Applications of Neutron Scattering
Prof. Thomas Brückel
IFF - Institute for Scattering Methods
& RWTH Aachen - Experimental Physics IV c
Neutron Laboratory Course 2009
Length- and Time Scales covered by
Research with Neutrons
Sword Swallower
Radiography / Tomography
Neutron Radiography and Tomography
running car engine
ki
t=0
kf
i j
Q r ( iQ r i t i
coh i, j
) ( )
0 t
S (Q, ) 1 e e e dt
2 N
∞ − ω − ⋅
−∞
ω =
⋅⋅
π h ∫ ∑
Scattering: Correlation Functions
Cross Section & Scattering Functions
self correlation function:
incoherent scattering function:
pair correlation function:
depends only on time dependent positions of atoms in sample!
coherent scattering function:
cross section:
for nuclear scattering ∂Ω∂∂2σω = kk'⋅ ⋅N ⎡⎢⎣
(
| |b 2−| |b 2)
Sinc( , ) | |Q ω + b 2 Scoh( , )Q ω ⎤⎥⎦phase space densityfactor
"interaction strength"
probe-sample interaction
propertyof system studied
( ) 3
( , ) 1 ( , )
2
i Q r t
Scoh Q ω G r t e ω d rdt π
= h
∫
⋅ −Fourier transform in space and time
3
i j
3
( , ) 1 ( ' r (0)) ( ' r ( )) ' 1 ρ( ',0) ρ( ' , ) '
ij
G r t r r r t d r
N
r r r t d r N
δ δ
= − ⋅ + −
= ⋅ +
∑∫
∫
( ) 3
( , ) 1 ( , )
2
i Q r t
inc s
S Q ω G r t e ω d rdt π
= h
∫
⋅ −3
j j
( , ) 1 ( ' r (0)) ( ' r ( )) '
s
j
G r t r r r t d r
N δ δ
=
∑∫
− ⋅ + −Point like scatterers
Particledensity
Elastic Scattering
intermediate scattering function: ( , ) : ( , ) 3 ( , ) '( , ) S Q t G r t eiQ rd r
S Q S Q t
= ⋅
=
∫
∞ + where ( , ) lim ( , )t
S Q S Q t
∞ = →∞
( , ) 1 ( , )
2
S Q ω S Q t e i tω dt π
+∞
−
−∞
⇒ = h
∫
1 ( , ) '( , ) 2
S Q S Q t e i tω dt π
+∞
−
−∞
⎡ ⎤
= h
∫
⎣ ∞ + ⎦1 1
( ) ( , ) '( , ) 2
i t
elastic scattering
inelastic scattering
S Q S Q t e ω dt
δ ω π
+∞
−
−∞
= h ∞ + h
∫
1442443 144424443 Elastic scattering: infinite time correlation function
e. g. "particles at rest"
(compare tennis ball reflected from wall!) Examples: - Bragg scattering from crystal: elastic
- scattering from liquids: purely inelastic
S(Q,t)
S’(Q,t) S(Q,∞)
S’(Q,t) S(Q,∞)
S(Q,ω)
(Quasi-) Static Approximation
• If the detection does not discriminate the final neutron energy, we measure an integral cross section for fixed direction k$'of k':
$ 2
,int '
coh k const
d d
d
σ σ ω
ω =
⎛⎜⎝ Ω⎞⎟⎠ =
∫
∂Ω∂∂ ⋅• Since Q = k - k' and , Q will vary
with E' or ω as this integral is performed (m ≠0 ⇒ neutron parabola!).
