Chapter 3. The structure of crystalline solids3.1. Crystal structures

3.1.1. Fundamental concepts

3.1.2. Unit cells3.1.3. Metallic crystal structures

3.1.4. Ceramic crystal structures

3.1.5. Silicate ceramics3.1.6. Carbon

3.1.7. Polymorphism

3.2. Crystallography

3.2.1. Crystal systems

3.2.2. Crystallographic directions and planes

3.2.3. Linear and planar density3.2.4. Closed-packed crystal structures

3.3. Crystalline and noncrystalline materials3.3.1. Single crystal

3.3.2. Polycrystalline materials

3.3.3. X-ray diffraction

3.3.4. Noncrystalline solids

3.1. Crystal structures

3.1.1. Fundamental concepts

Solid materials may be classified according to the regularity withwhich atoms or ions are arranged with respect to one another.A crystalline material is a material that the atoms are situated in aperiodic array over large atomic distances.A noncrystalline or amorphous material is a material that does nothave a long-range atomic order.

Crystal structure is a manner in which atoms, ions, or moleculesare spatially arranged.Some properties of crystalline solids depend on the crystal structure.

Atoms or ions are considered as atomic hard sphere model(the atoms touch one another).Lattice is a [3] array of points coinciding with atom positionsor sphere centers.

3.1.2. Unit cell

3.1. Crystal structures

Unit cell is the basic structural unit or building block of the crystalstructure.The unit cell defines the crystal structure by virtue of its geometryand the atom positions withinMost of unit cells are parallelepipeds of prisms having 3 sets ofparallel faces.A unit cell is a representation of the symmetry of the crystal structure

which is the highest level of geometrical symmetry.

The most relatively simple crystal structures are :•Face-centered cubic (FCC),•Body-centered cubic (BCC),•Hexagonal close-packed (HCP).

3.1.3. Metallic crystal structures

3.1. Crystal structures

FCC structure has a cubic geometry with atoms located at each ofthe corners and the centers of all the cube faces.Many metals have this FCC structure.

The relation between the cube edge length,a and the atomic radius, R:

22Ra =

The FCC structure has 4 whole atoms [= (8 x 1/8) + (6 x ½)]

The coordination number (CN) is a number that an atom touches the nearest neighbour atoms.The coordination number of FCC is 12

The atomic packing factor (APF)V

V

C

S

volumecellunit total

cellunit ain atoms vol.ofAPF ==

RRV 33S π

3

16π

3

4(4) == 216RV 3

c =For FCC: and

APF = 0.74; which is the maximum packing possible

BCC structure has a cubic unit cell with atoms located at each of the corners and a single atom at the cube center.

The relation between the cube edge length,a and the atomic radius, R:

3

4Ra =

The BCC structure has 2 whole atoms [= (8 x 1/8) + (1 x 1)]

The coordination number of BCC is 8

APF = 0.68

33s πR

3

8πR

3

4(2)V ==

33

64RV

3

c =For BCC: and

HCP structure is the common metallic crystal structure.It has 6 atoms in regular hexagones at the top and the bottom faces and surround a single atom in the center. Three additionalatoms are on a plane located between the top and the bottom planes.

The ideal c/a ratio is 1.633

The HCP structure has 6 whole atoms [= (2 x 6 x 1/6) + (2 x ½) + (1 x 3)]

The coordination number of HCP is 12 and the APF is 0.74

Metal density

where: n = number of atoms associated with each unit cell,A = atomic weight,Vc = volume of the unit cellNA = Avogadro’s number (6.023 X 1023 atoms/mol)

CAVN

nAρ =

ExampleCopper has an atomic radius of 0.18 nm, an FCC crystal structure,and an atomic weight of 63.5 g/mol. Compute its theoretical density and compare the answer with its measured density.

AnswerFCC: n = 4 atoms/unit cell; Acu = 63.5 g/mol; and 216RV 3

c =

3

233-8

AC

Cu

g/cm 8.89

atoms/mol)(6.02x10]2cell/unit cm)(1.28x10 [16

g/mol) cell)(63.5atom/unit (4

NV

nρ

A

=

=

=

The theoretical density is 8.94 g/cm3

3.1. Crystal structures

3.1.4. Ceramic crystal structures

Ceramics are more complex than metals because ceramics areconsisted of at least two elements.The atomic bonding is in the range of ionic to covalent.The degree of ionic character depends on the electronegativitiesof the atoms.(the electronegativity is usually shown in the Periodic Table)

For ceramics with predominantly ionic, the atoms are consideredas ions.

