006 Lect 20
11 Face Centered Cubic z y a 3 X One Primitive Cell Primitive lattice vectors = (1/2,1/2,0) a = (1/2,0,1/2) a a 3 = (0,1/2,1/2) a One atom per cell at position (0,0,0) Wigner-Seitz Cell Physics 460 F 2006 Lect 21
12 ody Centered Cubic z a 3 y X One Primitive Cell Primitive lattice vectors = (1/2,1/2,-1/2) a = (1/2, -1/2,1/2) a a 3 = (-1/2,1/2,1/2) a One atom per cell at position (0,0,0) Wigner-Seitz Cell Physics 460 F 2006 Lect 22
13 Lattice Planes - Index System s 3 a 3 s 1 s 2 Plane through the points s 1, s 2, s 3 a 3 Each s can be an integer or a rational fractions Define the plane by the reciprocals 1/s 1, 1/s 2, 1/s 3 Reduce to three integers with same ratio h,k,l Plane is defined by (h,k,l) Physics 460 F 2006 Lect 23
14 Schematic illustrations of lattice planes Lines in 2d crystals Lattice (01) φ (14) (02) (22) (11) asis Infinite number of possible planes Can be through lattice points or between lattice points Physics 460 F 2006 Lect 24
15 Schematic illustrations of lattice planes Lines in 2d crystals Lattice φ Equivalent parallel planes (14) (01) (02) Low index planes: more dense, more widely spaced High index planes: less dense, more closely spaced asis Physics 460 F 2006 Lect 25
16 Schematic illustrations of lattice planes Lines in 2d crystals (01) (02) (14) Planes slice through the basis of physical atoms Physics 460 F 2006 Lect 26
17 Lattice planes in cubic crystals z z y y X X (100) and (110) planes in a cubic lattice (illustrated for the fcc lattice) Physics 460 F 2006 Lect 27
18 (111) lattice planes in cubic crystals z y X Face Centered Cubic Lattice Lattice planes perpendicular to [111] direction Each plane is hexagonal close packed array of points Physics 460 F 2006 Lect 28
19 Stacking hexagonal 2d layers to make close packed 3-d crystal A C Each sphere has 12 equal neighbors 6 in plane, 3 above, 3 below Close packing for spheres Can stack each layer in one of two ways, or C above A Also see figure in Kittel Physics 460 F 2006 Lect 29
20 Stacking hexagonal 2d layers to make hexagonal close packed (hcp) 3-d crystal Each sphere has 12 equal neighbors Close packing for spheres See figure in Kittel for stacking sequence HCP is AAA.. Stacking asis of 2 atoms Physics 460 F 2006 Lect 20
21 Hexagonal close packed A a 3 A Hexagonal ravais Lattice Two atoms per cell Physics 460 F 2006 Lect 21
22 Stacking hexagonal 2d layers to make cubic close packed (ccp) 3-d crystal Cube AC stacking Each sphere has 12 equal neighbors Close packing for spheres See figure in Kittel for stacking sequence CCP is ACACAC.. Stacking asis of 1 atom Physics 460 F 2006 Lect 22
23 Stacking hexagonal 2d layers to make cubic close packed (ccp) 3-d crystal Cube [111] direction in cube Physics 460 F 2006 Lect 23
24 Recall from before z Face Centered Cubic (fcc) Also called cubic closed packed (ccp) [111] direction in cube y X Each atom has 12 equal neighbors The figure at the right shows the face centered character Physics 460 F 2006 Lect 24
25 (111) planes in an fcc crystal z X y [111] direction in cube C A C A [111] direction in cube ACAC... stacking of hexagonal planes fi fcc crystal fcc is a close packed crtsal cubic close packed - ccp Physics 460 F 2006 Lect 25
26 More on stacking hexagonal 2d layers A A A HCP C A C A CCP A A C A Other polytype Infinite number of ways to stack planes Polytypes occur in some metals, some compounds like silicon carbide (SiC) Physics 460 F 2006 Lect 26
27 Cubic crystals with a basis z y X NaCl Structure with Face Centered Cubic ravais Lattice ZnS Structure with Face Centered Cubic ravais Lattice C, Si, Ge form diamond structure with only one type of atom Physics 460 F 2006 Lect 27
28 NaCl Structure z y X NaCl Structure with Face Centered Cubic ravais Lattice Physics 460 F 2006 Lect 28
29 CsCl Structure z y X a 3 CsCl Structure Simple Cubic ravais Lattice From Physics 460 F 2006 Lect 29
30 Atomic planes in NaCl and ZnS crystals z y X (110) planes in NaCl crystal rows of the Na and Cl atoms (110) plane in ZnS crystal zig-zag Zn-S chains of atoms Physics 460 F 2006 Lect 2a 30
31 (110) plane in diamond structure crystal z y X (100) plane in ZnS crystal zig-zag Zn-S chains of atoms (diamond if the two atoms are the same) Calculated valence electron density in a (110) plane in a Si crystal (Cover of Physics Today, 1970) Physics 460 F 2006 Lect 2a 31
32 (111) planes in ZnS crystals z X y [111] direction in cube C A C A CCP Zn Zn Zn Zn Zn Zn S S S S S S (111) planes in cubic ZnS crystal Note: AA... stacking gives hexagonal ZnS Physics 460 F 2006 Lect 2a 32
33 Perovskite Structure AO 3 A O Simple Cubic ravais Lattice A atoms have 12 O neighbors atoms have 6 closer O neighbors Many compounds form the perovskite structure, SrTiO 3, atio 3, LaMnO 3,... Physics 460 F 2006 Lect 2a 33
34 Symmetries of crystals in 3 dimensions All Crystals can be classified by: 7 Crystal systems (triclinic, monoclinic, orthorhombic, tetragonal, cubic, hexagonal, trigonal) 14 ravais Lattices (primitive, face-centered or body-centered for each system 14 of the 7x3 possibilities describe all ravais lattices ) 32 Points groups (rotations, inversion, reflection) See references in Kittel Ch 1, G. urns, Solid State Physics Physics 460 F 2006 Lect 2a 34
35 Is a crystal really different from a liquid? Liquid Crystal Yes the crystal has order different directions are different Other crucial differences? Yes dislocations Example of a dislocation -a crystal with an extra plane of atoms on the left - The dislocation can move but Crystal with a dislocation it cannot disappear! Important for deformations, See Kittel Ch. 20 Physics 460 F 2006 Lect 2a 35
36 Next Time Diffraction from crystals Reciprocal lattice Read Kittel Ch 2 Physics 460 F 2006 Lect 2a 36
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