Post on 07-Aug-2020
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)