Mixed Dynamic Form Factors in Real Space'

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Mixed Dynamic Form Factorsand Stopping Powers

• Adam Sorini

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Outline

• Introduction Pictures, Experiments, and Theory
  
• Absorption and Scattering
• Meat Mixed Dynamic Form Factor
• Stopping Power
• Results
• Conclusions / Future Goals

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Scattering
Scattered probe particle
observer
Incident probe particle
Condensed matter sample
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X-ray Scattering Apparatus
Scattered X-rays
observer
Incident x-rays
sample
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Electron Scattering Apparatus
Video games
Incident electron
Sample
Scattered electron
Sample information is same regardless of probe
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Absorption
Transmitted x-ray beam
Incident x-ray beam
Sample (careful, its hot)
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Example Absorption Spectrum Cu
K edge
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Example Absorption Spectrum Cu(zoomed in on K
Edge)
K Edge
EXAFS wiggles
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Example Condensed Matter System Sodium Metal
So, how do we describe a sample such as this?
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We use a Greens functions

• Feffs Greens Function
• Sample described as collection of potentials
  centered at each lattice point

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Many body effects

• Must incorporate loss for photoelectrons
• Self-Energy
• Works well for deep core absorption

Screened Coulomb potential
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Spherical Potentials
Currently a necessary approximation for real
systems
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Muffin Tin Potentials
interstitial level
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Key Approximations

• Ion Cores condensed into a lattice and create
  periodic potential
• Electrons roughly independent particles
• Muffin tins replace unit cells
• Spherical symmetry within tins
• Good approximations for high energy (x-ray energy
  region) probes

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More AbsorptionPictures
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Multiple Scattering Greens Function

• Exploits spherical symmetry about sites
• Exploits flatness of interstitial region
• Physically motivated. Can be calculated (FEFF)

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Uses for FEFFs Greens function
Use it to sum over states in your golden rule
based calculations of
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Uses for FEFFs Greens function
1. X-ray absorption
Incoming photon polarization
A core state
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Approximations seem to work!
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Uses for FEFFs Greens function
2. Electron scattering my current interest
Double differential cross section
Dynamic form factor
Momentum transfer
Energy loss
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Uses for FEFFs Greens function
3. Coherent electron scattering
Mixed Dynamic Form Factor
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MDFF Experimental Motivation

• Application which Delights in MDFF Scattering
  perpendicular q and q Schattschneider et al.,
  Nature 441, 486 (2006) mimics dichroism signal
  of x-ray absorption.
• Application which Abhors yet needs MDFF Electron
  diffraction of Bloch waves in microscopy.

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MDFF GeCl4(dipole approximation good)
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MDFF Copper
Full MDFF
P-Projected MDFF
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MDFF Nickel
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Application to Stopping PowersMicroscopic
perspective

• The double differential cross section for
  electron scattering can be written in terms of
  the dynamic form factor.
• Mean free path is simply related to the total
  cross section
• Stopping power is the energy lost per unit path
  length. (A force, actually)

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Stopping PowersMacroscopic perspective (NR)

• Point charge (-e) in macroscopically continuous
  medium
• Response function (dielectric function, e)
  relates actual field to applied field
• Field gives force--stopping power is -dW/dx
  (force)

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Stopping powerBethe Formula (UR)

• Very high energy stopping power given by Bethes
  formula
• Parameter log(I) is roughly 10Z (eV)

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Relativistic Stopping PowerAb Initio DatA
Model (ADAM)

• Fernandez-Varea et al. Optical Data Model
  Nucl. Instr. and Meth. In Phys. Res. B 229
  (2005) 187.
• Current theory reviewed by Fano (1963)
• First ever stopping theory by Bethe (1930)

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Relativistic Stopping PowerAb Initio DatA
Model (ADAM)

• Data Model based on q0 broad spectrum
  measurements
• Calculation requires q-dependant dielectric
  function

Im(-1/e(q,?))
Bethe Ridge
QKE(q)
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Relativistic Stopping PowerAb Initio DatA
Model (ADAM)

• Data Model based on q0 broad spectrum
  measurements.
• Stopping power integral over all q and ?.
• In practice need a simple extension to finite q.

Bethe Ridge
Model preserves f-sum rule Z(8)2p2(N/V)
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Relativistic Stopping PowerAb Initio DatA
Model (ADAM)

• Fernandez-Varea et al. Optical Data Model
  Nucl. Instr. and Meth. In Phys. Res. B 229
  (2005) 187.
• Plug in model q-dependence
• Perform all integrals except one analytically
• Obtain stopping power (F) by numerical
  convolution with ab initio q0 dielectric
  loss-function.

