intro to neutron scattering

8/11/2019 Intro to Neutron Scattering

1/191

by

Roger Pynn

Los Alamos

National Laboratory

LECTURE 1: Introduction & Neutron Scattering Theory


2/191

Overview

1. Introduction and theory of neutron scattering

1. Advantages/disadvantages of neutrons

2. Comparison with other structural probes

3. Elastic scattering and definition of the structure factor, S(Q)

4. Coherent & incoherent scattering5. Inelastic scattering

6. Magnetic scattering

7. Overview of science studied by neutron scattering

8. References

2. Neutron scattering facilities and instrumentation3. Diffraction

4. Reflectometry

5. Small angle neutron scattering

6. Inelastic scattering


3/191

Why do Neutron Scattering?

To determine the positions and motions of atoms in condensed matter 1994 Nobel Prize to Shull and Brockhouse cited these areas

(see http://www.nobel.se/physics/educational/poster/1994/neutrons.html)

Neutron advantages:

Wavelength comparable with interatomic spacings Kinetic energy comparable with that of atoms in a solid

Penetrating => bulk properties are measured & sample can be contained

Weak interaction with matter aids interpretation of scattering data

Isotopic sensitivity allows contrast variation

Neutron magnetic moment couples to B => neutron sees unpaired electron spins

Neutron Disadvantages

Neutron sources are weak => low signals, need for large samples etc

Some elements (e.g. Cd, B, Gd) absorb strongly

Kinematic restrictions (cant access all energy & momentum transfers)


4/191

The 1994 Nobel Prize in Physics Shull & Brockhouse

Neutrons show where the atoms are.

and what the atoms do.


5/191

The Neutron has Both Particle-Like and Wave-Like Properties

Mass: mn = 1.675 x 10-27

kg Charge = 0; Spin =

Magnetic dipole moment: n = - 1.913 N Nuclear magneton: N = eh/4mp = 5.051 x 10-27 J T-1

Velocity (v), kinetic energy (E), wavevector (k), wavelength (),temperature (T).

E = mnv2/2 = kBT = (hk/2)2/2mn; k = 2 / = mnv/(h/2)

Energy (meV) Temp (K) Wavelength (nm)

Cold 0.1 10 1 120 0.4 3

Thermal 5 100 60 1000 0.1 0.4

Hot 100 500 1000 6000 0.04 0.1

(nm) = 395.6 / v (m/s)

E (meV) = 0.02072 k2 (k in nm-1)


6/191

Comparison of Structural Probes

Note that scattering methods

provide statistically averaged

information on structure

rather than real-space pictures

of particular instances

Macromolecules, 34, 4669 (2001)


7/191

Thermal Neutrons, 8 keV X-Rays & Low Energy

Electrons:- Absorption by Matter

Note for neutrons:

H/D difference

Cd, B, Sm

no systematic A

dependence


8/191

Interaction Mechanisms

Neutrons interact with atomic nuclei via very short range (~fm) forces.

Neutrons also interact with unpaired electrons via a magnetic dipole

interaction.


9/191

Brightness & Fluxes for Neutron &

X-Ray Sources

Brightness(s

-1m

-2ster

-1)

dE/E(%)

Divergence(mrad

2)

Flux(s

-1m

-2)

Neutrons 1015

2 10 x 10 1011

RotatingAnode

1016

3 0.5 x 10 5 x 1010

BendingMagnet

1024

0.01 0.1 x 5 5 x 1017

Wiggler 1026 0.01 0.1 x 1 1019

Undulator

(APS)

1033

0.01 0.01 x 0.1 1024


10/191

Cross Sections

dEd

dE&dintosecondperscatteredneutronsofnumber

d

dintosecondperscatteredneutronsofnumber

/secondperscatteredneutronsofnumbertotal

secondpercmperneutronsincidentofnumber

2

2

=

=

=

=

dEd

d

d

d

measured in barns:1 barn = 10-24 cm2

Attenuation = exp(-Nt)N = # of atoms/unit volume

t = thickness


11/191

Scattering by a Single (fixed) Nucleus

2

total

22

2

incident

2222

scat

4sobd

dbv

d

d

vv:isareasunitthroughpassingneutronsincidentofnumbertheSince

dbv/rbdSvdSv

:isscatteringaftersecondperdSareaanthroughpassing

neutronsofnumberthe),scatteringafterandbefore(sameneutrontheofvelocitytheisvIf

b

==

=

==

==

range of nuclear force (~ 1fm)

is scattering is

elastic

we consider only scattering far

from nuclear resonances where

neutron absorption is negligible


12/191

Adding up Neutrons Scattered by Many Nuclei

0

)(

,

)()'(

,

2

i

2

)'(

i

'.

2

)'.(

i

iscat

R.kii

'bydefinedisQr transferwavevectothewhere

gettodS/rdusecanweRrthatsoawayenoughfarmeasureweIf

R-r

1

R-r

b-iswavescatteredtheso

eisaveincident wtheRatlocatednucleusaAt

0

0

0

i0

kkQ

ebbebbdd

eebd

dS

vd

vdS

d

d

ee

jiji

i

ii

RR.Qi

j

ji

i

RR.kki

j

ji

i

R.kkirki

i

scat

RrkiR.ki

rrr

rr

rr

r

rrrrrrr

rrrrr

rrrrr

rr

=

==

=>>

=

=

=


13/191

Coherent and Incoherent Scattering

The scattering length, bi , depends on the nuclear isotope, spin relative to theneutron & nuclear eigenstate. For a single nucleus:

)(

jiunlessvanishesand0but

)(

zerotoaveragesb where

22

,

).(2

2222

2

Nbbebd

d

bbbbb

bbb

bbbbbbbb

bbb

ji

RRQi

ii

ji

jijiji

ii

ji

+=

==

==

+++=

+=

rrr

Coherent Scattering(scattering depends on the

direction of Q)

Incoherent Scattering(scattering is uniform in all directions)

Note: N = number of atoms in scattering system


14/191

Values of coh andinc

0.024.936Ar0.01.5Al

0.57.5Cu0.04.2O

5.21.0Co0.05.6C

0.411.5Fe2.05.62H

5.00.02V80.21.81H

inccohNuclideinccohNuclide

Difference between H and D used in experiments with soft matter (contrast variation) Al used for windows

V used for sample containers in diffraction experiments and as calibration for energy

resolution

Fe and Co have nuclear cross sections similar to the values of their magnetic cross sections

Find scattering cross sections at the NIST web site at:

http://webster.ncnr.nist.gov/resources/n-lengths/


15/191

Coherent Elastic Scattering measures the Structure

Factor S(Q) I.e. correlations of atomic positions

only.Roffunctionais)(where

.)(.1)(ie

)()(1

)'()(.'1

or

)(.1)(so

densitynumbernucleartheiswhere)(.)(.Now

1)(whereatomssimilarofassemblyanfor)(.

0

0

.

.)'.(

2.

N

...

ensemble,

).(2

rrrrr

rrr

rrrrrrrrrr

rrr

rrrrr

rr

rr

rrrrr

rr

rrrrrr

rrr

+=

+=

==

=

==

==

i

i

RQi

NN

RQi

NN

rrQi

NrQi

N

rQi

i

i

rQi

i

RQi

ji

RRQi

RRR)Rg(

eRgRdQS

RrrerdRdN

rrerdrdN

)QS(

rerdN

QS

rerdRrerde

eN

QSQSNbd

d

i

ji

g(R) is known as the static pair correlation function. It gives the probability that there is anatom, i, at distance R from the origin of a coordinate system at time t, given that there is also a

(different) atom at the origin of the coordinate system


16/191

S(Q) and g(r) for Simple Liquids

Note that S(Q) and g(r)/ both tend to unity at large values of their arguments The peaks in g(r) represent atoms in coordination shells

g(r) is expected to be zero for r < particle diameter ripples are truncation

errors from Fourier transform of S(Q)


17/191

Neutrons can also gain or lose energy in the scattering process: this is

called inelastic scattering


18/191

Inelastic neutron scattering measures atomic motions

The concept of a pair correlation function can be generalized:

G(r,t) = probability of finding a nucleus at (r,t) given that there is one at r=0 at t=0

Gs(r,t) = probability of finding a nucleus at (r,t) if the same nucleus was at r=0 at t=0

Then one finds:

dtrdetrGQSdtrdetrGQS

QNSk

kb

dEd

d

QNSk

kb

dEd

d

trQisi

trQi

iinc

inc

cohcoh

rrh

rrr

h

r

r

r

rrrr

==

=

=

).().(

22

22

),(21),(and),(

21),(

where

),('

.

),('

.

