Hydrogen and deuterium in shock wave experiments, ab initio simulations and chemical picture modeling

The European Physical Journal D (Impact Factor: 1.23). 05/2012; 66(4). DOI: 10.1140/epjd/e2012-20650-3
Source: arXiv


We present equation of state data of shock compressed hydrogen and deuterium.
These have been calculated in the physical picture by using {\it ab initio}
molecular dynamics simulations based on finite temperature density functional
theory as well as in the chemical picture via the Saha-D model. The results are
compared in detail with data of shock wave experiments obtained for condensed
and gaseous precompressed hydrogen and deuterium targets in a wide range of
shock compressions from low pressures up to megabars.


Available from: Igor Iosilevskiy
Eur. Phys. J. D (2012) 66: 104
DOI: 10.1140/epjd/e2012-20650-3
Regular Article
Hydrogen and deuterium in shock wave experiments, ab initio
simulations and chemical picture modeling
B. Holst
, and I.L. Iosilevskiy
Universit¨at Rostock, Institut f¨ur Physik, 18051 Rostock, Germany
Joint Institute for High Temperatures, Russian Academy of Sciences, 125412 Moscow, Russia
Institute of Problems of Chemical Physics, Russian Academy of Sciences, 142432 Chernogolovka, Moscow Region, Russia
Moscow Institute of Physics and Technology (State University), Joint Institute for High Temperatures, 141700 Dolgoprudny,
Moscow Region, Russia
Received 7 November 2011 / Received in final form 25 January 2012
Published online 30 April 2012
EDP Sciences, Societ`a Italiana di Fisica, Springer-Verlag 2012
Abstract. We present equation of state data of shock compressed hydrogen and deuterium. These have
been calculated in the physical picture by using ab initio molecular dynamics simulations based on finite
temperature density functional theory as well as in the chemical picture via the Saha-D model. The
results are compared in detail with data of shock wave experiments obtained for condensed and gaseous
precompressed hydrogen and deuterium targets in a wide range of shock compressions from low pressures
up to megabars.
1 Introduction
The equation of state (EOS) of hydrogen and its isotopes
has been in the focus of research for many years for several
reasons. In models of stellar and planetary interiors [13]
hydrogen is the most abundant element and its EOS is
the most important component for the results. Deuterium
and tritium are target materials in inertial confinement fu-
sion experiments [4]. Therefore, a lot of experimental and
theoretical efforts were done to understand the behavior
of hydrogen, deuterium, and tritium in a wide range of
densities and temperatures. Recent developments in shock
wave experiments have enabled an access to a precise
database in the megabar pressure range. Single or multiple
shock wave experiments have been performed for hydro-
gen (or deuterium) by using, e.g., high explosives [5], gas
guns [6], pulsed power [79], or high-power lasers [10,11].
The strongly correlated states of warm dense matter cover
a wide range, from an atomic-molecular mixture at low
temperatures to fully ionized weakly coupled plasma at
high temperatures. In particular, the characterization of
the transition region (partially ionized plasma) is a great
challenge both to theory and experiment since the bound
states exhibit a highly transient nature. This region of the
phase diagram is, however, of great relevance for plane-
tary interiors. Important experimental information is also
gained from helioseismology data [12] that allows to check
and correct theoretical models very accurately in the high
temperature limit.
Some theoretical methods yield accurate results for
limiting cases which then can be used to benchmark more
general but approximate methods. For instance, an ex-
act asymptotic expansion of thermodynamic functions can
be given in the limit of almost fully ionized, low den-
sity plasma [13,14]. Ab initio simulation techniques such
as path integral Monte Carlo (PIMC) [15,16], quantum
Monte Carlo (QMC) [1719] or finite-temperature den-
sity functional theory molecular dynamics (FT-DFT-MD)
simulations [20,21] which treat quantum effects and corre-
lations systematically have taken a great benefit from the
rapid progress in computing power. These methods pro-
vide very accurate and reliable results for a variety of prob-
lems and systems, especially for warm dense matter. In
addition to these approaches, advanced chemical models
developed for partially ionized plasmas [22] have also been
applied for warm dense matter for a long time [2332].
In the present work we compare the results of the
Saha-D model and of FT-DFT-MD simulations with shock
wave experiments for hydrogen and deuterium which were
performed for different initial densities in a wide range of
shock compressions. We find good agreement so that these
models can also be used to give detailed predictions for
high-pressure states that will be probed in future experi-
ments by varying the initial conditions accordingly.
