Mathematical Formulation and Numerical Modelling of the
Extraction of H
-
ions
Reinard Becker
Institut fr Angewandte Physik der Johann Wolfgang Goethe Universitt, Fach 180
D-60054 Frankfurt/M, Germany
ABSTRACT
In the past, ion extraction of volume produced H
-
ions has either been simulated with programs for the extraction
of positive ions [1] or by programs, which wrongly claime to have a real and genuine option for H
-
ions [2].
Although reasonable results have been obtained in both ways, the mathematical formulation of the physics at
the plasma sheath is wrong, and the modelling of electrodes near the sheath [3] must fail. In this paper a self
consistent formulation of the extraction problem for H
-
ions is presented, which takes into account any number of
positive ions, like fast or thermal protons, thermal cesium and molecular ions like H
2
and H
3
, which all are
essential for the generation of H
-
ions in the plasma volume. Equally important is the porting of this formulation
to a simulation program and the verification of experimental results. This has lead to the development of the
program nIGUN
.
INTRODUCTION
The present theory is the result of several steps of development: In 1997 the virtual cathode
behaviour of protons, entering the plasma sheath and being reflected by the field for H
-
extraction has been pointed out [4], however, without considering the role of thermal positive
ions, like molecules and cesium ions in the case of cesium seeding. This feature has been
added in 2002 [5] for one kind of thermal ions. In 2003 the attempt has been made, to include
the whole sheath from the plasma potential to the extraction in one theory [6]. This, however,
is too comprehensive, because the important effect of changing the electron to H
-
fraction by a
bias of the plasma electrode [7] cannot be included in a linear sheath model. The present
formulation therefore is an extension of the 2002 theory, allowing for any number and kind of
positive ions in front of the plasma electrode, like protons coming from the plasma potential
and protons, molecules, and cesium ions with thermal energy, being created in the neutralized
vicinity of the plasma electrode. Since H
-
are assumed to be thermal there, it will not matter, if
these will be produced by surface or by volume processes, as long as their birth potential is
the same. The new formulation only considers the transition from the plasma in front of the
plasma electrode to the beam region. Other work on the extraction of H
-
ions is reviewed in
ref. [6]. PBGUNS [2] uses wrong expressions for the electron space charge and ignores fast
positive ions as well as additional thermal ones, like cesium and molecules.
THE INVERTED SHEATH
In contrast to positive ion extraction, where the potential fall in the sheath is continued by the
acceleration field of the extraction potential [8], ion sources for H
-
production are more
complex: The natural potential fall of the plasma sheath needs to be reversed for the
extraction of negative ions (and electrons). In general this is provided by a transverse
magnetic field (filter) in front of the plasma electrode, which forces fast electrons to follow
the flux lines, while slow ones may have enough collisions to move across them by ambipolar
diffusion. This localized changes of the electron velocity distribution and of the electron
density will cause a fall of potential towards the plasma electrode, favouring the migration of
H
-
ions into this region [9]. According to the axial potential model shown in Fig. 1, we can
formulate the space charge term for each kind of particle:
Figure 1: Axial potential model for the definition of space charges in the extraction region
the density of electrons with energy U
e
will be reduced by acceleration in the same way as
the H
-
density:
the density of fast protons (and other fast ions) is dieing out by the virtual cathode process [4]:
other positive ions are considered to be thermal and trapped between the extraction field and
the plasma, hence will obey a Boltzmann distribution:
This problem now has 2 unknowns for each charged particle, the density and the energy
(either directed or thermal). While the energies must be known to obtain solutions, the
densities can be expressed by the definition for the electron to H
-
-current ()
and by the condition of quasineutrality at U=0
We then calculate from the balance of charged particle currents at the plasma electrode with
wall potential U
w
the relation of the density of the first kind of positive thermal ions to the
density of H
-
ions:
+
=
+
=
U
U
n
U n
U
U
n
U n
e
e
e
1
) 0 (
) ( ,
1
) 0 (
) (
|
|
.
|
\
|
=
P
P P
U
U
n U n 1 ) 0 ( ) (
)
`
=
i
i i
U
U
n U n exp ) 0 ( ) (
e
e e
U M
U m
n
n
|
|
.
|
\
|
(
|
|
.
|
\
|
|
|
.
|
\
|
+ +
=
U M
U M
U
U
U
U
U M
U M
n
n
U M
U M
U
U
U M
U m
n
n
P
P
P
w
i
w
i
i i
P
P
P
w
e
e
2
3
1
2
3
1
1 exp
1 1 1
+ =
1
1
1
n
n
n
n
n
n
n
n
i e P
(1)
(2)
(3)
(4)
(5)
(6)
PARAMETER RELATIONS OF SOLUTIONS
By solving eq. 6 for assumed values of the directed or thermal ion energies and for choosing
M
1
=1 or =133, relations will be obtained for the parameters of solutions either without or
with cesium seeding. The assumption of a negative wall potential, as used in a former
presentation [5] has been dropped, because the potential model (Fig. 1) and the associated
space charge terms (eq. 1-3) then will break down. In Fig. 2, the relation of parameters is
shown with cesium seeding for a directed proton energy of 10 times the H
-
, cesium, and
electron temperature. A positive wall potential always will reduce the electron to H
-
ratio, as
observed in experiments, and the cesium ion density always must be lower than the H
-
density.
