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Neutron scattering
Max Wolff
Division for Material PhysicsDepartment of Physics and Astronomy
Uppsala UniversityUppsalaSweden
Outline
Why is neutron scattering important?
Peculiarities of neutrons
How does it work?
What do we measure?How do we measure?
What is it good for?
Applications
Neutron scattering
3
Time and length scales
4
Why neutrons?
Neutrons are neutral: high penetation power, non-destructive
N
SNeutrons have a magnetic moment: magneticstructures and excitations
Neutrons have thermal energies: excitation of elemenatry modes, phonons, magnons, librons, rotons, tunneling, etc.
Neutrons have wavelengths similar to atomicspacings: structural information, short and long rangeorder, pore and grain sizes, cavities, etc.
Neutrons see nuclei: sensitive to light atoms, exploiting isotopic substitution, contrast variation withisotopes
Neutrons have a spin: polarized neutrons, coherentand incoherent scattering, nuclear magnetism
The neutron
�/2
8
Life time: T1/2= 890 s
Mass: m = 1.675 * 10 –27 kg
Charge: Q = 0
Spin: S =
Mag. Moment: µ/µn = -1.913
Properties:
Dispersion:mkmvE
221 22
2 ==
(Photon: E =
λ = 1.8 Å ⇒ E ≈ 7 keV)
Thermal neutron: T = 293 K
⇒ v = 2200 m/s
⇒ λ = 2π/k = 1.8 Å (atomic distances)
⇒ E = 25 meV (molecular/lattice excitations)
�ck
Energy regimes
Michael Marek Koza, koza@ill.fr
Quantum Mechanics and the Particle – Wave Duality
Energy – momentum relation :
E = 81.81 · λ-2 = 2.07 · k2 = 5.23 · v2
λ [Å] 1 2 4 6 10 E [meV] 80 20 5 2 1 T [K] 960 240 60 24 12 v [m/s] 3911 1956 978 620 437
Cold neutrons : 0.1-10 meV D2 at 25 K
Thermal neutrons : 10-100 meV D2O at room T
Hot neutrons : 100-500 meV graphite at 2400 K
Comparison
Michael Marek Koza, koza@ill.fr
Quantum Mechanics and the Particle – Wave Duality
Energy – momentum relation :
E = 81.81 · λ-2 = 2.07 · k2 = 5.23 · v2
X – ray properties : E = ħck, c = 3·109 m/s
Inelastic x-ray scattering : λ = 1 Å → E = 12.4 keV
Raman, infra-red, ... : 103-105 Å → E = 12.4 – 0.12 eV
λ [Å] 1 2 4 6 10 E [meV] 80 20 5 2 1 T [K] 960 240 60 24 12 v [m/s] 3911 1956 978 620 437
Heisenberg principle
Michael Marek Koza, koza@ill.fr
Quantum Mechanics and the Uncertainty Principle
Position Momentum
Heisenberg principle
Michael Marek Koza, koza@ill.fr
Quantum Mechanics and the Uncertainty Principle
Heisenberg’s heritage : nothing is certain !
ΔE·Δt ≥ ħπ Δν·∙Δt ≥ ½ Δω·∙Δt ≥ π
Δp·Δx ≥ ħπ Δλ-1·∙Δx ≥ ½ Δk·Δx ≥ π
Let’s get real : Spectrometer with ΔE/E of 5% (0.05%) utilized
with neutrons of E = 2 meV - what can we measure ?
ΔE = 0.1 meV → Δt ≥ 20·∙10-12 s = 20 ps
ΔE = 1 μeV → Δt ≥ 20·∙10-14 s = 2 ns
Interaction
11
Absorption
12
What can we measure?Φ = number of incident particles per area and timeσ = total number of particles scattered per time and Φdσ/dΩ = number of particles scattered per time into a certain direction per area and Φd2 σ/dΩdE = number of particles scattered per time into a certain direction per area with a certain energy per energy interval and Φ
⇤Q = ⇤kf � ⇤ki
Definitions
� =�k2
f
2m� �k2
i
2m
Qki
kf
Mathematical description
Schrödinger equation:
�i(⌃r,⇥) �= e�i(⌃k⌃x��t)
��t� = ( p̂2
2m + V̂ )�
⇥f (�r,⇤) �= e�i(�k�x��t) + f(�)e�i(�k�r��t)
�r
Kinematic approximation
Kinematic approximation means:
Combination of s-wave scattering (for neutrons) with the
Born approximation (single event scattering)
The following effects are neglected (dynamic theory):
- Attenuation of the beam. - Absorption.
