September 28, 2014: Cheiron School 2014 Small-Angle X-ray Scattering Basics & Applications Yoshiyuki Amemiya and Yuya Shinohara Graduate School of Frontier Sciences, The University of Tokyo
→ Introduction →  What’s SAXS ? → History → Theory →  Basic of X-ray scattering →  Structural Information obtained by SAXS → Experimental Methods →  X-ray Optics →  X-ray Detectors → Advanced SAXS →  Microbeam, GI-SAXS, USAXS, XPCS etc... 2
x-ray area detector Imaging Plate, CCD etc. Bragg’s law:  λ = 2d sin θ small angle  new 、large structure (1 – 100 nm) crystalline sample --> small-angle X-ray diffraction: SAXD solution scattering / inhomogeneous structure --> SAXS
Each electron in marerials vibrates and emits secondary spherical wave When there are two electrons, 。 。 interference between secondary waves from electrons when a distance between electrons is small --> scattering at large angle is intensified 。 when a distance between electrons is large --> scattering at small angle is intensified --> Incident X-ray
Krishnamurty (1930) Hendricks (1932) Mark (1932) Warren (1936) Observation of scattering from powders, fibers, and colloidal dispersions carbon black Molten silica - silica gel (below) (above ) courtesy to Dr. I.L.Torriani 6
Single crystals of Al- Cu hardened alloy
A.Guinier (1937,1939, 1943)
Interpretation of inhomogeneities in Al alloys “G-P zones”, introducing the concept of “particle scattering” and formalism necessary to solve the problem of a diluted system of particles. O. Kratky (1938,1942, 1962) G. Polod (1942, 1960, 1961) Description of dense systems of colloidal particles, micelles, and fibers. Macromolecules in solution. courtesy to Dr. I.L.Torriani Hemoglobin
Typical SAXS image Proteins in solution (Dr. Svergun, EMBL) Nanocomposite
→ Introduction →  What’s SAXS ? e History → Theory →  Basic of X-ray scattering 。 Structural Information obtained by SAXS → Experimental Methods → X-ray Optics →  X-ray Detectors → Advanced SAXS → Microbeam, GI-SAXS, USAXS, XPCS etc...
Scattering vector q = ks - ki ρ(r): electron density Complex number dr = dxdydz around  r momentum of a photon q = |q|= 4π sin θ /λ Amplitude of scattered X-ray Fourier transform of electron density Intensity of scattered X-ray :  I(q) = A(q)A*(q) =| A(q) |² (Extensive variable) A(q)A*(q) A(q) Intensity of scattered X-ray per volume:  I(q) : (Intensive variable) This doesn’t depend on sample volume, and is used to obtain the absolute intensity 10
Real Space Reciprocal Space = Patterson Function Wiener-Khinchin theorem
I 1 (q) =| A 1 (q) | 2 P2(r) I2(q) =| A2(q) |²
P1(r) I 1 (q) =| A 1 (q) | 2 P2(r) I2(q) =| A2(q) |²
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Long period changes. Thickness of crystal changes. p (r) r(r)
The average of the relative electron density fluctuations determines the magnitude of I(q).  The normalized autocorrelation function determines the shape of l(q). Not observable. 16
Invariant:   Ω = ſ ∞ I(q)dq ⇒ ſ ∞ I(q)4πq²dq (when isotropic) = (2π) 3 〈η 2 〉 ↑
= (2π)³<η²> ƒv γο(r)δ(0)dr = (2π)³<η²>γο(0) = (2π)³<η²>
Omitted. Parseval’s theorem Fourier Trans.
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→ Introduction →  What’s SAXS ? e History → Theory →  Basic of X-ray scattering → Structural Information obtained by SAXS → Experimental Methods → X-ray Optics →  X-ray Detectors → Advanced SAXS → Microbeam, GI-SAXS, USAXS, XPCS etc...
1.  Size and form of particulate system → Colloids, Globular proteins, etc... 2.  Correlation length of inhomogeneous structure Polymer chain, two-phase system etc. 3.  Lattice parameters of distorted crystals (para-crystal) 4.  Degree of crystallinity, crystal size, crystal distortion → Crystalline polymer
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ttering Intensity I
ρ(r) = Δρ r < R 0 else
I(q) isotropic scattering courtesy to Dr. I.L.Torriani     22 qR
Fixed particle Random orientation anisotropic scattering isotropic scattering courtesy to Dr. I.L.Torriani 23
Based on Gaussian distribution When the form has distribution, fringes are missed. f r²p(r)dr
ρ(r)dr -- Guinier Plot Guinier plot: log(l(q)) vs q² O. Glatter & O. Kratky ed., “Small Angle X-ray Scattering", Academic Press (1982).
I(q) = φVpar ticleS(q)F(q) Structure Factor   Form Factor ntra-particle structure inter-particle structure Separation of S(g) & F(g) Everlasting issue (especially, for non-crystalline sample) Proposed remedy: · GIFT (Generalized Inverse Fourier Trans.) by O. Glatter
Electron Density ρ(r) Autocorrelation Function Yo(r) Scattering Intensity Yo(r) = exp( ξ ) two phase system Autocorrelation Fourier trans. Yo(r) =  ξ exp(−ξ) polymer chain etc.
Phase 1:
Babinet’s principle Two complementary structures produce the same scattering.
Averaged square fluctuation of electron density
Invariant only on the volume fraction and the contrast between the two phases. 29
important for the characterization of porous materials
r dr Scattering intensity:   I(q) = 4π Yo(r (when isotropic) qr PDDF :  p(r) = r γ o(r) the set of distances joining the volume elements within a particle, including the case of non-uniform density distribution. Particle’s SHAPE and maximum DIMENSION pairs of volume elements i-j histogram of all intra-particle distances qr 31
courtesy to Dr. I.L.Torriani
courtesy to Dr. I.L.Torriani
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courtesy to Dr. I.L.Torriani

