neutron scattering - uppsala universitybasics of neutron scattering experiments kinematics of...

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