Nested Penning Trap as a Source of Singly Charged Ions
C. A. Ordonez
Department of Physics, University of North Texas, Denton, Texas 76203
Abstract. In the work reported, the possibility of using a nested Penning trap as a high purity source of low-charge-state ions
is studied. For the configuration considered, a relatively dense ion plasma is confined by a three-dimensional electric potential
well. The three-dimensional well is produced by the electric field generated by both the trap electrodes and a trapped electron
plasma. The ion and electron plasmas are each considered to have Maxwellian velocity distributions. However, it is shown
that the electron plasma must have a temperature that is higher than that of the ion plasma when the ions have low charge
states. The work reported includes a self-consistent prediction of a possible plasma equilibrium.
In various types of ion sources and traps, threedimensional electric potential wells are produced where
ions can be temporarily confined. For example, an electron beam ion trap/source produces a three-dimensional
electric potential well using a combination of the electric field generated by the trap electrodes and the electric
field generated by an electron beam [1, 2]. The inertial
electrostatic confinement approach to ion trapping has
been experimentally shown to be capable of producing a
three-dimensional electric potential well using the space
charge of a non-Maxwellian electron plasma [3]. There
also exists an ion confinement approach, referred to as
Penning fusion, in which a non-Maxwellian electron
plasma is confined within a Penning trap and produces
a three-dimensional electric potential well [4]. Here, a
theoretical study is presented on a “nested” Penning trap
configuration, which may also serve for temporarily trapping ions within a three-dimensional electric potential
well. The configuration incorporates a Maxwellian electron plasma for producing the electric potential well. In
principle, this aspect of the configuration should correspond to a reduced power requirement.
A Penning trap is typically used to confine charged
particles having the same charge sign that form a nonneutral plasma [5, 6]. Within a Penning trap, an electric
potential well is produced along a magnetic field. The
magnetic field provides plasma confinement in the two
dimensions perpendicular to itself, while the electric field
provides plasma confinement in the third dimension. The
plasma in a Penning trap can be kept away from surrounding solid structures for an indefinite amount of time
using the “rotating electric field” technique [7, 8]. Consequently, production of ultra-pure ion plasmas is possible in Penning traps because ion-solid interactions can
be minimal. When long-term plasma confinement is desired, Penning traps might be considered the best avail-
able plasma confinement approach. Similarly, when an
ultra-pure source of ions is desired, Penning traps might
be the best source. However, the plasma density can
be severely limited in Penning traps. The density limit,
which is named after L. Brillouin [9], is given (in SI
units) by nB = ε0 B2 /(2m), where B is the magnetic field
strength, m is the plasma particle mass, and ε0 is the permittivity of free space. Note the inverse mass dependence
of the Brillouin density limit. Even for a plasma of protons, the Brillouin density limit is exceedingly small. For
example, the Brillouin density limit for protons in a 0.3
T magnetic field is 2.4 × 1014 m−3 .
Plasma confinement in a nested Penning trap involves using nested, oppositely signed, electric potential
wells for confining oppositely signed overlapping plasma
species along a magnetic field. Much of the ongoing experimental [10, 11, 12, 13] and theoretical [14, 15] research on nested Penning traps is associated with experiments at CERN that are aimed at achieving overlap
of positron and antiproton plasmas such that recombination takes place, and trapping of antihydrogen atoms is
possible. Although the nested Penning trap configuration
was hypothesized some time ago [16], self-consistent
predictions of possible overlap equilibria have only recently been reported [14, 17, 18, 19]. For the equilibria
predicted in Refs. [14, 17, 18], an applied electric field
confined both plasma species along a uniform magnetic
field, which provided radial confinement of each species.
The overlap region was neutral, and radial ion losses
would be expected as a result of radial plasma diffusion. In Ref. [19], the predicted equilibrium consisted of
an overlap region with an overall negative space charge.
Three-dimensional electric confinement of ions having
moderately high charge states was found to be possible
as a result of the presence of a three-dimensional electric potential well created by the region of negative space
CP680, Application of Accelerators in Research and Industry: 17th Int'l. Conference, edited by J. L. Duggan and I. L. Morgan
© 2003 American Institute of Physics 0-7354-0149-7/03/$20.00
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FIGURE 1. A cross section of the electrode configuration
of a nested Penning trap (a), and an example of the electric
potential along the axis of the trap (b). A uniform magnetic
field parallel to the z axis provides radial confinement of the
electrons.
charge together with the applied electric field.
