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Chapter 3

Basic Crystallography and Electron Diffraction from Crystals

Lecture 14

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Announcement

Midterm Exam: Oct. 22, Wednesday, 2:30 4:30

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( ) ++=i

lzkyhxi

ihkliiiefF

2

HW#11: Prove the fcc factor rule: the three integers h,k,l must be all even or all odd. For

example, the lowest order diffractions are (111), (200), (220), (311), (222), (400), (331),

(420), but other diffractions such as the (100), (110), (210), (211), etc. are forbidden.Due day: 10/13/08

( ) ( )

{ }

oddandevenmixedarelk,h,if,0

oddallorevenallarelk,h,if,41so

2

1,

2

1,0,

2

1,0,

2

1,0,

2

1,

2

1,0,0,0zy,x,

isvectorbasisthefcc,for

)()()(

=

=+++=

=

+++

F

fFeeefF

lkilhikhi

D1

B2

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HW#12: Fe3AlC phase in Fe-C-A system has a cubic structure: Al is corner, C is in the cubic center, and Fe is in the center of

each face.

1. Derive an expression for the structure factor in terms of fAl, fFe, and fC

2. Sketch the (100)* section of the reciprocal structure for this Fe3AlC phase, labeling the low index diffractions and indicating

relative intensities.

C

Al

Fe

( ) ( ) ( )[ ]

obtainedbecanpatternndiffractiothefactor,structureon theBased

planesndiffractioorderlowas{020}and{011},{001},takestructure,reciprocal*(100)sketchTo

:(hkl)ineven1andodd2

:(hkl)inodd1andeven2

3:oddlk,h,

3:evenlk,h,

fF

1/2)1/2,(1/2,atC

and),(0,1/2,1/2),(1/2,0,1/2),(1/2,1/2,0atFe(0,0,0),atAl

)(

Al

FeCAl

FeCAl

FeCAl

FeCAl

klilhikhi

Fe

lkhi

C

fffF

fffF

fffF

fffF

eeefef

+=

=

+=++=

++++= +++++

000 010 020

001

002

011 0210-11

0-100-20

0-21

00-1

hkl all even strong intensity

hkl two odd and one even,

moderate intensity

hkl two even and one odd, low

intensity0-1-1 01-1 02-10-2-1

00-2

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Indexing Diffraction Patterns

(a) A single perfect

crystal

(b) A small number of

grains- note that

even with three

grains the spots

begin to form circle

(c) A large number pf

randomly orientedgrains-the spots

have now merged

into rings

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1. Analysis of polycrystalline diffraction pattern--- ring pattern

Incident beamSmall grains

Diffracted beamsfrom (hkl) planes in

each particles

(hkl) ring

g

The geometry of formation of a

single (hkl) ring by accumulation

of (hkl) beams from different

grains.

Ring pattern from a fine grained

polycrystalline sample is in effect the

superposition of many single crystalpatterns.

The rings occurs in the characteristic

sequence, regarding different dhkl

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The geometry of formation of a single (hkl) ring by accumulation (or superposition) of (hkl)beams from different grains.

If the grains in a polycrystalline material are randomly oriented or weakly textured, then the

reciprocal vectorg to each diffracting plane will be oriented in all possible direction.

Since the length of a particularg is a constant, these vectors g will describe a sphere withradius of |g|.

The intersection of such a sphere with Ewald sphere is a circle, and therefore the diffraction

pattern will consist of concentric rings.

If grains are sufficiently large, individual reflections can be seen in the rings as in Fig. a For fine grains the diffraction pattern would look more like that shown in Fig. b.

a b

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The following figures are some the most useful diffraction patterns for bcc and fcc crystal.

More diffraction patterns of other types of crystals can be found in crystallography handbook.

Keep the handbook in hand when you are using a TEM to study the crystal specimen.

In addition, we can use the reciprocal rule to assist understanding the bcc and fcc patterns.

