by

Roger Pynn

Los Alamos National Laboratory

LECTURE 3: Surface Reflection

Surface Reflection Is Very Different From Most Neutron Scattering

• We worked out the neutron cross section by adding scattering from different nuclei– We ignored double scattering processes because these are usually very weak

• This approximation is called the Born Approximation

• Below an angle of incidence called the critical angle, neutrons are perfectly reflected from a smooth surface– This is NOT weak scattering and the Born Approximation is not applicable to

this case

• Specular reflection is used:– In neutron guides– In multilayer monochromators and polarizers– To probe surface and interface structure in layered systems

This Lecture

• Reflectivity measurements– Neutron wavevector inside a medium– Reflection by a smooth surface– Reflection by a film– The kinematic approximation– Graded interface– Science examples

• Polymers & vesicles on a surface• Lipids at the liquid air interface• Boron self-diffusion• Iron on MgO

– Rough surfaces• Shear aligned worm-like micelles

What Is the Neutron Wavevector Inside a Medium?

[ ]

materials.most from reflected externally are neutrons 1,ngenerally Since2/1

:get we)A 10(~ small very is and ),definition(by index refractive Since

material ain r wavevecto theis and vevector neutron wa is where 4/)(2 Simlarly ./2 so )(

0)(/)(2

:equation sr'Schrodinge obeysneutron The(SLD)Density Length Scatteringnuclear thecalled is

1 where

2 :is medium theinside potential average theSo

nucleus. single afor )(2)( :bygiven is potentialneutron -nucleus

theRule,Golden sFermi'by given with thatS(Q)for expressionour Comparing

2

2-6-0

0

20

22220

.

22

2

2

<=

==

−=−===

=−+∇

==

=

∑

πρλ

ρ

πρψ

ψ

ρ

ρρπ

δπ

-n

nk/k

kin vacuokkVEmkmEkerin vacuo

rVEm

bvolumem

V

rbm

rV

rki

ii

o hh

h

h

rhr

rr

Typical Values

• Let us calculate the scattering length density for quartz – SiO2

• Density is 2.66 gm.cm-3; Molecular weight is 60.08 gm. mole-1

• Number of molecules per Å3 = N = 10-24(2.66/60.08)*Navagadro= 0.0267 molecules per Å3

• ρ=Σb/volume = N(bSi + 2bO) = 0.0267(4.15 + 11.6) 10-5 Å-2 = 4.21 x10-6 Å-2

• This means that the refractive index n = 1 – λ2 2.13 x 10-7 for quartz

• To make a neutron “bottle” out of quartz we require k= 0 I.e. k0

2 = 4πρ or λ=(π/ρ)1/2 . • Plugging in the numbers -- λ = 864 Å or a neutron velocity of

4.6 m/s (you could out-run it!)

Only Those Thermal or Cold Neutrons With Very Low Velocities Perpendicular to a Surface Are Reflected

α

α’nickelfor )(1.0)( :Note

quartzfor )(02.0)(

sin)/2(

A 1005.2 quartzFor

4 is reflection external for total of valuecritical The

4or 4sin'sin i.e.

4)sin(cos)'sin'(cos 4 Since

Law sSnell'obey Neutrons'/coscosn i.e 'cos'coscos

:so surface the toparallellocity neutron ve thechangecannot surface The

/

critical

critical

critical

1-3

00

20

2220

22

2220

22220

2

00

0

0

0

A

A

k

xk

kk

kkkk

kkkk

nkkk

nkk

o

o

critical

critical

zz

zz

zl

zl

λα

λα

αλπ

πρ

πρπραα

πρααααπρ

ααααα

≈

≈

⇒=

=

=

−=−=

−+=+−=

===

=

−

Reflection of Neutrons by a Smooth Surface: Fresnel’s Law

n = 1-λ2ρ/2π

TTRRII

TRI

kakaka

aaarrr

&

=+

=+⇒=

(1) 0zat & of

continuityψψ

)/()(/by given is ereflectanc sosinsin

sinsin

)()(

(3) & (1)

coscos :Law sSnell' (2) & (1)

(3) sinsin)((2) coscoscos

:surface the toparallel andlar perpendicu components

TzIzTzIzIR

Iz

Tz

RI

RI

TRI

TRI

kkkkaarkk

naaaa

n

nkakaankakaka

+−==

=′

≈′

=+−

=>

′==>

′−=−−

′=+

αα

αα

αααα

ααα

What Do the Amplitudes aR and aT Look Like?

