Material Science 01

1. The Structure of Crystalline
Solids
1

2. WHY STUDY The Structure of Crystalline Solids?
The properties of some materials are directly related to their crystal
structures. For example, pure and undeformed magnesium and beryllium,
having one crystal structure, are much more brittle (i.e., fracture at lower
degrees of deformation) than are pure and undeformed metals such as gold
and silver that have yet another crystal structure.
Furthermore, significant property differences exist between crystalline and
noncrystalline materials having the same composition.
For example, noncrystalline ceramics and polymers normally are optically
transparent; the same materials in crystalline (or semicrystalline) form tend
to be opaque or, at best, translucent.
2

3. Learning Objectives
1. Describe the difference in atomic/
molecular structure between crystalline
and non-crystalline materials.
2. Draw unit cells for face-centered cubic,
body-centered cubic, and hexagonal close-
packed crystal structures.
3. Derive the relationships between unit
cell edge length and atomic radius for
face-centered cubic and body-centered
cubic crystal structures.
4. Compute the densities for metals having
face-centered cubic and body-centered
cubic crystal structures given their unit
cell dimensions.
5. Given three direction index integers,
sketch the direction corresponding to these
indices within a unit cell.
6. Specify the Miller indices for a plane
that has been drawn within a unit cell.
7. Describe how face-centered cubic and
hexagonal close-packed crystal structures
may be generated by the stacking of close-
packed planes of atoms.
8. Distinguish between single crystals and
polycrystalline materials.
9. Sketch/describe unit cells for sodium
chloride, cesium chloride, zinc blende,
diamond cubic, fluorite, and perovskite
crystal structures.
10. Given the chemical formula for a
ceramic compound and the ionic radii of
its component ions, predict the crystal
structure.
3
After studying this chapter, you should be able to do the following:

4. Structure of Crystalline Materials
• Crystal Structure of Metallic and Semiconductor
Elements:
– Cubic, Face-centered cubic (FCC), Body-centered cubic
(BCC) and
– Diamond cubic structures;
• Crystals Structures of Ceramics:
– Rock Salt,
– Cesium Chloride, Zinc Blende (Sphalerite),
– Fluorite and Perovskite Structures;
– Fullerenes and Carbon Nano-tube Structure
4

5. Crystal Structures
• The arrangement of atoms, ions or molecules
in a material constitute the Crystal Structure
• If the arranged pattern repeats itself in 3-
dimension, the result is a crystal structure.
• The properties of some materials are directly
related to their crystal structures.
• Significant property differences exist between
crystalline and non-crystalline materials
having the same composition.
5

6. Unit Cell and Space lattice of ideal
Crystalline solid
6
Space Lattice:
A 3-D regular arrangements of
atoms characteristic of a
particular crystal structure
(points of intersection of a
network of lines in 3-D).
14 such arrangements exist
called Bravais Lattice
Unit Cell:
The smallest building block of a crystal, consisting of
atoms, ions, or molecules, whose geometric
arrangement defines a crystal's characteristic symmetry
and whose repetition in space produces a crystal lattice.

7. Lattice Constants (Lattice Parameters)
• Three lattice vectors a, b, c and the inter-axial
angles α, β, γ are called the lattice constants or
lattice parameters of a unit cell.
7
A unit cell showing
lattice constants

8. 8
• This results in the fact that, in 3 dimensions, there are only
• 7 different shapes of unit cell which can be stacked
together to completely fill all space without overlapping.
• This gives the 7 crystal systems, in which all crystal structures
can be classified. These are:
• The Cubic Crystal System (SC, BCC, FCC)
• The Hexagonal Crystal System (S)
• The Triclinic Crystal System (S)
• The Monoclinic Crystal System (S, Base-C)
• The Orthorhombic Crystal System (S, Base-C, BC, FC)
• The Tetragonal Crystal System (S, BC)
• The Trigonal (or Rhombohedral) Crystal System (S)
Classification of Crystal Structures
Crystallographers showed a long time ago that in 3-D, there are
7 CRYSTAL SYSTEMS with
14 BRAVAIS LATTICES

9. Seven Crystal Systems
9

12. Relation of the primitive cell in
the hexagonal system (heavy
lines) to a prism of hexagonal
symmetry. Here a1 = a2 ≠ a3.
Hexagonal System
In this Fig. the lattice vectors are:
a = a ≠ c
α = β = 90̊ and
γ = 120̊

