X-ray scattering by an
arbitrary structure
•
•
•
•
Coherence
Kinematic and static approximation
Interferences
Calculation of scattered intensity
Mutual correlation function
The mutual correlation function of the field 𝑈 𝒓, 𝑡 is defined by:
Γ 𝒓1 , 𝒓2 , 𝜏 = 𝑈 𝒓1 , 𝑡 𝑈 ∗ (𝒓2 , 𝑡 + 𝜏)
P1
𝒓1
𝜎
𝒓2
Γ 𝒓1 , 𝒓2 , 𝜏 =
𝐼 𝒓1
𝑡
Γ 𝒓, 𝒓, 0
=
𝐼(𝒓)
P2
𝐼 𝒓2 γ 𝒓1 , 𝒓2 , 𝜏 avec γ 𝒓1 , 𝒓2 , 𝜏
<1
For Gaussian source, plane quasi-monochromatic wave, small divergence :
𝛾 𝒓1 − 𝒓2 , 𝜏 =
𝜉𝑙
1 𝒓 −𝒓
−2 2⊥ 𝐺 1⊥
𝜉𝑡
𝑒 −𝑖(𝒌𝒊 ∙ 𝒓2−𝒓1 −𝜔𝜏) 𝑒
𝑐
𝜆2
= 𝑐𝜏𝑙 =
=
∆𝜈 Δ𝜆
Distance over which
the wave is monochromatic
𝜉𝑡
2
𝑒 −𝜏/𝜏𝑙
𝜆𝐷
=
2𝜋𝜎
Distance over which
the wave is planar
Coherence: effect on diffraction
Transverse coherence
𝜎
2𝐷
𝜆
2𝑎
𝑎
𝜎
𝜆
𝑎
𝜓
𝛿 ≅𝑎𝜓
𝐷
𝜉𝑡
𝜆𝐷
=
𝜎
Franges disappear if
𝜎
2𝐷
>
𝜆
2𝑎
𝑎 > 𝜉𝑡
Longitudinal coherence
Transverse coherence is the more important:
𝑎
𝜎
All sources are coherent enough to make atoms interfere
𝐷
but not enough or the whole crystal (Synch. XFEL)
2
𝜉𝑙
1𝜆
=
2 ∆𝜆
𝜹
𝜆
Franges disappear if 𝛥(2𝜋 ) > 𝜋
𝛿 > 𝜉𝑙
Classical approximations
• Kinematic approximation
• Scattered beam intensity is negligible compared to incident beam intensity
• No multiple scattering, no intensity decay (does not conserve energy! § Born)
• Good approximation for small crystals
• No approximation  Dynamical theory
• Static approximation
• X-ray frequencies: 1018 Hz
• Atomic vibrations frequencies: 1012 Hz (THz)
z
t
• Atomic displacements negligible compared to X-ray wavelengths.
t
Interferences (Th. Young)
Plane wave
Fresnel Fraunhofer
Two regimes of diffraction
Fresnel regime
Near-field
Augustin Fresnel
(1788-1827)
𝑎2
4𝜆
Fraunhofer regime
Far-field
Joseph von Fraunhofer
(1787-1826)
a
150 µm
10 µm
Fraunhofer regime: the phase difference
𝑟
𝒒
𝒌𝑖
𝒌𝑑
𝒒 = 𝒌𝑑 − 𝒌𝑖
Phase difference = −𝒒 ∙ 𝒓
Examples:
Speckles
Colloïd hard spheres (117 nm)
