Intro to Neutron Scattering
Transcript of Intro to Neutron Scattering
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by
Roger Pynn
Los Alamos
National Laboratory
LECTURE 1: Introduction & Neutron Scattering Theory
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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
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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)
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The 1994 Nobel Prize in Physics Shull & Brockhouse
Neutrons show where the atoms are.
and what the atoms do.
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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)
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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)
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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
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Interaction Mechanisms
Neutrons interact with atomic nuclei via very short range (~fm) forces.
Neutrons also interact with unpaired electrons via a magnetic dipole
interaction.
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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
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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
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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
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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
=
==
=>>
=
=
=
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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
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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/
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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
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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)
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Neutrons can also gain or lose energy in the scattering process: this is
called inelastic scattering
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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
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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
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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
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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.
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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
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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)
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by
Roger Pynn
Los Alamos
National Laboratory
LECTURE 2: Neutron Scattering Instrumentation & Facilities
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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
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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
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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
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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
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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.
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The ESRF* & ILL* With Grenoble & the Beldonne Mountains
*ESRF = European Synchrotron Radiation Facility; ILL = Institut Laue-Langevin
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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
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Neutron Sources Provide Neutrons for ManySpectrometers: Schematic Plan of the ILL Facility
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A ~ 30 X 20 m2 Hall at the ILL Houses About 30 Spectrometers.Neutrons Are Provided Through Guide Tubes
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Los Alamos Neutron Science Center
800-MeV LinearAccelerator
LANSCEVisitors Center
Manuel Lujan Jr.Neutron Scattering
Center
WeaponsNeutron Research
Proton StorageRing
ProtonRadiography
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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
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Components of a Spallation SourceA half-mile long proton linac
.a proton accumulation ring.
and a neutron production target
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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
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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
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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
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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
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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
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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
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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
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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)
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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
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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
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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.
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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)
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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
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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
==
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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)
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by
Roger Pynn
Los Alamos
National Laboratory
LECTURE 3: Diffraction
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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
Pair distribution functions (PDF)
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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 /
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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
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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
*
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*
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321332211
,
).(
MjiQalakahGQ
aa
aaV
a
aaaamamami
MjiQ
u
iuiRe
N
QS
hkl
ijji
iii
i
ii
ji
RRQi ji
=++==
=
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++=
=
+==
rrr
rrr
vr
rr
rrr
rrrrrrv
rrr
r
rr
rrr
vrr
a1
a2
a3
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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
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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
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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
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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
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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
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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.
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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*
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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.
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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
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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)
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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}
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How good is this function?
Protein Rietveld refinement - Very low angle fit1.0-4.0 peaks - strong asymmetry
perfect fit to shape
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Examples of Science using Powder Diffraction
Refinement of structures of new materials Materials texture
Strain measurements
Pair distribution functions (PDF)
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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
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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
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Texture Determination of Titanium Wire Plate
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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
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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*
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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 ?
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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
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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
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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
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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
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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
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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*
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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
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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
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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
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by
Roger Pynn
Los AlamosNational Laboratory
LECTURE 3: Surface Reflection
Surface Reflection Is Very Different From
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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
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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?
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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?
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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
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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
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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
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( ) 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
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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
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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*
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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*
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(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*
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1.3% PEG lipidin lipids
4.5% PEG lipidin lipids
9.0% 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
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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)
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Structure, Chemistry & Magnetism of Fe(001) on MgO(001)*
X R
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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)
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Qx-QzTransformation
Time-of-Flight, Energy-Dispersive Neutron Reflectometry
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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
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Observation of Hexagonal Packing of Thread-like MicellesUnder Shear: Scattering From Lateral Inhomogeneities
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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
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by
Roger Pynn
Los AlamosNational Laboratory
LECTURE 5: Small Angle Scattering
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Small Angle Neutron Scattering (SANS) Is Used to
Measure Large Objects (~1 nm to ~1 m)
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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
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Where Does SANS Fit As a Structural Probe?
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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?
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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
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The Scattering Cross Section for SANS
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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
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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
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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
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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
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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
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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
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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.
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Guinier Approximations: Analysis Road MapGeneralized Guinier approximation
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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
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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
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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)
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SANS Has Been Used to Study Bio-machines
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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
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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
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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
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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*
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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
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Viewgraphs describing the NIST 30-m SANS instrument http://www.ncnr.nist.gov/programs/sans/tutorials/30mSANS_desc.pdf
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by
Roger Pynn
Los Alamos
National Laboratory
LECTURE 6: Inelastic Scattering
We Have Seen How Neutron Scattering
Can Determine a Variety of StructuresX R
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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
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The Elastic & Inelastic Scattering Cross
Sections Have an Intuitive Similarity
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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
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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
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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
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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
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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*
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Spin waves collectiveexcitations
Local spin resonances (e.g. ZnCr2O4)
Crystal Field splittings(HoPd2Sn) local excitations
* Courtesy of Dan Neumann, NIST
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Measured Inelastic Neutron Scattering in Molecular
Systems Span Large Ranges of Energy
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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
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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
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).(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
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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
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kIkF
Q
scattering triangle
The Accessible Energy and Wavevector
Transfers Are Limited by Conservation Laws
Neutron cannot lose more than its initial kinetic energy &
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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)
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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
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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
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Roton dispersion in 4He
Phonons in 36Ar validation
of LJ potential
Phonons in 110Cd
Kohn anomalies
in 110Cd
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Much of the Scientific Impact of Neutron Scattering Has Involved
the Measurement of Inelastic ScatteringNeutrons in Condens ed Matter Res earch
10000
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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
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What Do We Need for a Basic Neutron Scattering Experiment?
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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
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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
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The Underlying Physics of Neutron Spin Echo (NSE)Technology is Larmor Precession of the Neutrons Spin
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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
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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?
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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
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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
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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
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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