Download Neutron Reflectivity

ISIS Neutron Scattering Training Course : Large Scale
Structures Module : May 2010
Introduction to Neutron Reflectivity
J.R.P.Webster
Detector
Sample
S1
Monitor 1
S2
Frame Overlap
Mirrors
S3
Supermirror
S4
Monitor 2
ISIS Facility, Rutherford Appleton Laboratory
Specular reflection of neutrons
from surfaces and interfaces
Analagous to optical interference,
ellipsometry
Equivalent to electromagnetic radiation with
electric vector perpendicular to the plane
of incidence
Depth Profiling : provides
information on concentration
or composition profile perpendicular
to the surface or interface
(Penfold, Thomas, J Phys Condens Matt, 2
(1990)1369,
T P Russell, Mat Sci Rep 5 (1990) 171 )
Reflectometry
Kinetics
Model Devices
Polymer Diffusion
Thin polymer films (finite size effects)
Critical exponents in SCF
Spin coating
Protein unfolding
Non equilibrium surfactant films
Temporal resolution of
Surfactants
Ion transfers
Parametric Studies
Solvent transfers
Liquid/Liquid Interface
Polymer structure
Reduce Label size in Structural Studies
Electrochemistry
Electrodeposition and Surface
nucleation
Self Assembly of systems
Metal Hydroxide
electroprecipitation
(batteries)
Novel templating
mechanisms
Self Assembly
Foams
Biology
•Protein adsorption
•Biocompatible polymers
•Drug transport
•Anaesthesia mechanisms
Atmospheric
chemistry
Protein
Resistant
Surfaces
Sustainable
Laundry
Detergents
Lung
Surfactant
Gene
delivery
Inorganic
Templating:
Organic
Light
Emitting
Diodes
Surfactant Adsorption
Biosensors
Neutron
Reflectivity
Ionic
Liquids
Specular reflection of neutrons
no
Refractive index defined
using the usual convention
in optics:
n = k
2
A = Nb 2 π
B=
k
0
n1
n = 1 − λ
N (σ a + σ i )
1
A − iλ B
X-rays
n = 1 − α − iβ
4π
α = Nλ2 Z re 2π
β = λ μ 4π
Refractive Index for neutrons
n =
k1
k0
n = 1 − λ 2 A − iλ B
A =
N b
2π
Extensively use H/D
isotopic substitution
to manipulate “ contrast “
or refractive index
H -0.374 x 10-12 cm
D 0.667 x 10-12
n < 1.0 hence
TOTAL EXTERNAL REFLECTION
Specular reflection of neutrons
( some basic optics )
no
From Snell’s Law,
θo
θ1
At total reflection
n1
λ2
cos θ c = 1 − Nb
2π
For
Nb
π
(
n1 sin θ1 = n − n cosθ
2
1
2
0
θ < θ c n1 sin θ1 is
θ > θ c n1 sin θ1
2
0
)
1
2
n1 cos θ 0
=
n0 cos θ1
θ0 = θc
θ 1 = 0 .0
Total reflection
Critical angle
θc = λ
n=
cos θ1 = 1.0
( R=1.0 ) for
θ < θc
θ > θ c Fresnel’s Law
n0 sin θ 0 − n1 sin θ1
R=
n0 sin θ 0 + n1 sin θ1
imaginary ( Evanescent wave )
Is real, and zero at
θ = θc
2
Specular reflection of neutrons
( some basic optics )
no
From Snell’s Law,
θo
θ1
At total reflection
n1
λ2
cos θ c = 1 − Nb
2π
For
Nb
π
(
n1 sin θ1 = n − n cosθ
2
1
2
0
θ < θ c n1 sin θ1 is
θ > θ c n1 sin θ1
2
0
)
1
2
n1 cos θ 0
=
n0 cos θ1
θ0 = θc
θ 1 = 0 .0
Total reflection
Critical angle
θc = λ
n=
cos θ1 = 1.0
( R=1.0 ) for
θ < θc
θ > θ c Fresnel’s Law
n0 sin θ 0 − n1 sin θ1
R=
n0 sin θ 0 + n1 sin θ1
imaginary ( Evanescent wave )
Is real, and zero at
θ = θc
2
Some typical values for θc and σa
Material
θc (deg / Å)
Ni
Si
Cu
Al
D2O
0.1
0.047
0.083
0.047
0.082
Material
σa(barns)
Si