courtesy to Dr. I.L.Torriani
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Diffraction from Unit cell (Crystalline structure factor)
f (q) : Atomic Form Factor
S Maximum ~ N FWHM ~ 2π/N FWHM --> Size of crystal
Laue function:  |G(q)|² : sin²(πNq・r) sin²(πq・r) N = 100 → Large crystal → High diffraction intensity → Narrow FWHM   Soft matter (crystal size: small)   → Wide FWHM low S/N Crystal size --> Intensity & FWHM of diffraction
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Imperfection of 1st kind Thermal fluctuation etc. Imperfection of 2nd kind in the case of soft matter
Imperfection of 1st kind Imperfection of 2nd kind
Perfect lattice  ✦─✦─✦─✦─✦─✦─✦─✦─✦─✦ Imperfection of 1 st kind ⭐ Imperfection of 2nd kind ✦─✦─✦─ 。 Effect of imperfections on diffraction ?
convolution z( r ) with imperfection  —> calculate Z(q)
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q / (2dπ) Decrease of diffraction intensity Increase of FWHM R. Hosemann, S. N. Bagchi, Direct Analysis of Diffraction by Matter
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→ Introduction →  What’s SAXS ? → History s Theory → Basic of X-ray scattering Structural Information obtained by SAXS → Experimental Methods →  X-ray Optics →  X-ray Detectors → Advanced SAXS → Microbeam, GI-SAXS, USAXS, XPCS etc...
Emittance -- Product of size and divergence of beam beam divergence
[photons/(s·mrad²·mm²·0.1% rel.bandwidth)] SAXS needs to use a low divergence and small beam High brilliance beam is required !
PF BL-15A PF BL-10C
Data acquisition system
→ Introduction →  What’s SAXS ? → History → Theory →  Basic of X-ray scattering → Structural Information obtained by SAXS → Experimental Methods →  X-ray Optics →  X-ray Detectors → Advanced SAXS → Microbeam, GI-SAXS, USAXS, XPCS etc...

Microbeam X-ray
- Inhomogeneity of nano-structure - local time evolution of structure
Time-resolved
- time evolution of structure GI-SAXS - surface, interface, thin films XPCS - structural fluctuation - dynamics Combined measurement with DSC, viscoelasticity 2D measurement wide-q (USAXS-SAXS-WAXS) - hierarchical structure - anisotropic structure
Collab. with Kao Itd. X-ray Microbeam ( 5 μm x 5 μm) Relationship between macroscopic form and nano structure? Local observation with an X-ray microbeam
H. Ito et al., Textile Res. J. 54, 397–402 (1986). R. D. B. Fraser et al., Proc. Int. Wool Text. Res. Conf., Tokyo, II, 37, (1985) partially changed. Relationship between IF distribution and hair curlness?
Scattering pattern 1 D intensity profile Real space structure
--> Structural difference in cortex.
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ROC: Radius of Curvature
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Stained with Methylene blue d • Diameter of Intermediate filament (IF) is almost constant • Distance between IFs increases from outer to inner sides.
Nanocomposite
TEM image Rubber filled with spherical silica Scattering pattern also shows hysteresis.
Y. Shinohara et al., J. Appl. Cryst., 40, s397 (2007). ,

F(q): form factor S(q): structure factor I(q): intensity 250 ％ Dave = 282.9 nm
Y. Shinohara et al., J. Appl. Cryst.,

Sample Courtesy to Dr.Hagita & Prof. Arai Visualization of structure change of fillers during elongation of rubber by using SAXS and RMC 0→150% : elongation process non-uniformity increases along the elongation direction.
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Measurement of fluctuation of X-ray scattering intensity -->  Structural fluctuation in sample speckle pattern
T θ Time-resőlved SAXS with coherent X-ray Fluctuation of intensity [（ relaxation time in system Autocorrelation
- Volume fraction of nano-particles - Vulcanization (cross-linking) - Type of nano-particles - Temperature etc. Dynamics of Filler in Rubber
s A. G Sons, New York. & O. Glatter and O. Kratky ed. (1982) “Small Angle X-ray Scattering” Academic Press, London. &  L. A. Feigin and D. A. Svergun (1987) "Structure Analysis by Small Angle X-ray and Neutron Scattering” Plenum Press. & P. Lindner and Th. Zemb ed. (2002) “Neutron, X-ray and Light Scattering: Soft Condensed Matter”, Elsevier. .  Proceedings of SAS meeting (2003 & 2006). Published in J. Appl. Cryst.   R-J. Roe (2000) “Methods of X-ray and Neutron Scattering in Polymer Science”, Oxford University Press.