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3. Beam extraction
3. Extraction of particle beams
3.1 The space charge limit and Child-Langmuirs law
3.2 External and internal fields in the extractor, laminar flow and
pierce angle
3.3 The beam emittance, the acceptance of the extraction
system and the conservation of phase space
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The extraction of particle beams
The plasma generator and the extraction systems are defining :
• The extracted beam current
• The quality of the particle beam (emittance)
Space charge forces in the extraction system and plasma properties of
the source limit the beam current.
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The space charge limit and Child-Langmuir law
anode
cathode
+Ze
0
(z=0)=0
d
z
(z=d)=-V0
1 2
mvz    e   ( z )
2
 ( z)
2
(Poisson)
 ( z )  
0
J  v    const .
 d 2 
 2   
 dz 

0
J
2e
m
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The space charge limit and Child-Langmuir law
0
d
0
z
space charge limited
J SC
z
 ( z )  V0  
d 
low current
no current
-V0
(z)
4
2e ( z ) 3 2
 0
9
m
d2
4
3
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Current limits given by the Child-Langmuir law
The total current extractable from an ion
source is given by :
100
electrons
2
j0*d (A)
10
•The area covered by the extraction
aperture (~ d2)
protons
•Extraction voltage (U3/2)
1
•Mass of particles (1/m)1/2
C+
•Charge state ()1/2
U2+
0.1
U
•The distance between the electrodes (d2)
+
0.01
1
10
100
V0(kV)
1000
Under the assumption that the particle
source is able to produce this current. For
electron sources this is usually valid, for
ion sources in general not !
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The Pierce method for the design of the extraction electrodes 1
cathode
anode
x
 ( x, y , z )  z 
 
V0
d 
unbalanced
space charge
forces
0
(z=0)=0
d
z
(z=d)=V0

0
x
4
3
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The Pierce method for the design of the extraction electrodes 2
cathode
anode
x
u  z  jx
with
j  1
2 f 2 f
 2 0
2
z
x
u  z  jx
Individually biased
electrodes to fulfil
border condition
0
(z=0)=0
d
z
(z=d)=V0
2
2 f

f
2
(j ) 2 0
2
u
u
  Re( f ) without space charge (   0)
u
f (u )  V0  
d 
4
3
 z  jx 
 V0 

 d 
4
3
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The Pierce method for the design of the extraction electrodes 3
cathode
anode
x
 z  jx 
 ( x, z )  V0  Re 

 d 
z   cos  ,
4
3
x   cos 
e j  cos   j sin 
0
(z=0)=0
d
z
(z=d)=V0

  j 
 Re  e 
V0
d

4
3
4
4 j


   Re  e 3
d 

3




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The Pierce method for the design of the extraction electrodes 4
cathode
  x2  z 2
anode
 x
  tan  
z
1  0
x
1
67.5
for
x0
4 
    67.50
3
2
and
 2  V0
4
0
(z=0)=0
d
(z=d)=V0
z
 3
 4 
  cos   1
d 
 3 
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Beam extraction, high voltage break down limit and aspect ratio
2qe U 3 2
4
J SC   0
m d2
9
U  E d
2qe E 2
4
J SC   0
m U
9
r
S
d
2qe S 3 2
4
E
J SC   0
m r
9
J SC
k*
Break down law:
S
maximum current density
for aspect ratio of :
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Phase space distribution of a particle beam
6 N
H (r1.......rN , p1....... p N , t )
H
H
  p i ,
 qi
qi
pri



r   qi ei   pi ei
i
i
i
i



v   qi ei   p i ei



divv  
qi  
p i
i qi
i pi

 H
  H
divv  

i qi pi
i pi qi

2H
2H
divv  

 0  0
qi pi
i qi pi
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The beam emittance, the phase space ellipse and the twist parameters
x
envelope
x’



 

z




single trajectory
E ( z )   ( z )
F=*
   ( z ) x 2 ( z )  2 ( z) x( z ) x' ( z )   ( z ) x'2 ( z )


 
x
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Phase space distribution and beam emittance
 RMS  x 2  x 2  x x 2
 x


i
x
2
i
i
i
i
 n  
 KV   RMS
 eff
, x
   x
   x x

, xx  


i
2
i
i
2
i
i
i
i
i i
i
i
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Conservation of phase space
x’
Drift
x’
x
x’
focusing
x’
x
x
x
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Influence of emittance on focal spot size
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plasma sheath
plasma
plasmagenerator extraction system
space charge
Ion beam extraction from a plasma
The extractable current from an ion source is
limited by :
•Space charge forces in the extraction region
•Plasma density in the source
•Production speed of ions in the plasma
•Diffusion speed of ions from the plasma into
the plasma sheath
Plasma sheath : While within the plasma the
charges neutralize each other, the plasma
itself is biased in respect to the walls to keep
an equilibrium of losses between the fast
electrons and slow ions. A thin boarder area
(plasma sheath) separates the plasma from
the outside by an electric field.
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Influence of particle density distribution at the extraction aperture on
initial phase space distribution
plasma
transversal current density profile
phase space
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Plasma density distribution at extraction aperture
Experimental result
Experimental set up
analysis
Extraction
aperture
window
lens system
aquisition
In praxi the particle density at the extraction aperture is not homogeneous. This
will lead to non linear space charge forces within the beam transport.
Redistributions of the beam particles within the beam cause growth of the
effective (RMS) beam emittance.
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The temperature of the source plasma, the potential depression in the
plasma sheath and wall effects (losses of particles) are influencing the
beam emittance.
resolution
The transversal (and longitudinal)
energy distribution of the beam ions is
defined by the plasma temperature and
the plasma potential (electric field in
the plasma sheath)
Losses of beam ions at the extraction
electrode further reduces the number of ions
to be extracted.
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Real extraction systems and the determination of beam formation
edge particles
Initial angle
radial lens
radial space charge
longitudinal
focus
focus without space charge
For real existing ion sources usually more than two extraction electrodes (here
triode) are used to maximize beam current and to influence the beam
emittance. To calculate beam formation self consistently numerical simulation
codes like EGUN, IGUN, PBGUN or Cobra are used.
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Influence of beam current and aspect ration on beam emittance
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Numerical simulation of H- extraction and transport in the LEBT for SNS using
PBGUN and comparison with measured data
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Numerical simulation of the extraction of a D+ beam for IFMIF using IGUN
and comparison with measured data
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