The Basics of Small-Angle X-Ray · PDF fileThe Basics of Small-Angle X-Ray Scattering...

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The Basics of Small-Angle X-Ray Scattering

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θ0E!

SE!

by Heimo Schnablegger


Imaging

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Fourier Optics

Object

Light Source

Detector

Picture

Scattering Reconstruction


Fourier Optics

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(1) Scattering Process:

(2) Reconstruction (Focusing):


Microscopy AND Scattering

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Ö  Mathematics replaces the lenses (phase problem) Ö  Small details are NOT visible Ö  Only the average over the whole sample is obtained Ö  The results are AMBIGOUS but representative Ö  NO preparation artefacts

Ö  Lenses produce the picture (no phase problem) Ö  Small details are visible Ö  Averages over large samples are hard to obtain Ö  The pictures are UNIQUE but NOT representative Ö  Preparation artefacts are inherent

Ø  In order to get the complete picture you need both !


The Scattering Process

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(1)

(2)

SE!

0E!

-  Electrons start oscillating -  Oscillating electrons emit radiation - Wavelength remains the same

Ö Every electron scatters: Constant Background Ö Contrast ~ Electron density

Ø  Fluorescence Ø  Inelastic (Compton) scattering

Be aware of:

+δ

−δ

−+ /δ


Sample Absorption

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(1) Check for the absorption at the available wavelength λ: dµeII ⋅−= )(

0λ µ(λ) = absorption coefficient from the

“International Tables For Crystallography” d = sample thickness

(2) The optimum sample thickness:

)(1

max)(

λ

λ

µd

edI

opt

dµscat

=

=⋅∝ ⋅−optd

(3) Usually for nm and organic materials: 1542.0)( =− αλ KCu

21−≈optd mm

d

0II


Interference Effect

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Incident direction

0E!

SE!

a

b

λ

,0=− ab

,2λ

=− abDetector:

light

dark

,λ ...

,23λ ... {

θ = Scattering Angle


The Form Factor

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)2sin(4θ

λπq =

θ = Scattering Angle λ = Wavelength P(q) = Form Factor q = Momentum Transfer

1.0

0.1

0.01

0.001

nsorientatioS qEqP 2)()(!

=

qθ

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SE!

D

q D = dimensionless number

(single-particle property)


Dilute Systems

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D

λ 12r23r

13rλ>>ijr

)()( qPNqI ⋅∝

Interparticle interference not visible inside the experimental window of angles.

The total intensity is the sum of all single-particle intensities !

For X-Ray Scattering diluted means: 5 - 15 g/L


Size Effects

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1.0

0.1

0.01

0.001

)(qI

q

)()( 2 qPVqI ∝

60 RI ∝

R = particle radius

= particle volume Vλ

Reciprocity Theorem: Big sizes scatter towards small angles, and vice versa.


Polydisperse Systems

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+ + +


Resolution

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If the scattering curve is sampled at increments Δq ≤ qmin starting at qmin,

the scattering data contain full information for all particles with maximum

dimension Dmax (Nyquist Theorem of Fourier Transforms).

qqq

D Δ≥= minmin

max ,π

and, vice versa, the particles can be resolved at increments Δr ≤ rmin starting at rmin, when the scattering data are available up to qmax :

minmax

, rrq

r ≤Δ=Δπ


Resolution vs. Bragg

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q

I(q)

qmin qmax qBragg

max22 Dq

dBragg

==π

minmax qD π

=

Do not mix up Bragg’s d-spacing with the resolution Dmax !


Contrast Effects

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(a) Molecules composed of different atoms (hydrocarbons in water) (b) Same molecules BUT different densities (crystalline/amorphous)

)()()( 2 qPqI ρΔ∝= particle-to-environment electron-density difference

ρΔ


Contrast Principle (Babinet)

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The resulting scattering intensity is the same for both systems, as long as the electron-density difference is the same.


12r

Lattice Particle

Dense Systems

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λ

S(q) x P(q) = I(q)

=

Interparticle interferences are not negligible ! λ≈ijr


Structure Factor

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qR0.01.53.04.56.07.59.010.5

lg P(qR)lg I(qR)

S(qR)

r/R0 1 2 3 4 5 6

p(r/R)i(r/R)

h(r/R)

Measurement (Reciprocal space)

Evaluation and interpretation (Real space)


Increasing Order

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q

)(qSDilute: Crystalline: Liquid:

q

)(qS

q

)(qS

1 1

0 0

No Order Near-Range Order Long-Range Order


Isotropic Samples

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θ

φ

),( φθI

)(θI

⇓ )lg(I

θ

I

φ


Samples With Orientation

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I

φ

I

φ

I

φ

Isotropic Partially Oriented Single-Crystalline

φ φ φ

Dispersions, Powders Gels, Liquid Crystals Fibers, Single Crystals

Orientation depends on the sample preparation !