𝝀/𝒂
of PMMA in decalin
Water
E=11 keV
E=8 keV
P. Wochner et al., PNAS 106, 11511 (2009).
T. Head-Gordon et al., Chem. Rev. 102, 2651 (2002).
1st max of 𝑆(𝒒) at 2 Å-1
First neighbor O-O distances ~3 Å
2 °C
77 °C
1st
Presence of speckles
max of 𝑆(𝒒) at 0.003 Å-1
Calculation of
scattered intensity
Kinematical aprroximation:
𝑑𝜎 𝑑𝜎
=
𝑑Ω 𝑑Ω
2
𝜌𝑡𝑜𝑡 (𝒓)𝑒 −𝑖𝒒∙𝒓 𝑑3 𝒓
𝑇ℎ
𝑉
volume of the crystal
A 𝐪 =
𝑉
𝜌𝑡𝑜𝑡 (𝒓)𝑒 −𝑖𝒒∙𝒓 𝑑3 𝒓 is the complex scattering amplitude
It is the 3D Fourier transform
of the total electron density 𝜌𝑡𝑜𝑡 (𝒓)
(1D FT of its projection on 𝒒)
Scattered intensity: 𝐼 𝒒 = A 𝐪
𝑇
T: time of experiment
Amplitude scattering
𝜌𝑡𝑜𝑡 (𝒓)𝑒 −𝑖𝒒∙𝒓 𝑑 3 𝒓
A 𝒒 =
𝑉
Electron density
𝜌𝑡𝑜𝑡 𝒓 =
𝜌𝑛 (𝒓 − 𝒓𝑛 )
𝑛
• Set of identical atoms
∗
=
𝜌𝑡𝑜𝑡 𝒓 =
𝜌𝑎 𝒓
∗
𝜌𝑎 (𝒓)𝑒 −𝑖𝒒∙𝒓 𝑑 3 𝒓
𝐴 𝒒 = 𝑓(𝒒)
𝑉
𝜌 𝒓
Intensity expression
Limited volume of scattering :
𝐴∗ 𝒒 𝐴 𝒒 = 𝑓 2
𝜌𝑎 𝒖 𝜌𝑎 𝒖′ 𝑒 𝑖𝒒∙
𝒖−𝒖′
𝜌𝑎 (𝒓)𝑒 −𝑖𝒒∙𝒓 𝑑3 𝒓
𝐴 𝒒 = 𝑓(𝒒)
𝑉
𝑑 3 𝒖𝑑 3 𝒖′
𝑉
= 𝑓2
𝜌𝑎 𝒖 𝜌𝑎 𝒖 + 𝒓 𝜎(𝒖)𝜎(𝒖 + 𝒓)𝑒 −𝑖𝒒∙𝒓 𝑑 3 𝒖𝑑 3 𝒓
𝜎 𝒖 is a window function (1 ou 0)
-r
𝜎 𝒖 𝜎 𝒖 + 𝒓 = 1 ⇒ 𝒖 ∈ 𝑉 et 𝒖 ∈ 𝑉 − 𝒓
𝒖 ∈ 𝑉(𝒓)
𝐴(𝒒)
2
= 𝑓2
𝜌𝑎 𝒖 𝜌𝑎 𝒖 + 𝒓 𝑑 3 𝒖 𝑒 −𝑖𝒒∙𝒓 𝑑 3 𝒓
𝑉(𝒓)
Density-density correlation function
1
𝑉(𝒓)
𝜌𝑎 𝒖 𝜌𝑎 𝒖 + 𝒓 𝑑 3 𝒖 = 𝑃 𝒓 +∆ 𝒓
𝑃(𝒓) = 𝜌𝑎 𝒖 𝜌𝑎 𝒖 + 𝒓
Statistical average
𝑉(𝒓)
𝜌𝑎2 𝑔 𝒓
𝜌𝑎 𝛿 𝒓
∆(𝒓) : fluctuation
𝒓
𝑃 𝑟 is related to the
pair distribution function 𝑔(𝒓) by
𝑃 𝒓 𝑑 3 𝒓 = 𝜌𝑎 𝑑𝑛 𝒓 = 𝜌𝑎 (𝛿 𝒓 + 𝜌𝑎 𝑔(𝒓)) 𝑑 3 𝒓
𝐴(𝒒)
2
= 𝑓2
𝑃 𝒓 𝑉 𝒓 + ∆ 𝒓 𝑉(𝒓) 𝑒 −𝑖𝒒∙𝒓 𝑑 3 𝒓
Role of coherence
𝐼 𝑞 = 𝐴(𝒒)
1.
2.
