Research with Neutrons Development of the Universe

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Applications of Neutron Scattering

Prof. Thomas Brückel IFF - Institute for Scattering Methods

& RWTH Aachen - Experimental Physics IV c

Neutron Laboratory Course 2007 Length- and Time Scales covered by

Research with Neutrons Development of the Universe

The Neutron - A Laboratory on the fm Scale

Phase transitions of the universe - and observables from neutron experiments

γd_n g

ιn nbar m_{WR, ζ}

ϕ q_n

aWM σv V_ud g_A/g_VN_Vσppαn

A_γ, P_γ α^-1α^,β, γ,

δ, ν, η

Sword Swallower

Radiography / Tomography

Neutron Tomography Disk Drive

Neutron Radiography and Tomography

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k_i t=0

k_f

i j

Q r ( iQ r i i t coh

i, j

) ()

0 t

S (Q, ) 1 e e e dt

2 N

∞ −ω − ⋅

−∞

ω = π ∫ ∑

⋅

Scattering: Correlation Functions 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 ² ^kk^'^N

(

^{| |}^b² ^{| |}^b²

)

^S^inc^{( , ) | |}^Q ^b²^S^coh^{( , )}^Q

σω ω ω

∂ = ⋅ ⋅⎡⎢⎣ − + ⎤⎥⎦

∂Ω∂

phase space density factor

"interaction strength"

probe-sample interaction property of system studied

( )3

( , ) 1 ( , )

2

i Q r t

ScohQω G r t e ^ωd rdt π

=

∫

⋅ −

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 rt

inc s

S Qω π G r t e ^ωd rdt

=

∫

⋅ −

3

j j

( , ) 1 ( ' r (0)) ( ' r ( )) '

s j

G r t r r r t d r

N δ δ

=

∑∫

− ⋅ + −

Elastic Scattering

intermediate scattering function: ( , ) : ( , ) ³ ( , ) '( , )

i Q r

S Q t G r t e d r S Q S Q t

= ⋅

=

∫

∞ + where ( , ) lim ( , )

S Q t S Q t

∞ =→∞

( , ) 1 ( , )

2

S Qω S Q t ei t^ωdt π

+∞

−

−∞

⇒ =

∫

1 ( , ) '( , )

2

S Q S Q t ei t^ωdt π

+∞

−

−∞

⎡ ⎤

=

∫

⎣ ∞ + ⎦

1 1

( ) ( , ) '( , )

2

i t

elastic scattering

inelastic scattering

S Q S Q t e^ωdt

δ ω π

+∞

−

−∞

= ∞ +

∫

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 directionk'of k':

2

,int '

coh kconst

d d

d

σ σω = ω

⎛ ⎞ = ∂ ⋅

⎜ Ω⎟ ∂Ω∂

⎝ ⎠

∫

• Since Q = k - k' and , Q will vary with E' orωas this integral is performed. ^ω^{= −}^E ^E^'⁼₂_m²

(

^k²⁻^k^'2

)

• The (quasi-) static approximation neglects this variation, uses Q₀forω= 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 π ω ω

π δ π

⋅ − ⋅

⋅ ⋅

⎛ ⎞ =

⎜ Ω⎟

⎝ ⎠

= =

∫ ∫

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

f

with collimators and “monochromatizers”

• inelastic scattering (spectroscopy):

determine change of neutron energy during scattering process

m E k

2

2 2

=

• 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

energy energy analysis analysis

detection detection

definition of definition of scattering angle scattering angle

select k

_i

select k

_f

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Diffraction

Diffraction: scattering without energy analysis

either trueelastic scattering(e. g. Bragg scattering from crystals) orquasistatic 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

(λ=10 Å)2θ

"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 ≈10Å÷10⁴Å=Q:10⁻¹Å⁻¹÷10⁻⁴Å⁻¹ wavelength: reasonable scattering angles→ λ ≈ ÷5 15 Å

Pin-hole SANS: definition of k_ithrough distant apertures

Focussing SANS: focus entrance aperture onto detector

SANS: Resolution

Resolution: "Smearing of signal due to finite performance of instrument"

Optimisation: the better the resolution (better angular collimation, Δθ, smaller wavelength spreadΔλ), the smaller the intensity

( ) ( )

2 2

2

4sin Q

Q Q

Q

π θ

λ

θ λ

= ⇒

∂ ∂

⎛ ⎞ ⎛ ⎞

Δ =⎜⎝∂⎟⎠ Δ +⎜⎝∂⎟⎠ Δ

2 ( ) 2

2 2 2

2

4 4 sin

π cosθ θ π θ λ

λ λ

⎛ ⎞ ⎛ ⎞

=⎜⎝ ⎟⎠ Δ +⎜⎝ ⎟⎠Δ ( )

2 2

0

4

θ

π θ θ λ

λ λ

↑→

⎡ Δ ⎤

⎛ ⎞ ⎛ ⎞

=⎜⎝ ⎟⎠⎢⎢⎣Δ + ⎜⎝ ⎟⎠⎥⎥⎦

2 2

2

12

S S

D E

D C C D

d d

k

L L L L

θ λλ

⎡⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛Δ⎞⎤

⎢ ⎥

= ⎢⎣⎜⎝ ⎟⎠ ⎝+⎜ ⎟ ⎜⎠ ⎝+ + ⎟⎠+ ⎜⎝ ⎟⎠⎥⎦ Optimised→all terms have similar values

→L_D= L_C; d_E= d_D= 2d_S typical: L_D= L_C= 10 m; d_D= d_E= 3 cm

Detector radius ~ 30 cm = : r_D

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

^-5

10

^-4

10

^-3

10

^-2

10

^-1

10

^-2

10

⁰

10

²

10

⁴

10

⁶

10

⁸

Q

^-2 0.5% sPP-P(E-co-P) 38.4K/106.6K in d-22

Q

^-1

Q

^-3

d

Σ

/d

Ω

[c m

-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 bundels 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

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)

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Monochromatization Velocity selector

Chopper Time-of-flight TOF

Crystal monochromator Primary collimation

collimation Sample table

detection

Large Scale Structures: Reflectometry

Schematics of a neutron reflectometer:

A C

d n0

n1 n2

α α

αt D

.

B

Path length difference:

AD n BC AB+ ⋅ −

=

Δ ( ) 1

dnsinαt

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 Spin

valve

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:

Solution:

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, …)

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

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• 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 superconductors; 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 s

v h m p

=

⋅

=

&

λ = ⋅ s ⋅ λ h t m

(typically 1 ms/m)

Path-Time Diagram

Example for an Application

Neutron spectroscopy from the molecular magnet Mn₁₂acetat:

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-T_Csuperconductors; 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

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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

G_hkl 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. Brückel et al

TAS-Applications

Chiral phase transitions

qchain[2π Å^-1] Low dimensional magnets

Phonons in High TC

Spin dynamics in frustrated spin systems

• 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

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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