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ESS-Bilbao Initiative Workshop. Low Energy Transport and space-charge compensation schemes
1. Low Energy Transport and Space Charge
Low Energy
Compensation Schemes
Transport
and Space
Charge
R. Duperrier
Romuald Duperrier
Front End
Ion source
Theory
More electrodes
Laboratoire d’Étude et de Développement pour les Accélérateurs
Codes
CEA/IRFU/SACM
LEBT
Electrostatic
Solenoids
sc neutralisation
Codes
RFQ
Basics
Beam dynamics
Current limits
Codes
Conclusions
Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 1 / 47
2. Outline
Main parts of a Front End
1
Low Energy
Transport
and Space
The ion source extraction system
2
Charge
R. Duperrier
Front End
The LEBT line
3
Ion source
Theory
More electrodes
Codes
The RFQs
4
LEBT
Electrostatic
Solenoids
sc neutralisation
Conclusions
Codes
5
RFQ
Basics
Beam dynamics
Current limits
Codes
Conclusions
Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 2 / 47
3. Outline
Main parts of a Front End
1
Low Energy
Transport
and Space
The ion source extraction system
2
Charge
R. Duperrier
Front End
The LEBT line
3
Ion source
Theory
More electrodes
Codes
The RFQs
4
LEBT
Electrostatic
Solenoids
sc neutralisation
Conclusions
Codes
5
RFQ
Basics
Beam dynamics
Current limits
Codes
Conclusions
Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 3 / 47
4. Typical scheme for the front end
For a H+ front end, ECRIS are now the preferred
solution and allow to reach a few 10s to more than
100 mA with a good emittance (< 0.2π µrad).
Low Energy
A LEBT line is used to match the beam into the RFQ.
Transport
It can also be used to pulse the beam with a slow
and Space
Charge
chopper (r. t. of ∼ 100 ns) instead of pulsing the
R. Duperrier
source (r. t. of ∼ 2 ms). Monitoring diagnostics are
Front End
sometimes inserted (CCD cams, DCCT).
Ion source
The RFQ creates the bunch structure et
Theory
More electrodes
pre-accelerates the beam up to a few MeV.
Codes
LEBT
Electrostatic
Solenoids
sc neutralisation
Codes
RFQ
Basics
Beam dynamics
Current limits
Codes
Conclusions
Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 4 / 47
5. Outline
Main parts of a Front End
1
Low Energy
Transport
and Space
The ion source extraction system
2
Charge
R. Duperrier
Front End
The LEBT line
3
Ion source
Theory
More electrodes
Codes
The RFQs
4
LEBT
Electrostatic
Solenoids
sc neutralisation
Conclusions
Codes
5
RFQ
Basics
Beam dynamics
Current limits
Codes
Conclusions
Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 5 / 47
6. Theoretical basics
Let us consider first a diode
system extractor.
To model the flow of ions in
Low Energy
the system, L. & B. proposed
Transport
and Space
to solve the Poisson
Charge
equation in a system of
R. Duperrier
delimited by concentric
[Langmuir & Blodgett, Phys. Rev. 24]
Front End
spheres. For obvious
Ion source
pratical reasons, the
Theory
More electrodes
solution is reduced to a
Codes
finite solid angle.
LEBT
Electrostatic
The limit current is then:
Solenoids
sc neutralisation
1/2
Codes
ˆ = 8πε0 2q ∆V 3/2 1−cosθ
I
RFQ m
9 −α 2
Basics
Beam dynamics
with α a series of the
Current limits
[Schneider et al, PAC’07]
Codes
function log(rb /ra ).
Conclusions
Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 6 / 47
7. The current dependancy
It turns out that the geometry is linked to the
current and the ratio q/m for a given voltage.
To illustrate, for heavy ions, it has been proposed to
Low Energy
adjust the gap with a moveable electrode.
Transport
The minimum of the divergence is then a strong
and Space
Charge
function of q/m or I. This has to be integrated for
R. Duperrier
the current ramp up during the commissioning.
Front End
Ion source
Theory
More electrodes
Codes
LEBT
Electrostatic
Solenoids
sc neutralisation
Codes
RFQ
Basics
Beam dynamics
Current limits
Codes
Conclusions [Zaim & Alton, PAC’01]
Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 7 / 47
8. A hollow beam
Integrating spherical aberrations in the motion
equation leads to solve D.E. like:
d2r
q
= m G(z)r + G3 (z)r 3 + ...
dt 2
Low Energy
It turns out that extreme particles are more focused
Transport
and Space
and that a particular radius is more populated.
Charge
Considering non linearities in the LEBT line, this point
R. Duperrier
is more advantage than a drawback.
Front End
Ion source
Theory
More electrodes
Codes
LEBT
Electrostatic
Solenoids
sc neutralisation
Codes
RFQ
Basics
Beam dynamics
Current limits
Codes
Conclusions
[Batygin et al, PAC’95]
Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 8 / 47
9. A electron barrier electrode
[Sherman, PAC’07]
Low Energy
Transport
and Space
Charge
R. Duperrier
Front End
Ion source
Theory
More electrodes
Codes
LEBT
In diode system, LEBT electrons tend to go back up
Electrostatic
Solenoids
to the plasma electrode and may induce sparks
sc neutralisation
Codes
and then voltage breakdowns.