• For reflection from a flat substrate, both aR and aT are complex when k0 < 4πρ I.e. below the critical edge. For aI = 1, we find:

Real (red) & imaginary (green) parts of aRplotted against k0. The modulus of aR is plotted in blue. The critical edge is at k0 ~ 0.09 A-1 . Note that the reflected wave is completely out of phase with the incident wave at the critical edge

Real (red) and imaginary (green) partsof aT. The modulus of aT is plotted inblue. Note that aT tends to unity at large values of k0 as one would expect

0.005 0.01 0.015 0.02 0.025

-1

-0.5

0.5

1

0.005 0.01 0.015 0.02 0.025

0.5

1

1.5

2

V(z

)

z

One can also think about Neutron Reflection from a Surface as a

1-d Problem

V(z)= 2 π ρ(z) h 2/mn

k2=k02 - 4π ρ(z)

Where V(z) is the potential seen by the neutron & ρ(z) is the scattering length density

substrate

Film Vacuum

Fresnel’s Law for a Thin Film

• r=(k1z-k0z)/(k1z+k0z) is Fresnel’s law

• Evaluate with ρ=4.10-6 A-2 gives thered curve with critical wavevectorgiven by k0z = (4πρ)1/2

• If we add a thin layer on top of thesubstrate we get interference fringes &the reflectance is given by:

and we measure the reflectivity R = r.r*

• If the film has a higher scattering length density than the substrate we get the green curve (if the film scattering is weaker than the substance, the green curve is below the red one)

• The fringe spacing at large k0z is ~ π/t (a 250 A film was used for the figure)

0.01 0.02 0.03 0.04 0.05

-5

-4

-3

-2

-1

Log(r.r*)

k0z

tki

tki

z

z

errerr

r1

1

21201

21201

1++

=

substrate

01 Film thickness = t2

Kinematic (Born) Approximation

• We defined the scattering cross section in terms of an incident plane wave & a weakly scattered spherical wave (called the Born Approximation)

• This picture is not correct for surface reflection, except at large values of Qz

• For large Qz, one may use the definition of the scattering cross section to calculate R for a flat surface (in the Born Approximation) as follows:

z

4222

22)'.(2

00

20

yx

yx

Q largeat form Fresnel theas same theis that thisshow easy to isIt

/16 so )()(4

'

:substratesmooth afor get section we cross a of definition theFrom. sin so cos because

sinsin1

sin1

sin

)sinLL( sampleon incident neutrons ofnumber

LL size of sample aby reflected neutrons ofnumber

zyxyxz

rrQi

xx

yx

yxyxyx

QRQQLLQ

erdrddd

dkdkkk

k

dkdk

dd

LLd

dd

LLLL

R

ρπδδπ

ρρσ

ααα

ασ

ασ

αασ

α

===Ω

−==

Ω=Ω

Ω==

Φ==

∫ ∫

∫∫

−rvrrr

Reflection by a Graded Interface

ion.justificat physical good havemust dataty refelctivifit torefinedmodels that means This n.informatio phaseimportant lack generally we

because (z),obtain ouniquely t inverted becannot dataty reflectivi :pointimportant an sillustrateequation eapproximat This (z).1/cosh as

such /dzd of forms convenient severalfor ly analytical solved becan This

)(

Q largeat formcorrect theas wellas interface,smooth afor answer right get the we,Rty reflectivi Fresnel by theprefactor thereplace weIf

parts.by ngintergratiafter followsequality

second the where)(16

)(16

:gives of

dependence-z thekeepingbut viewgraphprevious theof line bottom theRepeating

2

2

z

F

2

4

22

2

2

ρ

ρ

ρ

ρπρ

πρ

∫

∫∫

=

==

dzedz

zdRR

dzedz

zdQ

dzezQ

R

ziQF

ziQ

z

ziQ

z

z

zz

The Goal of Reflectivity Measurements Is to Infer a Density Profile Perpendicular to a Flat Interface

• In general the results are not unique, but independent knowledge of the system often makes them very reliable

• Frequently, layer models are used to fit the data• Advantages of neutrons include:

