2. Ionization
- Electron impact ionization &
Photo ionization -
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2.1
Overview
1. Atomic processes in plasma
2. Rate equation
Example EBIT, Charge Exchange, Radiative Recombination
3. Electron impact ionization
Ionization mechanism: Multistep ionization, Ionization factor,
approximation for cross section (Lotz formula), Carlson correction,
optimal electron energy, application (electron target)
4. Photo ionization
Basics, cross section, application (RILIS)
5. Summary
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2.2
2.1. Overview: Atomic processes in a plasma
In most ion sources, ions are produced in a plasma.
The basic atomic processes (selection) in plasmas are:
collisions with
electrons
Impact Ionization
Three-Body-Recombination (TBR)
AZ + + e ↔
Impact excitation
AZ + + e ↔
Photo ionization
collisions with
photons
A
Z+
+ hυ
Photo absorption
Z+
A
Impact disexcitation
Z+ ∗
(A )
+ e'
AZ+: Atom of species A
with charge state Z
e’ : electron changed
energy
Non-radiative transition
Radiative Recombination (RR)
↔
Excitation
AZ + + hυ
A( Z +1)+ + e' + e' '
A
( Z +1)+
+e
Line spectrum
Spontaneous emission
↔
+ hυ + e ↔
(A )
Z+ ∗
Bremsstrahlung
Z+
A
+ e'
Continuous spectrum
The electron changes from one free state to another free state with lower energy.
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2.3
2.2. Rate equation
From these multiple processes arise the dynamic balance quantities:
• Distribution of the abundance of all charge states Z (=0...Zmax), Ionization equilibrium
• Number of emitted and absorbed photons per time interval, Radiative equilibrium
The density of the particle species are determined from so-called rate equations:
dn
= souces − sinks
dt
Example: impact ionization:
dnz +1
= ne ⋅ nz ⋅ v eσ z → z +1 − ne ⋅ nz +1 ⋅ β z +1,TBR
dt
ß z+1, TBR : rate coefficient for Three-Body-Recombination (TBR)
The rate coefficients often not be calculated with sufficient precision; experimental data are only
available to a limited extent. Therefore one tries to obtain data from thermo dynamical equilibrium.
With decreasing electron density the TBR drops, so that the impact ionization is not in equilibrium with
the TBR anymore. The RR rate also decreases but not as strong. With decreasing ne also the photo
ionization becomes unlikely. As result the impact ionization and the RR-process dominate. The photons
leave the plasma without being re-absorbed.
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2.4
Example EBIT:
[
(
]
)
[
]
dn i
ion
RR
RR
chex
chex
coll
= n eυ e σ iion
−1→ i n i −1 − σ i → i +1 + σ i → i −1 n i + σ i +1→ i n i +1 − n oυ ion σ i → i −1 n i − σ i +1→ i n i +1 − ν i
dt
υion
σ
: averaged thermal ion velocity
υ ion = 2kTion Mion
: cross sections for impact ionization, RR and charge exchange
by collisions with residual gas
ν icoll : collision rate for all Coulomb collisions of ions with charge state i
UW
: depth of the electrostatic potential (radial and axial),
kTion : thermal energy of the ions heated by Coulomb collisions
neυe : electron density and velocity
no
: neutral particle density
The solutions of the linear system of equations
provides the charge state distribution in dependence
on the ionization factor jeτ
%
⎧ ieU w ⎫
exp ⎨−
⎬
⎩ kTion ⎭
ni
ieU w
kTion
0
80
60
8
40
20
-3
-2
-1
log(ni /nma x)
0
1
2
log(J*TAU)
ARGON
-1
16
TOP:
Ionization of pulsed injected argon gas by an electron beam with
an energy of 3.9 keV, which inhibits the ionization of the charge
state Ar17+ (EBIS-Mode with breeding at the shell closure).
BOTTOM:
Capture and ionization for the case of continuously injected argon
atoms and 6 keV electron beam energy (EBIT-Mode without RR
and charge exchange with residual gas).
