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1. INTERATOMIC BONDS AND CRYSTAL STRUCTURES OF CERAMICS

Preface

The purpose of this chapter is to acquaint the student with the crystalline structures on which ceramics are based. Properties of ceramics are substantially controlled by the type of the inter-atomic bonds and the crystal structure. Therefore understanding structures gives insight into the performance of the ceramics. Once the concepts of the structures in ceramics are understood, it is possible to predict the structure of new or modified compounds and to propose suitable substitutions and modifications in order to tailor desired properties to a considerable extent.

Introduction: Chemical Elements, Minerals, and Ceramic Raw Materials

Seven of the eight common elements in minerals are cations (Si+4, Al+3, Fe+3/+2, Ca+2, Na+1, K+1, Mg+2 ). Oxygen is the only common anion. Hence, most minerals (and ceramics) are oxides. Table 1.1: Major chemical elements in the earth’s crust and their abundance Element Weight% Atomic% Atomic # Atomic wt. Valence Oxygen (O) 46.6 62.6 8 16.00 -2

Silicon (Si) 27.7 21.2 14 28.09 +4

Aluminum (Al) 8.1 6.5 13 26.98 +3

Iron (Fe) 5.0 1.9 26 55.85 +2,+3

Calcium (Ca) 3.6 1.9 20 40.08 +2

Sodium (Na) 2.8 2.6 11 22.99 +1

Potassium (K) 2.6 1.4 19 39.1 +1

Magnesium (Mg) 2.1 1.8 12 24.31 +2

The important chemical elements and their oxides are listed below: Lithium is often used in glass-ceramic products with very low thermal expansion coefficients. !-spodumene (!-LiAlSi2O6) and !-eucryptite (!-LiAlSiO4) are the crystalline phases responsible for the thermal shock resistance. Important lithium minerals are "-spodumene ("-LiAlSi2O6) and petalite (LiAlSi4O10). Beryllium oxide (BeO) has a very high melting point and a very high thermal conductivity coefficient. It is sometimes used in aerospace applications because of its low density. High thermal conductivity and low electrical conductivity make it an attractive electronic substrate. Beryl

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(Be3Al2Si6O18) is the most important beryllium mineral. The gemstones aquamarine and emerald are colored varieties of beryl. Boron oxide (B2O3) is an important constituent of many glass and enamel compositions. Cubic boron nitride (BN) and boron carbide (B4C3) are hard abrasive materials. Borate minerals such as borax (Na2B4O7•10H2O) and kernite (Na2B4O7•4H2O) are the chief sources of boron. Carbon occurs in two crystalline states: diamond and graphite. Diamond is an important gemstone and is abrasive. Graphite, due to its high melting point, is used to fabricate crucibles. Important carbonate materials include calcite (CaCO3), dolomite (CaMg(CO3)2), and magnesite (MgCO3). All three readily decompose on heating to give reactive oxides. Sodium is usually extracted from halite (rocksalt or NaCl). Soda (Na2O) is an important constituent of common glass. Trona (Na3(CO3) (HCO3)•2H2O) is another mineral source of soda. Sodium forms 2.6 atomic % of the earth’s crust and is found in many common silicates such as the feldspar albite (NaAlSi3O8). Magnesium is a very common element too. Abundant to about 1.8 atomic %, Mg enters into a large number of rock-forming minerals including olivine (mostly Mg2SiO4) and pyroxenes such as diopside (CaMgSi2O6). Of greater economic importance are magnesite (MgCO3) and dolomite (CaMg(CO3)2) used in making ceramic refractories. Aluminum is the third most abundant element. The principal ore is bauxite, an impure mixture of diaspore (AlOOH) and gibbsite (Al(OH)3). Aluminum oxide (alumina, "-Al2O3) occurs in nature as the mineral corundum. Ruby and sapphire are gem varieties of corundum. Alumina ceramics are noted for their strength, hardness, and high melting point. China clay (kaolinite -Al2Si2O5(OH)4) is used in making whitewares. Kyanite (Al2SiO5) is an aluminosilicate used in refractories. Silicon dioxide (silica) occurs in nature as the mineral quartz (SiO2). Second only to oxygen in abundance, silicon occurs in a very large number of minerals. Silica glass (SiO2), silicon nitride (Si3N4) and silicon carbide (SiC) form important ceramic products. Phosphorus is a glass-forming element like Si and B. Apatite (Ca5(PO4)3OH), the most common phosphate mineral, is an important constituent of teeth and bones. Synthetic apatites are used as lamp phosphors. Sulfur occurs in elemental form but also as sulfates and sulfides. Gypsum (CaSO4•2H2O) is used to make plaster cement. Galena (PbS) and pyrite (FeS2) are sulfide minerals. Potassium is extracted from chloride minerals such as sylvite (KCl) and carnallite (KMgCl3•6H2O). The oxide potash (K2O) is a minor additive in many glass compositions. Orthoclase (KAlSi3O8) is potash feldspar. Calcium oxide (lime - CaO) is normally obtained by heating calcite (CaCO3). Apatite, gypsum, dolomite, and calcic feldspar (anorthite - CaAl2Si2O8) are other common calcium minerals. Lime is a key constituent of cement.

