Fundamentals of Small-Angle Neutron Scattering Charles Glinka ...

Center for Neutron Research

Fundamentals of Small-Angle Neutron Scattering

Charles GlinkaNIST Center for Neutron Research

Gaithersburg, Maryland

(http://www.ncnr.nist.gov)

SANS: A Tool for Relating NanoscaleStructure to Bulk Properties


Techniques for the Measurement of Microstructure

USANS

Neutron reflectivity


SANS or SAXS?

SANS SAXS

Sources Few and weak Many and strong

Source of scattering

Differences in scattering length density

Differences in electron density

Size scale 1 nm – 1000 nm 1 nm – 1000 nm


SANS or SAXS?

SANS SAXS

Special Features

• D labeling and H/D contrast variation

• Magnetic scattering

• Conducive to extreme envivonments

• nondestructive

• msec resolution for time-resolved measurements

•Superior Q-resolution

•Anomalous scattering

•Small sample size


SANS or SAXS?

SANS SAXS

Complications • Incoherent scattering

• H/D isotope effects

• Radiation damage to some samples

• Parasitic scattering

• Fluorescence

• Beam stability


Concepts Common to SANS and Neutron Reflectometry

Sn

vi pi kiscattering

angle

2θCollimated beam of monochromatic neutrons incident on sample, S

iik λ

π2=

rIncident neutron wave vector,

Scattered neutrons counted as a function of scattering angle, 2θ

Q

ki

Scattering Triangle ki kf Q- =

scattering vector2θ

Sample

For a steady state source


ki

kfQ

For elastic scattering (ki = kf = 2π/λ)

2θ

θλπ

θ

sin 4Q

sink2Q

kkQ fi

=

=

−=r

rrr

Recall Bragg’s Law θλ sin 2d=or

Q2

sin42

sin 2d π

θλππ

θλ

=

==

In general, diffraction (SANS or NR) probes length scale

θλπ2

d angles, scattering smallfor ,Q2d ≈≈


In general, diffraction (SANS or NR) probes length scale

θλπ2

d angles, scattering smallfor ,Q2d ≈≈

More specifically, diffraction (SANS or NR) probes

structure in the direction of Q, on a scale, Q2dr

π≈

2θki

kf2θki

kf

2θrQ

ki

kf2θ

rQ

ki

kf

same I(Q)r cf. Appendix A.


2 Q

k i k f Q2k i

k f

Qrrr

=− fi kkDiffraction Probes Structure in the Direction of Q

Specular Reflection Geometry SANS Geometry

Reflectivity probes structure perpendicular to surface (parallel to Q), and averages over structure in plane of sample.

SANS probes structure in plane of sample (parallel to Q), and averages over structure perpendicular to sample surface.


0.5 nm < λ < 2 nm (cold neutrons)

0.1° < θ < 10° (small angles)

1 nm < d < 300 nm2θλ≈d (wavelength)

(scattering angle)

Small-a scale d , where

Angle Neutron Scattering (SANS) probes structure on

SANS Instrument Schematic

2θ


SANS Fundamentals• For Length Scale Probed by SANS Can Use Continuum Approximation

• Therefore we can use material properties rather than atomic properties when doing small-angle scattering

( > 100 H2O molecules)

- consider H2O, v ~ 30 Å3, r ~ 2 ÅScattering Length Density

neutrons X rays

b Z re


SANS Fundamentals

•Inhomogeneities in scattering length density, ρ(r ), give rise to small-angle scattering

• Angular dependence of scattering, I(q), is given by:

∫2

)(1)(IV

i deV

rrq rq rrr rr⋅= ρ Rayleigh-Gans eqn.

