by

Roger Pynn

LECTURE 5: Small Angle Scattering

This Lecture

• The size of objects measured with SANS• Typical applications of SANS• Instrumental resolution – how well can size be determined?• Scattering length density• Formal expression for scattering cross section• Scattering from independent particles• Radius of gyration and Guinier approximation• Shapes of composite particles• Particle correlations• Contrast variation• Porod scattering• Scattering from fractals• Sample requirements for SANS• Tools for estimating scattering and interpreting data

Small Angle Neutron Scattering (SANS) Is Used to Measure Large Objects (~10 nm to ~1 μm)

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scattering elasticfor sin2

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00

≈==

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π/QdθdλQQk

kk'-kQrrr

i.e. small Q => large length scales

Scattering at small angles probeslarge length scales

Typical scattering angles for SANS are ~ 0.3º to 5º

Two Views of the Components of a Typical Reactor-based SANS Diffractometer

Note that SANS, like otherdiffraction methods, probesmaterial structure in thedirection of (vector) Q

The NIST 30m SANS Instrument Under Construction

Where Does SANS Fit As a Structural Probe?

• SANS resolves structures on length scales of 1 – 1000 nm

• Neutrons can be used with bulk samples (1-2 mm thick)

• SANS is sensitive to light elements such as H, C & N

• SANS is sensitive to isotopes such as H and D

Typical SANS Applications

• Biology– Organization of biomolecular complexes in solution– Conformation changes affecting function of proteins, enzymes, protein/DNA

complexes, membranes etc– Mechanisms and pathways for protein folding and DNA supercoiling

• Polymers– Conformation of polymer molecules in solution and in the bulk– Structure of microphase separated block copolymers– Factors affecting miscibility of polymer blends

• Chemistry– Structure and interactions in colloid suspensions, microemeulsions,

surfactant phases etc– Mechanisms of molecular self-assembly in solutions

Biological Applications of SANS

• Studying Biological Macromolecules in Solution– Proteins– Nucleic Acids– Protein-nucleic acid complexes– Multi-subunit protein complexes – Membranes and membrane components– Protein-lipid complexes

• One of the issues with studying bio-molecules is that most contain H which gives a large, constant background of incoherent scattering. To avoid this:– Use D2O instead of water as a fluid for suspension – May need to deuterate some molecular components

Instrumental Resolution for SANS

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θδθθ

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rms

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=

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Scattering Length Density

• Remember

• What happens if Q is very small? – The phase factor will not change significantly between neighboring atoms– We can average the nuclear scattering potential over length scales ~2π/10Q – This average is called the scattering length density and denoted

• How do we calculate the SLD?– Easiest method: go to www.ncnr.nist.gov/resources/sldcalc.html– By hand: let us calculate the scattering length density for quartz – SiO2– Density is 2.66 gm.cm-3; Molecular weight is 60.08 gm. mole-1

– Number of molecules per Å3 = N = 10-24(2.66/60.08)*Navagadro = 0.0267 molecules per Å3

– SLD=Σb/volume = N(bSi + 2bO) = 0.0267(4.15 + 11.6) 10-5 Å-2 = 4.21 x10-6 Å-2

• A uniform SLD causes scattering only at Q=0; variations in the SLD cause scattering at finite values of Q

2.2 )(. rnerdb

dd

nucrQi

cohrr rr

∫ −=Ωσ

)(rr

ρ

SLD Calculation

• www.ncnr.nist.gov/resources/sldcalc.html• Need to know chemical formula

and density

Not relevant for SLD

X-ray values

Background

Determine best sample thickness

Note units of the cross section – this is cross section per unit volume of sample

Enter

SANS Measures Particle Shapes and Inter-particle Correlations

norientatioparticle

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spaceP

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ρρσ

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These expressions are the same as those for nuclear scattering except for the additionof a form factor that arises because the scattering is no longer from point-like particles

Scattering from Independent Particles

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Scattering for Spherical Particles

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Radius of Gyration Is the Particle “Size” Usually Deduced From SANS Measurements

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Guinier Approximations: Analysis Road Map

• Guinier approximations provide a roadmap for analysis.

• Information on particle composition, shape and size.

• Generalization allows for analysis of complex mixtures, allowing identification of domains where each approximation applies.

