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XRD Line Broadening

With effects on Selected Area Diffraction (SAD) Patterns in a TEM

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In the example considered was far away (at a larger angular separation) from

(Bragg) and it was easy to see the destructive interference

In other words for incidence angle of the phase difference of is accrued just

by traversing one d.

If the angle is just away from the Bragg angle (Bragg), then one will have to go

deep into the crystal (many d) to find a plane (belonging to the same parallel

set) which will scatter out of phase with this ray (phase difference of) and

hence cause destructive interference

If such a plane which scatters out of phase with a off Bragg angle ray is absent

(due to finiteness of the crystal) then the ray will not be cancelled and diffraction

would be observed just off Bragg angles too line broadening!

(i.e. the diffraction peak is not sharp like a -peak in the intensity versus angle plot)

This is one source of line broadening of line broadening. Other sources include:

residual strain, instrumental effects, stacking faults etc.

When considering constructive and destructive interference we considered the

following points:

http://localhost/var/www/apps/conversion/tmp/scratch_13/constructive_interference.ppthttp://localhost/var/www/apps/conversion/tmp/scratch_13/constructive_interference.ppt

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Instrumental

Crystallite size

Strain

Stacking fault

XRD Line Broadening

Other defects

Unresolved 1 , 2 peaks

Non-monochromaticity of the source (finite width of peak)

Imperfect focusing

In the vicinity ofB the ve of Braggs equation not being satisfied

Residual Strain arising from dislocations, coherent precipitates

etc. leading to broadening

In principle every defect contributes to some broadening

Bi

BC

BS

...)( SFsci BBBBFWHMB

BSF

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...)( SFsci

BBBBFWHMB

The diffraction peak we see is a result of various broadening mechanisms at work

Full Width at Half-Maximum (FWHM) is typically used as a measure of the peak width

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Crystallite size

Size > 10 m Spotty ring

(no. of grains in the irradiated portion insufficient to produce a ring)

Size (10, 0.5) Smooth continuous ring pattern

Size (0.5, 0.1) Rings are broadened

Size < 0.1 No ring pattern

(irradiated volume too small to produce a diffraction ring pattern & diffraction occurs only at low angles)

Spotty ring

Rings

Broadened RingsDiffuse

10 m0.5

Rings

0.1 0.5

In a TEM Selected Area Diffraction (SAD) pattern, with decreasing crystallite size the

following effects are observed on the pattern obtained

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Effect of crystallite size on SAD patterns

Single crystal

Spotty pattern

Few crystals in the selected region

Schematics

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Effect of crystallite size on SAD patterns

Ring patternBroadened Rings

Schematics

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Subtracting Instrumental Broadening

Instrumental broadening has to be subtracted to get the broadening effects due to

the sample

1

Mix specimen with known coarse-grained (~ 10m), well annealed (strain free) does not give any broadening due to strain or crystallite size (the only

broadening is instrumental). A brittle material which can beground into powder form without leading to much stored strain is good.

If the pattern of the test sample (standard) is recorded separately then the

experimental conditions should be identical (it is preferable that one or morepeaks of the standard lies close to the specimens peaks)

2

Use the same material as the standard as the specimen to be X-rayed but with large

grain size and well annealed

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rsci BBBBB

...)( SFsci BBBBFWHMB

For a peak with a Lorentzian profile

222

ir BBB For a peak with a Gaussian profile

222 )( iir BBBBB A geometric mean can also used

Longer tail

The peaks are fitted to various profiles

12

2 210 2

1( )

( ) ( )L x

x x

2

2

1 ( )( ) exp

22

xf x

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Scherrers formula

( )c

B

kB

L Cos

Wavelength

L Average crystallite size ( to surface of specimen)

k 0.94 [k (0.89, 1.39)]

~ 1 (the accuracy of the method is only 10%)

For Gaussian line profiles and cubic crystals

0

2

4

6

8

10

12

14

0 30 60 90t

1/Cos(t)

The Scherrers formula is used for the determination of grain size from broadened peaks.

The formula is not expected to be valid for very small grain sizes (

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Strain broadening

( )s BB Tan

Strain in the material

Smaller angle peaks

should be used to

separate Bs and Bc

0

2

4

6

8

10

12

14

0 20 40 60 80 100

t

Tan(t)

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Separating crystallite size broadening and strain broadening

scr BBB )(

CosL

kBc )(TanBs

)()(

Tan

CosL

kBr

)()(

SinL

kCosBr

Plot of [BrCos] vs [Sin]

Crystallite size broadening

Strain broadening

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Annealed Al

Peak No. 2 () hkl Bi = FWHM () Bi = FWHM (rad)

1 38.52 111 0.103 1.8 103

2 44.76 200 0.066 1.2 103

3 65.13 220 0.089 1.6 103

Cold-worked Al

2 () Sin() hkl B() B (rad) BrCos (rad)

1 38.51 0.3298 111 0.187 3.3 103 2.8 103 2.6 103

2 44.77 0.3808 200 0.206 3.6 103 3.4 103 3.1 103

3 65.15 0.5384 220 0.271 4.7 103 4.4 103 3.7 103

222

ir BBB

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3107.1

L

knmLSizeGrain 90)(

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