Entire volume of sample


SANS Fundamentals: Coherent vs. Incoherent Scattering

Consider scattering from N atoms of a single element,

2

ibV1)qI( ∑ ⋅=

N

i

rqi ierrr

XAZ

X rays

scattering length

atomicnumber

atomform factor

electronradius

Neutrons

A1

A1

A1

A1

A2

A2

A2

A2

A3

A3

A3

A3

scattering lengths, b’s, depend on isotope and isotope spin

SANS Fundamentals: Coherent vs. Incoherent Scattering

Consider scattering from N atoms of a single element,

2

ibV1)qI( ∑ ⋅=

N

i

rqi ierrr

XAZ

scattering lengths: depend on isotope and isotope spin

A1

A2

A2

A2

A2

A3

A3

A3

A3

A1A1

A1

neutron

( )∑ −⋅=N

ji

rrqi jie,

jibbV1)qI(

rrrr

( ) ( ) ( ) IIbbV1)qI( IncohCoh

,ji qqe ji rrqi

N

ji

rrr rrr

+== −⋅∑

( )ji rrqiN

jie

rrrr −⋅∑=,

2Coh b

V1)q(I

( ) ( )jj rrqiN

jje

rrrr −⋅∑−=,

22Incoh bb

V1)q(I

Since b’s areuncorrelated with the atom positions

1no structural information

structuralinformation

where


General Results for a Two-Phase System

• Incompressible phases of scattering length density ρ1 and ρ2

‘phase’ in this contextrefers to scattering

length density


∫ ∫2

221121

1)(IV

i

V

i dedeV

rrq rqrq rrr rrrr⋅⋅ += ρρ

∫ ∫ ∫2

121111

1)(I

−+= ⋅⋅⋅

V

i

V

i

V

i dededeV

rrrq rqrqrq rrrr rrrrrr

ρρ

( ) ∫2

12

211

1)(IV

i deV

rq rq rr rr⋅−= ρρ

From the Rayleigh-Gans equation:

break total volumeinto two sub-volumes

Contrast Factor(depends on materialsand radiation properties)

Spatial arrangementof material


( ) ∫2

12

211

1)(V

i deVd

d rq rq rr rr⋅−=

ΩΣ ρρ

General Results for a Two Phase System

2

V

rqi

p

rd∫ ⋅ rrr

e

( ) rer ri

V

rr rr

d)(q)(I q221

⋅−∫−∝ γρρ

‘Particles’(i.e. discreteinhomogeneities) Non-particulate systems

contrast single particleshape

interparticlecorrelations

correlation function

( ) S(q)F(q)VN)q(I 22

21 ρρ −=


Scattering Invariant

10 % black90 % whitein each square

• Scattered intensity for each would certainly be different

• For an incompressible, two-phase system:

• Domains can be in any arrangement*Guinier and Fournet, pp. 75-81.


Porod Scattering• At large q:

S/V = specific surface area of sample


Porod Scattering

*Glatter and Kratky, pp. 30-31.


SANS and Thermodynamics

Thermal density fluctuations also produce small-angle scattering

T

2

TV

I(0) βρ k= Isothermal compressibility

Composition fluctuations also produce small-angle scattering

2

22

/TV

I(0)φ

ρ∂∂∆

= mGkCurvature of Free Energy of Mixing

Ref: Introduction to Polymer Physics, M. Doi, 1996


Multi-Phase Materials

• “contrast” and “structure” terms can still be factored as for 2 - phase systems



Multi-Phase Materials

• for ‘p’ different phases in a matrix ‘0’

• Scattering is now a sum of several terms with possibly many unknown Sij’s*Higgins and Benoit, pp. 121-122.


Solving Multi-Phase StructuresContrast Matching

reduce the number of phases “visible”

• The two distinct two - phase systems can be easily understood

(shell visible)

(core visible)

ρ solvent = ρ core

ρ solvent = ρ shell

or


Contrast Variation

• A set of scattering experiments can yield a set of equations of known contrasts and unknown ‘partial structure functions’

• Sturhmann AnalysisDetermine structure from Rg = f (contrast)


NIST Center for Neutron Research (NCNR)• 21 beam facilities (7 thermal, 14 cold)

• 6 irradiation facilities

• 20 MW Reactor (4 x 1014 n/cm2-s peak thermal flux)

• Large liquid H2 cold source (25 K)