( )
2
2 '2
' 2
E E k k
ω = − = hm − h
• The (quasi-) static approximation neglects this variation, uses Q0 for ω = 0, and is valid only, if the energy transfer is small compared to the initial energy (or if the movement of the
atoms is negligible during the propagation of the radiation wave from one atom to another):
(
0)
0 0
( ) 3
,
3 3
' ( , )
2
' '
( , ) ( ) ( ,0)
2 2
i Q r t coh QSA
iQ r iQ r
d k N
G r t e d rdt d
d k
k N k N
G r t e t d rdt G r e d r
k k
σ ω ω
π
π δ π
⋅ − ⋅
⋅ ⋅
⎛ ⎞ =
⎜ Ω⎟
⎝ ⎠
= =
∫ ∫
∫ ∫
h
h h
Integral scattering in quasistatic approximation: instantaneous spatial correlations; "snapshot".
k samplek'
detector
Summary: Correlation Functions
• coherent scattering:
• incoherent scattering:
• magnetic scattering:
• elastic scattering:
• integral scattering in (quasi-) static approximation:
pair correlation between
different atoms at different times one particle self correlation
function at different times spin pair correlation function;
vector quantity ↔ polarisation
infinite time correlation
"time averaged structure"
instantaneous correlations
"snapshot„
σ σ'
t t
t
inelastic elastic
• define k
i(k
i= 2π/λ
i) and k
fwith collimators and “monochromatizers”
• inelastic scattering (spectroscopy):
determine change of neutron energy during scattering process
m E k
2
2
h2
=
• two possibilities to define neutron energy E:
- diffraction from single crystal (Neutron as wave) - time-of-flight (Neutron as particle)
Principle of Scattering Experiment
collimation collimation
monomono-- chromatization chromatization
Scattering Scattering
@ sample
@ sample
energy energy
analysis analysis
detection detection
definition of definition of scattering angle scattering angle
select k
iselect k
fDirect geometry
Diffraction
Diffraction: scattering without energy analysis
either true elastic scattering (e. g. Bragg scattering from crystals) or quasistatic scattering (e. g. slow dynamics in polymer melts)
⇒ determination of the position of the scatterers the movement is neglected !
Relation between characteristic real space distance d and magnitude of scattering vector
4 2
sin :
Q Q
d
π θ π
= λ ≈ (compare Laue function: distance between maxima Q·d=2π)
example
Atom-atom distance in crystals
Co precipitates in Cu matrix
d 2 Å
400 Å
Q 3.14 Å-1
0.016Å-1
2θ (λ=10 Å)
"cut-off"
1.46°
technique
wide angle diffraction
small angle scattering (λ=1 Å)2θ
29°
0.14°
Small Angle Neutron Scattering SANS
SANS: large scale structures
wavelength: reasonable scattering angles → (! direct beam separation) Pin-hole SANS: definition of ki through distant apertures
Focussing SANS: focus entrance aperture onto detector
Å 15 Å
5
Å 10 Å
10 Å
10 Å
10 4 1 1 4 1
→
≈
→
≈
⇔
→
≈ − − − −
λ
Q d
SANS: Resolution
Resolution: "Smearing of signal due to finite performance of instrument"
Optimisation: the better the resolution (better angular collimation, Δθ, smaller wavelength spread Δλ), the smaller the intensity
( ) ( )
2 2
2 2
2
4 sin Q
Q Q
Q
π θ λ
θ λ
θ λ
= ⇒
∂ ∂
⎛ ⎞ ⎛ ⎞
Δ =⎜⎝ ∂ ⎟⎠ Δ +⎜⎝ ∂ ⎟⎠ Δ
2 ( ) 2
2 2 2
2
4 4 sin
π cos θ θ π θ λ
λ λ
⎛ ⎞ ⎛ ⎞
=⎜⎝ ⎟⎠ Δ +⎜⎝ ⎟⎠ Δ
2 ( ) 2
2 2
0
4
θ
π θ θ λ
λ λ
→↑
⎡ Δ ⎤
⎛ ⎞ ⎛ ⎞
= ⎜ ⎟ ⎢ Δ + ⎜ ⎟ ⎥
⎝ ⎠ ⎢⎣ ⎝ ⎠ ⎥⎦
2 2
2 2
2
2
12
S S
D E
D C C D
d d
d d
k
L L L L
θ λ λ
⎡⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛Δ ⎞ ⎤
⎢ ⎥
= ⎢⎣⎜⎝ ⎟⎠ +⎜⎝ ⎟⎠ +⎜⎝ + ⎟⎠ + ⎜⎝ ⎟⎠ ⎥⎦
Optimised → all terms have similar values
→ LD = LC; dE = dD = 2dS
typical: LD = LC = 10 m; dD = dE = 3 cm Detector radius ~ 30 cm = : rD
1 10%
10
E D E
C D D
d L d
L r r
λ λ
⇒ Δ = ⋅ ≈ ≈ = ⇒ velocity selector
velocity selector
“screw thread – principle”
10
-510
-410
-310
-210
-110
-210
010
210
410
610
8Q
-20.5% sPP-P(E-co-P) 38.4K/106.6K in d-22
Q
-1Q
-3d Σ /d Ω [cm
-1]
Q [Å
-1]
DKD KWS3 KWS2
A. Radulescu et al., Neutron News 16 (2005), 18
Correlations ?