The work reported here is a theoretical study in support of considering the plasma equilibrium identified in
Ref. [19] for possible applications. Possible applications
as an ion trap or ion source may open up if an ion plasma
can be generated with a density that exceeds the Brillouin ion density limit and with sufficient purity. The
configuration considered here might be referred to as an
“electron plasma ion trap/source” because of the similarity that the configuration has with the electron beam
ion trap/source [1, 2]. The primary difference is that a
trapped electron plasma replaces an electron beam. The
configuration is illustrated in Fig. 1. The electrode placement, which is shown in Fig. 1(a), is both cylindrically
symmetric and symmetric about a midplane. A cylindrical coordinate system with coordinates (r, z) is defined
such that the z axis is aligned with five cylindrical electrodes, which are placed end to end. The electrode placement midplane is located at z = 0, and a uniform magnetic field is present that is parallel to the z axis. The center electrode has a length L0 = 2z1 and is defined to have
an electric potential equal to zero, V0 = 0. Each electrode on either side of the center electrode has a length
L2 = z3 − z1 . For ion confinement, the two electrodes of
length L2 have positive applied potentials, V2l = V2r > 0.
Axial ion release in one direction is always possible by
switching either V2l or V2r to zero. Each outer electrode
has a length L4 = zw −z3 and a negative applied potential,
V4 < 0. All five electrodes have the same radius, rw .
An electron plasma and an ion plasma are illustrated
as being confined within the cylindrical electrodes in
Fig. 1(a). The electric potential along the z axis, which
would be responsible for electron and ion confinement,
is illustrated in Fig. 1(b). The electric potential forms
a well, referred to as the “inner well,” which is capable
of providing axial ion confinement. To either side of the
inner well are “end wells,” which are inverted with respected to the inner well. Thus, axial electron confinement is possible within each end well. The left end well
has a minimum axial depth of magnitude ∆φl , while the
right end well has a minimum axial depth of magnitude
∆φr . The magnitude of the minimum axial depth of the
inner well, denoted ∆φm , equals the smaller of ∆φl and
∆φr . It is also useful to define an “outer well,” which encompasses both end wells and the inner well. It can be
hypothesized that for an electron plasma with a temperature T− ≈ e∆φm , where e is the unit charge and temperature is in energy units, a large fraction of the electrons will not be trapped within one or both end wells.
Using a self-consistent computation, it is predicted that
such an electron plasma can be confined by the outer
well, while overlapping the inner well. In addition, the
predicted plasma equilibrium indicates that a three dimensional electric potential well of depth ∆φm can be
produced. In consideration of this, it can also be hypothesized that good ion confinement is possible in three dimensions within the inner well provided the condition
T+ Ze∆φm is met, where T+ is the ion temperature,
and Z is the ion charge state. A theory is presented in
an accompanying paper, which supports this hypothesis
[20]. The two conditions, T− ≈ e∆φm and T+ Ze∆φm ,
can be met simultaneously provided T− T+ /Z.
In Ref. [19], the electrons and ions were considered to
have the same temperature (T− = T+ = 3000 eV), and
the condition T− T+ /Z was met as a result of the
charge states of the ions (hZi = 17.2). The emphasis in
the present study is on considering low-charge-state ions,
with all ions approximated as having the same charge
state Z. For discussion purposes, it is assumed that the
condition T− T+ /Z can only be met by having an
electron temperature that is larger than that of the ions.
Thus, electron-ion collisions, which tend to cause the two
temperatures to approach the same value, will limit the
amount of time the condition T− T+ /Z will be met. It
is also assumed that the electron temperature and density
are kept temporally constant while the ions are confined
(e.g., as a result of external heating and use of the rotating
electric field technique).
Using a finite-difference method with simultaneous
over-relaxation [14, 21], a possible equilibrium has been
self-consistently computed. Singly charged ions with a
temperature T+ = 300 eV and a density ns+ = 1016 m−3
at the geometric center of the trap [at (r, z) = (0, 0)]
are assumed. The equilibrium obtained is essentially the
same for a lower temperature ion plasma. However, the
computation time increases with decreasing plasma temperature because the Debye length and associated grid
spacing decrease. The ions are overlapped by an electron
plasma with a temperature T− = 3 keV and a density
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of ns− = 2.4 × 1017 m−3 at (0, z2 ), where z2 is located
halfway between z1 and z3 . The central electrode has a
length L0 = 1.5 cm and a potential V0 = 0. Each electrode on either side of the central electrode has a length
L2 = 2 cm and a potential V2l = V2r = 26.4 kV. The outermost electrodes each have a length L4 = 0.5 cm and a
potential V4 = −30 kV. Longer electrode lengths would
not be expected to change the results aside from increasing the axial plasma dimensions. All electrodes have the
same inner wall radius rw = 0.5 cm.
