CHAPTER 3Crystalline

Structure–Perfection

The transmission electron microscope (Section4.7) can be used to image the regular arrange-ment of atoms in a crystalline structure. Thisatomic-resolution view is along individual columnsof gallium and nitrogen atoms in gallium ni-tride. The distance marker is 113 picometersor 0.113 nm. (Courtesy of C. Kisielowski, C.Song, and E. C. Nelson, National Center forElectron Microscopy, Berkeley, California.)

Unit cell

Figure 3-1 Various structural units that describe the schematic crystalline struc-ture. The simplest structural unit is the unit cell.

c

b

a

Figure 3-2 Geometry of a general unitcell.

Figure 3-3 The simple cubiclattice becomes the simplecubic crystal structure whenan atom is placed on eachlattice point.

(i) (ii)

(iv) (v)

(iii)

=

(a)

(c)

(b)

Structure: body-centered cubic (bcc)Bravais lattice: bccAtoms/unit cell: 1 + 8 × = 2Typical metals: α-Fe, V, Cr, Mo, and W

18

Figure 3-4 Body-centered cubic (bcc) structure for met-als showing (a) the arrangement of lattice points fora unit cell; (b) the actual packing of atoms (repre-sented as hard spheres) within the unit cell; and (c)the repeating bcc structure, equivalent to many adja-cent unit cells (Part (c) courtesy of Molecular Simu-lations, Inc.).

(b)(a)

(c)

Structure: face-centered cubic (fcc)Bravais lattice: fccAtoms/unit cell: 6 × + 8 × = 4Typical metals: γ-Fe, Al, Ni, Cu, Ag, Pt, and Au

12

18

Figure 3-5 Face-centered cubic (fcc) structure formetals showing (a) the arrangement of lattice pointsfor a unit cell; (b) the actual packing of atomswithin the unit cell; and (c) the repeating fcc struc-ture, equivalent to many adjacent unit cells (Part(c) courtesy of Molecular Simulations, Inc.).

(a)

(c)

2 atoms perlattice point

(b)

Atom in midplane

Atom in midplane Atom centeredin adjacent unit cell

One-twelfthof an atom

One sixth of an atom

Structure: hexagonal close-packed (hcp)Bravais lattice: hexagonalAtoms/unit cell: 1 + 4 × + 4 × = 2Typical metals: Be, Mg, α-Ti, Zn, and Zr

16

112

Figure 3-6 Hexagonal close packed (hcp) structure for metals showing (a) the ar-rangement of atom centers relative to lattice points for a unit cell. There are twoatoms per lattice point (note the outlined example). (b) The actual packing ofatoms within the unit cell. Note that the atom in the midplane extends beyond theunit cell boundaries. (c) The repeating hcp structure, equivalent to many adja-cent unit cells (Part (c) courtesy of Molecular Simulations, Inc.).

(c) Face-centered cubic (d) Hexagonal close-packed

(a) Stacking of close-packed planes

(b) Stacking of close-packed planes

Normal toclose-packed

planes

Normal toclose-packed planes

Close-packedplanes

A

A

A A A

A

A A

A

A

A

CB

B

BC

B

B

C C

A

A A A

A

A A

B

B

B

Close-packedplanes

Figure 3-7 Comparison of the fcc and the hcp structures. They are each efficient stackingsof close-packed planes. The difference between the two structures is the different stack-ing sequences. (After B. D. Cullity, Elements of X-Ray Diffraction, 2nd ed., Addison-Wesley Publishing Co., Inc., Reading, Mass., 1978.)

(a) (b)

2 ions perlattice point

Center ofunit cell

Structure: CsCl-typeBravais lattice: simple cubicIons/unit cell: 1Cs+ + 1Cl–

Cs+

Cl–

Figure 3-8 Cesium chloride (CsCl) unit cell showing (a) ion positions and the2 ions per lattice point, and (b) full-size ions. Note that the Cs+−Cl− pairassociated with a given lattice point is not a molecule because the ionic bond-ing is nondirectional and a given Cs+ is equally bonded to eight adjacentCl−, and vice versa. (Part (b) courtesy of Molecular Simulations, Inc.)