Collimation Types

φDetector θ

Point Collimation: Line Collimation:

Sample

Pinholes Slit System

X-Ray source X-Ray Source

θ

Sample

Detector


Smearing Effect

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1+ +n 2+ 3+............... Point Sources = 1 Slit Source

Ö  n-times bigger intensities => Shorter experiments Ö  Desmearing necessary (programs provided!) Ö  Beneficial only for isotropic samples


Application Check List

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(1) What is my incentive for the study ? [shape, size, order, surface per volume] (2) How big are the structures ? [0.2 < D < 45 nm] (3) Is there sufficient contrast ? [different atoms OR densities] (4) Is the sample absorbing ? [small atomic numbers] (5) Is it isotropic [line collimation] or oriented [point collimation] ? (6) What is the particle density (concentration) of the sample ? [complicated interpretation]


Data Preparation

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0 2 4 6 8 1

10

I(q)

q [nm -1 ]

0 2 4 6 8 1

10 I(q

)

q [nm -1 ]

0 2 4 0.1

1

I(q)

q [nm -1 ] 0.0 0.5 1.0 1.5 2.0 2.5 10 -3 10 -2 10 -1 10 0

I(q)

q [nm -1 ]

Sample

Water

Averaging: 2D => 1D Binning and Subtraction

Subtraction Constant Background


Domains of Scattering Functions

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lg[ P

(q) ]

lg(q)

2−q

1−q

4−q

0−q

Guinier Porod

Radius of Gyration Surface per Volume Internal Structure of cross-section


Guinier Analysis

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22

0 3)ln()ln( qRII G

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

Straight-line fitting with

delivers the radius of gyration, RG. RG is model independent.

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

=

62153

222

22

LRR

RR

G

GSphere (R):

Cylinder (R, L):

The same equations approximately apply also to smeared I(q)-s. (Error in RG ~ +4%)


Porod Limit (G.Porod 1951)

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3

4

)(

)(

qKBqI

qKBqI

SS +=∞→

+=∞→

dqqIqQ

dqqIqQ

Sq

S

q

)(

)(

0

0

2

∫

∫∞

=

∞

=

=

=

The final slope K (or KS):

Not smeared:

Smeared:

The invariant Q (or QS):

S

s

QK

QK

VS 4== πSurface per Volume:

Background: B


The (Inverse) Scattering Problem

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The Pair-Correlation Function p(r)

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p(r) contains exactly the same information as I(q) does !

(Other names: Pair-Distance Distribution Function, PDDF, Patterson Function, ...)


Aggregates: Dimers

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Enzyme Aggregates

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Comparison of two different models for a core enzyme. Experiment (points) and the theoretical PDDFs (full and dashed line).


Sphere vs. Cube

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Comparison of a sphere with a cube of same volume, scattering function and PDDF.


Homogeneous Particles

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)(qP1.0 0.1

0.01 0.001

q

1.0 0.1

0.01 0.001

1.0 0.1

0.01 0.001

)(rp1.0

r0.0

1.0

0.0

1.0

0.0

L

T

L

T

q

q

r

r

(a)

(b)

(c)

1

986.8q

D =

1q

qL

22 qDL

CD

T

)()( qPqLqP c=

)()( 2

2

qPqDqP tL=

CD

LD

D

LD

CD

D


Inhomogeneous Particles

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1.0

r0.0

1.0

r0.0

Homogeneous

Inhomogeneous

0.00 =ρ

0.11 =ρ

0.12 −=ρ

oD

D

D

oD

iDiD

)(rp

)(rp


Pathways of Interpretation

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)(qP )(rp

)(qE )(rρ

FT

FT

FT-1

FT-1

2(...) (...)± ⊗ 1−⊗

)(qI)(/)( qSqI

)()( qSqP ⋅

dense system dilute system

Model / Reconstruction

Scattering Function of a

Modelling Interpretation


Conclusion

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Ø SAXS can be used to study a large variety of structural

problems in the nanometer regime

Ø Most of the systems are non-oriented (isotropic)

Ø Some basic knowledge is required for a correct use of

the method

Ø This includes Fourier transformation and other non-trivial

numerical routines

Ø Such software modules are part of the SAXSess system

Ø User training is available from O.Glatter in Graz


Bibliography

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O.Glatter and R.May, “Small-angle techniques,” in International Tables For Crystallography, Volume C, (Kluwer Academic Publishers, Dortrecht, 1999). L.A.Feigin and D.I.Svergun, “Structure Analysis by Small-Angle X-Ray and Neutron Scattering,” (Plenum, New York, 1987). A.Guinier and G.Fournier, “Small-Angle Scattering of X-Rays,” (Wiley, New York, 1955).

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