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Introduction to Neutron Reflectivity

J.R.P.Webster

ISIS Facility, Rutherford Appleton Laboratory

ISIS Neutron Scattering Training Course : Large Scale Structures Module : May 2010

S1S2S3

S4

Supe

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Det

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Mon

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Mon

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Fram

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Specular reflection of neutronsfrom surfaces and interfaces

Analagous to optical interference,ellipsometry

Equivalent to electromagnetic radiation withelectric vector perpendicular to the plane

of incidence

Depth Profiling : providesinformation on concentration or composition profile perpendicularto the surface or interface

(Penfold, Thomas, J Phys Condens Matt, 2 (1990)1369,T P Russell, Mat Sci Rep 5 (1990) 171 )

ReflectometryKinetics

Polymer Diffusion

Critical exponents in SCF

Protein unfolding

Non equilibrium surfactant films

Temporal resolution of

Ion transfers

Solvent transfers

Polymer structure

SurfactantsParametric Studies

Liquid/Liquid Interface

Reduce Label size in Structural Studies

Self Assembly

FoamsElectrochemistry

Electrodeposition and Surface nucleation

Self Assembly of systems

Metal Hydroxide electroprecipitation (batteries)

Novel templating mechanisms

Model DevicesThin polymer films (finite size effects)

Spin coating

Biology•Protein adsorption

•Biocompatible polymers

•Drug transport

•Anaesthesia mechanisms

Neutron Reflectivity

Inorganic Templating:

Gene delivery

Sustainable Laundry

Detergents

Atmospheric chemistry

Protein Resistant Surfaces

Organic Light

Emitting Diodes

Lung Surfactant

Surfactant Adsorption Biosensors

Ionic Liquids

Specular reflection of neutrons

0

1k

kn =Refractive index definedusing the usual conventionin optics:

n A i B= − −1 2λ λ

A Nb= 2π

( )BN a i=

+σ σπ4 πμλβ

πλα

βα

42

12

==

−−=

reZNin

X-rays

n1

no

Refractive Index for neutrons

H -0.374 x 10-12 cm

D 0.667 x 10-12

Extensively use H/Disotopic substitution

to manipulate “ contrast “or refractive index

n A i B= − −1 2λ λ

n kk= 1

0

A N b= 2 π

n < 1.0 hence TOTAL EXTERNAL REFLECTION

Specular reflection of neutrons( some basic optics )

no

n1

oθ

1θ

From Snell’s Law,1

0

0

1

coscos

θθ

==nnn

At total reflection 0.01 =θ 0.1cos 1 =θ

πλθ

πλθ

Nb

Nb

c

c

=

−=2

1cos2

Critical angle Total reflection ( R=1.0 ) for

For Fresnel’s Law2

1100

1100

sinsinsinsin

θθθθ

nnnnR

+−

=( ) 2

120

20

2111 cossin θθ nnn −=

cθθ <

cθθ < is imaginary ( Evanescent wave )

cθθ >

cθθ >11 sinθn

11 sinθn Is real, and zero at

cθθ =0

cθθ =

Specular reflection of neutrons( some basic optics )

no

n1

oθ

1θ

From Snell’s Law,1

0

0

1

coscos

θθ

==nnn

At total reflection 0.01 =θ 0.1cos 1 =θ

πλθ

πλθ

Nb

Nb

c

c

=

−=2

1cos2

Critical angle Total reflection ( R=1.0 ) for

For Fresnel’s Law2

1100

1100

sinsinsinsin

θθθθ

nnnnR

+−

=( ) 2

120

20

2111 cossin θθ nnn −=

cθθ <

cθθ < is imaginary ( Evanescent wave )

cθθ >

cθθ >11 sinθn

11 sinθn Is real, and zero at

cθθ =0

cθθ =

Some typical values for θc and σa

Material θc (deg / Å)

Ni 0.1Si 0.047Cu 0.083Al 0.047D2O 0.082

Material σa(barns)

Si 0.17Cu 3.78Co 37.2Cd 2520Gd 29400

Specular Neutron Reflection(simple interface)

