Basic Crystallography and Electron Diffraction From Crystals

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CHEM 793, 2008 Fall Chapter 3 Basic Crystallography and Electron Diffraction from Crystals Lecture 14

Transcript of Basic Crystallography and Electron Diffraction From Crystals

Page 1: Basic Crystallography and Electron Diffraction From Crystals

CHEM 793, 2008 Fall

Chapter 3Basic Crystallography and Electron Diffraction from Crystals

Lecture 14

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CHEM 793, 2008 Fall

Announcement

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

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( )∑ ++=i

lzkyhxiihkl

iiiefF π2

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

( ) ( )

{ }

odd andeven mixed are lk,h, if,0odd allor even all are lk,h, if ,4

1 so21,

21,0,

21,0,

21,0,

21,

21,0,0,0zy,x,

is vector basis thefcc,for

)()()(

==

+++=

=

+++

FfF

eeefF lkilhikhi πππ

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 fC2. Sketch the (100)* section of the reciprocal structure for this Fe3AlC phase, labeling the low index diffractions and indicating

relative intensities.

C

Al

Fe

( ) ( ) ( )[ ]

obtained becan pattern n diffractio thefactor, structure on the Basedplanesn diffractioorder low as {020} and {011},{001}, takestructure, reciprocal *(100)sketch To

:(hkl)in even 1 and odd 2 :(hkl)in odd 1 andeven 23 :odd lk,h,3 :even l k,h,

fF1/2) 1/2, (1/2,at C

and),(0,1/2,1/2),(1/2,0,1/2 ),(1/2,1/2,0at Fe (0,0,0),at Al

)(Al

FeCAl

FeCAl

FeCAl

FeCAl

klilhikhiFe

lkhiC

fffFfffF

fffFfffF

eeefef

−+=−−=

+−=++=

++++= +++++ ππππ

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 oriented grains-the spots have now merged into rings

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

Incident beamSmall grains

Diffracted beams from (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 crystal patterns.

•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 vector g to each diffracting plane will be oriented in all possible direction.

• Since the length of a particular g is a constant, these vectors g will describe a sphere with radius 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 space fcc in reciprocal space

A1

C1B1

D1

D2A2

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 spacingsare 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 shown as well as angles between the principal plane normals. Forbidden reflection spots are indicated by 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 α-Fe, (b). The corresponding SAD pattern of α-Fe (bcc, a=2.866A, strong spots) and Fe2TiSi 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 and angles between spots

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

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

• These planes are therefore parallel and the lattice parameter of Fe2TiSi is twice that of α-Fe .

• Similarly, the (022) Fe2TiSi reflection is coincident with (011) α-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) α-Fe and zone axis: [0-11]FeiTiSi//[0-11] α-Fe

(022) Fe2TiSi coincident with (011) α-Fe

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(022) Fe2TiSi coincident with (011) α-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 electron beam 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 projectionswere 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 to crystallographic 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 the northern 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 of Stereographic 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 the equator 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 thestereographic 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 bcc crystal 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 matching stereographic projections

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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 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] projection

001

-112

Equator of Wulffnet

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 of longitude, 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 by cyclic 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 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

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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 original position, moving it along the equator or the Wulff Net. Point x” moves along a line of latitude to point x”’

100

X’

[100] projection

X” 100

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

X”100

[110] projection

X”X””

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Examples.

7. (*** extra information, you don’t 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

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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 <112> 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 =

⋅=θ

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 <-110>,<-212>,<1-2,2>,<-102>, and <0-12>. 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

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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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CHEM 793, 2008 Fall[211] Pole projection ( high order pattern). fcc pattern is obtained

excluding the forbidden spots

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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)