1. Beam – Specimen Interactions Electron optical system controls: beam voltage (1-40 kV) beam current (pA – μA) beam diameter (5nm – 1μm) divergence angle Small beam diameter is the first requirement for high spatial resolution Ideal:Diameter of area sampled = beam diameter Real:Electron scattering increases diameter of area sampled = volume of interaction

2. Scattering Key concept: Cross section or probability of an event Q = N/n t n i cm 2 N = # events / unit volume n t = # target sites / unit volume n i = # incident particles / unit area Large cross section = high probability for an event

3. From knowledge of cross sections, can calculate mean free path λ = A / N 0 ρ Q A = atomic wt. N 0 = Avogadro’s number (6.02 x10 23 atoms/mol) ρ = density Q = cross section Smaller cross section = greater mean free path

4. Scattering (Bohr – Mott – Bethe) Types of scattering Elastic Inelastic Elastic scattering Direction component of electron velocity vector is changed, but not magnitude Target atom EiEi ΦeΦe E i = E 0 E i = instantaneous energy after scattering Kinetic energy ~ unchanged <1eV energy transferred to specimen E0E0

5. Types of scattering Elastic Inelastic Inelastic scattering Both direction and magnitude components of electron velocity vector change Target atom EiEi ΦiΦi E i < E 0 E i = instantaneous energy after scattering significant energy transferred to specimen Φi << Φe E0E0

6. Elastic scattering Electron deviates from incident path by angle Φc (0 to 180 0 ) Results from interactions between electrons and coulomb field of nucleus of target atoms screened by electrons Cross section described by screened Rutherford model, and cross-section dependent on: atomic # of target atominverse of beam energy

7. Inelastic scattering Energy transferred to target atoms Kinetic energy of beam electrons decreases Note:Lower electron energy will now increase the probability of elastic scattering of that electron Principal processes: 1)Plasmon excitation Beam electron excites waves in the “free electron gas” between atomic cores in a solid 2) Phonon Excitation Excitation of lattice oscillations (phonons) by low energy loss events (<1eV) - Primarily results in heating

8. Inelastic scattering processes 3) Secondary electron emission Semiconductors and insulators Promote valence band electrons to conduction band These electrons may have enough energy to scatter into the continuum In metals, conduction band electrons are easily energized and can scatter out Low energy, mostly < 10eV 4) Continuum X-ray generation (Bremsstrahlung) Electrons decelerate in the coulomb field of target atoms Energy loss converted to photon (X-ray) Energy 0 to E 0 Forms background spectrum

9. Inelastic scattering processes 5) Ionization of inner shells Electron with sufficiently high energy interacts with target atom Excitation Ejects inner shell electron Decay (relaxation back to ground state) Emission of characteristic X-ray or Auger electron Background = continuum radiation

10. Total scattering probabilities: Elastic events dominate over individual inelastic processes 10 7 10 6 10 5 10 4 051015 Electron energy (keV) Total cross section (g cm -1 ) Elastic Plasmon Conduction L-shell

11. Bethe then uses Born’s quantum mechanical atomic collision theory as a starting point. He ascertained the quantum perturbation theory of stopping power for a point charge travelling through a three dimensional medium. Bethe ends up developing new methods for calculating probabilities of elastic and inelastic interactions, including ionizations. Found sum rules for the rate of energy loss (related to the ionization rate). From this, you can calculate the resulting range and energy of the charged particle, and with the ionization rate, the expected intensity of characteristic radiation. Max Born (1882-1970) -1926Schrödinger waves are probability waves. The theory of atomic collisions.