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16
ARGON
-2
-3
8
-4
-3
-2
-1
0
1
2
3
log(J*TAU)
2.5
Charge exchange:
For the approximation of the charge exchange cross section, commonly the equation by Müller and
Salzborn is used:
[
2
− 2.76
σ iex→ i −1 ≈ 1.43 × 10−12 i 1.17 Eion
cm
, gas
]
Therein i describes the charge state of the highly charged ion and Eion, gas is the ionization potential of the
gas atoms interacting with the ion.
Pressure (mbar)
Example: Reachable charge states depending on the current density and the pressure
10
-3
10
-6
100 A/cm
10
-9
10 A/cm
10
2
2
-12
0
20
40
60
80
P b Charge states
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2.6
Radiative Recombination (RR):
For the approximation of the RR cross sections the semi-classical expression by Kim und Pratt is used.
Their formula bases on the first theoretical description of the Radiative Recombination by Kramers (1923):
with
with:
• the effective nuclear charge
potential by the bound electrons
to take into account the shielding of the nuclear
• the effective main quantum number
to include the capture into states with
different angular momentums
Ry:
α:
ω0
Rydberg energy 13,6 eV
fine structure constant
ratio between the numbers of
occupied and unoccupied states
Z:
nuclear charge
q:
charge state
n:
main quantum number
(λe)r : reduced Compton-wavelength
E:
electron energy
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2.7
2.3. Electron impact ionization
The electron impact ionization is the most fundamental ionization process for the operation
of ion sources.
Why?
• The cross section for the impact ionization is by orders of magnitudes higher than the cross section for
the photo ionization.
• The cross section depends on the mass of the colliding particle. Since the energy transfer
of a heavy particle is lower, a proton needs for an identical ionization probability an ionization energy
three orders of magnitudes higher than an electron
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2.8
Mechanism for the impact ionization
There are two different possibilities to produce multiple charged ions:
• in a collision where many electrons are removed from the ion (double-ionization, triple-ionization, …)
• multistep or successive ionization, where only one electron is removed per collision and high charge
states are produced in different collisions
Electron impact
Double ionization
Cross section
Cross section
Electron impact
Single ionization
electron energy
electron energy
For energetic reasons the ionization releasing only one electron from the atomic shell is the most probable
process. To produce highly charged ions the kinetic energy of the projectile electrons has to be at least
equivalent to the n-th ionization potential.
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2.9
Ionization energies up to 100 keV (U91+ → U92+)
Ionisation energies by C. Moore
1E+6
eV
1E+5
1E+4
1E+3
1E+2
1E+1
1E+0
0
10
20
30
40
50
OXGAS2.XLS
R. Becker, 31.07.99
→ shell structure of the atomic shells is clearly visible
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60
70
80
90
100
Z
2.10
Successive ionization:
The probability for removing one electron and changing the charge state of the ion from q Æ q+1 is
determined by the cross section σqÆq+1 [cm2].
Are the cross sections for the successive ionization known, the average ionization time of the ions
with charge state q can be approximated from the collision frequency:
ν q → q +1 = ne nq v e σ q → q +1
⎡ 1 ⎤
⎢ m3 s ⎥
⎣
⎦
The time between to ionizing collisions is
τ q→q +1 =
nq
ν q→q+1
=
1
ne veσ q→q +1
=
e
jeσ q→q +1
This applies to electrons with a defined kinetic energy.
je ⋅ τ q → q +1 =
e
σ q → q +1
The average ionization time in the charge state q
depends only on the cross section and the current
density. This expression is called IONISATION
FACTOR. (Sometimes it is defined as only the inverse
cross section!)
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Principle of electron impact ionization
2.11
The average time necessary to reach the charge state q is therewith:
q −1
e q −1 1
τ q = ∑ τ i → i +1 = ∑
je i = 0 σ i → i +1
i =0
Example on the right side:
The ionization factor je*τ for different charge states of Xe depending
on the electron energy.