Structure: NaCl-typeBravais lattice: fccIons/unit cell: 4Na+ + 4Cl–

Typical ceramics: MgO, CaO, FeO, and NiO

(a) (b)

(c)

2 ions per lattice pointNa+

Cl–

Figure 3-9 Sodium chloride (NaCl) structure showing (a) ion positions in aunit cell, (b) full-size ions, and (c) many adjacent unit cells. (Parts (b) and(c) courtesy of Molecular Simulations, Inc.)

(a) (b)

Ca2+

F–

F– ions located at cornersof a cube(at one-quarter of the distance along the body diagonal)

Structure: flourite (CaF2)-typeBravais lattice: fccIons/unit cell: 4Ca2+ + 8F–

Typical ceramics: UO2, ThO2, and TeO2

Figure 3-10 Fluorite (CaF2) unit cell showing (a) ion positions and (b) full-sizeions. (Part (b) courtesy of Molecular Simulations, Inc.)

(a) (b) (c)

Si4+

O2–

Interior Si4+ located atpositions one-quarter of the distancealong the body diagonal

Structure: cristobalite (SiO2) -typeBravais lattice: fccIons/unit cell: 8Si4+ + 16O2–

Figure 3-11 The cristobalite (SiO2) unit cell showing (a) ion positions, (b) full-size ions, and (c) the con-nectivity of SiO4−

4 tetrahedra. In the schematic, each tetrahedron has a Si4+ at its center. In addition,an O2− would be at each corner of each tetrahedron and is shared with an adjacent tetrahedron. (Part(c) courtesy of Molecular Simulations, Inc.)

2000

1500

1000

500

0

High cristobalite fcc(shown in Fig. 3-11)

High tridymite Hexagonal

High quartz Hexagonal

Low quartz Hexagonal

1723 (melting point)

T(˚C)

1470

867

573

CrystallographicForm

Bravicelattice

Figure 3-12 Many crystallographic forms of SiO2 are stable as they are heatedfrom room temperature to the melting temperature. Each form represents adifferent way to connect adjacent SiO4−

4 tetrahedra.

Unit cell

Unit cell (6O2– layers high)Structure: corundum (Al2O3) -typeBravais lattice: hexagonal (approx)Ions/unit cell: 12Al3+ + 18O2–

Typical ceramics: Al2O3, Cr2O3, αFe2O3

Al3+

O2–

Top view Side view

Close-packed layer of O2– with ofoctahedral sites filled with Al3+

2

3

Figure 3-13 The corundum (Al2O3) unit cell is shown superimposed on the repeated stacking of layers ofclose-packed O2− ions. The Al3+ ions fill two-thirds of the small (octahedral) interstices between adja-cent layers.

(a) (b)

Ti4+: at the body center

Ca2+: at corners

O2–: at face centers

Structure: perovskite (CaTiO3)-typeBravais lattice: simple cubicIons/unit cell: 1Ca2+ + 1Ti4+ + 3O2−

Typical ceramics: CaTiO3, BaTiO3

Figure 3-14 Perovskite (CaTiO3) unit cell showing (a) ion positions and (b) full-size ions.(Part (b) courtesy of Molecular Simulations, Inc.)

Oxygen

Octahedral positions

Tetrahedral positions

Figure 3-15 Ion positions in the spinel (MgAl2O4) unit cell. The cir-cles in color represent Mg2+ ions (in tetrahedral or four-coordinatedpositions), and the black circles represent Al3+ ions (in octahedralor six-coordinated positions). (From F. G. Brockman, Bull. Am.Ceram. Soc. 47, 186 (1967).)

Octahedralcoordination

Tetrahedralcoordination

O2–

602–

4Si4+

+16

4Al3+

+12

6OH+

–6

4 O2– + 2 OH–

–10

–12

OH–

Al3+

Si4+

Anions Cations

0.893nm

0.73

7nm

0.51

4nm

α = 91˚48′β = 104˚30′γ = 90˚0′

c

a

b

Figure 3-16 Exploded view of the kaolinite unit cell, 2(OH)4Al2Si2O5.(After F. H. Norton, Elements of Ceramics, 2nd ed., Addison-WesleyPublishing Co., Inc., Reading, Mass., 1974.)

