small angle neutron scattering – sans mu-ping nieh cnbc 2009 summer school (june 16)

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


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



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



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



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


q =

sin 2

Current CNBC SANS Setup

M.-P. Nieh Y. Yamani, N. Kučerka and J. Katsaras Rev. Sci. Instrum. (2008) 79, 095102


qmin ~ 0.006 Å-1: Max. attainable length = 2/qmin ~ 1000 Å.


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


N

V22

= (p – o) · Vp· P( q ) S( q )dd( )v


Contrast Variation


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


= (p – o) · Vp· P( q ) S( q )dd( )v


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



Common form factors of particulate systems


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


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


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


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


= (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)


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

++++

++++



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


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)




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



“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



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


Dilute “Gaussian Chain” Solutions


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


Example – Polymer Solutions


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



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.



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



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



Model Independent Scaling Method


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



SANS Analysis – Model Independent Scaling Method


-CD3 -CH3

DOPC

Small Angle Diffraction



Example: Orientation of cholesterol in Biomembrane

Harroun, T. A. et al. Biochemistry 45 (2006) 1227-1233.



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.


small angle neutron scattering – sans mu-ping nieh cnbc 2009 summer school (june 16)

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