12. From first Born approximation: Bethe’s ionization equation – the probability of ionization of a given shell (nl) E 0 = electron voltage E nl = binding energy for the nl shell Z nl = number of electrons in the nl shell b nl = “excitation factor” c nl = 4E nl / B nl ( B is an energy ~ ionization potential) { Bethe parameters KαKα K L M

13. This is an expression of X- ray emission, so as X-ray production volume occurs deeper at higher kV, proportionally more absorbed… Binding energy = critical excitation potential Optimum overvoltage = 2- 3x excitation potential

14. Rate of energy loss of an electron of energy E (in eV) with respect to path length, x : There are a number of important physical effects to consider for EPMA, but perhaps the most significant for determination of the analytical spatial resolution is electron deceleration in matter… dE / dx

15. Low ρ and ave Z High ρ and ave Z

16. Or… E = electron energy (eV) x = path length e = 2.718 (base of ln) N 0 = Avogadro constant Z = atomic number ρ = density A = atomic mass J = mean excitation energy (eV) Bethe’s remarkable result: stopping power…

17. Joy and Luo… Modify with empirical factors k and J exp to better predict low voltage behavior Also: value of E must be greater than J /1.166 or S becomes negative!

18. 0 1 2 3 4 5 6 7 10 1 10 2 10 3 10 4 10 5 Bethe Joy et al. exp Bethe + Joy and Luo Stopping Power (eV/Å) Electron Energy (eV) Aluminum

19. backscattering

20. Bethe equation:

21. R KO Henoc and Maurice (1976) Bethe equation: j = mean ionization potential E i = 1.03 j EI = Exponential Integral Approximation of R… Electron range What is the depth of penetration of the electron beam? Bethe Range Kanaya – Okayama range (Approximates interaction volume dimensions) R KO = 0.0276AE 0 1.67 / Z 0.89 ρ E 0 beam energy (keV) ρdenstiy (g/cm 3 ) Aatomic wt. (g/mol) Zatomic #

22. Mean free path and cross section inversely correlated Mean free path increases with decreasing Z and increasing beam energy Results in volume of interaction Smaller for higher Z and lower beam energy However: Interaction volume and emission volume not actually equivalent

23. Interaction volume… Scattering processes operate concurrently Elastic scattering Beam electrons deviate from original path – diffuse through solid Inelastic scattering Reduce energy of primary beam electrons until absorbed by solid Limits total electron range Interaction volume = Region over which beam electrons interact with the target solid Deposits energy Produces radiation Three major variables 1)Atomic # 2)Beam energy 3)Tilt angle Estimate either experimentally or by Monte Carlo simulations

24. Monte Carlo simulations Can study interaction volumes in any target Detailed history of electron trajectory Calculated in step-wise manner Length of step? Mean free path of electrons between scattering events Choice of event type and angle Random numbers Game of chance

25. Casino (2002) Alexandre Real Couture McGill University Dominique Drouin Raynald Gouvin Pierre Hovington Paula Horny Hendrix Demers Win X-Ray (2007) Adds complete simulation of the X-ray spectrum and the charging effect for insulating specimens McGill University group and E. Lifshin SS MT 95 (David Joy – Modified by Kimio Kanda) Run and analyze effects of differing… Atomic Number Beam Energy Tilt angle Note changes in … Electron range Shape of volume BSE efficiency

26. Monte Carlo electron path demonstrations

27. 1  m Labradorite (Z = 11)Monazite (Z = 38) 15 kV 10 kV Electron trajectory modeling - Casino

28. Labradorite (Z = 11)Monazite (Z = 38) 30 kV 25 kV 20 kV 15 kV 10 kV 1m1m 5m5m 1m1m 5m5m =Backscattered

29. 5% 10% 25% 50% 75% Labradorite [.3-.5 (NaAlSi 3 O 8 ) –.7-.5 (CaAl 2 Si 2 O 8 ), Z = 11] 15 kV Electron energy 100%1% Energy contours

30. Labradorite (Z = 11)Monazite (Z = 38) 30 kV 25 kV 20 kV 15 kV 10 kV 1m1m 5m5m 1m1m 5m5m = Backscattered Electron energy 100%1%

31. Labradorite [.3-.5 (NaAlSi 3 O 8 ) –.7-.5 (CaAl 2 Si 2 O 8 ), Z = 11] 5% 10% 25% 50% 75% 100% 15 kV 10 kV 5 kV 1  m 5% 10% 25% 50% 1 kV (~ Na K ionization energy) (~ Ca K ionization energy)