Line Broadening - XRD
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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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