- Primary extinction (attenuation due to scattering). - Secondary extinction (attenuation due to multiple scattering).We will also not consider the finite size of the sample (lattice sum).
All these effects are important for perfect crystals that hardly exist is disordered materials.
Scattering function
This equation is also known as scattering law or dynamical structure factor. It completely describes the structural and dynamic properties of the
sample.
The scattering function:
Scattering potential for neutrons: V (�r, t) ⇥ (bn�(�r � �R) + bm(�r � �R))
S( ⇧Q, �) = 4�⇥s
kikf
⌅2⇥⌅�⌅⇤
S( ↵Q, ⌅) ⇥=�
�f ,�i⇥(Ef )|⇤kf , ⌅f , ⇤̂f |V̂ |⇤̂i, ⌅i, ki⌅|2�(⌅ � ⌅�f ,�i)
⇥s =� �
⌅2⇥
⌅�⌅⇤d�d⇤ = 4�b2
Potential
V ( r) =2⇥h
mb�( r � R)
For neutrons (scattering at the core) wave length (10-10 m) is much larger than the size of the scattering particle (10-15 m). This implies that no details of the core can be
seen and the scattering potential can be described by a single constant b, the scattering length:
This is called the Fermi pseudo potential. B is a phenomenological constant and has to be determined experimentally.
Cross section
The Fermi pseudo potential is true neutron-nucleus potential, but it is constructed to provide isotropic s-wave scattering in the Born approximation.
Inserting this potential into the scattering cross section results in:
The total scattering cross section is then:
Ensemble
V ( r) =2⇥h
m
�
i
bi�( r � Ri)
⇥�
⇥�=
�����⇥
i
biei ⌥Q⌥r
�����
2
=⇥
i,j
bibjei ⌥Q( ⌥Ri� ⌥Rj) =
⇥
ij
bibjei ⌥Q ⌥Rij
The potential for an ensemble of scatterers can be described by the sum of the scattering length:
The sum is taken over all nuclei at positions Ri.
For the cross section this results in:
Rij is the distance between the scatterers or the lattice parameter for a crystal.
Average
⇥�
⇥�=
�⇤
ij
bibjei ⇧Q⇧r
⇥
⇥�
⇥�=
�
ij
�bibj⇥ ei ⇧Q ⇧Rij
⇥�
⇥�=
�
i=j
�bibj⇥ ei ⇧Q⇧Rij +�
i�=j
�bibj⇥ ei ⇧Q⇧Rij
In a scattering experiment the ensemble average or time average is measured:
Assuming that the scatterers are fixed at their positions the average has to be taken only over the scattering length:
Formally we can divide up the double sum into to sums:
Averaging
As δbi averages to 0.
=0 =0 For no correlations i=j.
Inserting into the cross section gives:
bcoh=<b> and binc=(<b2>-<b>2)½ is then called coherent and incoherent scattering length, respectively.
Spin incoherent scattering
For a one atom with nuclear angular moment J the total moment of the neutron nucleus pair has to be considered.
The neutron is a Spin ½ particle. In an unpolarised neutron beam S= ½ and S=- ½ neutrons are present.
For the total moment of the system we get: M=J±S.
Accordingly two scattering length have to be considered:
b+ for M=J+ ½
andb- for M=J- ½
Generally, 2J+1 different values are possible for the angular moment J. This results in a total number of 2(2J+1) eigenstates (2(J+1) for b+ and 2J for b-).
Spin incoherent scattering
The probability for measuring b+ is then:
And for b-:
For J=0 we get 1 for b+ and 0 for b-.
