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