Inelastic coherent scattering measures correlatedmotions of atoms

Inelastic incoherent scattering measuresself-correlations e.g. diffusion

(h/2)Q & (h/2) are the momentum &energy transferred to the neutron during the

scattering process


19/191

Magnetic Scattering

The magnetic moment of the neutron interacts with B fieldscaused, for example, by unpaired electron spins in a material Both spin and orbital angular momentum of electrons contribute toB

Expressions for cross sections are more complex than for nuclear scattering

Magnetic interactions are long range and non-central

Nuclear and magnetic scattering have similar magnitudes

Magnetic scattering involves a form factor FT of electron spatial distribution

Electrons are distributed in space over distances comparable to neutron wavelength

Elastic magnetic scattering of neutrons can be used to probe electron distributions

Magnetic scattering depends only on component ofBperpendicular to Q

For neutrons spin polarized along a direction z (defined by applied H field):

Correlations involvingBzdo not cause neutron spin flip Correlations involvingBx orBy cause neutron spin flip

Coherent & incoherent nuclear scattering affects spin polarized neutrons

Coherent nuclear scattering is non-spin-flip

Nuclear spin-incoherent nuclear scattering is 2/3 spin-flip

Isotopic incoherent scattering is non-spin-flip


20/191

Magnetic Neutron Scattering is a Powerful Tool

In early work Shull and his collaborators: Provided the first direct evidence of antiferromagnetic ordering Confirmed the Neel model of ferrimagnetism in magnetite (Fe3O4)

Obtained the first magnetic form factor (spatial distribution of magneticelectrons) by measuring paramagnetic scattering in Mn compounds

Produced polarized neutrons by Bragg reflection (where nuclear and magnetic

scattering scattering cancelled for one neutronspin state) Determined the distribution of magnetic moments in 3d alloys by measuring

diffuse magnetic scattering

Measured the magnetic critical scattering at the Curie point in Fe

More recent work using polarized neutrons has:

Discriminated between longitudinal & transverse magnetic fluctuations Provided evidence of magnetic solitons in 1-d magnets

Quantified electron spin fluctuations in correlated-electron materials

Provided the basis for measuring slow dynamics using the neutron spin-echotechnique..etc


21/191

0 .0 01 0.01 0 .1 1 10 100

Q (-1

)

EnergyTransfer(meV)

Neutro ns in Co nde ns ed Matter Res e arch

Spa l la t io n - Chopp er

ILL - w ithout spin-echoILL - with s pin-ech o

CrystalFields

SpinWaves

Neutron

InducedExcitationsHydrogen

Modes

MolecularVibrations

LatticeVibra tions

andAnharmonicity

AggregateMotion

Polymersand

BiologicalSystems

CriticalScattering

DiffusiveModes

S low e r

Mo t i o n s

Reso l ved

MolecularReorientation

TunnelingSpectroscopy

Surface

Effects?

[L arger Obj ects Resolved]

Proteins in SolutionViruses

Micelles PolymersMetallurgical

Systems

Colloids

MembranesProteins

C rystal andMagnetic

Structures

AmorphousSystems

Accurate Liquid Structures

Precision Crystallography

Anharmonicity

1

100

10000

0.01

10-4

10-6

MomentumDistributions

Elastic Scattering

Electron-phonon

Interactions

Itine rantMagnets

MolecularMotions

CoherentModes inGlasses

and Liquids

SPSS - Chopper Spectrometer

Neutron scattering experiments measure the number of neutrons scattered at different values of

the wavevector and energy transfered to the neutron, denoted Q and E. The phenomena probed

depend on the values of Q and E accessed.


22/191

Next Lecture

2. Neutron Scattering Instrumentation and Facilities how is

neutron scattering measured?

1. Sources of neutrons for scattering reactors & spallation sources

1. Neutron spectra2. Monochromatic-beam and time-of-flight methods

2. Instrument components

1. Crystal monochromators and analysers

2. Neutron guides

3. Neutron detectors

4. Neutron spin manipulation

5. Choppers

6. etc

3. A zoo of specialized neutron spectrometers


23/191

References

Introduction to the Theory of Thermal Neutron Scattering

by G. L. Squires

Reprint edition (February 1997)

Dover Publications

ISBN 048669447

Neutron Scattering: A Primer

by Roger Pynn

Los Alamos Science (1990)

(see www.mrl.ucsb.edu/~pynn)


24/191

by

Roger Pynn

Los Alamos

National Laboratory

LECTURE 2: Neutron Scattering Instrumentation & Facilities


25/191


26/191

Recapitulation of Key Messages From Lecture #1

Neutron scattering experiments measure the number of neutrons scatteredby a sample as a function of the wavevector change (Q) and the energychange (E) of the neutron

Expressions for the scattered neutron intensity involve the positions and

motions of atomic nuclei or unpaired electron spins in the scattering sample

The scattered neutron intensity as a function of Q and E is proportional tothe space and time Fourier Transform of the probability of finding two atomsseparated by a particular distance at a particular time

Sometimes the change in the spin state of the neutron during scattering isalso measured to give information about the locations and orientations ofunpaired electron spins in the sample


27/191

What Do We Need to Do a Basic Neutron Scattering Experiment?

A source of neutrons A method to prescribe the wavevector of the neutrons incident on the sample

(An interesting sample)

A method to determine the wavevector of the scattered neutrons Not needed for elastic scattering

A neutron detector

Incident neutrons of wavevector k0 Scattered neutrons of

wavevector, k1

k0

k1Q

Detector

Remember: wavevector, k, &

wavelength, , are related by:k = mnv/(h/2) = 2/

Sample


28/191

Neutron Scattering Requires Intense Sources of Neutrons

Neutrons for scattering experiments can be produced either by nuclearfission in a reactor or by spallation when high-energy protons strike a heavymetal target (W, Ta, or U). In general, reactors produce continuous neutron beams and spallation sources produce

beams that are pulsed between 20 Hz and 60 Hz

The energy spectra of neutrons produced by reactors and spallation sources are different,

with spallation sources producing more high-energy neutrons

Neutron spectra for scattering experiments are tailored by moderators solids or liquids

maintained at a particular temperature although neutrons are not in thermal equilibrium

with moderators at a short-pulse spallation sources

Both reactors and spallation sources are expensive to build and requiresophisticated operation. SNS at ORNL will cost about $1.5B to construct & ~$140M per year to operate

Either type of source can provide neutrons for 30-50 neutron spectrometers Small science at large facilities


29/191

About 1.5 Useful Neutrons Are Produced by Each Fission Event ina Nuclear Reactor Whereas About 25 Neutrons Are Produced byspallation for Each 1-GeV Proton Incident on a Tungsten Target

Nuclear Fission

Spallation

Artists view of spallation


30/191

Neutrons From Reactors and Spallation Sources Must BeModerated Before Being Used for Scattering Experiments

Reactor spectra are Maxwellian Intensity and peak-width ~ 1/(E)1/2 at

high neutron energies at spallation

sources

Cold sources are usually liquid

hydrogen (though deuterium is alsoused at reactors & methane is some-

times used at spallation sources)

Hot source at ILL (only one in the

world) is graphite, radiation heated.


31/191

The ESRF* & ILL* With Grenoble & the Beldonne Mountains

*ESRF = European Synchrotron Radiation Facility; ILL = Institut Laue-Langevin


32/191

The National Institute of Standards and Technology (NIST) Reactor Is a20 MW Research Reactor With a Peak Thermal Flux of 4x1014 N/sec. It Is

Equipped With a Unique Liquid-hydrogen Moderator That Provides Neutrons

for Seven Neutron Guides


33/191

Neutron Sources Provide Neutrons for ManySpectrometers: Schematic Plan of the ILL Facility


34/191

A ~ 30 X 20 m2 Hall at the ILL Houses About 30 Spectrometers.Neutrons Are Provided Through Guide Tubes


35/191

Los Alamos Neutron Science Center

800-MeV LinearAccelerator

LANSCEVisitors Center

Manuel Lujan Jr.Neutron Scattering

Center

WeaponsNeutron Research

Proton StorageRing

ProtonRadiography


36/191

Neutron Production at LANSCE

Linac produces 20 H- (a proton + 2 electrons) pulses per second

800 MeV, ~800 sec long pulses, average current ~100 A

Each pulse consists of repetitions of 270 nsec on, 90 nsec off

Pulses are injected into a Proton Storage Ring with a period of 360 nsec

Thin carbon foil strips electrons to convert H-to H+ (I.e. a proton)

~3 x 1013

protons/pulse ejected onto neutron production target


37/191

Components of a Spallation SourceA half-mile long proton linac

.a proton accumulation ring.

and a neutron production target


38/191

A Comparison of Neutron Flux Calculations for the ESS SPSS (50 Hz, 5 MW)& LPSS (16 Hz, 5W) With Measured Neutron Fluxes at the ILL

0 1 2 3 4

1010

1011

1012

1013

1014

1015

1016

1017

ILL hot sourceILL thermal sourceILL cold source

average fluxpoisoned m.

decoupled m.coupled m.

and long pulse

Flux[n/cm

2/s/str/]

Wavelength []

0 1 2 3 4

1010

1011

1012

1013

1014

1015

1016

1017

ILL hot source

ILL thermal sourceILL cold source

peak flux

poisoned m.decoupled m.coupled m.

long pulse

Flux[n/cm

2/s/str/]

Wavelength []

0 500 1000 1500 2000 2500 3000

0.0

5.0x1013

1.0x1014

1.5x1014

2.0x1014

= 4

poisoned moderatordecoupled moderatorcoupled moderatorlong pulse

Instantaneousflux[n/cm

2/s/str/

]

Time [s]

Pulsed sources only make sense ifone can make effective use of the

flux in each pulse, rather than the

average neutron flux


39/191

Simultaneously Using Neutrons With Many Different WavelengthsEnhances the Efficiency of Neutron Scattering Experiments

Potential Performance Gain relative to use of a Single Wavelengthis the Number of Different Wavelength Slices used

ki

ki

kf

kf

res= bw

res T

bw T

I

I

I

NeutronSource Spectrum


40/191

The Time-of-flight Method Uses Multiple Wavelength Slicesat a Reactor or a Pulsed Spallation Source

When the neutron wavelength is determined by time-of-flight, T/Tp differentwavelength slices can be used simultaneously.

