1. 1 CHAPTER 5 Crystal Structure ： Crystal Structure of Ceramics

2. 2 I. Introduction to Ceramics  Chemical Composition mostly are compounds composed of metallic and nonmetallic elements, i.e., composed of at least two different elements, usually considering metallic element as cation, and nonmetallic element as anion. example：Al 2 O 3, SiO 2, TiO 2, AlN, BN, …… exceptions：diamond, graphite,……

3. 3 % ionic character = ( 1-e –(0.25)(X A -X B ) 2 )  100  Bonding mostly mixed ionic and covalent bonding  Coordination number (CN) ： 4, 6, and 8. exception ： diamond, silicon, graphite, …… considering the ceramics to be made up of cations and anions CN relative size of cation and anion  Crystal Structure considering the ceramics to be made up of cations and anions T 12.1

4. 4 II. General Features of Ceramic Crystal Structures The crystal sturctures may be thought of as being composed of cations, and anions. Two characteristics influencing the crystal structure: ˙magnitude of the electrical charge (electrically neutral) ˙relative sizes of the cations and anions ( CN). The chemical formula of a compound indicates the ratio of cations to anions, for example: C a F 2., C a +2 : F -1 =1:2. (the crystal must be electrically neutral) F 12.2F 3.7-4

5. 5  Basis (group) or lattice point ˙metals：one basis usually represents one atom.  all the atoms are located at the positions of lattice points, i.e., there are atoms only at the positions of lattice points lattice sites. ˙ceramics：one basis usually represents at least one cation and one anion. e.g., NaCl：one Na + and one Cl - ZnO：one Zn +2 and one O -2 CaF 2 ：one Ca +2 and two F -1  If the lattice point is assigned to the center of the anion, the cations will not be at the positions of lattice points. Where are the cations accommodated？  Interstices：the space among lattice points sublattice F 3.3-1F 3.7-3F 3.7-2 F 3.7-4 F 3.7-3 F 3.7-4 # 18F 3.3-1 F 24.3

6. 6 III. Interstices in Crystal Structure  location：center of the cube at  number：one per unit cell  shape：cubic  CN：8  r cation / r anion = 0.732~1.0  example：CsCl A. Interstices in SC Structure F 11.5 Interstices ≡ Interstitial site ≡Interstitial position ≡ sublattice  Shape of interstices：the geometric shape by connecting straight lines through all the nearest surrounding atoms (or ions). T 12.2 Interstices

7. 7  the largest hole in an FCC structure is at the center of the unit cell and at the center of each edge.  It has eight sides, celled an octahedral site. There are four octahedral sites per FCC unit cell.  CN = 6  r cation / r anion = 0.414~0.732  the size of the octahedral hole is defined as the radius of the largest sphere that can be placed within it. B. Interstices in the FCC Structure An atom roughly 40% of the size of the host atoms can “fit” into an octahedral interstitial position in the FCC structure. F 11.7 F 12.8 F 3.6-1 F 11.9 T 12.2

8. 8  the FCC sturcture also contains tetrahedral sites, in the l/4, m/4, n/4 positions, where l, m, and n are 1 or 3. Each cell contains eight of these ¼, ¼,, ¼-type tetrahedral sites. The k/r ratio for tetrahedral sites is 0.225. Atoms up to ~20% of the size of the host atoms can “fit” in the tetrahedral positions in FCC structures. F 12.7F 3.7-4F 3.7-3 F 11.10F 11.11

9. 9  The BCC structure also contains both octahedral and tetrahedral sites.  the octahedral sites are located in the center of each face and the center of each edge, giving a total of six sites per unit cell. C. Interstices in the BCC Structure F 3.6-1  The tetrahedral sites in BCC structures are located in the ¼, ½, 0-type positions, which are on the {100} faces, a total of 12 tetrahedral sites per unit cell, k/r =0.29 F 3.3-1

10. 10 D. interstices in the HCP Structure F 3.6-1  Also contains both octahedral and tetrahedral interstices.  6 octahedral sites per “big” cell or 2 sites per unit cell.  k/r = 0.414  12 tetrahedral sites per big cell or 4 per unit cell. Each small unit cell contains 2, each edge contains 2×(1/3) and 2 are located at the center line.  k/r = 0.225  Since both FCC and HCP are close-packed crystal structures, the relative sizes of the interstitial sites are the same in these two types of crystals. F 11.15T 3.6-1

