Small-angle scattering theory revisited: Photocurrent and ...
X School on Synchrotron Radiation “Small Angle X-ray ... · small -angle scattering appears to...
Transcript of X School on Synchrotron Radiation “Small Angle X-ray ... · small -angle scattering appears to...
X School on Synchrotron Radiation
“Small Angle X-ray Scattering”S.A.X.S.
STEFANO POLIZZI
DIP.CHIMICA FISICA
UNIVERSITÀ CA’ FOSCARI VENEZIA
X-Ray beam
Slits
Sample
Transmitted X-Ray
Beam stop
Detector
S.A.X.S.
W.A.X.S. (=XRD)
Why small angles?Why small angles?
All scattering phenomena are ruled by a reciprocal law through a Fourier transform: the larger the irradiated object, the smaller the scattering angle(if the wavelength of the incident radiation is constant)
As we approach the zero angle (the incoming direction in a transmission experiment), the dimensions we are exploring grow increasingly large.
λ [Å] 1° 0.1°0.15 nm (CuKα)α)α)α)
8 Κ8 Κ8 Κ8 ΚeV4.4 nm 44 nm
0.23 nm (CrKα)α)α)α)5.4 Κ5.4 Κ5.4 Κ5.4 ΚeV
6.8 nm 68 nm
……………… ……………….. ……………….400 nm (Visibile))))
3 3 3 3 eV11µm 110µm
How small is “small”?
So, if one is able to measure scattered intensity below 1°°°°from the incoming direction, one has a way to
investigate a range which spans from the atomic/molecular resolution of XRD to that of an
optical microscope.
Such dimensions are also called colloidal dimensions
Convolution
The convolution theorem:The convolution theorem:
The Fourier transform of the convolution of two functions is…
…the product of the Fourier transforms of the two functions (and viceversa)
?
Another example
Convolution
z(r )= ΣlΣmΣnδ(r -r lmn)ρc(r )CONVOLUTION
1/VcF(s)
σ(r )1 for |x|<L0 for |x|>L
z(r )= ΣlΣmΣnδ(r -r lmn)
Σ(s)
Z(s)= ΣlΣmΣnδ(s-slmn) CONVOLUZIONE
0.16 0.18 0.19 0.20 0.21 0.23 0.24 0.25 0.26 0.28
<D>=2π/∆h=1/∆s
200
111ZrO
2
h=4ππππsin( θθθθ )/λλλλ nm-1
XRP(owder)D: 1D section
Reciprocity of the Fourier Transform (2D)Reciprocity of the Fourier Transform (2D)
FTFTFTFTFTFTFTFT FTFTFTFT
Reciprocity of the Fourier Transform (1D)Reciprocity of the Fourier Transform (1D)
FTFTFTFT
FTFTFTFT FTFTFTFT
|F000(h)|2 | Σ(h)|2
S.A.X.S. W.A.X.S.
[|Fhkl(h)|2 | ΣhΣkΣl Σ(h-hhkl)|2]hkl≠000
<ρ2> | Σ(h)|2
0.2 0.3 0.3 0.4 0.4 0.5 0.6 0.60.0
0.2
0.4
0.6
0.8
1.0
1.2
222
311
220
200
111
h=4πsin(θ)/λ (nm-1)
Pd cuboctahedric cluster:Pd cuboctahedric cluster:Average diameter: 5 nmAverage diameter: 5 nm11 shells; 5083 atoms11 shells; 5083 atoms
nm
nmm
n mn rh
rhffhI
)sin()( ∑∑=
0.00 0.02 0.04 0.06
200
400
600
800
1000
inte
nsità
000
s [Å -1]
nm
nmm
n mn rh
rhffhI
)sin()( ∑∑=
2θ = 1.8° (λ=0.154 nm)
Pd cuboctahedric cluster:Pd cuboctahedric cluster:Average diameter: 5 nmAverage diameter: 5 nm11 shells; 5083 atoms11 shells; 5083 atoms
h [nm-1]
0.03 0.04 0.05
2
4
6
8
0.00 0.020
200
400
600
800
1000
inte
nsity
h[nm-1]
cubeoctaeder sphere
With low resolution the scatteringof the cubeoctaedric cluster is very similar to that of a sphere with radius 2.6 nm which contains the
same number of electrons
1E-3 0.01
inte
nsità
SFERA CLUSTER
s [ Å -1 ]
In logaritmic scale
h [nm-1]
( )matrixparticle
VI
ρρρρ
−=∆=∆=
electrons) ofnumber ()0( 222
( ) ( ) ( ) ( )( )
2
3
22 cossin3
−∆=hr
hrhrhrVhI ρ
SPHERE with Radius r
The electromagnetic radiation interacts with electrons. So, whatcauses small angle scattering are variations in the electron densityvariations in the electron densityof the irradiated matter on a scale which depends on the wavelenght of
the incoming radiation and on the scattering angle:
Dimensions of what?Dimensions of what?
