Diffraction Lineshapes (From “Transmission Electron Microscopy and Diffractometry of
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Transcript of Diffraction Lineshapes (From “Transmission Electron Microscopy and Diffractometry of
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Diffraction Lineshapes(From “Transmission Electron Microscopy and Diffractometry of
Materials”, B. Fultz and J. Howe, Springer-Verlag Berlin 2002. Chapter 8)
Peak form for X-ray peaks:Gaussian
LorentizianVoigt,
Psudo-Voigt:
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2
20 )(
exp)0(),(
xxIxI
Gaussian function
x0
)0(I
eI /)0(GB
2/)0(I
2
20)(
exp)0(2
)0(
xx
II
2
20)(
2ln
xx
2ln2GBFWHM
2ln0 xx
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Lorentzian function or Cauchy form
20 )(1
)0(),(
xx
IxI
2CBFWHM
x0
)0(I
GB2/)0(I
20 )(1
)0(
2
)0(
xx
II
1)( 20
xx 0xx
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Voigt: convolution of a Lorentzian and a Gaussian
)(erfiRe)0(),,( zIxI 2ix
z
Complex error function
)(erfc)(erfi2
izez z
)5145.41186.21245.21( 2 GV BBFWHM
most universal; more complex to fit.
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Lorentzian function or Cauchy form20 )(41
)0()(
C
C
Bxx
IxI
CBFWHM
2
20)(4
2lnexp)0()(G
GB
xxIxI
Gaussian functionFWHM
pseudo-Voigt:
GB
)()1()()0(),( xIxIIxI GCp
: Cauchy content, fraction of Cauchy form.
2ln22ln2 GG BB
22 CC BB
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2 = FWHM
FWHM2ln2
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Lineshapes: disturbed by the presence of K1 and K2.
Decouple them if necessary:
Rachinger Correction for K1 and K2 separation:
Assume: (1) K1 and K2 identical lines profiles (notnecessarily symmetrical); (2) Ip of K2 = ½ Ip of K1.
tan22
tan)/(22
)/sin2)(cos/(2
cos/2
cos2sin2
d
dd
)()( 12
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313
212
111
10
)(
)(
)(
0)(
II
II
II
I
2/)()( 11414 III
2/)()( 12515 III
2/)()( 131 iii III…
…
2/)()( 11 miii III General form
Example: Separated by 3 unitIi: experimental intensity at point iIi(1): part of Ii due to due to K1
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Diffraction Line Broadening and Convolution
Sources of Broadening: (1) small sizes of crystalline (2) distributions of strains within individual crystallites, or difference in strains between crystallites (3) The diffractometer (instrumental broadening)
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Size Broadening:
Interference function
3
12
222
sin
sin
i i
iitotal
NFAI
a
a
Define deviation vector
332211 bbb
3
12
222
sin
sin
i ii
iiitotal a
aNFAI
)()()()( 332211 IIII
11
2111
2
11 sin
sin)(
a
aNI
22
2222
2
22 sin
sin)(
a
aNI
…
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11
2111
2
11 sin
sin)(
a
aNI
I
21N2
11 )0(0 NI
Half width half maximum (HWHM): particular
k
'1
2)( 21
'1 NI
'1
'1 usually small
2)(
sin
sin
sin)(
21
21
'1
11'1
2
1'1
211
'1
2'11
N
a
aN
a
aNI
11'111
'1 sin2)( aNaN
Solve graphically
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Define 11'1 aNx
~ 1.392
Solution: x = 1.392
392.111'1 aN
LaNaN
443.0443.0392.1
1111
'1
Define1
k0k k
0kksink
sin2
k
kddd
kd
cos2
cos2'1
L
443.0
cos2
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FWHM '12 kd
cos2
89.0
L
cos2
89.0
L
In X-ray, 2 is usually used, define 2B
cos
89.0
BL B in radians
If the is used instead of 2, K should be divided by 2.
cosB
KL Scherrer equation, K is Scherrer constant
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Strain broadening:Uniform strain lattice constant change Bragg peaksshift.Assume strain = d0 change to d0(1+ ).
