Quasielastic Neutron Scattering

Franz Demmel

NTC 2013

Outline

• What is ‘quasielastic’ neutron scattering?

• Why we do it, illustrated with a few examples.

• What we want to do

• How we do it.

Double differential cross-section

Differential cross-section

Total cross-section

The cross-section is measured in the non-SI unit

of area the barn (10-28 m2)

Inelastic neutron scattering

2 2

2i f i f

i f

E Em

k k

Q k k

Energy and Momentum

Exchange with Target

Inelastic Neutron Scattering

N tiiti ee

NedtS

1,

01

2

1,

RQRQQ

,QS

Simulation

Theory

Experiment

The dynamical structure factor

tttdN

tG 11111 ,,1

, rrrrr

tttt ,,,, rrrr

The probability of finding a particle at position r at time t if there was a particle at r=0, t=0

An asymptotic (t→∞) non-zero value of F(Q,t) gives rise to a δ(ω)

term in S(Q,ω). But in a liquid

decays to zero at large times, and there is no elastic scattering!

,G tr

“Snapshot”:

The intermediate scattering function:

, ( , )exp( )S dtF Q t i t Q

FT

. ( ) . (0)( , ) i iiQ r t iQ r

sF Q t e e

Collective motion

Stochastic motions

t

trGtrGD s

s

),(),(2

)4/exp()4(

1),( 2

2/3Dtr

DttrGs

2( , ) exp( )F Q t Q Dt

Gs(r,t) is the solution of a diffusion equation

on long length scales

Fick’s diffusion

Qs > Ql

E 0

Half Width at Half Maximum=HWHM= = DQ2

Self diffusion (long range diffusion)

2 2

1( , )S Q

“A Lorentzian”

2 2 ( , ) 6sr r G r t dr Dt

Mean square displacement

2 2[ ( ) (0)]r r t r

21

6D r

t

2 2 ( , ) 6sr r G r t dr Dt

22 2 20

0

1( )

2 2

vr v t t

Free particle, ballistic, dilute gas

Diffusion, random walk: In the time domain:

t

r2

diffusion

free particle

oscillator

actual behaviour

Liquid Argon

So the FWHM will be:

1/2

02(2ln 2) Qv Free particle:

Diffusion: 22DQ

k2

22DQ

1/2

02(2ln 2) Qv

Models for Translational Diffusion

))sin(

1(1

)(Ql

QlQ

2 21( ) (1 exp( ))

6

Q lQ

2

2( )

(1 )

DQQ

DQ

Chudley Elliott: 2( )Q DQ

Gaussian distribution:

Random jump diffusion:

Jump diffusion by Chudley Elliott

Starting with a rate equation one gets:

( , ) exp( ( ( ))I Q t t f Q

0

1( , ) ( , )

( , ) s s

s l

G r l t G r tG r t fluctuation n

t relaxation time

l

lQin

Qf )1)(exp(1

)(0

is the residence time and l the jump length

6

2lD )/exp()( 0lllla

1/

Vibrations

• Well-defined vibrations lead to modulation of elastic-line intensities

• Debye-Waller factor (as in diffraction):

• “Elastic-window” scans provide a means of exploring the temperature

dependence of the mean-square displacement.

• Much biology and soft matter research begins with these

measurements.

2 2

3( )u Q

I Q e

Apoferritin

J. Phys Chem. B 112, 10873 (2008)

Adding Degrees of Freedom

For a particle (atom), we can write (approximately):

2 2

( )3( , ) ( , ) ( , )vib

trans

u Q

Q t

particle vib transI Q t I Q t I Q t e e

I ( , ) ( , ) ( , )molecule particle rotQ t I Q t I Q t

For a molecule (with a shape), need to add rotational modes:

( , ) ( , ) ( , )molecule particle rotS Q S Q S Q

Free isotropic rotations are treated as a

diffusing orientational probability P

( ) exp( ( 1) )F t l l D tl r Solution of the diffusion equation:

This gives a Lorentzian with a width:

which is Q independent!!

( 1)l l Dl r

2

0 2 21

( 1)1( , ) ( ) ( ) ( )

( ( 1) )

Rl

l R

l l DS Q j Qa A Q

l l D

2 ( , )( , )r

P tD P t

t

The EISF (= Elastic Incoherent Structure Factor)

( )

( ) ( )elastic

elastic QENS

I QEISF

I Q I Q

Elastic Scattering of 1 particle moving in a restricted space

2( ) exp( ) ( )S Q iQr p r drinc

Experimental definition of the EISF:

Trimethyloxosulphonium ion

Jump between two sites:

Jump probability 1/

1 2 /( , ) 1/ 2 ( )(1 ( ) 1/ 2(1 ( ))0 0 2 2(2 / )

S Q j Qd j Qd

d

Jumps among 3 sites:

r

Mean residence time

1 3/( , ) ( ) ( ) ( )0 1 2 2(3/ )

S Q A Q A Q

Example: methyl group jumps

Reorientations

Qd

Ao =½[1+sin(2Qd)/(2Qd)] EISF 1

½

EISF

2 2 2

02 2 2 21

1 ( ) 1 ( 1)( , ) e { ( ) ( ) ( ) } ....