Our paper is organized as follows. The FT-DFT-MD
simulations are explained in Section 2 and the Saha-D
model in Section 3.Wepresentourresultsforthe
Hugoniot curves in Section 4 and, finally, give a short sum-
mary in Section 5.
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Page 2 of 6 Eur. Phys. J. D (2012) 66: 104
2 FT-DFT-MD simulations
FT-DFT-MD simulations are a powerful tool to describe
warm dense matter [3346]. Correlation and quantum ef-
fects are considered by a combination of classical molec-
ular dynamics for the ions and density functional theory
for the electrons. We use the plane wave density functional
code VASP (Vienna ab initio simulation package) [4749]
to perform molecular dynamics simulations. VASP applies
Mermin’s finite temperature density functional theory [50]
which allows us to treat the electrons even at higher tem-
peratures on a quantum level. Projector augmented wave
potentials [51] were used and we applied a generalized gra-
dient approximation (GGA) within the parameterization
of PBE [52]. The plane wave cutoff E
has to be chosen
high enough to obtain converged EOS data [36,53]. A con-
vergence of better than 1% is secured for E
= 1200 eV
which was used in all calculations presented here. In the
MD scheme of VASP the Born-Oppenheimer approxima-
tion is used, i.e. the dynamics of the ions is treated within
a classical MD with inter-ionic forces obtained by FT-
DFT calculations via the Hellmann-Feynman theorem.
The electronic structure calculations were performed for
a static array of ions at each MD step. This was repeated
until the EOS measures were converged and a thermody-
namic equilibrium was reached.
The simulations were done for 256 atoms in a super-
cell with periodic boundary conditions. A Nos´ethermo-
stat [54] controlled the temperature of the ions, and the
temperature of the electrons was fixed by Fermi weighting
the occupation of the electronic states [48]. Sampling of
the Brillouin zone using up to 14 k-points showed that well
converged results were obtained using Baldereschi’s mean
value point [55] for 256 particles. The same convergence
behavior has previously been reported for water [56]. The
size of the simulated supercell fixed the density of the sys-
tem. The internal energy was corrected due to zero point
vibrations of the molecules at low temperatures, taking
into account quantum contributions of a harmonic oscilla-
tor for each molecule [57]. For this procedure the number
of molecules has to be known, which was obtained by eval-
uating the pair correlation function, see reference [53]for
details. The system was simulated 1000–1500 steps further
after reaching the thermodynamic equilibrium to ensure a
small statistical error. The EOS data were then obtained
by averaging over all particles and simulation steps in equi-
3 Saha-D model
The Saha-D model EOS is based on the chemical pic-
ture [22,30,58] which represents the plasma as a mixture
of interacting electrons, ions, atoms, and molecules. We
consider the following components for hydrogen and deu-
terium: e
, A, A
, A
, A
,(A : H, D). For this case the
Helmholtz free energy reads:
F ({N
. (1)
We shortly outline the approximations in which these
three contributions were treated; for details, see [30,59].
The first two terms of equation (1) are the ideal gas con-
tributions of heavy particles (atoms, ions, and molecules)
and of electrons. The latter corresponds to the partially
degenerate ideal Fermi gas [60]. The last two terms de-
scribe corrections due to Coulomb interactions and short-
range interactions between heavy particles. The Coulomb
interaction effects of charged particles are considered here
within a modified pseudopotential approach [30,61,62].
The electron-electron and ion-ion interaction is each
treated by using the Coulomb potential. For the effec-
tive electron-ion interaction we apply a pseudopotential
using the Glauberman form [63]. The parameters of the
pseudopotentials and of the corresponding pair correlation
functions were determined from the general conditions of
local electroneutrality and dipole screening, from the non-
negativity constraint for the pair correlation functions,
and from a relation between the screening cloud amplitude
and the depth of the electron-ion pseudopotential. In the
weak coupling limit, this approximation coincides with the
Debye model but in the strong coupling limit it is much
softer and demonstrates a quasi-crystalline behavior.
