THE STRUCTURE

1
CHAPTER 3

• THE STRUCTURE
• OF
• CRYSTALLINE SOLIDS

2
3.2 FUNDAMENTAL CONCEPTS
SOLIDS
AMORPHOUS
CRYSTALLINE
Atoms in a crystalline solid are arranged in a
repetitive three dimensional pattern Long Range
Order
Atoms in an amorphous solid are arranged
randomly- No Order
All metals are crystalline solids Many
ceramics are crystalline solids
Some polymers are crystalline solids
3
LATTICE  Lattice -- points arranged in a
pattern that repeats itself in three
dimensions. The points in a
crystal lattice coincides with atom centers  
4
3-D view of a lattice

5
3.3 UNIT CELL

• Unit cell -- smallest grouping which can be
  arranged in three dimensions to create the
  lattice. Thus the Unit Cell is basic structural
  unit or building block of the crystal structure

6
Unit cell Lattice
7
Unit Cell
Lattice
8
3.4 METALLIC CRYSTALS
Tend to be densely packed.
Have several 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 have lower bonding
energy.
Have the simplest crystal structures.
Let us look at three such structures...
4
9
FACE CENTERED CUBIC STRUCTURE (FCC)
10
FACE CENTERED CUBIC STRUCTURE (FCC)
Al, Cu, Ni, Ag, Au, Pb, Pt
11
BODY CENTERED CUBIC STRUCTURE (BCC)
12
BODY CENTERED CUBIC STRUCTURE (BCC)
Cr, Fe, W, Nb, Ba, V
13
HEXAGONAL CLOSE-PACKED STRUCTURE HCP
Mg, Zn, Cd, Zr,
Ti, Be
14
SIMPLE CUBIC STRUCTURE (SC)
Only Pd has SC structure
15
Number of atoms per unit cell

• BCC 1/8 corner atom x 8 corners 1 body
  center atom
• 2 atoms/uc
• FCC 1/8 corner atom x 8 corners ½ face atom x
  6 faces
• 4 atoms/uc
• HCP 3 inside atoms ½ basal atoms x 2 bases
  1/ 6 corner atoms x 12 corners
• 6 atoms/uc

16
Relationship between atomic radius and edge
lengths

• For FCC
• a 2Rv2
• For BCC
• a 4R /v3
• For HCP
• a 2R
• c/a 1.633 (for ideal case)
• Note c/a ratio could be less or more than the
  ideal value of 1.633

17
Face Centered Cubic (FCC)
r
2r
a0
r
a0
18
Body Centered Cubic (BCC)
a0
19
Coordination Number

• The number of touching or nearest neighbor atoms
• SC is 6
• BCC is 8
• FCC is 12
• HCP is 12

20
(No Transcript)
21
ATOMIC PACKING FACTOR
APF for a simple cubic structure 0.52
6
22
ATOMIC PACKING FACTOR BCC
APF for a body-centered cubic structure 0.68
a 4R /v3
8
23
FACE CENTERED CUBIC STRUCTURE (FCC)
Close packed directions are face diagonals.
--Note All atoms are identical the
face-centered atoms are shaded differently
only for ease of viewing.
Coordination 12
24
ATOMIC PACKING FACTOR FCC
APF for a face-centered cubic structure 0.74
a 2Rv2
25
3.5 Density Computations

• Density of a material can be determined
  theoretically from the knowledge of its crystal
  structure (from its Unit cell information)
• Density mass/Volume
• Mass is the mass of the unit cell and volume is
  the unit cell volume.
• mass ( number of atoms/unit cell) n x
  mass/atom
• mass/atom atomic weight A/Avogadros Number
  NA
• Volume Volume of the unit cell Vc

26
THEORETICAL DENSITY
27
Example problem on Density Computation

• Problem Compute the density of Copper
• Given Atomic radius of Cu 0.128 nm (1.28 x
  10-8 cm)
• Atomic Weight of Cu 63.5 g/mol
• Crystal structure of Cu is FCC
• Solution ? n A / Vc NA
• n 4
• Vc a3 (2Rv2)3 16 R3 v2
• NA 6.023 x 1023 atoms/mol
• 4 x 63.5 g/mol / 16 v2(1.28 x 10-8 cm)3 x 6.023
  x 1023 atoms/mol
• Ans 8.98 g/cm3
• Experimentally determined value of density
  of Cu 8.94 g/cm3

28
3.6 Polymorphism and Allotropy

• Polymorphism ? The phenomenon in some metals, as
  well as nonmetals, having more than one crystal
  structures.
• When found in elemental solids, the condition is
  often called allotropy.
• Examples
• Graphite is the stable polymorph at ambient
  conditions, whereas diamond is formed at
  extremely high pressures.
• Pure iron is BCC crystal structure at room
  temperature, which changes to FCC iron at 912oC.

