Lecture 2a - Structure of crystals - continued
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1 Lecture 2a - tructure of crstals - continued olid tate Phsics 460- Lecture 2a tructure of rstals (Kittel h. 1) From Last Time rstals crstal is a repeated arra of atoms rstal Lattice + asis ee man great sites like ob s rock shop with pictures and crstallograph info: Phsics 460 F 2006 Lect 2 Lattice of points rstal (ravais Lattice) asis of atoms rstals can be classified into a small number of tpes ee text for more details Phsics 460 F 2006 Lect 2 From Last Time Two Dimensional rstals Possible Two Dimensional Lattices From Last Time General oblique Hexagonal Φ = 60, = 6-fold rotation, reflections asis Lattice Infinite number of possible lattices and crstals Finite number of possible lattice tpes and crstal tpes Phsics 460 F 2006 Lect 2 quare 4-fold rot., reflect. Rectangular 2-fold rot., reflect. entered Rectangular 2-fold rot., reflect. These are the onl possible special crstal tpes in two dimensions Phsics 460 F 2006 Lect 2a 4 Three Dimensional Lattices Three Dimensional Lattices implest examples Ever 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 where the n s are integers Phsics 460 F 2006 Lect 2a 5 imple Orthorhombic ravais Lattice Hexagonal ravais Lattice Orthorhombic: angles 90 degrees, 3 lengths different Tetragonal: 2 lengths same ubic: 3 lengths same Hexagonal: different from, b smmetr Phsics 460 F 2006 Lect 2a 6 1
2 Lecture 2a - tructure of crstals - continued Length of each side - a ubic Lattices Length of each side - a ubic Lattices a a a imple ubic = (1,0,0) a = (0,1,0) a = (0,0,1) a One atom per cell at position (0,0,0) od entered ubic () onventional ell with 2 atoms at positions (000), (1/2,1/2,1/2) a Phsics 460 F 2006 Lect 2a 7 imple ubic = (1,0,0) a = (0,1,0) a = (0,0,1) a One atom per cell at position (0,0,0) Face entered ubic (F) onventional ell with 4 atoms at positions (000 ), (0,1/2,1/2), (1/2,0,1/2), (1/2,1/2,0)a Phsics 460 F 2006 Lect 2a 8 Face entered ubic Two views - onventional ubic ell Face entered ubic (fcc) lso called cubic closed packed (ccp) onventional ell of Face entered ubic Lattice 4 times the volume of a primitive cell Phsics 460 F 2006 Lect 2a 9 Each atom has 12 equal neighbors We will see later that this is a close packed lattice Phsics 460 F 2006 Lect 20 Face entered ubic od entered ubic One Primitive ell Wigner-eit ell One Primitive ell Wigner-eit ell = (1/2,1/2,0) a = (1/2,1/2,-1/2) a = (1/2,0,1/2) a = (1/2, -1/2,1/2) a = (0,1/2,1/2) a = (-1/2,1/2,1/2) a One atom per cell at position (0,0,0) Phsics 460 F 2006 Lect 21 One atom per cell at position (0,0,0) Phsics 460 F 2006 Lect 22 2
3 Lecture 2a - tructure of crstals - continued Lattice Planes - Index stem s 3 s 1 chematic illustrations of lattice planes Lines in 2d crstals Lattice (01) (02) s 2 Plane through the points s 1, s 2, s 3 Each s can be an integer or a rational fractions (22) (11) (14) asis Define the plane b the reciprocals 1/s 1, 1/s 2, 1/s 3 Reduce to three integers with same ratio h,k,l Plane is defined b (h,k,l) Phsics 460 F 2006 Lect 23 Infinite number of possible planes an be through lattice points or between lattice points Phsics 460 F 2006 Lect 24 chematic illustrations of lattice planes Lines in 2d crstals Lattice chematic illustrations of lattice planes Lines in 2d crstals (14) (01) (02) asis Equivalent parallel planes Low index planes: more dense, more widel spaced High index planes: less dense, more closel spaced Phsics 460 F 2006 Lect 25 (01) (02) (14) Planes slice through the basis of phsical atoms Phsics 460 F 2006 Lect 26 Lattice planes in cubic crstals (111) lattice planes in cubic crstals (100) and (110) planes in a cubic lattice (illustrated for the fcc lattice) Phsics 460 F 2006 Lect 27 Face entered ubic Lattice Lattice planes perpendicular to Each plane is hexagonal close packed arra of points Phsics 460 F 2006 Lect 28 3