The metallic ions or cations are positively chargedThe nonmetallic ions or anions are negatively charged.

The crystal structure depends on two characteristics:•the magnitude of the electrical charge on each of the component ions,•the relative sizes of the cations and anions

The 1st characteritic:the magnitude of the electrical charge on each of the component ions.The cations must be in balance with the anions.The chemical formula of a compound indicates the ratio of cations andanions or the composition that achieves this charge balance.Example:

A calcium ion has a +2 charge (Ca2+ )A fluorine ion has a –1 charge (F-)

CaF2

The 2nd characteristic:the relative sizes of the cations and anions.The sizes of ionic radii of the cations (rC) and anions (rA) determinedthe configuration of a compound.

Cations are ordinarily smaller than anions, rC/rA ≤ 1

3.1.4. Ceramic crystal structures

Most ceramics are composed by an equal number of cationsand anions and identified as AX compounds(A refers to the cation and X refers to anion).

A. Rock salt structuresThe most common AX crystal structures is the sodium chloride(NaCl) or rock salt.Two interpenetrating FCC lattices, one composed of the cations(Na+), and the other of anions (Cl-). The coordination number of both cations and anions is 6.The rC/rA is between 0.414 – 0.732.

3.1.4.1. AX-type crystal structures

Some common rock salt structures areNaCl, MgO, MnS, LiF, and FeO.

B. Cesium chloride structuresThe anions are situated a each of the corners of a cube, whereasthe cube center is a single cation.This is not a BCC crystal structure!!The coordination number of both cations and anions is 8.

C. Zinc blende or sphalerite structure(Mineralogical term for zinc sulfide (ZnS)S atoms are located in all corners and face positions of a cubic,while Zn atoms fill the interior in tetrahedral positions in covalentbonding.The coordination number for both cations and anions is 4 (tetrahedrally coordinated).

Some common zinc blendecrystals are ZnS, ZnTe, and SiC

If the Zn and S atom positions arereversed, the same structure is obtained.

3.1.4.2. AmXp-type crystal structures

This structure has dissimilar chargesof cations and anions.The chemical formula is AmXp

where m and/or p ≠ 1.Example : CaF2

The rC/rA is 0.8 and the coordinationnumber is 8.Calcium ions are located at the centerof a cube and fluorine ions at the corners.One unit cell consists of 8 cubes.Some other compounds are UO2,PuO2 , and ThO2.

3.1.4.3. AmBnXp-type crystal structures or perovskite crystal structure

Some compounds have more than one type of cations, they areidentified as A and B. The chemical formula is AmBnXp-Example: BaTiO3.

Ba2+ ions are located at eightcorners of a cube, a single To4+is at the cube center and O2- ionsare situated at the center of eachof the six faces.

Ceramic density

( )

AC

AC

NV

AAn'ρ

∑+∑=

where: n’ = the number of formula units within the unit cell,

ΣAC = the sum of the atomic weights of all cations in the formula unit,

ΣAA = the sum of the atomic weights of all anions in the formula unit,VC = the unit cell volume,NA = Avogadro’s number = 6.023 x 1023 formula units/mol.

ExampleCalculated the theoretical density for sodium chloride. The atomic weightof sodium and chloride, ANa and ACl are given as 22.99 g/mol and 35.45 g/mol respectively. The ionic radii of sodium and chloride, rNa+

and rCl- are given as 0.102 nm and 0.181 nm respectively.Compare the result with its measured density?

n’ = 4

ΣAC = ANa = 22.99 g/mol,

ΣAA = ACl = 35.45 g/mol,

a = (2 rNa+ + 2 rCl- ) mVC = a3 = (2rNa+ + 2rCl- )3 m3

= [(0.204 x 10-9)+0.362 x 10-9)]3 m3

= 559.276 10-9 m3

( )

A3

-ClNa

ClNa

N)2r(2r

AAn'ρ

+

+=

+

3

2323

2.14g/cmρ

))(6.023X100(559.276X1

35.45)4(22.99ρ

=

+=

−

3.1. Crystal structures

3.1.5. Silicate ceramics

Silicates are materials composed primarily of silicon and oxygen.The basic unit of silicates is SiO4

4- tetrahedron, which is each atom ofsilicon (located in the corners of tetrahedron) is bonded to four oxygenatoms (situated at the center of tetrahedron).Some common silicates are rocks, clays, sand, and a bulk of soils.