29
POLYMORPHISM AND ALLOTROPY

• BCC (From room temperature to 912 oC)
• Fe
• FCC (at Temperature above 912 oC)
• 912 oC
• Fe (BCC) Fe (FCC)

30
3.7 Crystal Systems

• Since there are many different possible crystal
  structures, it is sometimes convenient to divide
  them into groups according to unit cell
  configurations and/or atomic arrangements.
• One such scheme is based on the unit cell
  geometry, i.e. the shape of the appropriate unit
  cell parallelepiped without regard to the atomic
  positions in the cell.
• Within this framework, an x, y, and z coordinate
  system is established with its origin at one of
  the unit cell corners each x, y, and z-axes
  coincides with one of the three parallelepiped
  edges that extend from this corner, as
  illustrated in Figure.

31
The Lattice Parameters

• Lattice parameters
• a, b, c, ?, ?, ? are called the lattice
• Parameters.

32

• Seven different possible combinations of edge
  lengths and angles give seven crystal systems.
• Shown in Table 3.2
• Cubic system has the greatest degree of symmetry.
• Triclinic system has the least symmetry.

33
The Lattice Parameters

• Lattice parameters are
• a, b, c, ?, ?, ? are called the lattice
• Parameters.

34
3.7 CRYSTAL SYSTEMS
35
3.8 Point Coordinates in an Orthogonal
Coordinate System Simple Cubic
36
3.9 Crystallographic Directions in Cubic System

• Determination of the directional indices in cubic
  system
• Four Step Procedure (Text Book Method)
• Draw a vector representing the direction within
  the unit cell such that it passes through the
  origin of the xyz coordinate axes.
• Determine the projections of the vector on xyz
  axes.
• Multiply or divide by common factor to obtain the
  three smallest integer values.
• Enclose the three integers in square brackets
  .
• e.g. uvw
• u, v, and w are the integers

37
Crystallographic Directions
Algorithm
z
1. Vector repositioned (if necessary) to pass
through origin.2. Read off projections in
terms of unit cell dimensions a, b, and
c3. Adjust to smallest integer values4. Enclose
in square brackets, no commas uvw
y
x
ex 1, 0, ½
gt 2, 0, 1
gt 201
-1, 1, 1
families of directions ltuvwgt
38
Crystallographic Directions in Cubic System
111
120
110
39
Crystallographic Directions in Cubic System
40
Head and Tail Procedure for determining Miller
Indices for Crystallographic Directions

1. Find the coordinate points of head and tail
   points.
2. Subtract the coordinate points of the tail from
   the coordinate points of the head.
3. Remove fractions.
4. Enclose in

41
Indecies of Crystallographic Directions in Cubic
System
Direction A Head point tail point (1, 1, 1/3)
(0,0,2/3) 1, 1, -1/3 Multiply by 3 to get
smallest integers 3, 3, -1
A 33I
Direction B Head point tail point (0, 1, 1/2)
(2/3,1,1) -2/3, 0, -1/2 Multiply by 6 to get
smallest integers
_ _ B 403
C ???
D ???
42
Indices of Crystallographic Directions in Cubic
System
Direction C Head Point Tail Point (1, 0, 0)
(1, ½, 1) 0, -1/2, -1 Multiply by 2 to get the
smallest integers
? ? C 0I2
Direction D Head Point Tail Point (1, 0, 1/2)
(1/2, 1, 0) 1/2, -1, 1/2 Multiply by 2 to get the
smallest integers
? D I2I
A ???
B ???
43
Crystallographic Directions in Cubic System
210
44
Crystallographic Directions in Cubic System
45
CRYSTALLOGRAPHIC DIRECTIONS IN HEXAGONAL UNIT
CELLS  Miller-Bravais indices -- same as Miller
indices for cubic crystals except that there are
3 basal plane axes and 1 vertical axis.Basal
plane -- close packed plane similar to the (1 1
1) FCC plane.contains 3 axes 120o apart.
46
Direction Indices in HCP Unit Cells uvtw
where t-(uv) Conversion from 3-index system
to 4-index system
47
HCP Crystallographic Directions

• Hexagonal Crystals
• 4 parameter Miller-Bravais lattice coordinates
  are related to the direction indices (i.e.,
  u'v'w') as follows.