14. 14
• Non dense, random packing
• Dense, ordered packing
Dense, ordered packed structures tend to have lower energies.
Energy and Packing
Energy
r
typical neighbor
bond length
typical neighbor
bond energy
Energy
r
typical neighbor
bond length
typical neighbor
bond energy

15. 15
• atoms pack in periodic, 3D arrays
Crystalline materials...
-metals
-many ceramics
-some polymers
• atoms have no periodic packing
Noncrystalline materials...
-complex structures
-rapid cooling
crystalline SiO2
noncrystalline SiO2"Amorphous" = Noncrystalline
Materials and Packing
Si Oxygen
• typical of:
• occurs for:

16. 16
Metallic Crystal Structures
• How can we stack metal atoms to minimize
empty space?
2-dimensions
vs.

17. 17
• Tend to be densely packed.
• Reasons for dense packing:
-Typically, only one element is present, so all
atomic radii are the same.
-- Metallic bonding is not directional.
-- Nearest neighbor distances tend to be small in
order to lower bond energy.
- The “electron cloud” shields cores from each
other
• They have the simplest crystal structures.
Metallic Crystal Structures

18. 18

19. 19
• Rare due to low packing density only Polonium (Po, At. No. 84) has
this structure)
• Close-packed directions are cube edges.
Atoms per unit cell: 01,
• Coordination # = 6
(# nearest neighbors)
Simple Cubic Structure (SC)

20. 20
• APF for a simple cubic structure = 0.52
APF =
a3
4
3
p (0.5a) 31
atoms
unit cell
atom
volume
unit cell
volume
Atomic Packing Factor (APF)
APF =
Volume of atoms in unit cell*
Volume of unit cell
*assume hard spheres
close-packed directions
a
R=0.5a
contains 8 x 1/8 =
1 atom/unit cell

21. 21
• Coordination # = 8
• Atoms touch each other along cube diagonals.
All atoms are identical.
Body Centered Cubic Structure (BCC)
ex: Cr, W, Fe (), Tantalum, Molybdenum
2 atoms/unit cell: 1 center + 8 corners x 1/8

22. 22

23. 23
Atomic Packing Factor: BCC
a
APF =
4
3
p ( 3a/4)32
atoms
unit cell atom
volume
a3
unit cell
volume
length = 4R =
Close-packed directions:
3 a
• APF for a body-centered cubic structure = 0.68
a
R
a2
a3

24. 24
• Coordination # = 12
• Atoms touch each other along face diagonals.
--Note: All atoms are identical; the face-centered atoms are shaded
differently only for ease of viewing.
Face Centered Cubic Structure (FCC)
ex: Al, Cu, Au, Pb, Ni, Pt, Ag
4 atoms/unit cell: 6 face x 1/2 + 8 corners x 1/8

25. 25

27. 27
• APF for a face-centered cubic structure = 0.74
Atomic Packing Factor: FCC
maximum achievable APF
APF =
4
3
p ( 2a/4)34
atoms
unit cell atom
volume
a3
unit cell
volume
Close-packed directions:
length = 4R = 2 a
Unit cell contains:
6 x1/2 + 8 x1/8
= 4 atoms/unit cell
a
2 a

28. 28
Hexagonal Close-Packed Structure (HCP)

29. 29

30. 30

31. 31
• Coordination # = 12
• ABAB... Stacking Sequence
• APF = 0.74
• 3D Projection • 2D Projection
Hexagonal Close-Packed Structure (HCP)
6 atoms/unit cell
ex: Cd, Mg, Ti, Zn
• c/a = 1.633. Do it by yourself??
c
a
A sites
B sites
A sites Bottom layer
Middle layer
Top layer

32. ABAB... Stacking Sequence

33. Closed packed structures
33
A portion of a closed-packed plane of
atoms; A, B, and C position are indicated.
The AB stacking sequence for close-
packed atomic planes .
ABAB stacking sequence for hcp. ABCABC stacking
sequence for fcc.
A corner has been removed to show the
relation between the stacking of close-
packed planes of atoms and the fcc crystal
structure, the heavy triangle outlines a
(111) plane. .

34. Figure. Workflow for solving the structure of a molecule by X-ray crystallography.
Crystal
Diffraction pattern
Electron Density Map
Atomic Model

35. What are X-rays?
X-rays: Electromagnetic radiation with a wavelength from 0.1 Ǻ to
100 Ǻ (0.01 nm to about 10 nm).

36. What are X-rays
• Discovered in 1895 AD by Wilhelm Roentgen and got first Noble
prize in Physics in 1901 AD.
• X-rays are a part of the electromagnetic spectrum along with
visible light, Infra Red, Ultra Violet etc.
• The λ could be 10-8 to 10-11 meter (0.01 to 10 nm, which is almost
comparable to lattice parameter or interatomic distance).
• The frequencies are in the range of 3 x 1016 Hz to 3 x 1019 Hz and
energies in the range of 100 eV to 100 keV.
• The shorter λ’s are called hard x-rays while longer λ’s are known
as soft X-rays (the shorter the λ, more energy the radiation has).