This rule is very useful in practice. We can very quickly identify the diffraction direction, i.e.

beam direction.

bcc real space -- fcc reciprocal space

fcc real space -- bcc reciprocal space

e.g.

bcc in real spacefcc in reciprocal space

A1

C1B1

D1

D2

A2

B2

C2

So [001] diffraction pattern is to

extend the reciprocal plane of

reciprocal lattice unit cell,

A1B1C1D1, also see the standard

bcc [001] pattern.

A1

B1 C1

D1

The corresponding

reciprocal lattice is a fcc

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Four standard indexed

diffraction patterns for

bcc crystals in [001],

[010], [-111], and [-112]. Ratios of the

principal spot spacings

are shown as well as

angles between the

principal plane normals.Forbidden reflection

spots are indicated by

x.

The [001] pattern is

obtained by extending

A1B1C1D1 reciprocal

plane in reciprocal

lattice unit cell,

considering the

structure factor.

Similarly, the [110]

pattern is obtained by

extending B1B2D2D1

reciprocal plane in

reciprocal lattice unit

cell.

Reciprocal plane

A1B1C1D1 in unit cell

Reciprocal plane

B1B2D21D1 in unit cell

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Four standard

indexed diffraction

patterns for fcccrystals in [001],

[010], [-111], and [-

112]. Ratios of the

principal spot

spacings are shownas well as angles

between the principal

plane normals.

Forbidden reflection

spots are indicatedby x.

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Application of Electron Diffraction

Determining orientation relationship between crystals

Advantage of TEM: image and diffraction pattern can be obtained simultaneously

(a) A TEM Dark Field micrographs showing Fe2TiSi precipitated after ageing in -Fe,(b). The corresponding SAD pattern of-Fe (bcc, a=2.866A, strong spots) andFe2TiSi precipitates ( fcc, a=5.732A, weak spots) in a single crystallographic

orientation. The camera length is 31.5 Amm here.

(a) (b)

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fccfcc

bcc

Refer to the standard pattern

and measure the distances andangles between spots

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The patterns of-Fe and Fe2TiSi can be indexed as shown above, we can find:

(200) of Fe2TiSi is half distance of (200) -Fe , therefore twice d-spacing of-Fefrom center.

These planes are therefore parallel and the lattice parameter of Fe2TiSi is twicethat of-Fe .

Similarly, the (022) Fe2TiSi reflection is coincident with (011) -Fe

The zone axes, obtained by cross product of vectors, are both [0-11]

Therefore the orientation relationship may be specified by quoting the parallelisms:

(200)Fe2TiSi//(200) -Fe and zone axis: [0-11]FeiTiSi//[0-11] -Fe

(022) Fe2TiSi

coincident with

(011)

-Fe

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(022) Fe2TiSi

coincident with

(011)

-Fe

Other, more complicated, orientation relationships may be determined by

the same simple approach, but to go from the parallelism between the

planes and zone axes between planes not observed in the patterns (i.e.

those that are not on Laue condition or not nearly parallel to the electronbeam direction), requires a knowledge of the stereographic projection.

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Stereographic ProjectionNomenclature of Crystallographic Directions and Face normals / poles

Indices (no brackets, parenthesis) for directions

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(0-11)

(011)

(100)

(-100)

(01-1)

Stereographic Projection

Stereographic projections are 2-D

maps of the orientation

relationships between different

crystallographic directions.

They are useful for representing

the electron diffraction pattern,

although stereographic projections

were developed for representing 3-

D crystallography.

(001)

(00-1)

(010)

(0-10)

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3-D construction of Stereographic Projection

To construct a stereographic projection, begin

with a tiny crystal situated at the center of a large

sphere

Conventional terminology calls the normals tocrystallographic planes, poles. We specify the

poles pointing upwards to the north pole of the

sphere.

In figure, nine poles were extended from the

crystal and intersect the sphere.

We use the points of intersection to create a [001]stereographic projection.

To project these intersection points onto a 2-D

surface, first draw straight lines from the

intersection points to the south pole. Next, mark

with an X the points of intersection of these lines

on the equatorial plane of the sphere.

The stereographic projection is the equatorial

plane of the sphere with these marked

intersections, X points.

