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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)
θλθ
θπθππλ
θ
/2d smallfor or 2as sin2 law sBragg' rewritecan weso
/sin4)sin2//(2/2 thatand
scattering elasticfor sin2
: thatRecall
00
≈==
===
==
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
dominates. resolutionh wavelengtsample, thefrom distance and sizedetector by theset , of lueslargest va at the that Note
neutrons. Å 15 using instrument SANS m 40 ILL at the m 5about of maximum a achieves This
. /~Q/2~ isobject observablelargest The /)/2(~/4~)Q/(~Q
and dominates resolutionangular , of valueAt this/5.1~2 :size beamdirect by the determined is of aluesmallest v The
h/L.12/5 h, of sample and sourceat apertures
equal and L of distancesdetector -sample & sample-source equalFor
0025.0 so small, is and 5% ~ )/( SANS,For
sin.cos sin4
rms
rmsminminrmsrms
min
2
22
2
2
22
2
22
2
θ
μλδπ
λπλπδθθδθδθ
θθδθ
θδθδθλδλ
θδθθ
λδλδθ
λπ
hLLh
Lh
QQQ
rms
rms
=
+=
+=⇒=
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
xQi
RQiP
spaceP
norientatioparticle
xQiRRQi
space spacePP
rrQi
space spaceNN
exdQF
eRGRdNQFdd
exdeRnRnRdRd
ernrnrdrdbdd
2
.32
2P
.3220
2
.30
)'.(33
)'.(332
.)(
:factor form particle theis )QF( and origin) at the one s there'if Rat particle a is
rey that theprobabilit (thefunction n correlatio particle-particle theis G where
).(.)()(
.)()'()('
).'()('
∫
∫
∫∫ ∫
∫ ∫
=
−=Ω
−=
=Ω
−
−
vr
rr
vrrrr
rrr
v
vr
rr
rr
rr
ρρσ
ρρ
σ
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
2)(QFr
2
.220
2.
1)()(
particles identicalFor
)(1)( sample of eunit volumper intensity Scattered
∫
∫
−=
=Ω
==
particle
rQi
ppp
rQi
rdeV
VVNQI
rderVd
dQI
v
vrr
vv
vv
ρρ
ρσ
contrast factorparticle form factor
weightmolecular particle thefind way toa provides )()0( so
number s Avagadro'isN anddensity mass particle theis whereweight molecular particle and / ionconcentrat Particle
)()0( that Note
20
A
220
ρρρ
ρ
ρ
ρρ
−=
==
−=
pA
w
Apwp
pp
NcMI
NVMVNVc
VVNI
Scattering for Spherical Particles
particles. ofnumber thetimes volumeparticle theof square the toalproportion is )]r( )rG( when [i.e. particles
spherical eduncorrelat ofassembly an from scattering total the0,Q as Thus,
0Qat )(3)(
cossin3)(
:Q of magnitude on the dependsonly F(Q) R, radius of sphere aFor
shape. particle by the determined is )(factor form particle The
010
30
2
.2
vr
rr rr
δ→→
=→≡⎥⎦
⎤⎢⎣
⎡ −=
= ∫
VQRjQRV
QRQRQRQRVQF
erdQF
sphere
V
rQi
2 4 6 8 10
0.2
0.4
0.6
0.8
1
3j1(x)/x
xQ and (a) axismajor the
between angle theis where)cossin (
:by R replaceparticles ellipticalFor
2/12222
rϑ
ϑϑ baR +→
Radius of Gyration Is the Particle “Size” Usually Deduced From SANS Measurements
result. general thisof case special a is sphere a offactor form for the expression
that theifiedeasily ver isIt /3.r is Q of ueslowest val at the data theof slope The .Q vty)ln(Intensi plottingby or region)Guinier called-so (in the Q lowat data SANS to
fit a from obtainedusually isIt ./ isgyration of radius theis r where
...6
1....sin
.sincos
21
....).(21.)(
:Q smallat factor form theof definition in the lexponentia theexpand and particle theof centroid thefrom measure weIf
2g
2
332g
60
22
03
32
0
0
22
0
3230
.
22
∫∫
∫
∫
∫
∫
∫∫∫
=
≈⎥⎥⎦
⎤
⎢⎢⎣
⎡+−=
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
+−=
+−+≈=
−
VVg
rQg
V
V
VV
rQi
V
rdrdRr
eVrQ
Vrd
rdr
d
dQV
rdrQrdrQiVerdQF
r
g
π
π
θθ
θθθ
rrrrr
r
rr
2
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)()(
)()()'()()(
')()()(
20
220
.20
)'.(20
'..20
2
.20
max
γπ
γπρρ
γρρρρ
ρρρρ
≡
−=
−=−−=
−=−=
∫
∫∫
∫∫∫
−
−
drQrQrrrQI
RdeRVrrdeVNQI
rderdeVNrde
VNQI
D
p
norientatioV
RQipp
V
rrQip
V
rQi
V
rQip
V
rQip
pp
ppp
rrrr
rrr
rrrrr
rrrrrr
∫=≡ 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
lengthsegment theis where6/chain Gaussian aFor 22
==
=
ν
νNR
NR
g
g ll
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.
21.
1
)sin()()(
)()()(
)(1 2
QRQRQFQF
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
smooth. is surfaceparticle theprovided shape particleany for Q as holds and law sPorod' is This
surface. ssphere' theof area theisA where2)(2
9 Thus
)resolutionby out smeared be willnsoscillatio (the averageon 2/9 Q as cos9
cossin9)()(
cos.sin9
radius) sphere theis R where1/RQ (i.e. particlespherical afor Q of valueslargeat )(F(Q) ofbehavior theexamine usLet
2
44
22
2
22
224
2
3242
42
π
π
=++=
∞→
=→
=
∞→→
⎥⎦
⎤⎢⎣
⎡−=⎥
⎦
⎤⎢⎣
⎡ −=
>>
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/