Approximation of the cross section and the ionization time for the
production of bare ions from H-like ions using the Mosley’s law
for X-ray frequencies emitted in transitions from the continuum to the
K-shell:
Ei→k ( Z ) = 13.6 ⋅ Z [eV ] Î σ z −1→ z = 4.5 ⋅10
2
E = e ⋅ Ei → k ( Z )
j
1
ν z −1→ z = e σ z −1→ z =
e
τ z −1→ z
4
e
eZ 4
⎛Z⎞
≈
≈⎜ ⎟
Î jeτ z −1→ z =
−17
σ z −1→ z 9 ⋅10
⎝5⎠
wherein
−14
ln e
9 ⋅10 −17
⋅
=
2 4
e ⋅13.6 Z
Z4
Examples: Argon can be ionized by 10 keV electrons, ions of
the heavy elements by up to 100 keV electrons and Uranium by
150 keV electrons. The resulting values for the ionization
energy, cross section and ionization factor are summarized in
the table on the left side.
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2.12
Approximation of the cross section for the electron impact ionization
Approximation of the cross section in quantum-mechanical calculations done by Bethe (1930) using the
Born Approximation:
Scattering of a matter wave at a central potential V(r) for Ekin >> Eion (perturbation theory).
All electrons in an atom or ion contribute with their individual σei to the total cross section σ, as long as the
kinetic energy Ekin of the projectile is larger then the ionization energy Pi of these electrons.
N
N
σ q → q +1 = ∑ σ i = ∑ qiσ ei
i =1
N = number of subshells
i =1
All qi electrons of the subshell contribute to the σi of the shell. The cross section for the ionization of the
(n, l) - shell results from integration of the transition probabilities over all states n’, l’ Æ k und the
integration over the collision vector
r 2π
r v
q=
M (v − v ' )
h
with v before and v’ after the collision. One receives
σ inl = const ⋅ Z nl
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⎛E ⎞
1
ln ⎜⎜ kin ⎟⎟
E kin E nl ⎝ E nl ⎠
Bethe et al., Ann. Physik 5 (1930) 325
2.13
For practical reasons the semi-empirical formula developed by Lotz 1967 for the energy dependence of
the cross sections for the elements from H to Ca and for energies < 10 keV is commonly used. The error
is given by maximal 10%.
The Lotz formula for the case of high ionization energies Ekin >> Pi is:
E
ln ⎛⎜ kin ⎞⎟
Pi ⎠
σ q → q +1 = 4.5 ⋅ 10 −14 ⋅ ∑ ⎝
Ekin ⋅ Pi
i =1
N
[cm ]
2
Pi = Enl , N-subshells
This expressions is mostly used in calculations of the charge state distribution.
For E Æ Pi and with
follows
E
1 + x = kin
Pi
x2
(x Æ 0) , and with ln (1 + x ) ≈ x −
2
σ 0 →1 ≈ 4.5 ⋅ 10
−14
⋅
( EPkin1 − 1)
2
1
P
for x << 1
~ E kin
This means the cross section goes linear with EÆ Pi to zero.
Examples for the dependence of the cross sections on the energy are shown for the case of He. The
higher the initial charge state, the smaller is the cross section. Moreover the cross section is higher for
atoms in an excited state.
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2.14
Single-ionisation: He + e → He+ + 2e
10-17
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Multi-ionisation: He + e → He2+ + 3e
10-19
2.15
Ionization of the excited state
10-16
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He (2s) + e → He+ + 2e
Ionization of singly charged He
10-18
He+ + e → He2+ + 2e
2.16
Carlson-Correction for ionization energies:
The ionization energies Pq,i for ions with different charge states q, which does not describe the weakest
bound electron are sometimes difficult to find in literature. They can be approximated with the CarlsonCorrection.
The ionization energy Pq,i is calculated from the ionization energy Wi(q) of the ion with charge state q and
the atomic binding energies of the electrons as follows:
Pi = E0 i + Wi (q) − E0q
E0i
E0q
: binding energy of an electron in the i-th shell of an atom
: atomic binding energy of the electron, which is the weakest bound electron in the ion of the
charge state q
Wi(q) : ionization energy of the ion (describes always the weakest bound electron)
weakest bound electron q (ionization energy is known)
E0q-E0i
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Ion: Aq+
inner electron i,
to be removed
2.17
Optimal electron energy
Because σ(E) has a maximum at a certain energy, the ionization factor je*τ has a minimum there.
Basically the cross section for the last electron, which is removed, determines the ionisation time.