The coherent and incoherent scattering length can then be calculated:
Example 1H
This results in: p+=3/4 and p-=1/4.
Hydrogen scatters mainly incoherent!
Example 2H
This results in: p+=2/3 and p-=1/3.
Deuterium scatters mainly coherent!
Correlation functions
Coherent scattering:
Incoherent scattering:
⇒ Bragg reflections, liquid structure factor, phonon scattering etc.
⇒ Scatterers exist, self motion etc.
Example
(Random jump diffusion):
Example (Bragg law):
Real - reciprocal space
Michael Marek Koza, koza@ill.fr
Basics of Neutron Scattering Experiments
Observable in INS and Diff.
Observable in NSE, simple models
Simple models, computer simulations
Scattering cross section
Coherentscatteringcross section
Incoherentscatteringcross section
H D OC SiAl
Scattering length
Atomic number1 83
0
10
Scat
terin
g le
ngth
b [f
m]
Summary
Neutrons:
Instrumentation:
Weakly interactingSpinCore interaction
Low energy
Time of flightAngle dispersive
High penetrationMagnetic sensitivityIsotope sensitivityCoherent/Incoherent scatteringGood energy resolution
Scattering function describes a sample completely
Nuclear reactor
... discovered 1938 together with Fritz Strassmann the nuclear fission reaction of Uranium after irradiation with neutrons
Lise Meitner und Otto Hahn
neutron
Target
Access neutrons
Reaction products
n +23592 U ! 236
92 U !14456 Ba +89
36 Kr + 3n + 177MeV
Floor plan
Spallation
ESS - How it will look likeMAX IV
Instrumentation
Target station:SP: 50 HzLP: 16.6 HzPulse length 2 ms
Linear accelerator:NC: 769 mSC: 432 mEnergy: 1.334 GeVCurrent: 3.75 mAPower: 5 MW
Angle dispersive neutron scattering
Incident beam Monochromator/Polarizer
Spin !ipper
Analyzer
Exiting beam
Detector
SampleCollimator
Collimator
Source
Qki
kf
Spin !ipper
TOF neutron scattering
Qki
kf
Incident beamChopper
Spin !ipper
Analyzer
Exiting beam
Detector
SampleCollimator
Collimator
Source
Polarizer
Spin !ipper
Scattering geometry
Michael Marek Koza, koza@ill.fr
Basics of Neutron Scattering Experiments
Elastic scattering Rayleigh process
Neutron energy gain Anti-Stokes process
Neutron energy loss Stokes process
→ Constant directions of ki and kf but changing Q !
→ Constant directions of ki and Q but changing kf !
Kinematics
Michael Marek Koza, koza@ill.fr
Basics of Neutron Scattering Experiments
Kinematics of inelastically scattered neutrons :
ħω [meV]
ħQ [Å
-1]
energy gain energy loss
Ei = 100 meV
For 2θ = 0, 180° Q = Δk, E ~ Δk2
B.T.M. Willis & C.J. Carlile, ‘Experimental Neutron Scattering’, Oxford University Press:
Time of flight
Michael Marek Koza, koza@ill.fr
Basics of Neutron Scattering Experiments
Inelastic scattering instruments : Time-of-Flight Spectrometer
ħω [meV]
ħQ [Å
-1]
energy gain energy loss
Ei = 100 meV
Equipped with large detector arrays to cover a wide 2θ range with typically 1 detector tube per degree of 2θ. Data are to be interpolated to constant Q.
Three axis
Michael Marek Koza, koza@ill.fr
Basics of Neutron Scattering Experiments
Inelastic scattering instruments : Three-Axis Spectrometer
ħω [meV]
ħQ [Å
-1]
energy gain energy loss
Ei = 100 meV
Typically equipped with a single detector but with highly flexible positioning capabilities to map out, e.g., constant Q slices.
Backscattering
Michael Marek Koza, koza@ill.fr
Basics of Neutron Scattering Experiments
Inelastic scattering instruments : Back-Scattering Spectrometer
ħω [meV]
ħQ [Å
-1]
energy gain energy loss
Ei = 100 meV
Equipped with large detector arrays to cover a wide 2θ range. Dynamic range is limited and data are recorded approximately at constant Q.
