Introduction to Patterson Function
and its Applications
“Transmission Electron Microscopy and Diffractometry of
Materials”, B. Fultz and J. Howe, Springer-Verlag Berlin 2002. Chapter 9)
The Patterson function:
explain diffraction phenomena involving displacement
of atoms off periodic positions (due to temperature or
atomic size)  diffuse scattering
Phase factor: exp(ik  r ) instead of exp(i 2k  r )
Fourier transform prefactor ignored: 1 / 2
Supplement: Definitions in diffraction
Fourier transform and inverse Fourier transform
 F (u )   f ( x )e 2iux dx

System 1 : 
System 4

2iux
 f ( x )   F (u )e du
 F (u )   f ( x )e iux dx



System 2 : 
System
5
1 
iux
f
(
x
)

F
(
u
)
e
du


2 

1 

iux
 F (u )  2  f ( x )e dx
System 3 : 
System 6
1 
iux
 f ( x) 
F
(
u
)
e
du


2 
Relationship among Fourier transform, reciprocal
lattice, and diffraction condition
System 1
Reciprocal lattice
 




*
*
*
b c
c a
a b
a     ;b     ;c    
a  (b  c )
b  (c  a )
c  (a  b )
* *
*
*
Ghkl  ha  kb  lc
Diffraction condition
  *
S   S  Ghkl
 
*
k   k  2Ghkl
System 2, 3
Reciprocal lattice
 
*
2b  c  *
 
 
2c  a  *
2a  b
a     ;b     ;c    
a  (b  c )
b  (c  a )
c  (a  b )
* *
*
*
Ghkl  ha  kb  lc
Diffraction condition
 
*
S   S  2Ghkl
  *
k   k  Ghkl
Patterson function
Atom centers at Points in Space:
Assuming: N scatterers (points), located at rj.
The total diffracted waves is
N
 ( k )   f r exp( ik  r j )
j
rj
The discrete distribution of scatterers  f(r)
N

 ( k )   f r exp( ik  r j )   f (r )e ikr d 3r
j
rj

f(r): zero over most of the space, but at atom centers
such as r  ri , f (ri ) is a Dirac delta function times a
constant f r
f (ri )  f r  (r  ri )
i
i
N
N
j 1
j 1
f (r )   f (ri )   f ri  (r  ri )
Property of the Dirac delta function:

y ( x)    ( x  x) y ( x )dx




f (r )e ikr d 3r  


N
  f r j exp( ik  r j )
j 1
N

j 1
f r j  (r  r j )e ikr d 3r
Definition of the Patterson function:
P (r )  


f * (r ' ) f (r  r ' )d 3r '
Slightly different from convolution
called “autoconvolution” (the function is not inverted).
Convolution:
Autocorrelation:




f * (r ' ) f (r  r ' )d 3r '  f * (r )  f (r )


f * (r ' ) f (r  r ' )d 3r '  f * (r )  f (r )
Fourier transform of the Patterson function =
the diffracted intensity in kinematical theorem.
I ( k )   ( k ) ( k )  
*






 
*
'
"
( f (r ) f (r ) e
*
'
f (r ) e
ik r '
ik ( r" r ' )
d r
3 '


"
f (r ) e
ik r"
d 3r"
d 3r" )d 3r '
"
'
r

r

r
Define r  r  r 
"
I ( k )  


 
'

 
 
 
( f * (r ' ) f (r  r ' )e ikr d 3r )d 3r '
( f * (r ' ) f (r  r ' )d 3r ' )e ikr d 3r
P(r )

I ( k )   P(r )e ikr d 3r

 I ( k )  FP (r )
Inverse transform P(r )  F 1 I ( k )
d 3r"  d 3r
 The Fourier transform of the scattering factor
distribution, f(r)  (k)

 ( k )   f (r )e ikr d 3r

*
I
(

k
)


( k ) ( k )
and
 I ( k )  ( Ff (r ))* Ff (r ) | Ff (r ) |2
i.e.
I ( k )  FP (r ) | Ff (r ) |2
1D example of Patterson function
Properties of Patterson function comparing to f(r):
1. Broader Peaks
2. Same periodicity
3. higher symmetry
Case I: Perfect Crystals
much easier to handle f(r); the convolution of the atomic
form factor of one atom with a sum of delta functions
f (r )  f at (r )    (r  R n )  
Rn


f at (r  r ' )  * (r '  R n )d 3r '
Rn
  f at (r  R n )
Rn
P0 ( x )  f * ( x )  f (  x )
f * ( x )  f at* ( x ) 
N /2
'