This implies that what gives rise to small angle scattering are:
Amorphopus and/or crystalline particlesPores
BubblesCrazesetc …..
inside a homogeneous matrix
Structure 1 Structure 2
ρρρρ1111
ρρρρ2222
The two structures generate the same scattering:I(h)∝∝∝∝(∆ρ∆ρ∆ρ∆ρ)2 [Babinet’s principle]
h1
h2
Homogeneous particle (a portion of matter with a constant electronic density) dispersed in a matrix (a medium with a different electronic
density), e.g. a macromolecule in a solvent, a crystal phase-separated in a glass by thermal treatment, pores in a porous material.
2
3
222
0 3),,(3
4
−
∆=x
xcosxxsinabbaI πρh
22
221
2 hbhax +=
a
b
-0 .2 -0 .1 0 .0 0 .1 0 .21 E -9
1 E -8
1 E -7
1 E -6
1 E -5
1 E -4
1 E -3
0 .0 1
0 .1
1
1 0
h = 4 ππππs in ( θθθθ )/λλλλ
-0.2 -0.1 0.0 0.1 0.2
0.03125
0.0625
0.125
0.25
0.5
1
h
Different dimensions Different dimensions -- Same orientationSame orientation
I(h,a,b) = D(a,b)I0(h,a,b)dadb∫∫
Different dimensionsDifferent dimensionsIsotropic orientationIsotropic orientation
Small angles approximationSmall angles approximation
For dilutes systemsof particels at very low angles:
++−= ...)()(3
11)0()( 42
GG hROhRIhI
where Rg is the gyration radius.
This approximation allows one to easily determine particles dimensions
R g2 = 3
5r 2
R g2 = 3
5( a 2 + b 2 + c 2 )
R g2 = L 2
12+ r 2
2
scoordinatecenter -mass
)(
)(
)()( 22
2
=
−=
−−=
∫
∫
∫=
cm
V
cm
ρ
V
V
cmcm
g V
d
d
d
R
r
rrr
rr
rrrrr
costρ
ρ
What is the gyration radius?
r
ab
c
rL
Two approximations with the same first-order expansion series :
Zimmeq. 3/1
)0()(
Guinier eq. )0()(
220
3
22
0
hR
IhI
eIhI
gh
hgR
h
+≅
≅
→
−
→
Both approximations may be easily linearized by suitable plots
Zimm
Guinier
)0(3)0(
1
0)(
1
3)0(ln
0)(ln
22
22
I
hR
IhhI
hRI
hhI
g
g
+→≅
−→≅
0.0 5.0x10 -4 1.0x10 -3
e -2
e -1
e0
e1
e2
e3
e4
e5
e6
e7
Inte
nsità
h^2
cubo-ottaedro Guinier
Guinier approximation: 3
22
0)0()(
hgR
heIhI
−
→≅
0.1 1
3 6 9 12 15 18 21
h [nm -1]
2
1
<R>=6nm
nm
POLIDISPERSITY EFFECT on the scattering intensity
POLYDISPERSITY EFFECT on the gyration radiusPOLYDISPERSITY EFFECT on the gyration radius
For a polydispersion the radius of gyration is a weighed average wich largely overestimates the contribution of the
lager particles
RG2 = 3
5
R8
0
∞
∫ N(R)dR
R6 N(R)dR0
∞
∫
N(R) number of particles with radius between R and R+dR
0.0 0.1 0.2 0.3 0.4 0.5
e3
e4
e5
e6
e7
e8
e9
e10
6 12 18 24 30
h2[nm -2]
<R>=6nm
nm
2
3
5Gapp RR =
RG= 4.6 nmRapp= 6.0nm
RG= 6.5 nmRapp= 8.4 nm
RG= 19.3 nmRapp= 24.9nm
POLYDISPERSITY EFFECT POLYDISPERSITY EFFECT on the gyration radiuson the gyration radius
Asintoptic trend: the Porod’s Law
ShhI
hh
ShI
h∝
∞→∆∝
∞→
4
4
2
)(
2
)(ρπ
Holds for a “regular” separation interface.
