Scattering and resulting emission processes

Incident beamLight

(cathodoluminescence)

BremsstrahlungSecondary electrons

Backscattered electrons

heat

Elastically scattered electrons

Transmitted electrons

Specimen current

Auger electrons

Characteristic X-rays

Sample

Any of the collected signals can be displayed as an image if you either scan the beam or the specimen stage

5 mm

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 diameterReal: Electron scattering increases diameter of area sampled

= volume of interaction

Scattering

Key concept:Cross section or probability of an event

Q = N/ntni cm2

N = # events / unit volument = # target sites / unit volume

ni = # incident particles / unit area

Large cross section = high probability for an event

From knowledge of cross sections, can calculate mean free path

λ = A / N0 ρ Q

A = atomic wt.

N0 = Avogadro’s number (6.02 x1023 atoms/mol)

ρ = densityQ = cross section

Smaller cross section = greater mean free path

Scattering (Bohr – Mott – Bethe)Types of scattering

ElasticInelastic

Elastic scatteringDirection component of electron velocity vector is changed, but not magnitude

Target atom

Ei

Φe

Ei = E0

Ei = instantaneous energy after scatteringKinetic energy ~ unchanged<1eV energy transferred to specimen

E0

Types of scatteringElasticInelastic

Inelastic scatteringBoth direction and magnitude components of electron velocity vector change

Target atom

Ei

Φi

Ei < E0

Ei = instantaneous energy after scatteringsignificant energy transferred to specimenΦi << Φe

E0

Elastic scatteringElectron deviates from incident path by angle Φc (0 to 1800)Results from interactions between electrons and coulomb field of nucleus of target atoms screened by electronsCross section described by screened Rutherford model, and cross-section dependent on:

atomic # of target atom inverse of beam energy

Inelastic scatteringEnergy transferred to target atomsKinetic 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 ExcitationExcitation of lattice oscillations (phonons) by low energy loss events (<1eV) - Primarily results in heating

Inelastic scattering processes3) Secondary electron emission

Semiconductors and insulatorsPromote valence band electrons to conduction bandThese electrons may have enough energy to scatter into

the continuumIn metals, conduction band electrons are easily energized

and can scatter outLow energy, mostly < 10eV

4) Continuum X-ray generation (Bremsstrahlung)Electrons decelerate in the coulomb field of target atomsEnergy loss converted to photon (X-ray)

Energy 0 to E0

Forms background spectrum

Inelastic scattering processes5) Ionization of inner shells

Electron with sufficiently high energy interacts with target atomExcitation Ejects inner shell electronDecay (relaxation back to ground state) Emission of characteristic X-ray or Auger electron

Background = continuum radiation

Total scattering probabilities:

Elastic events dominate over individual inelastic processes

107

106

105

104

0 5 10 15Electron energy (keV)

Tota

l cro

ss s

ectio

n (g

cm

-1)

Elastic

Plasmon

Conduction

L-shell

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)-1926 Schrödinger waves are probability waves.

The theory of atomic collisions.

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

E0 = electron voltageEnl = binding energy for the nl shellZnl = number of electrons in the nl shellbnl = “excitation factor”cnl = 4Enl / Bnl (B is an energy ~ ionization potential){Bethe parameters

Kα

KL

M

ψp

ψ1sψp2

ψp1

0 5 10 15 20 25 30 35 400.00

0.50

1.00

1.50

2.00

Pb MVPb MIVU MIV

E0 (keV)

σ (1

0-20

cm

2)

Estimated ionization crosssection for Pb and UBethe (1930) model

0 5 10 15 20 25 30 35 400.00

0.10

0.20

0.30

0.40

0.50

E0 (keV)

PbM

α In

tens

iy/e

lect

ron

rang

e

Path length-normalized PbMα (MV ionization) intensity as a function of accelerating potential

PyromorphiteVLPET spectrometer

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

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

Low ρ and ave Z

High ρ and ave Z

Or…

E = electron energy (eV)x = path lengthe = 2.718 (base of ln)N0 = Avogadro constantZ = atomic numberρ = densityA = atomic massJ = mean excitation energy (eV)

Bethe’s remarkable result: stopping power…

Joy and Luo…

Modify with empirical factors k and Jexp to better predict low voltage behavior

Also: value of E must be greater than J /1.166 or S becomes negative!

0

1

2

3

4

5

6

7

101 102 103 104 105

BetheJoy et al. expBethe + Joy and Luo

Sto

ppin

g P

ower

(eV

/Å)

Electron Energy (eV)

Aluminum

eU backscattering

eU

Bethe equation:

RKO

Henoc and Maurice (1976)

Bethe equation:

j = mean ionization potential Ei = 1.03 j EI= Exponential Integral

Approximation of R…

Electron rangeWhat is the depth of penetration

of the electron beam?

Bethe Range

Kanaya – Okayama range (Approximates interaction volume dimensions)

RKO = 0.0276AE01.67 / Z0.89 ρ

E0 beam energy (keV)ρ denstiy (g/cm3)A atomic wt. (g/mol)Z atomic #

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 rangeInteraction volume =Region over which beam electrons interact with the target solid

Deposits energyProduces radiation

Three major variables1) Atomic #2) Beam energy3) Tilt angle Estimate either experimentally or by

Monte Carlo simulations

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 angleRandom numbersGame of chance

Casino (2002) Alexandre Real CoutureMcGill University Dominique Drouin

Raynald GouvinPierre HovingtonPaula HornyHendrix Demers

Win X-Ray (2007) Adds complete simulation of the X-ray spectrum and the charging effect for insulating specimensMcGill 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

1 mm

Labradorite (Z = 11) Monazite (Z = 38)

15 kV

10 kV

Electron trajectory modeling - Casino

Labradorite (Z = 11) Monazite (Z = 38)

30 kV30 kV 25 kV25 kV 20 kV20 kV 15 kV15 kV 10 kV10 kV

1mm

5mm

1mm

5mm

=Backscattered

5%

10%

25%

50%

75%

Labradorite [.3-.5 (NaAlSi3O8) – .7-.5 (CaAl2Si2O8), Z = 11]

15 kV

Electron energy100% 1%

Energy contours

Labradorite (Z = 11) Monazite (Z = 38)

30 kV30 kV 25 kV25 kV 20 kV20 kV 15 kV15 kV 10 kV10 kV

1mm

5mm

1mm

5mm

= Backscattered Electron energy100% 1%

Labradorite [.3-.5 (NaAlSi3O8) – .7-.5 (CaAl2Si2O8), Z = 11]

5%

10%

25%

50%

75%

100%

15 kV

10 kV

5 kV1 mm

5%

10%

25%

50%

1 kV(~ Na K ionization energy)

(~ Ca K ionization energy)