Within Born Approximation the Reflectivity is given as,

( ) ( )2

4

216∫

−′= dziQzezQ

QR ρπ

Q k k= − =1 2 4π θ λsin /

Reflectivity from a simple single interface is then given by Fresnels Law

( ) 24

216 ρπΔ=

QQR

2

1100

1100

sinsinsinsin

θθθθ

nnnnR

+−

=

Specular Neutron ReflectionFor thin films see interference effects that can be described using

standard thin film optical methods

2

21201

21201

1)( β

β

i

i

errerrQR −

−

++

=

( )( )

ϑsinii

ji

jiij

np

pppp

r

=

+

−=

iiii dn θλπβ sin2

=

For a single thin film at an interfaced

n0

n1

n2

0θ

1θ

For a single thin film :

( )1111201

212

201

11112012

122

01

2cos212cos2

dknrrrrdknrrrrQR

++++

=

For Q>>QC :

( ) ( ) ( )( ) ( )[ ]QdQ

QR cos216~)( 12012

122

014

2

ρρρρρρρρπ−−+−+−

FRINGE SPACING :

dQ π2=Δ

Fourier transform of 2 delta functions (young’s slits)

Rough or Diffuse Interface

( )2100 exp σqqRR −=

For a simple interface reflectivitymodified by,

λπ

θ2

sin2

=

=

k

kq ii

σ is rms Gaussian roughness

Gaussian factor ( like Debye-Waller factor ) resultsin larger that q-4 dependence in the reflectivity.

From specular reflectivity cannot distinquish betweenroughness and diffuse interface

Can be also applied to reflection coefficents in formulism for thin films,

( )( ) ( )( )25.0exp σji

ji

jiij qq

pppp

r −−

−=

( Nevot, Croce, Rev Phys Appl 15 (1980) 125, Sinha, Sirota, Garoff,Stanley, Phys Rev B 38 (1988) 2297)

Reflectivity from a simple interface

Glass optical flat35.0=θ

%533

1035.0 25

=ΔΑ=

Α= −−

θσ

xNb

Effect of roughnessand sld

Reflectivity from thin films

Effect of film thickness and refractive index

Reflectivity from thin films

Effect of interfacial roughness

Reflectivity from thin films

Effect of interfacial roughness

Reflectivity from a thin film

Deuterated L-B film on silicon

Α==Δ=Α=

Α=−−

20%,4,5.01074.0

119825

σθθxNb

d

NiC film on silicon

Α===Δ=Α=Α= −−

15,10%,4,5.01094.0,1194

21

25

σσθθxNbd

Reflection from more complex interfaces( multiple layers )

z

Nb

Airy’s fomula ( Parratt )

Combination of reflection and transmissioncoefficients give amplitude of successive beams reflected,

32

21

'11

221

'112

'111 ,,, rrttrrttrttr − and so on

More general matrix formulisms ( Born & Wolf, Abeles ) available

Phase change on traversing film, 1111 sin2 θλπδ dn=

K+−+= −− 11 4221

'11

22

'211

δδ ii errtterttrR

( Parratt, Phys Rev 95 91954) 359G B Airy, Phil Mag 2 (1833) 20)

Reflection from multiple layers n0

n1

nj

ns

Born and Wolf matrix formulism

Applying conditions that wave functions and theirgradients are continous at each boundary givesrise to a Characteristic matrix per layer,

( )

( ) jjjj

jjj

jjj

jjj

dn

np

ippi

Mj

θλπβ

θ

ββββ

sin2

sin

cossinsincos

=

=

⎥⎦

⎤⎢⎣

⎡−

−=

The resultant reflectivity is

[ ][ ] [ ]nR MMMM −−−−= 21 .

( ) ( )( ) ( )

2

22211211

22211211⎥⎦

⎤⎢⎣

⎡++++−+

=sas

sas

pMMppMMpMMppMMR

( Born & Wolf, ‘Principles in Optics’,6th Ed, Pergammon, Oxford, 1980)

Reflection from multiple layers n0

n1

nj

ns

In Born and Wolf approach can only includeroughness / diffusiveness at interfaces by further sub-division in small layers.

