Introduction to Patterson Functionand its Applications
“Transmission Electron Microscopy and Diffractometry ofMaterials”, 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: instead of Fourier transform prefactor ignored:
)exp( rk i )2exp( rk i2/1
Supplement: Definitions in diffraction
Fourier transform and inverse Fourier transform
System 1
dueuFxf
dxexfuF
iux
iux
2
2
)()(
)()(:
System 2
dueuFxf
dxexfuF
iux
iux
)(2
1)(
)()(:
System 3
dueuFxf
dxexfuF
iux
iux
)(2
1)(
)(2
1)(
:
System 4
System 5
System 6
Relationship among Fourier transform, reciprocallattice, and diffraction condition
System 1
Reciprocal lattice
)(;
)(;
)(***
bac
bac
acb
acb
cba
cba
**** clbkahGhkl
*
*
2 hkl
hkl
Gkk
GSS
Diffraction condition
System 2, 3
Reciprocal lattice
)(
2;
)(
2;
)(
2 ***
bac
bac
acb
acb
cba
cba
**** clbkahGhkl
*
*2
hkl
hkl
Gkk
GSS
Diffraction condition
Patterson functionAtom centers at Points in Space: Assuming: N scatterers (points), located at rj. The total diffracted waves is
N
j
j
jif
rr rkk )exp()(
The discrete distribution of scatterers f(r)
rrrkk rk
rr
3)()exp()( defif iN
j
j
j
N
jj
iN
jj
i
if
defdef
j
j
1
3
1
3
)exp(
)()(
rk
rrrrr
r
rkr
rk
N
ji
N
ji i
fff11
)()()( rrrr r
f(r): zero over most of the space, but at atom centers such as , is a Dirac delta function times a constant
irr )( if r
ifr )()( ii i
ff rrr r
Property of the Dirac delta function:
dxxyxxxy )()()(
'3''* )()()( rrrrr dffP
Slightly different from convolutioncalled “autoconvolution” (the function is not inverted).
)()()()( *'3''* rrrrrr ffdff
)()()()( *'3''* rrrrrr ffdff
Convolution:
Autocorrelation:
Definition of the Patterson function:
rrrrr rk 3'3''* ))()(( dedff i
Fourier transform of the Patterson function =the diffracted intensity in kinematical theorem.
"3"'3'** "'
)()()()()( rrrrkkk rkrk defdefI ii
'3"3)("'* ))()(('"
rrrr rrk ddeff i
Define '" rrr '" rrr
'33''* ))()(()( rrrrrk rk ddeffI i rr 3"3 dd
)(rP
rrk rk 3)()( dePI i )()( rk FPI
Inverse transform )()( 1 kr IFP
The Fourier transform of the scattering factor distribution, f(r) (k)
rrk rk 3)()( def i
)()()( * kkk Iand
2* |)(|)())(()( rrrk FfFfFfI
2|)(|)()( rrk FfFPI i.e.
1D example of Patterson function
Properties of Patterson function comparing to f(r):
1. Broader Peaks2. Same periodicity3. higher symmetry
Case I: Perfect Crystalsmuch easier to handle f(r); the convolution of the atomicform factor of one atom with a sum of delta functions
n
natffR
Rrrr )()()(
)()()( *0 xfxfxP
'3'*' )()( rRrrr
R
dfn
nat
n
natfR
Rr )(
2/
2/
'**
'
)()()(N
Nnat anxxfxf
2/
2/
"
"
)()()(N
Nnat xanxfxf
2/
2/
"2/
2/
'*0
"'
)()()()()(N
Nnat
N
Nnat xanxfanxxfxP
2/
2/
"2/
2/
'*0
"'
)()()()()(N
Nn
N
Nnatat xananxxfxfxP
Shape function RN(x): extended to
elsewhere 0
22 if 1 )(
Na/x-Na/xRN
2/
2/
"'2/
2/
'
"''
)()()()(N
NnnN
N
Nn
xananxxRanx
)()( "" xananx
''
)()()()(
)()()(
''
*0
nN
nN
atat
anxxRanxxR
xfxfxP
nnn
naxNanxanx )(?)()( '''
''
a2a
3a4a0-a
-2a-3a
-4a
N = 9
a 2a 3a 4a0-a-2a-3a
-4a
a2a
3a4a0-a
-2a-3a
-4a 5a6a
7a8a
9a-6a-7a
-8a-5a-9a
shift 8a
a triangleof twicethe totalwidth
)()()( 2 xTxRxR NNN
n
Nn
Nn
N naxxNTanxxRanxxR )()()()()()( 2''
''
n
Natat naxxTxfxfNxP )()()()()( 2*
0
F(P0(x)) I(k)
Convolution theorem: a*b F(a)F(b); ab F(a)*F(b)
2* )())()(( k atatat fxfxfF
2*
**
)()()(
))(())(())()((
kkk
atatat
atatatat
fff
xfFxfFxfxfF
))(())(()()( 2
2
n
Nat naxFxTFfNI kk
n
kain
n
kxi
n
kxi edxnaxedxnaxe )()(
dxnaxenaxFn
kxi
n
)())((
N: number of terms in the sum
Gn
kain
n
GkNenaxF )())((
If ka 2, the sum will be zero. The sum will have a nonzero value when ka = 2 and each term is 1.
