1. M. Anil Kumar Children’s club lecture, NCCR 28-04-10

2.  Matters (materials) are made of atoms  The atomic arrangement in a material determines to a large extent its properties crystalline If well organized and periodic: crystalline amorphous If randomly distributed: amorphous  The properties of the solids are mainly influenced by their lattice. For example, if you consider diamond and graphite, both contains the same atom-carbon. But the difference in their properties lies in how those atoms are arranged. 2/25/2016 2

3. 3 DIAMOND GRAPHITE 2/25/2016

4.  1-dimensional crystal › One parameter: a › Crystalline: › Amorphous: Atoms a a ? 2/25/2016 4

5. 2-dimensional crystal –Three parameters: a 1, a 2,  > crystalline: > Amorphous: a1a1 a2a2  a 1 ? a 2 ?  ? 2/25/2016 5

6.  3-dimensional crystal › Difficulty to draw › Six parameters: a 1, a 2, a 3, , ,  2/25/2016 6

7.  A mathematical description of crystal structures LATTICE  Definition of LATTICE › A lattice is a periodic array of points › The atoms in a crystal are in a regular repeating pattern called the crystalline lattice STM image showing the regular arrangement of Platinum atoms 2/25/2016 7

8. 8 CRYSTAL STRUCTURE Crystal structure is the periodic arrangement of atoms in the crystal. Association of each lattice point with a group of atoms(Basis or Motif). Lattice: Infinite array of points in space, in which each point has identical surroundings to all others. Space Lattice  Arrangements of atoms = Lattice of points onto which the atoms are hung. Elemental solids (Argon): Basis = single atom. Polyatomic Elements: Basis = two or four atoms. Complex organic compounds: Basis = thousands of atoms. + Space Lattice + Basis = Crystal Structure =

9. 9 ONE DIMENTIONAL LATTICE a a a or 2/25/2016

10. 10 TWO DIMENTIONAL LATTICE 2/25/2016

11.  Unit cell › cell dimensions › unit cell length (a, b, c) › cell angles (  ) The smallest repeating unit with all the symmetry of a crystal is called as unit cell. Three dimensional stacking of unit cells is a crystal lattice. 2/25/2016 11

12.  Cell relationships › edge › face-diagonal › body-diagonal › cell volume 2/25/2016 12

13.  Cell relationships › edge › face-diagonal › body-diagonal › cell volume 2/25/2016 13

14.  Cell relationships › edge › face-diagonal › body-diagonal › cell volume If the cell edge is a, how long is the face diagonal? If the cell edge is a, how long is the body diagonal? 2/25/2016 14

15.  Cell relationships › edge › face-diagonal › body-diagonal › cell volume 2/25/2016 15

16. The repeating pattern of the unit cell creates a lattice. Find the repeating pattern Choose the correct one among the possible unit cells 2/25/2016 16

17. Body center = 1 unit Face center = 1/2 unit Edge = 1/4 unit Corner = 1/8 unit Contribution of atoms in the unit cell to the lattice 2/25/2016 17

18. 18  Choices of unit cells are not unique.  Primitive cell: contains one lattice point A, B, and C are primitive unit cells D, E and F are not, in fact, they all have 2 lattice points. How many lattice point in cell A and D? A, every corner atom is shared by 4 neighboring A cells  ¼ * 4 = 1  primitive unit cell D, every corner atom is shared by 4 neighboring D cells and the mid-edge atom is shared by 2 neighboring D cells  ¼ * 4 + ½ *2 = 2  non-primitive unit cell G Which cells are primitive? H Are G and H unit cells? G, contains ½ lattice point, not a unit cell. H, is a lattice point, not a unit cell. 2/25/2016

19. 19 THREE DIMENTIONAL UNIT CELLS / UNIT CELL SHAPES 1 2 3 4 5 6 7 2/25/2016

20. 20 Primitive ( P )Body Centered ( I ) Face Centered ( F )C-Centered ( C ) LATTICE TYPES 2/25/2016

21. 21 BRAVAIS LATTICES 7 UNIT CELL TYPES + 4 LATTICE TYPES = 14 BRAVAIS LATTICES 2/25/2016

22. The three important types of unit cells are 1.Simple cubic, 2.Body centered cubic and 3.Face-centered cubic 2/25/2016 22

23. 23  If the atomic radius is r, the lattice constant a = 2r. r 2/25/2016

24. 24 Since there are 2 lattice points in this cubic unit cell and the volume of the cube is a 3, a primitive unit cell (containing one lattice point) must be a 3 /2. 2/25/2016 24

25. The famous Atomium…the BCC lattice 2/25/201625

26. The face-centered unit cell has 4 lattice points per unit cell with cell volume of a 3 /4 Number of nearest neighbours = 12 Nearest neighbour distance = a/  2 Packing fraction = (4/3)  (a/2  2) 3 a 3 /4 =  / 3  2 2/25/2016 26

