Introduction to Neutron Spin Echo Spectroscopy

NIST Center for Neutron research &

Antonio Faraone

NIST Center for Neutron research &

Dept. of Material Science and Engineering, U. of Maryland

Outline• Dynamic Neutron Scattering and NSE

– Coherent and Incoherent Dynamics

• The principles of Neutron Spin Echo (NSE)

• NSE Spectrometers Around the World

• NSE Exercise• NSE Exercise– Collective Dynamics of Microemulsion Droplet.

• NSE History

• Conclusion

Static and Dynamic Scatteringki, Ei

kf, Ef

Sample

Probe: photons (visible light, X-rays),neutrons, electrons, …

θ

Momentum Exchanged: Q=ki-kf

Q

ki

kf

Detector

Momentum Exchanged: Q=ki-kf

Energy Exchanged E=Ei-Ef

StaticScattering: )(QSd

d≈

Ω

σDynamic Scattering: ),(

2

EQSdEd

d≈

Ω

σ

))]0()((exp[),( rtrQiFTEQS −−= )](exp[)( ji rrQiFTQS −−=

Fourier Transform of the SpaceCorrelation Function

Fourier Transform of the Space-TimeCorrelation Function

Nuclear InteractionNeutrons are scattered by the nuclei.

Scattering power varies “randomly” from isotope to isotope.

The scattering also depends on nuclear spin state of the atom.

•If the scattered neutron waves from the different nuclei

Magnetic Scattering.

•If the scattered neutron waves from the different nuclei

have definite relative phases, they do interfere

Coherent Scattering

•If the scattered neutron waves from the different nuclei

have RANDOM relative phases, they don’t interfere

Incoherent Scattering

Scattering functionsS(Q,Ε) = Sinc(Q,Ε) + Scoh(Q,Ε)

S (Q,E) is the time and space Fourier transform of the SELF

Scoh(Q,E) is the time and space Fourier transform of the PAIR

correlation function

( )[ ] )0()(exp),( jicoh rtriQFTEQS −−=

Sinc(Q,E) is the time and space Fourier transform of the SELF

correlation function

( )[ ] )0()(exp),( iiinc rtriQFTEQS −−=

Hydrogen has a very high

Incoherent Scattering

Cross-Section

coherent

non-flip

Scattering Event andNeutron Moment

incoherent

1/3 non-flip 2/3 flip+

2/3 of the polarization signal is lost during the incoherent scattering event

Coherent Scattering

• Static Scattering– Diffraction

– Reflectometry

– SANS

• Inelastic/Quasielastic Scattering– Collective dynamics

• Phonons

• Shape Fluctuations

• Long Range coupled Domain Motions

• …

Incoherent Scattering

• Inelastic/Quasielastic Scattering– Single Particle Dynamics

• Diffusion

• Rotation• Rotation

• Vibrations

Neutron Scattering Techniques

NSE is the neutron scattering techniquesNSE is the neutron scattering techniquesthat gives acces to the largest length-

scales and longest time scales

What Does NSE Study?• Coherent Dynamics

Density Fluctuations corresponding to some SANS pattern

– Diffusion

– Shape Fluctuations (Internal Dynamics)

– Polymer Dynamics

– Glassy Systems

• Incoherent Dynamics• Incoherent Dynamics

Self-Dynamics (mostly of Hydrogen atoms)

• Magnetic Dynamics

Spin Glasses

Reptation motion Colldective DiffusionSegment motion

Local motionSide chain motion

coherent scattering incoherent

Dynamic Processes: Time and Length ScalesThe Dynamics Landscape of Polymers

NSEBS

DCSlarger scale objects: slower dynamics

coherent dynamics at low q and at q corresponding to relevant length scalesincoherent dynamics at high q

• Uncertainties in the neutron wavelength & direction of travelimply that Q and E can only be defined with a certain precision

• When the box-like resolution volumes in the figure are convolved,the overall resolution is Gaussian(central limit theorem) and has an

Instrumental Resolution

(central limit theorem) and has anelliptical shape in (Q,E) space

• The total signal in a scattering experiment is proportional to the phase space volume within the elliptical resolution volume

The better the resolution, the smaller the resolution volume and the lower the count rate

Neutron Spin Echo Breaks the Inverse Relationship between Intensity & Resolution

• Traditional – define both incident & scattered wavevectors in order to define E and Q accurately

• Traditional – use collimators, monochromators, choppers etc to define both

The Idea of Neutron Spin Echo

• Traditional – use collimators, monochromators, choppers etc to define both ki and kf

• NSE – measure as a function of the difference between appropriate components of ki and kf (original use: measure ki – kf i.e. energy change)

• NSE – use the neutron’s spin polarization to encode the difference between components of ki and kf

• NSE – can use large beam divergence &/or poor monochromatization to increase signal intensity, while maintaining very good resolution

Neutrons in magnetic fields:Precession

Neutron Properties

• Mass, mn = 1.675×10-27 kg

• Spin, S = 1/2 [in units of h/(2π)]

• Gyromagnetic ratio γ = µn/[S×h/(2π)] =

1.832×108 s-1T-1 (29.164 MHz T-1)

BωωωωL

S

In a Magnetic Field

• The neutron experiences a torque from a

magnetic field B perpendicular to its spin

direction.

