WDS'08 Proceedings of Contributed Papers, Part II, 25–30, 2008.
ISBN 978-80-7378-066-1 © MATFYZPRESS
Electron-Ion Recombination at 10 K
M. Jílek, T. Kotrík, R. Plašil, I. Korolov, M. Hejduk, J. Glosík
Charles University Prague, Faculty of Mathematics and Physics, Prague, Czech Republic.
Abstract. This work is focused on recombination processes in low temperature plasma
down to 10 K. When reaching this low temperature in atomic gases, Collisional
Radiative Recombination (CRR) where extra electron carries off energy, is dominant.
A few accomplished CRR experiments do not follow theoretical predictions of very
strong negative dependence. We propose to study transition from dissociative and
radiative recombination to CRR at cryogenic temperatures. For this study we are
constructing electron-ion trap. Electrons will be trapped in combination of
electrostatic and magnetic field (magnetic-mirror like). Ions will be trapped in 3D
potential minimum in the middle of the trap formed by space charge of trapped
electrons. The trapped electrons and ions will be cooled by collisions with helium
buffer gas. From the decay of the ion density recombination rate coefficient will be
obtained. PIC calculation is used to model trapping and cooling of electron-ion
plasma, FEMM program is used to model magnetic field in trap. Trap is currently
under construction, first problem, that will be studied with trap (in year 2009) is state
selective (ortho/para) recombination of H3+, which could not been measured by any
other technique yet (low temperature, low pressure).
Introduction
Actual and very fundamental problem of contemporary physics is plasma confinement and
processes related to the confinement. At high temperature plasmas it is connected with nuclear fusion.
At low and ultra low temperatures it is connected with study of fundamental properties of the matter at
conditions where de Broglie wavelength of interacting electrons is substantially larger then classical
dimension of atom. For these low temperatures, many recombination constants have never been
measured, in some cases they are not even theoretically calculated.
In plasma we propose to study (electron-ion plasma), the recombination involves capture of a free
electron by an ion. The decay rate of the ionization is determined by the physical and chemical nature
of the ionized media and in quantitative way it is given by cross section or in plasma in
thermodynamical equilibrium by the recombination rate coefficient α(Τ ). If the recombining species
are atomic ions and electrons, the only way the recombination energy can be dispersed is either
through the emission of radiation (Radiative Recombination, RR, αRR), or by transfer to a third body.
(Three Body Recombination, TBR). Neither process is very efficient under normal conditions
encountered in plasmas of moderate density; as a consequence atomic ion recombination is generally
slow. At low temperatures, 100 K and lower even at moderate electron density, electron can play role
of third body and TBR can be the dominant recombination process. Such process is called Collisional
Radiative Recombination (CRR, αCRR) [1]. The study of this process is the main objective of
proposed project.
Theory describes CRR as a two-step process: In electron-ion collision a highly excited weakly
bound Rydberg state is formed and in consequent collision with second electron recombination energy
is carried off and more stable particle is formed (see e.g. [2,3] and recent elaboration by Zygelman
[4]). The rate of CRR is controlled by a kinetic bottleneck at a binding energy of a few kBT (we
consider thermal plasma with temperature T). The effective binary rate coefficient (αCRR) of CRR is
dependent on T and on electron density ne. The theory predicts very strong negative temperature
dependence of αCRR, at low temperature αCRR ~ ne / T 4.5 (see [2]). Despite the fact that the CRR was
theoretically described over 40 years ago, there are hardly any direct experimental low temperature
data to support the theory [5]. In recent studies of expanding ultra cold neutral plasma serious
inconsistency with theory was observed [6]. The data from ATHENA experiment indicate decrease of
the rate of formation of antihydrogen with increasing temperature but a power law scaling, as
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JÍLEK ET AL.: ELECTRON-ION RECOMBINATION AT 10 K
predicted for CRR [4], was not observed [1]. The differences are rationalised by presence of strong
magnetic field used for the plasma confinement [7].
Two types of techniques are used for study of electron - ion recombination in general: Storage
ring experiments (single collision system, e.g. TSR Heidelberg) and plasma experiments (multiple
collision system, e.g. Flowing Afterglow - FALP experiments). Disadvantages of storage ring
experiments are high financial and personal demands and problems with relaxation and in situ
characterisation of stored and recombining ions. Note here, that standard storage ring techniques (e.g.
TSR[8] or CRYRING[9]) used for recombination studies are limited to binary processes only. The
disadvantage of afterglow experiments is limitation of studies to group of ions that at certain
conditions can be formed and stay dominant in an afterglow [10,11]. In these multi collision systems it
is nearly impossible or very difficult to study recombination of ions such as CH+, NH+, OH+ and ArH+.
Plasma experiments (e.g. FALP) can in principle be used for CRR study [5], but in reality it is difficult
and up to now they were not very successful. Apart from technical reasons, the main problem of
afterglow experiments is relaxation of electron energy distribution function in presence of process as
sensitive to collision energy as CRR.
