Chapter 3: Atomic & Ionic Arrangements

Week 2 Lecture Notes

R. Clemmer

ME 215 Winter 2011

References

D.R. Askeland, P.P. Fulay, W.J. Wendelin, “The Science and

Engineering of Materials,” 6th ed., Cengage Learning, 2011.

W.D. Callister, “Material Science and Engineering: An Introduction”,

5th ed.

Notes & Slides Outline

Short Range Order vs. Long Range Order

Amorphous Materials

Lattice, Basis, Unit Cells, & Crystal Structures

Lattice & Basis

Unit Cells

Lattice Parameters & Interaxial Angles

Number of Atoms per Unit Cell

Atomic Radius vs. Lattice Parameter

Coordination Number

Packing Factor

Hexagonal Close-Packed Structure

Allotropic or Polymorphic Transformations

Notes & Slides Outline

Points, Directions, & Planes in Unit Cells

Directions

Planes

Miller Indices for Hexagonal Unit Cells

Close-Packed Planes and Directions

Isotropic & Anisotropic behaviour

Interstitial Sites

Crystal Structures of Ionic Materials

Covalent Structures

Diffraction Techniques for Crystal Structure Analysis

Points, Directions, &

Planes in Unit Cells

38

Coordinates of Points

Some properties of a crystal are dependent on

directions and planes of atoms

To define directions and planes, a special

coordinate system is used – Miller Indices

Rather than using a Cartesian coordinate

system using length units, coordinates of atoms

are given as the number of lattice parameters

moved in the x, y, and z directions

For a unit cell, coordinates will range from 0,0,0

to 1,1,1 since a unit cell has dimensions of

1x1x139

Coordinates of Points

40(Askeland)

Miller Indices for Directions

Steps for Determining Miller Indices for Directions

1. Find the coordinates of 2 points that lie in the

direction

2. Subtract the “tail” coordinates from the “head”

coordinates (gives number of lattice points travelled)

3. Clear fractions and reduce to lowest terms

4. Enclose numbers in square brackets, e.g. [hkl]

5. Negatives are represented by a bar over the

number

41

Miller Indices of Directions

42(Askeland)

Points regarding Miller Indices (Directions)

Directions are vectors – a direction and its

negative are not equal i.e. [100] ≠ [-100]

Directions and their multiples are equal, i.e.

[100] = [200]

Families of directions are groups of

equivalent directions and indicated by < >

brackets

43

Equivalency of Directions

44(Askeland)

Directions of the Form <110>

45(Askeland)

Miller Indices for Planes

Steps for Determining Miller Indices for Planes

1. Identify the points where the plane intercepts

the x, y, z axes (may have to move origin if plane

passes through it)

2. Take reciprocals of the intercepts

3. Clear fractions but do NOT reduce to lowest

terms

4. Enclose numbers in parentheses, e.g. (hkl)

5. Negatives are represented by a bar over the

number

46

Miller Indices of Planes

47(Askeland)

Points regarding Miller Indices (Planes)

Planes and their negatives are equal i.e.

(100) = (-100)

Planes and their multiples are not equal, i.e.

[100] ≠ [200] (as demonstrated by planar

densities)

Families of planes are groups of equivalent

planes and indicated by { } brackets

48

face of area

atoms/face Density Planar

Planar Density of Different Planes

49(Askeland)

Planes of the Form {110}

50(Askeland)

Miller Indices for Hexagonal Unit Cells

A special set of Miller-Bravais indices for

HCP structures was devised to account for

the symmetry in the crystal structure

3 – axis system: Similar to before (hkl) or [hkl]

4 – axis system: A third a-axis (a3) is added

where a1 = h, a2 = k, and a3 = i and

51

i- khor 0 i k h

Miller Indices for HCP Unit Cells

52(Askeland)

Converting from (h’k’l’) to (hkil)

Clear fractions and reduce to lowest terms as

required

53

'

''

''

''

l l

)kh(3

1- i

)h2k(3

1 k

)k2h(3

1 h

Miller Indices for HCP Unit Cells

54(Askeland)

Close-Packed Planes & Directions

Close-packed directions are directions in which

atoms are in contact

Close-packed planes are planes of atoms all in

contact

HCP has two parallel close-packed planes

(called basal planes):

A (0001) and B (0002)

Stacking these planes produces the HCP structure

FCC crystals have 4 non-parallel close-packed

planes of the form {111}

55

Close-Packed Planes and

Directions

56(Askeland)

Isotropic vs. Anisotropic Behaviour

A material may behave differently in one

crystallographic direction or another because

of differences in atomic arrangement

Isotropic: Properties are identical in all

directions

Anisotropic: Properties are dependent on the

crystallographic direction

A polycrystalline material may act isotropic

since many crystals are randomly oriented

within the structure.

57

Interstitial Sites

In addition to the regular atom placement in a

crystal structure, other atoms (usually smaller)

may fit in the spaces between atoms

Interstitial sites are the locations between

regular atoms or ions in a crystal structure

SC – cubic interstitial sites

BCC – tetrahedral, octahedral sites

FCC – octahedral, tetrahedral sites

For an atom to fit in an interstitial site it cannot

be “too big or too small” (based on the radius ratio)

58

Simple Cubic Unit Cell

59

The Simple Cubic Unit Cell has 1 interstitial site

(Askeland)

BCC Unit Cell

60

The BCC Unit Cell has 24 Tetrahedral interstitial sites and

6 Octahedral interstitial sites

(Askeland)

FCC Unit Cell

61

The FCC Unit Cell has 8 Tetrahedral and

12 Octahedral interstitial sites(Askeland)

62

Octahedral Site in FCC

2r

2R

Ra2

40

Crystal Structures of Ionic

and Covalent Materials

63

Crystal Structures of Ionic Materials

Ionic crystals allow different sized ions to be

packed efficiently yet maintain electrical

neutrality

Typically the larger anions occupy the regular

lattice points with smaller cations occupying

interstitial sites

64

Sodium Chloride Structure

65(Askeland)

Zinc Blende

66(Askeland)

Crystal Structures of Covalent Materials

The directional qualities of covalent bonds

results in complex crystal structures

67

Diamond Cubic

68(Askeland)

Crystalline Silica

69(Askeland)

Crystalline Polymers

70(Askeland)

Diffraction Techniques for

Crystal Structure Analysis

71

How do we know the distance between

atoms and crystal structures?

X-ray diffraction (XRD-Max von Laue 1912 Nobel Prize winner)

Similar effects as diffraction grating in optics

X-rays have much smaller wavelength than visible light

Wavelengths have a similar order of magnitude to the spacing between atoms

72

X-Ray Diffraction Patterns

73(Askeland)

74

Bragg’s Law

n = 2d sin

= wavelength of x-rays

d = interplanar spacing

= angle of incidence and

diffraction of x-rays

n = an integer

75

Interplanar Spacing, dhkl

Distance between two parallel planes of

atoms with the same Miller indices

For cubic crystals:

dhkl a0

h2 k2 l2

76

X-ray

diffractometer

Diffraction

spectrum

for gold

X-Ray Detector

or photographic film strip

(Askeland)