RFQ
Basics
This effect can be suppressed by adding one
Beam dynamics
Current limits
electrode which is negatively polarized and a
Codes
second one at the ground to create a barrier.
Conclusions
Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 9 / 47
10. A fifth electrode?
In case of non
moveable
Low Energy
Transport
electrodes, it is
and Space
Charge
also to tune the
R. Duperrier
extraction with a
fifth intermediate
Front End
Ion source
electrode.
Theory
More electrodes
This could help
Codes
for tuning several
LEBT
Electrostatic
currents or other
Solenoids
sc neutralisation
changing
Codes
conditions.
RFQ
Basics
[Delferrière et al, Rev. Sci. Instr. 79]
Beam dynamics
Current limits
Codes
Conclusions
Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 10 / 47
11. Benchmark with experiments
(adjustment of unknown parameters)
In order to adjust parameters like the initial ion
temperature in the simulation, benchmarks with
Low Energy
experiences are performed with a certain degree
Transport
of success...
and Space
Charge
R. Duperrier
Front End
Ion source
Theory
More electrodes
Codes
LEBT
Electrostatic
Solenoids
sc neutralisation
Codes
RFQ
Basics
Beam dynamics
Current limits
Codes
[Delferrière et al, Rev. Sci. Instr. 75]
Conclusions
Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 11 / 47
12. Optimisation with PIC codes
Several commercial codes (2D, 2.5D or 3D) can be
used for extraction system optimisation: PBGUN,
IGUN, AXCEL, SCALA (see below), KOBRA.
Low Energy
Transport
and Space
Charge
R. Duperrier
Front End
Ion source
Theory
More electrodes
Codes
LEBT
Electrostatic
Solenoids
sc neutralisation
Codes
RFQ
Basics
Beam dynamics
Current limits
Codes
Conclusions
Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 12 / 47
13. Outline
Main parts of a Front End
1
Low Energy
Transport
and Space
The ion source extraction system
2
Charge
R. Duperrier
Front End
The LEBT line
3
Ion source
Theory
More electrodes
Codes
The RFQs
4
LEBT
Electrostatic
Solenoids
sc neutralisation
Conclusions
Codes
5
RFQ
Basics
Beam dynamics
Current limits
Codes
Conclusions
Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 13 / 47
14. LEBT line based on electrostatic lenses
LEBT lines based on electrostatic einzel lenses
permit very compact systems which can be
combined with the source extraction.
Low Energy
By splitting the lenses and playing with the different
Transport
and Space
polarization, it is possible to provide beam steering
Charge
and fast chopping.
R. Duperrier
Front End
Such system operates
Ion source
at SNS for a H− peak
Theory
More electrodes
current of 35 mA.
Codes
LEBT
To compensate the
Electrostatic
Solenoids
effect of the electron
sc neutralisation
Codes
extractor (dipole), the
RFQ
source is tilted with
Basics
Beam dynamics
respect to the LEBT
Current limits
Codes
axis.
[Reijonen et al, LINAC’00]
Conclusions
Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 14 / 47
15. Limits
[Han & Stockli, PAC’07]
Low Energy
Transport
and Space
Charge
R. Duperrier
Front End
For such currents, the beam size is very closed to
Ion source
Theory
the lenses apertures, this induces emittance
More electrodes
Codes
growths due to the high order terms and beam
LEBT
losses (sparks).
Electrostatic
Solenoids
If the current is greater than 100 mA, there is a
sc neutralisation
Codes
consensus that this scheme is not suitable.
RFQ
For the SNS power upgrade, the peak current has
Basics
Beam dynamics
to be increased up to 59 mA. It is planned to use a
Current limits
Codes
magnetic focusing system (better acceptance, ...).
Conclusions
Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 15 / 47
16. Solenoids and space charge
The effects of S. C. and solenoid aberrations on the
beam have been investigated theoretically and
experimentally by Loschialpo et al in 1984.
Low Energy
Due to the combined action of the nonlinear lens
Transport
and the S.C., the initially uniform density becomes
and Space
Charge
hollow or peaked (depending on distance).
R. Duperrier
Front End
Ion source
To cure this effect,
Theory
More electrodes
long lenses [Bailey,
Codes
LEBT
EPAC’98] or
Electrostatic
compact line to
Solenoids
sc neutralisation
get a small beam
Codes
RFQ
size are usual
Basics
Beam dynamics
techniques.
Current limits
Codes
Conclusions
[Loschialpo et al, J. Appl. of Phys. 57]
Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 16 / 47
17. A plasma lens
Low Energy
Transport
and Space
Charge
R. Duperrier
Front End
Ion source
If we integrate the residual gas presence (ex.: H2 ) in the
Theory
More electrodes
vacuum chamber, we can get the production of pairs
Codes
electrons / ions (H+ ) via the ionization process:
LEBT
2
Electrostatic
Solenoids
sc neutralisation
Codes
p+H2 → p+e− +H+
2
RFQ
Basics
Beam dynamics
Current limits
we assume that χ=Nbeam / Ngas 1 with Nbeam the
Codes
beam density.