Cu
Co
Cd
Gd
0.17
3.78
37.2
2520
29400
Specular Neutron Reflection
(simple interface)
Within Born Approximation the
Reflectivity is given as,
16π 2
− iQz
ρ ′(z )e
R (Q ) =
dz
∫
4
Q
Q = k − k = 4π sinθ / λ
1
2
Reflectivity from a simple single interface is then given by Fresnels Law
R=
n0 sin θ 0 − n1 sin θ1
n0 sin θ 0 + n1 sin θ1
16π 2 2
R(Q) = 4 Δρ
Q
2
2
Specular Neutron Reflection
For thin films see interference effects that can be described using
standard thin film optical methods
For a single thin film at an interface
− 2 iβ
rij
(
p
=
(p
r01 + r12 e
R (Q ) =
1 + r01 r12 e − 2 i β
i
i
− pj)
+ pj)
p i = n i sin ϑ
βi =
2π
λ
ni di sinθi
2
n0
θ0
θ1
n1
n2
d
For a single thin film :
r012 + r122 + 2r01r12 cos 2n1k1d1
R(Q ) =
1 + r012 r122 + 2r01r12 cos 2n1k1d1
For Q>>QC :
[
]
16π 2
2
2
(
)
(
)
R (Q ) ~
ρ
−
ρ
+
ρ
−
ρ
+ 2(ρ1 − ρ 0 )(ρ 2 − ρ1 ) cos(Qd )
1
0
2
1
4
Q
Fourier transform of 2 delta functions (young’s slits)
FRINGE SPACING :
2π
ΔQ =
d
Rough or Diffuse Interface
For a simple interface reflectivity
modified by,
(
R = R0 exp − q0 q1σ
σ
2
)
qi = 2k sin θ i
k = 2π
is rms Gaussian roughness
λ
( Nevot, Croce, Rev Phys Appl
15 (1980) 125, Sinha, Sirota, Garoff,
Stanley, Phys Rev B 38 (1988) 2297)
Gaussian factor ( like Debye-Waller factor ) results
in larger that q-4 dependence in the reflectivity.
Can be also applied to reflection coefficents in formulism for thin films,
rij
(
p −p )
=
exp(− 0.5(q q σ ))
(p − p )
i
j
2
i
i
j
j
From specular reflectivity cannot distinquish between
roughness and diffuse interface
Reflectivity from a simple interface
Effect of roughness
and sld
Glass optical flat
θ = 0.35
Nb = 0.35 x10 −5 Α −2
σ = 33Α
Δθ = 5%
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
d = 1198Α
Nb = 0.74 x10 −5 Α − 2
θ = 0.5, Δθ = 4%, σ = 20 Α
NiC film on silicon
d = 1194 Α, Nb = 0.94 x10 −5 Α −2
θ = 0.5, Δθ = 4%, σ 1 = 10, σ 2 = 15Α
Reflection from more complex interfaces
( multiple layers )
Airy’s fomula
( Parratt )
Nb
z
Combination of reflection and transmission
coefficients give amplitude of successive beams reflected,
r1 , t1t1' r2 ,−t1t1' r1r22 , t1t1' r12 r23
and so on
2π
δ
=
Phase change on traversing film, 1 λ n1d1 sin θ1
( Parratt, Phys Rev 95 91954) 359
G B Airy, Phil Mag 2 (1833) 20)
R = r1 + t1t 2' r2 e −2iδ1 − t1t1' r1r22 e −4iδ1 + K
More general matrix formulisms ( Born & Wolf, Abeles ) available
Reflection from multiple layers
n0
n1
Born and Wolf matrix formulism
nj
Applying conditions that wave functions and their
gradients are continous at each boundary gives
rise to a Characteristic matrix per layer,
⎡ cos β j
Mj = ⎢
⎣− ip j sin β j
p j = n j sin θ j
− (i p j )sin β j ⎤
⎥
cos β j
⎦
β j = (2π λ )n j d j sin θ j
ns
( Born & Wolf, ‘Principles in Optics’,
6th Ed, Pergammon, Oxford, 1980)
M R = [M 1 ][
. M 2 ] − − − −[M n ]
The resultant reflectivity is
⎡ (M 11 + M 12 ps ) pa − (M 21 + M 22 ) ps ⎤
R=⎢
⎥
(
)
(
)
+
+
+
M
M
p
p
M
M
p
12 s
21
22
a
s⎦
⎣ 11
2
Reflection from multiple layers
n0
In Born and Wolf approach can only include
roughness / diffusiveness at interfaces by
further sub-division in small layers.