For two phase systems S represents the surface area of separation between the tho phases
1E-3 0.01 0.1
0.01
0.1
1
10
100
1000
inte
nsity
h
CUBO-OTTAEDRO POROD GUINIER
GUINIER POROD
0.0 0.2 0.4 0.6 0.8 0.01 0.1 1
RG=2.1
RG=8.2
inte
nsity
a.u
.
h2(nm-2)
slope -4
h (nm-1)
Two groups of particles with distinct dimensions
GUINIER POROD
Q = 2π 2 V (∆ρ)2ϕ(1−ϕ) = I h( )0
∞
∫ h2dh
Other important equations: the Invariant
I 1(h) ≠≠≠≠ I 2(h) Q1 = Q2
π ϕ (1−ϕ) I(h) h4
I h( )0
∞
∫ h2dh
h →∞ → constant= S
V
Measure of the specific surface
1 2
V
S
V
S 21 >
12
Log(I(h))
Log(h)
2
1
h-4
In this case the total intensity is not the simple sum of the intensities of the individual
scattering particles
I(h) ≠ N I0(h)
I(h) ≠ P(r)∫ I0(h,r) dr
Interference effects between partciles must be taken into account
N Equal particles
Different particles
NON DILUTED SYSTEMS
0.00 0.02 0.04 0.06 0.08 0.100
1
2
3
0 3 6 9 12 15 18
h [nm-1]
N(r)
r [nm]
Rint<r>=6 nm<Rint>=2<r>=12 nm
Form factor P(h)
Structure factor S(h)
Scattering intensityI(h)=P(h) S(h)
int2
2
R
π
Measurements are usually carried out in transmission
The ideal beam is
� Monochromatic� Point-like� Well collimated
LOW INTESITY
� Increase the sample-detector distance� Increase λ
“From the experimental point of view […], small-angle scattering appears to have reached a steady value. The apparatus for small-angle scattering will certainly be continuously improved but no major change can be foreseen, unless the power of X-ray sources is increased by a factor 10 or 100, which is rather unlikely”.
GUINIER: 1969
Grenoble synchrotron radiation:1012 more brilliant than a conventional source
This opens new frontiers
� time-resolved measurements
� 2D- Detectors
� Local measurements (microdiffusion)
� Anomalous Scattering
The starting material is photochromic glass containing Ag(Cl,Br) crystallites
�Heating to 725 °°°°C ⇒⇒⇒⇒ Ag(Cl,Br) droplets
�Drawing at T>Tsoft ⇒⇒⇒⇒ cigar-like Ag(Cl,Br) particles
�Reduction at 430°°°°C in H2 ⇒⇒⇒⇒ cigar-like Ag particles
Example 1: Polarizing glasses S. Polizzi et al,
J. Appl. Cryst., 30, 487 (1997);J. Non-Cryst. Solids 232----234, 147 (1998)
VETRI POLARIZZATORI
200nm
ULTRA-SAXS (HASYLAB-DESY): Sample-detector distance: 12 m ; λλλλ=0.124nm
I(h,a,b) = D(a,b)I0(h,a,b)dadb∫∫
a = η r
b = r
η
r
ab
stretching
I(h,a,b) = D(r)I0(h,η r,r
η)dr∫
η (r) =1+ ηlim 1− exp −(r /rlim )m[ ]{ }
0.00
0.02
0.04
0.06
0 10 20 30 40 50 60
5
10
15
ηη ηη(R)
D(R
)
R [nm]
0 200 400 6000.000
0.004
0.007
0.011
0.014D
(2a)
length (2a) [nm]
0 10 20 300.00
0.01
0.02
0.03
0.04
0.05
D(2
b)
width(2b) [nm]
Length distribution
Width distribution
EXAMPLE 2:Aggregation of colloidal systems:
SULPHATE ZIRCONIA SOL-GEL
FRACTALITY
An object is called “fractals” when it shows a scale-invariance in a particular length range