The stereographic projection contains orientation

information about all poles that intersect thenorthern hemisphere of the sphere.

Poles such as (01-1) and (00-1), which intersect

the southern hemisphere of the sphere, are not

included in the [001] stereographic projection.

However, the entire southern hemisphere of the

crystal can be obtained by rotating the [001]stereographic projection by 180, and changing

the signs of all poles indices

(0-11)

(011)

(100)

(-100)

(01-1)

(001)

(00-1)

(010)

(0-10)

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Stereographic Projection

2-D description of construction ofStereographic Projection

Section through sphere of projection

showing relation of spherical poles (E, D) to

stereographic poles (E, D)

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Stereographic Projection

Relation of spherical

and stereographic

projections

Equatorial plane as

projection plane

South pole as projection pole

Face poles

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Stereographic projection

(equatorial plane) of

some cubic crystal faces[001] is zone axis, and all

poles on the great circle

(such as (010), (100), etc.)

belong to this zone axis,e.g. [-1-10]. [001]=0,

[110].[001]=0, etc., i.e.

(hkl) is normal to [vuw]

Stereographic Projection

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Relationship between stereographic projections and electron diffraction patterns

In the high energy electron diffraction, the Bragg

angles are so small that the incident electron beam

travels nearly parallel to the diffracting planes.

When the electrons travel down the crystal from the

north pole of a spherical projection, the diffractions

occurs from planes whose poles intersect theequator of the sphere, perhaps within a degree or

so (Zone Law).

-111 || -222

(-112)(002)

(000) (-110)(1-10)

(1-12)-112

001

110

1-12

1-10 -110

(-22-2)

(-222)

(00-2) (-11-2)(1-1-2)

1-1-1

-1121-1-2

00-1Orientation relationship between bcc [110] diffraction pattern at left, and [110] stereographic projection at right.

Angles between the vectors are the same on the left and right sides

The figures show a bcc crystal

oriented with its [110] direction

pointing upwards towards the electron

gun

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Relationship between stereographic projections and electron diffraction patterns

In relating stereographic projections to the diffraction planes, it is

important to remember that stereographic projections contain no

information about the distances between the diffraction spots,

and contain no information about structure rules. Nevertheless,

the angles between the vectors in diffraction pattern and in the

stereographic projection are the same, e.g. although {111}

diffraction are forbidden for bcc crystals, the (--222) diffraction

occurs at the angle of the [-111] direction

-111 || -222

(-112)(002)

(000) (-110)(1-10)

(1-12)-112

001

110

1-12

1-10 -110

(-22-2)

(-222)

(00-2) (-11-2)(1-1-2)

1-1-1

-1121-1-2

00-1Orientation relationship between bcc [110] diffraction pattern at left, and [110] stereographic projection at right.

Angles between the vectors are the same on the left and right sides

The figures show a bcccrystal oriented with its [110]

direction pointing upwards

towards the electron gun

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Manipulations of stereographic projections

The stereographic projection is a powerful tool for working problems that involve orientations

between two different crystals. We introduce a tool analogous to a protactor, called Wulff Net,

to do easily so. Wulff Net is a projection of lines of latitude (measuring north-south position)and longitude (measuring east to west position) obtained from a calibrated reference sphere.

The lines of latitude are arcs in the stereographic projection, as are the lines of longitude, but

the lines of longitude are concave inwards.

Wulff Net named after G.V. Wulff, Russian crystallographer (1863-1925)

Great cycles and small cycles are drawn at intervals of 2

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The Wulff Net

should be

photocopied onto

a transparency for

work with the

matchingstereographic

projections

809080

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0

10

20

30

40

50

60

708090

20

30

40

6070

80

10

50

90

0

10

20

30

40

50

60

7080

10

20

30

40

50

60

7080

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Wulff net is a tool to rotate a crystal into any 3-D orientation. Simple rotations include

rotation about the center of the projection and about the north-pole of the net

Examples:

1. Find the angle between two planes

(a). Poles are on the edge of the stereographic edge: 1 operation: just overlay the

Wulff Net in any orientation, and count the tick marks on the edge, Figure (a).

