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
Lorentizian
Voigt,
Psudo-Voigt:
Gaussian function
 ( x  x0 ) 2 

I ( x,  )  I (0) exp 
2



 ( x  x0 ) 2 
I ( 0)

 I (0) exp 
2
2



ln 2 
( x  x0 ) 2

2
x  x0   ln 2
FWHM BG  2 ln 2
I ( 0)
I ( 0) / 2
BG

x0
I ( 0) / e
Lorentzian function or Cauchy form
I ( x,  ) 
1 (
I ( 0)
x  x0

)2
I ( 0)
I ( 0)

x  x0 2
2
1 (
)
I ( 0)
BG


(
x  x0

x0
)2  1
x  x0  
FWHM BC  2
I ( 0) / 2
Voigt: convolution of a Lorentzian and a Gaussian
I ( x,  ,  )  I (0) Reerfi ( z )
x  i
z
 2
Complex error function
 z2
erfi( z )  e erfc( iz )
FWHM
BV  BG (1  2.1245   2  2.1186  4.5145 )
most universal; more complex to fit.
pseudo-Voigt:
I p ( x, )  I (0)I C ( x )  (1   ) I G ( x )

4( x  x0 ) 2 

I G ( x )  I (0) exp  ln 2
2
BG


Gaussian function
FWHM BG
BG  2 ln 2    BG 2 ln 2
I ( 0)
IC ( x) 
x  x0 2
1  4(
)
BC
Lorentzian function or Cauchy form
FWHM BC
BC  2    BC 2
: Cauchy content, fraction of Cauchy form.
2 ln 2  FWHM
2 = FWHM
Lineshapes: disturbed by the presence of K1 and K2.
Decouple them if necessary:
Rachinger Correction for K1 and K2 separation:
Assume: (1) K1 and K2 identical lines profiles (not
necessarily symmetrical); (2) Ip of K2 = ½ Ip of K1.
  
 2  2
 tan 
  
   ( 2 )   (1 )
2d sin       2d cos 
  2   / d cos 
  2  (  / cos  )( 2 sin  /  )
  2  2(  /  ) tan 
I 0 (1 )  0
I1 (1 )  I1
I 2 (1 )  I 2
I 3 (1 )  I 3
Example: Separated by 3 unit
Ii: experimental intensity at point i
Ii(1): part of Ii due to due to K1
I 4 (1 )  I 4  I1 (1 ) / 2
I 5 (1 )  I 5  I 2 (1 ) / 2
…
I i (1 )  I i  I i 3 (1 ) / 2
…
General form
I i (1 )  I i  I i m (1 ) / 2
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)
Size Broadening:
Interference function
I  Atotal
2
 3 sin 2   N i ai 
 F 

2


sin



a
i
 i 1

I  Atotal
2
2
Define deviation vector
  1b1   2b 2   3b3
2
3


sin
 i N i ai 
2
 F 

2


sin


a
i i
 i 1

I (  )  I1 (  1 ) I 2 (  2 ) I 3 (  3 )
sin 2 1 N1a1 
sin 2  2 N 2a2 
I1 (  1 ) 
I 2 ( 2 ) 
2
sin 1a1 
sin 2  2a2 
…
sin2 1 N1a1 
I1 (  1 ) 
sin2 1a1 
1  0  I (0)  N12
Half width half maximum
(HWHM): particular '1
I
I ( '1 )  N12 2
'1 usually small 
sin 2 '1 N1a1  sin 2 '1 N1a1  N12
I1 (  ) 


2
'
'
2
sin 1a1 
(1a1 )
2
'
1
 ('1 N1a1 )  2 sin '1 N1a1 
Solve graphically
N12
k
'1
x  '1 N1a1
Define
Solution: x = 1.392
'1 N1a1  1.392
~ 1.392
1.392 0.443 0.443
'
 1 


N1a1 N1a1
L
Define k 
1

k 
 dk 
2 sin 

2 cos d


k sin 
 k0
k
k
 0 k
 d 
0.443
  
2 cos  L

2 cos 
dk
 '1
FWHM
 dk  2
0.89
L
2  cos 
'
1
0.89
  
2 L cos 
In X-ray, 2 is usually used, define B  2 
0.89
L
B cos 
K
L
B cos 
B in radians
Scherrer equation, K is Scherrer constant
If the  is used instead of 2, K should be divided by 2.
Strain broadening:
Uniform strain  lattice constant change  Bragg peaks
shift.
Assume strain =   d0 change to d0(1+ ).
Diffraction condition:
1
1
 (1   )
d 0 (1   ) d 0
dk
1
   G
dk  Gd Peak shift
d
d0
2 cos d
2 sin 

d

k

k 
In terms of 
k  G 
2 cos 



d
d  Gd 

G
 k

d
2 cos 
d
 2 sin 


  tan  Larger shift for the diffraction
d
2 cos  
peaks of higher order
Distribution of strains 
diffraction peaks broadening
Strain distribution 
2



relate to
'

