Neutron reflectivity for the study of biological interfaces

Tim SaldittInstitut für Röntgenphysik, Universität Göttingen

ECNS'2003 Introductory course

φ f

α i

E in fa lle n d e rS tra h l

α f

φ i

6 2 n m

1 4 0 n m

G e b e u g te rS tra h l

Institut für Röntgenphysik Göttingen

X-ray nanobeams,interface-sensitive x-ray and neutron scattering

Inelastic scattering, anomalous scatteringnanostructures,

x-ray opticselectron lithography

macromolecules and assemblies, lipid bilayers, membrane peptides and proteins, spider silk

Sources of this talk

Course material:

Metin Tolan http://e1.physik.uni-dortmund.de/Guillaume Brotons thesis Univ. de Maine/ILL

Further reading:

Hercules course, X-ray and neutron reflectivity (A. Gibaud, J. Daillant ed., Springer, X-ray physics (J. Als-Nielsen)

Original work and data obtained in fruitful collaboration with:

Christian Münster, Ulrike Mennicke, Doru ConstantinFranz Pfeiffer, Christoph Ollinger LMU München / Uni Saarland/ Uni Göttingen

M. Rheinstaedter, L. Perrino-Gallice, G. Fragneto ILL Grenoble

outline

A. Motivation1. Lipid membranes as biological model systems2. Sample preparation and experiment

B. Foundations of neutron reflectivity1. From the wave equation to Fresnel reflectivity2. Neutron (and x-ray) optical indices 3. The effect of roughness or fluctuations4. Multiple interfaces

C. Applications to lipid bilayers1. Specular reflectivity 2. Non-specular (diffuse) reflectivity

protein crystallography

Numerous Nobel laureates: Laue, Bragg, Debey, Perutz&Kendrew, Hodgkin,Watson&Crick, Deisenhofer & Huber & Michel.... (a total of 21 !!!)

crystallography: biomolecules in a 3D cage

1. Why lipids are important: an electron microscopy study of cells, compartimentation of a cell: organelles and the abundance of membranes...

The building blocks of biological membranes: Lipids

hydrophil

hydrophob

1,2-Dimyristoyl-sn-Glycero-3-Phosphatidylcholin (DMPC)1,2-Dilauroyl-sn-Glycero-3-Phosphatidylcholin (DLPC)1-Palmitoyl,1-Oleoyl-sn-Glycero-3-Phosphatidylcholin (POPC)1-Palmitoyl,1-Oleoyl-sn-Glycero-3-Phosphatidylserin (POPS)1,2-Dimyristoyl-sn-Glycero-3-Phosphatidylglycerol (DMPG)

micellar, hexagonal, lamellar phases, inv. hexagonal phase...

Phase transitions within the lamellar phasesIsraelachvili Intramolecular forcesCevc Phospholipid bilayersSackmann/Lipowsky Handbook of Biological Physics

Lipid polymorphism

MolekulardynamikHeller et al., J-Phys.Chem 93

molecular snapshots

DPPC DPPE

Molecular dynamics Reciprocal space mapping (MD Heller et al.)

Multilamellar structures in the cell

Thykaloid-Membrane in Chloroplasten Rauhes Endoplasmisches Reticulum

biological membranes= lipid bilayers + membrane proteins

A small polypeptides in the bilayer: Alamethicin (from fungus)

hinge (Prolin14)polar

C-Terminus

N-Terminus

hydrophob

polar

Ac-MeA-Pro-MeA-Ala-MeA-Ala-Gln-MeA-Val-MeA-Gly-Leu-MeA-Pro-Val-MeA-MeA-Gln-Gln-Phol

Example 2: Magainin 2 (from frog, antibiotic)

PolarLysineLysine

N-Terminus

Hydrophob

C-Terminus

A

A

KK

K

K

S

H

G

G

G

I

VL

FF

F

S

E

N

M

1G

Gly-Ile-Gly-Lys-Phe-Leu-His-Ser-Ala-Lys-Lys-Phe-Gly-Lys-Ala-Phe-Val-Gly-Glu-Ile-Met-Asn-Ser

Isotropic lipid suspensions

Information loss due to ´powder´ averaging

qzqx

HochorientierteProbe

isotrope Lösung“Pulver”

Si - Wafer

Si - Wafer

Substrate

N

Lipid solutionsLipid/Peptide mixtures

2. Preparation of lipid membranes on solid surfaces

Hydrophilic surface

Mennicke & Salditt, Langmuir 02

Imaging of bilayer terraces by AFM ( DMPC )

L. Perrino-Galice et al EJPE 02

Measurement chambers for controlled T and humidity

Neutron beam can pass through wateror substrate (e.g. silicon ) !

