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CHAPTER 5CHAPTER 5

Crystal StructureCrystal Structure ：：Crystal Structure of CeramicsCrystal Structure of Ceramics

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I. Introduction to CeramicsI. 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 ： Al2O3, SiO2, TiO2, AlN, BN,

…… • exceptions ： diamond, graphite,……

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• % ionic character = ( 1-e –(0.25)(XA

-XB

)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

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II. General Features of Ceramic Crystal StructuresII. 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: CaF2., Ca

+2 : F-1=1:2. (the crystal must be electrically neutral)

F 12.2 F 3.7-4

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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

CaF2： 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 3.3-1 F 3.7-3 F 3.7-2 F 3.7-4

F 3.7-3

F 3.7-4

# 18 F 3.3-1

F 24.3

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III. Interstices in Crystal StructureIII. Interstices in Crystal Structure

location ： center of the cube at number ： one per unit cell shape ： cubic CN ： 8 rcation / ranion = 0.732~1.0 example ： CsCl

A. Interstices in SC Structure1 1 1

, ,2 2 2

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

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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 rcation / ranion = 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

114/112.,. ei

kra 22 2/4rFCCa 414.0/ rk

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

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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 12.7 F 3.7-4 F 3.7-3

F 11.10 F 11.11

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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

kra 22

3/4rBCCa 155.0/ rk

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

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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 11.15 T 3.6-1

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IV. Crystal Structures based on Number IV. Crystal Structures based on Number of Atoms of Atoms (Ions) per Lattice Site (Ions) per Lattice Site

One atom per lattice site

metals

Multiple atoms per lattice site

ceramics

# 8 # 9

F 3.7-3

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A simple cubic lattice with two ions, one of each type, per lattice position (i.e., the basis)anion ： lattice sitecation ： cubic site (center of the unit cell)

The coordination number is eight, a0(CsCl) = 2(r+R) /rCS+ / rCl = 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

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V. Ceramic Crystal Structure based on V. Ceramic Crystal Structure based on Number of Number of Atoms per Lattice Site 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 3.3-2 F 11.5

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B. The Sodium Chloride Structure

a0(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 sitecation ： 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).

2 2( ) / ( )r Mg R O

F 3.7-3 F 12.2

F 3.3-3

rNa+/rCl- = 0.012/0.181 = 0.56

CN = 6 octahedral siteT 12.3 T 12.2 F 12.2

F 11.7

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C. The Diamond Cubic Structure Diamond has an FCC lattice with two atoms per

site, there are eight atoms per unit cell.

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F 12.15

Why this structure ? rcation / ranion = 1 CN = 8Covalent bonding ： CN=4 the C-C-C bond angle = 109.50 tetrahedral sites

a0 (diamond cubic)=8r/ . Other materials with this structure: silicon and germanium.

one carbon ： lattice siteThe 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.

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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

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F 3.7-4

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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

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A. AX-TYPE CRYSTAL STRUCTURESAX 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 : FCCexamples: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.

VI. Ceramic Crystal Structure based on Chemical VI. Ceramic Crystal Structure based on Chemical FormulaFormula (Considering or looking at the packing of one (Considering or looking at the packing of one of the ions.)of the ions.)

(2) Cesium Chloride Structure

F 12.2

F 12.3

F 12.4F 3.7-4

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The charges on the cations and anions are not the same. Example ： fluorite structure (AX2) and antifluorite structure

(A2X).

fluorite (CaF2) ： rc/rA for CaF2: 0.8, coordination number: 8.

Crystal structure would be similar to CsCl except that only half the center cube positions are occupied by Ca2+ ions. One unit cell consists of eight cubes. Other compounds: UO2, PU2, and ThO2

C. AmBnXp – TYPE CRYSTAL STRUCTURES

A typical example ： barium titanate (BaTiO3), perovskite

crystal structure. At temperatures above 120℃: cubic, Ba2+ ions at all eight corners, single Ti4+ at the cube center, O2- ions at the center of each of the six faces.

B. AmXp— type Crystal Structures

F 3.7-5 F 3.5 F 12.5

F 12.6 T 12-4

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◎ 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. NaWP3, CaSnO3, YAIO3; AB3 structures, ReO3, WO3, NbO3, NbF3, TaF3; TiOF2, MoOF2.

