Introduction to X-ray and neutron scattering

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1 UNESCO/IUPAC Postgraduate Course in Polymer Science Lecture: Introduction to X-ray and neutron scattering Zhigunov Alexander Institute of Macromolecular Chemistry ASCR, Heyrovsky sq., Prague

2 Contents Examples of polymeric structures Common principles Wide-angle X-ray scattering Small-angle x-ray and neutron scattering Examples of structural studies

3 Examples of polymer structures Polymer chain (in solution) LENGTH SCALES Å Chains and particles Polymer particles LATEX MICELLE Networks ξ CRYST. AM. CRYST. Semicrystalline and organized structures LONG PERIOD: Å CRYSTAL STRUCTURE: Å CUBIC, LAMELLAR, HEXAGONAL

Wave characteristics are: amplitude, frequency,

$Interference Refraction Diffraction Polarization$

4 Wave characteristics are: amplitude, frequency, phase, speed Waves λ - Wavelength of a sinusoidal wave is the spatial period of the wave (the distance over which the wave's shape repeats). When passing through media: Absorption Reflection Interference Refraction Diffraction Polarization Picture by Spigget

5 Diffraction theory When a wave passes through an opening in a barrier, the wave spreads out, or diffracts. When two waves occupy the same location, they interfere. When this interference results in a larger wave, we call it constructive interference. When the size of the wave is reduced, it is called destructive interference.

6 Waves interactions When x-rays are incident on an atom, they make the electronic cloud move as does any electromagnetic wave. The movement of these charges re-radiates waves with the same frequency (blurred slightly due to a variety of effects); this phenomenon is known as Rayleigh scattering (or elastic scattering). These re-emitted wave fields interfere with each other either constructively or destructively (overlapping waves either add together to produce stronger peaks or subtract from each other to some degree), producing a diffraction pattern on a detector or film. The resulting wave interference pattern is the basis of diffraction analysis. Picture by Christophe Dang Ngoc Chan

7 Scattering Experiment Scattering vector r r r q = ( π / λ )( s s0 ) q = (4π / λ) sin θ S r Scattering intensity I(q) r X,N SOURCE COLL. θ SAMPLE S r 0 DETECTOR PLANE Wave should be coherent and collimated (parallel waves)

8 SAXS and WAXS WAS: Wide-Angle Scattering. Single crystal. Material with inhomogeneities with size of inter-atomic distances shows diffraction spots at angles -90º SAS: Small-Angle Scattering. Material, containing inhomogeneities from 10 to 1000 Ǻ scatters radiation into agngles 0- º

9 Small and Wide Angle Scattering

10 Scattering of materials Homogenous material Material with inhomogeneities Single crystal Polycrystalline material Primary beam only. No scattering. Vacuum is the only homogenous material. Inhomogenities A Scattering angle ~ 0- o Small Angle Scattering Interatomic distances Scattering angle ~-90 o Wide Angle Scattering Angle depends on disnatces

11 Polycrystalline sample

12 Examples of WAXS patterns AMORPHOUS SAMPLE SINGLE CRYSTAL POLYCRYSTALLINE POWDER

13 Crystal structure Unit cells Miller indices a3 Simple cubic a a1 Body-centered cubic Face-centered cubic

14 Crystal structure System Triclinic Monoclinic Orthorombic Tetragonal Hexagonal --- Hexagonal division --- Rhombohedral division Cubic Axial Translations (Unit-cell constants) a b c a b c a b c a = b c a = b c a = b = c a = b = c Angels between Crystal Axes (degrees) α β γ 90 β 90 α = γ = 90 α = β = γ = 90 α = β = γ = 90 α = β = 90 γ = 10 α = β = γ 90 α = β = γ = p-tsa 1000 WAXS of p-toluenesulfonic acid Intensity, a.u Θ, degree

15 Bragg s Law d sin Θ = n λ d = interplanar distance q d = π n n = integer The interference is constructive when the phase shift is a multiple of π Geometry of the Bragg reflection analogy: Lattice planes The waves reflected by the two adjacent planes are in phase at scattering angle Θ given by the Bragg equation. For all values of Θ that do not satisfy this equation the diffracted rays are out of phase with each other and no reflection is observed.