At high densities as typical for shock-compressed hy-
drogen the short-range repulsion between composite heavy
particles (A, A
, A
) becomes very important. This effect
is taken into account in the Saha-D model within a sim-
ple soft-sphere approximation [64] which is modified for a
mixture of soft spheres with different radii. In this case the
effective packing fraction Y is calculated via the individ-
ual diameters σ
of each particle species in correspondence
with the one-fluid approximation:
Y =
. (2)
is the diameter of the soft spheres in the respective
potential V
. The contribution of the in-
termolecular repulsion dominates the EOS of dense hy-
drogen and deuterium in a wide range of pressures at low
temperatures. The contribution of atom-atom and atom-
molecule repulsion becomes important at elevated tem-
peratures. The parameters for the soft-sphere repulsion
for A
, A
A,andA A are chosen according to
the spherically symmetric parts of the effective interac-
tion potentials of the non-empirical atom-atom approxi-
mation [65]. The key parameter of this approximation is
the ratio of corresponding soft-sphere diameters for atoms
and molecules, σ
. This ratio determines the change
of intrinsic volumes of two atoms in comparison with that
of a molecule (2V
); it determines the effective shift
of the dissociation and ionization equilibrium in warm
dense hydrogen and deuterium. All parameters of the soft-
sphere repulsion are given in Table 1. The parameter is
chosen such that our soft-sphere potential for molecule-
molecule repulsion will be close to the potential [65]ata
distance r =2a
(in this case =0.138 eV).
In order to take into account the existence of
condensed states (liquid and solid) for hydrogen and
Page 2
Eur. Phys. J. D (2012) 66: 104 Page 3 of 6
2.2 2.4
2.8 3 3.2 3.4
P [GPa]
Fig. 1. Shock compression of liquid hydrogen (blue) and deu-
terium (red). The Hugoniot curves obtained with the Saha-D
model (solid line) and the FT-DFT-MD (dashed line) are com-
pared with shock wave experiments of Nellis et al. [67](circles)
and Holmes et al. [68] (squares).
Table 1. Parameters of A
, A
, A A repulsion
in equation (2); a
is the Bohr radius.
A 63.2
deuterium, the attraction term in the free energy has to
be considered together with the soft-sphere repulsion. In
our case ΔF
= ΔF
+ ΔF
, (3)
= BN
The attractive corrections [66] are independent of temper-
ature. The choice of δ = 1 as in our case corresponds to a
van der Waals-like approximation. The parameter B sup-
plies the correct sublimation energy of a molecular system
in the condensed state.
We calculated Hugoniot curves based on the FT-DFT-
MD and Saha-D EOS data sets for hydrogen and deu-
terium. In Figure 1 we compare our calculations with
gas-gun experiments [67,68] on liquid hydrogen and deu-
terium. The experiments for deuterium and hydrogen are
reproduced by the Saha-D model within the uncertainties
of the experiments. The FT-DFT-MD Hugoniot curves
reproduce the experiments with less precision. In the
case of hydrogen the compression rate is slightly overesti-
mated above 5 GPa. For deuterium this occurs for pres-
sures above 15 GPa. Furthermore, the FT-DFT-MD curve
lies above the experiments for pressures below 10 GPa.
Between 10 GPa and 20 GPa the experimental data are
reproduced within error bars.
The less precision of the FT-DFT-MD data in the re-
gion of the gas gun experiments is connected with the rel-
atively abrupt onset of dissociation processes which lead
to an increase of the compression ratio at about 20 GPa
and 4200 K as it has already been reported [69]. The
reason for this behavior can be related to the under-
estimation of the fundamental band gap in DFT-based
electronic structure calculations. More accurate exchange-
correlation functionals than PBE, specifically derived for
finite temperatures, are urgently needed for warm dense
matter. QMC calculations treat the exchange-correlation
directly and are not affected by this approximation. Re-
cent QMC calculations find in fact a shift of the dissoci-
ation region compared to DFT [19] but show no different
results for the EOS for conditions relevant for planetary
interiors [18,70]. For instance, the Saha-D model yields
10% dissociation at 60 GPa and a temperature of 13 000 K
along the deuterium Hugoniot curve. Along the hydro-
gen Hugoniot curve dissociation occurs at about 15 GPa
and 3000 K within the FT-DFT-MD model. The Saha-D
model predicts 10% dissociation at 50 GPa and a temper-
ature of 14 000 K along the hydrogen Hugoniot curve.