(b). One pole is in the center of the projection, and the other is at an arbitrary position:

1 operation: Align the Wulff Net with its equator passing through the two points and

count the longitude tick marks along the equator.

001

-112

Angel between (-112)

and (001) or (002)=35

[110] projection [001] projection

001

-112

(a)

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Examples:

1. Find the angle between two planes

(b). One pole is in the center of the projection, and the other is at anarbitrary position: 1 operation: Align the Wulff Net with its equator passing

through the two points and count the longitude tick marks along the equator.

[001] projection

001

-112

Equator of Wulff

net

Angel between (-112)

and (001) or (002)=35

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Examples.

2. Find the angle between two arbitrary poles.

1 operation: Orient the Wulff Net so that the two points are intersected by a common line oflongitude, and count the latitude ticks along the line of longitude.

Pole 1

Pole 2

Angel between pole 1

and pole2 =20

Pole 1

Pole 2

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Examples.

3. Find a [010] stereographic projection from an [001] stereographic projection

When the indices of the new stereographic projection are obtained from the old bycyclic permutation, just make transformation xyz into yzx. E.g. the poles 100 and 010

on the edge of the old [010] projection become 001 and 100 in the new [010] projection.

We can confirm that [001]X[100]=[010], by right hand rule g3=g1Xg3

g1

g2

g3

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Examples.

4. Find a new [113] stereographic projection from an [001] stereographic projection

1 operation: Orient the Wulff net so that its equator passes through the 113 pole in the

[001] projection. Then move the 113 pole into center along equator, and move all otherpoles of the [011] projection along lines of latitude by same angle. Note the

appearance of the hkl pole at the bottom of the projection, and the disappearance of

the h-k-l at the top.

113

-h-k-l

hkl

113

-h-k-l

hkl

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Examples.

4. Find a new [113] stereographic projection from an [001] stereographic projection

1 operation: Orient the Wulff net so that its equator passes through the 113 pole in the[001] projection. Then move the 113 pole into center along equator, and move all other

poles of the [011] projection along lines of latitude by same angle. Note the

appearance of the hkl pole at the bottom of the projection, and the disappearance of

the h-k-l at the top.

113

-h-k-l

hkl

113

-h-k-l

hkl

-h-k-l is out and disappears

from new [113] projection

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Examples.

5. Rotation of a crystal about an arbitrary pole: You are given one crystal with a [110]

projection. A second crystal is then given a 10 rotation about its (100) pole. On the

projection of the first crystal, where is the poles of the second crystal after this rotation?

3 operations: 1). Move the pole (100) into center of the projection by moving it along the

equator of the Wulff Net. This generates a [100] projection, with the typical pole x moved

along a line of latitude to position x.

110100

x

[110] projection

100100

X

[100] projection

X

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Examples.

3 operations: 2). Rotate the [100] projection about its center by 10.

Point x moves to position x; 3). Rotate the (100) back to its originalposition, moving it along the equator or the Wulff Net. Point x moves

along a line of latitude to point x

100

X

[100] projection

X100

X

[110] projection

X

10

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Examples.

3 operations: 2). Rotate the [100] projection about its center by 10. Point x

moves to position x; 3). Rotate the (100) back to its original position, moving it

along the equator of the Wulff Net. Point x moves along a line of latitude to point

x

100

[100] projection

X100

[110] projection

X

X

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Examples.

7. (*** extra information, you dont need fully understand) Kurdjumov-Sachs (K-S)

orientation relationship between bcc and fcc crystals. The K-S relationship specifies the

parallel planes: (1-10)bcc || (-111)fcc and the parallel directions in these plans: [111]bbc ||[110] fcc

3 operations: 1). Use the [110] stereographic projection to the point the [110]fcc direction

upwards, and [111] stereographic projection to [111] bcc direction upwards.

