small angle neutron scattering – sans mu-ping nieh cnbc 2009 summer school (june 16)
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C anadian NeutronBeam Centre
Small Angle Neutron Scattering – SANS
Mu-Ping Nieh
CNBC 2009 Summer School (June 16)
C anadian NeutronBeam Centre Outline
• Common Features of SANS• Scattering Principle• SANS Instrument & Data Reduction• Contrast• Dilute Particulate Systems – Form Factor Data analysis Examples• Concentrated Particulate Systems – Structure Factor Data analysis Examples• Dilute Polymer Solutions Analysis for Gaussian Chain – Debye Function More Generalized Analysis – Zimm Plot• Small Angle Diffraction Analysis• Model Independent Scaling Method Analysis• Summary
CNBC Summer School, June 16, 2009
C anadian NeutronBeam Centre
1. The q range of SANS : between 0.002 and 0.6 Å-1 – achievable long wavelengths of neutrons and low detecting angles are important
2. A powerful technique for in-situ study on the global structures of isotropic samples
3. Easy to play contrast with isotope substitution4. Data analysis:
(A) Model dependent: well-defined particulate systems (possible non-unique solutions)
(B) Model independent: general feature of the structure
Common Features & Advantages of SANS
CNBC Summer School, June 16, 2009
C anadian NeutronBeam Centre
Neutron Scattering Principle - I
d
dddE
=
The number of neutrons scattered per second into a solid angle d with the final energy between E and E+dE
neutron flux of the incident beam • ddE
General Equation
For Elastic ScatteringWe do not analyze the energy but only count the number of the scattered neutrons
0
dd =
dddE dE
The number of neutrons scattered per second into a solid angle d
neutron flux of the incident beam • d=
[time-1area-1]
[time-1]
[area]
Differential cross section
CNBC Summer School, June 16, 2009
C anadian NeutronBeam Centre
Neutron Scattering Principle - IIScattering vector, q
ki
ko
q
ki
q = ko – ki = sin
2
dd( )v ·A ·t
Beam area on the sample
Path length
The measured intensity at q (or ), Im(q) can be expressed as
Im(q) = IF · o ·
·T ·
dd
Flux Solidangle
Detectorefficiency Sample
transmission
Differentialcross-section
dd( )v,sam = ( ) v,std
dd
Im,sam(q) · Astd · tstd · Tstd
Im,std(q) · Asam · tsam · Tsam
CNBC Summer School, June 16, 2009
C anadian NeutronBeam Centre
120
100
80
60
40
20
0
120100806040200
-0.3 -0.2 -0.1 0.0
-0.2
-0.1
0.0
0.1
0.2
4.5
4.0
3.5
3.0
2.5
2.0
1.5
Typical SANS Instrument
For = 8 Å and min~ 0.25o, qmin ~ 0.003 Å-1.Max. attainable length scale = 2/qmin ~ 2000 Å.
VelocitySelector
NeutronGuide
2-DDetector
Sampletable
circularly averaged into 1-D
102
2
4
103
2
4
I(q)
3 4 5 6 7 8 90.1
2
q
Neutron Beam from Reactor
CNBC Summer School, June 16, 2009
q =
sin 2
Current CNBC SANS Setup
M.-P. Nieh Y. Yamani, N. Kučerka and J. Katsaras Rev. Sci. Instrum. (2008) 79, 095102
CNBC Summer School, June 16, 2009
qmin ~ 0.006 Å-1: Max. attainable length = 2/qmin ~ 1000 Å.
C anadian NeutronBeam Centre
Contrast – A Key Parameter
Scattering intensity is proportional to “the square of the scattering length density difference between studied materials and medium” (known as “contrast factor”). low contrast 2 ways of increasing contrast
Ideally, we would like to increase the contrast yet without changing the chemical properties of the system. This is one of the greatest advantages using neutron scattering, since the neutron scattering lengths of isotopes can be very different.
-1.E-06
1.E-06
3.E-06
5.E-06
7.E-06
, Å
-2
CNBC Summer School, June 16, 2009
N
V22
= (p – o) · Vp· P( q ) S( q )dd( )v
C anadian NeutronBeam Centre
Contrast Variation
CNBC Summer School, June 16, 2009
J. Pencer, V. N. P. Anghel, N. Kučerka and J. Katsaras, J Appl Cryst 40 (2007) 513-525
C anadian NeutronBeam Centre SANS Analysis –
Dilute Particulate Systems
dd( )v
= (p – o) · Vp· P( q ) N
V22
orientational average
The form factor is determined by the structure of the particle.