T

T

Tp

Time

Distance

Detector

Moderator

res= 3956 Tp/ L

bw= 3956 T / L

L


41/191

The Actual ToF Gain From Source Pulsing

Often Does Not Scale Linearly With Peak Neutron Flux

ki

kf

QQ

k i

ki

k f

k f

I

I

k i k f

low rep. rate => large dynamic rangeshort pulse => good resolution neither may be necessary or useful

large dynamic range may result in intensitychanges across the spectrum

at a traditional short-pulse source the wave-length resolution changes with wavelength


42/191

A Comparison of Reactors & Spallation Sources

Neutron spectrum is MaxwellianNeutron spectrum is slowing downspectrum preserves short pulses

Energy deposited per useful neutron is~ 180 MeV

Energy deposited per useful neutron is~20 MeV

Neutron polarization easierSingle pulse experiments possible

Pulse rate for TOF can be optimizedindependently for different

spectrometers

Low background between pulses =>good signal to noise

Large flux of cold neutrons => verygood for measuring large objects and

slow dynamics

Copious hot neutrons=> very good formeasurements at large Q and E

Resolution can be more easily tailoredto experimental requirements, exceptfor hot neutrons where monochromatorcrystals and choppers are less effective

Constant, small / at large neutronenergy => excellent resolutionespecially at large Q and E

ReactorShort Pulse Spallation Source


43/191


44/191

Brightness & Fluxes for Neutron &X-ray Sources

Brightness(s

-1m

-2ster

-1)

dE/E(%)

Divergence(mrad

2)

Flux(s

-1m

-2)

Neutrons 1015 2 10 x 10 1011

RotatingAnode

1016 3 0.5 x 10 5 x 1010

BendingMagnet

1024 0.01 0.1 x 5 5 x 1017

Wiggler 1026

0.01 0.1 x 1 1019

Undulator(APS)

1033 0.01 0.01 x 0.1 1024


45/191

Instrumental Resolution

Uncertainties in the neutronwavelength & direction of travelimply that Q and E can only bedefined with a certain precision

When the box-like resolutionvolumes in the figure are convolved,the overall resolution is Gaussian(central limit theorem) and has anelliptical shape in (Q,E) space

The total signal in a scatteringexperiment is proportional to thephase space volume within theelliptical resolution volume the better the resolution, the lower the countrate


46/191

Examples of Specialization of Spectrometers:Optimizing the Signal for the Science

Small angle scattering [Q = 4 sin/; (Q/Q)2 = (/)2 + (cot )2] Small diffraction angles to observe large objects => long (20 m) instrument

poor monochromatization (/ ~ 10%) sufficient to match obtainable angular

resolution (1 cm2pixels on 1 m2 detector at 10 m => ~ 10-3 at ~ 10-2))

Back scattering [ = /2; = 2 d sin ; / = cot +] very good energy resolution (~neV) => perfect crystal analyzer at ~ /2

poor Q resolution => analyzer crystal is very large (several m2)


47/191

Typical Neutron Scattering Instruments

Note: relatively massive shielding; longflight paths for time-of-flight spectrometers;

many or multi-detectors

but small science at a large facility


48/191

Components of Neutron Scattering Instruments

Monochromators Monochromate or analyze the energy of a neutron beam using Braggs law

Collimators Define the direction of travel of the neutron

Guides

Allow neutrons to travel large distances without suffering intensity loss Detectors

Neutron is absorbed by 3He and gas ionization caused by recoiling particlesis detected

Choppers

Define a short pulse or pick out a small band of neutron energies Spin turn coils

Manipulate the neutron spin using Lamor precession

Shielding Minimize background and radiation exposure to users


49/191

Most Neutron Detectors Use 3He

3He + n -> 3H + p + 0.764 MeV

Ionization caused by triton and protonis collected on an electrode

70% of neutrons are absorbedwhen the product of gaspressure x thickness x neutronwavelength is 16 atm. cm.

Modern detectors are often positionsensitive charge division is usedto determine where the ionizationcloud reached the cathode. A selection of neutron detectors

thin-walled stainless steel tubes

filled with high-pressure 3He.


50/191

Essential Components of Modern NeutronScattering Spectrometers

Horizontally & vertically focusing

monochromator (about 15 x 15 cm2

)

Pixelated detector covering a wide range

of scattering angles (vertical & horizontal)

Neutron guide(glass coatedeither with Ni

or supermirror)


51/191

Rotating Choppers Are Used to Tailor NeutronPulses at Reactors and Spallation Sources

T-zero choppers made of Fe-Co are used at spallation sourcesto absorb the prompt high-energy pulse of neutrons

Cd is used in frame overlap choppers to absorb slowerneutrons

Fast neutrons from one pulse can catch-up with

slower neutrons from a succeeding pulse and spoil

the measurement if they are not removed.This is

called frame-overlap


52/191

Larmor Precession & Manipulation of the Neutron Spin

In a magnetic field, H, the neutronspin precesses at a rate

where is the neutrons gyromagnetic

ratio & H is in Oesteds This effect can be used to manipulate

the neutron spin

A spin flipper which turns the spin

through 180 degrees is illustrated

The spin is usually referred to as up

when the spin (not the magnetic

moment) is parallel to a (weak ~ 10

50 Oe) magnetic guide field

Hz4.29162/ HHL

==


53/191

Next Lecture

3. Diffraction1. Diffraction by a lattice

2. Single-crystal diffraction and powder diffraction

3. Use of monochromatic beams and time-of-flight to measure powder

diffraction

4. Rietveld refinement of powder patterns

5. Examples of science with powder diffraction

Refinement of structures of new materials

Materials texture

Strain measurements

Structures at high pressure Microstrain peak broadening

Pair distribution functions (PDF)


54/191

by

Roger Pynn

Los Alamos

National Laboratory

LECTURE 3: Diffraction


55/191

This Lecture

2. Diffraction1. Diffraction by a lattice

2. Single-crystal compared with powder diffraction

3. Use of monochromatic beams and time-of-flight to measure powder

diffraction

4. Rietveld refinement of powder patterns

5. Examples of science with powder diffraction

Refinement of structures of new materials

Materials texture

Strain measurements



56/191

From Lecture 1:

0

)(

,

)()'(

,

'bydefinedisQr transferwavevectothewhere

secondpercmsquareperneutronsincidentofnumber

dintosecondper2anglethroughscatteredneutronsofnumber

0

kkQ

ebbebbd

d

d

d

jiji RR.Qicoh

j

ji

coh

i

RR.kkicoh

j

ji

coh

i

cohrrrr

rrrrrrr

=

==

=

Incident neutrons of wavevector k0Scattered neutrons of

wavevector, k

k0

kQ

Detector

Sample

2

2For elastic scattering k0=k=k:

Q = 2k sin Q= 4 sin /


57/191

Neutron Diffraction

Neutron diffraction is used to measure the differential cross section, d/dand hence the static structure of materials Crystalline solids

Liquids and amorphous materials

Large scale structures

Depending on the scattering angle,structure on different length scales, d,

is measured:

For crystalline solids & liquids, use

wide angle diffraction. For large structures,

e.g. polymers, colloids, micelles, etc.

use small-angle neutron scattering

)sin(2//2 ==dQ


58/191

Diffraction by a Lattice of Atoms

.Gvector,latticereciprocalatoequalis

Q,r,wavevectoscatteringwhen theoccurs

onlylattice(frozen)afromscatteringSo

.2).(thenIf

.2.Then

ns.permutatiocyclicand2

Define

cell.unittheofon vectorstranslatiprimitivethe

are,,wherecan writewelattice,BravaisaIn

.2).(such thatsQ'forzero-nononlyisS(Q)s,vibrationthermalIgnoring

position.mequilibriuthefromthermal)(e.g.ntdisplacemeanyisandiatomof

positionmequilibriutheiswhere with1

)(

hkl

*

3

*

2

*

1

*

32

0

*

1

321332211

,

).(

MjiQalakahGQ

aa

aaV

a

aaaamamami

MjiQ

u

iuiRe

N

QS

hkl

ijji

iii

i

ii

ji

RRQi ji

=++==

=

=

++=

=

+==

rrr

rrr

vr

rr

rrr

rrrrrrv

rrr

r

rr

rrr

vrr

a1

a2

a3


59/191


60/191

Bragg Scattering from Crystals

Using either single crystals or powders, neutron diffraction can be used tomeasure F2 (which is proportional to the intensity of a Bragg peak) forvarious values of (hkl).

Direct Fourier inversion of diffraction data to yield crystal structures is notpossible because we only measure the magnitude of F, and not its phase =>models must be fit to the data

Neutron powder diffraction has been particularly successful at determiningstructures of new materials, e.g. high Tc materials

atomsofmotionslfor thermaaccountstfactor thaWaller-Debyetheisand

)(

bygivenisfactorstructurecell-unitthewhere

)()()2(

:findwebook),Squires'example,for(see,mathhethrough tWorking

.