Si

B. Neutron reflectivity from oriented membranes

Reciprocal space

Reflectivity and density profile

x-ray, noScattering length density from generalized Fresnel theory

-60-50-40-30-20-10

01020304050

ρ

Z

1. From the wave equation to the Fresnel equations

X-rays: Helmholtz equation

Neutron: wave equation = Schrödinger eq.

Solve equations with boundary conditions (continuity of wave field and its derivative !

The stationary Schrödinger equation for neutrons

A neutrons wavefunction can be described by the Schrödinger equation:

[ ] 0)(2 2

2

=−Ε+ ψψ rVrd

dmh

neutron energy :neutron wavevector:

λπ2

0 =kmk

2

20

2

0h

=Ε

Thermal neutrons as particle waves

Neutron at given energy: with scattering vector:

λmh

mk

22

20

2

0 ==Εh

if kkqv −=vv

izq θλπ sin4

=if αα =for

Similar to x-ray scattering formalism

Reflected amplitude and intensity from Schrödinger eq.

Solving the Schrödinger equation (V only z dependent):

[ ] 02 2

2

=−Ε+ zzzz V

zdd

mψψh zikzik

z BeAe ´´ −+=ψ

We get (with continuity of ψ and ∇ψ at the interface):

with

[ ] bkVEmk i πρ42´ 22

2 −=−=h

With the same formalism as in x-ray we can derive the fresnel equations:

´sinsin´sinsin

αααα

nnr

i

i

+−

= 2

0

0

trzz

trzz

kkkkR

+−

=And in terms of

scattering wave vectors:

´sinsinsin2

ααα

nt

i

i

+=

or more generally from a scalar wave and continuity conditions:

Wavevector: Incident: kI, Amplitude: aIReflected: kR aR

and Transmitted: kT; aT

Continuity ofthe wave andits derivative

at the interface

In vacuum:

In the material:RI kkk ==

Tkkn =

=+=+

TRI

R

kkk TRI

TI

aaaaaa (1)

(2)

ααααα

′=

′=+(nk)sin-a)ksina-(a

s)(coscos

TRI

conkakaka TRI

-

k componentsparallel:

and perpendicular:

Snell-Descartes law⇒ αα ′= scos nco

(3)

(4)

Snell‘s law (geometric construction)

Snell-Descartes law⇒ αα ′= scos nco

Case of Light reflectivity:nvacuum<nmaterial

Case of X-rays:nvacuum>nmaterial

Case of Neutrons: ?

Constant ν (elastic process, energy conservation)Speed v and wavelength (λ) changed in the medium

Refractive index: x-rays and neutrons

magneticpart+

magneticpart+

DispersionAbsorptionMinus!!

Refraction of waves at interfaces

n1<n2

n1

n2

n1>n2x-rays, neutronsn1 = 1 (air), n2 > 1

n1

n2

visible light, neutrons n1 = 1 (air), n2 > 1

Total external reflection occurs below the critical angle

cos αi = (1– δ) cos αt αt=0Critical Angle:αc ≈ √2δ ~ 0.3°

GRAZING ANGLES !!!

Critical angle

with snell’s law: ´coscos αα ni =

nc =αcosfor n<1 : α and α´ are small:

βαα

βδαα

ii

c 222

22

22

−+′=

−+′=λ

πρα b

c =with:

and the corresponding wavevector:

bq cc πρ

λαπ 4sin4

==

Fresnel coefficients (Single interface)

sinsin

αα

αα ′

≅′

=+−

naaaa

RI

RICombiningeqs (1) and (4):

In fact:

+=

+

−=

TzIz

Iz

TzIz

TzIz

kkk

t

kkkk

r

,,

,

,,

,,

2

⇒ Fresnel Reflectivity

and Transmittivity coefficients

in amplitudes (complex coefficients).