[110] fcc

[110] fcc projection

(1-11) fcc

(1-1-1) fcc

(-111) fcc

(-11-1) fcc

(1-12) fcc

(-11-2) fcc

(00-1) fcc

(001) fcc

[111]

bcc

[111] bcc projection

(01-1) bcc

(-110) bcc

(1-10) bcc

(0-11) bcc

(11-2) bcc

(-1-12) bcc

(10-1) bcc

(-101) bcc

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Examples.

3 operations: 2). Overlay it with the [111] stereographic projection so that [111] bcc is

parallel with [110]fcc direction.

[110] fcc

[110] fcc projection

(1-11) fcc

(1-1-1) fcc

(-111) fcc

(-11-1) fcc

(1-12) fcc

(-11-2) fcc

(00-1) fcc

(001) fcc

[111]

bcc

[111] bcc projection

(01-1) bcc

(-110) bcc

(1-10) bcc

(0-11) bcc

(11-2) bcc

(-1-12) bcc

(10-1) bcc

(-101) bcc

fcc

bcc

Examples

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Examples.

3). Rotate the two overlain projections so that the(-111)fcc pole on the edge of projection is

on top of the [1-10]bcc pole.

We see that a direction is parallel in both crystal

[111] bcc

(01-1) bcc

(-110) bcc

(1-10) bcc

(0-11) bcc

(11-2) bcc

(-1-12) bcc

(10-1) bcc

(-101) bcc

fcc

bcc

[110] fcc

(1-11) fcc

(1-1-1) fcc

(-111) fcc

(-111) fcc

(1-12) fcc

(-11-2) fcc

(001) fcc

(00-1) fcc

Some poles of overlain

[111]bcc and [110]fcc

stereographic projections

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Examples.

K-S orientation relationship between bcc and fcc crystal

An experimental result of Fe-9Ni steel shows the (002) fcc diffraction is isolated from the

bcc diffraction. We can locate small amounts of fcc phase within bcc matrix using this

diffraction spot for a DF image.

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Example

Using the [001] stereographic projection provided, sketch and label the (221)*

section of reciprocal space for fcc crystal.

1. First determine the necessary rotation to bring [221] to center. This can be

calculated as follows:

o701

]001[

9

]221[arccos =

=

2. To make the [221] projection, we need to rotate every point by 70. To

simplify the operation, we only select the points which will end up on the

outside edge of [221] projection, i.e. (hkl) satisfies 2h+2k+l=0. So we can

visually guess the following poles: [-110],[1-10],[-322],[-212],[1-22],[-102],[0-12] etc.

3. For fcc, h,k,l all even or odd, so we choose even multiples of ,,,, and . All these points in [001] projection should

be rotated 70

along their latitude to get [221] projection.

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Move 70

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(-110)

(1-10)

(-212)

(-102)

(0-12)

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Considering the structure factor, use even multiples all poles, and re-

arrange the spots according to ratios and angles

(-220)

(2-20)

(-424)

(0-24)

(-110)

(1-10)

(-212)

(-102)

(0-12)

: Forbidden spots

(4-40)

(-440)(-204)

71.6

Schematically drawing of [211] diffraction

Th t di W lff t ti b f d b l t

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The tedious Wulff net operation can be performed by several computer

programs ( such as EMS, Desktop Microscopist, and CrystalKit, etc.)

[001] Pole projection

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[211] Pole projection ( low order pattern)

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[211] Pole projection ( high order pattern). fcc pattern is obtainedexcluding the forbidden spots

HW# 12 Use two Wulff Nets to solve this problem

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HW# 12 Use two Wulff Nets to solve this problem.

In geoscience, one nautical mile is defined as one minute of arc along a great circle

of the earth. So one degree arc along a great circle is equal to 1x60 min.=60 nautical

mile. Based on the world map, we know Las Vegas, US, is at 36 degree north

latitude, 115 degree west longitude. Beijing, China, is at 40 degree north latitude,

116 degree east longitude. How many nautical miles is Beijing from Las Vegas?

Please briefly describe the operations you had to perform.

Due: Oct. 27, 08.

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Next Lecture:

Kikuchi Line and its indexing

Double diffraction

CBED pattern (convergent beam electron diffraction)