2

0

3

d

WdQi

d

dhkl

hkl

hklhkl

Bragg

W

eebQF

QFGQV

Nd

d

d

=

=

rrr

rrr


61/191

We would be better off

if diffraction measured

phase of scattering

rather than amplitude!

Unfortunately, naturedid not oblige us.

Picture by courtesy of D. Sivia


62/191

Reciprocal Space An Array of Points (hkl)

that is Precisely Related to the Crystal Lattice

a*

b*

(hkl)=(260)

Ghkl

a* = 2(b x c)/V0, etc.

OGhkl = 2/dhkl

k0k

A single crystal has to be aligned precisely to record Bragg scattering


63/191

Powder A Polycrystalline Mass

All orientations ofcrystallites possible

Typical Sample: 1cc powder of

10m crystallites - 109 particlesif 1m crystallites - 1012 particles

Single crystal reciprocal lattice- smeared into spherical shells

P d Diff ti i S tt i


64/191

Incident beamx-rays or neutrons

Sample(111)

(200)

(220)

Powder Diffraction gives Scattering on

Debye-Scherrer Cones

Braggs Law = 2dsinPowder pattern scan 2 or

Measuring Neutron Diffraction Patterns with a


65/191

Measuring Neutron Diffraction Patterns with a

Monochromatic Neutron Beam

Since we know the neutron wavevector, k,

the scattering angle gives Ghkl directly:

Ghkl = 2 k sin

Use a continuous beam

of mono-energetic

neutrons.


66/191

Neutron Powder Diffraction using Time-of-Flight

Sample

Detector

bank

Pulsed

source

Lo= 9-100m

L1 ~ 1-2m

2 - fixed

= 2dsin

Measure scattering asa function of time-of-

flight t = const*


67/191

Time-of-Flight Powder Diffraction

Use a pulsed beam with a

broad spectrum of neutron

energies and separate

different energies (velocities)

by time of flight.


68/191

10.0 0.05 155.9 CPD RRRR PbSO4 1.909A neutron data 8.8

Scan no. = 1 Lambda = 1.9090 Observed Profile

D-spacing, A

Counts

1.0 2.0 3.0 4.0

X10E

3

.5

1.0

1.5

2.0

2.5

10.000 0.025 159.00 CPD RRRR PbSO4 Cu Ka X-ray data 22.9.

Scan no. = 1 Lambda1,lambda2 = 1.540 Observed Profile

D-spacing, A

Counts

1.0 2.0 3.0 4.0

X10E

4

.0

.5

1.0

1.5X-ray Diffraction - CuKaPhillips PW1710

Higher resolution Intensity fall-off at small dspacings Better at resolving smalllattice distortions

Neutron Diffraction - D1a, ILL=1.909 Lower resolution Much higher intensity at small

d-spacings Better atomic positions/thermalparameters

Compare X-ray & Neutron Powder Patterns


69/191

Ic

= Ib

+Yp

Io

Ic

YpIb

Rietveld Model

TOF

Intensity

Theres more than meets the eye in a powder pattern*

*Discussion of Rietveld method adapted from viewgraphs by R. Vondreele (LANSCE)


70/191

The Rietveld Model for Refining Powder Patterns

Io - incident intensity - variable for fixed 2

kh - scale factor for particular phase

F2h - structure factor for particular reflection

mh - reflection multiplicity

Lh - correction factors on intensity - texture, etc.P(h) - peak shape function includes instrumental resolution,

crystallite size, microstrain, etc.

Ic = Io{khF2

hmhLhP(h) + Ib}


71/191

How good is this function?

Protein Rietveld refinement - Very low angle fit1.0-4.0 peaks - strong asymmetry

perfect fit to shape


72/191

Examples of Science using Powder Diffraction

Refinement of structures of new materials Materials texture

Strain measurements



73/191

High-resolution Neutron Powder Diffraction in CMR manganates*

In Sr 2LaMn2O7 synchrotron data indicated

two phases at low temperature. Simultaneous

refinement of neutron powder data at 2sallowed two almost-isostructural phases (one

FM the other AFM) to be refined. Only

neutrons see the magnetic reflections

High resolution powder diffraction

with Nd0.7Ca0.3MnO3 showed

splitting of (202) peak due to a

transition to a previously unknown

monoclinic phase. The data show-

ed the existence of 2 Mn sites with

different Mn-O distances. The

different sites are likely occupied

by Mn3+ and Mn4+ respectively

* E. Suard & P. G. Radaelli (ILL)

Texture: Interesting Preferred Orientation


74/191

Random powder - all crystallite orientations

equally probable - flat pole figure

Crystallites oriented along wire axis - pole figure

peaked in center and at the rim (100s are 90apart)

Orientation Distribution Function - probabilityfunction for texture

(100) wire texture(100) random texture

Texture: Interesting Preferred Orientation

Pole figure - stereographic projection of acrystal axis down some sample direction

Loose powder

Metal wire


75/191

Texture Determination of Titanium Wire Plate


76/191

Texture Determination of Titanium Wire Plate(Wright-Patterson AFB/LANSCE Collaboration)

Possible pseudo-single crystal turbine blade material

Ti Wire Plate - does itstill have wire texture?

Ti Wire -(100) texture

Bulk measurement

Neutron time-of-flight data

Rietveld refinement of texture

Spherical harmonics to Lmax = 16

Very strong wire texture in plate

TD

ND

reconstructed (100) pole figure

ND

RDTD

D fi iti f St d St i


77/191

Definitions of Stress and Strain

Macroscopic strain total strain measured by an extensometer Elastic lattice strain response of lattice planes to applied

stress, measured by diffraction

Intergranular strain deviation of elastic lattice strain from

linear behavior

Residual strains internal strains present with no applied force

Thermal residual strains strains that develop on cooling from

processing temperature due to anisotropic coefficients of

thermal expansion

hkl 0hkl

0

d d

d

=

appliedI hkl

hklE

=

Neutron Diffraction Measurements of Lattice Strain*


78/191

Neutron Diffraction Measurements of Lattice Strain

Neutron measurements :- Non destructive, bulk, phase sensitive Time consuming, Limited spatial resolution

Text box

Incident neutrons

-90 Detectormeasuresparallelto loading direction

+90 Detectormeasuresto loading direction

The Neutron Powder

Diffractometer at LANSCE

* Discussion of residual strain adapted from viewgraphs by M. Bourke (LANSCE)

Why use Metal Matrix Composites ?


79/191

Why use Metal Matrix Composites ?

Higher Pay Loads

High temperaturesHigh pressuresReduced Engine WeightsReduced Fuel ConsumptionBetter Engine Performance

RotorsFan Blades

Structural RodsImpellersLanding Gears

MMC ApplicationsNational AeroSpace PlaneF117 (JSF)

EngineLanding Gear

W-Fe :- Fabrication Microstructure & Composition


80/191

W-Fe :- Fabrication, Microstructure & Composition

200 m diameter continuous Tungsten fibers,

Hot pressed at 1338 K for 1 hour into Kanthal( 73 Fe, 21 Cr, 6 Al wt%) leads to residual strains after cooling

Specimens :- 200 * 25 * 2.5 mm3

10 vol.%

70 vol.%20 vol.%

30 vol.%W

K

K

W

d spacing (A)

NormalizedIntensity

Powder diffraction data includes Bragg peaks

from both Kanthal and Tungsten.

Neutron Diffraction Measures Mean Residual Phase


81/191

Strains when Results for a MMC are compared to an

Undeformed Standard

Symbols show data points

Curves are fits of the form =< 11>cos2 + < 22>sin

2

-4000

-3000

-2000

-1000

0

1000

2000

3000

4000

0 20 40 60 80 100

10 vol.% W

20 vol.% W

30 vol.% W

70 vol.% W

10 vol.% W

20 vol.% W

30 vol.% W

70 vol.% W

Angle to the fiber (deg)

ResidualStrain(e)

Kanthal

TungstenResiduals

train

(e)

Pressure Vessels and Piping, Vol. 373, Fatigue, Fracture & Residual Stresses, Book No H01154

Load Sharing in MMCs can also be


82/191

g

Measured by Neutron Diffraction

Initial co-deformation results

from confinement of W fibers

by the Kanthal matrix

Change of slope (125 MPa) is

the typical load sharing

behavior of MMCs:

Kanthal yields and ceases to bear

further load

Tungsten fibers reinforce and

strengthen the composite -1000 -500 0 500 1000 1500 2000 2500 30000

50

100

150

200

250

300

350

400

KanthalTungsten

Elastic lattice strain []

Appliedstress

[MPa

]

Deformation Mechanism in U/Nb Alloys


83/191

Important Programmatic Answers.1. Material is Single Phase Monoclinic ().2. Lack of New Peaks Rules out Stress Induced Phase Transition.3. Changes in Peak Intensity Indicate That Twinning is the Deformation Mechanism.

y

+ 90 Detector

-90 Detector

Tensile Axis

Q

Q||

0

100

200

300

400

500

0 0.02 0.04 0.06

U-6 wt% Nb

AppliedStress(M

Pa)

Macroscopic Strain

Typical Metal

Atypical Double Plateau Stress Strain Curve.