Figure 1 Left panel: The rotational states of H3+ with rotational quantum numbers J and K. The lowest
rotational state of H3+ is the (J=1,K=1) state, the next lowest state (J=1,K=0) lies 32 K higher.
However, the rotational quantum numbers are not the only difference between these two levels, they
also belong to different nuclear spin modifications. In H3+ the three proton spins can either align to a
total spin of 3/2 (ortho-H3+) or 1/2 (para-H3+).
Right panel: The present theoretical thermal rate coefficients for dissociative recombination of paraH3+ and ortho-H3+ [12]. Note large difference predicted for rate coefficients of H3+ with different
nuclear spin (para and ortho-H3+). The full line “present study” indicates rate calculated under
assumption that ortho to para ratio corresponds to plasma in thermodynamic equilibrium.
Neither of the mentioned techniques is really suitable for state selective studies of recombination
(e.g. recombination of para and ortho H3+, H2D+, HD2+, D3+, etc.), of course with exclusion of studies
of ground state ions [13]. In general there is lack of experimental data on state selective
recombination.
To understand processes in interstellar medium (ISM) it is crucial to know recombination
coefficients of ortho/para H3+. Despite the fact that this recombination is studied over fifty years it
seems that only very recently scientists approached very close to the solution of the problem. Recent
calculation (see [12]) of recombination cross sections of H3+ ions with electrons at low energies
predicts very large difference in recombination rate coefficients for para-H3+ and ortho-H3+ (species
with different nuclear spin, see Fig. 1 – left panel for notation of energy levels) at temperatures below
300 K, see Fig. 1 – right panel. According to these calculations, which are at this point in contradiction
with previous best results [14], at T < 20 K the recombination of para-H3+ is by factor of 10 faster than
recombination of ortho-H3+. This implies that for description of recombination processes in ISM two
26
JÍLEK ET AL.: ELECTRON-ION RECOMBINATION AT 10 K
rate coefficients for recombination of H3+ ions has to be used (by definition only one for plasma in
thermodynamic equilibrium).
Figure 2 Rate coefficients of CRR, RR and DR recombination as a function of temperature in sub
thermal region [15,16]. αCRR (CRR rate coefficient) is given for ne = 106 and 107 cm-3. Time constant of
recombination process, τ = 1/(α ne), calculated for ne = 107 cm-3 is plotted on the right axis.
The theoretical prediction of large difference in the rate of recombination of ortho and para-H3+
was not yet confirmed by experiments. There exists only qualitative agreement with storage ring
experiment (TSR) in Heidelberg [17].
Clearly, new approaches are needed to study electron interactions including electron ion
recombination at cryogenic temperatures. Typical values of rate coefficients for CRR (calculated for ne
= 106 and 107 cm-3) and dissociative recombination (DR) of selected ions are plotted in Figure 2. The
region where both processes overlap was never studied mainly because there is no suitable
experimental technique.
The problems are partly targeted by new techniques, such as cryogenic electrostatic ion storage
rings, which are under construction. New cryogenic electrostatic storage rings are constructed [18]:
CSR, DESIREE, TMU E-Ring and magnetic storage ring: China – Institute of Modern Physics,
Lanzhou. At Aarhus University in Denmark there is in operation the electrostatic storage ring ELISA. All these ‘high energy’ low temperature experiments were designed and they are constructed
to study low energy interactions of cold ions with electrons (merged beams), photons and neutrals.
Similar studies are also carried out in ‘low energy’ radio frequency (RF) traps (e.g. 22-pole traps)
and Malmberg-Penning traps cooled to cryogenic temperatures [19,20]. Malmberg-Penning traps use
magnetic field for radial confinement of electrons and/or light ions and electrostatic field is used for
axial confinement. In ‘nested traps’ particles of both charges can be trapped. In these experiments
electrons are cooled down by radiation in strong magnetic field. Because of magnetic confinement
mass of ions is limited. Very strong magnetic field used in these experiments can also affect
recombination coefficient.
With proposed electron-ion trap we will be able to study recombination at low (10 K)
temperatures, but we will not use as strong magnetic field as “nested traps” and we can also measure
with heavier ions. In contrary to storage rings, we can study more then 2-particles processes like CRR.
So this technique will be suitable for measuring state-selective recombination of o/p H3+ or transition
from dissociative recombination to CRR and many others.
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JÍLEK ET AL.: ELECTRON-ION RECOMBINATION AT 10 K
Figure 3 Outline of the electron-ion trap. The trap will be filled with pre-cooled helium buffer for
collisional cooling of trapped electrons and ions. 10 K cryostat will be used to cool the collision
chamber and helium.
New trap, calculation of electron confinement
We designed new experiment for study of recombination in cryogenic non-neutral electron-ion
plasma. In experiment plasma will be trapped in electron-ion trap (see Figure 3), relaxed and from the
plasma decay the rate of recombination process will be determined. The collision chamber will be
cooled down to 10 K. Electrons and ions will be cooled in collisions with pre-cooled He buffer gas.