(
x

n
a)

n'  N / 2
f (  x )  f at (  x ) 
N /2
"

(
n
 a  x)
n"   N / 2
N /2
N /2
 *



'
"
P0 ( x )   f at ( x )    ( x  n a )    f at (  x )    ( n a  x ) 
n'  N / 2
n"   N / 2

 

N /2
 N /2

'
"
P0 ( x )   f ( x )  f at (  x )      ( x  n a )    ( n a  x ) 
n"   N / 2
 n'  N / 2

*
at
Shape function RN(x): extended  to 
 1 if -Na/ 2  x  Na/ 2
RN ( x )  
elsewhere
 0
N /2

N /2
n'  N / 2
n '  
n"   N / 2
'
'

(
x

n
a
)

R
(
x
)

(
x

n
a) 


N
"

(
n
 a  x)
 ( x  n "a )   ( n "a  x )
P0 ( x )   f at* ( x )  f at (  x )  




'
'
 RN ( x )   ( x  n a )  RN ( x )   ( x  n a ) 
n '  
n '  



'

(
x

n
a) 

n '  




n '  
n  
'
'

(
x

n
a
)

?

N

  ( x  na )





N=9
-3a -a 0 2a 4a
-4a -2a
a
3a
shift 8a
-3a -a 0 2a 4a
-4a -2a
a
3a
-9a -7a -5a -3a -a 0 2a 4a 6a 8a
-8a -6a -4a -2a
a
3a 5a 7a 9a
a triangle
of twice
the total
width
RN ( x )  RN ( x )  T2 N ( x )



n '  
n '  
n  
RN ( x )   ( x  n 'a )  RN ( x )   ( x  n 'a )  NT2 N ( x )   ( x  na )

P0 ( x )  N  f at* ( x )  f at (  x )   T2 N ( x )   ( x  na )
n  
F(P0(x))  I(k)
Convolution theorem: a*b  F(a)F(b); ab  F(a)*F(b)
F ( f ( x )  f at (  x ))  f at ( k )
*
at
2
F ( f at* ( x )  f at (  x ))  F ( f at* ( x )) F ( f at (  x )) 
f (  k ) f at (  k )  f at ( k )
*
at
2

I ( k )  N f at ( k ) F (T2 N ( x ))  F (   ( x  na ))
2
n  

F (   ( x  na ))   e ikx

n  



e ikx



  ( x  na )dx 
n  


n  


  ( x  na )dx
n  
e ikx ( x  na )dx 

inka
e

n  
If ka  2, the sum will be zero. The sum will have a
nonzero value when ka = 2 and each term is 1.

F (   ( x  na )) 
n  
G  2n / a

inka
e
 N   ( k  G )

n  
G
1 D reciprocal lattice
N: number of
terms in the sum
F [T2 N ( x )]
2

sin
dkNa 
2
| S ( dk ) | 
sin 2 dka 
F.T.
2n
I ( k )  FP0 ( x )  N f at ( k ) S ( k )    ( k 
)
a
n  
2
2
2

2n
I ( k )  FP0 ( x )  N f at ( k ) S ( k )    ( k 
)
a
n  
2
2
2

A familiar result in a new form.
  -function  center of Bragg peaks
 Peaks broadened by convolution with the
shape factor intensity
 Bragg peak of Large k are attenuated by the
atomic form factor intensity
Patterson Functions for homogeneous disorder
and atomic displacement diffuse scattering
Deviation from periodicity:
Deviation function
f (r )  f (r )  f (r )
Perfect periodic function: provide sharp Bragg peaks
P( r )  f * (r )  f ( r )  [ f * (r )  f * (r )]  [ f ( r )  f ( r )]
 f * (r ) * f ( r )  f * (r )  f ( r )  f * (r )  f ( r )
Look at the second term
 f * (r )  f ( r )
 *