 2   1
|G |
k
'1 is the HWHM of the diffraction G along xˆ
Instrument broadening:
Main Sources:
Combining all these broadening
by the convolution procedure 
asymmetric instrument function
convolution
The Convolution Procedure:
instrument function f(x) and the specimen function g(x)
the observed diffraction profile, h().
The convolution steps are
4
f(x)
3
* Flip f(x) f(-x)
2
* Shift f(-x) with respect to g(x) by 
1
0
f(-x)  f(-x)
-2 -1 0 1 2
* Multiply f and g
f(-x)g(x)
4
g(x)
3
* Integrate over x



f (   x ) g ( x )dx  h(  )
Assume f and g are the functions on the
right, the h() that we will get is
2
1
0
4
3
2
1
0
-2 -1 0 1 2
f(-x)
-2 -1 0 1 2
4
3
2
1
0
4
3
2
1
0
4
3
2
1
0
 = -2
-2
4
3
2
1
0
0
2
0
=0
16/3
-2
2
0
=2
0
-2
2
0
h(  )  


4
3
2
1
0
6
5
4
3
2
1
0
 = -1
7/6
-2
0
2
=1
31/6
-2
0
2
h()
-2
0
2
f (   x ) g ( x )dx  f ( x )  g ( x )

Convolution of Gaussians:
 ( x  x0 ) 2 

I ( x,  )  I (0) exp 
2



B  2 ln 2
B 
Two functions
f(): breadth Bf
g(): breadth Bg
 h() = f()*g(); breadth Bh
Bh2  B 2f  Bg2
http://www.tina-vision.net/docs/memos/2003-003.pdf
Convolution of Lorentzians:
I ( x,  ) 
1 (
I ( 0)
x  x0

B  2
)2
Two Lorentzian functions:
f(): breadth Bf
g(): breadth Bg
 h() = f()*g(); breadth Bh
Bh  B f  Bg
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)
Fourier transform the above three functions (DFT)
f ( k )   F ( n )e 2ink / l
n
g ( k )   G ( n )e
'
2in ' k / l
n'
h ( K )   H ( n )e
''
n ''
2in ' ' K / l
l: [1/length], the range in k of
the Fourier series is the interval
–l/2 to l/2.
h( K )  


f ( K  k ) g ( k )dk
The function f and g vanished outside of the k range
 Integration from - to  is replaced by –l/2 to l/2
h( K )  
l /2
l / 2
 F ( n )e
 2in ( K  k ) / l
h( K )   G ( n ) F ( n )e
n'
 2in ' k / l
dk
n'
n
'
 G ( n )e
'
 2inK / l

l /2
e
2i ( n  n ' ) k / l
l / 2
n
dk
Orthogonality condition

l /2
e
2i ( n  n ' ) k / l
l / 2
l /2
l if n  n
dk  
'
0
if
n
n

l /2

l /2
dk  l
l / 2
'
'
cos(
2

(
n

n
)
k
/
l
)

i
sin(
2

(
n

n
)k / l )dk
l / 2
l / 2
vanishes by symmetry
l /2
'
  cos(2 ( n  n )k / l )dk
l / 2
l
'
'
'

0
if
n
n

[sin(

(
n

n
))