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Titanium dioxide (TiO2) has an exceptionally large refractive index and dielectric constant. It is an important constituent of paint and electroceramics. Rutile (TiO2) and ilmenite (FeTiO3) are the most common ore minerals. Chromium is obtained from the mineral chromite (FeCr2O4), which is widely used as a refractory. Oxides of chromium, cobalt, and other transition metals are used as pigments to provide color in ceramic bodies. Metallic chromium is an important constituent of stainless steel. Iron is the heaviest of the common elements. Hematite ("-Fe2O3) and magnetite (Fe3O4) are ore minerals used for steel-making. Magnetic ceramics are made from ferrites such as MnFe2O4 and BaFe12O19. Zinc oxide (ZnO) is an important constituent of glass, magnetic ferrites, and electroceramic varistors. Zincblende (ZnS) is the ore mineral.

Bonding trends: Electronegativity

Electronegativity is the measure of the tendency of an atom to attract an electron. Electronegativity difference between two atoms will have a strong influence on the polarity of their bond. The larger the difference in electronegativity between them, the more ionic the bond. The closer the electronegativities the more covalent the bond.

The most commonly used scale of electronegativity was proposed by Pauling. He scaled from 4.0 for the Fluorine, which is the most electronegative, down to Cesium with electronegativity 0.7. The scale is based on the observation that the energy of bonds between unlike atoms EAB is usually greater than the average of the homopolar bond energies, EAA and EBB. Pauling reasoned that the extra energy came from the difference between the ability of the A and B atoms to attract electrons. This energy difference, #, is calculated according to (1.1) and then is put into the empirical relationship in eq. (1.2), giving the electronegativity (X) of an atom in reference to another one (as is using [eV] - if using kJ/M a conversion factor applies, i.e. 0.102 ):

!

" = EAB #EAA + EBB

2 (1.1)

XA # XB = (eV #0.5) " (1.2)

The Pauling electronegativities are shown in Fig. 1.1.a

The trends are : Descending in the periodic chart means there is a larger separation between the valence electrons and the nucleus, hence the binding force is smaller (smaller electronegativity). When moving from right to left on the chart, the nuclear charge decreases – hence a decrease in the electron binding force and in the electronegativity (Fig. 1.1.b).

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Figure 1.1.a : Pauling electronegativities. The bold line marks an arbitrary boundary between metals (to the left) and non metals (to the right).

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Fig. 1.1.b: Trends in electronegativity following the periodic table.

The inter atomic bonds in ceramics

Electrostatic force is responsible for the cohesion of solids. Unlike metals that have mainly inter atomic metallic bonds and polymers that have mainly covalent and Van der Waals bonds, ceramics have all type of bonds. Many of the useful ceramics (most of the oxides) have ionic bonds or bonds with strong ionic character, but other important groups such as nitrides are covalently bonded. Van der Waals bonds appear often in ceramic compounds with layered structures, and hydrogen bonds appear in some ceramics that contain hydrogen, for example, important groups of minerals such as clay have these bonds. Metallic bonds exist also in some ceramics, for example in TiN. However, broadly speaking, ceramics are mainly ionic or covalent.

Ionic bonds

An ionic bond results from an electrostatic attraction between positive and negative ions (cations and anions) which are derived from the free atoms by the loss or gain of electrons. Metal atoms tend to loose the outer electrons which are loosely bound and become cations and non metal atoms tend to acquire electrons to complete unfilled orbitals and become anions. Namely, in the pure ionic bond, a complete transfer of electrons occurs. When the electronegativity of one ion is much larger than that of another ion the latter looses electrons to the first and ionic bond results. Ionic bonds are non-directional (isotropic).

