Chapter 3-

SUBJECTS

• Crystal Systems

• Crystallographic points, directions, and planes

• Linear and planar densities

1

CHAPTER 3_2: CRYSTAL

STRUCTURES & PROPERTIES

• X-ray diffraction

Chapter 3-

Crystal Systems

• There are seven different possible combinations of a, b, c,

and α, β, γ, each of which represents a distinct crystal

system.

• Cubic, tetragonal, hexagonal, orthorhombic, rhombohedral,

monoclinic, and triclinic.

Unit cell: smallest repetitive volume

which contains the complete lattice

pattern of a crystal.

Chapter 3- 2

Seven

Crystal

Systems

Cubic,

hexagonal

triclinic

monoclinic

orthorhombic

rhombohedral

tetragonal

a = b ≠ c and a = b = 90°, g = 120°;

a = b ≠ c and a = b = g = 90°;

a = b = c and a = b = g ≠ 90°;

a ≠ b ≠ c and a = b = g = 90°;

a ≠ b ≠ c and a = g =90° ≠ b

a ≠ b ≠ c and a ≠ b ≠ g ≠ 90°;

a = b =c and a = b = g= 90°;

Chapter 3-

Point Coordinates

• The position of any point located within a unit cell may be

specified in terms of its coordinates as fractional multiples

of the unit cell edges.

Chapter 3-

Example 3.4

• Locate the point having coordinates 1/4 1 1/2

Chapter 3-

Example 3.5

• BCC unit cell atom positions

Chapter 3-

Crystallographic Directions

• The length of the vector projection on each of the three

axes is determined; these are measured in terms of the unit

cell dimensions.

• These three numbers are reduced to the smallest integer

values by multiplying or dividing by a common factor.

• The three indices are represented as [uvw].

Chapter 3-

Crystallographic Directions

x y z

Projections a/2 b 0c

Projections in terms

of a, b,and c 1/2 1 0

Reduction to integers 1 2 0

Indices

[120]

Chapter 3-

Crystallographic Directions

Draw a direction within a cubic cell.]011[

]011[

where overbar represents a negative

index

Chapter 3-

Crystallographic Planes

• Crystallographic planes are specified by three

Miller indices as (hkl).

• Procedure:

– The length of the planar intercept for each axis is

determined in terms of the lattice parameters, a,b,c.

– The reciprocal of these numbers are taken.

– These three numbers are changed to the set of smallest

integers by a common factor.

– The indices are represented by (hkl).

Chapter 3-

Crystallographic Planes

x y z

Intercepts a b c

Intercepts in terms of a, b,

and c 1 1

Reciprocals of intercepts 1 1 0

Indices (110)

Chapter 3-12

Crystallographic Planesz

x

ya b

c

4. Miller Indices (110)

example a b cz

x

ya b

c

4. Miller Indices (200)

1. Intercepts 1 1

2. Reciprocals 1/1 1/1 1/

1 1 03. Reduction 1 1 0

1. Intercepts 1/2

2. Reciprocals 1/½ 1/ 1/

2 0 03. Reduction 2 0 0

example a b c

Chapter 3-13

Crystallographic Planes

z

x

ya b

c•

••

4. Miller Indices (634)

example1. Intercepts 1/2 1 3/4

a b c

2. Reciprocals 1/½ 1/1 1/¾

2 1 4/3

3. Reduction 6 3 4

(001)(010),

Family of Planes {hkl}

(100), (010),(001),Ex: {100} = (100),

Chapter 3-

Linear Density

vectordirection oflength

vectordirection on centered atoms ofnumber =LD

RatomsLD 4/2vectordirection [110] oflength

vectordirection [110]on centered atoms ofnumber = 110 =

Chapter 3-

Planar Density

plane a of area

plane aon centered atoms ofnumber =PD

28

2 =

2110R

atomsPD

Linear and planar densities are important due to their

relevance in the process of slip- that is, the mechanism by

which metals plastically deform.

For FCC

Chapter 3-16

Planar Density of (100) Iron

Solution: At T < 912ºC iron has the BCC structure.

(100)

Radius of iron R = 0.1241 nm

R3

34a =

Adapted from Fig. 3.2(c), Callister & Rethwisch 8e.

2D repeat unit

= Planar Density =a 2

1

atoms

2D repeat unit

= nm2

atoms12.1

m2

atoms= 1.2 x 1019

1

2

R3

34area

2D repeat unit

Chapter 3-18

X-Ray Diffraction

Chapter 3-19

X-Rays to Determine Crystal Structure

X-ray

intensity

(from

detector)

q

qc

d =nl

2 sin qc

Measurement of critical

angle, qc, allows

computation of planar

spacing, d.

• Incoming X-rays diffract from crystal planes.

Adapted from Fig. 3.20,

Callister & Rethwisch 8e.

reflections must

be in phase for

a detectable signal

spacing

between

planes

d

ql

q

extra

distance

travelled

by wave “2”

a = dhkl (h)2 + (k)2 + (l)2 = d110 (1)2 + (1)2 + (0)2

Chapter 3-20

X-Ray Diffraction Pattern

Adapted from Fig. 3.22, Callister 8e.

(110)

(200)

(211)

z

x

ya b

c

Diffraction angle 2q

Diffraction pattern for polycrystalline a-iron (BCC)

Inte

nsity (

rela

tive)

z

x

ya b

cz

x

ya b

c

Chapter 3-21

SUMMARY

• 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).

• Common metallic crystal structures are FCC, BCC, and

HCP.

• Crystallographic points, directions and planes are

specified in terms of indexing schemes.

Crystallographic directions and planes are related

to atomic linear densities and planar densities.

Chapter 3-22

• Some materials can have more than one crystal

structure. This is referred to as polymorphism (or

allotropy).

SUMMARY

• Materials can be single crystals or polycrystalline.

Material properties generally vary with single crystal

orientation (i.e., they are anisotropic), but are generally

non-directional (i.e., they are isotropic) in polycrystals

with randomly oriented grains.

• X-ray diffraction is used for crystal structure and

interplanar spacing determinations.

Chapter 3-

Classwork 13.31 Determine the indices for the directions shown in the following cubic unit

cell:

A x y z

Projections 0a –b –c

Projections in terms of a, b, and c0 –1 –1

Reduction to integers not necessary

Enclosure

x y z

Projections –a

b

2 0c

Proj in terms of a, b, and c –1

1

2 0

Reduction to integers –2 1 0

Enclosure

[2 10]

B

Chapter 3-

Classwork 23. 53 Derive linear density expressions for BCC [110] and [111] directions in terms

of the atomic radius R.

Chapter 3-

Classwork 3

3.54 (a) Derive planar density expressions for FCC (100) and (111) planes in terms of

the atomic radius R.

(100) plane for an FCC unit cell.

Hw (Ch 3): 30, 54, 55, 57, 58