37. Generation of X-radiation for diffraction
measurements
• Sealed X-ray tubes tend to operate at 1.8
to 3 kW.
• Rotating anode X-ray tubes produce much
more flux because they operate at 9 to 18
kW.
– A rotating anode spins the anode at
6000 rpm, helping to distribute heat
over a larger area and therefore
allowing the tube to be run at higher
power without melting the target.
• The Cu (or Co, Mo, Cr) source generates
X rays by striking the anode target with an
electron beam from a tungsten filament.
– The target must be water cooled.
– The target and filament must be
contained in a vacuum.
Cu
H2O In H2O Out
e-
Be
XRAYS
window
Be
XRAYS
FILAMENT
ANODE
(cathode)
AC CURRENT
window
metal
glass
(vacuum) (vacuum)

38. The Principle of Bremsstrahlung Generation
X-ray, (continuous or Bremsstrahlung)
Fast incident
electron
nucleus
Atom of the anode material
electrons
Ejected
electron
(slowed down
and changed
direction)

39. Generation of X-rays
K
L
M

40. An x-ray emission spectra for Cu
40

41. X-Ray Diffraction and Determination of
Crystal Structure
41
Using X-ray diffractometry.
• Crystal structure,
• Inter-planar distance,
• Phase composition and
• many more parameters can be determined.

42. Xray-Tube
Detector
Sample
Bragg´s Equation
λ = 2dsinθ
d – distance between the same atomic planes
λ – monochromatic wavelength
θ – angle of diffracto-meter
Xray-Tube
Detector
Metal Target
(Cu or Co)
X-Ray Diffractometer

43. Diffraction of x-rays from two planes
43
Diffraction of x-rays by planes of atoms (A – A’ and B-B);
Bragg’s Law

44. 44

45. 45

46. • Calculation of d-spacing and lattice parameters using
Bragg’s Law,
(220) plane in a unit cube

48. 48
3367
Atm)

49. 49
Theoretical Density, r
where
n = number of atoms/unit cell
A = atomic weight
VC = Volume of unit cell = a3 for a cubic structure
NA = Avogadro’s number
= 6.022 x 1023 atoms/mol
Density = r =
VC NA
n A
r =
CellUnitofVolumeTotal
CellUnitinAtomsofMass

50. 50
• Example: Cr (BCC)
A (atomic weight) = 52.00 g/mol
n = 2 atoms/unit cell
R = 0.125 nm
rtheoretical
a = 4R/ 3 = 0.2887 nm
ractual
a
R
r =
a3
52.002
atoms
unit cell
mol
g
unit cell
volume atoms
mol
6.022x1023
Theoretical Density, r
= 7.18 g/cm3
= 7.19 g/cm3

51. Ceramic Materials
• Name comes from Greek letter “Keramicos” mean burnt off. (so
these materials are fabricated at high temperature).
• Ceramic materials are inorganic and nonmetallic material,
• Metallic and non-metallic elements are bonded together,
• Materials could be totally ionic or predominantly ionic with
covalent character or totally covalent.
51

52. • Properties depend upon nature of bonding and crystal structure.
• In general, Ceramic are hard and brittle,
• Good electrical and thermal insulators.
• Have high melting point and high chemical stability.
• Being used in very harsh environments (say at high Temp.
52

53. 53

54. 54
• Bonding:
-- Can be ionic and/or covalent in character.
-- % ionic character increases with difference in
electronegativity of atoms.
• Degree of ionic character may be large or small:
Atomic Bonding in Ceramics
SiC: small
CaF2: large

55. Ceramics materials have two major
categories
• Traditional and Engineering ceramic materials.
• Traditional Ceramics are fabricted from “clay”, silica (flint)
and feldspar (rock forming silicate of Al together with Na, K.
Ca, and Ba).
• Examples are:
– Porcelain, bricks, tiles, and in addition, glasses and high temperature
ceramics.
55
• Engineering Ceramics;
• These are pure or nearly pure compounds of Aluminum
Oxide Al2O3, Silicon Carbide (SiC), and Silicon Nitrate
(Si3Ni4).
• They are being used in:
• I.T. Technology (Al2O3)
• In high temperature areas (SiC), furnaces and many more.