M∝R Df
Df =1, 2, 3 for euclidean objects1≤≤≤≤Df<3 for fractal objects
M=Object MassR= Object Radius
The fractal dimension of a surface ds
comes out to be
S∝RDs
S=Surfaceds=2 for non fractal “regular” surfaces2<ds<3 for fractal surfaces
One finds out that such trends translate in the reciprocal space so that the small angle scattering fractal dimensions are obtained by
I(h)∝h -Df
I(h)∝ h Ds-6
1E-3 0.01 0.1 11E-8
1E-7
1E-6
1E-5
1E-4
1E-3
0.01
0.1
1
10
h-Df
h-6+Ds
Raggregato
1/rmonomero
1/Raggregato
inte
nsity
h
I(h) = Nvo2 ∆ρ( )2
imon (h)
I(0) = Nvo2 ∆ρ( )2
imon (0) =1
I(h) = N
kk2vo
2 ∆ρ( )2iagg (h)
I(0) = k Nvo2 ∆ρ( )2
∝ k ∝ mass
iagg (0) =1
N identical colloidal particles (monomers) with volume vo
M =N/k aggregates of k particles
Thus one can measure the aggreates mass [I(0)]And dimension [Rg ] without any assumption on
their structure
I(0) ∝ M∝R Df
Measuring the scattering as a function of time, it is possible to calculate Df and thus
determine the growth mechanism
1
0.01
0.1
1
h (nm-1)
min
454
806683667
354
134
226
61
25
2
inte
nsity
(a.
u)
78.1gRM ∝
gRM ∝
DIFFUSION LIMITED CLUSTER AGGREGATION
P.Riello et al. J.Phys.Chem. 107, 15 (2003) 3390
1 10
0.1
1
RG(nm)
slope 1.78(6)
slope 0.98(6)
I(0)
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.00.00
0.02
0.04
0.06
0.08
0 20 40 60 80 600 800 10000.2
0.4
0.6
2
4
6
(b)I(0)
slope 0.0270±0.0005
<L> (nm)
Io/I
(a)
0.528±0.003
Radius Attenuation I/I
o
Ro (nm)
min
RoRo
L
EXAMPLE 3: Nanostructure of Pd/SiO2 catalysts
A. Benedetti et al. J. Catal. 171, 345 (1997)S Polizzi et al P.C.C.P., 2001, 3, 4614,4619
J. Synchrotron Rad. (2002). 9, 65±70
Catalysis is a surface phenomenon
� Efficient use of expensive metals
� Different electronic structure
� Increase of catalytical activity and selectivity
WHY NANOPARTICLES ?WHY NANOPARTICLES ?
Small particles→ high surface/volume ratio
20 40 60 80 100 1200
20
40
60
80
100
120
2θ°
Pd 6%
Pd 3%
Inte
nsity
[cou
nts/
s]
Wide-Angle X-ray Scattering
35 36 37 38 39 40 41 42 4312
16
20
Pd 3%
Peak intensities too small Peak intensities too small compared to those of 6 wt%compared to those of 6 wt%
Hint for very small particles (clusters)Hint for very small particles (clusters)
Particles smaller than 4Particles smaller than 4--5 nm5 nm
Very broad peak with superVery broad peak with super--lorentzian lorentzian shapeshape
0 5 10 15 20 25 30 35 40 45 500.0
0.1
0.2
0.3
0.4
0.5
0.0
0.1
0.2
0.3
0.4
0.5
Pd 3%
Pd 3%
Pd 6%
D
istr
ibut
ion
of p
artic
les
Diameter [nm]
23.0 23.2 23.4 23.6 23.8 24.0 24.2 24.4 24.6-12
-10
-8
-6
-4
-2
0
2
4
E0
E1
E2
E3
E4
f '
f "
f', f"
(el
ectr
ons)
Energy (keV)
Anomalous Small-Angle X-ray Scattering
f = f0 + f ´ (E) + f ́ ´(E)
0.1 1
101
102
103
104
Inte
nsity
[a.u
.]