• Precession with the Larmor frequency:

• The precession rate is predetermined by the

strength of the field only. LSBSdt

dSωγ ×=×=

BSN ×=

N

BL γω =

1.0

Spin Echo effect

l0 l1A B C

Px

x

y

zVn

S

B B

NSE measures the polarization: Px=⟨cosϕ⟩

l0=l1

-1.0

-0.5

0.0

0.5

Px

v

lBd∫= γϕ ∫

∫

== dv

v

BdlvfPx

γϕ cos)(cos

1cos)(cos =

+== ∫ ∫∫ dvvBdlBdlvfP

C

B

B

A

Cx γϕ ∫ ∫

== λγλλϕ dBdl

h

mFPx cos)(cos

Px is the Fourier transform of the wavelength distribution

Scattering Event: Single Neutron• elastic scattering

S

sampleB B

• inelastic scattering

( ) [ ] [ ]ÅmTJdlBh

mdl

h

BmN λ

π

λγλγ

πλ ×⋅×=== ∫∫ 7370

221

v

Bdl∫= γϕ ∫∫∫

∆=

∆=

−=∆ Bdl

v

v

h

mBdl

v

vBdl

vv

γλγγϕ

2'11

J field integral. At NCNR: Jmax = 0.5 T.mN (λ=8Å) ≈ 3×105∫= BdlJ

Scattering Event: Neutron Beam

Again:

S

f(λ)

B0B1 Analyzer

Again:The measured quantity is the Polarization, i.e.the spin component along x: Px=⟨cosϕ(λ)⟩:

Elastic scattering

( )λ

γγϕ

fv

dlB

v

dlB ∫∫−=

10Echo Condition: 10 JJ =

0=ϕ1=xPNote:

The requirement that ϕ=0 can be in some casesreleased. This treatment is valid for the mostcommon case of Quasi-Elastic Scattering

Scattering Event: Neutron Beam

Again:

S

f(λ)

B0B1 Analyzer

Again:The measured quantity is the Polarization, i.e.the spin component along x: Px=⟨cosϕ(λ)⟩:

Quasi-Elastic scattering

( ) ( ) vv

dlB

v

dlB

δλγ

λγϕ

+−=

∫∫ 10

( )

3

3

2

22

2

2

112

λπ

ωδλ

λ

δλ

δλλλω

mh

m

h

m

hE

=

≈

+−=∆=h

( )λγδλγϕ 100 JJh

mJ

h

m−+≈

( )λγωπ

λγϕ 1002

32

2JJ

h

mJ

h

m−+=

Series Expansion in δλ and δJ

0 at the echo condition

( ) ( ) ( ) ( )∫∫

×

−== ωω

π

λγωλλγλϕ dJ

h

mQSdJJ

h

mfPx 02

32

10 2cos,coscos

The Basic Equations of NSE( ) ( ) ( ) ( )∫∫

−+== ωλλγω

π

λγωλϕ ddJJ

h

mJ

h

mQSfPx 1002

32

2cos,cos

Even Function

FT of the wavelength FT of the Dynamic Structure2FT of the wavelengthdistribution

FT of the Dynamic StructureFactor

302

2

2λ

πγ J

h

mt =Fourier time

( ) ( )( ) [ ]

( )∫∫

∆=∆ωω

ωωω

dQS

dtQSJPtQJP ii ph

sph

x,

cos,,,

NSE measures the Fourier Transform of the Dynamic Structure Factor

How NSE works: summary• If a spin rotates anticlockwise & then clockwise by the same

amount it comes back to the same orientation

- Need to reverse the direction of the applied field

- Independent of neutron speed provided the speed is constant

• The same effect can be obtained by reversing the precession angle• The same effect can be obtained by reversing the precession angle

at the mid-point and continuing the precession in the same sense

- Use a π rotation

• If the neutron’s velocity is changed by the sample, its spin will not

come back to the same orientation

- The difference will be a measure of the change in the neutron’s

speed or energy.

NSE EffectElastic Scattering QuasiElastic Scattering

What does a NSE Spectrometer Look Like?