( ) ( ( 1) )

Q u Rl

l R

Q l l DS Q j Qa A Q

Q l l D

vibrations translation rotation

EISF

A simple starting point.......

Computer Simulations

• Molecular Dynamics simulations directly map onto timescales and

dynamics probed by QENS (from ps up to ns).

• Basic idea: simulate the outcome of neutron scattering experiments

taking as input the results of an atomistic simulation.

– Predictive power for large systems (experiment design).

– Beyond model-dependent data analysis (oftentimes too simplistic for complex systems)

– Common ground for comparison with other experimental techniques (e.g., light scattering, NMR, optical spectroscopy in THz region)

A Few Examples

Double perovskite structure:

Na2 (NaAl)F6

Cryolite Na3Al F6

“The elastic Scan”

))sin(

1(1

)(Ql

QlQ

Chudley Elliott:

H Conduction in Solid State

Mobility of H atoms is at the

heart of fuel cells or

hydrogen sensors

A very unusual combination of:

– Regular inorganic

framework.

– High proton density.

Are these hydride ions mobile?

If so, how do they move in the

inorganic lattice?

H-

Advanced Materials , 3304 (2006).

-0.4 -0.2 0.0 0.2 0.4

1E-6

1E-5

1E-4

1E-3

0.01

To

tal S

ca

tte

red

In

ten

sity

Energy Transfer [meV]

300K

675K

685K

700K

725K

(a)

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7 T=685K

Lo

ren

tzia

n F

WH

M [m

eV

]

Q [Å-1]

1.30 1.35 1.40 1.45 1.50

6

9

12

15

D [1

0+

5 c

m2s

-1]

1/T [10+3

K-1]

Ea = 230±45 meV

sin

1QR

QQR

1D hopping

mechanism

QENS Spectra Hydride Diffusion

Energetics of Diffusion

Quasielastic Neutron Experiments

Microscopic

Transport Coefficients

Dynamics in the multi-subunit Protein Apoferritin

Iron storage protein

24 peptide chains, hollow,

internal diameter = 80 Å

Shell thickness= 20 Å

simultaneous fit of elastic scattering data

to CH3 jump rotation model

Elastic window scans

IN16

Osiris

res

iommomm gQAppQApprQ

TQS arctan)))]((1(1[2

)))((1(3

exp),0,(22

S. Magazu et al;

Diffusion of Disaccharides

Bioprotectants

J Chem Phys B (2006)

Water

Local movements

in the Polymer

Large pore sizes

• Two biopolymers: HYA and HYDD

• HYADD produces stable hydrogels at much lower concentrations

• Polymer weight conc: 1 - 10%

QENS: 1 - 100ps

Human Blood on IRIS

Red Blood Cells

Haemoglobin

(oxygen carrier)

Volume fraction 25%

RBC = model systems for concentrated protein solutions

Haemoglobin activity is very sensitive

to physiological conditions –

cellular environment resembles a colloidal gel.

Self diffusion of Hb in red blood cells

• Agreement with predictions for hard sphere suspensions

• Hydration shell has to be taken into account

IRIS data

Water diffusion in Human Red Blood Cells

Translational diffusion of water Immobile fraction of water molecules: Bound water in the hydration layer of hemoglobin

Diffusion in liquid Al )(

)(2

2/1QS

QdQDE

“deGennes narrowing”

2 2( )( )

Bk TQ Q

MS Q

Cohen, deSchepper, PRL 1987

How OSIRIS works

Pulsed vs. Continuous Neutron Sources

• Time-of-flight spectrum is

trivially related to neutron

wavelength spectrum.

• Broad range of neutron

energies (from meV to eV).

• Multiplexing advantage:

broad range of

wavelengths can be used

simultaneously

n + 3He 3H + 1H + 0.764 MeV H.V.

3He gas

detector

3H

1H

θ

sin2dCrystal

Bragg reflection

How to measure the neutron energy?

Measuring the time of flight!

or

/h mv

Neutron detectors are energy

INsensitive

Direct Geometry Indirect Geometry

Sample

Chopper

Moderator

Detector

Crystal

Filter

Crystal

Two types of TOF spectrometer

ToF backscattering

Indirect geometry: kf=constant

bandwidth / frame-

overlap choppers

detectors

analyser crystals

near-backscattering

IRIS, OSIRIS

moderator

• Incident wavelengths: 2-20 Å (H2

moderator). Two disc choppers

select desired wavelength band

• Energy resolution

PG002: E=17/25µeV

PG004: E=55/95µeV

• Diffraction bank:

simultaneous structural

measurements

• High magnetic fields: 7 Tesla!