56. Crystal structure of Ceramics
• The metallic ion could be;
– Cation: Positively ion, (e- has left the atom)
– Anion: Negatively ion (e- is gained in the atom.
• In ionic compounds, packing of ions determine by:
– The relative size of the ions in the ionic solid.
– The need to balance electrostatics charges to maintain
electrical neutrality in the ionic solid.
56

57. 57
Table 12.3: Ionic Radii for Several Cations and Anions for a
Coordination Number of 6
• The size of an ion depends on
several factors.
• One of these is coordination
number: ionic radius tends to
increase as the number of
nearest-neighbor ions of
opposite charge increases.
• Ionic radii given in Table 12.3
are for a coordination number
of 6.
• Therefore, the radius is
greater for a coordination
number of 8 and less when
the coordination number is 4.
In addition, the charge on an ion will influence its radius. For example, from Table 12.3, the
radii for Fe2+and Fe3+ are 0.077 and 0.069 nm, respectively, which values may be contrasted to
the radius of an iron atom—0.124 nm.
When an electron is removed from an atom or ion, the remaining valence electrons become
more tightly bound to the nucleus, which results in a decrease in ionic radius.
Conversely, ionic size increases when electrons are added to an atom or ion.

58. 58
Factors that Determine Crystal Structure
1. Relative sizes of ions – Formation of stable structures:
--maximize the # of oppositely charged ion neighbors.
- -
- -
+
unstable
- -
- -
+
stable
- -
- -
+
stable
2. Maintenance of
Charge Neutrality :
--Net charge in ceramic
should be zero.
--Reflected in chemical
formula:
CaF2: Ca2+
cation
F-
F-
anions+
AmXp
m, p values to achieve charge neutrality

59. • For a Bravais Lattice,
The Coordinatıon Number 
The number of lattice points closest to a given point
(the number of nearest-neighbors of each point).
• Because of lattice periodicity, all points have the same
number of nearest neighbors or coordination number.
(That is, the coordination number is intrinsic to the lattice.)
Examples
Simple Cubic (SC) coordination number = 6
Body-Centered Cubic coordination number = 8
Face-Centered Cubic and HCP, coordination number = 12
Coordination Number

60. 60
• Coordination # increases with
Coordination # and Ionic Radii
2
rcation
ranion
Coord
#
< 0.155
0.155 - 0.225
0.225 - 0.414
0.414 - 0.732
0.732 - 1.0
3
4
6
8
linear
triangular
tetrahedral
octahedral
cubic
rcation
ranion
To form a stable structure, how many anions can
surround a cation?
(Zinc blende)

61. 2 <0.155
3 0.155 – 0.255
4 0.225 – 0.414
0.414 – 0.7326
0.732 – 1.08

62. 62
Computation of Minimum Cation-Anion
Radius Ratio
• Determine minimum rcation/ranion for an octahedral site
(C.N. = 6)
a = 2ranion
2ranion  2rcation = 2 2ranion

ranion  rcation = 2ranion

rcation = ( 2 1)ranion
arr 222 cationanion =
414.012
anion
cation ==
r
r
a
Measure the radii (blue and
yellow spheres)
Substitute for “a” in the
above equation

63. 63
• On the basis of ionic radii, what crystal structure would you
predict for FeO?
• Answer:
5500
1400
0770
anion
cation
.
.
.
r
r
=
=
Example Problem: Predicting the Crystal
Structure of FeO
Ionic radius (nm)
0.053
0.077
0.069
0.100
0.140
0.181
0.133
Cation
Anion
Al3+
Fe2+
Fe3+
Ca2+
O2-
Cl-
F-
based on this ratio,
-- coord # = 6 because
0.414 < 0.550 < 0.732
-- crystal structure is similar to NaCl

64. AX type Crystal Structure
64
• The ceramic materials with AX type Crystal Structure have;
• Number of Cation = Number of Anions
• Where ‘A’ denotes cations and ‘X’ the anion
• Several different AX compounds exist,
• Each is normally named after a common material that assumes the
particular structure
Examples:
• Rock Salt, NaCl
• Cesium Chloride, CsCl
• Zinc Blende (ZnS)

65. 65
Rock Salt Structure
Same concepts can be applied to ionic solids in general.
Example: NaCl (rock salt) structure
rNa = 0.102 nm
rNa/rCl = 0.564
 cations (Na+) prefer octahedral sites
rCl = 0.181 nm