n1
nj
Abeles method,
ns
using reflection
coefficients overcomes this limitation
Define characteristic matrix per layer, in optical terms from the relationship
between electric vectors in successive layers,
iβ j −1
⎡ e
C j = ⎢ −iβ j−1
⎢⎣rj e
iβ j −1
⎤
−iβ j −1 ⎥
e
⎥⎦
rj e
( Heavens, ‘Optical properties of solid thin films’,
Butterworths, London, 1955, F Abeles, Annale de
Phys 5 (1950) 596)
The resultant Reflectivity is
then,
⎡a b ⎤
C1 . C2 − − − − Cn +1 = ⎢
⎥
⎣c d ⎦
To include roughness,
[ ][ ]
p
(
r =
(p
j −1
[
]
) exp− 0.5q q
+p )
− pj
j
j
j −1
σ2
j −1
j
R = CC AA
*
*
Multiple Layer films
Region around 1st order Bragg peak for Ni/Ti multilayer
15 bilayers ( 46.7, 1.0 x 10-5 / 55.7,-0.13x10-5)
Effects of resolution
ΔQ 2
Q2
= Δt
2
t
Δθ
2 +
2
θ2
On SURF and CRISP resolution
is dominated by collimation
1000 Å film on Si , ΔQ/Q 2%, 6%
Damps interference fringes, rounds critical edge
Surface roughness and Waviness
Curvature >> coherence length
Curvature << coherence length
Rough
Waviness
This initially has an effect similar to resolution, and in the extreme can
be treated by geometrical optics.
Incoherent reflectivity from 2 surfaces, separated by an adsorbing media:
2
(
1 − R1 (Q )) R 2 (Q ) A (Q )
R tot (Q ) = R1 (Q ) +
1 − R1 (Q )R 2 (Q ) A (Q )
Model fitting Reflectivity data
Scattering length
reflectivity
density
Steepest decent, simplex,
simulated annealing,
genetic, cubic spline + fft,
etc etc
}
•Uniqueness ?
•Resolution ?
•Model dependent /
overinterpretation of data ?
•Does the scattering length
density profile give access to
the necessary physical
parameters (Intra molecular)
?
Lateral (z) and rotational
invariance
=
=
0.0
ρ(z)
=
Z=0?
z
=
Partial Structure Factors
R (κ ) =
16 π 2
κ
2
+∞
∫ ρ (z )e
2
− iκz
dz
−∞
ρ( z ) = bc nc ( z ) + bh nh ( z ) + bs ns ( z )
R(κ ) =
16π 2
κ
2
[b h
2
c cc
+ bh2 hhh + bs2 hss + 2bcbh hch + 2bcbs hcs + 2bhbs hhs
Self Partial Structure Factors :
hii = n$i
2
n$i is a one dimensional Fourier transform of ni ( z )
Cross partial structure factors:
{ }
hij = Re n$i n$ j
( Crowley, Lee, Simister, Thomas, Penfold, Rennie,
Coll Surf 52 (1990) 85 )
]
Cross Partial Structure Factors
If one distribution is shifted by
δ
nˆi' ( z ) = ni ( z − δ )
hij = Re{nˆi nˆ j }exp(iκδ ij )
nˆi' (κ ) = ni (κ ) exp(iκδ )
If both even functions
If even + odd functions
±
exp( iκδ )
Fourier transform is changed by phase factor
[
hij = ± hii h jj
δ
δ
]
1
cos iκδ
2
[
hij = ± hii h jj
because of phase uncertainty
Model Self-terms as Gaussian ( solvent as tanh )
]
1
2
sin iκδ
Effect of capillary wave and structural roughness
on cross-terms
Widths of individual
distributions affected
by roughness,
but separations
are NOT
Partial Structure Factor Analysis
haa
hss
Structure of binary
non-ionic mixtures
5 x10-5 M 30/70
C12E3 / C12E8
C12E3
C12E8
C12E8
C12E3
has
Example of simplest labelling scheme :
solvent, alkyl chain of each surfactant
Neutron Reflectivity at ISIS
Measure variation of reflectivity
with scattering vector, Qz,
perpendicular to the interface
Detector
Sample
INTER, POLREF, OFFSPEC,
SURF, CRISP reflectometers
( Penfold, Williams, Ward, J Phys E 20 91987) 1411; J Penfold et al,
J Chem Soc, Faraday Trans, 94 (1998) 955
S1
Monitor 1
S2
Frame Overlap
Mirrors
