Post on 21-Jun-2020
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