The presented theoretical predictions both agree with
the experiments within 10% accuracy in compression ra-
tio and can therefore describe the principal behavior of
the obtained results well. The compression reached in
shock compressed hydrogen is higher than in deuterium
at the same pressure. This is reproduced by the calculated
Hugoniot curves, see also [45].
The FT-DFT-MD hydrogen EOS data were also used
to calculate the deuterium Hugoniot curve. To adjust for
the deuterium initial conditions we considered the ini-
tial conditions of the hydrogen EOS at half the deu-
terium density given in the experiment (0.171 g/cm
), i.e.
0.0855 g/cm
. Plotting the resulting pressure versus the
compression ratio, both Hugoniot curves are almost the
same. On the other hand, calculating the pressure with
the deuterium EOS and adjusting for the hydrogen ini-
tial conditions in the same way, the resulting Hugoniot
curves are identical within a smaller error than the statis-
tical error of the FT-DFT-MD simulations. The different
compression ratios of hydrogen and deuterium as seen in
the experiment are, therefore, only slightly caused by dif-
ferences in the EOS data of both isotopes at warm dense
matter conditions. The difference in the compression ra-
tios is mainly due to the fact that the densities of the
liquid targets at 20 K do not scale exactly by a factor of
two. Scaling the deuterium density to that of hydrogen the
initial density of liquid deuterium would be 0.0855 g/cm
which differs from the value relevant for liquid hydrogen
which is 0.071 g/cm
. This deviation of about 20% entails
the different Hugoniot curves.
The temperature along the Hugoniot curves as pre-
dicted by the theoretical models is shown in Figure 2
and compared with gas-gun experimental data [68]. The
temperatures measured in the experiments are in general
higher than predicted by the Saha-D model and the FT-
DFT-MD simulations, except for two deuterium points
above 22 GPa which are below the Saha-D curve. Again,
onset of dissociation causes the slight kink in the FT-
DFT-MD curves at about 18 GPa (D
The general behavior indicated by the experiments can be
Page 3
Page 4 of 6 Eur. Phys. J. D (2012) 66: 104
P [GPa]
Fig. 2. Shock compression of liquid hydrogen (blue) and deu-
terium (red). Temperatures along the Hugoniot curves as ob-
tained with the Saha-D model (solid line) and FT-DTF-MD
(dashed line) are displayed as function of pressure and com-
pared with shock wave experiments (squares) [68].
0.6 0.7 0.8 0.9
ρ [g/cm
P [
Fig. 3. Deuterium single shock principal Hugoniot curves
starting from different initial densities (gaseous, liquid, and
solid) as derived from FT-DFT-MD (black), Saha-D [32](red),
RPIMC [73] (blue), DPIMC [74] (dark red). Dotted line:
=0.1335 g/cm
, dashed line: ρ
=0.153 g/cm
, solid
line: ρ
=0.171 g/cm
, dot-dashed line: ρ
=0.199 g/cm
Shock wave experiments: Nellis et al. [67] (blue circles), Holmes
et al. [68] (blue squares), Knudson et al. [7,8] (open green cir-
cles), Grishechkin et al. [71] (red triangles), Boriskov et al. [72]
(red squares and diamonds). The arrows at the top show the
limiting compression for ultra-high pressures (4 × ρ
principal Hugoniot.
reproduced: the temperature along the hydrogen Hugoniot
curve is higher than that for deuterium at the same pres-
sure. The maximum deviation of both theoretical models
from the experimental data is about 400 K.
We have also applied both theoretical EOS data sets
to calculate the Hugoniot curves for different initial con-
ditions in order to study the compression behavior of the
hydrogen isotopes for a wide range of densities off the
principal Hugoniot. Figure 3 shows Hugoniot curves with
respect to initial conditions as chosen in recent shock wave
Fig. 4. Temperature of shock compressed deuterium for dif-
ferent initial conditions predicted by FT-DFT-MD (black),
Saha-D [32](red),RPIMC[73] (blue), DPIMC [74] (dark red),
and the asymptotically strict Saha-S [59] (orange) model in
comparison with experiments using liquid (Holmes et al. [68]
(blue squares) and Bailey et al. [75] (green hexagons)) and
gaseous targets (Grishechkin et al. [71] (red stars)). Dotted,
dashed, and solid lines correspond to initial deuterium den-
sities of ρ
=0.1335, 0.153, 0.171 g/cm
, respectively. Arrows
indicate the limiting compression for ultra-high pressures for
each Hugoniot curve.