Abeles method, using reflectioncoefficients overcomes this limitationDefine characteristic matrix per layer, in optical terms from the relationshipbetween electric vectors in successive layers,

Ce r e

r e ej

ij

i

ji i

j j

j j=⎡

⎣⎢⎢

⎤

⎦⎥⎥

− −

− −− −

β β

β β

1 1

1 1

[ ] [ ] [ ]C C Ca bc dn1 2 1. − − − − =⎡

⎣⎢

⎤

⎦⎥+

The resultant Reflectivity is then,

R CC AA= * *

To include roughness,

( )( )rp p

p pq qj

j j

j jj j=

−

+−

−

−

−

1

11

20 5exp . σ( Heavens, ‘Optical properties of solid thin films’,Butterworths, London, 1955, F Abeles, Annale de Phys 5 (1950) 596)

Multiple Layer films

Region around 1st order Bragg peak for Ni/Ti multilayer15 bilayers ( 46.7, 1.0 x 10-5 / 55.7,-0.13x10-5)

Effects of resolution

1000 Å film on Si , ΔQ/Q 2%, 6%

Damps interference fringes, rounds critical edge

22

22

2

2

θθΔ+Δ=Δ

tt

QQ

On SURF and CRISP resolutionis dominated by collimation

Surface roughness and Waviness

Curvature << coherence length Waviness

This initially has an effect similar to resolution, and in the extreme can be treated by geometrical optics.

Curvature >> coherence length Rough

Incoherent reflectivity from 2 surfaces, separated by an adsorbing media:

( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )QAQRQR

QAQRQRQRQRtot21

22

11 1

1−−

+=

reflectivityScattering length

density

•Uniqueness ?•Resolution ?•Model dependent / overinterpretation of data ?•Does the scattering length density profile give access to the necessary physical parameters (Intra molecular) ?

Model fitting Reflectivity data

Steepest decent, simplex, simulated annealing, genetic, cubic spline + fft, etc etc

Lateral (z) and rotational invariance}

Z = 0 ?

= = =

=

z

ρ(z)

0.0

Partial Structure Factors

( ) ( )2

2

216∫+∞

∞−

−= dzezR ziκρκπκ

( ) ( ) ( ) ( )ρ z b n z b n z b n zc c h h s s= + +

( ) [ ]R b h b h b h b b h b b h b b hc cc h hh s ss c h ch c s cs h s hsκπκ

= + + + + +16

2 2 22

22 2 2

Self Partial Structure Factors : h nii i= $2

$ni is a one dimensional Fourier transform of ( )n zi

Cross partial structure factors: { }h n nij i j= Re $ $

( Crowley, Lee, Simister, Thomas, Penfold, Rennie,Coll Surf 52 (1990) 85 )

Cross Partial Structure Factors

If one distribution is shifted by δ Fourier transform is changed by phase factor ( )exp iκδ

( ) ( )( ) ( ) ( )κδκκ

δ

inn

znzn

ii

ii

expˆ

ˆ'

'

=

−= { } ( )ijjiij innh κδexpˆˆRe=

Model Self-terms as Gaussian ( solvent as tanh )

If both even functions

If even + odd functions

δ

δ

[ ] κδihhh jjiiij cos21

±=

[ ] κδihhh jjiiij sin21

±=

± because of phase uncertainty

Effect of capillary wave and structural roughnesson cross-terms

Widths of individualdistributions affected

by roughness,

but separationsare NOT

Structure of binarynon-ionic mixtures5 x10-5 M 30/70 C12E3 / C12E8

Example of simplest labelling scheme :solvent, alkyl chain of each surfactant

haa hss

has

C12E3

C12E3

C12E8

C12E8

Partial Structure Factor Analysis

Neutron Reflectivity at ISIS

INTER, POLREF, OFFSPEC,SURF, CRISP reflectometers

Measure variation of reflectivity with scattering vector, Qz,perpendicular to the interface

Using ‘white beam’ TOF methodwith fixed angle and range

of wavelengths

S1S2S3

S4

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rmirr

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Det

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Mon

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Mon

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Fram

e O

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( Penfold, Williams, Ward, J Phys E 20 91987) 1411; J Penfold et al,J Chem Soc, Faraday Trans, 94 (1998) 955

Instrumentation

( )( ) ( ) ( )[ ]( ) ( )[ ]

( )( )id

im

imim

ididi bI

bIfQRλελε

λλλλθλ

−−

=,

d,m refer to the detector and monitor

Correct for detector efficiency,spectral shape, background

Monitor Raw data

Corrected data

D2O

Specular, 1.5°

Background ( off-specular), 2°

Instrumentation

Si/D2O

Si/H2O

Si/30%D2O

0.35°0.8°

1.8°

Silicon / water interface

Reflectometry Village

InterDesigned for the study of chemical interfaces, with a particular emphasis on the air-water interface