VII. Ceramic Crystal Structures based on VII. Ceramic Crystal Structures based on building blocksbuilding blocksimagine the structure to be made of the various

building blocks.

F 12.6 F 3.9 F 3.8

F 3.7-8

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Named after the naturally occurring mineral MgAl2O4, general

formula is AB2O4, 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 TiO6 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, BaTiO3 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

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While SiO2 (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 (SiO4)4- tetrahedron. The crystobalite crystal

structure, can be envisioned as the diamond cubic structure with an (SiO4)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 Si3N4): Si tetrahedron, e.g., SiO4 in silicates, SiC4 in

SiC, SiN4 in Si3N4.

E. STRUCTURE OF COVALENT CERAMICS

F. The Crystobalite Structure

F 12.9 F 3.11

F 12.10F 12.9F 3.4-6

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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 VIII. Ceramic Crystal Structures From The Close Packing Packing of Anions of Anions

F 3.6-1

F 3.5-3 F 3.5-3

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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 (Al2O3): the oxygen ions are hexagonally close-packed, Al ions fill two-thirds of octahedral sites. Wurtzite: One-half the tetrahedral sites are filled.

F 12.2 F 3.7-4

F 11.15

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Other, but not all, ceramic crystal structures may be treated in a similar manner, included are the zinc blende and perovskite sturctures.

Spinel sturcture (AmBnXp): magnesium aluminate or spinel (MgAl2O4): the O2- ions form an FCC lattice, M2+ ions fill tetrahedral sites and Al3+ 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.

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IX. CERAMIC DENSITY IX. CERAMIC DENSITY COMPUTATIONSCOMPUTATIONS

Theoretical Density (perfect crystal) AC

AC

NV

AAn (12.1)

n’ = the number of formula units’ within the unit cell

AC = the sum of the atomic weights of all cations in

the formula unit

AA = the sum of the atomic weights of all anions in

the formula unit

VC = the unit cell volume

NA = Avogadro’s number, 6.023 1023 formula

units/mol

% theoretical density =measured densitytheoretical density

× 100%

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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

AC = ANa = 22.99g/mol

AA = ACl = 35.45g/mol

VC = a3

a = 2rNa+ + 2rcl

-

rNa+ and rCl

-: 0.102 and 0.181 nm, respectively.

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33 )22( ClNarraVc

3

23377

3

/14.2

10023.610181.0210102.02

45.3599.224

22

cmg

NArr

AAn

ClNa

ClNa

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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 SiO4

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

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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 SiO4

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

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THE SILICATES

One, two or three of the corner oxyge atoms of the SiO4-–4

thtrahedra are shared by other tetrahedra, examples: SiO44–,

Si2O76-and Si3O9

–6, positively charged cations such as Ca2+, Mg2+ , and Al3+ (1) compensate the negative charges from the SiO4

4- (2) ionically bond the SiO44- together.

Simple Silicates

For example, forsterite (Mg2SiO4): every Mg 2+ ion has six

oxygen nearest neighbors.

Akermanite (Ca2MgSi2O7) : Two Ca–2 and one Mg+2 bonded to each Si2O7

-6.

F 12.12

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Layered Silicates

Characteristic of the clays (黏土 ) and other minerals.

Kaolinite (高嶺土 ) clay has: Al2(Si2O5)(OH)4 , silica tetrahedral layer (Si2O5)2- is made electrically neutral by an adjacent Al2(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 <1m nearly hexagonal.

Other minerals also in this group are talc (滑石 ) [Mg3(Si2O5)2(OH)2] and the micas (雲母 )

[e.g., muscovite, KAl3Si3O10(OH)2].

F12-13 F12-14

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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.

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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.

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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

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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.

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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.

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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 C60 .

Each molecule is composed of both hexagon (six-carbon atom) and pentagon (five-carbon atom) One such molecule: 20 hexagons and 12 pentagons.

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C60 (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 C60 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.

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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 C60 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.

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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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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

Ti4+

Ba2+ or Ca2+

BaTiO3 , CaTiO3

O2-

o2-

89

Lattice

No. 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

rcation

ranion

SC

FCC

BCC

HCP