Structure and Form Factors

R. Beck, group meeting     10/4/07

General introduction

X‐ray Intensity ‐ I (q) Scattering amplitude ‐ A(q)

fn ‐Atomic scattering factor at position nρ(x) ‐ Electron densityλ −X‐ray wave length2θ − Scattering angle

2)()( qAqI =r

2)2(sin4||

)()(

θλπ

ρ

=

−=

→≡ ∫∑ ⋅−⋅−

q

qqq

xdexefqA

fi

xqixqin

n

rrr

rr rrrr

Form and structure factors

What is our electron density?

=⊗

444 3444 21

r

r

r

rr

rr

)(

2

*

)()(

)()()()(

qS

xqin

xqin

xdexxqF

xdexxxFxFqI

∫ ∑∫ ∑

⋅−

⋅−

−⋅=

−⊗=

δ

δ

Form and structure factor

The convolution theorem  

444 3444 21

rr

r

rr

)(

2 )()()(

qS

xqin xdexxqFqI ∫ ∑ ⋅−−⋅= δ

∫ ⋅−==

⊗=

xdeqHxhHFT

GHFTGFTHFTxqi rrr rr

)()()(

)()()(

Form factor – information about the individual building block 

Structure factor – information about the lattice

Form factor

Calculate a form factor

outin

V

xqi dxdydzexqF

ρρρ

ρ

−=Δ

Δ= ∫∫∫ ⋅−rrrr )()(

• Start with a simple model, and then make it more complex

•Use symmetry for simplification – reduced dimensionality  

Form factor

Calculate a form factor – cylinder

‐ “2d” object ρin

R

L=2H

ρout

( )

xxxJ

deddzeqqF

x

xdexqqF

qJ

iqR

HqJ

H

H

ziqz

V

xqiz

Z

z

sin)(

),(

)

)(),(

0

2

0

cos

0

)(

0

0

00

=

Δ∝

Δ=(Δ

Δ=

⊥

⊥∫∫∫

∫

−

−

⋅−⊥

⋅−⊥

44 344 2143421

rr rr

ρ

π θρ θρρρ

ρρ

ρ

qz

⊥q

x

Form factor

Calculate a form factor – cylinder

ρin

R

L=2H

ρout( )

( )∫∫

=

Δ∝ ⊥⊥

xdxxJxxJ

dqJHqJqqFR

Zz

01

0 000

)(

)(),( ρρρρ

qz

⊥q

( )RqRqJHqJVqqFF Zcyz

⊥

⊥⊥ Δ= 1

00 )(2),( ρ

x

http://www.ncnr.nist.gov/resources/

structure factor

Calculate a structure factor

∫ ∑ ⋅−−= xdexxqS xqin

rr rr

)()( δ

•Use symmetry and known information about the lattice for simplification

•In the continuum limit the structure factor is  a Fourier‐transform of the two‐point correlation function.

∫ ⋅−><= xdexqS xqi rr rr

)0()()( ρρ

Form factor

Calculate a Structure Factor – 1D lattice

interger,ˆ2)(

2)()()(

1 ==

⎟⎠⎞

⎜⎝⎛ −==−=−= ∑∫ ∑∫ ∑

∞

−∞=

−−⋅−

mqd

mqG

md

qexdendxxdexxqS

xD

xn

ndiqiqxxqin

x

π

πδδδ

r

rrr rr

Form factor

Calculate a Structure Factor – 2D lattice

( ) ( )

22

22

2

ˆ2ˆ2)(

34

ˆ3

2ˆ3

2)(

khd

G

qd

kqd

hqG

hkkhd

G

qd

khqd

khqG

hex

yxsqr

hex

yxhex

+=

+=

++=

++−=

π

ππ

π

ππ

r

r

Sample geometry

Sample geometry

Gd

GKKG outin

r

rrrr

=

ΨΦΨΦΨ=−=

π2

)cos,sinsin,cos(sin

χ

Sample orientation

Sample orientation

Powder average

Powder average

( ) ( ) ( )( )22

00

22

),,()(

),,(),,(),,(,,

)cos,sinsin,cos(sin

ΦΨΦΨ=

ΦΨ⋅ΦΨ=ΦΨ=ΦΨ

ΨΦΨΦΨ=

∫∫ qAddqI

qSFqFFqAqI

GG

ππ

rr

As always, use symmetry for simplification – reduced dimensionality

For example, a cylinder FF has a symmetry line upon which rotation will not change the problem  doesn’t depend on Φ.

Ψ=Ψ=ΨΨΨ⇒ΨΦΨΦΨ=

⊥ sincos)cos,sin,(sin)cos,sinsin,cos(sin

qqqqGGG

z

r

Powder average

Powder average

( )

( )

( )( )221

0

2

2

0

100

1,)(

]1...0[1

)sin,cos()(

)(2),(

sincos

xqqxFdqI

xxqqqxq

qqFdqI

RqRqJHqJVqqF

qqqq

z

Zcyz

z

−Ψ∝

=−==

ΦΦΨ∝

Δ=

Ψ=Ψ=

∫

∫⊥

⊥

⊥⊥

⊥

π

ρ

ρin

R

L=2H

ρout

⊥q

χ

Powder average

Powder average is a very intense smearing factorWill “hide” a lot of the single crystal information

Other considerations

Other considerations

•Finite size effects•Thermal fluctuations•Density fluctuations•Grain boundaries, texture and mixed orientations

Nematic order

Neurofilament‐Nematic order of flexible chains

1E-3 0.01

100

1000

I(a.

u.)

q(A-1)

Nematic order

Neurofilament‐Nematic order of flexible chains

Semi‐orientation

Semi‐orientated data – domain like structure 

( )2

0

2)(

),(),,(),(~

),,(

),(sincos

2

2

Ψ⋅Ψ−ΘΘ∝Ψ

=Ψ−Θ

Ψ=Ψ=

∫−

Ψ−Θ−

⊥⊥

qSFAPdqI

AeAP

qqSFqqqq

Osemi

zz

σ

σ

π

σ

Again convolution… 

( )∑∑Ψ⋅Ψ−Θ=Ψ−

N

iiii

ii

Osemi qSFAPP

qI 2),(),,(1),(~ σ

Semi‐orientation

Semi‐orientated data – domain like structure 

σ

Changing the alignment distribution (σ)

Semi‐orientation

Semi‐orientated data – domain like structure 

σ

Changing the alignment distribution (σ)

χ(deg)

104 105 106

10-2

10-1

100

q(cm-1)

inte

nsity

(a.u

.)

0.10.30.611.2

Semi‐orientation

Semi‐orientated data – domain like structure 

σ

Changing the alignment distribution (σ)