NPD Diffraction Geometry

Diffraction Results

Pair Distribution Functions


84/191

Pair Distribution Functions

Modern materials are often disordered.

Standard crystallographic methods lose the aperiodic

(disorder) information. We would like to be able to sit on an atom and look at

our neighborhood.

The PDF method allows us to do that (see next slide):

First we do a neutron or x-ray diffraction experiment Then we correct the data for experimental effects

Then we Fourier transform the data to real-space

Obtaining the Pair Distribution Function*


85/191

Obtaining the Pair Distribution Function

Raw data

Structure function

QrdQSQrG sin]1)([2

)(0

=

PDF

* See http://www.pa.msu.edu/cmp/billinge-group/

Structure and PDF of a High Temperature


86/191

Superconductor

The resulting PDFs look like this.

The peak at 1.9A is the Cu-Obond.

So what can we learn aboutcharge-stripes from the PDF?

The structure of La2-xSrxCuO4looks like this: (copper [orange]sits in the middle of octahedraof oxygen ions [shown shadedwith pale blue].)

Effect of Doping on the Octahedra


87/191

Effect of Doping on the Octahedra

Doping holes (positivecharges) by adding Srshortens Cu-O bonds

Localized holes in stripesimplies a coexistence ofshort and long Cu-O in-plane bonds => increase inCu-O bond distributionwidth with doping.

We see this in the PDF: 2is the width of the CuObond distribution whichincreases with doping then

decreases beyond optimaldoping

Polaronic

Fermi-

liquidlike

Bozin et al. Phys. Rev. Lett. Submitted;

cond-mat/9907017


88/191

by

Roger Pynn

Los AlamosNational Laboratory

LECTURE 3: Surface Reflection

Surface Reflection Is Very Different From


89/191

Most Neutron Scattering

We worked out the neutron cross section by adding scatteringfrom different nuclei We ignored double scattering processes because these are usually very weak

This approximation is called the Born Approximation

Below an angle of incidence called the critical angle, neutronsare perfectly reflected from a smooth surface This is NOT weak scattering and the Born Approximation is not applicable to

this case

Specular reflection is used: In neutron guides

In multilayer monochromators and polarizers

To probe surface and interface structure in layered systems

This Lecture


90/191

This Lecture

Reflectivity measurements Neutron wavevector inside a medium

Reflection by a smooth surface

Reflection by a film

The kinematic approximation

Graded interface Science examples

Polymers & vesicles on a surface

Lipids at the liquid air interface

Boron self-diffusion

Iron on MgO Rough surfaces

Shear aligned worm-like micelles

What Is the Neutron Wavevector Inside a Medium?


91/191

What Is the Neutron Wavevector Inside a Medium?

[ ]

materials.mostfromreflectedexternallyareneutrons1,ngenerallySince

2/1

:getwe)A10(~smallveryisand),definition(byindexrefractiveSincematerialainrwavevectotheisandvevectorneutron waiswhere

4/)(2Simlarly./2so)(

0)(/)(2

:equationsr'SchrodingeobeysneutronThe

(SLD)DensityLengthScatteringnuclearthecalledis

1where

2:ismediumtheinsidepotentialaveragetheSo

nucleus.singleafor)(2

)(:bygivenispotentialneutron-nucleus

theRule,GoldensFermi'bygivenwith thatS(Q)forexpressionourComparing

2

2-6-0

0

20

22220

.

22

2

2

==>

=

=+

What do the Amplitudes aR and aT Look Like?


95/191

What do the Amplitudes aR and aT Look Like?

For reflection from a flat substrate, both aR and aT are complex when k0 < 4I.e. below the critical edge. For aI = 1, we find:

0.005 0.01 0.015 0.02 0.025

-1

-0.5

0.5

1

Real (red) & imaginary (green) parts of aRplotted against k0. The modulus of aR isplotted in blue. The critical edge is atk0 ~ 0.009A

-1 .Note that the reflected wave iscompletely out of phase with the incident waveat k0 = 0

Real (red) and imaginary (green) partsof aT. The modulus of aT is plotted in

blue. Note that aT tends to unity atlarge values of k0 as one would expect

0.005 0.01 0.015 0.02 0.025

-1

-0.5

0.5

1

1.5

2

One can also think about Neutron Reflection from a Surface as a


96/191

V(z)

z

1-d Problem

V(z)= 2 (z) h 2/mn

k2=k02 - 4 (z)

Where V(z) is the potential seen bythe neutron & (z) is the scatteringlength density

substrate

Film Vacuum

Fresnels Law for a Thin Film


97/191

r=(k1z-k0z)/(k1z+k0z) is Fresnels law

Evaluate with =4.10-6A-2 gives thered curve with critical wavevector

given by k0z = (4)1/2

If we add a thin layer on top of thesubstrate we get interference fringes &

the reflectance is given by:

and we measure the reflectivity R = r.r*

If the film has a higher scattering length density than the substrate we get thegreen curve (if the film scattering is weaker than the substance, the green curve isbelow the red one)

The fringe spacing at large k0z is ~ /t (a 250 A film was used for the figure)

0.01 0.02 0.03 0.04 0.05

-5

-4

-3

-2

-1

Log(r.r*)

k0z

tki

tki

z

z

err

errr

1

1

21201

21201

1++

=

substrate

0

1 Film thickness = t

2

Kinematic (Born) Approximation


98/191

( ) pp

We defined the scattering cross section in terms of an incident plane wave & aweakly scattered spherical wave (called the Born Approximation)

This picture is not correct for surface reflection, except at large values of Qz For large Qz, one may use the definition of the scattering cross section to

calculate R for a flat surface (in the Born Approximation) as follows:

z

4222

22)'.(2

00

20

yx

yx

QlargeatformFresneltheassametheisthat thisshoweasy toisIt

/16so)()(4

'

:substratesmoothaforgetsection wecrossaofdefinitiontheFrom .sinsocosbecause

sinsin

1

sin

1

sin

)sinLL(sampleonincidentneutronsofnumber

LLsizeofsampleabyreflectedneutronsofnumber

zyxyx

z

rrQi

xx

yx

yxyxyx

QRQQLLQ

erdrdd

d

dkdkkk

k

dkdk

d

d

LLd

d

d

LLLL

R

===

==

=

==

==

rvrrr

Reflection by a Graded Interface


99/191

y

ion.justificatphysicalgoodhavemustdatatyrefelctivifittorefined

modelsthatmeansThisn.informatiophaseimportantlackgenerallywe

because(z),obtainouniquely tinvertedbecannotdatatyreflectivi

:pointimportantansillustrateequationeapproximatThis(z).1/coshas

such/dzdofformsconvenientseveralforlyanalyticalsolvedbecanThis

)(

Qlargeatformcorrecttheaswellasinterface,smoothaforanswerrightget thewe,RtyreflectiviFresnelby theprefactorthereplaceweIf

parts.byngintergratiafterfollowsequality

secondthe where)(16

)(16

:givesof

dependence-zthekeepingbutviewgraphprevioustheoflinebottomtheRepeating

2

2

z

F

2

4

22

2

2

=

==

dzedz

zdRR

dzedz

zd

Qdzez

QR

ziQ

F

ziQ

z

ziQ

z

z

zz

The Goal of Reflectivity Measurements Is to Infer a


100/191

The Goal of Reflectivity Measurements Is to Infer aDensity Profile Perpendicular to a Flat Interface

In general the results are not unique, but independentknowledge of the system often makes them very reliable

Frequently, layer models are used to fit the data

Advantages of neutrons include: Contrast variation (using H and D, for example)

Low absorption probe buried interfaces, solid/liquid interfaces etc

Non-destructive

Sensitive to magnetism

Thickness length scale 10 5000

Direct Inversion of Reflectivity Data is Possible*


101/191

Use different fronting or backing materials for two

measurement of the same unknown film E.g. D2O and H2O backings for an unknown film deposited on a quartz

substrate or Si & Al2O3 as substrates for the same unknown sample

Allows Re(R) to be obtained from two simultaneous equations for

Re(R) can be Fourier inverted to yield a unique SLD profile

Another possibility is to use a magnetic backing andpolarized neutrons

Unknown film

Si or Al2O3 substrate

SiO2

H2O or D2O

2

2

2

1 and RR

* Majkrzak et al Biophys Journal, 79,3330 (2000)

Vesicles composed of DMPC molecules fuse creating almost a perfectlipid bilayer when deposited on the pure, uncoated quartz block*


102/191

(blue curves)

When PEI polymer was added only after quartz was covered by the lipidbilayer, the PEI appeared to diffuse under the bilayer (red curves)

10-9

10 -8

0.00 0.02 0.04 0.06 0.08 0.10 0.12

Re

flec

tiv

ity,

R*Q

z

4

Qz[-1 ]

Lipid Bilayer on Quartz

Lipid Bilayer on Polymeron Quartz

Neutron Reflectivities

-8 10 -6

-6 10 -6

-4 10 -6

-2 10 -6

0

2 10 -6

Sca

ttering

Leng

thDens

ity

[-2]

Length, z []

HydrogenatedTails

Head

Head

Quartz

Quartz

Polymer

0.0 20.0 40.0 60.0 80.0 100.0

2O

D2

O

= 4.3

= 5.9

Scattering Length DensityProfiles

* Data courtesy of G. Smith (LANSCE)