P(Q) = 1; α = 0

απ Q–α; α = 1,2ΔMα0 exp –

Rα2 Q2

3 – α

d ln P(Q)d ln(Q) = Q

P(Q)d P(Q)

dQ = –α – 23 – α R α

2 Q2

Generalized Guinier approximation

Derivative-log analysis

* Viewgraph courtesy of Rex Hjelm

Radii of Gyration for Various Shapes*

Note that Guinier approximation works only when QRg <1

*Viewgraph from Charles Glinka

6

82

53R

RRg =

If particles are not mono-disperse, Rg is weighted towards larger particles

Beyond Guinier: Form Factors for Simple Shapes

*viewgraph from Charles Glinka (NIST)

Form Factors (cont’d)

Note that the slope of I(Q) v Q thatcorresponds to the particle shape occursover a region of Q bounded by the largestand smallest dimensions of the particles

Note also that at large Q the averageslope is -4. This is called the Porodregion. We will discuss it later. The slope is a manifestation of a smooth3-dimesnional surface of the particles

Calculating Form Factors• www.ncnr.nist.gov/resources/simulator.html• Note: T(1 mm H2O) = 0.5; T(1 mm D2O) = 0.9

dσ/dΩ (H2O) = 1 cm-1; dσ/dΩ (D2O) = 0.06 cm-1

No background H2O background

What Happens if Particles are Lined Up?

• Scattering probes structure in the direction of Q

Couette shear cell

Shape Determination for Dilute, Randomly Oriented, Uniform Particles

• If I(Q) is measured over a wide enough Q range then the inverse transform can be computed

rby separated particle in the points twofinding ofy probabilit theis (r)r4P(r) where

sin)(4)()(

)()()'()()(

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∫=≡ dQQrQQIrrrP )sin()(2)(4)( 2

πγπ

P(r) for Simple Models

Rg = 21.0Å

Rg = 29.0Å

Determining Particle Size From Dilute Suspensions

• Particle size is usually deduced from dilute suspensions in which inter-particle correlations are absent

• In practice, instrumental resolution (finite beam coherence) will smear out minima in the form factor

• This effect can be accounted for if the spheres are mono-disperse • For poly-disperse particles, maximum entropy techniques have been used

successfully to obtain the distribution of particles sizes

Correlations Can Be Measured in Concentrated Systems

• A series of experiments in the late 1980’s by Hayter et al and Chen et al produced accurate measurements of S(Q) for colloidal and micellar systems

• To a large extent these data could be fit by S(Q) calculated from the mean spherical model using a Yukawa potential to yield surface charge and screening length

Polymers Studied with SANS

• Typically Rg is 5 - 50 nm for most polymers – good for SANS

• In addition to examining chain conformation, SANS has been used to probe thermodynamics (e.g. of blends and block copolymers) and to challenge various theories (e.g. random phase approximation)

melt aor solvent thetaafor 1/2 solvent; good afor 3/5ν

where~ generally, More

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Probing Chain Conformation in Thin Films

Thin films of 25% d-PS & 75% PS spun on toSi wafers. 25 wafers => 10 nm total polymerthickness – 0.2 mg

Rg in the plane of the film isunchanged down to filmthickness of Rg/2

Contrast & Contrast Matching

Both tubes contain borosilicate beads + pyrex fibers + solvent. (A) solvent refractive index matched to pyrex;. (B) solvent index different from both beadsand fibers – scattering from fibers dominates

O

OW

ater

D2O

H2O

* Chart courtesy of Rex Hjelm

Contrast Variation

Deuterated ProteinDeuterated RNA

CD2

Deuterated Lipid Head Group

CONTRASTΔρ

RNA

DNA

Water

Protein

Lipid Head Group

CH2

Isotopic Contrast for Neutrons

HydrogenIsotope

Scattering Lengthb (fm)

1H -3.7409 (11)2D 6.674 (6)3T 4.792 (27)

NickelIsotope

Scattering Lengthsb (fm)

58Ni 15.0 (5)60Ni 2.8 (1)61Ni 7.60 (6)62Ni -8.7 (2)64Ni -0.38 (7)

Verification of of the Gaussian Coil Model for a Polymer Melt

• One of the earliest importantresults obtained by SANS was the verification of that rg~N-1/2

for polymer chains in a melt

• A better experiment was done3 years later using a smallamount of H-PMMA in D-PMMA(to avoid the large incoherentbackground) covering a MW range of 4 decades

Using Contrast Variation to Study Compound Particles

Viewgraph from Charles Glinka (NIST)

Examples include nucleosomes(protein/DNA) and ribosomes(poteins/RNA)

12

122121

22

22

21

21

2

2.