16 Examples of polymer structures Scattering from a single atom b = scattering length (s. amplitude) b X-ray = 0.8 x 10-1 cm x number of electrons b N = tabulated b N (H) = x 10-1 cm b N (D) = x 10-1 cm r r r q = ( π / λ)( s s0) Scattering vector B r Incident beam θ O Scattering from a group of atoms Scattered S r S r 0 Which technics to use? I = b r I q = TotalAmplitude rr ( ) = b exp( iqr ) k k k I = I(qr) {Short distances high q (WAXS) long distances small q (SAXS, SANS)}

17 WAXS on Polypropylene + 50 wt% Starch DEGREE OF CRYSTALLINITY x c = 0 0 q q I c ( q)dq I( q)dq CRYSTALLITE SIZE L = Kλ β cosθ β breadth of the reflection

18 WAXS Degree of crystallinity F Size of crystallites E A Distinguishing between ordered and disordered structures WAXS on Polymers Lattice parameters D C B Identification of crystalline phases Crystal structure (single crystals, fibres)

19 SCATTERING PATTERN Interpretation of SAS data I(q) q? STRUCTURE Scattering intensity: I(q) = P(q)S(q) Form factor of sphere: 4 3 sin( qr ) q R cos( qr ) P ( q, R ) = πr ρ ( qr ) Structure factor for N beads: 1 S( q) = N N n j= 1 k = 1 exp [ iq ( r j r k )] Distance distribution function: 0 r sin( qr ) p( r ) = I ( q ) q dq π qr Scattering length density: ρ n Zr i= = 1 V m e where Z is the atomic number r e =.81 x cm, is the classical radius of the electron V m is molecular volume

20 Radius of Gyration Radius of gyration is the name of several related measures of the size of an object, a surface, or an ensemble of points. It is calculated as the root mean square distance of the object s parts from its center of gravity. In polymer physics, the radius of gyration is proportional to the root mean square distance between the monomers: R g = 1 N ( ri i, j r j ) Rg = Sphere Thin rod Thin disc Cylinder 3 R 5 L R g = 1 R R g = R R g = + L 1

21 Interpretation of SAS data Experimental SAS curve I exp (q) Structure parameters (e.g., Rg, V, S) A priori information Other techniques Structure model NO I(experiment) = I(model)? YES STRUCTURE (? )

SAXS vs SANS Range of scattering vectors: q = 10-3 - 10-1 Å -1 Length scale: D = 10

Scattering intensity: I( q) = ρ(r)exp( iqr)dv V Contrast variation for

22 SAXS vs SANS Range of scattering vectors: q = Å -1 Length scale: D = Å Scattering density: ρ = b/v Scattering contrast: ρ(r) = ρ(r) - ρ 0 r r rr Scattering intensity: I( q) = ρ(r)exp( iqr)dv V Contrast variation for Multicomponent Particles???!!! ρ 0 ρ I(q) = I 1 (q) ρ 0 = ρ I(q) = I 1 (q) ρ ρ 0 ρ 1 1 ρ ρ 0 ρ 1 ρ ρ 0

23 Scattering from a polymer chain I I=I(0) Guinier exp(-q R g /3) Debye ~q - D: Size of chain L p : Persistence length Rod-like ~q -1 0 Length of scattering vector q q -1 qd«1 qd 1 qd ql p Magnification increases

$SAXS and SANS on polymers SANS: Contrast variation. Studying of multicomponent particles B Solid polymers: Characterization of heterogeneities (pores, domains in block copolymers, fractal structures,.$

24 SAXS and SANS on polymers SANS: Contrast variation. Studying of multicomponent particles B Solid polymers: Characterization of heterogeneities (pores, domains in block copolymers, fractal structures,...) A SAXS and SANS on polymers C Polymer particles: Shape, size (distribution), mass, surface, internal structure, degree of swelling Semicrystalline polymers: Degree of crystallinity, long period, size of crystallites. E D Polymer chains: Radius of gyration, mass, persistence length, cross-sectional parameters.

25 SANS Example

26 UNESCO/IUPAC Postgraduate Course in Polymer Science Thank you! and welcome to our laboratory. Institute of Macromolecular Chemistry ASCR, Heyrovsky sq., Prague

X-ray, Neutron and e-beam scattering

X-ray, Neutron and e-beam scattering Introduction Why scattering? Diffraction basics Neutrons and x-rays Techniques Direct and reciprocal space Single crystals Powders CaFe 2 As 2 an example What is the