Polymer-Decorated Lipids at a Liquid-Air Interface*


103/191

1.3% PEG lipidin lipids



10-3

10-2

10-1

100

101

0 0.1 0.2 0.3 0.4 0.5 0.6

R/RF

Qz

Pure lipid

Lipid + 9% PEG

X-Ray Reflectivities

Interface broadens as PEGconcentration increases - this ismain effect seen with x-rays

mushroom-to-brush transition

neutrons see contrast betweenheads (2.6), tails (-0.4),

D2O (6.4) & PEG (0.24)

x-rays see heads (0.65), but allelse has same electron densitywithin 10% (0.33)

10-9

10-8

10-7

0 0.05 0.1 0.15 0.2 0.25

Neutron Reflectivities

R*Qz4

Qz[-1]

Pure lipid

Lipid + 9% PEG

0

2 10-6

4 10-6

6 10-6

8 10-6

40 60 80 100 120 140 160

SLD Profiles of PEG-Lipids

SLD[

-

2]

Length []

Black - pure lipidBlue - 1.3% PEGGreen - 4.5% PEGRed - 9.0% PEG

*Data courtesy of G. Smith (LANSCE)

Non-Fickian Boron Self-Diffusion at an Interface*10 -4Boron Self Diffusion


104/191

10-11

10-10

10-9

10-8

10-7

10 -6

10-5

0.045 0.05 0.055 0.06 0.065 0.07 0.075 0.08

10-11

10-9

10-7

10-5

0.001

0.1

10

0 0.02 0.04 0.06 0.08 0.1 0.12

Unnanealed sample2.4 Hours5.4 Hours9.7 Hours

22.3 Hours35.4 Hours

Reflectivity

Q ( -1

)

0

1 10-6

2 10-6

3 10-6

4 10-6

5 10-6

6 10-6

7 10-6

8 10-6

-800 -600 -400 -200 0 200 400 600 800

Non-Standard Fickian Model

Scattering

Length

Density(

-2)

Distance from11

B-10

B Interface ()

0

1 10-6

2 10-6

3 10-6

4 10-6

5 10-6

6 10-6

7 10-6

8 10 -6

-600 -400 -200 0 200 400 600 800

Standard Fickian Model

Scattering

Length

Density(

-2)

Distance from 11 B-10 B Interface ()

11B

10B

Si

~650

~1350

Fickian diffusiondoesnt fit the data

Data requires densitystep at interface

*Data courtesy of G. Smith (LANSCE)


105/191

Structure, Chemistry & Magnetism of Fe(001) on MgO(001)*

X R


106/191

10-6

10-5

10-4

10-3

10-2

10-1

100

0 0.1 0.2 0.3 0.4 0.5

X-Ray

Re

flec

tiv

ity

Q [-1]

2.1(2)nm

1.9(2)nm

10.7(2)nm Fe

MgO

= 3.9(3)

= 5.2(3)

= 4.4(2)

= 3.1(3)

0.20(1)nm

0.44(2)nm

0.07(1)nm

0.03(1)nm

X-Ray

= 4.4(8)

= 8.6(4)

= 6.9(4)

= 6.2(5)

Neutron

-FeOOH

interface

X-Ray

Neutron

Co-Refinement

10 -5

10 -4

10 -3

10 -2

10 -1

100

0.00 0.02 0.04 0.06 0.08 0.10 0.12

R++

(obs)

R++

(fit)

R--(obs)

R--(fit)

RSF

(obs)

RSF

(fit)

Po

larize

dN

eu

tron

Re

flec

tiv

ity

Q [ -1]

H

-FeOOH (1.8B)

Fe(001) (2.2B)

H

fcc Fe (4B)

TEM

*Data courtesy of M. Fitzsimmons (LANSCE)


107/191


108/191

Qx-QzTransformation

Time-of-Flight, Energy-Dispersive Neutron Reflectometry


109/191

x

z

Q

=kf

-ki Qx =

2 *(cos f-cos i )

Q z =2 *(sin f+sin i )

Q

kf

-ki

ki

fi

where

For specular reflectivity f=i . Then,

Qz=4 sin( )

=2k z

Raw Data from SPRaw Data from SP

Vandium-Carbon Multilayer specular & diffuse scatteringin f-TOF space and transformed to Qx-Qz

Raw data in f-TOF space for a single layer.Note that large divergence does not imply poor Qz resolution

TOF . L / 4


110/191

Observation of Hexagonal Packing of Thread-like MicellesUnder Shear: Scattering From Lateral Inhomogeneities


111/191

TEFLON LIP OUTLET

TRENCH

INLETHOLES

TEST SECTION

NEUTRONBEAM

TEFLON

QUARTZ or SILICON

RESERVOIRS

46

Up to

Microns

H2O

Quartz

Singl

e

Crystal

z

Flowdirection

x

Scattering patternimplies hexagonal

symmetry

Thread-like micelle

Specularly reflected beam


112/191

by

Roger Pynn

Los AlamosNational Laboratory

LECTURE 5: Small Angle Scattering


113/191

Small Angle Neutron Scattering (SANS) Is Used to

Measure Large Objects (~1 nm to ~1 m)


114/191

g j ( )

Complex fluids, alloys, precipitates,biological assemblies, glasses,

ceramics, flux lattices, long-wave-

length CDWs and SDWs, critical

scattering, porous media, fractalstructures, etc

Scattering at small angles probes

large length scales

Two Views of the Components of a Typical

Reactor-based SANS Diffractometer


115/191


116/191

Where Does SANS Fit As a Structural Probe?


117/191

SANS resolves structureson length scales of 1 1000nm

Neutrons can be used withbulk samples (1-2 mm thick)

SANS is sensitive to lightelements such as H, C & N

SANS is sensitive toisotopes such as H and D

What Is the Largest Object That Can Be Measured by SANS?


118/191

Angular divergence of the neutron beam and its lack of monochromaticity contribute to finite

transverse & longitudinal coherence lengths that limit the size of an object that can be seenby SANS

For waves emerging from slit center,

path difference is hd/L if d


119/191

The Scattering Cross Section for SANS


120/191

Since the length scale probed at small Q is >> inter-atomic spacing we may use thescattering length density (SLD), , introduced for surface reflection and note thatnnuc(r)b is the local SLD at position r.

A uniform scattering length density only gives forward scattering (Q=0), thus SANSmeasures deviations from average scattering length density.

If is the SLD of particles dispersed in a medium of SLD 0, and np(r) is theparticle number density, we can separate the integral in the definition of S(Q) intoan integral over the positions of the particles and an integral over a single particle.

We also measure the particle SLD relative to that of the surrounding medium I.e:

densitynuclear

theisandlengthscatteringnuclearcoherenttheiswhere

)(.1)(where)(thatrecall

2

2

.2

)r(nb

rnerdN

QSQNSbdOds

nuc

nucrQi

r

rr

rr rr

==

SANS Measures Particle Shapes and Inter-particle Correlations


121/191

2

.32

2

P

.32

2

0

2

.30

)'.(33

)'.(332

.)(

:factorformparticletheis)QF(andorigin)at theonesthere'ifRatparticleais

rey that theprobabilit(thefunctionncorrelatioparticle-particletheisGwhere

).(.)()(

.)()'()('

).'()('

=

=

=

=

parti cle

xQi

RQi

Pspace

P

parti cle

xQiRRQi

space space

PP

rrQi

space space

NN

exdQF

eRGRdNQFd

d

exdeRnRnRdRd

ernrnrdrdb

d

d

v

r

rr

v

rrrr

rr

r

v

vr

rr

rr

rr

These expressions are the same as those for nuclear scattering except for the additionof a form factor that arises because the scattering is no longer from point-like particles

Scattering for Spherical Particles


122/191

particles.ofnumberthetimes

volumeparticletheofsquarethetoalproportionis)]r()rG(when[i.e.particlessphericaleduncorrelatofassemblyanfromscatteringtotalthe0,QasThus,

0Qat)(3

)(

cossin3)(

:Qofmagnitudeon thedependsonlyF(Q)R,radiusofsphereaFor

shape.particleby thedeterminedis)(factorformparticleThe

010

30

2

.

2

vr

r

r rr

=

=

=

VQRjQR

V

QR

QRQRQRVQF

erdQF

sphere

V

rQi

2 4 6 8 10

0.2

0.4

0.6

0.8

1

3j1(x)/x

xQand(a)axismajorthe

betweenangletheiswhere

)cossin(:byRreplace

particlesellipticalFor

2/12222

r

baR +

Examples of Spherically-Averaged Form Factors


123/191

H

R

R

r

t = R - r

Form Factor for a Vesicle

)(

)()(3),,(

3321

21

2

0rRQ

QrjrQRjRVrRQF

=

Form Factor for Cylinder with Q at angle to cylinder axis

210

)sin(cos

)sin()cos]2/sin([4),,,(

QRQH

QRJQHVRHQF =

Determining Particle Size From Dilute Suspensions


124/191

Particle size is usually deduced from dilute suspensions in which inter-particle

correlations are absent In practice, instrumental resolution (finite beam coherence) will smear out

minima in the form factor

This effect can be accounted for if the spheres are mono-disperse

For poly-disperse particles, maximum entropy techniques have been used

successfully to obtain the distribution of particles sizes

Correlations Can Be Measured in Concentrated Systems

A i f i i h l 1980 b H l d Ch l


125/191

A series of experiments in the late 1980s by Hayter et al and Chen et al

produced accurate measurements of S(Q) for colloidal and micellar systems

To a large extent these data could be fit by S(Q) calculated from the mean

spherical model using a Yukawa potential to yield surface charge and screening

length

Size Distributions Have Been Measured

for Helium Bubbles in Steel


126/191

The growth of He bubbles under neutron irradiation is a key factor limitingthe lifetime of steel for fusion reactor walls

Simulate by bombarding steel with alpha particles

TEM is difficult to use because bubble are small

SANS shows that larger bubbles grow as the steel is

annealed, as a result of coalescence of small bubbles

and incorporation of individual He atoms

SANS gives bubble volume (arbitrary units on the plots) as a function of bubble sizeat different temperatures. Red shading is 80% confidence interval.


