Polymorphism or Allotropy Many elements or compounds exist in more than one crystalline form under different conditions of temperature and pressure. This phenomenon is termed polymorphism and if the material is an elemental solid is called allotropy . Example: Iron (Fe – Z = 26) liquid above 1539 C. δ-iron (BCC) between 1394 and 1539 C. γ-iron (FCC) between 912 and 1394 C. α-iron (BCC) between -273 and 912 C. 912oC 1400oC 1539oC α iron γ iron δ iron liquid iron BCC FCC BCC Another example of allotropy is carbon. Pure, solid carbon occurs in three crystalline forms – diamond, graphite; and large, hollow fullerenes. Two kinds of fullerenes are shown here: buckminsterfullerene (buckyball) and carbon nanotube. Crystallographic Planes and Directions Atom Positions in Cubic Unit Cells A cube of lattice parameter a is considered to have a side equal to unity. Only the atoms with coordinates x, y and z greater than or equal to zero and less than unity belong to that specific cell. z 0 ≤ x, y, z <1 0,0,1 0,1,1 1,0,1 1,1,1 ½, ½, ½ y 0,1,0 0,0,0 1,0,0 1,1,0 x Directions in The Unit Cell For cubic crystals the crystallographic directions indices are the vector components of the direction resolved along each of the coordinate axes and reduced to the smallest integer. z Example direction A 0,0,1 0,1,1 a) Two points origin coordinates 0,0,0 and final position 1,0,1 1,1,1 coordinates 1,1,0 ½, ½, ½ b) 1,1,0 - 0,0,0 = 1,1,0 y 0,1,0 0,0,0 A c) No fractions to clear 1,0,0 1,1,0 d) Direction [110] x Example direction B a) Two points origin coordinates 1,1,1 and final position coordinates 0,0,0 b) 0,0,0 - 1,1,1 = -1,-1,-1 z c) No fractions _to_ clear_ d) Direction 1[ 1 ]1 0,0,1 Example direction C C 1,1,1 a) Two points origin coordinates B ½,1,0 and final position y coordinates 0,0,1 0,0,0 b) 0,0,1 - ½,1,0 = -½,-1,1 ½, 1, 0 c) There are fractions to clear. Multiply times 2. 2( -½,-1,1) = x -1,-2,2 _ _ d) Direction 1[ 2 ]2 Notes About the Use of Miller Indices for Directions A direction and its negative are not identical; [100] is not equal to [bar100]. They represent the same line but opposite directions. direction and its multiple are identical: [100] is the same direction as [200]. We just forgot to reduce to lowest integers. Certain groups of directions are equivalent; they have their particular indices primarily because of the way we construct the co- ordinates. For example, a [100] direction is equivalent to the [010] direction if we re-define the co-ordinates system. We may refer to groups of equivalent directions as directions of the same family . The special brackets < > are used to indicate this collection of directions. Example: The family of directions <100> consists of six equivalent directions < 100 > ≡ [100],[010],[001],[010],[001],[100] Miller Indices for Crystallographic planes in Cubic Cells Planes in unit cells are also defined by three integer numbers, called the Miller indices and written (hkl). Miller’s indices can be used as a shorthand notation to identify crystallographic directions (earlier) AND planes . Procedure for determining Miller Indices locate the origin identify the points at which the plane intercepts the x, y and z coordinates as fractions of unit cell length. If the plane passes through the origin, the origin of the co-ordinate system must be moved! take reciprocals of these intercepts clear fractions but do not reduce to lowest integers enclose the resulting numbers in parentheses (h k l). Again, the negative numbers should be written with a bar over the number. z Example : Miller indices for plane A a) Locate the origin of coordinate. b) Find the intercepts x = 1 , y = 1 , z = 1 c) Find the inverse 1/x=1, 1/y=1, 1/z=1 A d) No fractions to clear y e) (1 1 1) x More Miller Indices - Examples c c c 1/5 2/3 b b b 0.5 a a a 0.5 c c c b b b a a a Notes About the Use of Miller Indices for Planes A plane and its negative are parallel and identical . Planes and its multiple are parallel planes: (100) is parallel to the plane (200) and the distance between (200) planes is half of the distance between (100) planes. Certain groups of planes are equivalent (same atom distribution); they have their particular indices primarily because of the way we construct the co-ordinates. For example, a (100) planes is equivalent to the (010) planes. We may refer to groups of equivalent planes as planes of the same family. The special brackets { } are used to indicate this collection of planes. In cubic systems the direction of miller indices [h k l] is normal o perpendicular to the (h k l) plane. in cubic systems, the distance d between planes (h k l ) is given by the formula a where a is the lattice constant. d = h2 + k 2 + l 2 Example: The family of planes {100} consists of three equivalent planes (100) , (010) and (001) A “family” of crystal planes contains all those planes are crystallo- graphically equivalent. • Planes have the same atomic packing density • a family is designated by indices that are enclosed by braces. - {111}: (111), (111), 1( 11), 1( 11), (111), (111), 111( ), )111( • Single Crystal • Polycrystalline materials • Anisotropy and isotropy Two Types of Indices in the Hexagonal System a1 ,a2 ,and c are independent, a3 is not! c a3 = - (a1 + a2) a3 Miller: (hkl) (same as before) a2 Miller-Bravais: (hkil) → i = - (h+k) a1 (001) = (0001) c c (110)- = (1100)- a3 a3 a2 a2 a1 a1 (110)- = (1100)- (100) = (1010)- Structure of Ceramics Ceramics keramikos - burnt stuff in Greek - desirable properties of ceramics are normally achieved through a high temperature heat treatment process (firing). Usually a compound between metallic and nonmetallic elements Always composed of more than one element (e.g., Al 2O3, NaCl, SiC, SiO 2) Bonds are partially or totally ionic, can have combination of ionic and covalent bonding (electronegativity) Generally hard, brittle and electrical and thermal insulators Can be optically opaque, semi-transparent, or transparent Traditional ceramics – based on clay (china, bricks, tiles, porcelain), glasses. “New ceramics” for electronic, computer, aerospace industries. Crystal Structures in Ceramics with predominantly ionic bonding Crystal structure is defined by The electric charge: The crystal must remain electrically neutral. Charge balance dictates chemical formula (Ca 2+ and F - form CaF 2). Relative size of the cation and anion. The ratio of the atomic radii (rcation /r anion ) dictates the atomic arrangement. Stable structures have cation/anion contact. Coordination Number: the number of anions nearest neighbors for a cation. As the ratio gets larger (that is as rcation /r anion ~ 1) the coordination number gets larger and larger. Holes in sphere packing Triangular Tetrahedral Octahedral Calculating minimum radius ratio O for a triangle: B B O A C C A 1 1 AB r AB = ra = a 2 2 for an octahedral hole AO r r AO = ra + rc = a + c 1 AB 1 AB 2 = cosα ( α = 30°) 2 = cosα ( α = 45°) AO AO r 3 ra o 2 a = cos30 = = cos45 = ra + rc 2 ra + rc 2 r r c = 0.155 c = 0.414 r a ra C.N. = 2 rC/r A < 0.155 C.N. = 3 0.155 < rC/r A < 0.225 C.N. = 4 0.225 < rC/r A < 0.414 C.N. = 6 0.414 < rC/r A < 0.732 C.N. = 8 0.732 < rC/r A < 1.0 Ionic (and other) structures may be derived from the occupation of interstitial sites in close-packed arrangements. Comparison between structures with filled octahedral and tetrahedral holes o/t fcc(ccp) hcp all oct. NaCl NiAs all tetr. CaF 2 (ReB 2) o/t (all) (Li 3Bi) (Na 3As) ½ t sphalerite (ZnS) wurtzite (ZnS) (½ o CdCl 2 CdI 2) Location and number of tetrahedral holes in a fcc (ccp) unit cell - Z = 4 (number of atoms in the unit cell) - N = 8 (number of tetrahedral holes in the unit cell) Crystals having filled Interstitial Sites Octahedral, Oh, Sites Ionic Crystals prefer the NaCl Structure: • Large interatomic distance NaCl structure has Na+ ions • LiH, MgO, MnO, AgBr, PbS, at all 4 octahedral sites KCl, KBr Na+ ions Cl- ions Crystals having filled Interstitial Sites Tetrahedral, Th, Sites Covalently Bonded Crystals Prefer this Structure • Shorter Interatomic Distances than ionic • Group IV Crystals (C, Si, Ge, Sn) • Group III--Group V Crystals (AlP, GaP, GaAs, AlAs, InSb) • Zn, Cd – Group VI Crystals Both the diamond cubic structure And the Zinc sulfide structures (ZnS, ZnSe, CdS) have 4 tetrahedral sites occupied • Cu, Ag – Group VII Crystals and 4 tetrahedral sited empty. (AgI, CuCl, CuF) Zn atoms S atoms AX Type Crystal Structures Rock Salt Structure (NaCl) NaCl structure: rC = rNa = 0.102 nm, rA = rCl = 0.181 nm rC/r A = 0.56 Coordination = 6 Cl Na NaCl, MgO, LiF, FeO, CoO Cesium Chloride Structure (CsCl) CsCl Structure: rC = rCs = 0.170 nm, rA = rCl = 0.181 nm → rC/r A = 0.94 Coordination = 8 Cl Cs Is this a body centered cubic structure? Zinc Blende Structure (ZnS) radius ratio = 0.402 Coordination = 4 S Zn ZnS, ZnTe, SiC have this crystal structure AmXp-Type Crystal Structures If the charges on the cations and anions are not the same, a compound can exist with the chemical formula AmXp , where m and/or p ≠ 1.