= N
V(r1)(r2e-iq· (r
1- r
2) > dr1dr2
vp
p
oVp
V
N particles
r1 r2
Or1 and r2 are vectors of scattering center from one particle
1 kqk=1V
=N Particle Form Factor
Amplitude of the Form Factor
( r )·<e-iq · r > drvp
2N
V=
( r )·<e-iq · r > drparticle k
kq
CNBC Summer School, June 16, 2009
= (p – o) · Vp· P( q ) S( q )dd( )v
C anadian NeutronBeam Centre
Simulation of a particular system - Spheres
Since spheres are isotropic, there is no need to do orientational average( r )·e-iq · r drP(q) =
1
Vsphere vsphere
2
2
= e-iqr·cos r2 sin d d dr1
Vsphere
2
2
r= 0
r= R
= 0
=
= 0
= 2
= [sin(qR) - qR·cos(qR)]29
(qR)6
10-5
10-3
10-1
101
arb
itra
ry u
nit
6 80.01
2 4 6 80.1
2
q, Å-1
R= 100 Å = 1x10-6 Å-2
= 0.1
dd( )v = (sphere – o) · Vsphere· P( q )
N
V
22
=sphereVsphere 2 [sin(qR) - qR·cos(qR)]29
(qR)6
SANS Analysis –Dilute Particulate Systems
r
CNBC Summer School, June 16, 2006
C anadian NeutronBeam Centre
Common form factors of particulate systems
CNBC Summer School, June 16, 2009
SANS Analysis –Dilute Particulate Systems
Cylinders(radius: Rlength: L)
Spherical shells (outer radius: R1 inner radius: R2)
Morphologies P(q)
Triaxial ellipsoids (semiaxes: a,b,c)
Morphologies
Spheres(radius :R)
[sin(qR) - qR·cos(qR)]2 =Asph(qR) 9
(qR)62
(R13 – R2
3)2
[R13·Asph(qR1)– R2
3·Asph(qR2)]2
Asph[q a2 cos2(x/2) + b2sin2(x/2)(1-y2)1 + c2y2 ] dx dy0 0
1 12
1
0
J12[qR 1-x2 ]
4[qR 1-x2 ]2
dxsin2(qLx/2)
(qLx/2)2
Thin disk (radius: R)
Long rod (length: L)
By setting L = 0 2 - J1(2qR)/qR
q2R2
By setting R = 0 qL 0
qL2 sin(t)
tdt -
sin2(qL/2)
(qL/2)2
“Structure Analysis by Small Angle X-Ray and Neutron Scattering” L. A. Feigen and D. I. Svergun
J1(x) is the first kind Bessel function of order 1
C anadian NeutronBeam Centre
Procedure of Data Analysis
Select a model forpossible structure of the aggregates
Fit the experimental data using the selected model
(Fix the values of any“known” physical parameters as
many as possible)
Is it a good fit
? No
Change the model
Yes
A PossibleStructure
CNBC Summer School, June 16, 2009SANS Analysis –
Dilute Particulate Systems
C anadian NeutronBeam Centre
Examples of A Dilute Particulate SystemSample: A mixture of dimyristoyl and dihexanoyl phosphatidylcholine (DMPC, DHPC), and dimyristoyl phosphatidylglycerol (DMPG) in D2O; total lipid conc. = 0.1 wt.%
10-2
10-1
100
101
102
103
104
I (a
rbitr
ary
unit)
4 6 80.01
2 4 6 80.1
2
q (Å-1
)
10 oC
45 oC
10 oC
Path
In these cases, the known (or constrained) physical parameters are SLDs of lipid and D2O and the bilayer thickness. Thus, there are only 2 parameters to be determined through fitting in each case.
Oblate shells: a=180 Å, b=62 Å
Spherical shells: R=133 Å, p= 0.15
Bilayer disks: R=156 Å, L= 45 Å
CNBC Summer School, June 16, 2009
Demo of SANS data analysis in three afternoons!