Conclusions
Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 17 / 47
18. Illustration
Example for a uniform beam of 100 mA @ 100 keV
Low Energy
Transport
800
and Space 14000
700
Charge 12000
champ
puits de 600
10000
électrique
potentiel 500
R. Duperrier 8000
E(V/m )
V(v)
400
6000
300
4000
200
Front End
100 2000
0
Ion source 0
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
r(m)
Theory r(m )
More electrodes
Codes
LEBT
The e− are trapped in the beam and the ions H+ are
Electrostatic
Solenoids
2
sc neutralisation
repelled to the pipe. An electrical neutralization is
Codes
obtained
RFQ
Basics
⇒ space charge compensation
Beam dynamics
Current limits
Codes
Conclusions
Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 18 / 47
19. Illustration
Example for a uniform beam of 100 mA @ 100 keV
Low Energy
Transport
800
and Space 14000
700
Charge 12000
champ
puits de 600
10000
électrique
potentiel 500
R. Duperrier 8000
E(V/m )
V(v)
400
6000
300
4000
200
Front End
100 2000
0
Ion source 0
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
r(m)
Theory r(m )
More electrodes
Codes
LEBT
The e− are trapped in the beam and the ions H+ are
Electrostatic
Solenoids
2
sc neutralisation
repelled to the pipe. An electrical neutralization is
Codes
obtained
RFQ
Basics
⇒ space charge compensation
Beam dynamics
Current limits
Codes
Conclusions
Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 18 / 47
20. Illustration
Example for a uniform beam of 100 mA @ 100 keV
Low Energy
Transport
800
and Space 14000
700
Charge 12000
champ
puits de 600
10000
électrique
potentiel 500
R. Duperrier 8000
E(V/m )
V(v)
400
6000
300
4000
200
Front End
100 2000
0
Ion source 0
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
r(m)
Theory r(m )
More electrodes
Codes
LEBT
The e− are trapped in the beam and the ions H+ are
Electrostatic
Solenoids
2
sc neutralisation
repelled to the pipe. An electrical neutralization is
Codes
obtained
RFQ
Basics
⇒ space charge compensation
Beam dynamics
Current limits
Codes
Conclusions
Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 18 / 47
21. Illustration
Example for a uniform beam of 100 mA @ 100 keV
Low Energy
Transport
800
and Space 14000
700
Charge 12000
champ
puits de 600
10000
électrique
potentiel 500
R. Duperrier 8000
E(V/m )
V(v)
400
6000
300
4000
200
Front End
100 2000
0
Ion source 0
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
r(m)
Theory r(m )
More electrodes
Codes
LEBT
The e− are trapped in the beam and the ions H+ are
Electrostatic
Solenoids
2
sc neutralisation
repelled to the pipe. An electrical neutralization is
Codes
obtained
RFQ
Basics
⇒ space charge compensation
Beam dynamics
Current limits
Codes
Conclusions
Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 18 / 47
22. Illustration
Example for a uniform beam of 100 mA @ 100 keV
Low Energy
Transport
800
and Space 14000
700
Charge 12000
champ
puits de 600
10000
électrique
potentiel 500
R. Duperrier 8000
E(V/m )
V(v)
400
6000
300
4000
200
Front End
100 2000
0
Ion source 0
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
r(m)
Theory r(m )
More electrodes
Codes
LEBT
The e− are trapped in the beam and the ions H+ are
Electrostatic
Solenoids
2
sc neutralisation
repelled to the pipe. An electrical neutralization is
Codes
obtained
RFQ
Basics
⇒ space charge compensation
Beam dynamics
Current limits
Codes
Conclusions
Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 18 / 47
23. Illustration
Example for a uniform beam of 100 mA @ 100 keV
Low Energy
Transport
800
and Space 14000
700
Charge 12000
champ
puits de 600
10000
électrique
potentiel 500
R. Duperrier 8000
E(V/m )
V(v)
400
6000
300
4000
200
Front End
100 2000
0
Ion source 0
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
r(m)
Theory r(m )
More electrodes
Codes
LEBT
The e− are trapped in the beam and the ions H+ are
Electrostatic
Solenoids
2
sc neutralisation
repelled to the pipe. An electrical neutralization is
Codes
obtained
RFQ
Basics
⇒ space charge compensation
Beam dynamics
Current limits
Codes
Conclusions
Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 18 / 47
24. Illustration
Example for a uniform beam of 100 mA @ 100 keV
Low Energy
Transport
800
and Space 14000
700
Charge 12000
champ
puits de 600
10000
électrique
potentiel 500
R. Duperrier 8000
E(V/m )
V(v)
400
6000
300
4000
200
Front End
100 2000
0
Ion source 0
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
r(m)
Theory r(m )
More electrodes
Codes
LEBT
The e− are trapped in the beam and the ions H+ are
Electrostatic
Solenoids
2
sc neutralisation
repelled to the pipe. An electrical neutralization is
Codes
obtained
RFQ
Basics
⇒ space charge compensation
Beam dynamics
Current limits
Codes
Conclusions
Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 18 / 47
25. Illustration
Example for a uniform beam of 100 mA @ 100 keV
Low Energy