Silicates are not considered to be ionic because interatomic covalentSi-O bonds are very strong.Silicate structures vary in different arrangements as each oxygen atom requires an extra electron to achieve a stable electronic structure.The arrangements could be [1] or [2] or [3] dimensional structures.

3.1.5.1. Silica

Silica or silicon dioxide (SiO2) is the most simple silicate.Oxygen atom at every corner of each tetrahedron shares theoxygen atoms with the adjacent tetrahedral and this configurationbuilds a [3] network structure.This material is electrically neutral and all atoms have stable electronic structures.

A crystalline structure is formed if these tetrahedra are arrayed in aregular and ordered manner. There are 3 primary polymorphiccrystalline forms of silica, they are: quarts, cristobalite, and tridymite.

The structures are complicated and relatively open so that they have

low densities. (eg ρquartz 2.65 g/cm3).The Si-O interatomic bonding is strong so that they have relatively high melting temperature. (eg Tmquartz 1710 ºC).

A noncrystalline solid or glass structurecan also be formed from silica.

3.1.5.2. The Silicates

Oxygen atom at every corner of each tetrahedron shares the other

tetrahedral to form complex structures.Some positively charged cations such as Ca2+ ,Mg , and Al canbe part of the structures. These cations compensate the negativecharges of SiO4

4- to form a neutral compound andto build an ionic bonding between the SiO4

4-

tetrahedral together.

A. Simple silicates

B. Layered silicates

A [2] sheet or layered structure can also be produced by the sharingof 3 oxygen ions in each of tetrahedral.The repeating unit formula is (Si2O5)

2-

The net negative charge isassociated with the unbound

oxygen atoms ( ⊥ layer).The excess of cations isassociated with the 2nd layer.Net charge: neutral.Basic structure of clays andother minerals.

One of the most common clay minerals and simple two-layer silicatesheet structure is kaolinite.Its formula is Al2(Si2O5)(OH)4

The OH- and the O2- ions in the anion midplane has a strong ionic-covalent bonding, whereas the adjacent sheets have weak van der Waalsbondings.

3.1. Crystal structures

3.1.6. Carbon

Carbon is an element that exsist in various forms, from polymorphicto amorphous sates.Carbon is not classified as metal, ceramic or polymer, however sometimes it is classified as a ceramic polymorphic

3.1.6.1. Diamond

Diamond is a metastable carbon polymorphat RT and atmospheric pressusre.The crystal structure is a variant of the zincblende.It is called the diamond cubic crystal structure.

3.1.6.2. Graphite

Graphite is more stable than diamond at ambient temperature and pressure.The crystal structure is composed of layers of hexagonally arranged carbonatoms. In the layers, each carbon atom is bonded to three coplanarneighbour atoms by strong covalent bonds.Between layers the electron is bondedby a weak van der Waals bond.

3.1.6.3. Fullerenes and carbon nanotubes

Fullerenes is a polymorphic of carbon (1985).The structure consists of a hollow spherical cluster of 60 carbon atoms and is called a C60 molecule. A molecule is composed of groups of carbon atoms that are bonded to One another to from both hexagon (six-carbon atom) and pentagon (five-carbon atom) geometrical configurations.

Its commercial name is buckminsterfullerene.Invented by R. Buckminster Fuller (1985)

“Soccer ball symmetry”20 hexagon and 12 pentagon are arrayedin such way that no two pentagons sharea common side.

Carbon nanotube structure consists of a single sheet of graphite, rolled into a tube & both ends are capped with fullerene hemispheres.The nano represents that the tube diameter is on the order of a nanometer. It has a very good aspect ratio; which the length of the molecule is muchgreater than its diameter.

Special properties of CNT:- Extremely strong and stiff, relatively ductile- Relatively low density- Unique and structure-sensitive electrical characteristics

Some properties:Tensile strength 50 – 200 GPa

Elastic modulus ≥ 103 GPaFracture strain 5% - 20%

Another form:Multiple-walled CNT(consists of concentric cylinders)

3.1.7. Polymorphism

Polymorphism is a phenomenon when some metals and nonmetals may have more than one crystal structure.(In elemental solids, it is called allotropy).Example:�Graphite is stable at ambient condition, whereas diamond is formed

at extremely high pressures.

�Iron has a BCC structure at RT and FCC structure at 912 °C.