27. Structure Along which dimensions of a cube do the atoms touch Length simple (primitive) cubic body-centered cubic face-centered cubic edge2 r body-diagonal face-diagonal 4 r 2/25/2016 27

28. 28 CLOSE-PACKING OF SPHERES 2/25/2016

29. A crystal is built up by placing close packed layers of spheres on top of each other. There is only one place for the second layer of spheres. There are two choices for the third layer of spheres: –Third layer eclipses the first (ABAB arrangement). This is called hexagonal close packing (hcp); –Third layer is in a different position relative to the first (ABCABC arrangement). This is called cubic close packing (ccp). 2/25/2016 29

30. 30 Close-packing-HEXAGONAL coordination of each sphere SINGLE LAYER PACKING SQUARE PACKINGCLOSE PACKING 2/25/2016

31. Square packing: Not most space efficient Hexagonal packing: Most space efficient 2/25/201631

32. 32 TWO LAYERS PACKING THREE LAYERS PACKING 2/25/2016

33. 33 2/25/2016

34. 34 Hexagonal close packing Cubic close packing 2/25/2016

35. 35

36. Close Packing of Spheres  Each sphere is surrounded by 12 other spheres (6 in one plane, 3 above and 3 below).  Coordination number: the number of spheres directly surrounding a central sphere.  If unequally sized spheres are used, the smaller spheres are placed in the interstitial holes. 2/25/2016 36

37. Eclipsed Staggered 2/25/2016 37 HCP Vs CCP

38. The Holes that Exist Among Closest Packed Uniform Spheres 2/25/201638

39. ccphcpbcc 2/25/2016 39

40. 40 Radius ratioCoordinate number Holes in which positive ions pack 0.225 – 0.4144Tetrahedral holes 0.414 – 0.7326Octahedral holes 0.732 – 18Cubic holes Hole Occupation - RADIUS RATIO RULE Radius of the positive ion Radius ratio = Radius of the negative ion 2/25/2016

41. One type of ion occurs at the corners of a cube the centers of a each face The other ion occurs at the center of the each edge the center of the cell 2/25/2016 41

42. 2/25/2016 42

43. One type of ion occurs at the corners of a cube The other ion occurs at the center of the cell 2/25/2016 43

44. Stacking of CsCl unit cells into a crystal lattice 2/25/2016 44

45. Properties of Solids Solids are classified into 3 groups namely, (i) Conductors e.g., Metals. (ii) Semiconductors e.g. Semimetals. (iii) Insulators e.g. Non metals.  In most of the solids conduction is through electron movement under an electric field, however, in some ionic solids the conduction is through ions.  In metals conductivity strongly depends upon the no. of valence electrons available per atom. The atomic orbitals form molecular orbitals which are so close to each other as to form a BAND.  The conductivity of solids can be better explained on the basis of energy gap present between the conduction band (HIGHER UNOCCUPIED BAND) and the valence band. 2/25/2016 45

46. In metals the conduction band is almost overlapping with the valence band i.e., there is no energy gap present between these two bands or valence band is not completely filled. Then electrons can flow easily under the influence of electric field in both the cases. Conduction band Valance band overlap Hence, metals have high conductivity Conductors 2/25/2016 46

47.  Intrinsic semiconductors – pure material having semiconductive properties.  Doped semiconductors – semiconductors that are fabricated by adding a small amount of another element with energy levels close to the pure state material. › n-type semiconductors › p-type semiconductors (look at figure) Fermi level 2/25/2016 47

48.  Fermi-level (semiconductor) – the energy at which an electron is equally likely to be in each of two levels.  Effects of dopants on the Fermi level. › n-type and p-type. Insulators  In the case of non metals (insulators) the energy gap between valence band and conduction band is so large that it cannot even covered up by supplying energy in the form of heat. 2/25/2016 48

49. 2/25/2016 49

50. Besides conductivity solids also show magnetic properties and dielectric properties. On the basis of magnetic properties solids can be categorized to - (a) Diamagnetic i.e., which are feebly repelled by magnetic field. For Example - Non metallic elements (except O 2 and S) inert gases and species with paired electrons e.g. TiO 2, V 2 O 5, C 6 H 6, NaCl etc. (b) Paramagnetic i.e., which are attracted by magnetic field due to the presence of atoms, ions or molecules with impaired electron in them (e.g., O 2, Cu 2+, Fe 3+ etc.). These are used in electronic appliances. (c) Ferromagnetic i.e., which show magnetism even in the absence of magnetic field, Fe, Co and Ni are 3 elements which show ferromagnetism at room temperature. (d) Antiferromagnetic i.e., those which have net magnetic moment zero due to compensatory alignment of magnetic moments. For example MnO, MnO 2, FeO,NiO, Cr 2 O 3 etc. 2/25/2016 50

51. All the best 2/25/2016 51