127/191

Guinier Approximations: Analysis Road MapGeneralized Guinier approximation


128/191

2

Guinier approximations provide aroadmap for analysis.

Information on particlecomposition, shape and size.

Generalization allows for

analysis of complex mixtures,allowing identification of domainswhere each approximationapplies.

P(Q) = 1; = 0 Q; = 1,2 M0 exp R2

Q2

3

d ln P(Q)d ln(Q)

=Q

P(Q)

d P(Q)dQ

= 23

R2 Q2

Ge e ed Gu e pp o o

Derivative-log analysis

* Viewgraph courtesy of Rex Hjelm

Contrast & Contrast Matching


129/191

Both tubes contain borosilicate beads +pyrex fibers + solvent. (A) solventrefractive index matched to pyrex;. (B)solvent index different from both beadsand fibers scattering from fibersdominates

O

O

Water

D2O

H2O

* Chart courtesy of Rex Hjelm

Isotopic Contrast for Neutrons


130/191

HydrogenIsotope

Scattering Lengthb (fm)

1H -3.7409 (11)

2D 6.674 (6)

3T 4.792 (27)

NickelIsotope Scattering Lengthsb (fm)58Ni 15.0 (5)

60Ni 2.8 (1)

61Ni 7.60 (6)

62Ni -8.7 (2)

64Ni -0.38 (7)


131/191

SANS Has Been Used to Study Bio-machines


132/191

Capel and Moore (1988) used the fact that

prokaryotes can grow when H is replaced

by D to produce reconstituted ribosomes

with various pairs of proteins (but not

rRNA) deuterated

They made 105 measurements of inter-

protein distances involving 93 30S protein

pairs over a 12 year period. They also

measured radii of gyration

Measurement of inter-protein distances

is done by Fourier transforming the form

factor to obtain G(R)

They used these data to solve theribosomal structure, resolving ambiguities

by comparison with electron microscopy

Porod Scattering

42


133/191

function.ncorrelatiofor theform(Debye)h thisfactor witformtheevaluatetoand

rsmallatV)]A/(2a[with..brar-1G(r)expandtoisitobtainy toAnother wa

smooth.issurface

particletheprovidedshapeparticleanyforQasholdsandlawsPorod'isThis

surface.ssphere'theofareatheisAwhere2

)(2

9Thus

)resolutionbyoutsmearedbewillnsoscillatio(theaverageon2/9

Qascos9

cossin

9)()(

cos.sin9

radius)spheretheisRwhere1/RQ(i.e.particle

sphericalaforQofvalueslargeat)(F(Q)ofbehaviortheexamineusLet

2

44

22

2

22

2

24

2

3242

42

=++=

=

=

=

=

>>

Q

A

QR

VF(Q)

V

QRV

QRQR

QRVQR

QR

QRQRQRV(QR)F(Q)

QR

Scattering From Fractal Systems


134/191

Fractals are systems that are self-similar under a change of scale I.e. R -> CR

For a mass fractal the number of particles within a sphere of radius R is

proportional to RD where D is the fractal dimension

2.dimensionofsurfacessmoothforform

Porodthetoreduceswhich)(thatprovecanonefractal,surfaceaFor

sin..1

2

)4/.(sin..2

)(.)(and

)4/()(

dRRandRdistancebetweenparticlesofnumber)(.4

Thus

6

2

3.

3

12

sD

D

D

D

DRQi

D

D

Q

constQS

Q

constxxdx

Q

c

RcQRRdRQ

RGeRdQS

RcRG

dRcRRGdRpR

==

==

=

=+=

rr

rr

Typical Intensity Plot for SANS From Disordered

Systems


135/191

ln(I)

ln(Q)

Guinier region (slope = -rg2/3 gives particle size)

Mass fractal dimension (slope = -D)

Porod region - gives surface area and

surface fractal dimension{slope = -(6-Ds)}

Zero Q intercept - gives particle volume if

concentration is known

Sedimentary Rocks Are One of the Most

Extensive Fractal Systems*


136/191

SANS & USANS data from sedimentary rockshowing that the pore-rock interface is a surfacefractal (Ds=2.82) over 3 orders of magnitude inlength scale

Variation of the average number of SEMfeatures per unit length with feature size.

Note the breakdown of fractality (Ds=2.8to 2.9) for lengths larger than 4 microns

*A. P. Radlinski (Austr. Geo. Survey)

References


137/191

Viewgraphs describing the NIST 30-m SANS instrument http://www.ncnr.nist.gov/programs/sans/tutorials/30mSANS_desc.pdf


138/191

by

Roger Pynn

Los Alamos

National Laboratory

LECTURE 6: Inelastic Scattering

We Have Seen How Neutron Scattering

Can Determine a Variety of StructuresX R


139/191

2.1(2)nm

1.9(2)nm

10.7(2)nm Fe

MgO

= 3.9(3)

= 5.2(3)

= 4.4(2)

= 3.1(3)

0.20(1)nm

0.44(2)nm

0.07(1)nm

0.03(1)nm

X-Ray

= 4.4(8)

= 8.6(4)

= 6.9(4)

= 6.2(5)

Neutron

-FeOOH

interface

but what happens when the atoms are moving?

Can we determine the directions and

time-dependence of atomic motions?

Can well tell whether motions are periodic?

Etc.

These are the types of questions answered

by inelastic neutron scattering

crystals

surfaces & interfacesdisordered/fractals biomachines

The Neutron Changes Both Energy & Momentum

When Inelastically Scattered by Moving Nuclei


140/191

The Elastic & Inelastic Scattering Cross

Sections Have an Intuitive Similarity


141/191

The intensity of elastic, coherent neutron scattering is proportional to thespatial Fourier Transform of the Pair Correlation Function, G(r) I.e. the

probability of finding a particle at position r if there is simultaneously a

particle at r=0

The intensity of inelastic coherent neutron scattering is proportional to

the space and time Fourier Transforms ofthe time-dependentpaircorrelation function function, G(r,t) = probability of finding a particle atposition r at time twhen there is a particle at r=0 and t=0.

For inelastic incoherentscattering, the intensity is proportional to the

space and time Fourier Transforms ofthe self-correlation function, Gs(r,t)I.e. the probability of finding a particle at position r at time t when the

same particle was at r=0 at t=0

The Inelastic Scattering Cross Section


142/191

Lovesey.andMarshalorSquiresbybooksin theexample,forfound,

becanDetailsus.ally tediomathematicisoperators)mechanicalquantumcommuting-nonas

treatedbetohavefunctions-ands'the(in whichfunctionsncorrelatiotheofevaluationThe

))(())0((1

),(and),()0,(1

),(

:casescatteringelasticfor thethosesimilar toyintuitivelarethatfunctionsncorrelatiotheand

),(2

1),(and),(

2

1),(where

),('.

and),('.

thatRecall

).().(

22

22

rdtRRrRrN

trGrdtRrrN

trG

dtrdetrGQSdtrdetrGQS

QNSkkb

dEddQNS

kkb

dEdd

j

jjsNN

trQi

si

trQi

iinc

inc

coh

coh

rrrrrrrrrrrr

rr

h

rrr

h

r

rr

rrrr

+=+=

==

=

=

Examples of S(Q,) and Ss(Q,)

Expressions for S(Q,) and Ss(Q,) can be worked out for a


143/191

Expressions for S(Q,) and Ss(Q,) can be worked out for a

number of cases e.g: Excitation or absorption of one quantum of lattice vibrational

energy (phonon)

Various models for atomic motions in liquids and glasses

Various models of atomic & molecular translational & rotationaldiffusion

Rotational tunneling of molecules

Single particle motions at high momentum transfers

Transitions between crystal field levels Magnons and other magnetic excitations such as spinons

Inelastic neutron scattering reveals details of the shapes of

interaction potentials in materials

A Phonon is a Quantized Lattice Vibration


144/191

Consider linear chain of particles of mass M coupled bysprings. Force on nth particle is

Equation of motion is

Solution is:

...)()( 2221110 +++++= ++ nnnnnn uuuuuF

First neighbor force constant displacements

)2

1(sin

4 with)( 22)( qa

MeAtu

q

tqnai

qn

== nn uMF &&=

a

-1 -0.5 0.5 1

0.2

0.4

0.6

0.8

1

1.2

1.4

qa/2

u

Phonon Dispersion Relation:

Measurable by inelastic neutron scattering

L

N

LLq

2

2,......

