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: Real: Diameter of area sampled = beam diameter Electron scattering increases diameter of area sampled = volume of interaction

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

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

Scattering (Bohr – Mott – Bethe)

Types of scattering Elastic Inelastic

Elastic scattering

Direction component of electron velocity vector is changed, but not magnitude E i E 0 Φ e Target atom E i = E 0

E i = instantaneous energy after scattering Kinetic energy ~ unchanged <1eV energy transferred to specimen

Types of scattering Elastic Inelastic

Inelastic scattering

Both direction and magnitude components of electron velocity vector change Φ i E 0 E i Target atom E i < E 0

E i = instantaneous energy after scattering significant energy transferred to specimen

Φi << Φe

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 atom inverse of beam energy

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

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

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

Total scattering probabilities: 10 7 Elastic events dominate over individual inelastic processes 10 6 10 5 10 4 0 Elastic Plasmon Conduction L-shell 5 10 Electron energy (keV) 15

Max Born (1882-1970) -1926 Schrödinger waves are probability waves.

The theory of atomic collisions.

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.

From first Born approximation: Bethe’s ionization equation – the probability of ionization of a given shell (

nl

) M L K Bethe parameters { E 0 E nl Z nl b nl c nl

= electron voltage = binding energy for the = number of electrons in the = “excitation factor” =

4E nl / B nl

(

B nl

shell

nl

shell is an energy ~ ionization potential)

K α

ψ p1 ψ p2 ψ p ψ 1s

Optimum overvoltage = 2 3x excitation potential

2,00 1,50 1,00 0,50

Binding energy = critical excitation potential

0,00 0 0,50 0,40 0,30 0,20 0,10 0,00 0

Estimated ionization cross section for Pb and U Bethe (1930) model

10 20 E 0 (keV)

Pb MV Pb MIV U MIV

30 40 Path length-normalized PbMα (MV ionization) intensity as a function of accelerating potential Pyromorphite VLPET spectrometer 10 20 E 0 (keV) 30 40

This is an expression of X ray emission, so as X-ray production volume occurs deeper at higher kV, proportionally more absorbed…

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 … Rate of energy loss of an electron of energy E to path length, x : (in eV) with respect

dE / dx

Low ρ and ave Z High ρ and ave Z

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

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

J /1.166

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

U e backscattering

U e Bethe equation:

Electron range What is the depth of penetration of the electron beam?

Bethe Range Bethe equation : R KO Approximation of

R…

Henoc and Maurice (1976) EI

= Exponential Integral

j

= mean ionization potential

Kanaya – Okayama range (Approximates interaction volume dimensions) R KO = 0.0276AE

0 1.67

/ Z 0.89

ρ

E 0

ρ

beam energy (keV) denstiy (g/cm 3 ) A Z atomic wt. (g/mol) atomic # E

i

= 1.03 j

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

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

Monte Carlo simulations

Length of step?

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

Casino (2002) McGill University Alexandre Real Couture Dominique Drouin Raynald Gouvin Pierre Hovington Paula Horny Hendrix Demers Win X-Ray (2007) of the X-ray spectrum and the charging effect for insulating specimens Adds complete simulation 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

Monte Carlo electron path demonstrations

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

Labradorite (Z = 11) 1 m m 5 m m =Backscattered Monazite (Z = 38) 1 m m 5 m m

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

Labradorite (Z = 11) 1 m m 5 m m Monazite (Z = 38) 1 m m 5 m m = Backscattered Electron energy 100% 1%

1

m

m (~ Ca K ionization energy)

50%

10 kV

25%

5 kV

10% 5%

1 kV (~ Na K ionization energy)

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