Figure 3-17 Transmission electron micrograph(see Section 4.7) of the structure of clay platelets.This microscopic-scale structure is a mani-festation of the layered crystal structure shownin Figure 3–16. (Courtesy of I. A. Aksay)

1C

2C

1C

0.67nm

0.25nm

C R = 0.08nm

(a) (b)

Figure 3-18 (a) An exploded view of the graphite (C) unit cell. (From F. H. Norton, Elementsof Ceramics, 2nd ed., Addison-Wesley Publishing Co., Inc., Reading, Mass., 1974.) (b) Aschematic of the nature of graphite’s layered structure. (From W. D. Kingery, H. K. Bowen,and D. R. Uhlmann, Introduction to Ceramics, 2nd ed., John Wiley & Sons, Inc., New York,1976.)

(a)

(b)

Figure 3-19 (a) C60 molecule, orbuckyball. (b) Cylindricalarray of hexagonal rings ofcarbon atoms, or buckytube.(Courtesy of Molecular Sim-ulations, Inc.)

Figure 3-20 Arrangement of polymeric chains in the unit cell of polyethylene.The dark spheres are carbon atoms, and the light spheres are hydrogen atoms.The unit cell dimensions are 0.255 nm × 0.494 nm × 0.741 nm. (Courtesyof Molecular Simulations, Inc.)

Top viewof fold plane

Side viewof fold plane

Orientation of unit cell

b

a

Figure 3-21 Weaving-like pattern of folded polymeric chains that occursin thin crystal platelets of polyethylene. (From D. J. Williams, Poly-mer Science and Engineering, Prentice Hall, Inc., Englewood Cliffs,N.J., 1971.)

b

a

CH2

O

O

O

O

O

O

C

CH2

CH2

CH2

NH

NHC

CH2

CH2

CH2

CH2

CH2

NH

NH

NH

NH

NH

NH

CH2

1

2

4

3

Figure 3-22 Unit cell of the α-form of polyhex-amethylene adipamide or nylon 66. (FromC. W. Bunn and E. V. Garner, “Packing ofnylon 66 molecules in the triclinic unit cell:α form, Proc. Roy. Soc. Lond. 189A, 39(1947).)

Courtesy of SEAMATECH.

Seed crystal

Crystal

Melt

Heatingelements

Crucible

Container

Pull

Rotate

Schematic of growth of single crystals using the Czochralskitechnique. (After J. W. Mayer and S. S. Lau, Electronic Mate-rials Science: For Integrated Circuits in Si and GaAs, Macmil-lan Publishing Company, New York, 1990.)

(a)

(b)

Interior atoms located atpositions one-quarter of thedistance along the body diagonal

2 atoms perlattice point

Structure: diamond cubicBravais lattice: fccAtoms/unit cell: 4 + 6 × + 8 × = 8Typical semiconductors: Si, Ge, and gray Sn

12

18

Figure 3-23 Diamond cubic unit cell show-ing (a) atom positions. There are two atomsper lattice point (note the outlined exam-ple). Each atom is tetrahedrally coordi-nated. (b) The actual packing of full-sizeatoms associated with the unit cell. (Part(b) courtesy of Molecular Simulations,Inc.)

(a)

(b)

Two ions perlattice point

Zn2+ S2–

Structure: Zinc blende (ZnS)-typeBravais lattice: fccIons/unit cell: 4Zn2+ + 4S2–

Typical semiconductors:GaAs, AlP, InSb (III-V compounds),ZnS, ZnSe, CdS,HgTe (II-VI compounds)

Figure 3-24 Zinc blende (ZnS) unit cellshowing (a) ion positions. There aretwo ions per lattice point (note theoutlined example). Compare thiswith the diamond cubic structure (Fig-ure 3–23a). (b) The actual packingof full-size ions associated with theunit cell. (Part (b) courtesy of Molec-ular Simulations, Inc.)

Zn2+

S2–

S2– ion

at of

height ofunit cell

(a) (b)

Structure: wurtzite (ZnS)-typeBravais lattice: hexagonalIons/unit cell: 2Zn2+ + 2S2–

Typical semiconductors: ZnS, CdS, and ZnO.

3

8

S2– ion

at of

height ofunit cell

7

8

Zn2+ ion in midplane (tetrahedrally coordinated by S2–)

Figure 3-25 Wurtzite (ZnS) unit cell showing (a) ion positions and (b) full-size ions.