experiments with precompressed targets [71,72]. Two ex-
periments were performed with gaseous targets at 1.5 kbar
=0.1335 g/cm
) and 2.0 kbar (ρ
=0.153 g/cm
Two other data sets were obtained with liquid (ρ
0.171 g/cm
) and solid (ρ
=0.199 g/cm
The two theoretical results and the experiments show
the same general behavior: the attained absolute density
is higher the more precompressed the target is. It has to be
pointed out that the maximum compression ratio shows
the inverse behavior, it decreases with higher precompres-
sion. Even so, the maximum density that can be probed
in single shock experiments increases with precompres-
sion. The theoretical predictions of the two methods for
the maximum density agree at the conditions of the ex-
periments with gaseous and liquid targets and range from
0.65 g/cm
to 0.775 g/cm
. For the initial condition in
the solid, there is a slight difference: FT-DFT-MD pre-
dicts 0.83 g/cm
and Saha-D 0.87 g/cm
. The pressure at
this maximum compression density is also different for the
two models; it ranges from 30 to 50 GPa within FT-DFT-
MD and from 80 to 150 GPa according to Saha-D. These
values cannot be discriminated via the few experimental
Figure 4 shows the temperatures along the Hugoniot
curves of deuterium using the initial conditions of the
experiments with liquid and precompressed gaseous tar-
gets [68,71,75].
We have to note again that with a higher pre-
compression also a higher density can be reached. The
measurements indicate a temperature of about 2 eV for
all three initial conditions. These results can be repro-
duced by the theoretical models within the error bars. The
Page 4
Eur. Phys. J. D (2012) 66: 104 Page 5 of 6
densest states are reached for the liquid deuterium target.
The experiments of Bailey et al. [75]showamaximum
density between 0.7 and 0.8 g/cm
which is reproduced
by both theories. The temperatures measured in the ex-
periments are underestimated by FT-DFT-MD and over-
estimated by Saha-D, with increasing deviation for higher
temperatures. The limit of the Saha-D model at high tem-
peratures can be checked by comparing with results ob-
tained by the Saha-S model [59] which is asymptotically
exact for Γ
1. Such a comparison shows
that Saha-D (together with PIMC [73,74]) yields the cor-
rect high-temperature limit. The deviation of the Saha-S
from the Saha-D curves at lower temperatures is due to
the fact that the Saha-S model is no longer applicable for
these parameters. In particular, the Saha-S model does
not take into account the short-range repulsion effects of
composite particles (A, A
). Interestingly, the temper-
ature at maximum compression is almost independent of
the initial conditions. The predictions of the theoretical
models show that the curves are shifted only to higher
densities while the temperature remains almost constant.
This finding is supported by the experimental results.
5 Conclusions
We have calculated the EOS of deuterium and hydrogen
with FT-DFT-MD simulations in the physical picture and
the Saha-D model in the chemical picture over a wide
range of density and temperature which enabled us to
compare those results with recent shock wave experiments.
The theories predict, in agreement with the experiments,
that higher compressed densities can be reached using pre-
compressed targets, while the maximum compression ra-
tio decreases. We compare also the temperature along the
Hugoniot curve of deuterium with experimental data and
find that only the density is affected by precompression,
while the temperature remains almost the same along the
different pathways. This leads to an increased pressure
with higher precompression along the Hugoniot.
A check of EOS models against experiments within
the WDM regime is available mostly for the relatively
limited density and temperature range along the princi-
pal Hugoniot curve. Experiments producing shock waves
within precompressed targets enable to check the quality
of EOS models in a wider range in the phase diagram.
Both EOS (based on FT-DFT-MD simulations and the
Saha-D model) could reproduce the experimental data.
On the other hand, neither the Saha-D model, which uses
effective two-particle potentials with parameters that have
been chosen to match physical constraints, nor the FT-
DFT-MD method, which has no adjustable parameters,
can reproduce all experimental features precisely. Nev-
ertheless, experimental data is still not available in the
needed quantity and precision to allow for a definite deci-
sion of which model has to be used to describe all quanti-
ties in all ranges of the phase diagram. We therefore look
forward to future high-pressure experiments, especially o
the principal Hugoniot curve.