4 µm
Networks of bundels with mass-fractal
aspect: “snow flake”
Rods on medium length scale
2-d structure on shortest length scale Rods organize as bundles
Selforganisation of crystalline- amorphous diblock-copolymer
SANS: Example
• Polymers and colloids, e.g.
Micelles Dendrimers Liquid crystals Gels
Reaction kinetics of mixed systems
• Materials Science
Phase separation in alloys and glasses Morphologies of superalloys
Microporosity in ceramics
Interfaces and surfaces of catalysts
• Biological macromolecules
Size and shape of proteins, nucleic acids and of macromolecular complexes
Biomembranes Drug vectors
• Magnetism
Ferromagnetic correlations
Flux line lattices in superconductors Magnetic nanoparticles
Micelles
Superalloy
Flux line lattice Protein
shape
SANS: Applications
Large Scale Structures: Reflectometry
soap-bubbles: colours due to interference:
destructive (here: blue)
constructive (here: red)
⇒ determination of film thickness
(soap bubble ~ µm)
Monochromatization
Velocity selector
Chopper
Time-of-flight TOF
Crystal monochromator
Large Scale Structures: Reflectometry
Schematics of a neutron reflectometer:
primary collimationslits monitor
thin film sample on goniometer reflected beam
PSD: position
sensitive detector monochromator
white beam from source
A C
d n0
n1 n2
α α
αt
D
.
B
Path length difference:
AD n
BC
AB+ ⋅ −
=
Δ ( ) 1
dn sinαt
2 1
=
Distance of interference maxima (neglect refraction on top surface):
Q d
d α π
λ = 2 ⋅(Δ )⇒ Δ ≈ 2
Layer Thickness
0.00 0.02 0.04 0.06 0.08
Q = 4π/λ∗sinθ (A−1)
−5
−4
−3
−2
−1 0 1
Log(R)
Nickel on Glass
Points measured on HADAS
fit : d = 837,5 A ; σ = 14,5 A ; δQ = 2,08e−3 A−1 simulation : d = 837,5 A ; σ = 14,5 A ; δQ = 0 simulation : d = 837,5 A ; σ = 0 ; δQ = 0 simulation : substrate only
Example: Reflectivity of neutrons from a Ni layer on glass substrate
(Neutron guide):
“Kiessig fringes”
Q 2
d Δ = π
• Soft Matter:
Thin films, e.g. polymer films: polymer diffusion, selforganization of
diblockcopolymers;
surfactants; liquid-liquid interfaces,…
• Life science:
Structure of biomembranes;
• Materials Science:
Surfaces of catalysts; Kinetic studies of interface
evolution; structure of buried interfaces
• Magnetism:
Thin film magnetism, e.g.
exchange bias, laterally structured systems for magnetic data storage, multilayers of highly
correlated electron systems
polymers Biomembranes
Catalyst surfaces
Buried interfaces
Magnetic Random Access Memory
MRAM
Spinvalve
Reflectometry: Applications
Atomic Structures: Single Crystal Diffraction
Example: D9 at ILL:
Monochromator instrument:
2 sind θ λ=
Monochromatization by Bragg diffraction from a single crystal
Atomic Structures: Powder Diffraction
Example: D2B at ILL/Grenoble
Overlap of Reflections: Rietveld-Refinement
Example:
Solutions:
Rietveld-Refinement (profile refinement)
- refine structural parameters (unit cell metric, atom positions and site occupations, Debye- Waller-factors, …)
together with instrument parameters (2θ0, U, V, W, …)
Pair Distribution Function analysis (PDF)
How to determine structural parameters?