– Contrast variation (using H and D, for example)– Low absorption – probe buried interfaces, solid/liquid interfaces etc– Non-destructive– Sensitive to magnetism– Thickness length scale 10 – 5000 Å

Direct Inversion of Reflectivity Data is Possible*

• Use different “fronting” or “backing” materials for two measurement of the same unknown film– E.g. D2O and H2O “backings” for an unknown film deposited on a quartz

substrate or Si & Al2O3 as substrates for the same unknown sample– Allows Re(R) to be obtained from two simultaneous equations for

– Re(R) can be Fourier inverted to yield a unique SLD profile

• Another possibility is to use a magnetic “backing” and polarized neutrons

Unknown film

Si or Al2O3 substrateSiO2

H2O or D2O

22

21 and RR

* Majkrzak et al Biophys Journal, 79,3330 (2000)

Vesicles composed of DMPC molecules fuse creating almost a perfectlipid bilayer when deposited on the pure, uncoated quartz block*

(blue curves)

When PEI polymer was added only after quartz was covered by the lipidbilayer, the PEI appeared to diffuse under the bilayer (red curves)

10 -9

10 -8

0.00 0.02 0.04 0.06 0.08 0.10 0.12

Ref

lect

ivity

, R*Q

z4

Qz [Å-1 ]

Lipid Bilayer on Quartz

Lipid Bilayer on Polymer on Quartz

Neutron Reflectivities

-8 10 -6

-6 10 -6

-4 10 -6

-2 10 -6

0

2 10 -6

Sca

tterin

g Le

ngth

Den

sity

[Å-2

]

Length, z [Å]

Hydrogenated Tails

Hea

d

Hea

d

Quartz

Quartz

Polymer

0.0 20.0 40.0 60.0 80.0 100.0

2O

D2O

σ = 4.3 Å

σ = 5.9 Å

Scattering Length DensityProfiles

* Data courtesy of G. Smith (LANSCE)

1.3% PEG lipid in lipids

4.5% PEG lipid in lipids

9.0% PEG lipid in lipids

10-3

10 -2

10 -1

10 0

101

0 0.1 0.2 0.3 0.4 0.5 0.6R

/RF

Qz

Pure lipid

Lipid + 9% PEG

X-Ray Reflectivities

Polymer-Decorated Lipids at a Liquid-Air Interface*

Interface broadens as PEG concentration increases - this is main effect seen with x-rays

mushroom-to-brush transition

neutrons see contrast betweenheads (2.6), tails (-0.4),D2O (6.4) & PEG (0.24)

x-rays see heads (0.65), but allelse has same electron densitywithin 10% (­0.33)

10 -9

10 -8

10-7

0 0.05 0.1 0.15 0.2 0.25

Neutron Reflectivities

R*Q

z4

Qz [Å -1]

Pure lipid

Lipid + 9% PEG

0

2 10 -6

4 10 -6

6 10 -6

8 10 -6

40 60 80 100 120 140 160

SLD Profiles of PEG-Lipids

SLD

[Å-2

]

Length [Å]

Black - pure lipidBlue - 1.3% PEGGreen - 4.5% PEGRed - 9.0% PEG

*Data courtesy of G. Smith (LANSCE)

Non-Fickian Boron Self-Diffusion at an Interface*

10 -11

10 -10

10-9

10 -8

10-7

10 -6

10-5

10 -4

0.045 0.05 0.055 0.06 0.065 0.07 0.075 0.08

10-11

10-9

10-7

10-5

0.001

0.1

10

0 0.02 0.04 0.06 0.08 0.1 0.12

Boron Self Diffusion

Unnanealed sample2.4 Hours5.4 Hours9.7 Hours22.3 Hours35.4 Hours

Ref

lect

ivit

y

Q (Å-1

)

0

1 10 -6

2 10 -6

3 10 -6

4 10 -6

5 10 -6

6 10 -6

7 10 -6

8 10 -6

-800 -600 -400 -200 0 200 400 600 800

Non-Standard Fickian Model

Scat

teri

ng L

engt

h D

ensi

ty (

Å-2

)

Distance from 11

B-10

B Interface (Å)

0

1 10 -6

2 10 -6

3 10 -6

4 10 -6

5 10 -6

6 10 -6

7 10 -6

8 10 -6

-600 -400 -200 0 200 400 600 800

Standard Fickian Model

Scat

teri

ng L

engt

h D

ensi

ty (

Å-2

)