IN-11 @ ILLThe firstThe first

IN-11c @ ILL; Wide Detector Option

IN-15 @ ILL

NSE @ NCNR

NSE @ SNS

Surfactant molecules

A surfactant (“Surface Active Agent”) is soluble both in

•Oils and water do not mix! Why?

Water is a polar liquid, ε = 81;

Oils are non polar, ε ~ 2

Hydrophobic tailCH3-CH2-CH2- …CF3-CF2-CF2- …

Hydrophilic headSO3

-Na+

NH3+Cl-

-(OCH2CH2)n-

A surfactant (“Surface Active Agent”) is soluble both in

water and in organic liquids (oils)

Spherical micelles

When surfactants are dissolved in water they:- reduce the surface tension because they are adsorbed on the surfaces - form variety of aggregates – micelles, lamellae, bicelles, vesicles, etc

Cylindrical micelles Lamellae Vesicles

Surfactant aggregates in water

H2O

H2O

Vesicles

Ås

nm

Wormlike micelles canbe as long as fewmicrons

Surfactants are everywhereSurfactants are very useful to:

• Reduce the interfacial tension

• Solubilize oils in water

• Stabilize liquid films and foams

• Modify the interparticle interactions

• Stabilize dispersion• Stabilize dispersion

• Modify the contact angle and wetting

• …

Surfactants in our daily life:

• Cosmetics – moisturizers, lotions, healthcare products, soap, …

• Food – mayonnaise, margarine, ice cream, milk, …

• Industry – lubricants, stabilizers, emulsifiers, detergents, …

• Medicine – drugs, bio applications, …

• Agriculture – aerosols, fertilizers, …

Surfactants are everywhere II

• Agriculture – aerosols, fertilizers, …

• …

Micelles and MicroemulsionsOils and water do not mix?!? The surfactants help them mix.

Microemulsion dropletoil

Solubilized organicmolecules (oil)

H2O

Micelle

H2O

When surfactants are dissolved in oils they form “inverse” micelles, …Inverse microemulsion droplet

H2O

4-50 nm

H2OInverse micelle

oil oil

Microemulsion Properties

• Thermodynamically stable, isotropic, and optically transparent solutions

• The diameters range between 2 and 50 nm

decane

water

AOT/water/decane

Properties of the surfactant film

Properties of the surfactant

film change with:

• Molecular structure

Surfactant film Properties of the surfactant film:

• Interfacial tension

• Lateral elasticity

• Spontaneous curvature• Molecular structure• Additives• Ionic strength• Co-surfactant• Temperature, pressure etc.

• Spontaneous curvature

• Bending elasticity

• Saddle splay elasticity

∫

+

−++= dS

RR

k

RRR

kE

s 2121

2112

γHelfrich Free Energy

An Example: Microemulsion

• Light Scattering• Small Angle Scattering (Neutrons: SANS; x-rays: SAXS)

– Large length scales (10 Å-1000 Å)– ‘Low resolution diffraction technique’

Structure

SANS:

The intensity is the FT of the contrast distribution.

Contrast: Difference in Scattering Length Density

Contrast Matching Technique

∑=i

cohiA

w

bNM

dρ

Microemulsion:How to study them

Microemulsions move in solution because of thermal energy.– Diffusion– Shape fluctuations

Experimental techniques:

Dynamics

Experimental techniques:• Dynamic Light Scattering• Nuclear magnetic resonance• Neutron Spin-Echo (NSE)

NSE: T scale from ≈0.01 to 100 ns, L scale from ≈1 to 100 Å

ExperimentalShape fluctuations in AOT/water/hexane microemulsion

AOT

Inverse Microemulsion droplet

•Translational Diffusion

•Shape FluctuationsAOT

D2O

C6D14

σS (barn) bcoh (fm) bincoh (fm)

H 82.03 -3.741 25.274

D 2.05 6.671 4.04

4

6

1

2

4

6

10

I/cm

-1

AOT micelles AOT/D2O/C10D22

SANS dataAOT micellesAOT microemulsion

AOT/D2O/C6D14

Vol. fraction 0.078

Avg. radius (Å) ~ 30

polydispersity ~ 0.2

SLD (×10-6 Å-2)

n-hexane -0.67

H2O -0.56

d-hexane 6.14

D2O 6.35

AOT 0.10

0.1

2

2 3 4 5 6 7 8 90.1

2 3 4

Q/Å-1

Data Analysis: DiffusionFick Law, Diffusion Equation

[ ]tDQtQI 2exp),( −=

( )tQIDt

tQI,

),( 2∇−=∂

∂

φφ 2∇−=

∂

∂D

t

NSE measures coherent dynamics.