f (r )  f ( r )   f at    (r  R n )  f ( r )
Rn


*
  (r  R n )  f (r )   f (R n )  0
Rn
Rn
Mean value for
deviation is zero
The same argument for the third term  0
P( r )  f * (r )  f ( r )  f * (r )  f ( r )  f * (r )  f ( r )
 Pavge (r )
 Pavge (r )  Pdevs (r )
 Pdevs (r )
1st term: Patterson function from the average crystal,
2nd term: Patterson function from the deviation crystal.
I ( k )  F Pavge (r )  Pdevs (r )  F [ Pavge (r )]  F [ Pdevs (r )]
2
F [ Pavge (r )]  F f (r ) ; F [ Pdevs (r )]  F [ f (r )]
I ( k )  F f (r )
2
 F [ f (r )]
Sharp diffraction peaks
from the average crystal
2
2
often a broad
diffuse intensity
 Uncorrelated Displacements:
Types of displacement: (1) atomic size differences in an
alloy  static displacement, (2) thermal vibrations 
dynamic displacement
Consider a simple type
of displacement
disorder: each atom has
a small, random shift, ,
off its site of a periodic lattice
Consider the overlap of
the atom center
distribution with itself
after a shift of x  na  
 a
12
0
No correlation in   probability of overlap of two atom
centers is the same for all na   shift except n = 0
When n = 0, perfect overlap at  = 0, at   0: no overlap
+
P( x )  Pavge ( x )  Pdevs ( x )
Pdevs ( x )  Pdevs1 ( x )  Pdevs 2 ( x )
=
=
area of Pdevs1 ( x )  area of Pdevs 2 ( x )
 The same number of atomatom overlap
+
constant
deviation
F[Pdevs1(x)]
increasingly
dominates over
F[Pdevs2(x)] at
larger k.

I ( k  0)   Pdevs ( x )e


i 0 x

dx   Pdevs ( x )dx

  [ Pdevs1 ( x )  Pdevs 2 ( x )]dx  0

The diffuse scattering increases with k !
Correlated Displacements: Atomic size effects
a big atoms locate
Overall effect: causes an asymmetry in the shape of
the Bragg peaks.
Diffuse Scattering from chemical disorder:
Concentration of A-atoms: cA;
Concentration of B-atoms: cB.
Assume cA > cB 
f ( x)  f A  f ( x)  f B
f  c A f A  cB f B
*
'
'

f
(
x
)

f
(
x

x
)
When the product
is summed over x.
# positive > # negative
H positive < H ones negative
Pdevs(x  0) = 0; Pdevs(0)  0
fA  f
Let’s calculate Pdevs(0): cAN peaks of
cBN peaks of
Pdevs ( x )  ( c A N f A  f
2
f  fB
2
2
2
 cB N f  f B ) (0)
 ( c A N f A  c A f A  cB f B  cB N c A f A  cB f B  f B ) (0)
2
2
 ( c A N f A (1  c A )  cB f B  cB N c A f A  (1  cB ) f B ) (0)
2
cB
2
 ( c A Nc f A  f B  cB Nc f  f B ) (0)
2
B
2
 ( c A Nc  cB Nc ) f A  f
2
B
2
A
2
A A
2
B
2
cA
 ( 0)  c A c B N f A  f B  ( 0)
2
P( x )  Pavge ( x )  Pdevs ( x )
I total ( k )  F [ Pavge ( x )]  F [ Pdevs ( x )]
F [ Pdevs ( x )]  F [c AcB N | f A  f B |2  (0)]
I devs ( k )  c AcB N | f A  f B |2
Just like the case
F [ Pavge ( x )]  N f ( k )   ( k  2n / a )
of perfect crystal
n
2
Total diffracted intensity
I total ( k )  c AcB N f A  f B  N f ( k )
2
2
  (k  2n / a )
n
 as k   f  as k 
The diffuse scattering part is: the difference
between the total intensity from all atoms and the
2
intensity N f in the Bragg peaks

 N c
 N c

|  N | c f
|   Nc | f
I devs  N c A | f A |2  cB | f B |2  N | f |2
2
|
f
|
 cB | f B
A
A
2
|
f
|
 cB | f B
A
A
c A  c  c A (1  c A )  c AcB
2
A

2
A
2
2
A
2

c
f
|
A
B B
2
*
*
|

Nc
c
f
f

Nc
c
f
A
A B A B
A B A fB
 NcB2 | f B |2
 N c AcB | f A |2 c AcB f A* f B  c AcB f A f B*  c AcB | f B |2
 Nc AcB | f A  f B |2