11. 11 IV. Crystal Structures based on Number of Atoms (Ions) per Lattice Site  One atom per lattice site  metals  Multiple atoms per lattice site  ceramics # 8# 9 F 3.7-3

12. 12  A simple cubic lattice with two ions, one of each type, per lattice position (i.e., the basis) anion：lattice site cation：cubic site (center of the unit cell)  The coordination number is eight, a0(CsCl) = 2(r+R) / r CS + / r Cl =  CN = 8  Other ionic solids with the CsCl structure : CsBr, and CsI. V-1. Crystals with Two Atoms per Lattice Site A. The Cesium Chloride Structure V. Ceramic Crystal Structure based on Number of Atoms per Lattice Site No. of cubic site No. of lattice site ＝ 1 1 No. of Cs + ＝ No. of Cl - F 3.7-2 F 3.3-2F 11.5

13. 13 B. The Sodium Chloride Structure  a 0 (NaCl)=2(r+R). Ions touch along the cube edge.  Other compounds with this sturcture: MgO, CaO, SrO, FeO, BaO,MnO, NiO and KCl  NaCl has an FCC lattice with a basis of two different atoms. anion：lattice site cation：octahedral site No. of octahedral site No. of lattice site ＝ 4 4 No. of Na + ＝ No. of Cl - 1 1 ＝ How does the material “ choose ” its crystal structure? The key concepts are the r/R ratio (  CN) and stoichiometry (No. of cation/ No. of anion). For example, consider MgO: the ratio is 0.59, the most stable coordination number is 6. Consequently, MgO Forms crystals of the NaCl-type (Mg +2 at octahedral). F 3.7-3F 12.2 F 3.3-3  r Na +/r Cl - = 0.012/0.181 = 0.56  CN = 6  octahedral site T 12.3T 12.2F 12.2 F 11.7

14. 14 C. The Diamond Cubic Structure  Diamond has an FCC lattice with two atoms per site, there are eight atoms per unit cell. F 12.15  Why this structure ? r cation / r anion = 1  CN = 8 Covalent bonding：CN=4 the C-C-C bond angle = 109.50  tetrahedral sites  a 0 (diamond cubic)=8r/. Other materials with this structure: silicon and germanium. one carbon：lattice site The other carbon：tetrahedral site No. of tetrahedral site No. of lattice site ＝ 8 4 ＝ 2 1  Only half of the tetrahedral sites are occupied and the other half are empty.

15. 15  The zinc-blende structure is similar to the diamond cubic structure but with two different elements: zinc and sulfur.  Other materials with this structure：GaAs, CdTe.  Why are only half of the tetrahedral sites filled？ The answers are the stoichiometry of the compound： there are four FCC sites per cell and eight tetrahedral sites per cell.  Coordination number：four；a0(zinc-blende)=4(r+R)/ D. The Zinc-Blende Structure F 3.7-4

16. 16  M2X, including Li2O, Na2O, and K2O, simple the inverse of the fluorite structure with the X ions at the FCC positions and the M ions filling all of the tetrahedral positions.  The cations are smaller than the anions as ordinary cases. D-2. Crystals with Three Atoms per Lattice Site ◎ Generally, with a basis of three atoms. A. Fluorite Structure B. Antifluorite Structure  MX2, e.g., CaF2, UO2, ThO2 and ZrO2, M ions are located in the FCC positions and the X ions fill all the terrahedral sites. CN(M)=8, CN(X)=4.  The cations are relatively large compared to ordinary cases. F 3.7-5 F 3.5