2
𝑇
= 𝑓2
𝑃 𝒓 𝑉 𝒓 + ∆ 𝒓
No coherence ⇔ statistical average
Coherent beam:
a) Ergodicity on 𝑻: ∆ 𝒓 𝑇 = ∆ 𝒓 = 0
b) Non-ergodicity on 𝑻: Speckles
Nano-particles 390 nm-Si
in water-lutidin solution
Phase transitions
 33°C
‘‘Repulsive’’ glass
 33.39°C Liquid: smoothed speckles
 33.6°C ‘‘Attractive’’ glass
X.Lu et al. Soft Matter, 6, 2010, 6160
𝑇 𝑉(𝒓)
𝑒 −𝑖𝒒∙𝒓 𝑑 3 𝒓
Role of the volume
𝐼(𝒒) = 𝑓 2
𝐼 𝒒 = 2𝜋
𝑉 𝒓 𝑃 𝒓 𝑒 −𝑖𝒒∙𝒓 𝑑 3 𝒓
−3 2
𝑓
𝑇𝐹 𝑉 𝒓
∗ 𝑇𝐹(𝑃 𝒓 )
Separation of the effects of the volume and of the microscopic structure
𝑉 𝒓 𝑒 −𝑖𝒒∙𝒓 𝑑3 𝒓 = 𝛴(𝒒)
𝟐
where 𝛴 𝒒 =
𝜎(𝒖)𝑒 −𝑖𝒒∙𝒓 𝑑3 𝒖
𝑉2
Cube 𝒂 : 𝛴 𝒒 = 𝑉
sin 𝑎𝑞𝑥 /2
𝑞𝑥 /2
…
𝛴(𝒒) 𝟐 𝑑 3 𝒓 = (2𝜋)3 𝑉
...
0,888𝜋
𝑎
𝐼(𝒒) = (2𝜋)−3 𝑓 2 𝛴(𝒒)
2𝜋/𝑎
𝑞𝑥
𝟐
∗
𝑃 𝒓 𝑒 −𝑖𝒒∙𝒓 𝑑 3 𝒓
Total intensity
Introducing the
pair distribution function 𝑔 𝒓
𝑃 𝒓 = 𝜌𝑎 (𝛿 𝒓 + 𝜌𝑎 𝑔 𝒓 − 1 + 𝜌𝑎 )
Intensity scattered by a arbitrary
homogenous body is:
𝐼 𝒒 = 𝑓 2 𝜌𝑎2 𝛴 𝒒
𝟐
+ 𝑁𝑓 2 1 + 𝜌𝑎
(𝑔 𝒓 − 1)𝑒 −𝑖𝒒∙𝒓 𝑑 3 𝒓
Interpretation
𝐼𝑆𝐴𝑋𝑆 𝒒 = 𝑓 2 𝜌𝑎2 𝛴 𝒒
𝟐
Small angle scattering
Size, shape and long distance density fluctuations
I(q)
Large/wide angle scattering
Atomic structure of sample
q
𝐼𝐿𝐴𝑋𝑆 𝒒 = 𝑁𝑓 2 1 + 𝜌𝑎
(𝑔 𝒓 − 1)𝑒 −𝑖𝒒∙𝒓 𝑑 3 𝒓
Scattering function
Scattering function 𝑆(𝒒) is:
𝑆 𝒒 =
𝐼 𝒒
= 1 + 𝜌𝑎
𝑁𝑓 2
(𝑔 𝒓 − 1)𝑒 −𝑖𝒒∙𝒓 𝑑3 𝒓
For a isotropic system:
𝑆 𝒒 = 1 + 4𝜋𝜌𝑎
1
𝑔 𝑟 =1+ 2
2𝜋 𝑟𝜌𝑎
𝑔 𝒓 −1
sin 𝑞𝑟 2
𝑟 𝑑𝑟
𝑞𝑟
(𝑆 𝒒 − 1) sin(𝑞𝑟) 𝑞𝑑𝑞
𝑆 𝒒 can be measured
by X-ray or neutron scattering
→ 𝒈(𝒓)
Example 1
Liquid argon
85 K
Neutrons
Solid argon
f.c.c
6.5 Å
3.75 Å
5.31 Å
Example 2: water
Ice Ih (P63/mmc)
(Bernal-Fowler Ice)
Oxygen tetrahedra
E.D. Isaacs et al., Phys. Rev. Lett. 82 (1999) 600
Low density ( r = 0.0295 mol.A3)
O-O Correlation function
Extrapolation
26 MPa
209 MPa
400 MPa
Breaking
of H bonding
Extrapolation
High density ( r = 0.0402 mol.A3)
A. Soper et al., Phys. Rev. Lett. 84 (2000) 2881
• Ferroelectric
KNbO3/BaTiO3
• Perovskite ABO3, ferroelectric
• T > 120 °C, Cubic Pm3m, paraelectric
• 0°C < T < 120 °C, Tetragonal P4mm, ferroelectric
P4mm  Pm3m, 1st order transition (domains).