The combination of an advanced chemical model with
an ab initio approach yields a reasonable description of
warm dense hydrogen because the low-density molecu-
lar liquid, the strongly correlated warm dense fluid, as
well as the hot plasma can be described adequately, see
also [76]. Simultaneously this combination saves computa-
tional power as the treatment of a low-density molecular
liquid is increasingly demanding when using FT-DFT-MD
simulations. The treatment of a free energy minimiztion
model like Saha-D is much less expensive regarding com-
putational time. Accordingly, this combination provides
an opportunity to construct a wide-range EOS for plan-
etary interior modelling for which a database from ambi-
ent conditions up to pressures of several tens of megabar
and temperatures up to about 100 000 K is needed. This
project is going to be compiled for and will be be applied
to model the interior of Jupiter [77].
This work was supported by the Deutsche Forschungsgemein-
schaft within the SFB 652, the High Performance Computing
Center North (HLRN), and the Program of the Presidium of
the Russian Academy of Sciences “Research of Matter at Ex-
treme Conditions”. We acknowledge support from the com-
puter center of the University of Rostock and of the Educa-
tion Center – Physics of High Energy Density Matter – of the
Moscow Institute of Physics and Technology. We thank Eugene
Yakub for helpful discussions and for providing us with the re-
sults of calculations for the deuterium Hugoniot.
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    • ". By scaling the initial density, theoretical hydrogen EOSs can reasonably be compared to deuterium experimental data, although some differences between D and H become non-negligible in the molecular region, which are probed by the gas gun data, due to differences in the molecular vibrational states; see Holst et al. (2012) for a detailed discussion. Figure 1 also shows the theoretical hydrogen Hugoniot curves for H- REOS.2 (Holst et al., 2012, with additional data points by A. Becker pers. comm), for the H-SCvH-i EOS, and for the H-Sesame EOS. "
    [Show abstract] [Hide abstract] ABSTRACT: The core mass of Saturn is commonly assumed to be 10-25 ME as predicted by interior models with various equations of state (EOSs) and the Voyager gravity data, and hence larger than that of Jupiter (0-10 ME). We here re-analyze Saturn's internal structure and evolution by using more recent gravity data from the Cassini mission and different physical equations of state: the ab initio LM-REOS which is rather soft in Saturn's outer regions but stiff at high pressures, the standard Sesame-EOS which shows the opposite behavior, and the commonly used SCvH-i EOS. For all three EOS we find similar core mass ranges, i.e. of 0-20 ME for SCvH-i and Sesame EOS and of 0-17 ME for LM-REOS. Assuming an atmospheric helium mass abundance of 18%, we find maximum atmospheric metallicities, Zatm of 7x solar for SCvH-i and Sesame-based models and a total mass of heavy elements, MZ of 25-30 ME. Some models are Jupiter-like. With LM-REOS, we find MZ=16-20 ME, less than for Jupiter, and Zatm less than 3x solar. For Saturn, we compute moment of inertia values lambda=0.2355(5). Furthermore, we confirm that homogeneous evolution leads to cooling times of only about 2.5 Gyr, independent on the applied EOS. Our results demonstrate the need for accurately measured atmospheric helium and oxygen abundances, and of the moment of inertia for a better understanding of Saturn's structure and evolution.
    Preview · Article · Apr 2013 · Icarus
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    [Show abstract] [Hide abstract] ABSTRACT: The thermodynamic parameters—pressure and density—of quasi-isentropically compressed helium have been measured in a pressure range of 100–500 GPa. A thermodynamic model that satisfactorily describes the behavior of strongly compressed helium in a wide range of compression parameters has been proposed.
    Full-text · Article · Dec 2012 · JETP Letters
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    [Show abstract] [Hide abstract] ABSTRACT: Model SAHA-S based on chemical picture for equation of state of solar plasma is presented. Effects of Coulomb interaction, exchange and diffraction effects, free electron degeneracy, relativistic corrections, radiation pressure contributions are taken into account. Solar model based on SAHA-S taking into account extended element composition and variation of heavy element abundance is represented and discussed. Comparison of SAHA-S equation of state data for hydrogen plasma with the results of other models applicable to description of solar plasma equation of state and results obtained with first principle methods is demonstrated and discussed.
    Full-text · Article · May 2013 · Contributions to Plasma Physics
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