The trap will use magnetic field from permanent magnets (NdFeB) for radial confinement of electrons.
Strong gradient of magnetic field (magnetic mirror like) and electrostatic field will be used for axial
confinement of electrons.
The numerical simulation was used for calculation of magnetic field inside the proposed trap. It is
important to achieve high gradient of B on the ends of confinement volume. The finite element solver
FEMM developed by D.C. Meeker was used for optimal design of magnetic field. Calculated lines of
magnetic force are depicted in Figure 4. Magnetic field along the collision chamber is nearly constant
(~0.05 Tesla), maximum intensity at position of “magnetic mirror” is B = 0.6 Tesla
(see Figure 5: Left-Top).
Figure 4 The calculated magnetic field of the “magnetic mirror”. The collision chamber is on the left
side. The field is generated by permanent magnets, NdFeB. The maximum value of magnetic field
intensity on the axis is roughly 0.6 Tesla.
The ions will be trapped in three dimensional potential minimum created by electrodes and space
charge of trapped electrons (sort of nested trap – Figure 5: Left). The electron and ion trapping and
cooling were simulated using code XOOPIC (see discussion below). The trap is axially symmetric. In
the calculation we are using calculated magnetic field (see Figure 4). The calculation of magnetic field
was confirmed by measurement of the magnetic field in the prototype of the trap. In the calculations
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JÍLEK ET AL.: ELECTRON-ION RECOMBINATION AT 10 K
105 – 108 electrons are injected into the trap and time evolutions of the plasma cloud are calculated.
Helium buffer fills centre of the trap or the whole trap. Example of obtained electric potential and
intensity of magnetic field are depicted in Figure 5. Distribution of trapped electrons is plotted in
Figure 6 together with trap proposal scheme for better orientation. As can be seen, the minimum of the
electric potential is formed in the centre of the trap. The depth of minimum depends on ne, geometry of
the trap and potentials on electrodes. This potential minimum will be used to trap ions under condition
nion<< ne.
We have carried out kinetic numerical simulation of confined plasma in proposed trap. Contrary
to hydrodynamic (fluid description) that is appropriate to dense plasma, in the kinetic technique
equations of motion for all charged particles in plasma are solved individually. For numerical
simulation of the electron confinement we chose Particle-In-Cell (PIC) technique [21]. In this
technique the simulation region is divided into cells for calculation of potential. Coulomb interactions
among charged particles are converted to the solving of Poisson equation in the centres of
computational mesh. Computed electrical field influences charged particle. Collisions with neutral
particles are reduced to elastic collisions in this case. They are usually realised using Monte Carlo
method.
|B| [T]
0.60
0.30
0.00
-90
-60
-30
0
30
60
90
-0.30
-0.90
U [V]
-0.60
-1.20
Figure 5 Left-Top: Calculated intensity of magnetic field along the axis of symmetry (FEMM). Ions
trapped by potential minimum are visualised by circles.
Left-Bottom: Calculated electric field of electrodes plus trapped electrons (done by XOOPIC).
Right: Calculated 3-D electric potential near the “magnetic mirror”.
Figure 6 Upper panel: Radial and axial distribution of trapped electron cloud.
Lower panel: Trap proposal scheme. Stronger, NdFeB magnets are used to create big magnetic-field
gradient on both sides, SmCo magnets are used to maintain constant field along axis.
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JÍLEK ET AL.: ELECTRON-ION RECOMBINATION AT 10 K
Great advantage of the proposed technique is a self-consistency. It is not necessary to assume any
parameters of plasma. External parameters of trap as geometry, potentials and magnetic field are
defined only. Then we can compute time-dependence of internal parameters of plasma for various
buffer gas temperatures and pressure and also electron (ion) initial temperature and concentration.
Our model space has axial symmetry thus we can use two-dimensional approach. We can model
smaller region close to high gradient of magnetic field to find characteristics of the trap.
For our PIC simulations we used the XOOPIC code, which was developed at the University of
California, Berkeley [22]. The Dynamic Alternate-Direct-Implicit method [23] was used for the
Poisson equation solving. Neutral particles interaction was calculated by Monte Carlo technique using
Null Collision method [24]. As we are modelling up to 108 electrons at the same time, electrons in our
trap were replaced by superparticles in PIC model. Each superparticle represents 10 - 100 electrons.
The total charge and mass were found to be conserved indeed.
Conclussion
Electron-ion trap for recombination studies at cryogenic temperatures was designed. We carried
out calculation of configuration of magnetic field generated by permanent magnets. For given
configuration of the trap and calculated magnetic field we simulated electron and ion trapping. Further
calculations and final design of electron-ion trap are in progress.
Using these calculations, electron-ion trap (ELECTRA) is constructed. First measurements should
be done in year 2009.
Acknowledgements. This work is a part of the research plan MSM 0021620834 financed by the Ministry
of Education of the Czech Republic and was partly supported by GACR (202/07/0495, 202/08/H057) by GAUK
53607, GAUK 124707, GAUK 86908 and Vakuum Praha
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