• -90°C < T < 0 °C, Orthorhombic Cmm2
Cmm2  P4mm, 1st order transition .
• T < -90 °C, Trigonal R3m
R3m  Cmm2, 1st order transition .
Ba2+/K+, Ti4+/Nb3+, O24 Å
O
1st
1st
1st
Ti
Trigonal
Ba
Orthorhombic
Tetragonal
Ferroelectric perovkite: KNbO3
Measure of 𝑆 𝑞 D4 ILL (Grenoble, France)
1
𝑔 𝑟 =1+ 2
2𝜋 𝑞𝜌𝑎
(𝑆 𝑞 − 1) sin(𝑞𝑟) 𝑞𝑑𝑞
3
5
KNbO3
3
2
(R)
(R)
(O)
(O)
(O)
(T)
(T)
(C)
FT
2
g(r)
20 K
100 K
200 K
RT
180 C
250 C
380 C
450 C
4
S(Q)
20 K (R)
200 K (R)
TA
(O)
180 C (O)
250 C (T)
380 C (T)
450 C (C)
KNbO3
1
1
0
0
0
5
10
15
20
25
0
-1
Q (Å )
Coll. Claire Laulhé, Françoise Hippert, Gabriel Cuello
2
4
6
r (Å)
8
10
Pair distribution function in solids
3
K-O
O-O
2
20 K (R)
200 K (R)
TA
(O)
180 C (O)
250 C (T)
380 C (T)
450 C (C)
KNbO3
Nb-O

g(r)
a
a
1
K-Nb
0
0
2
4
6
r (Å)
8
10

Determination of Nb-O distances
R
1.8
1.6
1.4
O
1.8
Data: GRKN20K_B
Model: Gauss2
Weighting:
y
No weighting
1.6
1.4
Data: GRKNTACRYO_B
Model: Gauss2
Weighting:
y
No weighting
1.6
1.4
g(r)
0.8
0.6
0
±0
0.13455
0.20314
1.87538
0.24541
2.12257
1.0
0.8
0.6
y0
A
w1
xc1
w2
xc2
±0.00036
±0.00108
±0.00056
±0.00142
±0.00076
±0.01038
0.21259
1.88239
0.29319
2.11497
±0.00206
±0.0006
±0.00301
±0.00136
0.8
0.6
0.4
0.2
0.2
0.2
0.0
0.0
0.0
1.6
2.0
r (Å)
2.2
2.4
1.6
Chi^2/DoF
= 0.00014
R^2
= 0.99956
1.8
2.0
r (Å)
Longue
[111]
2.2
2.4
Data: GRKN
Model: Gaus
Weighting:
y
No w
1.4
0.07593
0.4
1.8
1.6
y0
A
w1
xc1
w2
xc2
1.0
0.1259 ±0.00149
0.4
1.6
1.8
1.2
g(r)
y0
A
w1
xc1
w2
xc2
C
Data: GRKN250C_B
Model: Gauss2
Weighting:
y
No weighting
Chi^2/DoF
= 0.00037
R^2
= 0.99885
1.2
1.0
2.0
T
1.8
Chi^2/DoF
= 0.00054
R^2
= 0.99855
1.2
g(r)
2.0
2.0
0.06103
0.13033
0.24388
1.89078
0.33784
2.10711
1.2
g(r)
2.0
1.0
0.8
Chi^2/DoF
R^2
= 0.