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-NaCl is a good example of an ionic material; before bonding, the electronic configuration is as follows:

Na: 1s2 2s2 2p6 3s1

Cl: 1s2 2s2 2p6 3s2 3p5

-Na has one electron outside the closed shell, while Cl needs one more electron to gain the inert gas

configuration. As a result, Na looses an electron and Cl gains one:

Na+: 1s2 2s2 2p6

Cl-: 1s2 2s2 2p6 3s2 3p6

Covalent bonds

Whereas ionic bonds involve electron transfer to produce oppositely charged species, covalent bonds arise as a result of electron sharing. Namely, the electrons spend more time in the area between the nuclei than near to one or the other nucleus. The attraction between the electrons and the nuclei lowers the potential energy of the system forming a bond.

In covalently bonded solids the bonds between atoms form 3-dimensional structures. Silicon based

solids are a good example for this. The ground state of Si is 3s23p2, leading to the formation of two

bonds only. However, the outer 3s2 and 3p2 electrons participate in the bonding; in Si there is a so

called orbital hybridization between the s and the p orbitals to form sp3 hybrid orbitals. The

character of these new orbitals is such that they form a tetrahedral arrangement around the Si, each

of the orbital being populated with one electron of the given Si (Fig. 1.2). Consequently, each Si

can bond to 4 other atoms to form 3-dimensional structures.

Many III-V compounds (compounds formed between atoms of the III group and atoms of the V group) are covalent and thus tetrahedrally coordinated (have 4 neighbors around each atom): BN, AlP, GaAs, InP. Also many II-VI compounds have a covalent character and are tetrahedrally coordinated: CdS, BeO, ZnS.

Figure 1.2: Ground state of Si atom (a), electronic state after hybridization (b) and the directionality of sp3 bonds (c).

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Mixed bonds

Often a given bond has partial ionic nature and partial covalent nature. The degree of ionicity of the bond between two ions A and B is defined by their electronegativity difference (XA-XB)

ionicity = 1 ! e!XA!XB( )2

4 (1.3)

Oxides of rock-salt structure (NaCl), for example, will have metal-oxygen bond of different ionicity depending on the cation. The bond is considered predominantly ionic when its ionicity > 0.5, namely #X>1.7. In ternary compounds (having 3 cations in 3 distinct lattice sites) and more complex compounds the fractional ionicity is determined by using stoichiometrically weighted averages for the values of XA and/or XB.

Secondary bonds Van Der Waals bond is a secondary bond (weaker than the primary ionic, covalent, and metallic bonds). It exists due to electrostatic attraction between dipoles. Dipoles are created due to oscillation of the electron cloud around the nucleus of the atom. Therefore the centers of the positive and negative charges do not coincide permanently and a weak fluctuating dipole is produced. A force then exists between opposite ends of dipoles in adjacent atoms and tends to draw them together. The bonds produced by these fluctuating dipoles are non-directional and very weak. These bonds exist very often in ceramic layer structures, in some silicates, for example, between their layers, and in the CdI2 structure (Fig. 1.3).

I

Cd

Figure 1.3: Van der Waals bonds in CdI2 (MoS2) between the CdI2 (MoS2) layers.

Hydrogen bonds - When hydrogen loses an electron and becomes H+, it essentially becomes a proton. The H+ acts as a bridge between two strongly electronegative anions such as O2- or F-. In water, due to the asymmetry of the molecule a dipole is created, with the oxygen acting as the negative end of it. The hydrogens, being the positive end of the dipole tend to attach also to neighboring oxygens, forming weak intermolecular bonds. The hydrogen bond constitutes also the bonding between layers in some clays (Fig. 1.4).

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Oxygen

2 oxygenssuperimposed

Silicon

Si-Al

Figure 1.4: Hydrogen bond in clay

Heterodesmique structures Many compounds have structures where bonds of different nature coexist. These compounds are said to have heterodesmic structures. In a large number of layer structures the bonds within the layer plan are ionic or covalent and perpendicular to the plan they have Van der Waals nature. Some examples of heterodesmic structures are shown in Fig. 1.5.