Transport
800
and Space 14000
700
Charge 12000
champ
puits de 600
10000
électrique
potentiel 500
R. Duperrier 8000
E(V/m )
V(v)
400
6000
300
4000
200
Front End
100 2000
0
Ion source 0
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
r(m)
Theory r(m )
More electrodes
Codes
LEBT
The e− are trapped in the beam and the ions H+ are
Electrostatic
Solenoids
2
sc neutralisation
repelled to the pipe. An electrical neutralization is
Codes
obtained
RFQ
Basics
⇒ space charge compensation
Beam dynamics
Current limits
Codes
Conclusions
Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 18 / 47
26. Illustration
Example for a uniform beam of 100 mA @ 100 keV
Low Energy
Transport
800
and Space 14000
700
Charge 12000
champ
puits de 600
10000
électrique
potentiel 500
R. Duperrier 8000
E(V/m )
V(v)
400
6000
300
4000
200
Front End
100 2000
0
Ion source 0
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
r(m)
Theory r(m )
More electrodes
Codes
LEBT
The e− are trapped in the beam and the ions H+ are
Electrostatic
Solenoids
2
sc neutralisation
repelled to the pipe. An electrical neutralization is
Codes
obtained
RFQ
Basics
⇒ space charge compensation
Beam dynamics
Current limits
Codes
Conclusions
Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 18 / 47
27. Illustration
Example for a uniform beam of 100 mA @ 100 keV
Low Energy
Transport
800
and Space 14000
700
Charge 12000
champ
puits de 600
10000
électrique
potentiel 500
R. Duperrier 8000
E(V/m )
V(v)
400
6000
300
4000
200
Front End
100 2000
0
Ion source 0
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
r(m)
Theory r(m )
More electrodes
Codes
LEBT
The e− are trapped in the beam and the ions H+ are
Electrostatic
Solenoids
2
sc neutralisation
repelled to the pipe. An electrical neutralization is
Codes
obtained
RFQ
Basics
⇒ space charge compensation
Beam dynamics
Current limits
Codes
Conclusions
Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 18 / 47
28. Illustration
Example for a uniform beam of 100 mA @ 100 keV
Low Energy
Transport
800
and Space 14000
700
Charge 12000
champ
puits de 600
10000
électrique
potentiel 500
R. Duperrier 8000
E(V/m )
V(v)
400
6000
300
4000
200
Front End
100 2000
0
Ion source 0
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
r(m)
Theory r(m )
More electrodes
Codes
LEBT
The e− are trapped in the beam and the ions H+ are
Electrostatic
Solenoids
2
sc neutralisation
repelled to the pipe. An electrical neutralization is
Codes
obtained
RFQ
Basics
⇒ space charge compensation
Beam dynamics
Current limits
Codes
Conclusions
Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 18 / 47
29. Illustration
Example for a uniform beam of 100 mA @ 100 keV
Low Energy
Transport
800
and Space 14000
700
Charge 12000
champ
puits de 600
10000
électrique
potentiel 500
R. Duperrier 8000
E(V/m )
V(v)
400
6000
300
4000
200
Front End
100 2000
0
Ion source 0
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
r(m)
Theory r(m )
More electrodes
Codes
LEBT
The e− are trapped in the beam and the ions H+ are
Electrostatic
Solenoids
2
sc neutralisation
repelled to the pipe. An electrical neutralization is
Codes
obtained
RFQ
Basics
⇒ space charge compensation
Beam dynamics
Current limits
Codes
Conclusions
Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 18 / 47
30. Illustration
Example for a uniform beam of 100 mA @ 100 keV
Low Energy
Transport
800
and Space 14000
700
Charge 12000
champ
puits de 600
10000
électrique
potentiel 500
R. Duperrier 8000
E(V/m )
V(v)
400
6000
300
4000
200
Front End
100 2000
0
Ion source 0
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
r(m)
Theory r(m )
More electrodes
Codes
LEBT
The e− are trapped in the beam and the ions H+ are
Electrostatic
Solenoids
2
sc neutralisation
repelled to the pipe. An electrical neutralization is
Codes
obtained
RFQ
Basics
⇒ space charge compensation
Beam dynamics
Current limits
Codes
Conclusions
Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 18 / 47
31. Illustration
Example for a uniform beam of 100 mA @ 100 keV
Low Energy
Transport
800
and Space 14000
700
Charge 12000
champ
puits de 600
10000
électrique
potentiel 500
R. Duperrier 8000
E(V/m )
V(v)
400
6000
300
4000
200
Front End
100 2000
0
Ion source 0
0 0.02 0.04 0.06 0.08 0.1
0 0.02 0.04 0.06 0.08 0.1
r(m)
Theory r(m )
More electrodes
Codes
LEBT
The e− are trapped in the beam and the ions H+ are
Electrostatic
Solenoids
2
sc neutralisation
repelled to the pipe. An electrical neutralization is
Codes
obtained
RFQ
Basics
⇒ space charge compensation
Beam dynamics
Current limits
Codes
Conclusions
Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 18 / 47
32. Time scale
In a first approach, it can be estimated with the
classical formulation (DC beam):
1
τn = σ Ngas β c
Low Energy
with σ the ionization cross section, Ngas = P/kTroom
Transport
and Space
and β the beam reduced speed.