S3
Supermirror
S4
Monitor 2
Using ‘white beam’ TOF method
with fixed angle and range
of wavelengths
Instrumentation
Correct for detector efficiency,
spectral shape, background
R (Q (λ i , θ
)) =
f
[I d (λ i ) −
[I m (λ i ) −
b d (λ i )] ε
b m (λ i )] ε
m
d
(λ i )
(λ i )
d,m refer to the detector and monitor
Corrected data
Monitor
Raw data
D 2O
Specular, 1.5°
Background ( off-specular), 2°
Instrumentation
Silicon / water interface
Si/D2O
Si/H2O
Si/30%D2O
0.35°
0.8°
1.8
°
Reflectometry Village
Inter
Designed 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
Scientific Opportunities
Biology
Cell adhesion using synthetic polymer
analogues
Kinetics of action of interfacial enzymes
Interfacial structure of designed peptides
(folding)
Biofouling and adsorption kinetics
Interlayer forces in polymer and biological
systems
Supported bilayers
Polymer diffusion
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
Inter
Designed 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
PolRef
Uses polarised neutrons to study the inter an intra-layer magnetic
ordering in thin films and surfaces
>20 times the flux of CRISP
b = bN ± bM
Much wider dynamic range
Flexible polarisation
POLREF
Dual Geometry
High precision sample stage
CRISP
Scientific Opportunities
Spin Electronics
Spin-Injection
Spin Torque
Dilute magnetic semiconductors
Giant/Tunnelling magneto-resistance
Model Magnetic Systems
Ultrathin films (finite size effects)
Exchange springs (domain walls, surface
magnetic phase transitions)
Stabilise new single-crystal phases (Ce,
Mn,..)
wavelength range
0.9 – 16 Å
Moderator
Coupled s-CH4 grooved – 26K
Primary flight path
23m
Secondary flight path
3m
Beam size
60(h) x 30(v) mm
Flux at sample
~107 n/s/cm2
PolRef
Uses polarised neutrons to study the inter an intra-layer magnetic
ordering in thin films and surfaces
>20 times the flux of CRISP
b = bN ± bM
Much wider dynamic range
Flexible polarisation
POLREF
Dual Geometry
High precision sample stage
CRISP
wavelength range
0.9 – 16 Å
Moderator
Coupled s-CH4 grooved – 26K
Primary flight path
23m
Secondary flight path
3m
Beam size
60(h) x 30(v) mm
Flux at sample
~107 n/s/cm2
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 Layer
Silicon
Polarised Neutrons for Biology
n = 1 − λ A − iλ B
Nb
protein
A –=Extract
thickness and
orientation
2
π
– Better resolution
than conventional
btotalAFM=studies
bnuclear ± bm
•
Magnetic Layer
Silicon
Use polarised 2
neutrons to provide
additional
information for
protein absorption
In-plane dynamic range
of 50Å-42μm
Scientific Opportunities
In-plane Structures
Patterned Storage Media
Mesoporous films
Polymers
Biological membranes
Surfactants
Grazing Incidence Diffraction
Surface crystalline structure
Surface phase transitions
Magnetic surface structure
Neutron Spin-Echo
Simulated,
measured
signal.
Simulated,
measured
reflectivity.
θ0
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)
sample
Larmor precession codes
scattering angle
Qz
θ
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
Spin echo small angle scattering in
transmission (SESANS)
Classical Spin echo in transmission or
reflection of inelastic samples
Realisation/Design
Nov 05
Realisation/Design
Aug 08
Z
Realisation/Design
Z
May 09