For the equilibrium that is computed, the electron
plasma is assumed to follow the Boltzmann density
relation axially (i.e., along each magnetic field line),
n− (r, z) = n− (r, z2 )ee[φ (r,z)−φ (r,z2 )]/T− . Considering a
magnetic field of 0.3 T, the electrons are magnetically
confined, having a cyclotron radius of 0.4 mm. In contrast, the ion cyclotron radius is 0.6 cm for protons, and
it would not be possible to magnetically confine the ions
within the 0.5 cm radius trap electrodes. Hence, for computing the equilibrium, the ions are assumed to follow the
Boltzmann density relation in three dimensions within
the inner well, n+ (r, z) = n+ (0, 0)e−Ze[φ (r,z)−φ (0,0)]/T+ ,
where n+ (0, 0) = ns+ . The density along the z axis is
plotted for each plasma species in Fig. 2(a), while the
density along the midplane is plotted for each plasma
species in Fig. 2(b). The plots in Fig. 2 show clearly that
the electron plasma overlaps the inner well. For a 0.3 T
magnetic field, the ion density exceeds the Brillouin ion
density limit by a factor of forty two at (0, 0).
The Brillouin density limit occurs in single-well Penning traps because the magnetic field must apply a radially inward force on the confined nonneutral plasma
to balance both self-electric and centrifugal radially outward forces. Consequently, the Brillouin density limit
does not directly apply to an ion plasma in a nested Penning trap if the ion plasma is confined radially by an
electric field produced by an electron plasma. However,
within each end well, where the plasma is completely
unneutralized, the electron density will be limited either
by the Brillouin limit, which is at least a factor of 1836
larger for electrons than for ions, or by some other density limit. Various mechanisms limit the density of the
nonneutral electron plasma in each end well, thereby indirectly limiting the density of the ion plasma. The most
restrictive density limit appears to be associated with the
size of the applied voltage difference that is required
for providing axial confinement of the electron plasma
within the outer well [22].
The axial electric potential profile shown in Fig. 1(b)
is actually the self-consistently computed potential along
the axis of the trap. Debye shielding causes the selfconsistent value for ∆φm , the depth of the inner well
along the axis, to be reduced to a much smaller value than
the vacuum value. Without the plasma, the axial depth of
the inner well is 25 kV, while with the plasma present,
FIGURE 2. The electron and ion densities along the z axis
(a) and along the midplane (b). Values are in SI units.
the axial depth of the inner well is 3.1 kV. Overlap of
the inner well by the electron plasma causes there to be
a radially inward electric field. The value for V2l and
V2r was chosen such that the radial well depth would
equal the axial well depth. This is illustrated in Fig. 3,
where contours of constant potential are plotted on the
x-z plane. The three-dimensional electric potential well
has a total depth of 3.1 kV.
For a plasma trapped in a potential energy well, if the
well depth is much larger than the plasma temperature
then the loss regions in velocity space become negligibly
small, and the plasma can be reasonably approximated
as following a Maxwell-Boltzmann phase-space distribution (see, for example, Eq. (7) of Ref. [18] and the
remarks thereafter). For the equilibrium reported here,
the potential energy well depth is about an order of magnitude larger than the plasma temperature both axially
for the electrons and in three dimensions for the ions.
This is consistent with using the Boltzmann density relation axially for the electrons and axially and radially
for the ions and signifies that both plasma species will
have essentially full Maxwellian velocity distributions.
Consequently, neither plasma species should be affected
by the instabilities that arise in magnetic mirrors from
the presence of loss cones. However, although the ions
do not experience cyclotron orbits and should have no
free energy for instabilities themselves, the presence of
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ACKNOWLEDGMENTS
The author would like to thank Dr. D. D. Dolliver for
early contributions to this material. This material is based
upon work supported by the National Science Foundation under Grant No. PHY-0099617 and the Texas Advanced Research Program under Grant No. 3594-00032001.
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FIGURE 3. Inner-well contours of constant potential. Shown
are contour lines for a cross section of constant-potential surfaces of revolution. The diameter of the outer contour is just
less than the inner diameter of the electrodes (1 cm). The axial
length of the outer contour is approximately 2 cm. The difference in potential between adjacent contours is 300 V. The inner
contour represents a 300 V increase in potential with respect
to the bottom of the inner well at the geometric center, while
the outer contour represents a 3000 V increase in potential with
respect to the bottom of the inner well.
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