Scattering and resulting emission
processes
Incident
Light
beam Auger
(cathodoluminescence)
electrons
Secondary
electrons
Bremsstrahlung
Characteristic
X-rays
Backscattered
electrons
Sample
heat
Any of the collected signals can be
displayed as an image if you either scan
the beam or the specimen stage
Specimen
current
5 mm
Elastically
scattered
electrons
Transmitted
electrons
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
Scattering
Key concept:
Cross section or probability of an event
Q = N/ntni
cm2
N = # events / unit volume
nt = # 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 = 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
Ei
Φe
E0
Target atom
Ei = E0
Ei = 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
Ei
Φi
E0
Target atom
Ei < E0
Ei = instantaneous energy
after scattering
significant energy
transferred to specimen
Φi << Φe
Elastic scattering
Electron 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 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 E0
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:
Elastic events dominate over
individual inelastic processes
Total cross section (g cm-1)
107
106
Elastic
Plasmon
105
Conduction
L-shell
104
0
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)
Bethe parameters
{
E0 = electron voltage
Enl = binding energy for the nl shell
Znl = number of electrons in the nl shell
bnl = “excitation factor”
cnl = 4Enl / Bnl (B is an energy ~ ionization potential)
Kα
L
M
ψp1
ψp
ψp2
ψ1s
K
2.00
Estimated ionization cross
section for Pb and U
Bethe (1930) model
Optimum overvoltage = 23x excitation potential
σ (10-20 cm2)
1.50
1.00
Pb MV
0.50
Pb MIV
U MIV
0.00
0
Binding energy = critical
excitation potential
10
20
E0 (keV)
30
40
PbMα Intensiy/electron range
0.50
0.40
This is an expression of Xray emission, so as X-ray
production volume occurs
deeper at higher kV,
proportionally more
absorbed…
0.30
0.20
Path length-normalized PbMα (MV ionization)
intensity as a function of accelerating potential
0.10
Pyromorphite
VLPET spectrometer
0.00
0
10
20
E0 (keV)
30
40
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 (in eV) with respect
to path length, x:
dE / dx
Low ρ
and
ave Z
High ρ
and
ave Z
Bethe’s remarkable result:
stopping power…
E = electron energy (eV)
x = path length
e = 2.718 (base of ln)
N0 = Avogadro constant
Z = atomic number
ρ = density
A = atomic mass
J = mean excitation energy (eV)
Or…
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!
7
Aluminum
6
Stopping Power (eV/Å)
5
Bethe
Joy et al. exp
Bethe + Joy
and Luo
4
3
2
1
0
101
102
103
Electron Energy (eV)
104
105
U
e
backscattering
U
e
Bethe equation:
Electron range
What is the depth of penetration
of the electron beam?
Bethe Range
RKO
Bethe equation:
Approximation of R…
Henoc and Maurice
(1976)
EI= Exponential Integral
j = mean ionization potential
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 #
Ei = 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
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
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
Run and analyze effects of differing…
Kanda)
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 mm
Electron trajectory modeling - Casino
Labradorite (Z = 11)
1mm
Monazite (Z = 38)
1mm
5mm
30 kV
25
20
15
10
=Backscattered
5mm
30 kV
25
20
15
10
75%
50%
25%
10%
5%
Energy contours
Electron energy
100%
Labradorite [.3-.5 (NaAlSi3O8) – .7-.5 (CaAl2Si2O8), Z = 11]
15 kV
1%
Labradorite (Z = 11)
Monazite (Z = 38)
1mm
1mm
5mm
5mm
30 kV
25
20
15
10
= Backscattered
30 kV
25
20
15
10
Electron energy
100%
1%
10 kV
50%
1 mm
5 kV
25%
(~ Ca K ionization energy)
10%
5%
5%
1 kV
(~ Na K ionization energy)
10%
25%
50%
75%
100%
Labradorite [.3-.5 (NaAlSi3O8) – .7-.5 (CaAl2Si2O8), Z = 11]
15 kV