Charge
R. Duperrier
Evolution of neutralization degree, as a function of
time for a proton beam of 100 mA and 100 keV in a
Front End
drift (1,5D PIC code computations):
Ion source
Theory
More electrodes
Codes
Non linear
LEBT
transcient phase:
Electrostatic
Solenoids
ion inertia, Te− .
sc neutralisation
Codes
This rise time has
RFQ
Basics
to be evaluated
Beam dynamics
Current limits
for pulsed
Codes
operation. [Ben Ismail et al, Phys. Rev STAB 10]
Conclusions
Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 19 / 47
33. The ion slowness
Proton beam of 100 mA and 100 keV in a drift for
several pressures:
Low Energy
Transport
and Space
Charge
R. Duperrier
Front End
Ion source
Theory
More electrodes
Codes
LEBT
Electrostatic
Solenoids
sc neutralisation
Codes
Non linear transcient due to the ion inertia is
RFQ
Basics
non-existent at the beginning if P < 10−5 hPa.
Beam dynamics
Current limits
Codes
Conclusions
Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 20 / 47
34. The solenoid combined with the space
charge neutralisation
(H+ beam, 100 mA, 100 keV)
Low Energy
Transport
and Space
Charge
R. Duperrier
Front End
Ion source
Theory
More electrodes
Codes
LEBT
Electrostatic
Solenoids
sc neutralisation
Codes
Ion density Electron density
RFQ
Basics
Magnetic mirror at the edges.
Beam dynamics
Current limits
Transversal drift inside.
Codes
Conclusions
Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 21 / 47
35. Gas : nature and pressure (experiment)
Low Energy
Transport
and Space
Charge
R. Duperrier
Front End
Ion source
Theory
More electrodes
Codes
LEBT
Electrostatic
Solenoids
sc neutralisation
Codes
RFQ
Basics
[Gobin et al, Rev. Sci. Instr.,99]
Beam dynamics
Current limits
Codes
Conclusions
Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 22 / 47
36. Gas : nature and pressure (experiment)
Low Energy
Transport
and Space
Charge
R. Duperrier
Front End
Ion source
Theory
More electrodes
Codes
LEBT
Electrostatic
Solenoids
sc neutralisation
Codes
RFQ
Basics
Beam dynamics
[Gobin et al, Rev. Sci. Instr.,99]
Current limits
Codes
Conclusions
Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 22 / 47
37. Electrical field comparison
(PIC code computation)
12,50
Low Energy
10,00
Transport
and Space
Mass 100 @ 4e-4 hPa
7,50
Charge
Mass 100 @ 4e-5 hPa
Mass 4 @ 4e-4 hPa
5,00
R. Duperrier
Mass 4 @ 4e-5 hPa
2,50
Ex (kV/m)
Front End
0,00
Ion source
Theory
-2,50
More electrodes
Codes
-5,00
LEBT
-7,50
Electrostatic
Solenoids
-10,00
sc neutralisation
Codes
-12,50
RFQ
-80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80
Basics
R (mm)
Beam dynamics
Current limits
Codes
Conclusions
Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 23 / 47
38. Emittance evolution
(PIC code computation)
3
Low Energy
Transport 2,8
Masse 100 @ 4e-4 hPa
and Space Masse 100 @ 4e-5 hPa
Charge 2,6 Masse 4 @ 4e-4 hPa
Masse 4 @ 4e-5 hPa
R. Duperrier 2,4
Grossissement émittance
2,2
Front End
2
Ion source
Theory
1,8
More electrodes
Codes
1,6
LEBT
1,4
Electrostatic
Solenoids
1,2
sc neutralisation
Codes
1
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7
RFQ
z (m)
Basics
Beam dynamics
Current limits
Codes
Conclusions
Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 24 / 47
39. Gas : nature and pressure (conclusion)
Low Energy
Transport
and Space
Charge
R. Duperrier
Front End
Ion source
Theory
More electrodes
Codes
LEBT
Emittance enhancement with an increasing of the
Electrostatic
Solenoids
pairs production rate (pressure and/or cross
sc neutralisation
Codes
section).
RFQ
The enhancement with the heavy gas is due to a
Basics
Beam dynamics
cross section which is multiplied by a factor 5 and a
Current limits
Codes
greater mass.
Conclusions
Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 25 / 47
40. Gas : nature and pressure (conclusion)
Low Energy
Transport
and Space
Charge
R. Duperrier
Front End
Ion source
Theory
More electrodes
Codes
LEBT
Emittance enhancement with an increasing of the
Electrostatic
Solenoids
pairs production rate (pressure and/or cross
sc neutralisation
Codes
section).
RFQ
The enhancement with the heavy gas is due to a
Basics
Beam dynamics
cross section which is multiplied by a factor 5 and a
Current limits
Codes
greater mass.
Conclusions
Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 25 / 47
41. Gas : nature and pressure (conclusion)
Low Energy
Transport
and Space
Charge
R. Duperrier
Front End
Ion source
Theory
More electrodes
Codes
LEBT
Emittance enhancement with an increasing of the
Electrostatic
Solenoids
pairs production rate (pressure and/or cross
sc neutralisation
Codes
section).
RFQ
The enhancement with the heavy gas is due to a
Basics
Beam dynamics
cross section which is multiplied by a factor 5 and a
Current limits
Codes
greater mass.
Conclusions
Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 25 / 47
42. The recombination
Low Energy
Transport
and Space
Charge
R. Duperrier
Front End
Ion source
Theory
More electrodes
Codes
LEBT
Electrostatic
Solenoids
sc neutralisation
Codes
Above, the transmission of 2 m LEBT line with
RFQ
10−5 hPa of H2 and 4.10−5 hPa of Kr.