17. 17 A. AX-TYPE CRYSTAL STRUCTURES AX compounds, A: cation X: anion (1) Rock Salt Structure Sodium chloride (NaCl), or rock salt type, coordination number: 6, cation-anion radius ratio: 0.414―0.732, unit cell : FCC examples:NaCl, MgO, MnS, LiF, and FeO. CsCl, coordination number: 8, crystal sturcture: SC (not a BCC ) (3) Zinc Blende Structure Coordination number: 4; tetrahedrally coordinated. Zinc blende, or sphalerite, structure, e.g., zinc sulfide (ZnS): sach Zn atom is bonded to four S atoms, and vice versa. Examples: ZnS, ZnTe, and SiC. F12- 4 VI. Ceramic Crystal Structure based on Chemical Formula (Considering or looking at the packing of one of the ions.) (2) Cesium Chloride Structure F 12.2 F 12.3 F 12.4F 3.7-4

18. 18 The charges on the cations and anions are not the same. Example ： fluorite structure ( AX 2 ) and antifluorite structure (A 2 X). fluorite (C a F 2 ) ： r c /r A for CaF 2 : 0.8, coordination number: 8. Crystal structure would be similar to CsCl except that only half the center cube positions are occupied by Ca 2+ ions. One unit cell consists of eight cubes. Other compounds: UO 2, PU 2, and ThO 2 C. A m B n X p – TYPE CRYSTAL STRUCTURES A typical example ： barium titanate (BaTiO 3 ), perovskite crystal structure. At temperatures above 120℃: cubic, Ba 2+ ions at all eight corners, single Ti 4+ at the cube center, O 2- ions at the center of each of the six faces. B. AmXp— type Crystal Structures F 3.7-5F 3.5F 12.5 F 12.6T 12-4

19. 19 ◎ Calcium titanate, CaTiO3 ◎ Barium titanate, BaTiO3: simple tetraggonal, a=b=0.398nm, c=0.403nm. The central Ti4+ ion does not lie in the same plane as the four oxygen atoms in the side faces of the tetragonal unit cell. A. Perovskite Structure Perovskite is a naturally occurring mineral CaTiO3, general formula is ABX3; larger A cations surrounded by 12 oxygens, smaller B(Ti4+ ) ions by 6 oxygens. NaWP 3, CaSnO 3, YAIO 3 ; AB 3 structures, ReO 3, WO 3, NbO 3, NbF 3, TaF 3 ; TiOF 2, MoOF 2. VII. Ceramic Crystal Structures based on building blocks imagine the structure to be made of the various building blocks. F 12.6F 3.9F 3.8 F 3.7-8

20. 20 Named after the naturally occurring mineral MgAl 2 O 4, general formula is AB 2 O 4, FCC stacking of the oxygen, the cations occupy one-eighth of the tetrahedral sites and one-half of the octahedral. An idealized version consisting of TiO 6 octahedra, each oxygen is shared by three octahedra. Actual structure comprises distorted octahedra rather than the regular ones.  Important electrical properties arising from local electric dipoles:The strength of the dipole can be altered by either an applied force or electric field. Thus, BaTiO 3 can be used as a transducer to convert electrical voltages into mechanical energy and vice versa.  Applications: telephone receivers, phonograph cartridges, and etc. B. Antifluorite Structure C. Spinel Structure D. Rutile Structure F 3.8 F 3.10 F 3.6

21. 21 While SiO 2 (silica) has three atoms per lattice site, it is much easier to visualize the structure of crystobalite in a different fashion: The basic building block for all Si-O compounds is the negatively charged (SiO 4 ) 4 - tetrahedron. The crystobalite crystal structure, can be envisioned as the diamond cubic structure with an (SiO 4 ) 4 - tetrahedron positioned on each lattice site. Thus, crystobalite has an FCC lattice with six atoms, or two tetrahedra, perlattice site. The building block of silicon-based covalent ceramics (silicates, SiC and Si 3 N 4 ): Si tetrahedron, e.g., SiO 4 in silicates, SiC 4 in SiC, SiN 4 in Si 3 N 4. E. STRUCTURE OF COVALENT CERAMICS F. The Crystobalite Structure F 12.9F 3.11 F 12.10F 12.9F 3.4-6

22. 22  A number of ceramic crystal structures may be considered in terms of close-packed planes of ions, (the large anions), the cations may reside small interstitial sites.  Interstitial positions, two different types: tetrahedral position and octahedral position, the coordination numbers for cations: 4and 6, respectively.  Two factors: (1) the stacking of the close-packed anion layers: FCC or HCP (ABCABC……or ABABAB…… ); (2) the interstitial sites: for example, the rock salt crystal structure. VIII. Ceramic Crystal Structures From The Close Packing of Anions F 3.6-1 F 3.5-3

