Introduction to Energy Loss Spectrometry Helmut Kohl Physikalisches

Introduction to Energy Loss Spectrometry Helmut Kohl Physikalisches Institut Interdisziplinäres Centrum für Elektronenmikroskopie und Mikroanalyse (ICEM) Westfälische Wilhelms-Universität Münster, Germany Contents: 1. 2. 3. 4. 5. 6. Introduction The scattering process Inner shell losses The low-loss regime Relativistic effects Summary and conclusion

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

2. The scattering process Assumptions: - weak scattering - non-relativistic - object initially in the ground state Fermis golden rule (1. order Born approximation)

Scattering geometry

plane wave state of the incident initial and final and outgoing electron state of the object interaction between the incident electron the electrons in the object and

After some calculations (Bethe, 1930) kinematics object function Scattering vector Å Bohrs radius Fourier transformed density (operator) dynamic form factor (van. Hove, 1954)

More general case: coherent superposition of two incident waves Scattering of two coherent waves Mixed dynamic form factor (MDFF; Rose, 1974) P. Schattschneider, Thursday How can one calculate the dynamic form factor?

3. Inner-shell losses Approximations: - free atoms - describe initial and final state as a Slater-determinant of single-electron atomic wave functions (not valid for open shells 3 d, 4 d: transition metals; 4 f, 5 f: lanthanides, actinides) single-electron matrix element. SIGMAK (Egerton, 1979), SIGMAL (Egerton, 1981) Hartree-Slater model (Rez et al. )

For small scattering angles dipole approximation small scattering vectors geometry: ; scattering angle

oscillator strength photo absorption generalized oscillator strength (GOS): Example: - Ionisation of hydrogen - experiment for carbon In solids the final states are not completely free. near-edge structure (ELNES) analogous to XANES extended 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. Hébert, Wednesday

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

4. Low loss spectra For relatively low frequencies ( low energy losses) the free electron gas can partly follow the field of the incident electron shielding Electron causes -field div Acting field: Absorption: Imaginary part Relation to dynamic structure factor ?

Formally: describes fluctuations in the object (density-density correlation); is response function Dissipation-fluctuation theorem: peaks for : volume plasmons Why don‘t we use that for higher energy losses ? For In addition: surface plasmon losses O. Stephan, Thursday

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

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

5. Relativistic effects Non-relativistic: Relativistic: Incident electron causes Coulomb field is instantaneously everywhere in space Incident (moving) electron causes an additional magnetic 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 conclusions - quantitative 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 one–electon model - large errors may occur when 3 d, 4 f, 5 f 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