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

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

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

After some calculations (Bethe, 1930) kinematics object function Scattering vector Å Fourier transformed density (operator) Bohrs radius dynamic 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 losses Approximations: - 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 angle For small scattering angles small scattering vectors dipole approximation

Example: - Ionisation of hydrogen - experiment for carbon photo absorption oscillator strength generalized oscillator strength (GOS): 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 Acting field: Absorption: Imaginary part Relation to dynamic structure factor ? div

For In addition: surface plasmon losses O. Stephan, Thursday is response function Dissipation-fluctuation theorem: peaks for : volume plasmons Why don‘t 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 effects Non-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 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 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