introduction to x-ray and neutron scattering · introduction to x-ray and neutron scattering...
TRANSCRIPT
UNESCO/IUPAC Postgraduate Course in Polymer Science
Introduction to X-ray and neutron scattering
Zhigunov Alexander
Institute of Macromolecular Chemistry ASCR, Heyrovsky sq. 2, Prague -162 06http://www.imc.cas.cz/unesco/index.html
Lecture:
Contents
Examples of structural studies
Small-angle x-ray and neutron scattering
Wide-angle X-ray scattering
Common principles
Examples of polymeric structures
Examples of polymer structuresChains and particles
Networks
Semicrystalline and organized structures
Polymer chain (in solution)
LENGTH SCALES
100 - 103Å
Polymer particles
LATEX MICELLE
102 - 103 101 - 102
101 - 103ξξξξ
CRYST.AM.CRYST.
CRYSTAL STRUCTURE: 10-2 - 100 Å
101 - 102
CUBIC, LAMELLAR, HEXAGONAL
LONG PERIOD: 101-102 Å
WavesWave characteristics are: amplitude, frequency, phase, speed
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:AbsorptionReflectionInterferenceRefractionDiffractionPolarization
Picture by Spigget
Diffraction theoryWhen 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.
Waves interactionsWhen 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
Scattering Experiment
X,N2θθθθ
Sr
0Sr
Scatteringintensity
Scatteringvector
COLL. SAMPLE DETECTORPLANE
)(qIr
)()/2( 0ssqrrr
−−−−==== λλλλππππ
θλπ sin)/4(=q
SOURCE
Wave should be coherent and collimated (parallel waves)
SAXS and WAXS
SAS: Small-Angle Scattering. Material, containing inhomogeneities from 10 to 1000 Ǻscatters radiation into agngles 0-2º
WAS: Wide-Angle Scattering. Single crystal. Material with inhomogeneities with size of inter-atomic distances shows diffraction spots at angles 2-90º
Small and Wide Angle Scattering
Scattering of materials
Homogenous material Material with inhomogeneities Single crystal Polycrystalline material
Primary beam only.No scattering.Vacuum is the only homogenous material.
Inhomogenities 10-1000 AScattering angle ~ 0-2 o
Small Angle Scattering
Interatomic distancesScattering angle ~2-90 o
Wide Angle Scattering
Angle depends on disnatces
Polycrystalline sample
Examples of WAXS patterns
AMORPHOUS SAMPLE
POLYCRYSTALLINE POWDER
SINGLE CRYSTAL
Crystal structure
Simple cubic
Body-centered cubic
Face-centered cubic
Unit cells Miller indices
a1
a2
a3
Crystal structure
5 10 15 20 25 30 35 400
200
400
600
800
1000
1200
1400
Inte
nsity
, a.u
.
2Θ, degree
p-TSA
WAXS ofp-Toluenesulfonic acid
α = β = γ = 90a = b = cCubic
α = β = γ ≠ 90a = b = c--- Rhombohedral division
γ = 120
α = β = 90a = b ≠ c--- Hexagonal division
Hexagonal
α = β = γ = 90a = b ≠ cTetragonal
α = β = γ = 90a ≠ b ≠ cOrthorombic
α = γ = 90
β ≠ 90a ≠ b ≠ cMonoclinic
α ≠ β ≠ γ ≠ 90a ≠ b ≠ cTriclinic
Angels between CrystalAxes (degrees)
Axial Translations(Unit-cell constants)
System
Bragg’s Law2 d sin ΘΘΘΘ = n λλλλ d = interplanar distance
q d = 2ππππ n n = integer
Lattice planes
Geometry of the Bragg reflection analogy:
The waves “reflected” by the two adjacent planes are in phase at scattering angle 2Θgiven 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.