SANS Instrumentation at the NCNR

8-m Pinhole SANS0.05 nm-1 < Q < 0.5 nm-1

CHRNS 30-m SANS0.010 nm-1 < Q < 6 nm-1

NIST / ExxonMobil 30-m SANS

0.008 nm-1 < Q < 7 nm-1

30 mPerfect-Xtal USANS5 x 10-4 nm-1 < Q < ~ .01 nm-1


SANS Instruments at the NCNR

NIST/NSF 30-m SANS

1992

John Barker at Perfect-Crystal USANS Instrument

Triple-bounceanalyzer xtal

Detector

shielding


POLYMERS:• Conformation of Polymer Molecules in Solution and in the bulk• Structure of Microphase-Separated Block Copolymers• Factors Affecting Miscibility of Polymer Blends

BIOLOGY:• Organization of Biomolecular Complexes in Solution• Conformation Changes Affecting Function of Proteins, Enzymes, DNA/Proteincomplexes, Membranes, etc.

• Mechanisms and Pathways for Protein Folding and DNA Supercoiling

CHEMISTRY:• Structure and Interactions in Colloidal Suspensions, Microemulsions,Surfactant Micelles, etc.

• Microporosity of Chemical Absorbents• Mechanisms of Molecular Self-Assembly in Solutions and on Surfacesof Microporous Media

SANS APPLICATIONS


High Concentration Labeling- if labeled chains are randomly dispersed

(Q))b(b x)-(1x I(Q) 2DH P−∝

homopolymer

( )

Factor" Form"Chain Single -- )Q(

)Q(bb c I(Q) 2DH

<−∝

PP

∑∑ ⋅=N

i

z

j

rqi jiez

P ,2

1(Q)rr

Using Deuterium Labeling to Reveal Polymer Chain Conformation

bH

bD

(Q))b(b x)-(1 x

(Q)S )x)b-(1bx (I(Q)2

DH

T2

HD

P−

++∝blend

Can determine ST and P(Q) from 2 measurements, with different fractions (x) of labeled chains


METALS AND CERAMICS:

• Kinetics and Morphology of Precipitate Growth in Alloys and Glasses

• Defect Structures (e.g. microcracks, voids) Resulting from Creep, Fatigue or Radiation Damage

• Grain and Defect Structures in Nanocrystalline Metals and Ceramics

MAGNETISM:

• Magnetic Ordering and Phase Transitions in Ferromagnets, Spin Glasses,Magnetic Superconductors, etc.

• Flux-Line Lattices in Superconductors

SANS APPLICATIONS


General References: SANS and SAXS

Methods of X-Ray and Neutron Scattering in Polymer ScienceBy R.-J. Roe, Oxford University Press (2000).

Small-Angle Scattering of X-Rays, A. Guinier and G. Fournet, John Wiley & Sons (1955).

Polymers and Neutron Scattering, J.S. Higgins and H.C. Benoit, Clarendon Press-Oxford (1994).

Small Angle X-Ray Scattering, O. Glatter and O. Kratky, Academic Press (1982).

X-Ray Diffraction in Crystals, Imperfect Crystals, and Amorphous Bodies, A. Guinier, Dover Books (1994).


r

k i

k f

Q

2

1incidentneutrons

d

Appendix A. Scattering from Two Nuclei

fi krkrrrrr ⋅−⋅=∆φ

Q

)(rr

rrr

⋅=∆

⋅−⋅=∆

r

kkr fi

φ

φ

λπφ d∆

=∆ 2

)(21Rb

Rb φ∆+−−=Ψ kRiikR

scat ee

for N nucleiSAS of X rays - Guinier (1955), Ch.1

Polymers and Neutron Scattering,Higgins & Benoît (1994), p.11

scatΨ

2N

1

rQi

ib1dd ∑ ⋅=Ω

rrieN

σ

Scattered wave function


Appendix A. Scattering from N Nuclei

2N

1

rQi

ib1dd ∑ ⋅=Ω

rrieN

σScattering cross section: # neutrons scattering in direction corresponding to Q, divided by # incident per unit area

//i

//i

rQrQ

rrr

=⋅

+= ⊥rr

rrr

Therefore, diffraction probes structure in the direction of Q only!!

rri

rQ

Only components of ri parallel to Q contribute to summation

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