VAN DER WAALSMETALIQUE

COVALENT

IONIQUE

MoS 2Graphite

Argiles

SilicatesNiobates, Titanates

Oxydes avec défauts

Ionic

Metallic

ClaysDefected oxides

Figure 1.5: Heterodesmic ceramics

Crystal bond - property relation: relationship between bond type and hardness “Materials resistance to scratching” is the simplest definition of “Hardness”. To analyze hardness,

a material is simply scratched with another and the material where the scratch marks appear will

have the lower hardness. This is commonly done in mineralogy, and the Moh’s hardness scale is

widely used (table 1.2). The scratch means “breaking of the crystal bonds”. Covalent bonds make

the hardest materials (e.g. diamond). In a given material is the weakest bond which breaks first.

Therefore, in heterodesmic materials their hardness is dermined by the weaker bond. A typical

example is graphite. The C-C bonds within the layers are as strong as those in diamond. However,

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the weak bonds between the layers determine its hardness of only 2 as compared to 10 for diamond. The bonding in hard materials must be 3-dimensional. Remark: Metals do not appear on Moh’s

scale because they seldom occur as minerals. Pure metals have approximately the same hardness of

ionic crystals (3-5). Some carbon-reinforced metals (steel) have a higher hardness of about 6-8.

Table 1.2: Moh’s hardness scale Hardness Materials Formula Broken bonds Bond type 10 Diamond C C - C Covalent 9 Corundum Al2O3 Al - O 50% ionic 8 Topaz Al2SiO4F2 Al - O 50% ionic Al – F ionic 7 Quartz SiO2 Si - O 50% ionic 6 Orthoclase KAlSi3O8 Al - O 50% ionic Si – O 50% ionic K - O ionic 5 Apatite Ca5(PO4)3F Ca -O Ionic Ca - F Ionic 4 Fluorite CaF2 Ca - F Ionic 3 Calcite CaCO3 Ca - O Ionic 2 Gypsum CaSO4•2H2O OH—O Hydrogen-Bond 1 Talc Mg3Si4O10(OH)2 O---O Van der Waals

Ionically bonded ceramics

Ceramics based on crystals with ionic bonds (anions and cations) show in general high melting temperatures. This is due to the strong bonds.

In order to disrupt an ionic bond, Coulomb attraction between the anion and cation has to be overcome. The repulsive forces that prevent the ions from collapsing into each other (related to Pauli exclusion principle) have to be taken into account too. Altogether the bonding energy (for a single bond) is:

!

Ebond = "e 2Z1Z24#$oR

+BRn

(1.4)

Coulomb Repulsion energy energy

Where e is the charge of an electron, $0 is the permittivity of vacuum, Z - the valence of the ions, R is the distance between the ions, B is a material constant, and n - Born exponent ( ~10). The minimum energy state is attained for the equilibrium distance (R = R0). This can be shown to be:

!

Ebond = "e 2Z1Z24#$oR0

(1" 1n) (1.5)

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For a crystal, the ensemble of electrostatic forces has to be taken into account (attraction between cations and anions that are not first neighbors, repulsion between cations and other cations, etc.). This has been done for different crystalline structures to give the lattice energy:

!

Ela t t i c e= "Ne 2Z1Z2#4$%oR0

(1" 1n) (1.6)

where N is Avogadro number, and " is a constant (called Madelung constant) that represents the electrostatic energy of the crystal relative to the energy of the same number of isolated molecules. " is the summation of the electrostatic interactions given by

!

"=Zi /Zi( ) Z j /Zj( )

xi j# (1.7)

where x is the distance between the ions divided by the cation-anion nearest distance. Table 1.3 shows the values of the Madelung constant for common ceramic structures. It can be seen that it does not vary strongly between zinc-blend and wurtzite. It will not be surprising that some materials having zincblende structure can have also wurtzite structure , as will be seen later (MnS, ZnS, CdS).

An important result of the above discussion is that the lattice energy is proportional to the ionic charge and inversely proportional to the distance between the ions:

!

E "Zcation Zanion

Ro (1.8)

Table 1.3: Madelung constants for common ceramic structures *

Structure "

Rocksalt 1.748

Cesium Chloride 1.763

Zincblende 1.638

Wurtzite 1.641

Fluorite 2.519

Corundum 4.040

* the structures are described later on in this chapter

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Relation between ionic radius, ionic charge and material’s properties

Knowing the ionic radius and the charge, various properties can be related to the bond strength.

