introduction the scattering process inner shell losses the low-loss regime relativistic effects
DESCRIPTION
Introduction to Energy Loss Spectrometry Helmut Kohl Physikalisches Institut Interdisziplinäres Centrum für Elektronenmikroskopie und Mikroanalyse (ICEM) Westfälische Wilhelms-Universität Münster , Germany. Introduction The scattering process Inner shell losses The low-loss regime - PowerPoint PPT PresentationTRANSCRIPT
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Introduction to Energy Loss SpectrometryHelmut Kohl
Physikalisches Institut Interdisziplinres Centrum fr Elektronenmikroskopie und Mikroanalyse (ICEM) Westflische Wilhelms-Universitt Mnster, GermanyIntroductionThe scattering processInner shell lossesThe low-loss regimeRelativistic effectsSummary and conclusion
Contents:
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1. Introduction
integrated over the energy window and up to the acceptance angleSpectrum of BN (Ahn et al., EELS Atlas 1982)
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2. The scattering process
Assumptions:-weak scattering non-relativisticobject initially in the ground state
Fermis golden rule (1. order Born approximation)
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Scattering geometry
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plane wave state of the incident and outgoing electron
initial and final state of the objectinteraction between the incident electron and
the electrons in the object
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After some calculations (Bethe, 1930) kinematics object functionScattering vectorFourier transformed density (operator)Bohrs radiusdynamic form factor (vanHove, 1954)
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More general case: coherent superposition of two incident waves Scattering of two coherent waves
How can one calculate the dynamic form factor?Mixed dynamic form factor (MDFF; Rose,1974)P. Schattschneider, Thursday
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3. Inner-shell lossesApproximations: - free atoms - describe initial and final state as a Slater-determinant of single-electron atomic wave functions (not valid for open shells 3d, 4d: transition metals; 4f, 5f: lanthanides, actinides)single-electron matrix element.SIGMAK (Egerton, 1979), SIGMAL (Egerton, 1981)Hartree-Slater model (Rez et al.)
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geometry:; scattering angleFor small scattering angles small scattering vectors dipole approximation
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Example: - Ionisation of hydrogen
- experiment for carbonphoto absorptionoscillator strengthgeneralized oscillator strength (GOS):In solids the final states are not completely free.near-edge structure (ELNES) analogous to XANESextended fine structure (EXELFS) analogous to EXAFS
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generalized oscillator strength for hydrogen (Inokuti, Rev. Mod. Phys. 43, (1971) 297)
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double differential cross-section for carbon (Reimer & Rennekamp, Ultramicr. 28, (1989) 256)
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C. Hbert, Wednesday
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Spectrum of BN (Ahn et al., EELS Atlas 1982)
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4. Low loss spectraFor relatively low frequencies ( low energy losses) the free electron gascan partly follow the field of the incident electron shielding
Electron causes -fieldActing field:Absorption: Imaginary part
Relation to dynamic structure factor ?div
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For In addition: surface plasmon losses O. Stephan, Thursdayis response functionDissipation-fluctuation theorem:peaks for : volume plasmons
Why dont we use that for higher energy losses ? Formally: describes fluctuations in the object (density-density correlation);
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dielectric function of Ag (Ehrenreich & Philipp, Phys. Rev. 128 (1962) 1622)
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dielectric functions of Cu (Ehrenreich & Philipp, Phys. Rev. 128 (1962) 1622)
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5. Relativistic effectsNon-relativistic: Incident electron causes Coulomb field field is instantaneously everywhere in space Relativistic: Incident (moving) electron causes an additional magnetic field fields move in space with the speed of light c ( retardation)
Matrix elements are sums of an electric and a magnetic term In Coulomb gauge: electric term corresponds to the non-relativistic term, but with relativistic kinematics Double-differential cross-section in dipole-approximation
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(Kurata at al., Proc. EUREM-11 (1996) I-206)
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6) Summary and conclusionsquantitative interpretation of EEL-spectra requires knowledge of cross-sections
-cross-section related to dynamic form factor
for inner-shell ionization these can be calculated using a oneelecton model
large errors may occur when 3d, 4d, 4f, 5f shells are involved
for small scattering angles (dipole approximation) one obtains a Lorentzian angular shape
in dipole approximation the cross-section is closely related to the photoabsorption cross-section
near-edge and extended fine structures can be interpreted as in the X-ray case
the low-loss spectrum permits to determine the dielectric function
WARNING: relativistic effects are not included in the commonly used equations