′+≈=

′+′−

≈=

ααα

αααα

2

I

T

T

R

aat

aar

Only depend on kz

Where:( )2

0,2

0,,c

lzzlz kkk −= λπ /2, 0 =k

( ) ( ) ( )020

220

20, 4 bbknnk ll

critlz ρρπ −=−=Angle :

( ) ( )

( ) ( )

=

=

=

−

nzk

ekk

tT

rR

Tz

zk

I

Tz Tz

ImIm

Re

,

Im2,2

2

,

TR −= 1

Critical

( ) 02

0,, Re4 kibkk llzlz µρπ −−=i.d.

total external reflection regime in the Fresnel reflectivity RF(αi)

Fresnel transmission function TF = |t|2

Penetration depth

Λ

Λ = (kp–)-1

In scattering, it is conveniant to define the wavevector transfer :

ccC kkQkkQ αααα 2sin2;2sin2 ≈=≈=

In terms of their dimensionless counterparts:

αα ′

≈=′

≈=

c

c

cc Qk

QQq

Qk

QQq 2;2

Therefore eq. (6) can be rewritten as:

µibqq 2122 −+′=

µ22

cµ Q

kb =Where bµ is related to the adsorption coefficient µ via:

Where the wavevector Qc at the critical angle:

)1(42 0 ZfrkQ cc

′+== πρα

In terms of the newly defined variables, the Fresnel reflectivity and transmittivity amplitudes are:

)qIm(Q

1(q)

,2)(

)(

c ′=Λ

′+=

′+′−

=

qqqqt

qqqqqr

and

α´ is complex number,

Intensity falls off

like:

)Im(21

α′=Λ

k

zket )Im(α′−∝

1/e penetration depth Λ:

Re-phrased in a different notation

ReflectedAmplitude

TransmittedAmplitude

Wave-Vectors

Fresnel reflectivity (of a single sharp interface)

−+ += ppt iα

2. Neutron optical indices (index of refraction)Mean potential independent of x- and y- coordinates

∫ ==v

bm

rdrVv

V ρπ 23 2)(1 h

Solving the Schrödinger eqation:

[ ] 0)(2

22

2

2

2

=+=−Ε+ ψψψψ krd

drVrd

dmh

With:

[ ]VEmk −= 22 2

hLeads to the definition:

bEV

kkn ρ

πλ2

20

22 11 −=−==

The index of refraction is linked to the scattering length

ikr

ibi : scattering length (m)

of the chemical element idefines the degree of interaction between the element i and the beam

deb

AA uki

ii

drr

.

0

∝

; X-rays : For one electronbi=2.85 10-15 m

A0

dkr

One point

rdV erbqA r.qi rrr rr

∫= −)()( ρ

ij ii brrb )()( rr ρρ ∑=

Density of scattering length (m-2)k

Summation over the volume

Scattering amplitude: length (m)

id kkqrrr

−=λ

θπ )sin(4 nq =

2θ

dkr

ikr

rrPhase differencebetween volumes

1 and 2 :r.q rr1

2

Two points

( ) ( )2 20 0

4 21 1l l ln b b

k kπ πρ ρ= − ≈ −

Refractive index: x-rays and neutrons

magneticpart+

magneticpart+

DispersionAbsorptionMinus!!

X-rays: the scattering length = classical electron radius

Electron DensityProfile !

E = 8 keV λ = 1.54 Å

The scattering length for neutrons varies from nucleus to nucleus

Negative!

Absorption:µn ≈ 0

Scattering Length Densityof the

Nuclei !