Resolution function:
(
Δ2θ)
2 =U tan2θ +V tanθ +W- Bragg reflections overlap for larger unit cells e. g. due to finite peak width.
CMR
Manganite
• Life science:
Structure of biological
macromolecules, e.g. water in protein structures
• Chemistry:
Structure determination of new compounds, position of light atoms; Time resolved reaction kinetics
• Materials science:
Stress / Strain in structure materials; texture
• Geoscience:
Phase and texture analysis
• Solid state physics:
Structure-function relations, e.g. in high-Tc super-
conductors; magnetic
structures and spin densities e.g. in molecular magnets
Water in proteins
Zelolite structure
reaction kinetics
strain analysis
Spin density in molecular magnets
Phase and texture analysis Spin structures
Diffraction: Applications
neutrons
Generic TOF Spectrometer
t v d
v h m
p
=
⋅
=
=
&
λ = ⋅ d ⋅ λ h
t m
(typically 1 ms/m)
Path-Time Diagram
Example for an Application
Neutron spectroscopy from the molecular magnet Mn12 acetat:
determination of magnetic interaction parameters (Güdel et al.) Molecular structure Excitation spectrum
Measured with TOF
Energy level diagram
Applications TOF Spectroscopy
Membrane dynamics
• Soft Matter and Biology:
dynamics of gels, proteins and
biological membranes; diffusion of liquids, polymers; dynamics in confinement
• Chemistry:
vibrational states in solids and adsorbed molecules on surfaces; rotational
tunnelling in molecular crystals
• Materials Science:
molecular excitations in materials of technological interest (e.g. zeolites) and especially in diluted systems (matrix isolation); local and long-range
diffusion in superionic glasses, hydrogen-metal systems, ionic conductors.
• Solid State Physics:
quantum liquids; crystal field splitting in magnetic systems; spin dynamics in
high-TC superconductors; phase transitions and quantum critical
phenomena; phonon density of states
vibrational modes in nanotubes, peapods
hydrogen in metals
magnetic excitations in high TC
Neutron Spin Echo Spectroscopy
Δλ/λ=
10%
ds s B dt = γ ×
Larmor- precession Problem:
Conventional TOF: high resolution requires good monochromatization
→ low intensity Solution:
Neutron Spin Echo NSE: each individual neutron carries its own clock
to measure its individual time of flight
Example NSE: Polymer Dynamics
• • • • •
• • • • •
• • • • •
• • • • •
• • • • •
• • • • •
. . . and in reciprocal space
monochromator
axis 1
sample
axis 2
analyzer
axis 3
detector
in real space . . .
reactor shielding
monochrom.
shielding
φ
ki
000 Q
Ghkl q kf
φ
Q = k
f– k
i= G
hkl+ q
inelastic scattering !
Triple-Axis Spectroscopy
monochromator shielding
sample table
analyzer shielding
detector shielding
glass floor air pads
TAS-Example: SV-30 / FZJ
thermal
motion eigenmode
(optical phonon)
Lattice & Spin Dynamics
ferromagnetic spin wave (magnon)
antiferromagnetic spin wave (magnon)
Determination of Spin Wave Dispersions
Determination of interaction (exchange) parameters
Dispersion relations for the garnet Fe2Ca3Ge3O12:
eigenmodes
magnetic structure
E [THz]
counts per 10 min
constant Q-scans
Brillouin zone
Th. Brückelet al
TAS-Applications
Chiral phase transitions
qchain[2πÅ-1]
Low dimensional magnets Phonons in
High TC
• Phonon dispersions
→ interatomic forces
• Spin wave dispersions
→ exchange and anisotropy parameters
• Dynamics of biological model membranes
• Lattice and spin excitations:
Quantum magnets, superconductors, …
• Phase transitions:
critical behaviour
Spin dynamics in frustrated systems
Experimental techniques with spatial resolution:
Neutron Diffraction compared to other experimental techniques
Experimental techniques
with time / energy resolution:
Neutron spectroscopy compared to other experimental techniques
Comparison of Techniques
TAS