Diffraction condition:
)1(1
)1(
1
00
dd
Gk
Gdd
kd
0
1
Gdkd
sin2
kIn terms of d
kdcos2
Gdd
cos2
Peak shift
Gd
d
cos2 k
tansin2
cos2
d
d Larger shift for the diffractionpeaks of higher order
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Distribution of strains diffraction peaks broadeningStrain distribution relate to
k
2
||
'12
G
'1 is the HWHM of the diffraction G along x̂
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Instrument broadening:
Main Sources:
Combining all these broadening by the convolution procedure asymmetric instrument function
convolution
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The Convolution Procedure:instrument function f(x) and the specimen function g(x)the observed diffraction profile, h(). The convolution steps are * Flip f(x) f(-x) * Shift f(-x) with respect to g(x) by f(-x) f(-x) * Multiply f and g f(-x)g(x) * Integrate over x
0 1 2-1-201234
0 1 2-1-201234
f(x)
g(x)
)()()( hdxxgxf
Assume f and g are the functions on the right, the h() that we will get is
1 2-1-201234
f(-x)
0
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01234
0 2-2
= -1
01234
0 2-2
= 1
01234
0 2-2
= -2
01234
0 2-2
= 0
01234
0 2-2
= 2
01234
2-2 0
31/616/3
07/6
0h()
)()()()()( xgxfdxxgxfh
56
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Convolution of Gaussians:
2
20 )(
exp)0(),(
xxIxI
2ln2B
Two functionsf(): breadth Bf
g(): breadth Bg h() = f()*g(); breadth Bh
222gfh BBB
http://www.tina-vision.net/docs/memos/2003-003.pdf
B
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Convolution of Lorentzians:
Two Lorentzian functions: f(): breadth Bf g(): breadth Bg
h() = f()*g(); breadth Bh
gfh BBB
20 )(1
)0(),(
xx
IxI
2B
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Fourier Transform and Deconvolutions:
Remove the blurring, caused by the instrument function:deconvolution (Stokes correction).
Instrument broadening function: f(k) (*k is function of )True specimen diffraction profile: g(k)Measured by the diffractometer: h(K)
n
linkenFkf /2)()(
'
' /2' )()(n
lkinenGkg
''
'' /2'' )()(n
lKinenHKh
l: [1/length], the range in k ofthe Fourier series is the interval–l/2 to l/2.
Fourier transform the above three functions (DFT)
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dkkgkKfKh )()()(
dkenGenFKhl
ln
lkin
n
lkKin
2/
2/
/2'/)(2
'
'
)()()(
The function f and g vanished outside of the k range Integration from - to is replaced by –l/2 to l/2
dkeenFnGKhl
l
lknni
n n
linK
2/
2/
/)(2/2' '
'
)()()(
Orthogonality condition
nn
nnldke
l
l
lknni
'
'2/
2/
/)(2
if 0
if '
dklknnilknndkel
l
l
l
lknni
2/
2/
''2/
2/
/)(2 )/)(2sin()/)(2cos('
vanishes by symmetrydklknn
l
l
2/
2/
' )/)(2cos(
))](sin())([sin()(2
'''
nnnnnn
l
nn ' if 0
ldkl
l
2/
2/
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n
linKenFnGlKh /2)()()( ''
'' /2'' )()(n
lKinenHKh
)()()( nHnFnlG Convolution in k-space is equivalent to a multiplicationin real space (with variable n/l). The converse is alsotrue. Important result of the convolution theorem!
Deconvolution:)(
)()(
nlF
nHnG
{G(n)} is obtained from n
linkenGkg /2)()(
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Data froma perfect specimen
Data fromthe actualspecimen
RachingerCorrection(optional)
RachingerCorrection(optional)
f(k)Stokes
CorrectionG(n)=
H(n)/F(n)h(k)
F.T.
