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:

1. Introduction

integrated over the energy window and up to the acceptance angleSpectrum of BN (Ahn et al., EELS Atlas 1982)

2. The scattering process

Assumptions:-weak scattering non-relativisticobject initially in the ground state

Fermis golden rule (1. order Born approximation)

Scattering geometry

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

After some calculations (Bethe, 1930) kinematics object functionScattering vectorFourier transformed density (operator)Bohrs radiusdynamic form factor (vanHove, 1954)

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

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.)

geometry:; scattering angleFor small scattering angles small scattering vectors dipole approximation

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

generalized oscillator strength for hydrogen (Inokuti, Rev. Mod. Phys. 43, (1971) 297)

double differential cross-section for carbon (Reimer & Rennekamp, Ultramicr. 28, (1989) 256)

C. Hbert, Wednesday

Spectrum of BN (Ahn et al., EELS Atlas 1982)

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

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);

dielectric function of Ag (Ehrenreich & Philipp, Phys. Rev. 128 (1962) 1622)

dielectric functions of Cu (Ehrenreich & Philipp, Phys. Rev. 128 (1962) 1622)

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

(Kurata at al., Proc. EUREM-11 (1996) I-206)

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