sin(

(
n

n
))]
'
2 ( n  n )
e
2i ( n  n ' ) k / l
dk  
'
 h ( K )  l  G ( n ) F ( n )e
2inK / l
n
 h ( K )   H ( n )e
''
2in ' ' K / l
n ''
 lG ( n ) F ( n )  H ( n )
Convolution in k-space is equivalent to a multiplication
in real space (with variable n/l). The converse is also
true. Important result of the convolution theorem!
H (n)
Deconvolution: G ( n ) 
lF ( n )
{G(n)} is obtained from
g ( k )   G ( n )e 2ink / l
n
Data from
a perfect
specimen
Rachinger
Correction
(optional)
Data from
the actual
specimen
Rachinger
Correction
(optional)
f(k)
Stokes
-1
F.T.
Correction
F.T. G(n)=
H(n)/F(n)
h(k)
Corrected
data free
of
instrument
broadening
g(k)
“Perfect” specimen: chemical composition, shape,
density similar to the actual specimen ( specimen
roughness and transparency broadening are similar)
* E.g.: For polycrystalline alloy, the specimen is usually
obtained by annealing
f(k), g(k), and h(k): asymmetric  F.T. complex coeff.
1 H r ( n )  iH i ( n )
Gr ( n )  iGi ( n ) 
l Fr ( n )  iFi ( n )
1 H r ( n )  iH i ( n ) Fr ( n )  iFi ( n )
Gr ( n )  iGi ( n ) 
l Fr ( n )  iFi ( n ) Fr ( n )  iFi ( n )
1 H r ( n ) Fr ( n )  H i ( n ) Fi ( n )
Gr ( n ) 
l
Fr2 ( n )  Fi 2 ( n )
1 H i ( n ) Fr ( n )  H r ( n ) Fi ( n )
Gi ( n ) 
l
Fr2 ( n )  Fi 2 ( n )
g(k) is real and can be reconstructed as
g ( k )   [Gr ( n )  iGi ( n )]e 2ink / l
n
  2nk 
 2nk 
  [Gr ( n )  iGi ( n )]cos
  i sin

 l 
n
  l 
 2nk 
 2nk 
g ( k )   Gr ( n ) cos
  Gi ( n ) sin

 l 
 l 
n
real part
Simultaneous Strain and Size Broadening:
True sample diffraction profile:
strain broadening and size broadening effect
Usually, know one to get the other
Both unknown
Take advantage of the following facts:
Crystalline size broadening is independent of G
Strain broadening depends linearly on G
Williamson-Hall Method
Easiest way!
Requires an assumption of the shape of the peaks:
sin (Na) 1

I ( ) 
 exp(  2 )
2
sin (a) G
G
2
2
Gaussian function
characteristic of the
strain broadening
convolution
Kinematical crystal shape factor intensity
Assume a Gaussian strain distribution (quick falloff for
strain larger than the yield strain) ()
 2 
relate  2 to  2G
 ( )d  exp  2 d
  






k  G0 (1   )    G0  k  G0
G0 G
2

1

 ( )d   ()d  exp  2 2
 G 
G

 G  G
2

d


Approximate the size broadening part with a Gaussian
function
1.392
Na
(see page 9)
1.392
1
 

Na ln 2
 Na
characteristic width
 B  2 ln 2  2 
Good only when strain broadening >> size broadening
I ( )  N 2 exp( 
(Na ) 2  2

1
2
)  exp(  2 )
G
G
 2G  G
2
The convolution of two Gaussians

N2
2 

I ( ) 
exp 
2 
G
 (k ) 
k  2
sin 

1
1
2
2
2
2
(k ) 

G



G

N 2a 2
L2
2
1
G
d hkl
Plot k2 vs G2
(k)2
(HWHM)
1
L2
G2
2

Slope =
Approximate the size broadening and strain broadening
: Lorentzian functions
1.392
0.443
 B  2  2 
 
Na
L
Size:
Strain:
I ( 0)
I ( ) 
L 2
1 (
)
0.443
1
1
1
1

2

G
G 1  (  )2
1 ( 2 )

G 2
N2
1
1
I ( ) 

L 2 G
 2
1 (
)
1 ( )
0.443
G
G  G
2
The convolution of two Lorentzian
N2
1
I ( ) 
G 1  (  )2
k
0.443
 k 
G
L
N2
1
1
I ( ) 

L 2 G

1 (
)
1  ( )2
0.443
G
2
Plot k vs G
k
(HWHM)
Slope =  2
0.443
L
G
The following pages are from:
http://www.imprsam.mpg.de/nanoschool2004/lecturesI/Lamparter.pdf
Ball-milled Mo from P. Lamparter
L
G
 (FWHM)
2
2
Nanocrystalline CeO2 Powder
from P. Lamparter
Nb film, WH plot
from P. Lamparter
from P. Lamparter
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!
Ek 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
Warren and Averbach Method
Fourier Methods with Multiple Orders
I (Q)   A( L) exp( 2iQL)dQ
A( L)  AD ( L) A ( L)
size strain
How to interpret A(L)?
QG
from P. Lamparter
from P. Lamparter
from P. Lamparter
from P. Lamparter
from P. Lamparter
from P. Lamparter
Williamson-Hall Method
Easy to be done
Only width of peaks needed
Warren-Averbach Method
More mathematics
Precise peak shapes needed
Distributions of size and microstrain
Relation to other properties(dislocations)