The interference is constructive when the phase shift is a multiple of 2π
Examples of polymer structuresScattering from a single atom
Scattering from a group of atoms
Which technics to use?
b = scattering length (s. amplitude)
bX-ray = 0.282 x 10-12 cm x number of electrons
bN = tabulated bN(H) = - 0.374 x 10-12 cm
bN(D) = + 0.667 x 10-12 cm
I = b2
O2θ
B
Sr
0Sr
rr
Incident beam
Scattered))(/2( 0ssq
rrr−−−−==== λλλλππππ
Scattering vector
∑ −==k k
rqik
bqI 2|)exp(|2|AmplitudeTotal|)(rrr
I = I(qr) ⇒⇒⇒⇒ {Short distances ⇔⇔⇔⇔ high q (WAXS)
long distances ⇔⇔⇔⇔ small q (SAXS, SANS)}
WAXS on Polypropylene + 50 wt% Starch
∫
∫∞
∞
=
0
2
0
2
d)(
d)(
qqIq
qqIqx
c
cDEGREE OF CRYSTALLINITY
Θ=
cosβλK
LCRYSTALLITE SIZE
ββββ ≡≡≡≡ breadth of the
reflection
WAXS
WAXS onWAXS onPolymersPolymers
FF
DD BB
CC
EE
Size of crystallites
Degree of crystallinity
Distinguishing between ordered
and disordered structures
Lattice parameters
Identification of crystalline phases
AA
Crystal structure (single crystals, fibres)
Interpretation of SAS data
∫∞
=0
22
2 )sin()(
2)( dqq
qr
qrqI
rrp
π
[ ]∑∑= =
−⋅−=N
j
n
kkj rrqi
NqS
1 1
)(exp1
)(
SCA
TT
ER
ING
PA
TT
ER
N STRU
CTU
REI(q)
q
?
Scattering intensity: I(q) = P(q)S(q)
Distance distribution function:
Structure factor for N beads:
Form factor of sphere:2
33
)(
)cos()sin(3
3
4),(
⋅⋅−⋅∆=qR
qRRqqRRRqP ρπ
m
n
ie
V
Zr∑== 1ρ
Scattering length density:
where Z is the atomic numberre
= 2.81 x 10-13 cm, is the classical radius of the electronV
mis molecular volume
Radius of Gyration
∑ −=ji
jig rrN
R,
22
2 )(2
1
22
5
3RRg =
12
22 L
Rg =
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:
Sphere Thin rod Thin disc Cylinder
2
22 R
Rg =122
222 LR
Rg +=
Interpretation of SAS dataExperimental SAS curve
I exp(q)
Structure parameters
(e.g., Rg, V, S)A priori information
Structure model
I(experiment) =
I(model) ?
STRUCTURE ( ? )
Other techniques
YES
NO
SAXS vs SANS
??? !!!
Range of scattering vectors: q = 10-3 - 10-1 Å -1
Length scale: D = 101 - 103 Å
Scattering density:ρρρρ = b/V
Scattering contrast: ∆ρ∆ρ∆ρ∆ρ(r) = ρρρρ(r) - ρρρρ02
V)dVrqi)exp(r∆ρ()qI( ∫ −=rrrr
Scattering intensity:
ρρρρ0
ρρρρ
ρρρρ0
ρρρρ1ρρρρ1
ρρρρ2
ρρρρ0 ≠≠≠≠ ρρρρ2 ⇒⇒⇒⇒ I(q) = I 12(q)
ρρρρ0
ρρρρ1
ρρρρ2
ρρρρ0 = ρρρρ2 ⇒⇒⇒⇒ I(q) = I 1(q)
Contrast variation for Multicomponent Particles
Scattering from a polymer chain
q-1
I
I=I(0)~q-2
~q-1
Length of scattering vector q
qD«1 qD≅≅≅≅1 qD≥≥≥≥2
∼∼∼∼ exp(-q2Rg2/3)
qL p≥≥≥≥2
D: Size of chainL p: Persistence
length
0
Guinier Debye
Rod-like
„Magnification” increases
SAXS and SANS on polymers
SAXS and SANSSAXS and SANSon polymerson polymers
BB
EE
CC
DD
AA
Solid polymers:Characterization of
heterogeneities (pores, domains in block copolymers, fractal structures,
....)
SANS: Contrast variation. Studying of multicomponent
particles
Polymer particles:Shape, size
(distribution), mass, surface, internal
structure, degree of swelling
Semicrystallinepolymers: Degree of
crystallinity, long period, size of
crystallites.
Polymer chains: Radius of gyration, mass, persistence
length, cross-sectional parameters.
SANS Example
Thank you!and welcome to our laboratory.
UNESCO/IUPAC Postgraduate Course in Polymer Science
•Institute of Macromolecular Chemistry ASCR, Heyrovsky sq. 2, Prague -162 06•http://www.imc.cas.cz/unesco/index.html•[email protected]