>10 times the flux of SURF

Much wider dynamic range

Tuneable resolution

wavelength range 1 – 16 (22) Å

Moderator Coupled s-CH4 grooved – 26K

Primary flight path 19m (m=3 supermirror guides)

Secondary flight path 3-8 m

Beam size 60(h) x 30(v) mm

Flux at sample ~107 n/s/cm2

Scientific Opportunities

BiologyCell adhesion using synthetic polymer analogues Kinetics of action of interfacial enzymesInterfacial structure of designed peptides (folding)Biofouling and adsorption kinetics Interlayer forces in polymer and biological systemsSupported bilayers

Polymer diffusion

InterDesigned for the study of chemical interfaces, with a particular emphasis on the air-water interface

>10 times the flux of SURF

Much wider dynamic range

Tuneable resolution

wavelength range 1 – 16 (22) Å

Moderator Coupled s-CH4 grooved – 26K

Primary flight path 19m (m=3 supermirror guides)

Secondary flight path 3-8 m

Beam size 60(h) x 30(v) mm

Flux at sample ~107 n/s/cm2

Kinetic data from INTER

SURF

PolRefUses polarised neutrons to study the inter an intra-layer magnetic ordering in thin films and surfaces

>20 times the flux of CRISP

Much wider dynamic range

Flexible polarisation

Dual Geometry

High precision sample stage

wavelength range 0.9 – 16 Å

Moderator Coupled s-CH4 grooved – 26K

Primary flight path 23m

Secondary flight path 3 m

Beam size 60(h) x 30(v) mm

Flux at sample ~107 n/s/cm2

Scientific Opportunities

Spin ElectronicsSpin-InjectionSpin TorqueDilute magnetic semiconductorsGiant/Tunnelling magneto-resistance

Model Magnetic SystemsUltrathin films (finite size effects)Exchange springs (domain walls, surface magnetic phase transitions)Stabilise new single-crystal phases (Ce, Mn,..)

CRISP

POLREF

b = bN ± bM

PolRefUses polarised neutrons to study the inter an intra-layer magnetic ordering in thin films and surfaces

>20 times the flux of CRISP

Much wider dynamic range

Flexible polarisation

Dual Geometry

High precision sample stage

wavelength range 0.9 – 16 Å

Moderator Coupled s-CH4 grooved – 26K

Primary flight path 23m

Secondary flight path 3 m

Beam size 60(h) x 30(v) mm

Flux at sample ~107 n/s/cm2

CRISP

POLREF

b = bN ± bM

Polarised Neutrons for Biology• Use polarised

neutrons to provide additional information for protein absorption– Extract protein

thickness and orientation

– Better resolution than conventional AFM studies

Magnetic LayerSilicon

Polarised Neutrons for Biology• Use polarised

neutrons to provide additional information for protein absorption– Extract protein

thickness and orientation

– Better resolution than conventional AFM studies

Magnetic LayerSilicon

mnucleartotal bbb

NbA

BiAn

±=

=

−−=

π

λλ

2

1 2

Neutron Spin-Echo

Simulated, measured signal.

Simulated, measured reflectivity.

In-plane dynamic range of 50Å-42μm

Scientific Opportunities

In-plane StructuresPatterned Storage MediaMesoporous filmsPolymersBiological membranesSurfactants

Grazing Incidence DiffractionSurface crystalline structureSurface phase transitionsMagnetic surface structure

Larmor precession codes scattering angle

Unscattered beam gives spin echo (net precession) Independent of height and angle

Scattering by sample over angle q results in a net precession

Proportional to the spin echo length z

Measure polarisation

Keller et al. Neutron News 6, (1995) 16 Rekveldt, NIMB 114, 366 (1996)

θ0 sam

ple

θQz

5 modes of operation

SE reflection measurements to probe in plane structure

SE reflectivity with “high resolution”at low q and “wavy surface”

Spin-echo reflection “separation”of specular and off-specular reflection

Classical Spin echo in transmission or reflection of inelastic samples

Spin echo small angle scattering in transmission (SESANS)

Realisation/DesignNov 05

Realisation/Design

ZAug 08

Realisation/Design

ZMay 09