Contrast (∆ρb) ? Neutrons & X-rays Ex: polystyrene (Latex) in water

bT: X-ray scattering length for an electron = 2.82 10-15 m2bT =

0

2

4 cme

eπε( )molecular

iraysX V

beNb

−

− =ρX-rays

CH2

CH

CH2

CH

Br

Mstyrène = 104.15 g/moldstyrène = 1.06 g/cm3

N e- = 8*6+8 = 56ρbpolystyrène = 0.343 e-/Å3 = 9.633 1010 cm-2

ρbbromostyrène = 0.444 e-/Å3 = 12.47 1010 cm-2 ∆ρb = 0.11 e-/Å3

∆ρb multiplied by 12(∆ρb)2 multiplied by 150

Meau = 18.0152 g/mold = 1 g/cm3

N e- = 8+2 = 10ρbwater = 0.334 e-/Å3 = 9.38 1010 cm-2Water

∆ρb = 0.253 1010 cm-2

= 0.009 e-/Å3Styrene, PS

Mbromo = 183 g/moldbromo = 1.5 g/cm3

N e- = 8*6+7+35 = 90

Bromo-styrene

b(1H) = -3.74 10-15 mb(2D) = +6.67 10-15 m

b(C) = +6.6 10-15 mb(O) = +5.8 10-15 mNeutrons ρ ∑= iiNeutrons brb )(rρ

Styrene PSH ρ Styrene PSDbPS-H = +1.41 1010 cm-2 ρbPS-D = +6.46 1010 cm-2

ρbH2O = -0.56 1010 cm-2 ρbD2O = +6.38 1010 cm-2Heavy Water

D2OWater H2O

Fresnel Reflectivity

Measurement

X-Ray Reflectivity:Water Surface

Difference Experiment-

Theory:RoughnessRoughness !!!!

3. Real Interfaces: the effect of roughness

Braslau et al. PRL 54, 114 (1985)

Structure of a rough or undulated interface

Integration

ProbabilityDensity

Refractive Index Profile n(z)

ElectronElectron DensityDensity Profile Profile ρρ(z)(z)

Structure affects averaging procedure

Same Roughness σ & Refractive Index Profile n(z) !Lateral Structure Different

Different Averaging Procedures: σ/ξP < 1 or σ/ξP >1

At small kz: subtle differences ...

Beckmann-Spizzichino Result (1963):

Nevot-Croce Result (1980):

Reflection Coefficient

TransmissionCoefficient

TransmissionCoefficient

Reflection Coefficient

σj Exponential Dampingof Reflectivity!

σ/ξP <1

σ/ξP >1

Roughness damps reflectivity

σj = 10 Åλ = 1.54 Å

4. Multiple interfaces

0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,910-14

10-12

10-10

10-8

10-6

10-4

10-2

100

102

104

106

108

22 bilayers 10 bilayers 9 bilayers simulation

X-r

ay re

flect

ivity

[a.u

.]

qz [Å-1]

10-6

Kiessig Oszillationen

Parrat-Algorithm (1954)

Fresnel CoefficientsSingle Interfaces Iteration- Start

Reflectivity

n(x,y,z) = n(z) + δn(x,y,z)

Lateral Distortions

DiffuseScattering

Refractive Index Profile

Reflectivity

Refractive Indexof the

samplen(x,y,z)

5. Diffuse (nonspecular scattering)

ki

kfq qz

qx

Wave-Vector: q = q = kkff –– kkii

Diffuse Scattering:qx , qy ≠ 0qz= (4π/λ)sin(αi+αf)/2

Reflectivity: qx= qy = 0qz= (4π/λ)sinαi

αi , αf < 5°

Scattering geometry

C. Applications to oriented membranes

Neutron reflectomer ADAM at ILL

Neutron reflectivity of DMPC in Lα phase –effect of hydration

0,0 0,1 0,2 0,3 0,4 0,5

101

102

103

104

105

106

DMPC fully hydrated DMPC partially hydrated

In

t. [a

rb.u

.]

Qz [Å-1]

X-ray reflectivity (OPPC)

( ) 2)(25.0 ),,()()()( 00

22

NLqSeqFepeeqRFqR zdiq

zdiqqlq

zzzzzz ρδ ϖσ ∆+∆+∆= −−

Parametrisierung durch Fourierkoeffizienten. / therm. Unordnung

0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,910-8

10-6

1x10-4

10-2

100

R

efle

ktiv

ität

Experiment Simulation

qz (Å-1)

ρelectron

Z

substrate

membraneswater

Thermal fluctuations in solid supported bilayers are subject to boundary conditions

0=∂

∂

= Lzz)z,q(u II00 =),q(u II

∑−

=

δ=1

1

N

nnn )z(f)q(u)z,q(u IIII

22 dTkB

λπ

=ηB B

K=λ

• Fouriertransformation: rII -> qIIElimination of surface terms By boundary conditions