• The diffusion coefficient measured is the collective diffusioncoefficient.

• At finite concentration inter-particle interaction make the measured(effective) diffusion coefficient be Φ and Q dependent: Dc(Φ,Q).

• In the limit of infinite dilution the diffusion coefficient coincide with theself diffusion coefficient.

Data Analysis: Shape Fluctuations

∫∫ +

−+=

2121

12112 RR

dSkRRR

dSk

Es

bend

Expansion of r in spherical harmonics with amplitude a:

I(Q) Deff(Q)

( ) ( )

ΩΥ+=Ω ∑

mllmlmarr

,0 1

Frequency of oscillations of a droplet:

( )ηη

ηφ

πηλ

32'2324

4

334 0

30

2+

−−= f

k

Tk

k

k

R

R

R

k B

s4 5 6 7 8 9

0.12

Q

Dtr

Ddef

Data Analysis III

Translational Diffusion( )( )

[ ]tDQQI

tQI 2exp0,,

−=

( )( )

( )[ ]tQQDQI

tQIeff

2exp0,,

−=AOT/D2O/C6D14 MicroemulsionTranslational Diffusion + shape fluctuations

( ) ( )QDDQD deftreff +=

( )

( )[ ] ( )

+

+=2

2022

002

22022

54

5)(

aQRfQRjQ

aQRfDQD treff

π

λ

The two dynamical processes are statistically independent.

( ) ( ) ( )[ ]20300202 45 QRjQRQRjQRf −=

The bending modulus, k( )

( )[ ] ( )

+

+=2

2022

002

22022

54

5)(

aQRfQRjQ

aQRfDQD treff

π

λ

++=

η

ηηηλ

π 332'23

481 3

022R

p

Tkk B

ηπ 348 2p

λ2 – the damping frequency – frequency of deformation

<|a|2> – mean square displacement of the 2-nd harmonic – amplitude of deformation

p2 – size polydispersity, measurable by SANS or DLS

η and η’ are the solvent and core viscosities

Results

[ ] 2

2022

002

2

2022

)(5)(4

)(5)(

aQRfQRjQ

aQRfDQD treff

++=

π

λ

The beginning

F. Mezei, Z. Physik. 255, 146 (1972).

NG5-NSE

(SNS-NSE)

MUSESIN11

IN15J-NSERESEDASPAN

iNSE

NSE Today in the World

+ NSE option for triple-axis spectrometers+ developing ones

60

50

40

30

Num

ber of Pap

ers

total number of Papers Spectrometer polymer glass membrane SolidStatePhysics

P. G. de GennesNobel Prize in Physics 1991

Further improvement and development of the NSE technique!

NSE Publications

30

20

10

0

Num

ber of Pap

ers

2005200019951990198519801975

Yearpublication record; searched using web of science: keyword=neutron spin echo

discovery of the principleby F. Mezei in 1972

Nobel Prize in Physics 1991

polymer dynamics researchesby D. Richter in 1980’s

Still increasing the technique papers

15

20

25

30

35

proposals and BTR

NG5-NSE proposals

Instrumentation

SmallMolecules

ComplexFluids

Magetism

Polymer

Bio

Biomodel bio-membraneprotein motion

Polymersegment dynamicshydro gelspolymer complex

NG5 NSE proposal statistics

0

5

10

11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

Cycle Call

Magnetismfrustrated magnetsspin glass

Complex Fluidsmicroemulsionssurfactant membranescluster dynamics

Molecular Liquidsconfined waterionic liquids

1999.4 2010.5

Take Home Messages• NSE gives you access to dynamics in the ps to ns

time range (covering four orders of magnitude in time)

• NSE Resolution is independent of the beam resolution

• NSE data is measured as S(Q,t), i.e. in the time • NSE data is measured as S(Q,t), i.e. in the time domain (the resolution can be simply divided out)

Caveat

Spin Echo is neutron intensive.

A successful experiment is due in large part to the sampleselection (quality, amount, etc.). Make sure you:

• choose and characterize your sample well

• take advantage of the isotopic substitution technique

Further ReadingsF. Mezei (ed.): Neutron Spin-Echo, Lecture Notes in Physics,128, Springer, Heidelberg, 1980

F. Mezei, C. Pappas, T. Gutberlet (Eds.): Neutron Spin-EchoSpectroscopy (2nd workshop), Lecture Notes in Physics, 601,Springer, Heidelberg, 2003.Springer, Heidelberg, 2003.

D. Richter, M. Monkenbusch, A. Arbe, and J. Colmenero,“Neutron spin echo in polymer systems” Adv. in polym. Sci, 174,1 (2005).