The optimal energy is given by
dσ z → z +1
dE
N
1
∑
2
i =1 Pi E
⎛ ln ⎛ E kin ⎞ ⎞
Pi ⎟⎠ ⎟
d ⎜ ⎜⎝
−14
⎟=0
⎜
= 4.5 ⋅ 10 ⋅ ∑
E kin ⋅ Pi ⎟
i =1 dE ⎜
⎜
⎟
⎠
⎝
N
⎛
⎛ E ⎞⎞
⎜1 − ln ⎜ ⎟ ⎟ = 0 ⇒
⎜ P ⎟⎟
⎜
⎝ i ⎠⎠
⎝
E max
⎛ N 1 + ln Pi
⎜∑
P
= exp ⎜ i =1 N i
⎜
⎜ ∑ 1P
i
⎝ i =1
⎞
⎛ N ln Pi
⎟
⎜∑
⎟ = e ⋅ exp ⎜ i =1 Pi
⎟
⎜ N
⎟
⎜ ∑ 1P
i
⎠
⎝ i =1
⎞
⎟
⎟
⎟
⎟
⎠
For the optimal energy of the last electron, which is removed, follows:
E max
⎛ ln Pz
⎜
P
≈ e ⋅ exp ⎜ z
⎜ 1
⎜ Pz
⎝
⎞
⎟
⎟ = e⋅P
z
⎟
⎟
⎠
Therewith the optimal energy is nearly e-times the ionization energy of the last electron that is removed
from the ion with the charge state z.
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2.18
electron energy
Ionization factor and optimal electron energy
ionization factor
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2.19
Application of the electron impact ionization for the production of ions
Slow electrons
Stripper foil
ECR plasma
MI
+ + ++ ++
Slow ions
Fast electrons
MI
+ ++
Slow ions
+++
+++
Fast electrons
+++
+++
EBIS beam
Fast ions
MI
• Plasma confinement by magnetic fields of different structure (cusp, magnetic mirror, …)
(see next lectures)
• The efficiency of the electrons in the plasma and the charge state can be increased by using
the electron several times, by:
- reflection of the electrons, e.g. reflection discharge or penning discharge
- magnetic confinement, e.g. by magnetic mirrors
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2.20
Further application of electron impact ionization
Example: electron target
Investigation of electron-ion-collisions:
• Investigation of electron impact ionization
• Investigation of recombination processes
• Investigation of excitation processes
Measurement of rate coefficients R
r
r
R = α ∫ ne (r ) ni (r ) d 3r
An+
and therewith cross sections:
α = σ v r = ∫ σ (v r ) v r f (v r ) d 3v r
Important is the overlap of ion and
electron beam
σ: cross section
ne: electron density
ni: ion density
vr: collision velocity / relative velocity
f(vr): distribution function of the relative velocity
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An+
r
vi
r
vr
r
ve
2.21
Transversal electron target
Longitudinal electron target
• „Merged beams“.
• also used as cooler for ion beams
• 2-2.5m long interaction region
• “crossed beams technique”
• better energy resolution then gas target
• 10-15 cm long interaction region
• spectroscopic access is possible
• electrostatic focusing
BUT:
• limited access for spectroscopy
BUT:
• lower interaction rate as longitudinal electron
target or gas target
0.061A/cm, R=8.19 mesh units, J=0.2 A/cm2
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2.22
2.4. Photo ionization
The reaction is:
A + hν = A + + e −
Atoms of a gas can be ionized by an intensive beam of photons with the adequate
energy (photo ionization). Therefore the photon energy has to be h ⋅ ν > e ⋅ ϕ q, i
The energy of a photo electron is:
Cross section σp:
.