• N-1 independent Eigen-modes in fourier space

• equipartition theorem -> correlation functions

• specular reflectivityr rII= 0

with and

)z,z,C(q)z,r(u)z,r(u ′=′ IIIIII

( ) ( )∑−

=

−−π π⋅η=

1

12

12212

120N

nLzn

nd sin,z)C(

0.0 0.2 0.4 0.6 0.8 1.00.0

0.5

1.0

1.5

2.0

2.5

C(0

,z)/η

[ Å

2 ]

z/L

N=20

The lamellar periodicity „d“ is determined by interaction potentials

∂∂

= 2w

2

dVdB

Total potential for DMPC

20 30 40 50 60 70 80

-4.0x10-6

0.0

4.0x10-6

[ Jm

-2 ]

dw [ Å ]

Gesamtpotential Vhyd+VVan der Waals

harmonische Näherung Hydration Van der Waals

attractive interaction :

Van der Waals potential

Repulsive Interaction:

Hydration-Interaction

Electrostatic repulsion (solution of the Poisson-Boltzmann-equation)

Undulation-interaction(according to Helfrich)

( ) hwd

eHd wλ−

= 0HydV

( ) ( )[ ]2114 1

2

wewewe dLdLdLwd σσπ +−⋅= Tk

elBV

( )

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

dTkT,d

hn

hn

n d/axexpd/axexp

nr

h

Bh

−−

π=

−∆−

−−∆∞′

=

∞

∑ ∫2

11

0

2

21

8VdWV

dK κ

=

V.A. Parsegian und R.P. Rand, (1989), Biochim. Biophys. Acta 988: 351W. Fenzl, (1995),Z. Phys. B 97, 333-336

Si

N bilayersH2O

master formula

form factor and structure factor

+=ρ σ−

∞

∫22

2

0

zqiqz edzedzd

∑−

=

−+−+−−

⋅+⋅+

σ 1

0

22

2

02

2

2

02

22 N

n

u)ndd(iqz

u)ndd(iqz

* nzq

znzq

zzq

ee)q(Ffee)q(Ffe

( )∑ ∑−

=

−

=

+−−−

⋅+

1

0

1

0

2 222

2N

m

N

n

uu)nm(diqz

nmzq

z ee)q(Ff

( ) ( )∫−

⋅ρ

=2

2

0

D

D

ziqz dzez

dzdqFf z

( )n

N

nSi udnz),d(Erf)z( +⋅−ρ+σ−−ρ=ρ ∑

−

=

1

000

2

∫∞

∞

⋅⋅=0

1 dzedzd ρ

ρ(q)RR(q) iqz

F

Reflectivity of bilayer can be modelled in semi-kinematic theory

Fluctuation amplitudeun=C(0,z=nd,z´=nd)

Modelling with increasing fluctuation amplitude

0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,710-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

experimental data semikinematic fit

x-ra

y re

flect

ivity

[a.u

.]

qz [ Å-1 ]

16 DMPC Membranes on Silicon in PEG solution 3.6% wt.

0 2 4 6 8 10 12 14 160,0

0,2

0,4

0,6

0,8

1,0n

n

coverage functionf(n) α=1.7

0 2 4 6 8 10 12 14 16

0

10

20

30

40

[Å2 ]

fluctuation amplitudeσ2(n), η = 0.065

Density profile OPPC, T=45° C

-20 -10 0 10 20

0,20

0,25

0,30

0,35

0,40

0,45

MD sim (Heller et al.) reflectivity fit

ρ(z)

[e- /Å

3 ]

z [Å]

Position and conformation of peptides

-40 -20 0 20 40 60

0,25

0,30

0,35

0,40

0,45

P/L=0 P/L=1/100 P/L=1/30

Elek

trone

ndic

hte

[Å-3]

Z (Å)

Konformation aus Dichteprofil ?Konzentrationsgetriebener Phasenübergang ?

Contrast variation by selective deuteration

0 10 20 30 40 50 60-0,2

0,0

0,2

0,4

0,6

0,8

1,0

deuterated chains / H2O

z [Å]

scat

terin

g le

ngth

den

sity

0 10 20 30 40 50 60-0,2

0,0

0,2

0,4

0,6

0,8

1,0

deuterated heads / H2O

0 10 20 30 40 50 60-0,2

0,0

0,2

0,4

0,6

0,8

1,0

all deuterated lipid / H2O

0 10 20 30 40 50 60

-0,2

0,0

0,2

0,4

0,6

0,8

1,0

all deuterated lipid / D2O

0,00 0,05 0,10 0,15 0,20 0,25 0,301E-11

1E-9

1E-7

1E-5

1E-3

0,1

10

1000

Alamethicin in DMPC 1/25 molar ratio

chains deut. in H2O heads deut. in H2O all deut. lipid in H2O all deut. lipid in D2O

neut

ron

refle

ctiv

ity [a

.u.]