• 40 cm high array of 8750 graphite crystals

• Solid angle: 1.1 ster, 10° < 2 < 150°

TDS

Energy Transfer (meV)

-0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4

Nor

mal

ised

Inte

nsity

0.00

0.01

0.02

0.03

0.04

0.05

Analyser 8K

Analyser 290K

Energy Transfer (meV)

-0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4

Nor

mal

ised

Inte

nsity

0.00

0.01

0.02

0.03

0.04

0.05

Analyser 8K

Analyser 290K

Improving the Energy Resolution: Mica

Fluorophlogopite:

Mica004 not active

Natural Mica – “Muscovite” OH- replaced by F-

Mica006: E=10 µeV

Mica004: E=4.5 µeV

Mica002: E=1 µeV

Energy and Momentum Transfers

kinematic plane

i fE E E

2

22 cosn

inc scat inc scat scat

mQ E E E E

direct

0 2 4 6 8 10 12

-60

-40

-20

0

20

40

60

170o

120o

90o

60o

45o

En

erg

y T

ran

sfe

r [m

eV

]

Momentum Transfer [Å-1]

= 10o

0 2 4 6 8 10 12

-60

-40

-20

0

20

40

60

170o

120o90

o60

o45

o

Momentum Transfer [Å-1]

= 10o

Neutron energy loss

Neutron energy gain

Neutron guides

21

2

cohbNn

ncrit cos

3107.1

Nicrit

The refractive index is given

by

Which will have a critical angle

θ

Neutrons will reflect from a polished surface of

most materials, but Ni (or 58Ni) are particularly

good

Deviation from the Ideal Instrument: Resolution

Two major contributions:

• Timing uncertainties:

– In neutron production for a given wavelength.

– Path differences between source – sample – detector, etc.

• How good your crystal analyser is.

Typically expressed as uncertainties in incident and scattered

energies Einc

and Escat

inc scatE E E Remembering that:

2 2E t

E t

IN16, ILL

How to do a QENS experiment?

2

s, ,4 4

f coh inc

i

kdS S

d d k

Q Q

ki

kf

Ei

Ef

detector

sample

What should we use as a sample?

Hydrogen: A Very Special Case

Isotope substitution!

Contrast variation

1

2

0 ,, kdd

dNCI

In the ideal experiment, all relevant quantities

are exactly defined. Thus one has:

• - no container

• - no detector

• - no beam

Going through a material...

The beam intensity is attenuated

exponentially

Two removal processes:

•scattering

•absorption

slab texptransIT L

I

How much water do we need?

L

( )t s absn

Transmitted fraction for a flat plate geometry:

Try answering the following questions:

• Calculate optimal thickness L to attain a 15% scattered intensity.

• How thick should the plate be for a 10% a 20% and a 50% water sample?

slab texpT L

( )t s absn

ln( )

t

TL

15 % Scatterer: L= 0.3 mm!!

So which sample container should we use??

Beam size????

A fluid sample needs a container (nearly always).

The most common shapes are slabs and cylinders.

cylinder:

•suited for high pressures with thin walls

•good for all scattering angles

•nearly θ-independent attenuation (see later)

slab (rectangular or circular):

•good beam coverage with very thin or very thick samples

•good for small-angle experiments

•good for non-metal windows (e.g. single-crystal sapphire)

Hollow cylinder

So what do we really

measure?

I=Isample+Ican+Ibackground+Imultiple scattering +.....

1

2

0 ,, kdd

dNCI

Ñ/degreesA(Ñ)

2 ,

exp-1

2 ,0

20 ,exp

exp-1

0 ,exp

A

θ

Attenuation depends on: θ, even for isotropic scattering

shape and size of sample

property of material(s)

cor

s c s cI I I

Sample +container

rrr stinctexp1

dV

A

Generalizing to any sample shape...

usually referred to as the "As,s Paalman & Pings attenuation coefficient"

astt nn

sm 2200 absorption "1" ; ref

ref

refa

refrefaa

vv

k

kk

,

, , ,

1cor c sc

s c s c

s sc s sc c c

AI I I

A A A

Multiple scattering causes both the loss of "good neutrons" and the

detection of "bad neutrons".

Multiple scattering is an unwanted feature in neutron spectroscopy.

Maximizing the single-scattering intensity is incompatible with

minimizing the multiple-to-single scattering ratio.

Multiple scattering

Software to analyse the data: Mantid!