4,

2,0 =

Inelastic Magnetic Scattering of Neutrons

I th i l t t i i i f t


145/191

In the simplest case, atomic spins in a ferromagnet precess

about the direction of mean magnetization

==

+== +

l

lqi

qqq

qq

q

qll

ll

elJJJJS

bbHSSllJH

rrrh

hrrrr

.

0

0'

',

)(re whe)(2

with

.)'(

exchange couplingground state energy

spin waves (magnons)

tferromagneaforrelationdispersiontheis2

Dqq =hFluctuating spin is

perpendicular to mean spin

direction => spin-flipneutron scattering

Spin wave animation courtesy of A. Zheludev (ORNL)

Measured Inelastic Neutron Scattering Signals in Crystalline

Solids Show Both Collective & Local Fluctuations*


146/191

Spin waves collectiveexcitations

Local spin resonances (e.g. ZnCr2O4)

Crystal Field splittings(HoPd2Sn) local excitations

* Courtesy of Dan Neumann, NIST


147/191

Measured Inelastic Neutron Scattering in Molecular

Systems Span Large Ranges of Energy


148/191

Vibrational spectroscopy(e.g. C60)

Molecular reorientation(e.g. pyrazine)

Rotational tunneling(e.g. CH3I)

Polymers Proteins

Atomic Motions for Longitudinal & Transverse Phonons

)0,0,1.0(2

aQ

=

r


149/191

Transverse phonon Longitudinal phonon

Q Q

).(

0

tRQi

sllleeRR +=

rrrrraeT )0,1.0,0(=

raeL )0,0,1.0(=

r

a

a

Transverse Optic and Acoustic Phonons


150/191

).(0 tRQi

slklkleeRR +=

rrrrr

aeae

blue

red

)0,14.0,0()0,1.0,0(

==r

r

aeae

blue

red

)0,14.0,0()0,1.0,0(

==r

rAcoustic Optic

Q

Phonons the Classical Use for Inelastic Neutron Scattering

Coherent scattering measures scattering from single phonons


151/191

Note the following features:

Energy & momentum delta functions => see single phonons (labeleds)

Different thermal factors for phonon creation (ns+1) & annihilation (ns) Can see phonons in different Brillouin zones (different recip. lattice vectors, G)

Cross section depends on relative orientation of Q& atomic motions (es)

Cross section depends on phonon frequency (s) and atomic mass (M) In general, scattering by multiple

excitations is either insignificant

or a small correction (the presence of

other phonons appears in the Debye-

Waller factor, W)

)()()2

11(

).('2

2

0

2

1

2

GqQneQ

eMVk

k

dEd

dss

s G s

sW

coh

coh

rrrm

rr

+=

The Workhorse of Inelastic Scattering Instrumentation at

Reactors Is the Three-axis Spectrometer


152/191

kIkF

Q

scattering triangle

The Accessible Energy and Wavevector

Transfers Are Limited by Conservation Laws

Neutron cannot lose more than its initial kinetic energy &


153/191

Neutron cannot lose more than its initial kinetic energy &

momentum must be conserved

Triple Axis Spectrometers Have Mapped Phonons

Dispersion Relations in Many Materials

Point by point measurement in (Q,E)


154/191

y p ( , )

space

Usually keep either kI or kF fixed

Choose Brillouin zone (I.e. G) to maximize

scattering cross section for phonons

Scan usually either at constant-Q

(Brockhouse invention) or constant-E

Phonon dispersion of 36Ar

What Use Have Phonon Measurements Been?

Quantifying interatomic potentials in metals rare gas solids ionic


155/191

Quantifying interatomic potentials in metals, rare gas solids, ionic

crystals, covalently bonded materials etc

Quantifying anharmonicity (I.e. phonon-phonon interactions)

Measuring soft modes at 2nd order structural phase transitions

Electron-phonon interactions including Kohn anomalies

Roton dispersion in liquid He

Relating phonons to other properties such as superconductivity,

anomalous lattice expansion etc

Examples of Phonon Measurements

Soft mode


156/191

Roton dispersion in 4He

Phonons in 36Ar validation

of LJ potential

Phonons in 110Cd

Kohn anomalies

in 110Cd


157/191

Much of the Scientific Impact of Neutron Scattering Has Involved

the Measurement of Inelastic ScatteringNeutrons in Condens ed Matter Res earch

10000


158/191

0.001 0.01 0.1 1 10 100

Q (-1)

EnergyTransfer(meV)

Spallation - Choppe r

ILL - withou t s pin-ech o

ILL - with s pin-ech o

CrystalFields

SpinWaves

NeutronInduced

ExcitationsHydrogenModes

MolecularVibrations

LatticeVibrations

andAnharmonicity

AggregateMotion

Polymersand

BiologicalSystems

CriticalScattering

DiffusiveModes

Slower

Motions

Resolved

MolecularReorientation

TunnelingSpectroscopy

SurfaceEffects?

[L arger Objects Resolved]

Proteins in SolutionViruses

Micelles PolymersMetallurgical

Systems

Colloids

MembranesProteins

Crystal andMagneticStructures

AmorphousSystems

Accurate Liquid StructuresPrecision Crystallography

Anharmonicity

1

100

0.01

10-4

10-6

MomentumDistributions

Elastic Scattering

Electron-phonon

Interactions

ItinerantMagnets

Molecular

Motions

CoherentModes inGlasses

and Liquids

SPSS - Chopper Spectrometer

Energy & Wavevector Transfers accessible to Neutron Scattering


159/191

What Do We Need for a Basic Neutron Scattering Experiment?


160/191

A source of neutrons A method to prescribe the wavevector of the neutrons incident on the sample

(An interesting sample)

A method to determine the wavevector of the scattered neutrons A neutron detector

ki

kfQWe usually measure scatteringas a function of energy (E) andwavevector (Q) transfer

Incident neutrons of wavevectorki Scattered neutrons ofwavevector, kf

Detector

Sample

)(2

222

fi kkmE ==

hh

Instrumental Resolution


161/191

Uncertainties in the neutronwavelength & direction of travelimply that Q and E can only bedefined with a certain precision

When the box-like resolutionvolumes in the figure are convolved,

the overall resolution is Gaussian(central limit theorem) and has anelliptical shape in (Q,E) space

The total signal in a scatteringexperiment is proportional to thephase space volume within theelliptical resolution volume the better the resolution, the smaller theresolution volume and the lower the count rate


162/191

The Underlying Physics of Neutron Spin Echo (NSE)Technology is Larmor Precession of the Neutrons Spin


163/191

The time evolution of the expectation value of the spin of aspin-1/2 particle in a magnetic field can be

determined classically as:

The total precession angle of the spin, , depends on the timethe neutron spends in the field: tL=

11.2*2913

=

==

sGauss

BBsdt

sdL

rr

r

B

s

~29

Turns/m for

4 neutrons

2918310

N (msec-1)L (103 rad.s-1)B(Gauss)

Larmor Precession allows the Neutron Spin to be Manipulatedusing or /2 Spin-Turn Coils: Both are Needed for NSE


164/191

The total precession angle of the spin, , depends on the timethe neutron spends in the B field

v/BdtL ==

Neutron velocity, v

d

B

][].[].[.135.65

1

turnsofNumber AngstromscmdGaussB =

How does a Neutron Spin Behave when theMagnetic Field Changes Direction?


165/191

Distinguish two cases: adiabatic and sudden

Adiabatic tan() > 1 large spin precesses around new

field direction this limit is used to design spin-turn devices

When H rotates withfrequency , H

0

->H1->H2, and the spin

trajectory is describedby a cone rolling on the

plane in which H moves

Neutron Spin Echo (NSE) uses Larmor Precession toCode Neutron Velocities


166/191

A neutron spin precesses at the Larmor frequency in amagnetic field, B.

The total precession angle of the spin, , depends on the time

the neutron spends in the field

BL =

v/BdtL ==

d

Neutron velocity, vB

][].[].[.135.65

1turnsofNumber AngstromscmdGaussB =

rr

,H

The precession angle is a measure of the neutrons speed v

The Principles of NSE are Very Simple


167/191

If a spin rotates anticlockwise & then clockwise by the sameamount it comes back to the same orientation Need to reverse the direction of the applied field

Independent of neutron speed provided the speed is constant

The same effect can be obtained by reversing the precession

angle at the mid-point and continuing the precession in thesame sense Use a rotation

If the neutrons velocity, v, is changed by the sample, its spinwill not come back to the same orientation The difference will be a measure of the change in the neutrons speed or

energy

In NSE*, Neutron Spins Precess Before and After Scattering & aPolarization Echo is Obtained if Scattering is Elastic

/2

y

x


168/191

Initially,neutronsare polarizedalong z

Rotate spins intox-y precession plane

Allow spins to

precess around z:slower neutronsprecess further overa fixed path-length

Rotate spinsthrough aboutx axis

ElasticScatteringEvent

Allow spins to precessaround z: all spins are inthe same direction at the

echo point if E = 0

Rotate spins

to z andmeasure

polarization

SF

SF

/2

)cos(, 21 =PonPolarizatiFinal

z

* F. Mezei, Z. Physik, 255 (1972) 145

For Quasi-elastic Scattering, the EchoP

intro to neutron scattering

Documents