Kinematics
Michael Marek Koza, koza@ill.fr
Basics of Neutron Scattering Experiments
Kinematics of inelastically scattered x rays and visible light:
Energy of interest : ħω = 10 meV
X rays : Ei ~ 10 keV → ki = kf → ΔE/E ~ 10-7
→ Q = 2kisin(θ) (~ 2θki)
Raman : ki = kf ~ 10-3 Å-1
→ Q = 0
ħω [meV]
ħQ [Å
-1]
energy gain energy loss
Neutron Spin Echo
OSNS 2007 - Neutron Spin-Echo
The principle of spin echo
Velocity Spin Spin
Selector Polarizer Analyzer
�/2 B0 � B
1 �/2 Detector
n beam
magnetic field profile
spin precession
spin defocusing
0 5 10 15 200 5 10 15 20
Sample
NSE
OSNS 2007 - Neutron Spin-Echo
Beyond Elastic Scattering
Detector
sample
Velocity selector
Axial Precession field,
B1
Axial Precession field,
Bo
Supermirror
polarizer
Supermirror
analyser
�/2 � �/2
Mezei flipper coils
A B C
So far we have assumed that the neutrons enter and leave the spectrometer with the
same distribution of velocities (i.e. any scattering by the sample is elastic)
Quasi-elastic scattering process will change the energy and therefore the velocity of the
neutrons. The accumulated precession angle is now
�L= �n[(LoBo)/vo- (L1B1)/v1]
Thus the accumulated precession angle �L can be used as a measure of the energy
transfer associated with the scattering process
Elastic/inelastic scattering
Michael Marek Koza, koza@ill.fr
Inelastic and Quasielastic Signal – How does it look like ?
Elastic line Inelastic discrete lines
Quasielastic signal
Sample Resolution
Example - clathrate
Michael Marek Koza, koza@ill.fr
Generic Signal in a Scattering Experiment
CH4 in H2O-clathrate: Enormous amounts trapped in perma-frost soils and deep-sea sedimants! Environmental risk through global warming!
CO2 in H2O-clathrate : Opportunity for dumping our combustion junk!
Composition – decomposition mechanism? Stability regime and conditions?
Example - clathrate
Michael Marek Koza, koza@ill.fr
Generic Signal in a Scattering Experiment
Tetrahydrofuran (THF) in H2O hydrate-clathrate :
Example - clathrate
Michael Marek Koza, koza@ill.fr
Generic Signal in a Scattering Experiment
Phonon Density of States :
Sum up all detectors !
Text
M. Koza et al.
Example - clathrate
Michael Marek Koza, koza@ill.fr
Generic Signal in a Scattering Experiment
Elastic Structure Factor:
Sum up intensity in the energy channels around the elastic line only !
Static Structure Factor:
Sum up intensity in all energy channels ! DIFFRACTION!!!
Example - detailed balance
Michael Marek Koza, koza@ill.fr
Detailed Balance Principle and Bose-Einstein Factor
Neutron energy gain Anti-Stokes line
Neutron energy loss Stokes line Clathrate :
Ba8ZnxGe46-x
Example - ballistic motion
Michael Marek Koza, koza@ill.fr
Diffusion in Lipid Bilayers
Particular interest : Is there a change in diffusion dynamics from ballistic at short distances (time scales) to Brownian like motion at long distances (time scales) ? Brownian → Lorentz line
Ballistic → Gaussian line
Example - balistic motion
Michael Marek Koza, koza@ill.fr
Diffusion in Lipid Bilayers
Instrument : IN13 with cooled/heated monochromator λ = 2.2 Å ΔE = 8 μeV E range = -100...+100 μeV
At r <2.4 Å indication of ballistic motion?