112

001

000

100

111 121

010

1100

00

0 –1

1

2

1

2

1

2

1

2

1

2

1

2

1

2

b

c

a

Figure 3-26 Notation for lattice positions.

1 × b

2 × c

b

c

a

Figure 3-27 Lattice translations connect structurallyequivalent positions (e.g., the body center) invarious unit cells.

[112]

[111]

[111]

111

[111]

[111]

[111]

1

2

1

2

1

2

b

c

a

Figure 3-28 Notation for lattice directions. Note that parallel [uvw] direc-tions (e.g., [111]) share the same notation because only the origin is shifted.

[111] [111]

[111]

[111][111][111]

[111]

[111]

a1

a3

a2

<111>

Figure 3-29 Family of directions, 〈111〉, representing all body diago-nals for adjacent unit cells in the cubic system.

Intercept at ∞

Intercept at b

Miller indices (hkl):

Intercept at a

a

aa

b

b

c

c

a

b

c

a

b

c

b

c

(a)

(b)

(210)

(010) (111)(111)(020)

1

1/2

1

1

1

∞

1

2

Figure 3-30 Notation for lattice planes. (a) The (210) plane illustrates Miller indices (hkl). (b) Additional exam-ples.

120˚

Miller-Bravais indices (hkil): (0110)

120˚a1

c

a3 a2

1

1

1

–1

1

∞1

∞→, , ,

Note: h + k = −i

Figure 3-31 Miller-Bravais indices, (hkil), for the hexago-nal system.

(100) on back face(010) on side face

(100)

(001)

(010)

(001) on bottom face

a1

a3

a2

{100}

Figure 3-32 Family of planes, {100}, representing all facesof unit cells in the cubic system.

000

110

b

c

a[110]

01

2

1

2

Anions

Cations

Glass plate

Incidentrays

ab

d

Figure 3-33 Diffraction grating for vis-ible light. Scratch lines in the glassplate serve as light-scattering centers.(After D. Halliday and R. Resnick,Physics, John Wiley & Sons, Inc., NewYork, 1962.)

X-radiation Visible light Microwaves

γ-radiation

Wavelengh (nm)

UV ir Radio waves

10–6 10–3 1 103 106 109 1012

Figure 3-34 Electromagnetic radiation spectrum. X-radiation represents that por-tion with wavelengths around 0.1 nm.

Incident X-ray beam (in phase) Diffracted beam (in phase)

ABC = nλ (for constructive interference)

nλ = 2d sin θ

Therefore

AB = BC = d sin θ

A CB

d

θ

θ

θ

θ

λ

Figure 3-35 Geometry for diffraction of x-radiation. The crystal struc-ture is a three-dimensional diffraction grating. Bragg’s law (nλ =2d sin θ ) describes the diffraction condition.

X-ray source

Sample

X-ray detector

Figure 3-36 Relationship of the Bragg angle (θ ) and the experimentally measureddiffraction angle (2θ ).

Figure 3-37 Diffraction pattern of a single crystal of MgO(with the NaCl structure of Figure 3–9). Each spot onthe film represents diffraction of the x-ray beam froma crystal plane (hkl).

180 – 2θ

Sample

Holder

Collimator

X-ray source

(b)(a)

Figure 3-38 (a) Single-crystal diffraction camera (or Laue camera). (Courtesy of Blake Industries, Inc.)(b) Schematic of the experiment.

2θ (degrees)

λ = 0.1542nm (CuKα-radiation)

Inte

nsit

y (a

rbit

rary

uni

ts)

200

20

40

60

80

100(111)

(200)

(220) (311)

(222) (400) (331)(420)

30 40 50 60 70 80 90 100 110 120

Figure 3-39 Diffraction pattern of aluminum powder. Each peak (in theplot of x-ray intensity versus diffraction angle, 2θ ) represents diffrac-tion of the x-ray beam by a set of parallel crystal planes (hkl) in vari-ous powder particles.

Sample

(b)

(a)

2θ

Collimator Collimator

Detector

Computer display

Scandirections

X-ray source

Figure 3-40 (a) An x-ray diffractometer. (Courtesy of Scintag,Inc.) (b) A schematic of the experiment.

X-ray beamSample

3cm

θ

ϕθ

2θ

1cm

Film