qz [Å-1]

H20 / D2O exchange

0,00 0,05 0,10 0,15 0,20 0,25 0,301E-9

1E-8

1E-7

1E-6

1E-5

1E-4

1E-3

0,01

0,1

1

10

D16 Okt 00

Alamethicin in DMPC d63P/L = 1/25 40°C / 100 % r.h. in H2O

in D2O

neut

ron

refle

ctiv

ity a

.u.

qz [Å-1]

2. Diffuse scattering due to fluctuationsz

x

n du(x,y,z)

∫

∇+

∂∂

= 222

][ uKzuBdVH

Thermal fluctuations

surface tension =0

12

,2

2)/ln(2

,2

232

222

==><=><

==><=><

dLTNLTh

daLTNLTh

κπδ

κπ

κπδ

κπ

Principles of condensed matter physics, Chaikin & Lubensky

Elasticity coefficients K and B

[ ]3/ mJd

dB

∂Π∂

−=[ ]mJd

K /

=

κ

Reciprocal space: diffuse Bragg-sheets

uncorrelated vs. correlated (conformal) fluctuations

Reciprocal space: diffuse Bragg-sheets

0,000 -0,001 -0,002 -0,003 -0,004 -0,0050,085

0,090

0,095

0,100

0,105

0,110

0,115

0,120

Qx

Qz

Vogel, Fenzl,Münster, Salditt Phys.Rev.Lett. 00Münster, Vogel, Salditt, Euro.Phys.Lett 99Salditt, Münster, Vogel, Fenzl PRE 99

The diffuse scattering reflects the positional correlation functions

Diffuse scattering of DMPC fluid phase, partiall hydrationAdam/ILL

0.1 0.01 1E-3 1E-4 1E-5

103

104

105

106

Inte

nsity

[arb

.u.]evanescent

wave

diffusespecular

qII [Å-1]

100

αi

αf

ki

ka

qz

qx

Vogel, Fenzl,Münster, Salditt Phys.Rev.Lett. 00Münster, Vogel, Salditt, Euro.Phys.Lett 99

The effect of membrane-active biomolecules on thermal fluctuations and elasticity: the example DMPC/antibiotic peptide Magainin 2

0,0002 0,0000 -0,0002 -0,0004 -0,0006

0,095

0,100

0,105

0,110

0,115

0,120

0,125

-0,010 -0,015 -0,020 -0,025 -0,030 -0,035 -0,040

0,000 -0,002 -0,004 -0,006

0,080

0,085

0,090

0,095

0,100

0,105

0,110

-0,010 -0,015 -0,020 -0,025 -0,030 -0,035 -0,040

0,000 -0,002 -0,004 -0,0060,105

0,110

0,115

0,120

0,125

0,130

0,135

0,140

-0,010 -0,015 -0,020 -0,025 -0,030 -0,035 -0,040

200

100

30

Outlook: time of flight (TOF) specular and non-specular neutronreflectometry (D17/ILL)

Analysis of diffuse Bragg sheet -> elasticity constants

1E-5 1E-4 1E-3 0,01

10-2

10-1

100

101

diffuse

TOF, full hydration monochr., full hydration monochr., partial hydration discrete smectic model Λ=40 Å In

tegr

. int

ensi

ty (I

int λ

) [no

rm]

qx [Å-1]

specularDMPC, Lα-phase, S(qx ,qz=2π/d)

1 2 3 4 5 6 7 8

0,0

0,2

0,4

0,6

0,8

1,0

HW

HM

[Å]

θ [deg]

λ-width parabolic fit resol. corr. smectic model

T. Salditt, C. Münster, C. Ollinger, G. Fragneto, Langmuir (2003)

Appendix: further neutron techniques for the study of lipid membranes and biological systems: inelastic neutron scattering

The IN12- a triple axis spectrometer at a cold guide of ILL

Inelastic an elastic scans at the same time !

Dispersion relation of collective modes in acyl chains

M. Rheinstädter, C. Ollinger, G. Fragneto, T. Salditt, in preparation