Introduction to Neutron Reflectivity
J.R.P.Webster
ISIS Facility, Rutherford Appleton Laboratory
ISIS Neutron Scattering Training Course : Large Scale Structures Module : May 2010
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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
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
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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