4 Lecture 2a - tructure of crstals - continued tacking hexagonal 2d laers to make close packed 3-d crstal tacking hexagonal 2d laers to make hexagonal close packed (hcp) 3-d crstal Each sphere has 12 equal neighbors 6 in plane, 3 above, 3 below lose packing for spheres an stack each laer in one of two was, or above lso see figure in Kittel Phsics 460 F 2006 Lect 29 Each sphere has 12 equal neighbors lose packing for spheres ee figure in Kittel for stacking sequence HP is.. tacking asis of 2 atoms Phsics 460 F 2006 Lect 20 Hexagonal close packed tacking hexagonal 2d laers to make cubic close packed (ccp) 3-d crstal ube stacking Hexagonal ravais Lattice Two atoms per cell Phsics 460 F 2006 Lect 21 Each sphere has 12 equal neighbors lose packing for spheres ee figure in Kittel for stacking sequence P is.. tacking asis of 1 atom Phsics 460 F 2006 Lect 22 tacking hexagonal 2d laers to make cubic close packed (ccp) 3-d crstal ube Face entered ubic (fcc) lso called cubic closed packed (ccp) Recall from before Each atom has 12 equal neighbors The figure at the right shows the face centered character Phsics 460 F 2006 Lect 23 Phsics 460 F 2006 Lect 24 4
5 Lecture 2a - tructure of crstals - continued (111) planes in an fcc crstal More on stacking hexagonal 2d laers... stacking of hexagonal planes fi fcc crstal fcc is a close packed crtsal cubic close packed - ccp HP P Other poltpe Infinite number of was to stack planes Poltpes occur in some metals, some compounds like silicon carbide (i) Phsics 460 F 2006 Lect 25 Phsics 460 F 2006 Lect 26 ubic crstals with a basis Nal tructure Nal tructure with Face entered ubic ravais Lattice tructure with Face entered ubic ravais Lattice, i, Ge form diamond structure with onl one tpe of atom Nal tructure with Face entered ubic ravais Lattice Phsics 460 F 2006 Lect 27 Phsics 460 F 2006 Lect 28 sl tructure tomic planes in Nal and crstals sl tructure imple ubic ravais Lattice From Phsics 460 F 2006 Lect 29 (110) planes in Nal crstal rows of the Na and l atoms (110) plane in crstal ig-ag - chains of atoms Phsics 460 F 2006 Lect 20 5
6 Lecture 2a - tructure of crstals - continued (110) plane in diamond structure crstal (111) planes in crstals P (111) planes in cubic crstal (100) plane in crstal ig-ag - chains of atoms (diamond if the two atoms are the same) alculated valence electron densit in a (110) plane in a i crstal (over of Phsics Toda, 1970) Phsics 460 F 2006 Lect 21 Note:... stacking gives hexagonal Phsics 460 F 2006 Lect 22 Perovskite tructure O 3 imple ubic ravais Lattice O atoms have 12 O neighbors atoms have 6 closer O neighbors Man compounds form the perovskite structure, rtio 3, atio 3, LaMnO 3,... Phsics 460 F 2006 Lect 23 mmetries of crstals in 3 dimensions ll rstals can be classified b: 7 rstal sstems (triclinic, monoclinic, orthorhombic, tetragonal, cubic, hexagonal, trigonal) 14 ravais Lattices (primitive, face-centered or bod-centered for each sstem 14 of the 7x3 possibilities describe all ravais lattices ) 32 Points groups (rotations, inversion, reflection) ee references in Kittel h 1, G. urns, olid tate Phsics Phsics 460 F 2006 Lect 24 Is a crstal reall different from a liquid? Next Time Diffraction from crstals Reciprocal lattice Liquid rstal Read Kittel h 2 Yes the crstal has order different s are different Other crucial differences? Yes dislocations Example of a dislocation -a crstal with an extra plane of atoms on the left - The dislocation can move but rstal with a dislocation it cannot disappear! Important for deformations, ee Kittel h. 20 Phsics 460 F 2006 Lect 25 Phsics 460 F 2006 Lect 26 6
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