1
2
mvmax
= hν − e ⋅ ϕ q , i
2
Z 4 −5
σp ∝
(h ⋅ ω ) 7 / 2
• σp has a strong dependence on the photon energy and the nuclear charge Z:
• For a given atomic shell the cross section σp is the largest close to threshold, meaning where the photon
energy reaches the ionization energy I (resonance/ threshold behavior):
σ max : h ⋅ ω ≈ I K , I L , I M
σpn : cross section for RR into the K-shell
n:
main quantum number
cross section (barn)
• For high photon energies h⋅ ω >> IK the ionization of
the s-orbital is most probable and the K-shell ionization
delivers the dominant contribution:
∞
1
1
σ p n = 3 ⋅ σ K → σ p = σ p K ⋅ ∑ 3 = 1.2021 ⋅ σ p K
n
n =1 n
1E7
1000000
100000
10000
1000
100
10
0,1
• Description of σp as time-inverse effect
to the radiative recombination by the
2
(
hω )
Milne-formula:
gq +1 ⋅ σ RR =
gq ⋅ σ P
2mec 2E
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1
10
100
photon energy (keV)
with
g: statistical weights
2.23
Application of photo ionization for the production of ions
1 eV Î l = 1.24 mm (IR-radiation),
5 eV Î l = 248 nm (close UV)
For direct ionization photons from the UV – x-ray region are necessary.
RILIS (Resonant ionization laser ion source)
Ionization via resonant excitation with three laser beams of frequencies f1 – f3, as shown
in the following scheme.
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2.24
In reality there a many more effects to be takeen into account:
• Excitation into auto-ionizing state (AIS) with
typical lifetimes of 10-15 – 10-10 s
• Excitation in Rydberg-states n = n*
Lifetime
Binding energy
τ = τ 0 ⋅ (n* )3
M
Z2
E = R⋅
M + me (n* ) 2
R = Rydberg constant
Radius of Rydberg atoms
r = a0 (n* )2
Orders of magnitudes for the cross sections:
• non-resonant (direct ionization):
σ = 10-19 – 10-17 cm2
• AIS: σ = 1.6 x 10-14 cm2
• Rydberg states: σ ~ 10-14 cm2
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Excitation schemes used for
resonant laser ionization
2.25
A prominent example for a RILIS is at
ISOLDE/CERN
(see picture).
Advantages of a RILIS:
• high selectivity
• separation of surface-ionizing
contaminations by adjusting the
temperature of the cavity
• high efficiency
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2.26
2.5. Summary: Ion production
Processes increasing the charge state
Processes decreasing the charge state
• Ionization
- single-ionization
- double-ionization
• Recombination
- radiative recombination
The production of higher charge states is a
successive process
The cross section is larger for lower
electron temperatures
- dielectronic recombination
(resonant process)
The ionization has energy threshold
→ higher charge states need higher
projectile energies (electron energies)
• Charge exchange
(for low charge states)
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• Charge exchange
(for high charge states)
depending on the neutral particle density
(residual gas)
cross section are larger for higher charge
states
2.27
10. Spouses never need to worry about what their ion source physicist husband or wife is up to when they still aren’t home by
midnight...they know with great confidence that they’re just at the lab changing a source!
9. Between
–
–
–
–
–
–
–
the International Conference on Ion Sources,
the Workshop on Ion Source Issues Relevant to a Pulsed Spallation Neutron Source,
the ECR Ion Source Workshop,
The International Conference on Negative Ions, Beams and Sources,
the International Symposium on the Production and Neutralization of Negative Ions and Beams,
the Workshop on Negative Ion Formation and Beam Handling,
the International Conference on Sources of Highly Charged Ions,
the ion source community has the greatest number of conferences and workshops of any scientific discipline measured
per linear inch of subject matter
8. Endless hours can be devoted to the important philosophical meaning of “pi” as in “the output emittance is 0.2 pi-mm-mrad”
7. Believe it or not, sometimes the SNS ion source antenna picks up free Satellite Television!
6. High-voltage, hydrogen gas, antennas and power supplies: it’s every little kids dream!
5. In what other field can you acquire a Ph.D., and then spend your professional life practicing the occult?
4. When the phone rings during dinner time, you always know who it is: no its not a tele-marketer, it just the lab...the ion
source went down!
3. In what other field could you advertise a workshop on “RF-driven, multicusp, Cesium-enhanced H-sources for the SNS”,
and have 50 people actually show up?
2. Job security...when was the last time you heard a manager say, “the ion source is running so well, we’re going to let the
entire group go.”
1. It’s a subject in which anyone can make a contribution, and everyone has an opinion!
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Courtesy of Stuart Henderson/ORNL
2.28