R. Beck, group meeting 10/4/07
Structure and Form Factors
r
2
I (q ) = A(q )
X‐ray Intensity ‐ I (q) Scattering amplitude ‐ A(q)
A(q ) ≡ ∑ f n e
r r r
q = qi − q f
rr
− iq ⋅ x n
→∫
r −iqr⋅ xr r
ρ ( x ) e dx
4π
(2θ )
| q |=
sin
λ
2
fn ‐Atomic scattering factor at position n
ρ(x) ‐ Electron density
λ −X‐ray wave length
2θ − Scattering angle
General introduction
What is our electron density?
=
⊗
I (q) = ∫ F ( x) F ( x) ⊗ ∑ δ ( x − xn )e
*
rr
− iq ⋅ x
r
= F (q) ⋅ ∫ ∑ δ ( x − xn )e dx
14442r 4443
rr
− iq ⋅ x
2
S (q )
Form and structure factors
r
dx
The convolution theorem FT ( H ) FT (G ) = FT ( H ⊗ G )
r
r −iqr⋅ xr r
FT ( H ) = h( x ) = ∫ H (q )e dx
rr
r
2
− iq ⋅ x r
I (q ) = F (q) ⋅ ∫ ∑ δ ( x − xn )e dx
14442r 4443
S (q )
Form factor – information about the individual building block Structure factor – information about the lattice
Form and structure factor
Calculate a form factor
r
r − iqr⋅ xr
F (q ) = ∫∫∫ Δρ ( x )e dxdydz
V
Δρ = ρ in − ρ out
• Start with a simple model, and then make it more complex
• Use symmetry for simplification – reduced dimensionality Form factor
qz
Calculate a form factor – cylinder
L=2H
R
‐ “2d” object
F ( q⊥ , q z ) = ∫
V
ρin
ρout
r −iqr⋅ xr r
Δρ ( x )e dx
q⊥
Δρ ( x) = Δρ 0
H
2π
R
F (q⊥ , q z ) ∝ Δρ 0 ∫ e
dz ∫ ρdρ ∫ e −iq⊥ ρ cosθ dθ
−H
0
1
4243 0
1
44244
3
−iq z ⋅ z
J 0 ( q⊥ ρ )
J 0 ( qZ H )
sin x
J 0 ( x) =
x
Form factor
x
qz
Calculate a form factor – cylinder
L=2H
F (q⊥ , q z ) ∝ Δρ 0 J 0 (qZ H ) ∫ J 0 (q⊥ ρ )ρdρ
R
R
ρin
ρout
0
xJ1 ( x) = ∫ J 0 (x )xdx
q⊥
x
J1 (q⊥ R )
FF (q⊥ , q z ) = 2Δρ 0Vcy J 0 (qZ H )
q⊥ R
http://www.ncnr.nist.gov/resources/
Form factor
Calculate a structure factor
r
S (q ) = ∫
∑ δ ( x − x )e
n
rr
− iq ⋅ x
r
dx
• 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.
rr
r
− iq ⋅ x r
S (q ) = ∫ < ρ ( x) ρ (0) > e dx
structure factor
Calculate a Structure Factor – 1D lattice
r
S (q ) = ∫
∑ δ ( x − xn )e
rr
− iq ⋅ x
r
dx = ∫
r
2π
G1D (q ) = m
qˆ x , m = interger
d
Form factor
∑ δ ( x − nd )e
−iqx
r
dx =
2π ⎞
⎛
−iq x nd
e
q
m⎟
δ
=
−
⎜ x
∑
d ⎠
⎝
n = −∞
∞
Calculate a Structure Factor – 2D lattice
r
2π
2π
Ghex (q ) = (h − k )
qˆ x + (h + k )
qˆ y
3d
3d
4π
Ghex =
h 2 + k 2 + hk
3d
r
2π
2π
Gsqr (q ) = h
qˆ x + k
qˆ y
d
d
2π
Ghex =
h2 + k 2
d
Form factor
Sample geometry
χ
r r
r
r
G = Kin − Kout = G (sinΨ cosΦ, sinΨ sinΦ, cosΨ)
2π r
=G
d
Sample geometry
Sample orientation
Sample orientation
Powder average
r r
G = G (sinΨ cosΦ, sinΨ sinΦ, cosΨ)
I (q, Ψ, Φ) = ( A(q, Ψ, Φ)) = (FF(q, Ψ, Φ) ⋅ SF(q, Ψ, Φ))
2
π
2π
0
0
2
I (q) = ∫ dΨ∫ dΦ( A(q, Ψ, Φ))
2
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 Æ doesn’t depend on Φ.
r
G = G(sinΨ cosΦ, sinΨ sinΦ, cosΨ) ⇒ G(sinΨ, sinΨ, cosΨ)
qz = q cosΨ q⊥ = q sinΨ
Powder average
χ
Powder average
qz = q cosΨ q⊥ = q sinΨ
π
I (q) ∝ ∫ dΨ(F(q cosΦ, q sinΦ))
2
R
L=2H
J1(q⊥ R)
F(q⊥ , qz ) = 2Δρ0Vcy J0 (qZ H )
q⊥ R
ρin
ρout
0
qz = qx q⊥ = q 1− x x = [0...1]
2
1
((
I (q) ∝ ∫ dΨ F qx, q 1− x
0
Powder average
2
)
2
q⊥
Powder average is a very intense smearing factor
Will “hide” a lot of the single crystal information
Powder average
Other considerations
•Finite size effects
•Thermal fluctuations
•Density fluctuations
•Grain boundaries, texture and mixed orientations
Other considerations
Neurofilament‐ Nematic order of flexible chains
I(a.u.)
1000
100
1E-3
-1
q(A )
Nematic order
0.01
Neurofilament‐ Nematic order of flexible chains
Nematic order
Semi‐orientated data – domain like structure qz = q cosΨ q⊥ = q sinΨ SF(q⊥ , qz )
−(Θ−Ψ)2
P( A,σ , Θ − Ψ) = Ae
2σ 2
π
~
2
I semi−O (q, Ψ) ∝ ∫ dΘP( A,σ , Θ − Ψ) ⋅ (SF(q, Ψ))
0
Again convolution… ~
I semi −O (q, Ψ ) =
1
∑
i
Semi‐orientation
N
P ( A ,σ , Θ
∑
P
i
i
i
i
− Ψ ) ⋅ (SF (q, Ψ ) )
2
i
Semi‐orientated data – domain like structure σ
Changing the alignment distribution (σ)
Semi‐orientation
Semi‐orientated data – domain like structure σ
χ(deg)
Changing the alignment distribution (σ)
Semi‐orientation
Semi‐orientated data – domain like structure 0
intensity (a.u.)
10
-1
10
0.1
0.3
0.6
1
1.2
-2
10
4
10
σ
5
10
q(cm-1)
Changing the alignment distribution (σ)
Semi‐orientation
6
10