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Stopping powerBethe Formula (UR, special case)

• Very high energies model of Fernandez-varea et
  al. reduces to the Bethe formula
• Parameter log(I) is calculable material dependant
  property

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Sample Calculation
1. Start with an Energy Loss Function (courtesy
of Micah)
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SampleCalculation
2. Plug energy-loss function into stopping power
functional.
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SampleCalculation
3. End up with a stopping power
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Stopping Power Silver
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Conclusions (Stopping Powers)

• Calculated over wide energy range
• Ab initio relativistic stopping powers and mean
  free paths compare well with experiment.
• Further results in Sorini et al. Phys. Rev. B
  74, 165111 (2006).

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Future Work (MDFFs)

• Currently have many pretty pictures but not yet
  ready to compare with Ni L2,3 experiments
  Schattschneider et al..
• Finish MDFF code for all core initial states.
• Combine with Optical Code Goal 10eV.
• Combine with TDDFT code to account for many body
  effects which become more important at low
  energies.

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Thanks

• Group Members John Rehr, Kevin Jorissen, Josh
  Kas, Hadley Lawler, Micah Prange, Yoshi Takimoto,
  Fernando Vila.
• Others Aleksi Soininen, Zachary Levine, Tim
  Fister.
• Committee Members Jerry Miller, Bruce Robinson,
  Larry Sorenson, Larry Yaffe.
• Useful comments Rob J, Eric, Matt, Rob S, Andy.

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BEGIN BACKUP SLIDES
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Backup slide. Fluc-diss theorem

• Relates density fluctuation to dissipative part
  of loss function (i.e. s(q,w) to eps(q,w))
• Easy to see proportionality via sumrules

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Meet the particles.
Vortex Rings?
Rotons?

• Electrons Pointlike, Massive (m1), Charged
  (q-1), Spin ½.

Plasmons?
2. Photons Pointlike, Massless, Chargeless, Spin
1.
2. Instantaneous Coulomb Interaction Only.
Excitons?
Polaritons?
3. Nuclei Pointlike, Massive (mgtgt1000), Charged
(qZA), Classical.
3. Ion Core Small, Massive (mgtgt1000), Charged
(qZ), Classical.
Cooper Pairs?
4. Thats it!
Phonons?
Holes?
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Basic Hamiltonian.
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Periodic Effective muffin tin Potential
Periodic Potential of Ion Cores

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t-matrix properties

• Vanish if either spatial argument is outside of
  muffin tin.
• Are diagonal in angular momentum expansion about
  site.
• Made be chosen to be dimensionless.

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Mixed Dynamic Form Factor

• Thus we can write the MDFF as

Angular projected density.
Atomic matrix elements
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Scattering and Dynamic Form Factors

• Use Fermis Golden Rule

• Separate Sample and Probe.

DFF def. on next page.
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Mixed Dynamic Form Factor

• But waitwho cares about the MIXED dynamic form
  factor?
• Scattering of plane wave versus coherent
  superpositions.

Def. on next slide really
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Mixed Dynamic Form Factor
Sum over final states is a pain. We need a good
way to preform this sum We need a Greens
Function FEFFs Greens function!
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Central Atom Greens Function

• It is useful to single out one site as the
  central atom.
• Then solve for the central atom Greens function
  exactly.
• The full Greens function can then be written as
  an expansion about the central atom.

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Central and Scattering Wiggles
Scattering Part Added.
Central Atom Only.
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RSMS Punchline

• We now use all of our cute definitions and some
  useful site/angular-momentum matrix notation.
• And find the RSMS Greens function.

Coefficients. See Rehr and Albers, Phys. Rev. B
41, 8139 (1990).
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Mixed Dynamic Form Factor

• Or, with a little massaging

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What is Condensed Matter Physics?

• Its tough to say.
• We study the response of crystals (the sample) to
  various probes (photon, electron, neutron).
• The particles are well known and the forces are
  well known.
• The sample is the difficulty--it is really
  intractably difficult, so

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MDFF Copper
Atomic BG only.
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The Many-Body Hamiltonian
a number
Replace it with an average potential.
Periodic Potential of Ion Cores
This term is a problem.
Periodic Effective muffin tin Potential
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Muffin Tin Potential.
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Example One-Body Hamiltonian

• One body effective potential
• Independent electron states
• Neglects exchange and correlation (i.e., all
  but mean effects of electron-electron interaction)

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Relativistic Stopping PowerAb Initio DatA
Model (ADAM)

• Fernandez-Varea et al. Optical Data Model
  Nucl. Instr. and Meth. In Phys. Res. B 229
  (2005) 187.
• Extend q0 calculation of loss function (or
  optical data) to finite q via simple model.
• Perform all integrals except one analytically.

Bethe Ridge
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Many-Body Hamiltonian of Sample
Periodic Potential of Ion Cores
electron-electron interaction term
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Example One-Body Hamiltonian

• Electron sees effective one-body potential
• Hartree approximation (product wavefunction)
• Next step include exchange and correlation