Basics
Beam dynamics
The choice is then a compromise between loss in
Current limits
Codes
the LEBT, loss in the RFQ and the rise time.
Conclusions
Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 26 / 47
43. The electron repeller
Let’s consider a dual
solenoid LEBT.
A computation of the
Low Energy
s. c. potential with
Transport
and Space
the correct boundary
Charge
conditions leads to
R. Duperrier
this steady state.
Front End
For beam tuning, a
Ion source
DCCT is located at
Theory
More electrodes
the RFQ entrance
Codes
LEBT
and it may be
Electrostatic
perturbed by a
Solenoids
sc neutralisation
electron flow.
Codes
RFQ
A cleaning electrode
Basics
Beam dynamics
may help too for
Current limits
Codes
beam tuning.
Conclusions
Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 27 / 47
44. The electron repeller
Let’s consider a dual
solenoid LEBT.
A computation of the
Low Energy
s. c. potential with
Transport
and Space
the correct boundary
Charge
conditions leads to
R. Duperrier
this steady state.
Front End
For beam tuning, a
Ion source
DCCT is located at
Theory
More electrodes
the RFQ entrance
Codes
LEBT
and it may be
Electrostatic
perturbed by a
Solenoids
sc neutralisation
electron flow.
Codes
RFQ
A cleaning electrode
Basics
Beam dynamics
may help too for
Current limits
Codes
beam tuning.
Conclusions
Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 27 / 47
45. The electron repeller
Let’s consider a dual
solenoid LEBT.
A computation of the
Low Energy
s. c. potential with
Transport
and Space
the correct boundary
Charge
conditions leads to
R. Duperrier
this steady state.
Front End
For beam tuning, a
Ion source
DCCT is located at
Theory
More electrodes
the RFQ entrance
Codes
LEBT
and it may be
Electrostatic
perturbed by a
Solenoids
sc neutralisation
electron flow.
Codes
RFQ
A cleaning electrode
Basics
Beam dynamics
may help too for
Current limits
Codes
beam tuning.
Conclusions
Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 27 / 47
46. The electron repeller
Let’s consider a dual
solenoid LEBT.
A computation of the
Low Energy
s. c. potential with
Transport
and Space
the correct boundary
Charge
conditions leads to
R. Duperrier
this steady state.
Front End
For beam tuning, a
Ion source
DCCT is located at
Theory
More electrodes
the RFQ entrance
Codes
LEBT
and it may be
Electrostatic
perturbed by a
Solenoids
sc neutralisation
electron flow.
Codes
RFQ
A cleaning electrode
Basics
Beam dynamics
may help too for
Current limits
Codes
beam tuning.
Conclusions
Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 27 / 47
47. Codes
The WARP code developed at Berkeley can be
used to simulate the sc neutralisation in a LEBT. This
code is also used for e-clouds modeling.
Low Energy
The SOLMAXP code developed at Saclay which is
Transport
based on a classical algorithm for modeling of
and Space
Charge
plasma coupled with a Maxwell solver permits such
R. Duperrier
simulations too.
Front End
Ion source
Theory
More electrodes
Codes
LEBT
Electrostatic
Solenoids
sc neutralisation
Codes
RFQ
Basics
Beam dynamics
Current limits
Codes
Conclusions
Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 28 / 47
48. Outline
Main parts of a Front End
1
Low Energy
Transport
and Space
The ion source extraction system
2
Charge
R. Duperrier
Front End
The LEBT line
3
Ion source
Theory
More electrodes
Codes
The RFQs
4
LEBT
Electrostatic
Solenoids
sc neutralisation
Conclusions
Codes
5
RFQ
Basics
Beam dynamics
Current limits
Codes
Conclusions
Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 29 / 47
49. A bit of history
In the early days, the injection of ions was
performed with high voltage systems which
typically produced continuous beam of ∼ 700 keV.
Low Energy
Transport
The bunch structure was
and Space
Charge
made with one or several
R. Duperrier
bunchers. The efficiency
was between 60 to 70 %
Front End
Ion source
(Beijing proton linac).
Theory
More electrodes
Codes
LEBT
Electrostatic
Solenoids
sc neutralisation
Codes
RFQ
Basics
Beam dynamics
Current limits
Codes
Conclusions
Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 30 / 47
50. The principle
RFQ was invented by Kapchinsky from ITEP in the
late 60s. Teplyakov of the same institute
constructed a first cavity.
Low Energy
Important contributions to the RFQ have also been
Transport
and Space
made by the LANL (POP in 1980). Since then, this
Charge
structure has become very popular.
R. Duperrier
Front End
Ion source
Theory
More electrodes
Codes
LEBT
Electrostatic
Solenoids
sc neutralisation
Codes
RFQ
The features of the RFQ are that it bunches,
Basics
Beam dynamics
focuses and accelerates charged particles by
Current limits
Codes
using RF fields only.
Conclusions
Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 31 / 47
51. The cavity
A TE210 mode is used.
The equivalent circuit
of a 4 vanes RFQ.
Low Energy
Typical view of a 4
Transport
and Space
vanes RFQ (TRASCO).
Charge
For a better stability,
R. Duperrier
quadrants may be
Front End
coupled (more RF
Ion source
power cons.).