Correcteddata free
of instrumentbroadening
F.T.-1
g(k)
“Perfect” specimen: chemical composition, shape,density similar to the actual specimen ( specimenroughness and transparency broadening are similar)* E.g.: For polycrystalline alloy, the specimen is usually obtained by annealing
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f(k), g(k), and h(k): asymmetric F.T. complex coeff.
)()(
)()(1)()(
niFnF
niHnH
lniGnG
ir
irir
)()(
)()(
)()(
)()(1)()(
niFnF
niFnF
niFnF
niHnH
lniGnG
ir
ir
ir
irir
)()(
)()()()(1)(
22 nFnF
nFnHnFnH
lnG
ir
iirrr
)()(
)()()()(1)(
22 nFnF
nFnHnFnH
lnG
ir
irrii
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nir l
nknG
l
nknGkg
2sin)(
2cos)()( real part
g(k) is real and can be reconstructed as
n
linkir eniGnGkg /2)]()([)(
nir l
nki
l
nkniGnG
2sin
2cos)]()([
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Simultaneous Strain and Size Broadening:
True sample diffraction profile: strain broadening and size broadening effect
Take advantage of the following facts:Crystalline size broadening is independent of GStrain broadening depends linearly on G
Usually, know one to get the other
Both unknown
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Williamson-Hall MethodEasiest way!Requires an assumption of the shape of the peaks:
)exp(1
)(sin
)(sin)(
2
2
2
2
GGa
NaI
Kinematical crystal shape factor intensity
Gaussian functioncharacteristic of thestrain broadening
convolution
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dd
2
2
exp)( 22 to relate G
)1(0 Gk 00 GkG GG
0
Assume a Gaussian strain distribution (quick falloff forstrain larger than the yield strain) ()
dGG
dd
22
2
exp1
)()(
2GG
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Approximate the size broadening part with a Gaussian function
Good only when strain broadening >> size broadening
)exp(1
))(
exp()(2
2222
GG
NaNI
NaB
392.1
22ln2 (see page 9)
NaNa 1
2ln
392.1 characteristic width
22 GG
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The convolution of two Gaussians
2
22
)(exp)(
kG
NI
222
2222
2 11)(
G
LG
aNk
hkldGk
1sin2
Plot k2 vs G2
(k)2
G2
2
1
L
2Slope =(HWHM)
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Approximate the size broadening and strain broadening: Lorentzian functions
LNaB
443.0392.122
Size:
2)443.0
(1
)0()(
LI
I
Strain:
22
2
)(1
11
)443.0
(1)(
G
GLN
I
2
22
2
)(1
11
)(1
11
G
GG
2GG
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The convolution of two Lorentzian
22
2
)(1
11
)443.0
(1)(
G
GLN
I
2
2
)(1
1)(
kG
NI
2443.0 GL
k
Plot k vs G
k
GL
443.02Slope =
(HWHM)
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The following pages are from:
http://www.imprs-am.mpg.de/nanoschool2004/lectures-I/Lamparter.pdf
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Ball-milled Mo from P. Lamparter
(FWHM)
GL
2
2
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Nanocrystalline CeO2 Powder from P. Lamparter
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Nb film, WH plot from P. Lamparter
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from P. Lamparter
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anisotropy of shape or elastic constants, strains. and sizes k2 vs G2 or k vs G not linear
Using a series of diffraction e.g. (200), (400){(600) overlap with (442), can not be used} provide a characteristic size and characteristic mean-square strain for each crystallographic direction!
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Ek fit better than k in this case elastic anisotropic is the main reason for the deviation of k to G.
Ball-milled bcc Fe-20%Cu
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Warren and Averbach MethodFourier Methods with Multiple Orders
Q)2exp()()Q( diQLLAI
)()()( LALALA D
size strain
How to interpret A(L)?
GQ
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from P. Lamparter
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from P. Lamparter
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from P. Lamparter
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from P. Lamparter
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from P. Lamparter
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from P. Lamparter
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Williamson-Hall MethodEasy to be doneOnly width of peaks needed
Warren-Averbach MethodMore mathematicsPrecise peak shapes neededDistributions of size and microstrainRelation to other properties(dislocations)