48
Crystallographic Directions in Cubic System
49
Indices of a Family or Form 
50
3.10 MILLER INDICES FOR CRYSTALLOGRAPHIC PLANES 

• Miller Indices for crystallographic planes are
  the reciprocals of the fractional intercepts
  (with fractions cleared) which the plane makes
  with the crystallographic x,y,z axes of the three
  nonparallel edges of the cubic unit cell.
• 4-Step Procedure
• Find the intercepts that the plane makes with the
  three axes x,y,z. If the plane passes through
  origin change the origin or draw a parallel plane
  elsewhere (e.g. in adjacent unit cell)
• Take the reciprocal of the intercepts
• Remove fractions
• Enclose in ( )

51
Crystallographic Planes
4. Miller Indices (110)
4. Miller Indices (100)
52
Crystallographic Planes
example
a b c
4. Miller Indices (634)
53
Miller Indecies of Planes in Crystallogarphic
Planes in Cubic System
54
Drawing Plane of known Miller Indices in a cubic
unit cell
Draw ( ) plane
55
Miller Indecies of Planes in Crystallogarphic
Planes in Cubic System
Origin for A
Origin for B
Origin for A
A (II0) B (I22)
A (2II) B (02I)
56
CRYSTALLOGRAPHIC PLANES AND DIRECTIONS IN
HEXAGONAL UNIT CELLS  Miller-Bravais indices
-- same as Miller indices for cubic crystals
except that there are 3 basal plane axes and 1
vertical axis.Basal plane -- close packed plane
similar to the (1 1 1) FCC plane.contains 3 axes
120o apart.
57
Crystallographic Planes (HCP)

• In hexagonal unit cells the same idea is used

Adapted from Fig. 3.8(a), Callister 7e.
58
Miller-Bravais Indices for crystallographic
planes in HCP
_ (1211)
59
Miller-Bravais Indices for crystallographic
directions and planes in HCP
60
Atomic Arrangement on (110) plane in FCC
61
Atomic Arrangement on (110) plane in BCC
62
Atomic arrangement on 110 direction in FCC
63
3.11 Linear and Planar Atomic Densities

• Linear Density LD
• is defined as the number of atoms per unit
  length whose centers lie on the direction vector
  of a given crystallographic direction.

64
Linear Density

• LD for 110 in BCC.
• of atom centered on the direction vector 110
• 1/2 1/2 1
• Length of direction vector 110 ?2 a
• a 4R/ ? 3

110
? 2a
65
Linear Density

• LD of 110 in FCC
• of atom centered on the direction vector 110
  2 atoms
• Length of direction vector 110 4R
• LD 2 /4R
• LD 1/2R
• Linear density can be defined as reciprocal of
  the repeat distance r
• LD 1/r

66
Planar Density

• Planar Density PD
• is defined as the number of atoms per unit area
  that are centered on a given crystallographic
  plane.
• No of atoms centered on the plane
• PD
• Area of the plane

67
Planar Density of (110) plane in FCC

• of atoms centered on the plane (110)
• 4(1/4) 2(1/2) 2 atoms
• Area of the plane
• (4R)(2R ? 2) 8R22

(111) Plane in FCC
a 2R ? 2
4R
68
Closed Packed Crystal Structures

• FCC and HCP both have
• CN 12 and APF 0.74
• APF 0.74 is the most efficient packing.
• Both FCC and HCP have Closed Packed Planes
• FCC ----(111) plane is the Closed Packed Plane
• HCP ----(0001) plane is the Closed Packed Plane
• The atomic staking sequence in the above two
  structures is different from each other

69
Closed Packed Structures
70
Closed Packed Plane Stacking in HCP
71
Hexagonal Close-Packed Structure (HCP)
ABAB... Stacking Sequence
3D Projection
2D Projection
Adapted from Fig. 3.3(a), Callister 7e.
6 atoms/unit cell
Coordination 12
ex Cd, Mg, Ti, Zn
APF 0.74
c/a 1.633
72
FCC Stacking Sequence
ABCABC... Stacking Sequence 2D Projection
FCC Unit Cell
73
Closed Packed Plane Stacking in FCC
74
Crystalline and Noncrystalline Materials3.13
Single Crystals

• For a crystalline solid, when the periodic and
  repeated arrangement of atoms is perfect or
  extends throughout the entirety of the specimen
  without interruption, the result is a single
  crystal.
• All unit cells interlock in the same way and have
  the same orientation.
• Single crystals exist in nature, but may also be
  produced artificially.
• They are ordinarily difficult to grow, because
  the environment must be carefully controlled.
• Example Electronic microcircuits, which employ
  single crystals of silicon and other
  semiconductors.

75
Polycrystalline Materials

• 3.13 Polycrytalline Materials
• Polycrystalline ? crystalline solids composed of
  many small crystals or grains.
• Various stages in the solidification
• Small crystallite nuclei Growth of the
  crystallites.
• Obstruction of some grains that are adjacent to
  one another is also shown.
• Upon completion of solidification, grains that
  are adjacent to one another is also shown.
• Grain structure as it would appear under the
  microscope.

76
3.15 Anisotropy

• The physical properties of single crystals of
  some substances depend on the crystallographic
  direction in which the measurements are taken.
• For example, modulus of elasticity, electrical
  conductivity, and the index of refraction may
  have different values in the 100 and 111
  directions.
• This directionality of properties is termed
  anisotropy.
• Substances in which measured properties are
  independent of the direction of measurement are
  isotropic.

77
(No Transcript)