2) Ionisation: Electron impact ionisation and photo ionisation
Slow electrons
Fast ions
Stripper foil
MI
+ + ++ ++
Fast electrons
ECR plasma
MI
+ ++
Slow ions
+++
+++
Fast electrons
+++
+++
EBIS beam
Slow ions
MI
The electron impact ionisation is the most fundamental ionisation process and most important for ion
sources.
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2.1
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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2.2
There are two different possibilities to produce multiple charged ions:
• in a single collision where many electrons are removed from the ion (double-ionization, tripleionization, etc.
• 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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2.3
• Ionisation energies up to 100 keV (U91+ --> U92+) (see graphic below)
• shell structure of the atomic shells is clearly visible
Ionisation energies by C. Moore
1E+6
eV
1E+5
1E+4
1E+3
1E+2
1E+1
1E+0
0
OXGAS2.XLS
R. Becker, 31.07.99
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10
20
30
40
50
60
70
80
90
100
Z
2.4
A + e − = A + + 2e −
Successive Ionisation:
The probability for removing one electron and changing the charge state of the ion from q
determined by the cross section σq q+1 [cm2].
q+1 is
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:
 1 
ν q → q +1 = ne ⋅ nq veσ q → q +1  3 
m s
(2.1)
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
(2.2)
This applies to electrons with a well-defined
kinetic energy.
je ⋅ τ q → q +1 =
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e
σ q → q +1
(2.3)
2.5
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!)
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
(2.4)
Example on the right side:
The ionization factor je*t 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 ] ⇒ σ z −1→ z = 4.5 ⋅10
2
Wherein
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E = e ⋅ Ei→k (Z )
jeτ z −1→ z =
−14
ln e
9 ⋅10−17
⋅
=
2 4
e ⋅13.6 Z
Z4
e
σ z −1→ z
eZ 4
Z
≈
≈
 
−17
9 ⋅10
5
(2.5)
4
(2.6)
2.6
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.
Approximation of the cross section in quantummechanical 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 than the ionization energy Pi of these electrons.
N
N
i =1
i =1
σ q → q +1 = ∑ σ i = ∑ qiσ ei
N = Number of subshells
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’ k und the
integration over the collision vector
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2.7
r 2π
r v
q=
M (v − v ' )
h
σ
nl
i
with v before and v’ after the collision. On receives
 Ekin 
1

= const ⋅ Z nl
ln 
Ekin Enl  Enl 
Bethe et al., Ann. Physik 5 (1930) 325
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
(2.7)
This expression is mostly used in calculations of the charge state distribution.
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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2.8
Single ionisation
10-17
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Multi-ionisation of He
10-19
2.9
Ionization of the excited state
10-16
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Ionisation of singly charged He
10-18
2.10
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 (T. A. Carlson et al., ORNL-4562 UC-34-Physics).
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 ) − E0 q
(2.8)
E0i = Binding energy of an electron in the i-th shell of the atom
E0q = atomic binding energy of the electron, which is the weakest bound electron in the ion of the
charge state q.
weakest bound electron q (ionization energy is known)
E0q-E0i
inner electron i,
to be removed
Ion: Aq+
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2.11
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
 ln  E kin  
N
Pi  
dσ z → z +1
d  
−14

=0
= 4.5 ⋅ 10 ⋅ ∑
dE
E kin ⋅ Pi 
i =1 dE 




 N 1 + ln Pi 
 N ln Pi
∑

∑
N
 E 
1 
P
i =1
i =1 Pi



i




1 − ln    = 0 ⇒ E max = exp
= e ⋅ exp N
∑
N
2 



P
E
P
i =1 i
 i 
1

 ∑ P 
 ∑ 1P
i
i
 i =1

 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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2.12
Ionization factor and optimal electron energy
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2.13
Investigation of electron impact ionisation: Electron targets
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 3
R = α ∫ ne (r )ni (r ) d r
An
+
(2.9)
An
And there with the cross sections
α = σ v r = ∫ σ (v r )v r f (v r ) d 3 v 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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+
(2.10)
r
vr
r
vi
r
ve
2.14
Transversal electron target
•
•
•
•
•
BUT:
•
“crossed beams technique”
better energy resolution then gas target
10-15 cm long interaction region
spectroscopic access is possible
electrostatic focusing
•
•
•
BUT:
•
„Merged beams“.
also used as cooler for ion beams
2-2.5m long interaction region
limited access for spectroscopy
lower interaction rate as longitudinal
electron target or gas target
0.061A/cm, R=8.19 mesh units, J=0.2 A/cm
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Longitudinal electron target
2
2.15
Photoionisation
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 i . The energy of a photo electron is:
1 2
mvmax = hν − eϕi
2
Ionisation energy
of atoms:
1 eV
λ = 1.24 µm
(Indra red IR)
5 eV
λ = 248 nm
(near UV)
In case of direct ionisation
photons from the UV or
X-ray region would be
required.
Therefore the concept is
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2.16
RILIS (Resonant ionisation laser ion source): Ionization via resonant excitation with three laser
beams of frequencies f1 – f3, as shown in the following scheme
A prominent example for a RILIS is at ISOLDE/CERN (see picture).
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2.17
Advantages of a RILIS: high selectivity, separation of surface-ionizing contaminations by adjusting the
temperature of the cavity, high efficiency
In reality there a many more effects:
• Excitation into auto-ionizing state (AIS) with
typical lifetimes of 10-15 – 10-10 s
• Excitation into Rydberg-states n = n*
* 3
τ
=
τ
⋅
(
n
)
Lifetime
0
Binding energy
M
Z2
E = R⋅
M + me (n* ) 2
R = Rydberg constant
Radius of Rydberg atoms
r = a0 (n* ) 2
Cross sections (orders of magnitude):
• Non-resonant (direct ionisation)
σ = 10-19 – 10-17 cm2
• AIS
σ = 1.6*10-14 cm2
• Rydberg states
σ ~ 10-14 cm2
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2.18
Cross section σp:
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
Milne-formula:
gq +1 ⋅ σ RR =
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1
10
100
photon energy (keV)
(hω )2
2me c 2 E
gq ⋅ σ P
with
g: statistical weights
2.19