23. 23 A. Cubic close-Packed  The structure in which the anions are in an FCC arrangement：rock salt, rutile, zinc blende, antifluorite, perovskite and spinel.  Rock salt structure：cations on each of the octahedral sites Zinc blende structure：half the tetrahedral sites are filled. B. Hexagonal close-packed  The anion arrangement is HCP: Wurtzite, nickel arsenide, cadmium odide, corundum, illmenite, and olivine.  For example, corundum (Al 2 O 3 ): the oxygen ions are hexagonally close-packed, Al ions fill two-thirds of octahedral sites. Wurtzite: One-half the tetrahedral sites are filled. F 12.2F 3.7-4 F 11.15

24. 24 Other, but not all, ceramic crystal structures may be treated in a similar manner, included are the zinc blende and perovskite sturctures. Spinel sturcture (A m B n X p ): magnesium aluminate or spinel (MgAl 2 O 4 ): the O 2- ions form an FCC lattice, M 2+ ions fill tetrahedral sites and Al 3+ reside in octahedral positions. Magnetic ceramics, or ferrites, have a crystal structure that is a slight variant of this spinel structure, and the magnetic characteristics are affected by the occupancy of tetrahedral and octahedral positions.

25. 25 IX. CERAMIC DENSITY COMPUTATIONS Theoretical Density (perfect crystal) (12.1) n’ = the number of formula units’ within the unit cell  A C = the sum of the atomic weights of all cations in the formula unit  A A = the sum of the atomic weights of all anions in the formula unit V C = the unit cell volume N A = Avogadro’s number, 6.023  10 23 formula units/mol % theoretical density = measured density theoretical density × 100%

26. 26 Example Problem On the basis of crystal structure, compute the theoretical density for sodium chloride. How does this compare with its measured density? Solution the number of NaCl units per unit cell, is 4  A C = A Na = 22.99g/mol  A A = A Cl = 35.45g/mol V C = a 3 a = 2r Na + + 2r cl - r Na + and r Cl - : 0.102 and 0.181 nm, respectively.

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28. 28 12.3 Silicate Ceramics Silicates are materials composed primarily of silicon and oxygen: soils, rocks, clays, and sand. Rather than unit cells, it is more convenient to use various arrangements of an SiO 4 4- tetrahedron (Figure 12.9) SILICA Every corner oxygen atom in each tetrahedron is shared by adjacent tetrahedra. Three primary polymorphic crystalline forms: quarttz, cristobalite, and tridymite. The atoms are not closely packed to gether, silicas have relatively low densities. F12-9 F12-10

29. 29 Silica Glasses Noncrystalline solid or glass, called fused silica, or vitreous silica. Other oxides (e.g., B2O3 and GeO2) may also form glassy structures these materials, as well as SiO2, are termed network formers. Common inorganic glasses: silica glasses with added other oxides such as CaO and Na2O. These oxides do not form polyhedral networks, rather modify the SiO 4 4- network: network modifiers Other oxides, such as TiO2 and Al2O3, while not network formers, substitute for silicon and become part of and stabilize the network; these are called intermediates. These modifiers and intermediates lowers the melting point and viscosity of a glass, and makes it easier to form at lower temperatures. F12-11

30. 30 THE SILICATES One, two or three of the corner oxyge atoms of the SiO 4- –4 thtrahedra are shared by other tetrahedra, examples: SiO 4 4–, Si 2 O 7 6- and S i3 O 9 –6, positively charged cations such as Ca 2+, Mg 2+, and Al 3+ (1) compensate the negative charges from the SiO 4 4- (2) ionically bond the SiO 4 4- together. Simple Silicates For example, forsterite (Mg 2 SiO 4 ): every Mg 2+ ion has six oxygen nearest neighbors. Akermanite (Ca 2 MgSi 2 O 7 ) : Two Ca –2 and one Mg +2 bonded to each S i2 O 7 -6. F 12.12