Theory
More electrodes
Codes
For heavy ions
LEBT
machine, low
Electrostatic
Solenoids
frequencies are
sc neutralisation
Codes
required (a few 10s of
RFQ
MHz), inductance
Basics
Beam dynamics
based on stems are
Current limits
Codes
preferred (Tokyo RFQ).
Conclusions
Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 32 / 47
52. The cavity
A TE210 mode is used.
The equivalent circuit
of a 4 vanes RFQ.
Low Energy
Typical view of a 4
Transport
and Space
vanes RFQ (TRASCO).
Charge
For a better stability,
R. Duperrier
quadrants may be
Front End
coupled (more RF
Ion source
power cons.).
Theory
More electrodes
Codes
For heavy ions
LEBT
machine, low
Electrostatic
Solenoids
frequencies are
sc neutralisation
Codes
required (a few 10s of
RFQ
MHz), inductance
Basics
Beam dynamics
based on stems are
Current limits
Codes
preferred (Tokyo RFQ).
Conclusions
Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 32 / 47
53. The cavity
A TE210 mode is used.
The equivalent circuit
of a 4 vanes RFQ.
Low Energy
Typical view of a 4
Transport
and Space
vanes RFQ (TRASCO).
Charge
For a better stability,
R. Duperrier
quadrants may be
Front End
coupled (more RF
Ion source
power cons.).
Theory
More electrodes
Codes
For heavy ions
LEBT
machine, low
Electrostatic
Solenoids
frequencies are
sc neutralisation
Codes
required (a few 10s of
RFQ
MHz), inductance
Basics
Beam dynamics
based on stems are
Current limits
Codes
preferred (Tokyo RFQ).
Conclusions
Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 32 / 47
54. The cavity
A TE210 mode is used.
The equivalent circuit
of a 4 vanes RFQ.
Low Energy
Typical view of a 4
Transport
and Space
vanes RFQ (TRASCO).
Charge
For a better stability,
R. Duperrier
quadrants may be
Front End
coupled (more RF
Ion source
power cons.).
Theory
More electrodes
Codes
For heavy ions
LEBT
machine, low
Electrostatic
Solenoids
frequencies are
sc neutralisation
Codes
required (a few 10s of
RFQ
MHz), inductance
Basics
Beam dynamics
based on stems are
Current limits
Codes
preferred (Tokyo RFQ).
Conclusions
Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 32 / 47
55. 4 subsections
At the RFQ entrance, a short section which ramps
1
the field amplitude performs the transition static to
time focusing.
Low Energy
A delicate section called “gentle buncher”
2
Transport
bunches adiabatically the beam.
and Space
Charge
Once the bunch is made, it is accelerated by
3
R. Duperrier
decreasing the synchronous phase and ramping
Front End
the modulation factor. Sometimes, the voltage is
Ion source
also increased.
Theory
More electrodes
To help the matching in the MEBT line, the length of
4
Codes
the last part of the cavity (“Fringe Field Section”)
LEBT
Electrostatic
can be adjusted.
Solenoids
sc neutralisation
Codes
RFQ
Basics
Beam dynamics
Current limits
Codes
Conclusions
Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 33 / 47
56. The potential
A general solution of the
Laplace equation which
Low Energy
obeys to the RFQ
Transport
and Space
symmetries is detailed by
Charge
Weiss in CAS proceedings
R. Duperrier
[CAS 95-06].
Front End
Ion source
Theory
More electrodes
Codes
This solution contains all the harmonics in infinite
LEBT
series but only a few harmonics are necessary to
Electrostatic
Solenoids
well describe a real RFQ. To facilitate the analysis,
sc neutralisation
Codes
we shall consider a two terms potential:
RFQ
Basics
V
A01 r 2 cos2θ + A10 I0 (kr)cos(kz)
U(r, θ , z) =
Beam dynamics
2
Current limits
Codes
with k = 2π/β λ
Conclusions
Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 34 / 47
57. The transverse focusing
By linearizing the transverse field components
derived from the previous potential (small
amplitude), the equation of motion can be
Low Energy
Transport
simplified to the following form:
and Space
Charge
d2x
+ [Bsin2πτ + ∆rf ] x = 0
R. Duperrier
dτ 2
with 2πτ = ωt + φ and :
Front End
Ion source
qπ 2 |sinφs |A10 V
λ 2 qV
B= and ∆rf =
Theory
2 2mc 2 βs2
mc 2 R0
More electrodes
Codes
At first order, the solution of this Mathieu equation
LEBT
Electrostatic
is:
Solenoids
sc neutralisation
x(τ) = C0 ejσt τ (1 + Csin2πτ)
Codes
RFQ
with:
Basics
Beam dynamics
B2 B
σt2 ∼ and C∼
+ ∆rf
Current limits
8π 2 4π 2
Codes
Conclusions
Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 35 / 47
58. The longitudinal focusing
With the same technique (linearization for small
amplitude), Weiss show in the same reference that
Low Energy
the second order equation of the evolution of
Transport
and Space
∆φ = φ − φs can be written:
Charge
R. Duperrier
2 qA V |sinφ |
d d
βs2 dτ ∆φ + π s
∆φ = 0
10
dτ mc 2
Front End
Solving this oscillator D.E., one finds that the phase
Ion source
Theory
advance per period is:
More electrodes
Codes
1/2
LEBT
π 2 qA10 V |sinφs |
σl (τ) =
Electrostatic
mc 2 βs2
Solenoids
sc neutralisation
Let us note that:
Codes
RFQ
σl (τ) ∝ βs−1
Basics
Beam dynamics
Current limits
Codes
Conclusions
Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 36 / 47
59. Acceptances
In the same reference, it is given the expression of
the hamiltonian for large amplitude oscillations
from which it is extracted the limit of the separatrix:
Low Energy
Transport
∆Wmax = ± mc 2 βs2 qA10 V (φs cosφs − sinφs ) ∝ βs
and Space
Charge
It has to be noticed that adding the space charge
R. Duperrier
contribution will lead to a smaller acceptance but
Front End
also provide a smaller emittance!