C anadian NeutronBeam Centre SANS Analysis –
Concentrated Particulate Systems
p
oVp
V
r1 r2
ON particles
R2
dd( )v =
N
V(r1)(r2e-iq· (r
1- r
2) > dr1dr2
Vp
= (p – o) · Vp· P( q ) S( q )N
V22
1 kqk=1V
=N
1 kqj*q
k=1V+
N
jk
Ne-iq· (R
k- R
j)
~1
kq{1+ }V
k=1
N
jk
Ne-iq· (R
k- R
j)
Structure Factor
In dilute solution,S( q ) = 1
R1
CNBC Summer School, June 16, 2009
= (p – o) · Vp· P( q ) S( q )dd( )v
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
Str
uct
ure
Fac
tor
0.300.250.200.150.100.050.00q (Å
-1)
C anadian NeutronBeam Centre
Models for S(q)
u(rjk)Coulomb RepulsionQ = 20, csalt=0.01M
HardSphere
SqaureWell Attraction
R= 50 ÅConc. = 5% 2R
2.4R-3kT
S(q=0) = kT(N/)
Osmotic compressibilityS(0)>1 more compressibleS(0)<1 less compressible
++++
++++
CNBC Summer School, June 16, 2009SANS Analysis –
Concentrated Particulate Systems
10-3
10-2
10-1
100
101
102
I(cm
-1)
4 6 80.01
2 4 6 80.1
2
q (Å-1
)
0.10.050.020.005
qmax
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Simulation Result using Coulomb Repulsive Model
R= 50 ÅQ = 50Csalt = 0.001M
2x10-2
3
4
qmax
5 6 70.01
2 3 4 5 6 70.1
qmax ~
Fixed, due to P(q)
CNBC Summer School, June 16, 2009SANS Analysis –
Concentrated Particulate Systems
C anadian NeutronBeam Centre
Examples of Concentrated Particulate Systems
“Bicelles” composed of DMPC/DHPC and small amount of Tm3+
10-3
10-2
10-1
100
101
102
I (cm
-1)
2 4 6 80.01
2 4 6 80.1
2
q (Å-1
)
0.25 g/mL
0.05
0.025
0.0025
56
0.01
2
3
4
q ma
x (Å
-1)
0.001 0.01 0.1
Total Lipid Conc. (g/mL)
0.37
CNBC Summer School, June 16, 2009
C anadian NeutronBeam Centre
“Gaussian Chain”
“Gaussian Chain” Model : The effective bond length, a, of one step (composed of several segments of the chain) has a Gaussian distribution.
R1
R2
R3R4
RN
a2 = |Rn - Rn-1|2n=1
N
N
The distribution vector between any two “steps” m, n, i.e., (Rn - Rm) is
Gaussian
(Rn - Rm, n-m) =[ ]3/2 exp[ - ]a2|n-m|
a2|n-m|
Rn - Rm2
This model is adequate for describing long polymer chains in a theta solvent, where the segment-segment and segment-solvent interaction and the excluded volume effect are canceled out. Such polymer chains are also known as “phantom” chains.
“The Theory of Polymer Dynamics” M. Doi and S. F. Edwards
CNBC Summer School, June 16, 2009
C anadian NeutronBeam Centre
Scattering from a Gaussian Chain
dd( )v
= Npp(p – o)2VM (e
-q2RG2 – 1 + q2RG
2)2q4RG
4
Debye Function
Radius of Gyration, RG2= <(Rn – Rm)2>
2N2
1
m,n
10-4
10-3
10-2
I, cm
-1
6 80.01
2 4 6 80.1
2 4 6 81
q, Å-1
q-2RG = 80 Å
VM: molecular volume of monomerNp : degree of polymerization
CNBC Summer School, June 16, 2009SANS Analysis –
Dilute “Gaussian Chain” Solutions
C anadian NeutronBeam Centre
Debye function is not always adequate to describe the polymer conformation! An example is that as polymers are in a good solvent, the intensity at high q regime will decay with q-5/3 instead of q-2.
