Scattering

Alan Goldman

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Section Outline

• March 31 Introduction to Scattering (Goldman)

• April 2 e-beam measurements (Kramer)

• April 4 Advanced e-beam measurements (Kramer)

• April 7 Single crystal x-ray and neutron diffraction (Goldman)

• April 9 Magnetic neutron and x-ray diffraction (Kreyssig)

• April 11 Powder diffraction methods (Jesche)

• April 14 Inelastic neutron scattering (Tucker)

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• atoms pack in periodic, 3D arrays• typical of:

Crystalline materials...

-metals-many ceramics-some polymers

What is the atomic scale structure?

• atoms have no “regular” packing• occurs for:

Noncrystalline materials...

-complex structures-rapid cooling

"Amorphous" = Noncrystalline

Distance between atoms ~ Å (10-9 m)

Si Oxygen

crystalline SiO2

noncrystalline SiO2Adapted from Fig. 3.18(b),Callister 6e.

Adapted from Fig. 3.18(a),Callister 6e.

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Interference of two waves Double slit diffraction

2 slits 2 slits and 5 slits

You can also do this with light (as well as neutrons and electrons)!

Diffraction

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Visible light

X-rays

Resolution ~ wavelength

So, 10-9 m resolution Requires λ ~ 10-9 m

How do we use x-rays to study crystal structures?

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Length scales

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What are the atomic scale dynamics?

Transverse acoustic mode

Lynn, et al., Phys. Rev. B 8, 3493 (1973).

Linear spin waves

Perring et al., Phys. Rev. Lett. 77, 711 (1996).

Ferromagnetic

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Dynamical (time) scalesFa

ster

pro

cess

es

Shorter length scales

Elasticscattering

Cold neutrons

Thermal neutrons

Epithermalneutrons

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How are these changed by….

• Temperature• Magnetic Field• Electric Field• Pressure

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Different probes

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Different probes

NEUTRONS X-RAYS ELECTRONS

Wavelength range 0.4 - 10 Å 0.1 - 5 Å 0 .04 - 0.2 Å

Energy range 0.001 - 0.5 eV 3000 - 100000 eV 6000 - 120000 eV

Cross-section 10-25 barns 10-25 Z2 barns ~10-22 barns

Penetration depth ~ cm ~ m - mm ~ nm

Typical flux 1011 s-1 m-2 1024 s-1 m-2 1026 s-1 m-2

Beam size mm-cm m-mm nm-m

Typical sample Any bulk sample Small crystals, powders, surfaces

Surfaces, thin films, grains, gases

TechniquesDiffraction

Inelastic scatteringReflectivity

DiffractionPhoton absorption

PhotoemissionInelastic scattering

MicroscopyDiffraction

Emission spectroscopyEELS

Phenomena

Magnetic/crystal structurescollective excitations

(phonons, spin waves)electronic excitations (crystal-

field, spin-orbit)

Crystal structures, electronic transitions

(photoemission, absorption)

microstructurecrystal structures

electronic transitions

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Atomic scattering cross-sections

Abundance (%)

Cross-section (bn)

Absorption (bn)

Gd --- 180 49700152Gd 0.2 13 735154Gd 2.1 13 85155Gd 14.8 66 61100156Gd 20.6 5 1.5157Gd 15.7 1044 259000158Gd 24.8 10 2.2160Gd 21.8 10.52 0.77

Neutrons Random with Z Depends on isotope Depends on nuclear spin Absorption can be problem

X-rays: Atomic form factor means that scattering decreases with scattering angle.Neutrons: constant scattering as a function of scattering angle

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Why neutrons??

Penetration ~ centimeters, bulk probe Sensitivity to low-Z Isotopic contrast Wavelength range ~ interatomic spacing (1-2 Å) → diffraction

interacts with nuclei Interacts with magnetic moment of unpaired e-

Can measure crystal and magnetic structures Thermal neutron energies are comparable to elementary

excitations → natural probe for atomic-scale dynamics

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Neutron magnetism

Spin-1/2 particle Magnetic moment n

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Nuclear (lattice) excitations

Commonly studied excitations Phonons Librations and vibrations in molecules Diffusion Collective modes in glasses and liquids

Neutron scattering measures simultaneously the wavevector and energy of collective excitations dispersion relation, (q)In addition, local excitations can of course be observed

Excitations can tell us about Interatomic potentials & bonding Phase transitions & critical phenomena (soft modes) Fluid dynamics Momentum distributions & superfluids (eg. He) Interactions (eg. electron-phonon coupling)

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Spin excitations

Spin excitations Spin waves in ordered magnets Paramagnetic & quantum spin fluctuations Crystal-field & spin-orbit excitations

Magnetic inelastic scattering can tell us about Exchange interactions Single-ion and exchange anisotropy (determine Hamiltonian) Phase transitions & critical phenomena Quantum critical scaling of magnetic fluctuations Other electronic energy scales (eg. CF & SO) Interactions (eg. spin-phonon coupling)

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Why x-rays??

Easily accessible in the laboratory Penetration ~ microns to millimeters (tuned with energy)

Low-energies can be used to probe surface and near surface regions High energies can probe the bulk of a sample

High flux at National Facilities Allows you to throw a lot of photons away → high energy and

momentum resolution Is highly polarized and highly collimated Is continuously tunable across a wide range of energies (1 keV →

200 keV) Wavelength range ~ interatomic spacing (1-2 Å) → diffraction

interacts with electrons Magnetic interactions are higher order

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Single-crystal diffraction

Wide angle diffraction: Get an overview of everythingSingle-crystal: more detail than powders

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Powder diffraction

For powder: Qhkl = 2kisin

ki = 2/ 2Qhkl

ki

kf

ki

kf

2

Qhkl = 2/dhkl

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X-ray resonant magnetic scattering***

Magnetic structures of neutron absorbing materials with very high resolution

Pump-probe experiments Dynamics on picosecond to millisecond timescales

Inelastic x-ray scattering Approaching neutron energy resolution

X-ray absorption Spectroscopy***

Studies of local atomic environments and chemistry

Circular magnetic x-ray dichroism***

Element specific magnetization measurements

More than “just diffraction”

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Small–angle scattering Studying mesoscopic-scale structures

X-ray radiography and tomography 3D views of structural features with submicron resolution

Surface scattering Studies of both solid and liquid surfaces

Coherent x-ray diffraction Probing coherence effects on the atomic scale

More than “just diffraction”

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Producing neutrons

SpallationParticle accelerator

FissionNuclear reactor

neutrons

Moderators →Cold-EpithermalModerators →Cold-Thermal

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Neutrons by reactor fission

High flux isotope reactor - ORNL

NIST

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Neutrons by pulsed spallation

Spallation Neutron Source (ORNL)

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Target-moderator system

SNS liquid Hg target

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Production of x-rays

X-ray tube Rotating anode

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Synchrotron radiation

Static charge ---- Electric field Charge moving at constant v ---- Magnetic field Accelerating charge --- Electromagnetic radiation

v

a

v << c v ≈ c

Synchrotron radiation is highly collimated highly linearly polarizedhighly brilliantcontinuous wavelength distribution (beyond Cu, Mo, etc..) 29

Advanced Photon Source

Synchrotron

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Insertion Devices

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Main Undulator Line

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Places to go

HFIRSNS

MURR

LANSCEEurope

AMES

NISTSSRL

APSNSLS

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