Distance from 11 B- 10 B Interface (Å)

11 B

10B

Si

~650Å

~1350Å

Fickian diffusiondoesn’t fit the data

Data requires densitystep at interface

*Data courtesy of G. Smith (LANSCE)

Polarized Neutron Reflectometry (PNR)

Non-Spin-Flip Spin-Flip++ measures b + Mz- - measures b - Mz

+- measures Mx + i My-+ measures Mx – i My

Structure, Chemistry & Magnetism of Fe(001) on MgO(001)*

10-6

10-5

10-4

10-3

10-2

10-1

100

0 0.1 0.2 0.3 0.4 0.5

X-R

ay R

efle

ctiv

ity

Q [Å-1]

2.1(2)nm

1.9(2)nm

10.7(2)nm Fe

MgO

β = 3.9(3)

β = 5.2(3)

β = 4.4(2)

β = 3.1(3)

0.20(1)nm

0.44(2)nm

0.07(1)nm

0.03(1)nm

X-Ray

β = 4.4(8)

β = 8.6(4)

β = 6.9(4)

β = 6.2(5)

Neutron

α-FeOOH

interface

X-Ray

Neutron

Co-Refinement

10 -5

10 -4

10 -3

10 -2

10 -1

100

0.00 0.02 0.04 0.06 0.08 0.10 0.12

R++(obs)R

++(fit)

R--(obs)

R--(fit)RSF(obs)

RSF(fit)

Pol

ariz

ed N

eutr

on R

efle

ctiv

ity

Q [Å -1]

H

α-FeOOH (1.8µB)

Fe(001) (2.2µB)

H

fcc Fe (4µB)

TEM

*Data courtesy of M. Fitzsimmons (LANSCE)

Reflection from Rough Surfaces

• diffuse scattering is caused by surface roughness or inhomogeneities in the reflecting medium

• a smooth surface reflects radiation in a single (specular) direction• a rough surface scatters in various directions• specular scattering is damped by surface roughness – treat as graded

interface. For a single surface with r.m.s roughness σ:

k1k2

kt1

z

x

θ1 θ2z = 0

212 σt

zIzkkFeRR −=

When Does a “Rough” Surface Scatter Diffusely?

• Rayleigh criterion

path difference: ∆r = 2 h sinγ

phase difference: ∆φ = (4πh/λ) sinγ

boundary between rough and smooth: ∆φ = π/2

that is h < λ/(8sinγ) for a smooth surface

γ

γ γ

γ

h

where g = 4 π h sin γ / λ = Qz h

Qx-Qz Transformation

x

z

Q =kf -kiQ x =

2 πλ *(cos θf-cos θi )

Q z =2 πλ *(sin θf +sin θi )

Qkf

-kiki

θfθi

where

For specular reflectivity θ f =θi . Then,

Qz=4 πsin( θ)

λ =2k z

Raw Data from SPEARRaw Data from SPEAR

Vandium-Carbon Multilayer — specular & diffuse scatteringin θf -TOF space and transformed to Qx-Qz

Time-of-Flight, Energy-Dispersive Neutron Reflectometry

Raw data in θf -TOF space for a single layer. Note that large divergence does not imply poor Qz resolution

TOF ­ λ . L / 4

The Study of Diffuse Scattering From Rough Surfaces Has Not Made Much Headway Because Interpretation Is Difficult

down.break toon theory perturbatifor ion wavefunctsurface" average"an using a ofion approximat expect the also wouldone surfaces) faceted (e.g. cases someIn

surface in the nscorrelatio of range theis where

,1/ down when breaksbut cases somein workssurface,rough a from

scattering describe toused ion)ApproximatBorn Wave(Distorted theory The22

ξξ >>kkz

220

210

2

2

sksmoothfacet

skksmoothroughmicro

eRR

eRRt

−

−−

=

=

Observation of Hexagonal Packing of Thread-like Micelles Under Shear: Scattering From Lateral Inhomogeneities

TEFLON LIP OUTLET TRENCH

INLET HOLES

TEST SECTION

NEUTRON BEAM

TEFLON

QUARTZ or SILICON

RESERVOIRS

46Å

Up to Microns

H2O

Quartz Single

Crystal

z

Flow direction

x

Scattering patternimplies hexagonalsymmetry

Thread-like micelle

Specularly reflected beam