Example - confined water
Michael Marek Koza, koza@ill.fr
Examples of Quasielastic Scattering Experiments
Example - confined water
Michael Marek Koza, koza@ill.fr
Examples of Quasielastic Scattering Experiments
Size effect of micelles on diffusion properties of confined water
Instruments : IN5 and IN16 at ILL BASIS at SNS, USA
Rouse modelRouse DynamicsThe current description of the Rouse dynamics is based on the bead and spring model for polymer chains.
Winter School on Neutron Scattering in Soft MatterBjörkliden, 3/25-30/’12
( )
!
!= 442
2
9exp
3exp, QWl
ttQ
N
WltQI self
Rouse "
( ) [ ]{ }
( ) ( ) ( )[ ] WlQ
xx
xydxyh
tuhtudulQ
tQI
R
RR
pair
Rouse
36 ;exp1
cos2
exp12
,
44
0
22
022
=#!!=
##!!=
∫
∫$
$
"
NSE - rouse - selfRouse Dynamics NSE results: SelfNSE results validate the theory based on Rouse dynamics:
PDMS, T=373 K
Winter School on Neutron Scattering in Soft MatterBjörkliden, 3/25-30/’12D. Richter, et al., Phys. Rev. Lett., 62, 2140 (1989) .
NSE - rouse - pairRouse Dynamics NSE results: PairNSE results validate the theory based on Rouse dynamics:
Winter School on Neutron Scattering in Soft MatterBjörkliden, 3/25-30/’12H. Montes, et al., J. Chem. Phys., 110, 10188 (1999) .
Example - Reptation
OSNS 2007 - Neutron Spin-Echo
Examples of neutron spin echo studies
Reptation in polyethylene
The dynamics of dense polymeric systems are
dominated by entanglement effects which
reduce the degrees of freedom of each chain
de Gennes formulated the reptation hypothesis
in which a chain is confined within a “tube”
constraining lateral diffusion - although several
other models have also been proposed
Schleger et al, Phys Rev Lett 81, 124 (1998)
The measurements on IN15 are in agreement with
the reptation model. Fits to the model can be
made with one free parameter, the tube diameter,
which is estimated to be 45Å
Example - glass
OSNS 2007 - Neutron Spin-Echo
Examples of neutron spin echo studies
Glassy Dynamics in Polybutadiene
The first peak in the structure factor
corresponds to interchain correlations (weak
Van der Waals forces), the second peak is
dominated by intrachain correlations
(covalent bonding)
at the first peak the spectra follow a single,
stretched exponential (�=0.4) universal curve,
indicating that the interchain dynamics very
closely follow the �-relaxational behaviour
associated with macroscopic flow of
polybutadiene.
Each NSE spectrum was rescaled by the
experimentally determined characteristic time,
th, for bulk viscous relaxation for the
corresponding temperature.
Scattering vector, Q, (nm-1)
0 10 20 30
S(Q
) ( 3
4
5
6
7
8
9
4K160K270K
(a)
S(Q
)
tf /��
10-4 10-3 10-2 10-1 100 1010.0
0.2
0.4
0.6
0.8
1.0
220K230K240K260K280K
(b) Q=14.8nm-1
S(Q
,t)/
S(Q
,0)
Example - glass
OSNS 2007 - Neutron Spin-Echo
Examples of neutron spin echo studies
Glassy Dynamics in Polybutadiene
Those correlation functions measured
at the second peak of the structure
factor do not rescale. They are
characterised by a simple exponential
function, with an associated
temperature dependent relaxation
rate which follows an Arrhenius
dependence with the same activation
energy as the dielectric �-process in
polybutadiene, and which is
unaffected by the glass transition.
A. Arbe et al Phys. Rev. E 54, 3853 (1996)
Scattering vector, Q, (nm-1)
0 10 20 303
4
5
6
7
8
9
4K160K270K
(a)
S(Q
)
10-12 10-10 10-8 10-6 10-4 10-2 100 1020.0
0.2
0.4
0.6
0.8
1.0
170K180K190K205K220K240K260K280K300K
tf /��
(c) Q=27.1nm-1
S(Q
,t)/
S(Q
,0)
Summary
Why is neutron scattering important?
Peculiarities of neutrons
How does it work?
What do we measure?How do we measure?
What is it good for?
Applications
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