±0.00882
±0.00136
±0.00201
±0.00064
±0.0037
±0.00114
y0
A
w1
xc1
w2
xc2
0.6
0.4
0.2
1.8
2.0
2.2
r (Å)
Courte
2.4
0.0
1.6
1.8
2.0
r (Å)
2.2
2.4
0.05
0.13
0.25
1.89
0.35
2.10
Lead liquid
structure
H. Reichert, Nature 408, (2000) 839
Local order:
5-fold symmetry
Icosahedral?
Diffraction by thermotropic liquid crystals
Isotropic liquid
Smectic A
Nematic
Smectic C
Small Angle Scattering
SAXS-SANS is used to
determine
• the shape
• the size
• the organisation…
…of small objects
(clusters, macromolecules, precipitates, bubbles)
of nano(micro)metric size (20–1000 Å)
Applications :
• Polymer science, colloids, soft matter
• Metallurgy, earth science
• Biology
SAXS
At small angles 𝑓2 = 𝑍2
𝐼𝑆𝐴𝑋𝑆 𝒒 = 𝜌𝑒2 𝛴 𝒒
𝐼𝑆𝐴𝑋𝑆 𝒒 = 𝑓 2 𝜌𝑎2 𝛴 𝒒
𝟐
Set of small objects,
of electron density densité 𝜌𝑒,
in a solvent of density 𝜌0
𝜌𝑒
Scattered intensity/object:
𝐼𝑆𝐴𝑋𝑆 𝒒 = 𝜌𝑒 − 𝜌0
𝜌0
Effective density
2
𝛴 𝒒
𝟐
𝑜𝑟.
Average on
all orientations
𝟐
Guinier’s law
The curvature of 𝛴 𝒒 𝟐 at the origin
doesn’t depend on the shape of a molecule
but on its radius of gyration RG:
𝑅𝐺2
𝛴 𝒒
1
=
𝑉
𝑟 2 𝑑𝑣
𝟐
L6
0.88𝜋/𝐿
2𝜋/𝐿
Guinier ’s law:
𝐼𝑆𝐴𝑋𝑆
2 𝑅2
𝑞
𝐺
𝒒 = 𝜌𝑒 − 𝜌0 2 𝑉 2 exp −
3
Guinier’s law
Example of a sphere
𝜑=𝜋
𝜃=𝜋
𝑟=𝑎
𝑅𝐺2
𝑒 −𝑖𝑞𝑟 cos 𝜃 𝑟 2 sin 𝜃𝑑𝜃𝑑𝜑𝑑𝑟
𝛴 𝒒 =
𝜑=0
𝜃=0
𝑟=0
4
sin 𝑞𝑎 − 𝑞𝑎 cos 𝑞𝑎
𝛴 𝒒 = 𝜋𝑎 3 3
3
𝑞𝑎3
I(q)/Vf
2
1,0
Sphère
Guinier
0,8
0,6
0,4
RG/a ~ 0.77
0,2
0,0
0
1
2
-1
q (nm )
3
4
5
1
=
𝑉
3 2
𝑟 𝑑𝑣 = 𝑎
5
2
Porod’s law
Set of particles of
total surface S
lim 𝐼𝑆𝐴𝑋𝑆 (𝑞) = 2𝜋 𝜌𝑒 − 𝜌0
𝑆
𝑞4
2
10
I(q)/Vf
𝑞𝑎≫1
2
10
10
0
-3
-6
Sphère
Guinier
10
Porod
-9
0,1
1 -1
q(nm )
Deviation to the Porod regime:
interface roughness...
10
100
𝑔(𝑟) ~ 𝑟 𝑑𝑓 −𝐷
S 𝑞 ~
~
~
1
𝑞 𝑑𝑓
Fractals
𝑔 𝑟 𝑒 𝑖𝒒∙𝒓 𝑑𝐷 𝑟
𝑟 𝑑𝑓 −𝐷 𝑒 𝑖𝒒∙𝒓 𝑑 𝐷 𝑟
𝑥 𝐷−𝑑𝑓 𝑒 𝑖𝑥𝑐𝑜𝑠𝜃 𝑑 𝐷 𝑥
Measure of the fractal
dimension 𝑑𝑓
SANS on
sedimentary rock
Valid on 3 orders of magnitude in 𝒒