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1

)sin()()(

)()()(

)(1 2

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QFQFQI

rderdeVNQI

V V

rQirQi

ρρ

ρρ

ρρ

ΔΔ

+Δ+Δ=

Δ+Δ= ∫ ∫rr rrrr

SANS Has Been Used to Study Bio-machines

• Capel and Moore (1988) used the fact that prokaryotes can grow when H is replacedby D to produce reconstituted ribosomeswith various pairs of proteins (but not rRNA) deuterated

• They made 105 measurements of inter-protein distances involving 93 30S proteinpairs over a 12 year period. They alsomeasured radii of gyration

• Measurement of inter-protein distancesis done by Fourier transforming the formfactor to obtain G(R)

• They used these data to solve the ribosomal structure, resolving ambiguitiesby comparison with electron microscopy

A Recent Example of using Contrast Variation:Substrate Interactions with Molecular Chaperonin GroEL*

• Molecular chaperonins, including GroEL– Protect proteins against stress conditions that cause denaturation– Binds to unfolded proteins & helps them fold to active conformations– 14 subunits, Mw = 57,400 each; total Mw > 800,000

• Use deuterated substrate molecule and protiated GroEL

• Measure under different solvent conditions to separate the scattering into GroEL and substrate components

• Make use of previous SANS work on GroEL as well as GroEL/GroES complexes and available crystal structures

GroEL viewgraphs courtesy of Susan Krueger (NIST)

Contrast Variation of GroEL Complexes

Deuterated ProteinDeuterated RNA

CD2

Deuterated Lipid Head Group

RNA

DNA

Water

Protein

Lipid Head Group

CH2

Protiated GroEL

86% deuterated substrate

GroEL Analysis and Results

• New information obtained:– Shape of GroEL complex compared with shape of GroEL in solution– Shape and position of substrate molecule in complex with GroEL– Shape change of substrate molecule in complex with GroEL/GroES in

presence of ADP and ATP

Scattered intensity from the two components can be separated

I(q) = Δρ12 I1(q) + Δρ1 Δρ2I12(q) + Δρ2

2 I2(q)

Δρ1, Δρ2: contrast for components 1 and 2

I1(q), I2(q): intensity for components 1 and 2

I12(q): cross-term between components 1 and 2

sρ-ρΔρ =

Porod Scattering

function.n correlatio for the form (Debye) h thisfactor wit form theevaluate toandr smallat V)]A/(2 a[with ..brar-1G(r) expand toisit obtain y toAnother wa

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9 Thus

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2

44

22

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22

224

2

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π

π

=++=

∞→

=→

=

∞→→

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⎤⎢⎣

⎡ −=

>>

QA

QRVF(Q)

VQRV

QRQRQRVQR

QRQRQRQRV(QR)F(Q)

QR

Scattering From Fractal Systems

• Fractals are systems that are “self-similar” under a change of scale I.e. R -> CR• For a mass fractal the number of particles within a sphere of radius R is

proportional to RD where D is the fractal dimension

2.dimension ofsurfacessmooth for form

Porod the toreduces which )( that provecan one fractal, surface aFor

sin..12

)4/.(sin..2)(.)( and

)4/()( dRR and R distancebetween particles ofnumber )(.4

Thus

6

2

3.

3

12

sD

DD

D

DRQi

D

D

QconstQS

Qconstxxdx

Qc

RcQRRdRQ

RGeRdQS

RcRGdRcRRGdRπR

−

−

−

−

−

∝

==

==

=∴

=+=

∫

∫ ∫ πππ

rrrr

Typical Intensity Plot for SANS From Disordered Systems

ln(I)

ln(Q)

Guinier region (slope = -rg2/3 gives particle “size”)

Mass fractal dimension (slope = -D)

Porod region - gives surface area andsurface fractal dimension {slope = -(6-Ds)}

Zero Q intercept - gives particle volume if concentration is known

Sedimentary Rocks Are One of the Most Extensive Fractal Systems*

SANS & USANS data from sedimentary rockshowing that the pore-rock interface is a surfacefractal (Ds=2.82) over 3 orders of magnitude in length scale

Variation of the average number of SEMfeatures per unit length with feature size.Note the breakdown of fractality (Ds=2.8to 2.9) for lengths larger than 4 microns

*A. P. Radlinski (Austr. Geo. Survey)

Sample Requirements for SANS

• Monodisperse, non-interacting (i.e. dilute)• Concentration: 1-5 mg/ml• Volume: 350-700 μl per sample • Data collection time: 0.5-6 hrs per sample• Typical biology experiment: 2-4 days• Deuterated solvent is highly desirable• Multiple concentrations are usually necessary.• Specific deuteration may be necessary.• Multiple solvents of different deuteration are

highly desirable → contrast variation.

References

• Viewgraphs describing the NIST 30-m SANS instrument– www.ncnr.nist.gov/programs/sans/tutorials/30mSANS_desc.pdf

• SANS data can be simulated for various particle shapes using theprograms available at: – www.ncnr.nist.gov/resources/simulator.html

• To choose instrument parameters for a SANS experiment at NIST go to:– www.ncnr.nist.gov/resources/sansplan.html

• Several routines for analyzing SANS data are available at:– http://sasdap.lanl.gov/webGuinfit/– http://sasdap.lanl.gov/webExtrap/– http://sasdap.lanl.gov/webGnom/