BeO MgO CaO SrO BaO

Distance (Å) 1.65 2.10 2.40 2.57 2.77

Hardness 9.0 6.5 4.5 3.5 3.3

NaF (1) MgO (2) ScN (3) TiC (4)

Distance (Å) 2.31 2.10 2.23 2.23

Hardness 3.0 6.5 7-8 8-9

NaF NaCl NaBr NaI

Distance (Å) 2.31 2.79 2.94 3.18

Melting T(°C) 988 801 740 660

NaF CaO

Distance (Å) 2.31 2.40

Melting T(°C) 988 2570

NaF NaCl NaBr NaI CaF2

Distance (Å) 2.31 2.79 2.94 3.18 2.43

Thermal expansion(10-6) 39 40 43 48 19

From the above and from Table 1.2 it can be concluded that bond type bond strength, bond length and valence are all important to the hardness. Same parameters control also the melting point of the material and its thermal expansion coefficient.

The ionic radius

The ionic radius has been shown above to be very important for determining the stability of crystalline structures, the structure of ionic solids, and their properties.

The inter atomic distance is easily found from X ray diffraction data. By comparing a large number of compounds it was possible to determine the ionic radii. The radius of the oxygen ion is taken as 1.26 Angstrom. The full table of ionic radii is given in appendix 1. The ionic radius can be determined more precisely from electron density maps obtained from X-ray diffraction data. The radius is defined from the volume of a sphere in which the electron cloud is effectively found. The radius depends therefore on the number of electrons in the ion, and therefore the valence of the ion will change the ionic radius. The number of nearest ionic neighbors influences the density of the electrons cloud too and hence the radius is affected.

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The different factors that influence the ionic size are:

1. When a new layer of electrons is added, the ionic radius increases.

2. The charge of the nucleus attracts the electron cloud, therefore an increase of the nucleus charge increases the attraction and decreases the ionic radius.

3. Electron clouds of inner orbitals screen the outer electrons from the attraction of the nucleus, therefore, the larger the number of inner orbital electrons, the larger is the ionic radius.

4. The larger the valence of an ion, the stronger is the inter ionic bond, hence larger overlap of electron clouds with the neighboring ions and the electrons in each ions are more "compressed", hence smaller radius.

Based on these factors the changes in ionic radii along rows and columns of the periodic table can be analyzed:

1. Cations are smaller than atoms, and the higher the valence of the cation, the smaller is the radius - this is because of the loss of electrons from the outer shell. In addition, the remaining cloud is drawn inward because its outer part is subjected to greater effective nuclear charge. Moreover, the higher the valence, the stronger is the bond to the neighboring anion, hence an even smaller radius.

2. Anions are bigger than atoms because of the addition of electrons in the outer shell. However the increase is not very large because the higher the valence, the stronger is the bond to the neighboring cation, and hence a "compression" of the electron-cloud results.

3. For a column of isovalent ions (isovalent =equal valence) or atoms, the radius increases upon increase of the atomic number, due to the addition of electron shells. However, for the transition metals the increase is less sharp: The additional electrons enter inner shells and the attraction to the nucleus is stronger, but the outer electrons can expand due to the better screening provided by the inner electrons, hence a mild increase of the radius.

4. For a row of atoms, the radius decreases drastically at the beginning, due to the increase of attraction by the nuclei (more charges). The additional electrons are in the outer shell, far away from the attractive forces of the nuclei. For the transition metals, the situation is different: The additional electrons enter the inner shells and therefore there is an increase in screening power and the outer electrons can even slightly expand.

5. An increase in the number of first neighbors around an ion (increase in the co-ordination number, CN) results in an increase in the radius. The bond strength, defined as the valence of the ion divided to number of neighbors decreases and hence the ionic radius increases.

RC.N.=4 ! 0.95 RC.N.=6 ; RC.N.=8 ! 1.09 RC.N.=6

Fig. 1.5 shows the ionic radii of selected ions, when surrounded by 6 ion neighbors (co-ordination number =6), thus illustrating, in part, the above discussion.

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Figure 1.5 : The ionic radii (in nanometers) of selected ions, at CN=6

Pauling's Rules on the packing of ions into crystalline structures

The structure of crystals of predominantly ionic character can be predicted following some basic empirical rules that are related to the ionic dimensions, to the valence of the ions and to the electrostatic forces between the ions. These simple rules were formulated by Pauling. They treat the ions as hard spheres, and this is correct only in the first approximation, therefore deviation or exceptions from and to the rules are found too.