Figure 2: Relation of parameters for solutions of the inverted sheath with cesium seeding
Figure 3: Relation of parameters for solutions of the inverted sheath without cesium seeding
IMPLEMENTATION INTO nIGUN
The 2D simulation program for positive ion extraction, IGUN
[8] has been modified to
simulate negative ion extraction on the basis of the theory presented in this paper and given
the name nIGUN
. The evaluation of eq. 6 is part of the input procedure. For the "real"
solution, however, the densities of the different particles must be known absolutely. This is
achieved by the integration of Poisson's equation with the space charge terms eq. 1-3,
resulting in eq. 7:
0.0 0.2 0.4 0.6 0.8 1.0
0
10
20
30
40
50
60
70
16
20
25
13
10
1
1.3
1.6
2
2.5 3.2
4
5.0
6.3
7.9
U
-
, U
1
, U
e
=1
U
p
=100, M
1
=1
Parameter:
U
w
n
1
/n
-
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
0
1
2
3
4
5
1
1.3
1.6
2
2.5
3.2
4
5.0
U
-
, U
1
, U
e
=1
U
p
=10, M
1
=133
Parameter:
U
w
n
1
/n
-
For any combination of the potential and the field strength in the extraction region the
absolute H
-
density can be determined from eq. 7. The other densities are following then from
eq. 4-6. It has been found that this determination of densities is done reasonably at U=U
p,
where the space charge of fast positive ions has disappeared. As a result, the densities of a
cesium seeded plasma are shown in fig. 4 from the quasineutral plasma (on the left) towards
the extraction region (right side) together with the increase of the potential function for a
planar diode of 20 mesh thickness.
Figure 4: Self-consistent densities of electrons, H
-
ions, fast protons and thermal cesium ions in an inverted
plasma sheath according to the presented theory and its implementation into nIGUN
CONCLUSIONS
The presented theory for H
-
extraction is self-consistent and can take into account any number
of fast and of thermal positive ions. The axial potential function starts with vanishing field
strength at the potential of the wall electrode (if there is no bias on the PE). This model
accepts birth of H
-
ions in the volume around the extraction aperture as well as on the wall
electrode. In agreement with measurements, the proton energy is much lower with added
cesium than without. The mathematical formulations have been implemented into a new
program, called nIGUN
[10], for the simulation of the extraction of H
-
ions.
REFERENCES
[1] Leitner, M.A., D.C.Wutte, K.N.Leung, Nucl. Instrum. and Meth., A 427, 242 (1999)
[2] Boers, J.E., http://thunderbirdsimulations.com/
[3] Welton, R.F. et al., Rev. Sci. Instrum. 73, 1013 (2002)
[4] Becker, R., K.N.Leung, W.Kunkel, Rev. Sci. Instrum., 69, 1107 (1998)
[5] Becker, R., AIP conference Proceedings 639, 82 (2002)
[6] Becker, R., Rev. Sci. Instrum. 75, 1687 (2003)
[7] Leung, K.N., S.R. Walther, and W.B. Kunkel, Phys. Rev. Lett. 62,7, 764 (1989)
[8] Becker, R., W.B.Herrmannsfeldt, Rev. Sci.Instrum. 63, 2756 (1993)
[9] Bacal, M., J.Bruneteau, P.Devynck, Rev. Sci. Instrum., 59,10, 2152 (1988)
[10] http://www.egun-igun.com
|
|
.
|
\
|
+
|
|
.
|
\
|
(
+ +
(
+
=
1 exp
2
1 1 2 1 1 2
2
1
1
2
2
i
i
i
P
P
e
e
e
o
U
U
U
n
n
n
n
U
U
U
n
n
U
U
U
U
U
U
n
n
n
e
U
0 5 10 15 20
-0.6
-0.4
-0.2
0.0
0.2
0.4
0
2
4
6
8
10
right scale
V
sum of all
electrons
H
-
ions
fast protons
thermal Cs ions
particle
densities
Z (mesh units)
U(Z)
(7)