h [nm-1]
E1
E5
1570
1580
1590
1600
1610
1620
1630
1640
1650
2.252712.255 2.26 2.265 2.27 2.275 2.28 2.285 2.29
46
47
48
49
50
Both samples show a double distribution of particles
The total surface area is the same for both
samples
0.1 1102
103
104
105
106
Pd 3% Pd 6%
h [nm-1]
inte
nsity
[el2 /n
m3 ]
0 5 10 15 20 25 300.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Pd 6%
Pd 3%
Dis
trib
utio
n of
par
ticle
s
Diameter [nm]
Smaller particles are
80% in the 3 wt% sample80% in the 3 wt% sample
56% in the 6 wt% sample56% in the 6 wt% sample
The size of the clusters increases with increasing metal content
EXAMPLE 4: SiO2-PEG Hybrid materials
Obtained by hidrolysis of the precursor(OEt)3Si-(PEG)-Si(OEt)3
Suitably doped they can be:
•Ionic conductors•Photocromic materials• Luminescent materials•……..
Karim Dahmouche et al. J. Phys. Chem. B 1999, 103, 4937-4942
Microstructure determine by SAXS analysis
EFFECT OF PEG MOLECULAR WEIGTH Mw
1900 g/mole
800 g/mole
500 g/mole
200 g/mole
hmax
h[Å -1]
Average distance of SiO2 clusters in the polimeric matrix:
max
2
hd s
π=
( )3
1
ws Md ≅
( )
3223
2
0
22
3
4 )(2
3
4
radius with particles spherical of system aFor
1)-(1 systems diluitedFor
clusters SiO2by occupied volume
)1()(2
cc
c
RNQrRNV
RN
V
dhhhIVQ
πρππϕ
ϕϕ
ϕϕρπ
∆=⇒=
≅=
=−∆= ∫∞
INVARIANT
224
4
2
42)(
2
)(
:particles spherical of system aFor
cRhhI
hh
ShI
limh
N
πρπ
ρπ
∆=
∞→∆∝
∞→
POROD’S LAW
cRhhI
Q
limh
3)( 4
π=
∞→
From whichRc can be calculated:
ds=20-60Å
Rc=3-6 Å
EXAMPLE 5:
Grazing Incidence Small-Angle X-ray Scattering (GISAXS)of Cu-Ni alloys clusters obtained by implantation in a glassy matrix
E. Cattaruzza et al. J. Appl. Cryst. (2000). 33, 740-743,
In-detph view Plan view
Courtesy of: M. Buljan et al.Vacuum 71 (2003)65-70
Grazing Incidence Small-Angle X-ray Scattering (GISAXS)
0.0 0.2 0.4 0.6 0.80.0
0.2
0.4
0.6
0.8
1.0
Materiale SiO2
Enegia Rx: 8Kev, (0.154nm)
Rifl
ettiv
ità
angolo di incidenza (deg)
( ) ( ) ( )( )
b)a,on (dependenton distributi radia ),,(
clustersbetween radiusn Interactio
Yevich)-(Percus PYfactor Structure ),(
cossin3
3
4 ),(
),,(),(),()(
int
int
2
3
23
0
int
=•==•
=•
−
∆=Φ•
Φ= ∫∞
baRP
RR
RhS
hr
hrhrhrRRh
dRbaRPRhSRhKhI
α
ρπ
RR α=int
intR
R
<R> = 21 Åα = 1.2
Atoms/cm3= 1.5 1022 (GISAXS)2.0 1022 (RBS)
Volume fraction occupied by particles = 0.16
0.04 0.06 0.08 0.10 0.12 0.14
exp. data
LMA fit
scat
tere
d in
tens
ity (
arb.
u.)
q(Å-1)
1
3
2
0
EXAMPLE 6: Low resolution
structure of macromolecules
Dmitri I. Svergun Michel H. J.Current Opinion in Structural Biology 2002, 12, 654-660
Svergun D.I. J. Appl. Cryst.
(1997). 30, 792-797;
Dmitri I. Svergun Michel H. J.Current Opinion in Structural Biology 2002, 12, 654-660
,
B
Yeats Hexokinase: the monomere structure is known, but the biologicaly active form is a dimer, whose quaternary structure in solution is uncertain
A B
A
Peter Fratzl
J. Appl. Cryst. (2003). 36, 397±404
ESEMPIO 7:
MICRODIFFUSION