31. 31 Layered Silicates Characteristic of the clays ( 黏土 ) and other minerals. Kaolinite ( 高嶺土 ) clay has: Al 2 (Si 2 O 5 )(OH) 4, silica tetrahedral layer (Si 2 O 5 ) 2- is made electrically neutral by an adjacent Al 2 (OH) 4 2+ layer, the bonding within this two layered sheet is strong and intermediate ionic-covalent, adjacent sheets are only loosely bound to one another by weak van der waals forces. A crystal of kaolinite is made of a series of these double layers or sheets stacked parallel to each other, flat plates <1  m nearly hexagonal. Other minerals also in this group are talc ( 滑石 ) [Mg 3 (Si 2 O 5 ) 2 (OH) 2 ] and the micas ( 雲母 ) [e.g., muscovite, KAl 3 Si 3 O 10 (OH) 2 ]. F12-13F12-14

32. 32 DIAMOND A metastable carbon polymorph at room temperature and atmospheric pressure. Crystal structure: a variant of the zinc blende, carbon atoms occupy all positions (both Zn and S). Each carbon bonds to four other carbons and totally covalent: diamond cubic crystal structure [also: germanium, silicon, and gray tin, below 13 ℃ (55 ℉ )]. F12-15 12.4 CARBON Various polymorphic forms: graphite, diamond, fullerenes, carbon nanotubes, as well as in the amorphous state.

33. 33 Diamond thin films, for example, the surfaces of drills, dies, bearings, knives, and other tools have been coated with diamond films to increase surface hardness; some lenses and radomes. Potential applications: gears, optical recording heads and disks, and as substrates for semiconductor devices. F12-16 Physical properties: extremely hard (the hardest known material ), a very low electrical conductivity, an unusually high thermal conductivity, optically transparent in the visible and infrared regions, high index of refraction. Industrial applications: to grind or cut other softer materials. Synthetic diamonds beginning in the mid-1950s, today a large proportion of the industrial-quality materials are man-made.

34. 34 GRAPHTTE Crystal structure: more stable than diamond at ambient temperature and pressure. Layers of hexagonally arranged carbon atoms; within the layers: strong covalent bonds; between the layers: van der waals type of bond. Weak interplanar bonds: excellent lubricative properties of graphite. Electrical conductivity is relatively high in crystallographic directions parallel to the hexagonal sheets. Other desirable properties: high strength, and good chemical stability at elevated temperatures and in nonoxidizing atmospheres, high thermal conductivity, low coefficient of thermal expansion, high resistance to thermal shock, high adsorption of gases, good machinability. Applications: heating elements, electrodes for arc welding, metallurgical crucibles, insulations in rocket nozzles, chemical reactor vessels, electrical contacts, brushes and resistors, electrodes in batteries in air purification devices. F12-17

35. 35 Today a large proportion of the industrial-quality materials are man-made,diammond thin films. For example, the surfaces of drills, dies, bearings, knives, and other tools have been coated with diamond films to increase surface hardness; some lenses and radomes.Potential applications: gears, to optical recording heads and disks, and as substrates for semiconductor devices. F12-16 GRAPHITE Crystal structure more stable than diamond at ambient temperature and pressure.layers of hexagonally arranged carbon atoms; within the layers: strong covalent bonds. Van der Waals type of bond between the layers. Weak interplanar bonds, excellent lubricative properties of graphite. Electrical conductivity is reatively high in crystallographic directions parallel to the hexagonal sheets.

36. 36 Other desirable properties high strength and good chemical stability at elevated temperatures and in nonoxidizing atmospheres, high thermal conductivity, low coefficient of thermal expansion high resistance to thermal shock,high adsorption of gases, good machinability. Applications: heating elements electrodes for arc welding, metallurgical crucibles, Casting molds high-temperature refractories insulations, in rocket nozzles, chemical reactor vessels, electrical contacts, brushes and resistors, electrodes in Batteries in air purification devices.