anG /2 1 D reciprocal lattice
-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0
0
1
2
3
4
5
6
7
)]([ 2 xTF N
kad
kNadkdS
2
22
sin
sin|)(|
-10 -5 0 5 10
0
2
4
6
8
10
F.T.
n
at a
nkkSkfNxFPkI )
2()()()()(
2220
n
at a
nkkSkfNxFPkI )
2()()()()(
2220
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 disorderand atomic displacement diffuse scattering
)()()( rrr fff
Perfect periodic function: provide sharp Bragg peaks
Deviation function
)]()([)]()([)()()( *** rrrrrr ffffffrP
)()()()()(*)( *** rrrrrr ffffff
)()(* rr ffLook at the second term
)()()()( ** rRrrrR
ffff
n
nat
0)()()( nn
nn ffRR
RrRr Mean value fordeviation is zero
Deviation from periodicity:
The same argument for the third term 0)()()()()()()( *** rrrrrr ffffffrP
1st term: Patterson function from the average crystal,2nd term: Patterson function from the deviation crystal.
)(ravgeP )(rdevsP)()( rr devsavge PP
)]([)]([)()()( rrrrk devsavgedevsavge PFPFPPFI 22
)]([)]([;)()]([ rrrr fFPFfFPF devsavge
22)]([)()( rrk fFfFI
Sharp diffraction peaksfrom the average crystal
often a broaddiffuse 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 nax
a
0
12
No correlation in probability of overlap of two atom centers is the same for all shift except n = 0na
When n = 0, perfect overlap at = 0, at 0: no overlap
+ =
)()()( xPxPxP devsavge )()()( 21 xPxPxP devsdevsdevs
=
+)( ofarea )( ofarea 21 xPxP devsdevs The same number of atom-atom overlap
constantdeviation
dxxPdxexPkI devsxi
devs )()()0( 0
0)]()([ 21
dxxPxP devsdevs
The diffuse scattering increases with k !
F[Pdevs1(x)]increasingly dominates over F[Pdevs2(x)] atlarger 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
BA fxffxf )()(
BBAA fcfcf
When the productis summed over x.
)()( ''* xxfxf
# positive > # negativeH positive < H ones negative
Pdevs(x 0) = 0; Pdevs(0) 0
Let’s calculate Pdevs(0): cAN peaks of cBN peaks of
2ff A
2
Bff
)0()()(22BBAAdevs ffNcffNcxP
)0()(22 BBBAABBBAAAA ffcfcNcfcfcfNc
)0())1()1((22 BBAABBBAAA fcfcNcfccfNc
cAcB
)0()(2222 BAABBABA ffNccffNcc
)0()0()(2222 BABABAABBA ffNccffNccNcc
)()()( xPxPxP devsavge
)]0(||[)]([ 2 BABAdevs ffNccFxPF 2||)( BABAdevs ffNcckI
Total diffracted intensity
)]([)]([)( xPFxPFkI devsavgetotal
n
avge ankkfNxPF )/2()()]([2
Just like the caseof perfect crystal
n
BABAtotal ankkfNffNcckI )/2()()(22
as as kfk
222 |||||| fNfcfcNI BBAAdevs
222 |||||| BBAABBAA fcfcNfcfcN
BABABABAAABBAA ffcNcffcNcfNcfcfcN **2222 |||||| 22 || BB fNc
2**2 |||| BBABABABABAABA fccffccffccfccN 2|| BABA ffcNc
BAAAAA cccccc )1(2
The diffuse scattering part is: the differencebetween the total intensity from all atoms and theintensity in the Bragg peaks
2fN