66. 66
MgO and FeO
O2- rO = 0.140 nm
Mg2+ rMg = 0.072 nm
rMg/rO = 0.514
 cations prefer octahedral sites
So each Mg2+ (or Fe2+) has 6 neighbor oxygen atoms
MgO and FeO also have the NaCl structure

67. 67
CsCl Crystal Structures
939.0
181.0
170.0
Cl
Cs ==


r
r
Cesium Chloride structure:
 Since 0.732 < 0.939 < 1.0,
cubic sites preferred
So each Cs+ has 8 neighbor Cl-
CsCl is an AX-type Crystal Structures

68. Zinc blende (ZnS) Crystal Structure
68
A unit cell for the zinc blende (ZnS)
crystal structure
• Zinc blende or Sphalerite structure
• Coordination No. = 4,
• All corner's and face positions occupied by S
atoms,
• Zn atoms fill interior tetrahedral position.
• Reverse structure is also possible (Zn, S
position reversed)
• Each Zn atom is bonded to 4 S atoms,
• Atomic bonding is highly covalent;
• Include:
• ZnS,
• ZnTe,
• SiC

69. 69
AX2 Crystal Structures
• Calcium Fluorite (CaF2)
• Calcium ions in cubic sites
• Fluorine ions at the corners
• Similar to CsCl, except that only half the
center cube positions occupied by Ca2+ ions
• Charges on cation ≠ anions
• UO2, ThO2, ZrO2, CeO2, PuO2
• Coordination No. = 8
• Antifluorite structure also exists
• in which positions of cations and
anions are reversed.
Fluorite structure

70. 70
ABX3 Crystal Structures
• Perovskite structure
Ex: Complex oxide
BaTiO3

71. ABX3 Crystal Structures
• More than one type of cation.
• For two types of Cations (represented by A and B), chemical
formula is listed above.
• Examples are;
– CaTiO3, BaTiO3, SrZrO3, SrSnO3 etc.
• Perovskite Crystal Structure is shown on R.H.S.
• Ca2+ and O2- ions at the center of the faces of the unit cell.
• The highly charged Ti4+ ion is located at the octahedral interstitial
site at the center of the unit cell and is coordinated to six O2- ions.
• BaTiO3 has the perovskite structure above 1200 C, but below, it is
slightly changed.
71

72. Structure of some ceramics
72

73. 73
• Atoms may assemble into crystalline or amorphous structures.
• We can predict the density of a material, provided we know the
atomic weight, atomic radius, and crystal geometry (e.g., FCC,
BCC, HCP).
SUMMARY
• Common metallic crystal structures are FCC, BCC and HCP.
Coordination number and atomic packing factor are the same
for both FCC and HCP crystal structures.
• Ceramic crystal structures are based on:
-- maintaining charge neutrality
-- cation-anion radii ratios.
• Interatomic bonding in ceramics is ionic and/or covalent.

74. Silicate Ceramics
74
The arrangement of silicon and oxygen atoms in
a unit cell of cristobalite, a polymorph of SiO2.

75. Silica (SiO2)
75
• Most simple silicate material
• 3-D network generated when every corner O2 atom of one
tetrahedron is shared by adjacent tetrahedron.
• Electrically, material is neutral, so stable electronic
structure is formed.
• Three primary polymorphic structures of silica,
• a). Quartz, b). Cristobalite, and c). Tridymite.
• Complicated and relatively open structure
• Low densities,
• Due to strong Si-O interatomic bonds, melting
• point is relatively high 17100 C.
The arrangement of silicon
and oxygen atoms in a unit
cell of cristobalite, a
polymorph of SiO2.

76. Silica Glass
• A form of silica which is non-crystalline solid or glass.
• Having high degree of atomic randomness.
76
Search out what type of
materials are:
Network Formers
Network modifiers,
Intermediates

77. Simple Silicate
77
Fig. 12.12. Five silicate ion structures
formed SiO4-
4 tetrahedron

78. 78
A unit cell for the diamond cubic crystal structure
Carbon Structures
• Diamond, Graphite,
Fullerene, Carbon Nanotube
(CNT)
• In diamond, each C atom is
bonded to 4 other C atoms.
• 4 tetrahedral sites with
covalent bonding
• Ge, Si and Gray tin (below
13̊ C) have this structure.

79. 79
C60 or
fullerene
Molecule
20 hexagons,
12 pentagons
The structure of Graphite
C atom
Different Structures of C

80. 80