CHAPTER 7: The Stereographic Projection -...

Department of Materials Engineering at Isfahan University of Technology

ISSUES TO ADDRESS...

• How to make a stereographic for a crystal

structure?

• What is a stereographic projection?

• What is application of the stereographic

projection in crystallography?

CHAPTER 7: The Stereographic Projection

Department of Materials Engineering at Isfahan University of Technology

Crystals are made of lattice planes and angles

Crystals information can be illustrated from their planes and angles

between planes.

Stereography is a graphical method to find angular relationship

between lattice planes and directions

With stereographic projection, a 3-Dimension perspective of a crystal

can be illustrated in a 2-Dimension plaper.

The Stereographic Projection

Stereography

Department of Materials Engineering at Isfahan University of Technology

The angle between two crystal faces

as the angle between lines that are

perpendicular to the faces. Such a

lines are called the poles to the crystal

face.

The Stereographic Projection

Crystallographic Angles

Department of Materials Engineering at Isfahan University of Technology

The Stereographic Projection

Crystallographic Angles

Department of Materials Engineering at Isfahan University of Technology

From each crystal face (in the

centre of sphere) we draw a line

perpendicular to the face

The Stereographic Projection

Stereographic Projection

We define this face (010) as having a φ angle of

0o. For any other face, the φ angle will be

measured from the b axis in a clockwise sense in

the plane of the equator.

We define the ρ angle, as the angle between the c

axis and the pole to the crystal face, measured

downward from the North pole of the sphere. In the

diagram shown here, a crystal face has a ρ

angle measured in the vertical plane containing the

axis of the sphere and the face pole, and a ρ angle

measured in the horizontal equatorial plane. Note

that the (010) face has a ρ angle = 90o.

Department of Materials Engineering at Isfahan University of Technology

These angular measurements are

similar to those we use for latitude

and longitude to plot positions of

points on the Earth's surface. For the

Earth, longitude is similar to the φ

angle, except longitude is measured

from the Greenwich Meridian, defined

as φ = 0o. Latitude is measured in

the vertical plane, up from the

equator, shown as the angle ρ.

The Stereographic Projection

Stereographic Projection

Department of Materials Engineering at Isfahan University of Technology

Example, the ρ and φ angles for the

(111) crystal face in a crystal model is

shown here. Note again that the ρ

angle is measured in the vertical

plane containing the c axis and the

pole to the face, and the φ angle is

measured in the horizontal plane,

clockwise from the b axis.

The Stereographic Projection

Stereographic Projection

Department of Materials Engineering at Isfahan University of Technology

The Stereographic Projection

Angles between two planes

Department of Materials Engineering at Isfahan University of Technology

The Stereographic Projection

Stereographic Projection Method

Department of Materials Engineering at Isfahan University of Technology

The Stereographic Projection

Stereographic Projection Method

1. The plane C is represented by its normal CP.

2. The normal CP is represented by its pole P, which

is its intersection with the reference sphere.

3. The pole P is represented by its stereographic

projection P’

Department of Materials Engineering at Isfahan University of Technology

Stereographic Projection Method

The Stereographic Projection

Department of Materials Engineering at Isfahan University of Technology

Stereographic Projection of Main

Planes

The Stereographic Projection

Department of Materials Engineering at Isfahan University of Technology

The Stereographic Projection

The Wulff Net

Department of Materials Engineering at Isfahan University of Technology

The Stereographic Projection

Wulff net Drawn for 2°intervals

The Wulff Net

Department of Materials Engineering at Isfahan University of Technology

Applications:

Example 1: Finding zone circle of two known poles

The Stereographic Projection

Reminder:

A family of faces all parallel to some particular line

is called a zone, and the line is called the zone axis

Department of Materials Engineering at Isfahan University of Technology

Applications:

Example 2: Finding pole of plane of a given trace

The Stereographic Projection

Department of Materials Engineering at Isfahan University of Technology

Applications:

Example 3: Finding Angle between two planes

The Stereographic Projection

Department of Materials Engineering at Isfahan University of Technology

Applications:

Example 4: Finding area of all planes of α angle with P

The Stereographic Projection

Department of Materials Engineering at Isfahan University of Technology

Applications:

Example 5: Finding a plane with α and β angle with P1 and P2

The Stereographic Projection

Department of Materials Engineering at Isfahan University of Technology

Example 6: Plotting F and rUsing a Wulff NetPlotting F and r

Suppose you measured r = 60o and F = 30o

The pole for (010) is on the eastern edgeMeasure F = +30o clockwise along the circumference x

Rotate tracing paper so x falls over E-W diameterPlot r = 60o in from center

x

Tracing paper over a Wulff Net,

rotating about a thumb tack at the center

The Stereographic Projection

Department of Materials Engineering at Isfahan University of Technology

Miller indices of a poleMiller indices are a convenient way to represent a direction or a plane

normal in a crystal. Directions are simply defined by the set of multiples of

lattice repeats in each direction. Plane normals are defined in terms of

reciprocal intercepts on each axis of the unit cell.

When a plane is written with

parentheses, (hkl), this indicates a

particular plane normal: by

contrast when it is written with

curly braces, {hkl}, this denotes a

the family of planes related by the

crystal symmetry. Similarly a

direction written as [uvw] with

square brackets indicates a

particular direction whereas

writing within angle brackets ,

<uvw> indicates the family of

directions related by the crystal

symmetry.

Department of Materials Engineering at Isfahan University of Technology

Miller indices of Unknown Poles

N

S

WE

Department of Materials Engineering at Isfahan University of Technology

Miller indices of Unknown Poles

N

S

WE

Procedure:

1. Draw great circles which pass

through at least three points

2. Draw great circle (zone) of each

pole

3. 3. Find degree of (Fold)

symmetry

2-fold symmetry: PA=PB

Department of Materials Engineering at Isfahan University of Technology

Miller indices of Unknown Poles

3-fold symmetry:

PA=PB=PC

AB=BC=CA

4-fold symmetry:

PA=PB=PC=PD

AB=BC=CD=DA

Department of Materials Engineering at Isfahan University of Technology

Miller indices of Unknown Poles

Procedure:

4. Find a triangle which the highest

number of great circles passed

through the poles

Department of Materials Engineering at Isfahan University of Technology

Cube Component = {001}<100>

{110}

{100}

{111}

Department of Materials Engineering at Isfahan University of Technology

Rotation of a Crystal

Department of Materials Engineering at Isfahan University of Technology

Standard Projection of cubic

crystals

The Stereographic Projection

Department of Materials Engineering at Isfahan University of Technology

Standard Projection of cubic

crystals

The Stereographic Projection

Department of Materials Engineering at Isfahan University of Technology

We have 8 tetrahedral position per an fcc unit cell.

We have (6 · 4)/2 = 12 positions per an BCC unit cell

Tetrahedral Positions

Department of Materials Engineering at Isfahan University of Technology

We have 12/4 +1 = 4 positions per an fcc unit cell.

We have 12/4 + 6/2 = 6 positions per an bcc unit cell

Octahedral Positions