Ion source
For the transverse plane, the mean beam size is
Theory
More electrodes
given by:
Codes
LEBT
εt,g βs λ
R beam =
Electrostatic
σt
Solenoids
sc neutralisation
Replacing by the expression for σt and setting
Codes
R beam = R0 , one finds for a pure RF quadrupole
RFQ
Basics
(A10 = 0):
Beam dynamics
Current limits
λ qV
Codes
ˆ = f (R0 )
εt,n = mc 2
Conclusions
Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 37 / 47
60. Space charge field
The space charge field for a uniformly charge
ellipsoidal bunch can be calculated analytically
[Lapostolle, CERN Report SG 65-15, 1965]:
Low Energy
Transport
and Space
Charge
R. Duperrier
3Iλ (1−f ) x
Esx = 4πε0 c(rx +ry )rz rx
Front End
Ion source
3Iλ (1−f ) y
Theory
Esy = 4πε0 c(rx +ry )rz ry
More electrodes
Codes
LEBT
3Iλ f z
Esz =
Electrostatic
4πε0 crx ry rz
Solenoids
sc neutralisation
Codes
with f(p) for p<1 and f(1/p)
RFQ
Basics
when p>1 and p = γrz /rx ry
Beam dynamics
Current limits
Codes
Conclusions
Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 38 / 47
61. Transverse current limit
Once these new field components are included in
Low Energy
the motion equation, it can be find that the new
Transport
transverse phase advance per focusing period is:
and Space
Charge
1/2
R. Duperrier
π 2 qA10 V |sinφs | 3qIλ 3 (1−f )
q2 λ 4 V 2
− −
σt = 4 2mc 2 βs2 4πε0 mc 3 (rx +ry )rx rz
m2 c 2 R0 8π 2
Front End
Ion source
Solving for σt = 0, one finds:
Theory
More electrodes
Codes 2 qA V |sinφ |
qλ V 2 4πε0 c(rx +ry )rx rz
ˆ= −π s
It 10
LEBT 4 2βs2 λ 3 3(1−f )
mc 2 R0 8π 2
Electrostatic
Solenoids
See Wangler’s book for a more detailed analysis
sc neutralisation
Codes
(Wiley series).
RFQ
Basics
Beam dynamics
Current limits
Codes
Conclusions
Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 39 / 47
62. Longitudinal current limit
The same approach allows to find the longitudinal
current limit expression:
ˆ = 8π 3 ε0 cA10 V |sinφs |rx ry rz
I l 3βs2 λ 3 f
Low Energy
Transport
If we use the approximation for f = 1/3p, assume
and Space
Charge
that:
R. Duperrier
rz ∼ 3|φs |β λ /4π
Front End
and maximize the transverse beam size rx/y ∼ R0 ,
Ion source
Theory
one can find:
More electrodes
Codes
ˆ = 3πε0 cA10 V |sinφs |φs2 R0
I
LEBT
l 2λ
Electrostatic
Solenoids
Usually, the longitudinal current limit is lower than
sc neutralisation
Codes
the transverse one. This induces that the
RFQ
bottleneck in a RFQ uses to be in the longitudinal
Basics
Beam dynamics
plane. But this bottleneck does not always occur
Current limits
Codes
at the end of the gentle buncher.
Conclusions
Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 40 / 47
63. Current Limit for several RFQs
To illustrate this
point, here a
graph which
shows the
Low Energy
Transport
longitudinal
and Space
Charge
current limit
R. Duperrier
normalized by
the design
Front End
peak current
Ion source
Theory
for several
More electrodes
Codes
RFQs.
LEBT
Electrostatic
RFQs with a constant voltage give a minimum for ˆl I
Solenoids
sc neutralisation
at the end of the acceleration section.
Codes
RFQ
The reduction of the product A10 V |sinφs |φs2 is not
Basics
Beam dynamics
sufficiently damped by increasing the modulation
Current limits
Codes
factor when the voltage is kept constant.
Conclusions
Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 41 / 47
64. Safety factors for several RFQs
Below, a report of these minimums (values before
the gentle buncher end are ignored).
Low Energy
Transport
and Space
Charge
R. Duperrier
Front End
Ion source
Theory
More electrodes
Codes
LEBT
Electrostatic
Solenoids
sc neutralisation
Codes
It has to be emphasized that the current limit is not
RFQ
the only figure of merit in a RFQ (RF power, length,
Basics
Beam dynamics
cost, ...) and the requirements for the beam loss
Current limits
Codes
tolerance are a strong function of the duty cycle.
Conclusions
Low Energy Transport and Space Charge March 17, 2009- Bilbao workshop 42 / 47