1. To satisfy the electrostatic forces, cations are surrounded by anions and vice verse. Each ion prefers to be surrounded by the maximum number of ions of the opposite charge as first nearest neighbors. Each ion prefers to be as far separated as possible from ions of the same charge who are the second nearest neighbors.

Each cation is surrounded by a polyhedron of anion, the number of which is determined by the cation / anion size ratio. The sum of the cation and anion radii gives the distance between them. The size ratio cation / anion determines the number of nearest neighbors of each ions, the so-called co-ordination number of the ion (CN).

Assuming the cation size to be smaller than the anion size, a stable structure occurs when the co-ordination number is such that the cation fills exactly or is bigger than the minimum size of the interstice within the anion polyhedron (Fig. 1.6). If the anion size is smaller than the cation size, the same argument holds for the cation interstice. In this way the critical radius ratios for various co-ordination numbers can be calculated (Fig. 1.7).

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Figure 1.6: Stability criterion for structures: (a) stable, (b) stable (c) unstable.

Arrangement

Close-packed

Cube

Octahedron

Tetrahedron

Triangle

Linear

Cation/anion ratio

>1.0

>0.732

>0.414

>0.225

>0.155

>0

b'

c'

a

Figure 1.7: Critical radius for various packaging of ions.

2. Ions are charged and charge neutrality has to be preserved. The arrangement of the polyhedra has to take care of this. The charge of the ion divided to the co-ordination number is defined as the bond strength. In stable structures the sum of the bond strength of each ion is equal its valence.

3. The linkage between co-ordination polyhedra is most stable when they share corners, less stable when they share edges and least stable when they share faces.

This rule originates from the fact that cations prefer to maximize their distance from neighboring cations, due to electrostatic repulsion reason.

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4. Cations with small size and large valence prefer to be linked in corners (example is SiO2). This rule is in fact a stronger statement for rule 3, as small ions with large charge will have a larger repulsive force between them.

5. Simple structures are usually preferred over more complicated arrangements. Namely, in a stable structure there will be a limited number of polyhedra types. There are a number of exceptions to this last rule.

The limitations of crystal chemistry approach for predicting structures and properties:

It is important to emphasize that crystal chemistry can not predict ‘everything’ although it helps for first approximations and is a good useful starting point. In particular the prediction of electrical properties are not successful: for example the YBa2Cu3O7 superconductor was not ‘predicted’ based on its structure. On the contrary, its ionic bonding may lead to the conclusion that it will be an insulator. Also the fact that carbon can take the form of diamond (covalent bond) and graphite (van der Waals bond) and C60 (molecular structure with a shape of a ball) could not have been predicted by a crystal chemistry approach.

The crystalline structure in ceramics

Most of the structures in ceramic are based on close packing of the large ions with the small ions occupying the interstitial sites between the large ions.

Equally sized balls on a plane will be arranged as a layer with contiguous filled exagons in order to obtain the highest number of spheres per surface unit (see Fig. 1.8 a).

There are two basic ways to put layers of closely packed equally sized balls one on top of the other in a closed packed manner.

The ABA arrangement is that of [001] of hexagonal close packed (HCP) structure and

the ABC arrangement is that of [111] of face centered cubic (FCC) structure.

In both cases the interstitial sites are tetrahedral or octahedral (Fig. 1.8). Most of the oxides are arranged indeed with oxygens having the HCP or FCC structures, and with the interstitial sites partly or fully occupied by smaller cations. The site occupancy is tightly related with the ion size and its valence.

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Figure 1.8: Description of the ABA and ABC closely packed arrangement of hard spheres. The first layer of hard spheres (A) is drawn (a). The crosses (b) indicate the centers of these spheres. Layer B is on top of the A layer with centers of the spheres indicated by squares (b). The diamonds indicate the centers of the spheres in the C layer which lies above the B layer. The HCP structure consists of ABA layers. Interstitial tetrahedral sites are located directly bellow the centers of the B layer spheres. The octahedral interstices (indicated by the diamonds in (c)) are located between the A and B layers at equal distance to centers of 3 neighboring spheres of the A layer and 3 neighboring spheres situated above them in the B layer. The FCC structure consists of ABC layer arrangement. The tetrahedral and octahedral co-ordination polyhedra are shown in fig. 1.8.d.

The next figure depicts both, the Hexagonal and Cubic Compact- or Close-Packed structures. Please, identify the tetragonal and octahedral sites in both structures and compare the size of “a” in both illustrations. Identify the equivalent a-triangle of the hcp within the ccp picture.