37. 37 FULLERENES AND CARBON NANOTUBES Fullerenes Another polymorphic form of carbon discovered in1985. Discrete molecular form consisting of a hollow spherical cluster of sixty carbon atoms: a single molecule denonted by C 60. Each molecule is composed of both hexagon (six-carbon atom) and pentagon (five-carbon atom) One such molecule: 20 hexagons and 12 pentagons. F12-18

38. 38 C 60 (soccer ball.) : buckminsterfullerene, (in honor of R. Buck-minster Fuller,) Often referred to as “buckyball” or fullerene. Diamond and graphite: network solids; buckminsterfullerene : molecular solids In the solid state, the C 60 units form a crystalline structrue and pack together in a face-centered cubic array. As a pure crystalline solid: electrically insulating. However, with proper impuity additions: highly conductive and semi- conductive.

39. 39 Carbon Nanotubes Another molecular form of carbon. Its structure consists of a single sheet of graphite, rolled into a tube, both ends of which are capped with C 60 fullerene hemispheres. Tube diameters are of a nanometer(i.e.,100nm or less). Each nanotube is a single molecule composed of millions of atoms; Multiple-walled carbon nanotubes also exist. These nanotubes are extremely strong and stiff, relatively ductile, and have low densities. For single-walled nanotubes, tensile strengths range between 50 and 200 Gpa (approximately an order of magnitude greater than for carbon fibers); this is the strongest known material. F12-19

40. 40 The carbon nanotube has been termed the “ultimate fiber” and is extremely promising as a reinforcement in composite materials. Carbon nanotubes have unique and structrue-sensitive electrical charac-teristics: may behave electrically as either a metal or a semiconductor. Reported applications: flat-panel and full-color displays(i.e.,TV and computer monitors) Future electronic applications: diodes and transistors.

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44. 44 3.3 CRYSTALS WITH ONE ATOM PER LATTICE SITE AND HEXAGONAL CRYSTALS (usually metals) 3.3.1 Body-centered Cubic Crystal ◎ body-centered cubic (BCC) structure :e.g., tungsten chromium, iron, molybdenum, and vanadium. ◎ a 0 /2=2r, or a 0 (BCC)=4r/ (3.3) (a0 a, r R) ◎ The total number of atoms per unit cell is two {[8× (1/8)]+(1× 1)} ; CN(BCC)=8 F302f3.3-1 f3.3-2

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57. 57 4.3 CRYSTALS WITH ONE ATOM PER LATTICE SITE AND HEXAGONAL CRYSTALS (usually metals) 4.3.1 Body-centered Cubic Crystal ◎ body-centered cubic (BCC) structure :e.g., tungsten chromium, iron, molybdenum, and vanadium. ◎ a 0 /2=2r, or a 0 (BCC)=4r/ (3.3) (a0 a, r R) ◎ The total number of atoms per unit cell is two {[8× (1/8)]+(1× 1)} ; CN(BCC)=8 F302 f3.3-1f3.3-2

58. 58 4.3.2 Face-centered Cubic Crystals ◎ Face-centered Cubic (FCC) structure : e.g., aluminum, calcium, cooper, gold, lead, platinum, and silver. ◎ (FCC)=4r/ (3.1) ◎ There are four atoms per FCC cell {[8× (1/8)]+[(6× 1/2)]}, CN(FCC)=12 4.3.3 Hexagonal Close-packed structure ◎ Hexagonal close-packed (HCP) structure : e.g., cadmium, cobalt, magnesium, titanium, yttrium, and zinc. F301 F303 f3.3-3 f3.3-4

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71. 71 Figure 3.8 Examples of composite crysral structures. (b) Perovskite structure (CaTiO3), At the center of each cuboctahefron is a Ca ion. Each Ca cuboctahedron is surrounded by eight titania octahedra. Also see Fig. 3.9 Ti 4+ Ba 2+ or Ca 2+ BaTiO 3, CaTiO 3 O 2- o 2-

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82. 82 BaTiO 3, CaTiO 3 Based on crystallography  Simple cubic (the schematic shoron is just s lattice point) Other ways of description: Considering Ba 2+ packing  Simple cubic (actually : simple tetragonal) Considering anion O 2- packing  FCC building block view

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89. 89 LatticeNo. of lattice point per unit cell No. of Octahedral sites per unit cell No. of tetrahedral sites per cell ratio of interstices to lattice point CN of cation CN of anion r cation r anion SC FCC BCC HCP