10-3
2
4
68
10-2
2
4
68
10-1
I, c
m-1
2 3 4 5 6 70.1
2 3
q, Å-1
4%
2%
1%
Polylactide in CD2Cl2
p Np RG (Å)1%
2%
4%
111
74
42
39
32
27
CNBC Summer School, June 16, 2009
Example – Polymer Solutions
C anadian NeutronBeam Centre
Zimm Plot
For dilute polymer solutions, the scattering intensity at low q regime (i.e., q·RG < 1) can be expressed as a function of RG, MW (molecular weight), A2 (second virial coefficient) and p (volume fraction of the polymer)
I(q)
Vmp
= (1 + q2RG2/3 + …)( + 2pA2 + …)
Np
1
Vm is the molecular volume of the monomerNp is the degree of polymerization
I(q)
Vmp
is plotted against (q2 +cp) and two extrapolation lines
The slope of the extrapolation line for p=0 is RG2/3Np.
RG, Np and A2 can be obtained by Zimm plot, where
for q=0 and p=0 are also obtained (c is an arbitrary number.).
The slope of the extrapolation line for q=0 is 2A2/c.The intercept of both extrapolation lines is 1/Np.
q2 + cp
I(q)
Vmp
q=0
p=0 1 2 43
q1
q2
q3
1/Np
CNBC Summer School, June 16, 2009
C anadian NeutronBeam Centre
Application of Zimm Plot
Np = 387RG = 79 ÅA2= 88.6
Polylactide in CD2Cl2
50x10-3
40
30
20
10
0
pV
M
40x10-3
3020100q
2 + cp
p = 0
q = 0
1/Np
Compared to the result obtained from Debye fitting
p Np RG (Å)1%
2%
4%
111
74
42
39
32
27
It requires at least two concentrations to make up a Zimm plot.
CNBC Summer School, June 16, 2009
C anadian NeutronBeam Centre
SANS Analysis – Model Independent Scaling Method
Guinier regime – low-q scattering
~ p(p – o)2Vpe-q2RG2/3
dd( )vln[ ] = ln(Io) – q2RG
2/3
dd( )v
N
V
22[1- (qRG)2/3 + …]= (p – o) · Vp·
dd( )vln[ ]
q2
-RG2/3
“Polymers and Neutron Scattering” J. S. Higgins and H. C. Benoit
CNBC Summer School, June 16, 2009
C anadian NeutronBeam Centre
Porod regime – high-q (with respect to the length scale) scattering
=sphereVsphere 2 [sin(qR) - qR·cos(qR)]29
(qR)6dd( )v
For qR>>1, i.e, focusing at smooth interface between two bulks 3-dimensionally
The form factor of a sphere (radius of R)
dd( )v
NsphereVsphere2 2
~V
9
2(qR)4
2ST
Vq4=
dd
q4( )v =2ST
V
total surface area
log I
log q
-4
“Polymers and Neutron Scattering” J. S. Higgins and H. C. Benoit
CNBC Summer School, June 16, 2009SANS Analysis –
Model Independent Scaling Method
C anadian NeutronBeam Centre
If Im(q) scales with q-n over a wide range of q. The “possible” structure can be obtained with some knowledge of the systems.
•Long Rigid rod 1•Smooth 2-D Objects 2•Linear Gaussian Chain 2•Chain with Excluded Volume 5/3•Interfacial Scattering from 3-D Objectswith Smooth Surface (Porod regime) 4with fractal Surface 3 ~ 4
nparticles
“Polymers and Neutron Scattering” J. S. Higgins and H. C. Benoit
CNBC Summer School, June 16, 2009
SANS Analysis – Model Independent Scaling Method
C anadian NeutronBeam Centre
-CD3 -CH3
DOPC
Small Angle Diffraction
CNBC Summer School, June 16, 2009
C anadian NeutronBeam Centre
Example: Orientation of cholesterol in Biomembrane
Harroun, T. A. et al. Biochemistry 45 (2006) 1227-1233.
CNBC Summer School, June 16, 2009
C anadian NeutronBeam Centre
Take Home Messages
• SANS is a powerful tool to study the global structures of isotropic systems in length scales ranging from 10 to 1000 Å.
• Isotope replacement could enhance/vary the contrast without changing the chemistry of the systems.
• The scattering function is proportional to the product of contrast factor, form factor (intraparticle scattering function) and structure factor (interparticle interaction).
• Structural parameters can be obtained through model dependent approaches, while model independent scaling law can also reveal possible morphology of the studied system.
CNBC Summer School, June 16, 2009
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